diff --git a/public/LICENSE-text.html b/public/LICENSE-text.html new file mode 100644 index 0000000000000000000000000000000000000000..d0f44d4b9eeead70f076ac4f93144576aaa48e46 --- /dev/null +++ b/public/LICENSE-text.html @@ -0,0 +1,163 @@ + + + + + + + + +License • spam + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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+ + +
YEAR: 1996-2008
+COPYRIGHT HOLDER: D.C. Sorensen, R.B. Lehoucq, C. Yang, and K. Maschhoff
+ORGANIZATION: Rice University
+
+ +
+ +
+ + + +
+ + + + + + diff --git a/public/articles/figures/fig_ch2_factors.png b/public/articles/figures/fig_ch2_factors.png new file mode 100644 index 0000000000000000000000000000000000000000..4e033c04c9d252efd0a11403dfa2d03b196c5e32 Binary files /dev/null and b/public/articles/figures/fig_ch2_factors.png differ diff --git a/public/articles/figures/ill.png b/public/articles/figures/ill.png new file mode 100644 index 0000000000000000000000000000000000000000..8b6e3fb167268fa443c41a789e0bcbef3d326fb1 Binary files /dev/null and b/public/articles/figures/ill.png differ diff --git a/public/articles/figures/tree.png b/public/articles/figures/tree.png new file mode 100644 index 0000000000000000000000000000000000000000..d6afa309cce5b9ee7ce72230ee458057b9934654 Binary files /dev/null and b/public/articles/figures/tree.png differ diff --git a/public/articles/index.html b/public/articles/index.html new file mode 100644 index 0000000000000000000000000000000000000000..e28f893bad970c15a5c768693b7e807f2b6fdc00 --- /dev/null +++ b/public/articles/index.html @@ -0,0 +1,164 @@ + + + + + + + + +Articles • spam + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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+

All vignettes

+

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+ + + + + + diff --git a/public/articles/jss15.pdf b/public/articles/jss15.pdf new file mode 100644 index 0000000000000000000000000000000000000000..803af7f7e5ce0fb08bcb6f65f5655110ee2048dd Binary files /dev/null and b/public/articles/jss15.pdf differ diff --git a/public/articles/spam.html b/public/articles/spam.html new file mode 100644 index 0000000000000000000000000000000000000000..6033e1299373801ef9d4750980c24a8f2fb5167a --- /dev/null +++ b/public/articles/spam.html @@ -0,0 +1,381 @@ + + + + + + + +Illustrations and Examples • spam + + + + + + + + + +
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+ + + + +
+

+Rational for spam

+

At the core of drawing multivariate normal random variables, calculating or maximizing multivariate normal log-likelihoods, calculating determinants, etc., we need to solve a large (huge) linear system involving a variance matrix, i.e., a symmetric, positive definite matrix.

+

Assume that we have such a symmetric, positive definite matrix \(\boldsymbol{Q}\) of size \(n\times n\) that contains many zeros (through tapering, Furrer, Genton, and Nychka (2006), Furrer and Bengtsson (2007), or through a Markovian conditional construction). Typically, \(\boldsymbol{Q}\) contains only \(\mathcal{O}(n)\) non-zero elements compared to \(\mathcal{O}(n^2)\) for a regular, full matrix. To take advantage of the few non-zero elements, special structures to represent the matrix are required, i.e., only the positions of the non-zeros and their values are kept in memory. Further, new algorithms work with these structures are required. The package spam provides this functionality, see Furrer and Sain (2010) for a detailed exposition.

+
+
+

+A Simple Example

+

This first section illustrates with a simple example how to work with spam. Within a running R we install and load the current spam version from CRAN.

+
install.packages("spam")
+
+library("spam")
+

We create a trivial matrix and “coerce” it to a sparse matrix.

+
Fmat <- matrix(c(3, 0, 1, 0, 2, 0, 1, 0, 3), nrow = 3, ncol = 3)
+Smat <- as.spam(Fmat)
+

spam is conceptualized such that for many operations, the user proceeds as with ordinary full matrices. For example:

+ +

Hence, the user should not be worried which objects are sparse matrices and which are not. Of course not all operations result in sparse objects again,

+ +

However, other operations yield to different results when applied to full or sparse matrices

+ +
+
+

+Creating Sparse Matrices

+

The implementation of spam is designed as a trade-off between the following competing philosophical maxims. It should be competitively fast compared to existing tools or approaches in R and it should be easy to use, modify and extend. The former is imposed to assure that the package will be useful and used in practice. The latter is necessary since statistical methods and approaches are often very specific and no single package could cover all potential tools. Hence, the user needs to understand quickly the underlying structure of the implementation of spam and to be able to extend it without getting desperate. (When faced with huge amounts of data, sub-sampling is one possibility; using spam is another.) This philosophical approach also suggests trying to assure S3 and S4 compatibility, Chambers (1998), see also Lumley (2004). S4 has higher priority but there are only a handful cases of S3 discrepancies, which do however not affect normal usage.

+

To store the non-zero elements, spam uses the “old Yale sparse format”. In this format, a (sparse) matrix is stored with four elements (vectors), which are (1) the nonzero values row by row, (2) the ordered column indices of nonzero values, (3) the position in the previous two vectors corresponding to new rows, given as pointers, and (4) the column dimension of the matrix. We refer to this format as compressed sparse row (CSR) format. Hence, to store a matrix with \(z\) nonzero elements we thus need \(z\) reals and \(z+n+2\) integers compared to \(n \times n\) reals. Section Representation describes the format in more details.

+

The package spam provides two classes, first, spam representing sparse matrices and, second, spam.chol.NgPeyton representing Cholesky factors. A class definition specifies the objects belonging to the class, these objects are called slots in R and accessed with the @ operator, see for a more thorough discussion. The four vectors of the CSR representation are implemented as slots. In spam, all operations can be performed without a detailed knowledge about the slots. However, advanced users may want to work on the slots of the class spam directly because of computational savings, for example, changing only the contents of a matrix while maintaining its sparsity structure, see Section Tips. The Cholesky factor requires additional information (e.g., the used permutation) hence the class spam.chol.NgPeyton contains more slots, which are less intuitive. There are only very few, specific cases, where the user has to access these slots directly. Therefore, user-visibility has been disregarded for the sake of speed. The two classes are discussed in the more technical Section Representation.

+
+
+

+Displaying

+

As seen in Section A Simple Example, printing small matrices result in an expected output, i.e., the content of the matrix plus a line indicating the class of the object:

+ +

For larger objects, not all the elements are printed. For example:

+ +

The size of the matrix when switching from the first printing format to the second is a spam option, see Section Options. Naturally, we can also use the str command which gives us further insight into the individual slots of the spam object:

+ +

Alternatively, calling summary gives additional information of the matrix.

+ +

summary itself prints on the standard output but also returns a list containing the number of non-zeros (nnz) and the density (density) (percentage of nnz over the total number of elements).

+

Quite often, it is interesting to look at the sparsity structure of a sparse matrix. This is implemented with the command display. Again, depending on the size, the structure is shown as proper rectangles or as points. Figure is the result of the following code.

+
+\label{fig:display_spam}Sparsity structure of sparse matrices.

+Sparsity structure of sparse matrices. +

+
+

Additionally, compare Figures of this document. Depending on the cex value, the image may issue a warning, meaning that the dot-size is probably not optimal. In fact, visually the density of the matrix Smat1 seems to be around some percentage whereas the actual density is 0.024.

+

The function image goes beyond the structure of the matrix by using a specified color scheme for the values. Labels can be added manually or with image.plot from the package fields.

+
+
+

+Solving Linear Systems

+

To be more specific about one of spam’s main features, assume we need to calculate \(\boldsymbol{A}^{-1}\boldsymbol{b}\) with \(\boldsymbol{A}\) a symmetric positive definite matrix featuring some sparsity structure, which is usually accomplished by solving \(\boldsymbol{A}\boldsymbol{x}=\boldsymbol{b}\). We proceed by factorizing \(\boldsymbol{A}\) into \(\boldsymbol{R}^T\boldsymbol{R}\), where \(\boldsymbol{R}\) is an upper triangular matrix, called the Cholesky factor or Cholesky triangle of \(\boldsymbol{A}\), followed by solving \(\boldsymbol{R}^T\boldsymbol{y}=\boldsymbol{b}\) and \(\boldsymbol{R}\boldsymbol{x}=\boldsymbol{y}\), called forwardsolve and backsolve, respectively. To reduce the fill-in of the Cholesky factor \(\boldsymbol{R}\), we permute the columns and rows of \(\boldsymbol{A}\) according to a (cleverly chosen) permutation \(\boldsymbol{P}\), i.e., \(\boldsymbol{U}^T\boldsymbol{U}=\boldsymbol{P}^T\boldsymbol{A}\boldsymbol{P}\), with \(\boldsymbol{U}\) an upper triangular matrix. There exist many different algorithms to find permutations which are optimal for specific matrices %tridiagonal matrices finite element/difference matrices defined on square grids or at least close to optimal with respect to different criteria. Note that \(\boldsymbol{R}\) and \(\boldsymbol{U}\) cannot be linked through \(\boldsymbol{P}\) alone. Figure illustrates the factorization with and without permutation. For solving a linear system the two triangular solves are performed after the factorization. The determinant of \(\boldsymbol{A}\) is the squared product of the diagonal elements of its Cholesky factor \(\boldsymbol{R}\). Hence the same factorization can be used to calculate determinants (a necessary and computational bottleneck in the computation of the log-likelihood of a Gaussian model), illustrating that it is very important to have a very efficient integration (with respect to calculation time and storage capacity) of the Cholesky factorization. In the case of GMRF, the off-diagonal non-zero elements correspond to the conditional dependence structure. However, for the calculation of the Cholesky factor, the values themselves are less important than the sparsity structure, which is often represented using a graph with edges representing the non-zero elements or using a “pixel” image of the zero/non-zero structure, see Figure .

+ +
+\label{fig:tree}On the left side the associated graph to the matrix $\boldsymbol{A}$ is visualized. The nodes of the graph are labeled according to $\boldsymbol{A}$ (upright) and $\boldsymbol{P}^T\boldsymbol{A}\boldsymbol{P}$ (italics). On the right side the sparsity structure of $\boldsymbol{A}$ and $\boldsymbol{P}^T\boldsymbol{A}\boldsymbol{P}$ (top row) and the Cholesky factors $\boldsymbol{R}$ and $\boldsymbol{U}$ of $\boldsymbol{A}$ and $\boldsymbol{P}^T\boldsymbol{A}\boldsymbol{P}$ respectively are given in the bottom row. The dashed lines in $\boldsymbol{U}$ indicate the supernode partition.\label{fig:tree}On the left side the associated graph to the matrix $\boldsymbol{A}$ is visualized. The nodes of the graph are labeled according to $\boldsymbol{A}$ (upright) and $\boldsymbol{P}^T\boldsymbol{A}\boldsymbol{P}$ (italics). On the right side the sparsity structure of $\boldsymbol{A}$ and $\boldsymbol{P}^T\boldsymbol{A}\boldsymbol{P}$ (top row) and the Cholesky factors $\boldsymbol{R}$ and $\boldsymbol{U}$ of $\boldsymbol{A}$ and $\boldsymbol{P}^T\boldsymbol{A}\boldsymbol{P}$ respectively are given in the bottom row. The dashed lines in $\boldsymbol{U}$ indicate the supernode partition.

+On the left side the associated graph to the matrix \(\boldsymbol{A}\) is visualized. The nodes of the graph are labeled according to \(\boldsymbol{A}\) (upright) and \(\boldsymbol{P}^T\boldsymbol{A}\boldsymbol{P}\) (italics). On the right side the sparsity structure of \(\boldsymbol{A}\) and \(\boldsymbol{P}^T\boldsymbol{A}\boldsymbol{P}\) (top row) and the Cholesky factors \(\boldsymbol{R}\) and \(\boldsymbol{U}\) of \(\boldsymbol{A}\) and \(\boldsymbol{P}^T\boldsymbol{A}\boldsymbol{P}\) respectively are given in the bottom row. The dashed lines in \(\boldsymbol{U}\) indicate the supernode partition. +

+
+

The Cholesky factor of a banded matrix is again a banded matrix. But arbitrary matrices may produce full Cholesky factors. To reduce this so-called fill-in of the Cholesky factor \(\boldsymbol{R}\), we permute the columns and rows of \(\boldsymbol{A}\) according to a (cleverly chosen) permutation \(\boldsymbol{P}\), i.e., \(\boldsymbol{U}^T\boldsymbol{U}=\boldsymbol{P}^T\boldsymbol{A}\boldsymbol{P}\), with \(\boldsymbol{U}\) an upper triangular matrix. There exist many different algorithms to find permutations which are optimal for specific matrices or at least close to optimal with respect to different criteria. The cost of finding a good permutation matrix \(\boldsymbol{P}\) is at least of order \(\mathcal{O}(n^{3/2})\). However, there exist good, but suboptimal, approaches for \(\mathcal{O}(n\log(n))\).

+

A typical Cholesky factorization of a sparse matrix consists of the steps illustrated in the following pseudo code algorithm.

+ ++++ + + + + + + + + + + + + + + + + + + +
StepDescription
[1]Determine permutation and permute the input matrix \(\boldsymbol{A}\) to obtain \(\boldsymbol{P}^T\boldsymbol{A}\boldsymbol{P}\) +
[2]Symbolic factorization, where the sparsity structure of \(\boldsymbol{U}\) is constructed
[3]Numeric factorization, where the elements of \(\boldsymbol{U}\) are computed
+

When factorizing matrices with the same sparsity structure Step 1 and 2 do not need to be repeated. In MCMC algorithms, this is commonly the case, and exploiting this shortcut leads to very considerable gains in computational efficiency (also noticed by Rue and Held (2005), page 51). However, none of the existing sparse matrix packages in R (SparseM, Matrix) provide the possibility to carry out Step 3 separately and spam fills this gap.

+

As for Step 1, there are many different algorithms to find a permutation, for example, the multiple minimum degree (MMD) algorithm, Liu (1985), and the reverse Cuthill-McKee (RCM) algorithm, George (1971). The resulting sparsity structure in the permuted matrix determines the sparsity structure of the Cholesky factor. As an illustration, Figure shows the sparsity structure of the Cholesky factor resulting from an MMD, an RCM, and no permutation of a precision matrix induced by a second-order neighbor structure of the US counties.

+

How much fill-in with zeros is present depends on the permutation algorithm, in the example of Figure @ref{fig:ch2:factor} there are \(146735\), \(256198\) and \(689615\) non-zero elements in the Cholesky factors resulting from the MMD, RCM, and no permutation, respectively. Note that the actual number of non-zero elements of the Cholesky factor may be smaller than what the constructed sparsity structure indicates. Here, there are \(14111\), \(97565\) and \(398353\) zero elements (up to machine precision) that are not exploited.

+
+\label{fig:ch2:factor}Sparsity structure of the Cholesky factor with MMD, RCM and no permutation of a precision matrix induced by a second-order neighbor structure of the US counties. The values *nnzR* and *fillin* are the number of non-zero elements in the sparsity structure of the factor and the fill-in, respectively.

+Sparsity structure of the Cholesky factor with MMD, RCM and no permutation of a precision matrix induced by a second-order neighbor structure of the US counties. The values nnzR and fillin are the number of non-zero elements in the sparsity structure of the factor and the fill-in, respectively. +

+
+
+
+

+More about Methods

+

For both sparse classes of spam standard methods like plot, dim, backsolve/forwardsolve, determinant (based on a Cholesky factor) are implemented and behave as in the case of full matrices. Print methods display the sparse matrix as a full matrix in case of small matrices and display only the non-zero values otherwise. The corresponding cutoff value as well as other parameters can be set and read via spam.options.

+
+

+Methods with Particular Behavior

+

For the spam class additional methods are defined, for examples rbind/cbind, dim<-, etc. The group generic functions from Math, Math2 and Summary are treated particularly since they operate only on the nonzero entries of the spam class. For example, for the matrix A presented in the introduction range(A) is the vector c(0.5, 1), i.e. the zeros are omitted from the calculation. The help lists further available methods and highlights the (dis-)similarities compared to when applied to regular matrices or arrays.

+
+
+

+Particular Methods with Ordinary Behavior

+

Besides the two sparse classes mentioned above, spam does not maintain different classes for different types of sparse matrices, such as symmetric or diagonal matrices. Doing so would result in some storage and computational gain for some matrix operations, at the cost of user visibility. Instead of creating more classes we consider additional specific operators. As an illustration, consider multiplying a diagonal matrix with a sparse matrix. The operator \%d*\% uses standard matrix multiplication if both sides are matrices or multiplies each column according the diagonal entry if the left hand side is a diagonal matrix represented by vector.

+
+
+

+Importing Foreign Formats

+

spam is not the only R package for sparse matrix algebra. The packages SparseM Koenker and Ng (2003) and Matrix Bates and Maechler (2006) contain similar functionalities for handling sparse matrices, however, both packages do not provide the possibility to split up the Cholesky factorization as discussed previously. We briefly discuss the major differences with respect to spam; for a detailed description see their manual.

+

SparseM is also based on the Fortran Cholesky factorization of Ng and Peyton (1993) using the MMD permutation and almost exclusively on SparseKit. It was originally designed for large least squares problems and later also ported to S4 but is in a few cases inconsistent with existing R methods. It supports different sparse storage systems.

+

Matrix incorporates many classes for sparse and full matrices and is based on C. For sparse matrices, it uses different storage formats, defines classes for different types of matrices and uses a Cholesky factorization based on UMFPACK, Davis (2004).

+

spam has a few functions that allow to transform matrix formats of the different packages.

+

spam also contains functions that download matrices from MatrixMarket, a web side that stores many different sparse matrices. The function read.MM(file), very similar to the function readMM from Matrix, opens a connection, specified by the argument, and reads a matrix market file. However, as entries of spam matrices are of mode double, integers matrices are coerced to doubles, patterns lead to matrices containing ones and complex are coerced to the real part thereof. In these aforementioned cases, a warning is issued.

+

MatrixMarket also defines an array format, in which case a (possibly) dense spam object is return (retaining only elements which are larger than getOption('spam.eps')), a warning is issued.

+

Similarly to read.MM(file), the function read.HB(file) reads matrices in the Harwell-Boeing format. Currently, only real assembled Harwell-Boeing can be read with read.HB. Reading MatrixMarket formats is more flexible.

+

The functions are based on readHB and readMM from the library Matrix to build the connection and read the raw data. At present, read.MM(file) is more flexible than readMM.

+

For many operations, spam is faster than Matrix and SparseM. It would also be interesting to compare spam and the sparse matrix routines of Matlab (see Figure 6 of Furrer, Genton, and Nychka (2006) for a comparison between SparseM and Matlab).

+
+
+
+

+Bibliography

+
+
+

Bates, Douglas, and Martin Maechler. 2006. Matrix: A Matrix Package for R.

+
+
+

Chambers, John M. 1998. Programming with Data: A Guide to the S Language. Secaucus, NJ, USA: Springer-Verlag New York, Inc.

+
+
+

Davis, Timothy A. 2004. “Algorithm 832: UMFPACK V4.3—an Unsymmetric-Pattern Multifrontal Method.” ACM Transactions on Mathematical Software (TOMS) 30 (2): 196–99. https://doi.org/10.1145/992200.992206.

+
+
+

Furrer, R., and T. Bengtsson. 2007. “Estimation of High-Dimensional Prior and Posteriori Covariance Matrices in Kalman Filter Variants.” J. Multivariate Anal. 98 (2): 227–55. https://doi.org/10.1016/j.jmva.2006.08.003.

+
+
+

Furrer, R., M. G. Genton, and D. Nychka. 2006. “Covariance Tapering for Interpolation of Large Spatial Datasets.” J. Comput. Graph. Statist. 15 (3): 502–23.

+
+
+

Furrer, R., and S. R. Sain. 2010. “spam: A Sparse Matrix R Package with Emphasis on MCMC Methods for Gaussian Markov Random Fields.” Journal of Statistical Software 36 (10): 1–25. http://www.jstatsoft.org/v36/i10/.

+
+
+

George, John Alan. 1971. “Computer Implementation of the Finite Element Method.” PhD thesis, Stanford, CA, USA: Stanford University.

+
+
+

Koenker, Roger, and Pin Ng. 2003. SparseM: Sparse Matrix Package for R.

+
+
+

Liu, Joseph W. H. 1985. “Modification of the Minimum-Degree Algorithm by Multiple Elimination.” ACM Transactions on Mathematical Software (TOMS) 11 (2): 141–53. https://doi.org/10.1145/214392.214398.

+
+
+

Lumley, Thomas. 2004. “Programmers’ Niche: A Simple Class, in S3 and S4.” R News 4(1) (1): 33–36.

+
+
+

Ng, Esmond G., and Barry W. Peyton. 1993. “Block Sparse Cholesky Algorithms on Advanced Uniprocessor Computers.” SIAM Journal on Scientific Computing 14 (5): 1034–56. https://doi.org/10.1137/0914063.

+
+
+

Rue, H., and L. Held. 2005. Gaussian Markov Random Fields: Theory and Applications. Chapman & Hall, London.

+
+
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+
+ + + +
+ + + +
+ + + + + diff --git a/public/articles/spam_files/figure-html/disp6-1.png b/public/articles/spam_files/figure-html/disp6-1.png new file mode 100644 index 0000000000000000000000000000000000000000..ad498bbdee1e186415a71ed56b58da4fb0d28338 Binary files /dev/null and b/public/articles/spam_files/figure-html/disp6-1.png differ diff --git a/public/authors.html b/public/authors.html new file mode 100644 index 0000000000000000000000000000000000000000..05fac6550d7b639f48a586a687b78054d7d3141d --- /dev/null +++ b/public/authors.html @@ -0,0 +1,269 @@ + + + + + + + + +Citation and Authors • spam + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + + +
+ +
+
+ + +

Furrer R, Sain SR (2010). +“spam: A Sparse Matrix R Package with Emphasis on MCMC Methods for Gaussian Markov Random Fields.” +Journal of Statistical Software, 36(10), 1–25. +doi: 10.18637/jss.v036.i10, http://www.jstatsoft.org/v36/i10/. +

+
@Article{,
+  title = {{spam}: A Sparse Matrix {R} Package with Emphasis on {MCMC} Methods for {G}aussian {M}arkov Random Fields},
+  author = {Reinhard Furrer and Stephan R. Sain},
+  journal = {Journal of Statistical Software},
+  year = {2010},
+  volume = {36},
+  number = {10},
+  pages = {1--25},
+  url = {http://www.jstatsoft.org/v36/i10/},
+  doi = {10.18637/jss.v036.i10},
+}
+

Gerber F, Furrer R (2015). +“Pitfalls in the Implementation of Bayesian Hierarchical Modeling of Areal Count Data: An Illustration Using BYM and Leroux Models.” +Journal of Statistical Software, Code Snippets, 63(1), 1–32. +doi: 10.18637/jss.v063.c01, http://www.jstatsoft.org/v63/c01/. +

+
@Article{,
+  title = {Pitfalls in the Implementation of {B}ayesian Hierarchical Modeling of Areal Count Data: An Illustration Using {BYM} and {L}eroux Models},
+  author = {Florian Gerber and Reinhard Furrer},
+  journal = {Journal of Statistical Software, Code Snippets},
+  year = {2015},
+  volume = {63},
+  number = {1},
+  pages = {1--32},
+  url = {http://www.jstatsoft.org/v63/c01/},
+  doi = {10.18637/jss.v063.c01},
+}
+

Gerber F, Moesinger K, Furrer R (2017). +“Extending R packages to support 64-bit compiled code: An illustration with spam64 and GIMMS NDVI3g data.” +Computer & Geoscience, 104, 109–119. +ISSN 0098-3004, doi: 10.1016/j.cageo.2016.11.015. +

+
@Article{,
+  title = {Extending {R} packages to support 64-bit compiled code: An illustration with spam64 and {GIMMS} {NDVI3g} data},
+  author = {Florian Gerber and Kaspar Moesinger and Reinhard Furrer},
+  journal = {Computer & Geoscience},
+  year = {2017},
+  volume = {104},
+  pages = {109--119},
+  issn = {0098-3004},
+  doi = {10.1016/j.cageo.2016.11.015},
+}
+ + +
    +
  • +

    Reinhard Furrer. Author, maintainer. +

    +
  • +
  • +

    Florian Gerber. Author. +

    +
  • +
  • +

    Roman Flury. Author. +

    +
  • +
  • +

    Daniel Gerber. Contributor. +

    +
  • +
  • +

    Kaspar Moesinger. Contributor. +

    +
  • +
  • +

    Youcef Saad. Contributor. +
    SPARSEKIT http://www-users.cs.umn.edu/~saad/software/SPARSKIT/

    +
  • +
  • +

    Esmond G. Ng. Contributor. +
    Fortran Cholesky routines

    +
  • +
  • +

    Barry W. Peyton. Contributor. +
    Fortran Cholesky routines

    +
  • +
  • +

    Joseph W.H. Liu. Contributor. +
    Fortran Cholesky routines

    +
  • +
  • +

    Alan D. George. Contributor. +
    Fortran Cholesky routines

    +
  • +
  • +

    Lehoucq B. Rich. Contributor. +
    ARPACK

    +
  • +
  • +

    Maschhoff Kristi. Contributor. +
    ARPACK

    +
  • +
  • +

    Sorensen C. Danny. Contributor. +
    ARPACK

    +
  • +
  • +

    Yang Chao. Contributor. +
    ARPACK

    +
  • +
+ +
+ +
+ + + +
+ + + + + + diff --git a/public/index.html b/public/index.html new file mode 100644 index 0000000000000000000000000000000000000000..24e45132e916b065fa43cc1ed20a122a8d5cc4d1 --- /dev/null +++ b/public/index.html @@ -0,0 +1,184 @@ + + + + + + + +SPArse Matrix • spam + + + + + + + + + +
+
+ + + +
+
+ + + + + +

A set of functions for sparse matrix algebra. Differences with other sparse matrix packages are:

+
    +
  • we only support (essentially) one sparse matrix format,
  • +
  • based on transparent and simple structure(s),
  • +
  • tailored for MCMC calculations within G(M)RF.
  • +
  • and it is fast and scalable (with the extension package spam64).
  • +
+
+\label{fig:sparsematrix}

+ +

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+ + + +
+ + + + + diff --git a/public/index_files/figure-html/sparsematrix-1.png b/public/index_files/figure-html/sparsematrix-1.png new file mode 100644 index 0000000000000000000000000000000000000000..e2d0b3e20bf2bf0d10e9a06822714744865dc4b1 Binary files /dev/null and b/public/index_files/figure-html/sparsematrix-1.png differ diff --git a/public/news/index.html b/public/news/index.html new file mode 100644 index 0000000000000000000000000000000000000000..af5d5f65db983cbddabe2954ee109355415cfa5e --- /dev/null +++ b/public/news/index.html @@ -0,0 +1,669 @@ + + + + + + + + +Changelog • spam + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + + +
+ +
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+ + +
+

+spam 2.3-0 2019-09-14 +

+

NEW FEATURES

+ +

SIGNIFICANT USER-VISIBLE CHANGES

+ +

INTERNAL CHANGES

+
    +
  • Fortran modification to address LTO issues.
  • +
  • Not exported depreceated function subset.rows.spam() is now defunct.
  • +
+

BUG FIXES

+
    +
  • Dataset UScounties.ndorder contained no-zeros on the diagonal. Now entire diagonal is zero.
  • +
+
+
+

+spam 2.2-2 2019-03-08 +

+

BUG FIXES

+ +
+
+

+spam 2.2-1 2018-12-20 +

+

SIGNIFICANT USER-VISIBLE CHANGES

+
    +
  • New vignette including illustrations and examples
  • +
  • Improved documentation for covariance functions like cov.exp().
  • +
+

BUG FIXES

+ +

INTERNAL CHANGES

+
    +
  • Spam fit for pkgdown website.
  • +
  • Replacing some internal functions by their primitive equivalent.
  • +
+
+
+

+spam 2.2-0 2018-06-19 +

+

SIGNIFICANT USER-VISIBLE CHANGES

+
    +
  • Implementation of ‘eigen.spam’ and ‘eigen_approx’ to calculate eigenvalues and eigenvectors for sparse matrices.
  • +
+
+
+

+spam 2.1-4 2018-04-12 +

+

SIGNIFICANT USER-VISIBLE CHANGES

+
    +
  • ‘germany.plot’ has new argument ‘cex.main’.
  • +
+

BUG FIXES

+
    +
  • ‘NAOK=TRUE’ did not properly dispatch in all matrix multiplications.
  • +
  • ‘.newSpam()’ was not always properly called, causing possible errors when creating empty ‘spam’ objects.
  • +
+

INTERNAL CHANGES

+
    +
  • Code cleaning and coherence improvements.
  • +
+
+
+

+spam 2.1-2 2017-12-22 +

+

SIGNIFICANT USER-VISIBLE CHANGES

+
    +
  • In case 64bits are required, issues an error to load package ‘spam64’ first.
  • +
  • Minor fixes in help files and other documentation.
  • +
+

INTERNAL CHANGES

+
    +
  • Improved registering of compiled functions.
  • +
  • Compatibility with new versions of package ‘testthat’.
  • +
+
+
+

+spam 2.1-1 2017-07-03 +

+

SIGNIFICANT USER-VISIBLE CHANGES

+
    +
  • With the addon of the package spam64, we have truly 64 bits available!
  • +
  • Dependency on dotCall64.
  • +
  • Different option handling, we use now the classical R setting. For the moment, the old functionality throws a message but in upcoming versions we use .Deprecated. ‘spam.options’ -> ‘options’, ‘spam.getOption’ -> ‘getOption’
  • +
  • ‘validate_spam’ superseeds ‘validspamobject’.
  • +
  • Added the list of Fortran contributors.
  • +
+

INTERNAL CHANGES

+
    +
  • Many… many. More still to come.
  • +
  • Unit testing with testthat.
  • +
  • Change of archaic trig functions to standard. Minor edits to great circle dist.
  • +
+
+
+

+spam 1.4-0 2016-08-30 +

+

INTERNAL CHANGES

+
    +
  • Fortran modification to address more pedantic compilations.
  • +
+
+
+

+spam 1.3-0 2015-10-29 +

+

BUG FIXES

+
    +
  • Adding nonsparse to sparse matrices did not always dispatch properly and caused some errors. Thanks to Johan Lindström for pointing out.
  • +
+

SIGNIFICANT USER-VISIBLE CHANGES

+
    +
  • Help improvements
  • +
  • Added additional method dispatching for ‘all.equal’.
  • +
+
+
+

+spam 1.2-0 and 1.2-1 Unreleased +

+

SIGNIFICANT USER-VISIBLE CHANGES

+
    +
  • Renaming of demos from JSS10 paper.

  • +
  • Continuing to implement ‘spam_xxx’ function names: Like ‘spam_rdist’ there is now ‘spam_diag’.

  • +
  • ‘subset.rows.spam’ deprecated. There is an internal function ‘subset_rows.spam’.

  • +
+

INTERNAL CHANGES

+
    +
  • Ample minor modifications in ‘DESCRIPTION’ and ‘NAMESPACE’ for CRAN check conformity.
  • +
  • Elimination of a few unnecessary functions (not exported).
  • +
  • Minor modifications in tests.
  • +
+
+
+

+spam 1.1-0 Unreleased +

+

SIGNIFICANT USER-VISIBLE CHANGES

+
    +
  • Upload of vignette linked to JSS15 paper. Related demos and tests.
  • +
  • “DUP=FALSE” is deprecated and will be disabled in future versions of R. All “DUP” arguments have been eliminated. You have to expect a slightly slower version of spam.
  • +
+
+
+

+spam 1.0 and 1.0-1 Unreleased +

+
    +
  • This version is up to ‘DESCRIPTION’ and its implied changes identical to 0.90-1. With the upcoming JSS article “Pitfalls in the implementation of Bayesian hierarchical modeling of areal count data. An illustration using BYM and Leroux models.” a “major” version jump is adequate.
  • +
  • Referencing to spam data through spam::…
  • +
  • Implemented 1.1.3.1 Suggested packages approach.
  • +
+
+
+

+spam 0.70/0.80/0.90/0.90-1 Unreleased +

+

SIGNIFICANT USER-VISIBLE CHANGES

+
    +
  • Introduction of many as(‘spam’,‘…’) functions.
  • +
  • Coercion function as.vector for spam objects.
  • +
  • Wrapper functions spam_rdist and ’spam_rdist.earthfor smooth use infields`.
  • +
  • The use of update(A, B) without assignment has been eliminated. This is one way to address the change in memory handling changes from R 3.0.2 to R 3.1.0. There is a slight overhead in memory. If this causes problems, let me know.
  • +
  • Adjustment of the license.
  • +
+

NEW FEATURES

+
    +
  • Arguments diag and eps in nearest.dist cause now an error.
  • +
  • Further augmented help pages.
  • +
+

BUG FIXES

+
    +
  • The demo now points to the new JSS article.
  • +
+

INTERNAL CHANGES

+
    +
  • Set ‘structurebased=TRUE’ for the demos.
  • +
  • Link to upcoming JSS article in one of the demos.
  • +
  • ‘update.spam.chol.NgPeyton’ preserves the structure (pointed out by Chris Paciorek), see above.
  • +
  • Using similar License approach as SparseM. New files README, inst/0LICENSE.
  • +
  • File renaming (OChangeLog -> 0ChangeLog)
  • +
  • Adjusted error messages for precmat.RW2
  • +
+
+
+

+spam 0.60-0 Unreleased +

+

SIGNIFICANT USER-VISIBLE CHANGES

+
    +
  • Using the flag ‘structurebased’, the behavior of spam is now more consistent.
  • +
  • “Arith”, “Compare”, “Logic” (getGroupMembers(“Ops”)) have now a consistent behavior.
  • +
+

NEW FEATURES

+
    +
  • Few new S3 functions for simplicity: ‘var.spam’, ‘eigen.spam’, …
  • +
  • New constructor functions ‘colindices<-’ etc. Maybe additional tests may be required.
  • +
  • Operators from ‘Arith’ obey now the structure based calculation.
  • +
+

BUG FIXES

+
    +
  • ‘inefficiencywarning’ passes message correctly.
  • +
+

INTERNAL CHANGES

+
    +
  • many more spam/tests/*.
  • +
  • Consistent use of ‘spam’ and ‘vector’ siglist for ‘Ops’.
  • +
  • Minor cleaning of Fortran code.
  • +
  • Renaming/restructuring/cleaning of files…
  • +
  • Fortran arguments are copied when updating the cholesky structure.
  • +
+
+
+

+spam 0.50-0 Unreleased +

+

SIGNIFICANT USER-VISIBLE CHANGES

+
    +
  • Using the flag ‘structurebased=FALSE’, the behavior of spam is now much, much closer to regular matrix calculations. This is illustrated when calculating gamma of a sparse matrix.
  • +
  • Along the same lines, the flag ‘NAOK=TRUE’ allows the use of the “not finite numbers” (NA, NaN, Inf). We have tested many, many functions but full fledged use is not yet guaranteed.
  • +
  • Currently, we can still guarantee backwards compatibility…
  • +
+

NEW FEATURES

+
    +
  • New functions ‘crossprod’ and ‘tcrossprod’ as well as according method definitions.
  • +
  • New constructor functions ‘rowpointers<-’ etc.
  • +
  • Better option handling. The option ‘safemode’ is now ‘safemodevalidity’. Additionally, new option ‘NAOK’.
  • +
  • Help pages have been improved.
  • +
  • Operators from ‘Summary’ and ‘Math’ obey now the structure based calculation. (‘Math2’ inherently does).
  • +
+

BUG FIXES

+
    +
  • rmvnorm.[].const now work properly for any number of constraints and n.
  • +
  • Assignment handles properly recycling.
  • +
  • todo() now works properly.
  • +
+

INTERNAL CHANGES

+
    +
  • eliminated {d,i}check by equivalent coercion.
  • +
  • Consistent use of NAOK in Fortran calls.
  • +
  • Minor cleaning of Fortran code.
  • +
  • Renaming/restructuring of files…
  • +
+
+
+

+spam 0.42-0 Unreleased +

+

NEW FEATURES

+
    +
  • More consistent handling of subsetting. Warning is issued if subsetting with NA
  • +
+

BUG FIXES

+
    +
  • Fixed several issues when rowsubsetting…
  • +
+

INTERNAL CHANGES

+
    +
  • Additional tests for positive definiteness in ‘chol’.
  • +
+
+
+

+spam 0.41-0 2014-02-26 +

+

NEW FEATURES

+
    +
  • Functions grid_trace2() received more functionality.
  • +
+

BUG FIXES

+
    +
  • Eliminated bug in cov.mat(). Pointed out by Joshua French.
  • +
+

INTERNAL CHANGES

+
    +
  • Updated DESCRIPTION: added Florian Gerber [ctb].
  • +
  • Minor code and help cleanup. Additional testing files. File header edits.
  • +
  • Addressed Rdevel CMD check –as-cran Notes, especially workaround for DUP=FALSE.
  • +
+
+
+

+spam 0.40-0 2013-09-12 +

+

BUG FIXES

+
    +
  • A severe bug in subsetting a spam object with a nx2 matrix crept in spam in version 0.29-3. Thanks to Andrew Hong and Beat Briner for pointing out.
  • +
  • To simplify communication, we have switched increased the tenth version number.
  • +
  • All other changes are of cosmetic nature.
  • +
+
+
+

+spam 0.30-x Unreleased +

+

SIGNIFICANT USER-VISIBLE CHANGES

+
    +
  • Added several plots to visualize several MCMC chains (‘grid_trace2’, ‘grid_zoom’, …).
  • +
+

NEW FEATURES

+
    +
  • New function ‘germany.plot’ to draw the landkreise. (‘map.landkreis’ is now obsolete).
  • +
  • Switched from ‘tim.colors’ to ‘colorRampPalette’ in ‘germany.plot’.
  • +
  • Metadata in ‘germany.info’, polygon definitions in ‘germany.poly’ (‘germany’ kept for backwards compatibility).
  • +
+

INTERNAL CHANGES

+
    +
  • Switched to mercurial for maintaining the package.
  • +
  • Updated ChangeLog file (hg log).
  • +
  • Increased dependency to >= R 2.15.
  • +
  • Minor code and help cleanup.
  • +
+
+
+

+spam 0.29-0, 0.29-1, 0.29-2, 0.29-3 2012-05-01 +

+

SIGNIFICANT USER-VISIBLE CHANGES

+
    +
  • There is a generic conflict with ‘backsolve’ between spam and other packages (e.g., bdsmatrix). To avoid the issue, we use the standard generic implemented in ‘methods’ which requires an additional argument for version 0.29-0 (see also PR#14883). However to maintain backwards compatibility with packages that depend on spam, this was reverted in 0.29-1. Currently, this conflict is not properly solved. I propose to load ‘spam’ first then the other packages, followed by manually calling:

  • +
  • setMethod(“backsolve”,“spam.chol.NgPeyton”,backsolve.spam) setMethod(“backsolve”,“spam”,backsolve.spam)

  • +
+ +
    +
  • Calls like:

  • +
  • mat <- diag.spam(4) diag(mat[-1, ]) <- 3 diag.spam(mat[ , -1]) <-2

  • +
+ +
    +
  • The Gibbs sampler in the demo article-jss-example2 contains several bugs, pointed out by Steve Geinitz and Andrea Riebler. I’ll post an updated sampler in a future release.
  • +
+

NEW FEATURES

+
    +
  • New functions ‘rmvnorm.const’, ‘rmvnorm.prec.const’ and ‘rmvnorm.canonical.const’ to draw constrained multivariate normal variates.
  • +
  • New functions ‘precmat’ (wrapper to), ‘precmat.RW1’, ‘precmat.RW2’, ‘precmat.season’, ‘precmat.IGMRFreglat’ and ‘precmat.IGMRFirreglat’ to create precision matrices for IGMRF.
  • +
  • New methods ‘rowSums’, ‘colSums’ and ‘rowMeans’, ‘colMeans’ for ‘spam’ objects.
  • +
  • New methods ‘head’ and ‘tail’ for ‘spam’ and ‘spam.chol.NgPeyton’ objects.
  • +
  • New method ‘chol2inv’ for ‘spam’ object.
  • +
  • New option ‘inefficiencywarning’: handling of warnings issued in case of an inefficient calculation.
  • +
  • New option ‘structurebased’: should operations be performed on the nonzero entries or on including the zeros. Classical example: what should the cosine of a sparse matrix look like? In the near future, all operations from Math and Ops will include this option. Some loss of backwards compatibility might be lost in the future.
  • +
+

INTERNAL CHANGES

+
    +
  • New much faster approach to extract rows. For not too sparse large matrices improvements over two orders of magnitudes are achieved.
  • +
  • Elininated ‘.Internal()’ calls that induce a ‘Note’ on CRAN checks. This also implied a minor rewrite of ‘image.spam’.
  • +
  • Minor code improvements.
  • +
  • Eliminated non-API calls (29.1).
  • +
  • Rewritten .C(“bincode”,…) call as suggested by Brian Ripley (29.2).
  • +
+

BUG FIXES

+
    +
  • Bug fix that occures when multiplying elementwise matrices that have non-intersecting structures (pointed out by Corentin Barbu).
  • +
  • Bug fix in triangular backsolves involving ‘spam’ objects and rhs matrices.
  • +
  • Bug fix in triangular backsolve causing errors on some architectures.
  • +
+
+
+

+spam 0.28 Unreleased +

+

NEW FEATURES

+
    +
  • New function ‘cleanup’ (suggested by Simon Barthelme).
  • +
  • Extending help files.
  • +
  • Improved functionality of ‘isSymmetric’.
  • +
+

INTERNAL CHANGES

+
    +
  • Proper storage of data files.
  • +
  • Cleaning up argument names within spam functions.
  • +
  • Cleaning up old Fortran code, i.e., eliminating unnecessary subroutines and write calls.
  • +
+

BUG FIXES

+
    +
  • Bug fix that may occure when extracting zero elements (pointed out by Corentin Barbu).
  • +
+
+
+

+spam 0.27 Unreleased +

+

NEW FEATURES

+
    +
  • Requires now R2.10 and higher.
  • +
  • Functions to create Toeplitz and circulant matrices.
  • +
  • Function to create precision matrices for gridded GMRF.
  • +
  • Improvements in the mle.* functions.
  • +
  • Method diff for sparse matrices (suggested by Paul Eilers).
  • +
  • Improvement of help pages.
  • +
  • Eliminated some help aliases to base functions (for which no ‘usage’ is given).
  • +
+

INTERNAL CHANGES

+
    +
  • Change to iL coding.
  • +
  • Start to using ‘identical’.
  • +
  • Code cleaning due to requirement of R2.10 and higher.
  • +
+

BUG FIXES

+
    +
  • Bug fix in as.spam.list (thanks to Paul Eilers).
  • +
  • Bug fix in demo(spam) (thanks to Thomas Gsponer).
  • +
+
+
+

+spam 0.24, 0.25 and 0.26 Unreleased +

+
    +
  • Devel versions, not released.
  • +
+
+
+

+spam 0.23 Unreleased +

+

NEW FEATURES

+
    +
  • Further improved versions of demos.
  • +
  • Some improvements to meet Rd standards. Adjustments for future R versions.
  • +
+
+
+

+spam 0.22 Unreleased +

+

NEW FEATURES

+
    +
  • Improved versions of demos. Synchronized with the JSS article.
  • +
  • Additional changes and improvements in the help files (thanks to Steve Geinitz).
  • +
+
+
+

+spam 0.21 Unreleased +

+

NEW FEATURES

+
    +
  • New NEWS file, to work better with news() command. The previous is available under ONEWS.
  • +
  • New functions bandwidth, permutation, mle[.nomean][.spam], neg2loglikelihood[.spam].
  • +
  • Renamed adiag.spam to bdiag.spam.
  • +
  • Cleaned up argument naming with the rmvnorm.* suite.
  • +
+

INTERNAL CHANGES

+
    +
  • Various Fortran code, R code and help file improvements.
  • +
+

BUG FIXES

+
    +
  • Minor change in one of the demos (solves a 64bit issue).
  • +
+
+
+ + + +
+ + +
+ + + + + + diff --git a/public/reference/Oral.html b/public/reference/Oral.html new file mode 100644 index 0000000000000000000000000000000000000000..3982555a512470cd4e3ceb18a8a1ce8efa3561bb --- /dev/null +++ b/public/reference/Oral.html @@ -0,0 +1,218 @@ + + + + + + + + +Oral Cavity Cancer — Oral • spam + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + + +
+ +
+
+ + +
+ +

Oral cavity cancer counts in 544 districts in Germany over 1986-1990.

+ +
+ + +

Format

+ +

Oral is a dataframe with 3 columns.

+
Y

observed counts

+
E

expected counts

+
SMR

standardized mortality ratios

germany is a +list of 544 elements, each describing an individual polygon of the district.

+ +

Details

+ +

The expected counts depend on the number of people in the + region and their age distribution.
+The regions are ordered according the supplied polygon description and +adjacency graph.

+

There is a similar dataset data(Germany) with larynx cancer cases from the package + INLA as well, with an additional smoking covariate.

+ +

Source

+ +

The data is available from the package INLA + distributed from http://www.r-inla.org.

+ +

References

+ +

Knorr-Held, L. and Rasser, G. (2000) Bayesian Detection of Clusters and Discontinuities in Disease Maps, + Biometrics, + 56, + 13--21.

+ +

See also

+ + + + +
+ +
+ + +
+ + + + + + diff --git a/public/reference/UScounties.html b/public/reference/UScounties.html new file mode 100644 index 0000000000000000000000000000000000000000..6227dc5687a25180d0d8a1a1c23eb35bcb08655f --- /dev/null +++ b/public/reference/UScounties.html @@ -0,0 +1,209 @@ + + + + + + + + +Adjacency structure of the counties in the contiguous United States — UScounties • spam + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + + +
+ +
+
+ + +
+ +

First and second order adjacency structure of the counties + in the contiguous United States. We consider that two counties + are neighbors if they share at + least one edge of their polygon description in maps.

+ +
+ + +

Format

+ +

Two matrices of class spam

+
+
UScounties.storder

Contains a one in the i and j element if county + i is a neighbor of county j.

+
UScounties.ndorder

Contains a one in the i and j element if counties + i and j are a neighbors of county k and counties + i and j are not neighbors.

+
+ +

See also

+ +

map from maps.

+ + +

Examples

+
# number of counties: +n <- nrow( UScounties.storder)
# NOT RUN { +# make a precision matrix +Q <- diag.spam( n) + .2 * UScounties.storder + .1 * UScounties.ndorder +display( as.spam( chol( Q))) +# }
+
+ +
+ + +
+ + + + + + diff --git a/public/reference/USprecip.html b/public/reference/USprecip.html new file mode 100644 index 0000000000000000000000000000000000000000..24d1bf46e901aeaea108ac17b7112fec7b8e957c --- /dev/null +++ b/public/reference/USprecip.html @@ -0,0 +1,225 @@ + + + + + + + + +Monthly total precipitation (mm) for April 1948 in the contiguous United States — USprecip • spam + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + + +
+ +
+
+ + +
+ +

This is a useful spatial data set of moderate to large size consisting of 11918 +locations. See https://www.image.ucar.edu/GSP/Data/US.monthly.met/ for the source of these data.

+ +
+ + +

Format

+ +

This data set is an array containing the following columns:

+
+
lon,lat

Longitude-latitude position of monitoring stations

+
raw

Monthly total precipitation in millimeters for April 1948

+
anomaly

Preipitation anomaly for April 1948.

+
infill

Indicator, which station values were observed (5906 out of the 11918) +compared to which were estimated.

+
+ +

Source

+ +

https://www.image.ucar.edu/GSP/Data/US.monthly.met/

+ +

See also

+ + + +

References

+ +

Johns, C., Nychka, D., Kittel, T., and Daly, C. (2003) +Infilling sparse records of spatial fields. +Journal of the American Statistical Association, +98, 796--806.

+ + +

Examples

+
# plot +
# NOT RUN { +library(fields) + +data(USprecip) +par(mfcol=c(2,1)) +quilt.plot(USprecip[,1:2],USprecip[,3]) +US( add=TRUE, col=2, lty=2) +quilt.plot(USprecip[,1:2],USprecip[,4]) +US( add=TRUE, col=2, lty=2) +# }
+
+ +
+ + +
+ + + + + + diff --git a/public/reference/adjacency.html b/public/reference/adjacency.html new file mode 100644 index 0000000000000000000000000000000000000000..d47bf3615b1611a096b4c5359c41d82f090b058a --- /dev/null +++ b/public/reference/adjacency.html @@ -0,0 +1,227 @@ + + + + + + + + +Administrative districts of Germany — adjacency • spam + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + + +
+ +
+
+ + +
+ +

Constructing the adjacency graph + of the administrative districts of Germany

+ +
+ +
adjacency.landkreis( loc)
+ +

Arguments

+ + + + + + +
loc

location of the graph structure, can be an URL.

+ +

Details

+ +

The function is included as an example on how to +construct adjacency matrices form a (common) adjacency structure. +For the particular example, note that the nodes are not numbered +consecutively and that they start from zero.

+ +

Value

+ +

a sparse matrix in spam format.

+ +

References

+ +

The adjacency data has been provided by Havard Rue and is + also available in INLA.

+ +

See also

+ +

germany.plot super-seeding map.landkreis + for plotting.
+ Oral.

+ + +

Examples

+
# NOT RUN {
+loc <- system.file("demodata/germany.adjacency", package="spam")
+display( adjacency.landkreis( loc))
+# }
+
+ +
+ + +
+ + + + + + diff --git a/public/reference/allequal.html b/public/reference/allequal.html new file mode 100644 index 0000000000000000000000000000000000000000..019a71b04b9600f8dcb037c9faa1735b7c85ace6 --- /dev/null +++ b/public/reference/allequal.html @@ -0,0 +1,277 @@ + + + + + + + + +Test if Two 'spam' Objects are (Nearly) Equal — allequal • spam + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + + +
+ +
+
+ + +
+ +

Utility to compare two spam objects + testing 'near equality'. Depending on the type of difference, comparison is + still made to some extent, and a report of the differences is + returned.

+ +
+ +
# S3 method for spam
+all.equal(target, current, tolerance = .Machine$double.eps^0.5,
+    scale = NULL, check.attributes = FALSE,...)
+ +

Arguments

+ + + + + + + + + + + + + + + + + + + + + + + + + + +
target

a spam object.

current

another spam object to be compared with target.

tolerance

numeric >= 0. Differences smaller than + tolerance are not considered.

scale

numeric scalar > 0 (or NULL). See ‘Details’.

check.attributes

currently not yet implemented.

...

Further arguments for different methods.

+ +

Value

+ +

Either TRUE or a vector of 'mode' "character" describing the + differences between + target and current.

+ +

Details

+ +

Numerical comparisons for + scale = NULL (the default) are done by + first computing the mean absolute difference of the two numerical + vectors. If this is smaller than tolerance or not finite, + absolute differences are used, otherwise relative differences + scaled by the mean absolute difference.

+

If scale is positive, absolute comparisons are made after + scaling (dividing) by scale.

+

Do not use all.equal.spam directly in if + expressions: either use isTRUE( all.equal.spam(...)) or + identical if appropriate.

+

Cholesky decomposition routines use this function to test for + symmetry.

+

A method for matrix-spam objects is defined as well.

+

There is the additional catch of a zero matrix being represented by + one zero element, see ‘Examples’ below.

+ +

See also

+ + + + +

Examples

+
obj <- diag.spam(2) +obj[1,2] <- .Machine$double.eps + +all.equal( diag.spam(2), obj)
#> [1] "Lengths (2, 3) differ"
+all.equal( t(obj), obj)
#> [1] "Column-sparsity structure differ (at least 1 instance(s))" +#> [2] "Row-sparsity structure differ (at least 1 instance(s))"
+all.equal( t(obj), obj*1.1)
#> [1] "Column-sparsity structure differ (at least 1 instance(s))" +#> [2] "Row-sparsity structure differ (at least 1 instance(s))" +#> [3] "Mean relative difference: 0.1"
+# We can compare a spam to a matrix +all.equal(diag(2),diag.spam(2))
#> [1] TRUE
+# the opposite does often not make sense, +# hence, it is not implemented. +all.equal(diag.spam(2),diag(2))
#> [1] TRUE
+ +# A zero matrix contains one element: +str(spam(0))
#> Formal class 'spam' [package "spam"] with 4 slots +#> ..@ entries : num 0 +#> ..@ colindices : int 1 +#> ..@ rowpointers: int [1:2] 1 2 +#> ..@ dimension : int [1:2] 1 1
# hence +all.equal.spam(spam(0,3,3), diag.spam(0,3) )
#> [1] "Lengths (1, 3) differ"
norm(spam(0,3,3) - diag.spam(0,3) )
#> [1] 0
+
+ +
+ + +
+ + + + + + diff --git a/public/reference/apply.html b/public/reference/apply.html new file mode 100644 index 0000000000000000000000000000000000000000..a6941a6ed6ea527ab1f72ab443300c8d02794fe3 --- /dev/null +++ b/public/reference/apply.html @@ -0,0 +1,306 @@ + + + + + + + + +Apply Functions Over Sparse Matrix Margins — apply • spam + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + + +
+ +
+
+ + +
+ +

Returns a vector or array or list of values obtained by applying a + function to margins of a sparse matrix.

+ +
+ +
apply.spam(X, MARGIN=NULL, FUN, ...)
+ +

Arguments

+ + + + + + + + + + + + + + + + + + +
X

the spam matrix to be used.

MARGIN

a vector giving the subscripts which the function will be + applied over. 1 indicates rows, 2 indicates columns, + NULL or c(1,2) indicates rows and columns.

FUN

the function to be applied.

...

optional arguments to FUN.

+ +

Details

+ +

This is a handy wrapper to apply a function to the (nonzero) + elements of a sparse matrix. + For example, it is possible to apply a covariance matrix to a distance + matrix obtained by nearest.dist, see Examples.

+

A call to apply only coerces the sparse matrix to a regular one.

+

The basic principle is applying the function to @entries, or to + the extracted columns or rows ([,i,drop=F] or + [i,,drop=F]). It is important to note that an empty column + contains at least one zero value and may lead to non intuitive + results.

+

This function may evolve over the next few releases.

+ +

Value

+ +

Similar as a call to apply with a regular matrix. The most + important cases are as follows. The + result is a vector (MARGIN is length 1 and FUN is + scalar) or a matrix (MARGIN is length 1 and FUN returns + fixed length vectors, or MARGIN is length 2 and FUN is + scalar) or a list (if FUN returns vectors of different lengths).

+ +

See also

+ +

base:apply for more details on Value.

+ + +

Examples

+
S <- as.spam(dist(1:5)) +S <- apply.spam(S/2, NULL, exp) +# instead of +# S@entries <- exp( S@entries/2) + +# Technical detail, a null matrix consists +# of one zero element. +apply.spam(S,c(1,2),pmax)
#> [,1] [,2] [,3] [,4] [,5] +#> [1,] 0.000000 0.000000 0.000000 0.000000 0 +#> [2,] 1.648721 0.000000 0.000000 0.000000 0 +#> [3,] 2.718282 1.648721 0.000000 0.000000 0 +#> [4,] 4.481689 2.718282 1.648721 0.000000 0 +#> [5,] 7.389056 4.481689 2.718282 1.648721 0 +#> Class 'spam' (32-bit)
apply.spam(S,1,range)
#> [,1] [,2] [,3] [,4] [,5] +#> [1,] 0 1.648721 1.648721 1.648721 1.648721 +#> [2,] 0 1.648721 2.718282 4.481689 7.389056
+ +# A similar example as for the base apply. +# However, no dimnames else we would get warnings. +x <- as.spam(cbind(x1 = 3, x2 = c(0,0,0, 5:2))) +apply.spam(x, 2, mean, trim = .2)
#> [1] 3.0 3.5
col.sums <- apply.spam(x, 2, sum) +row.sums <- apply.spam(x, 1, sum) +rbind(cbind(x, row.sums), c(col.sums, sum(col.sums)))
#> [,1] [,2] [,3] +#> [1,] 3 0 3 +#> [2,] 3 0 3 +#> [3,] 3 0 3 +#> [4,] 3 5 8 +#> [5,] 3 4 7 +#> [6,] 3 3 6 +#> [7,] 3 2 5 +#> [8,] 21 14 35 +#> Class 'spam' (32-bit)
+apply.spam(x, 2, is.vector)
#> [1] TRUE TRUE
+# Sort the columns of a matrix +# Notice that the result is a list due to the different +# lengths induced by the nonzero elements +apply.spam(x, 2, sort)
#> [[1]] +#> [1] 3 3 3 3 3 3 3 +#> +#> [[2]] +#> [1] 2 3 4 5 +#>
+# Function with extra args: +cave <- function(x, c1, c2) c(mean(x[c1]), mean(x[c2])) +apply(x,1, cave, c1=1, c2=c(1,2))
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7] +#> [1,] 3.0 3.0 3.0 3 3.0 3 3.0 +#> [2,] 1.5 1.5 1.5 4 3.5 3 2.5
+ma <- spam(c(1:4, 0, 0,0, 6), nrow = 2) +ma
#> [,1] [,2] [,3] [,4] +#> [1,] 1 3 0 0 +#> [2,] 2 4 0 6 +#> Class 'spam' (32-bit)
apply.spam(ma, 1, table) #--> a list of length 2
#> [[1]] +#> +#> 1 3 +#> 1 1 +#> +#> [[2]] +#> +#> 2 4 6 +#> 1 1 1 +#>
apply.spam(ma, 1, stats::quantile)# 5 x n matrix with rownames
#> [,1] [,2] +#> 0% 1.0 2 +#> 25% 1.5 3 +#> 50% 2.0 4 +#> 75% 2.5 5 +#> 100% 3.0 6
+
+ +
+ + +
+ + + + + + diff --git a/public/reference/bandwidth.html b/public/reference/bandwidth.html new file mode 100644 index 0000000000000000000000000000000000000000..300b547286e41823a0ec2e500ef153eeea814c7a --- /dev/null +++ b/public/reference/bandwidth.html @@ -0,0 +1,214 @@ + + + + + + + + +Bandwidth of a Sparse Matrix — bandwidth • spam + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + + +
+ +
+
+ + +
+ +

Returns the lower and upper bandwidth of a sparse matrix

+ +
+ +
bandwidth(A)
+ +

Arguments

+ + + + + + +
A

spam object

+ +

Details

+ +

The matrix does not need to be diagonal. Values can be negative + indicating the the matrix contains a band cinfined in the + upper or lower triangular part.

+ +

Value

+ +

Integer vector containing the lower and upper bandwidth

+ +

See also

+ + + + +

Examples

+
bandwidth(spam(c(0, 1), 3, 2))
#> [1] 1 1
+bandwidth(spam(c(0, 0, 1, rep(0, 9)), 4, 3))
#> [1] 2 -2
+
+
+ +
+ + +
+ + + + + + diff --git a/public/reference/bdiag.html b/public/reference/bdiag.html new file mode 100644 index 0000000000000000000000000000000000000000..d37f4594158c55cd2cdea6c7c28c5e0644e0dcbe --- /dev/null +++ b/public/reference/bdiag.html @@ -0,0 +1,237 @@ + + + + + + + + +Binds Arrays Corner-to-Corner — bdiag • spam + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + + +
+ +
+
+ + +
+ +

Creates a sparse block-diagonal matrix.

+ +
+ +
bdiag.spam(...)
+ +

Arguments

+ + + + + + +
...

Arrays to be binded together

+ +

Details

+ +

This is a small helper function to create block diagonal sparse matrices. In the two matrix case, +bdiag.spam(A,B), this is equivalent to a complicated rbind(cbind(A, null), cbind(B, t(null))), +where null is a null matrix of appropriate dimension.

+

It is recursively defined.

+

The arrays are coerced to sparse matrices first.

+

This function is similar to the function bdiag from the package +Matrix. It is also similar to the function adiag from the package +magic. However, here no padding is done and all the dimnames are +stripped.

+

+ +

Value

+ +

Returns a spam matrix as described above.

+ +

See also

+ + + + +

Examples

+
A <- diag.spam(2, 4) # 2*I4 +B <- matrix(1,3,3) +AB <- bdiag.spam(A,B) + +# equivalent to: +ABalt <- rbind(cbind( A, matrix(0,nrow(A),ncol(B))), + cbind( matrix(0,nrow(B),ncol(A)), B)) + +norm(AB-ABalt)
#> [1] 0
+ +# Matrices do not need to be square: +bdiag.spam(1,2:5,6)
#> [,1] [,2] [,3] +#> [1,] 1 0 0 +#> [2,] 0 2 0 +#> [3,] 0 3 0 +#> [4,] 0 4 0 +#> [5,] 0 5 0 +#> [6,] 0 0 6 +#> Class 'spam' (32-bit)
+
+ +
+ + +
+ + + + + + diff --git a/public/reference/chol.html b/public/reference/chol.html new file mode 100644 index 0000000000000000000000000000000000000000..287da453bd8b1d28def5e456de83ae343b59bbaf --- /dev/null +++ b/public/reference/chol.html @@ -0,0 +1,341 @@ + + + + + + + + +Cholesky Factorization for Sparse Matrices — chol • spam + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + + +
+ +
+
+ + +
+ +

chol performs a Cholesky +decomposition of a symmetric positive definite sparse matrix x +of class spam.

+ +
+ +
# chol(x, \dots)
+
+# S4 method for spam
+chol(x, pivot = "MMD", method = "NgPeyton",
+       memory = list(), eps = getOption("spam.eps"), Rstruct=NULL, …)
+
+
+# update.spam.chol.NgPeyton(object, x,...)
+# S4 method for spam.chol.NgPeyton
+update(object, x,...)
+
+ +

Arguments

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
x

symmetric positive definite matrix of class spam.

pivot

should the matrix be permuted, and if, with what + algorithm, see ‘Details’ below.

method

Currently, only NgPeyton is implemented.

memory

Parameters specific to the method, see ‘Details’ below.

eps

threshold to test symmetry. Defaults to getOption("spam.eps").

Rstruct

sparsity structure of the factor, see ‘Details’ below.

further arguments passed to or from other methods.

object

an object from a previous call to chol, i.e., + sparsity structure of the factor.

+ +

Value

+ +

The function returns the Cholesky factor in an object of class + spam.chol.method. Recall that the latter is the Cholesky + factor of a reordered matrix x, see also ordering.

+ +

Details

+ +

chol performs a Cholesky decomposition of a symmetric +positive definite sparse matrix x of class +spam. Currently, there is only the block sparse Cholesky +algorithm of Ng and Peyton (1993) implemented (method="NgPeyton").

+

To pivot/permute the matrix, you can choose between the multiple minimum +degree (pivot="MMD") or reverse Cuthill-Mckee (pivot="RCM") +from George and Lui (1981). It is also possible to furnish a specific +permutation in which case pivot is a vector. For compatibility +reasons, pivot can also take a logical in which for FALSE +no permutation is done and for TRUE is equivalent to +MMD.

+

Often the sparsity structure is fixed and does not change, but the +entries do. In those cases, we can update the Cholesky factor with +update.spam.chol.NgPeyton by suppling a Cholesky factor and the +updated matrix. For U <- chol(A), update(U, + Anew) and chol(Anew, Rstruct=U) are equivalent.

+

The option cholupdatesingular determines how singular matrices +are handled by update. The function hands back an error +("error"), a warning ("warning") or the value NULL +("null").

+

The Cholesky decompositions requires parameters, linked to memory +allocation. If the default values are too small the Fortran routine +returns an error to R, which allocates more space and calls the Fortran +routine again. The user can also pass better estimates of the allocation +sizes to chol with the argument memory=list(nnzR=..., + nnzcolindices=...). The minimal sizes for a fixed sparsity +structure can be obtained from a summary call, see ‘Examples’.

+

The output of chol can be used with forwardsolve and +backsolve to solve a system of linear equations.

+

Notice that the Cholesky factorization of the package SparseM is also +based on the algorithm of Ng and Peyton (1993). Whereas the Cholesky +routine of the package Matrix are based on +CHOLMOD by Timothy A. Davis (C code).

+ +

References

+ +

Ng, E. G. and Peyton, B. W. (1993) Block sparse Cholesky algorithms on +advanced uniprocessor computers, SIAM J. Sci. Comput., 14, +1034--1056.

+

Gilbert, J. R., Ng, E. G. and Peyton, B. W. (1994) An efficient + algorithm to compute row and column counts for sparse Cholesky + factorization, SIAM J. Matrix Anal. Appl., 15, + 1075--1091.

+

George, A. and Liu, J. (1981) +Computer Solution of Large Sparse Positive Definite Systems, +Prentice Hall.

+ +

Note

+ +

Although the symmetric structure of x is needed, only the upper +diagonal entries are used. By default, the code does check for +symmetry (contrarily to base:::chol). However, +depending on the matrix size, this is a time consuming test. +A test is ignored if +options("spam.cholsymmetrycheck") is set to FALSE.

+

If a permutation is supplied with pivot, +options("spam.cholpivotcheck") determines if the permutation is +tested for validity (defaults to TRUE).

+ +

See also

+ + + + +

Examples

+
# generate multivariate normals: +set.seed(13) +n <- 25 # dimension +N <- 1000 # sample size +Sigma <- .25^abs(outer(1:n,1:n,"-")) +Sigma <- as.spam( Sigma, eps=1e-4) + +cholS <- chol( Sigma) +# cholS is the upper triangular part of the permutated matrix Sigma +iord <- ordering(cholS, inv=TRUE) + +R <- as.spam(cholS) +mvsample <- ( array(rnorm(N*n),c(N,n)) %*% R)[,iord] +# It is often better to order the sample than the matrix +# R itself. + +# 'mvsample' is of class 'spam'. We need to transform it to a +# regular matrix, as there is no method 'var' for 'spam' (should there?). +norm( var( as.matrix( mvsample)) - Sigma, type='m')
#> [1] 0.1119671
norm( t(R) %*% R - Sigma)
#> [1] 0.6665039
+ +# To speed up factorizations, memory allocations can be optimized: +opt <- summary(cholS) +# here, some elements of Sigma may be changed... +cholS <- chol( Sigma, memory=list(nnzR=opt$nnzR,nnzcolindices=opt$nnzc))
+
+ +
+ + +
+ + + + + + diff --git a/public/reference/circulant.html b/public/reference/circulant.html new file mode 100644 index 0000000000000000000000000000000000000000..32a43256b98d82e44fb3a04e070d430aa2574809 --- /dev/null +++ b/public/reference/circulant.html @@ -0,0 +1,220 @@ + + + + + + + + +Create Circulant Matrices — circulant • spam + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + + +
+ +
+
+ + +
+ +

Creates a circulant matrix in spam format.

+ +
+ +
circulant.spam(x, n = NULL, eps = getOption("spam.eps"))
+ +

Arguments

+ + + + + + + + + + + + + + +
x

the first row to form the circulant matrix or a list + containing the indices and the nonzero values.

n

if x is a list, the dimension of the matrix.

eps

A tolerance parameter: elements of x such that abs(x) + <= eps set to zero. Defaults to eps = getOption("spam.eps")

+ +

Value

+ +

The circulant matrix in spam format.

+ +

See also

+ +

circulant from package magic, toeplitz.spam

+ + +

Examples

+
circulant.spam(c(1,.25,0,0,0))
#> [,1] [,2] [,3] [,4] [,5] +#> [1,] 1.00 0.25 0.00 0.00 0.00 +#> [2,] 0.00 1.00 0.25 0.00 0.00 +#> [3,] 0.00 0.00 1.00 0.25 0.00 +#> [4,] 0.00 0.00 0.00 1.00 0.25 +#> [5,] 0.25 0.00 0.00 0.00 1.00 +#> Class 'spam' (32-bit)
+
+ +
+ + +
+ + + + + + diff --git a/public/reference/cleanup.html b/public/reference/cleanup.html new file mode 100644 index 0000000000000000000000000000000000000000..7d7f0d939ba775c9f268b82775e3744cdba7ec1c --- /dev/null +++ b/public/reference/cleanup.html @@ -0,0 +1,216 @@ + + + + + + + + +Cleaning up sparse matrices — cleanup • spam + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + + +
+ +
+
+ + +
+ +

Eliminates an zeros in a sparse matrix.

+ +
+ +
cleanup(x, eps = getOption("spam.eps"))
+ +

Arguments

+ + + + + + + + + + +
x

a sparse matrix of class spam.

eps

numeric scalar > 0. Smaller entries are coerced to zero.

+ +

Details

+ +

A sparse matrix may still contain zeros. This function (aliased + to as.spam) filters these values.
+ This often causes confusion when testing such matrices for symmetry + or comparing apparently equal matrices with all.equal + (see ‘Examples’ below.

+ +

See also

+ + + + +

Examples

+
A <- diag.spam(2) +A[1,2] <- 0 + +all.equal(A, t(A))
#> [1] "Column-sparsity structure differ (at least 1 instance(s))" +#> [2] "Row-sparsity structure differ (at least 1 instance(s))"
#> [1] TRUE
all.equal(cleanup(A), diag.spam(2))
#> [1] TRUE
+
+ +
+ + +
+ + + + + + diff --git a/public/reference/coerce.html b/public/reference/coerce.html new file mode 100644 index 0000000000000000000000000000000000000000..5a1a0edcab92de23478a6a7565aa2be8dfb507d5 --- /dev/null +++ b/public/reference/coerce.html @@ -0,0 +1,196 @@ + + + + + + + + +Force a <code>spam</code> Object to Belong to a Class — coerce-methods • spam + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + + +
+ +
+
+ + +
+ +

These functions manage the relations that allow coercing a spam object + to a given class.

+ +
+ + +

Methods

+ +
+
signature(from = "spam", to = "matrix")

this is essentially equivalent to as.matrix(object).

+
signature(from = "spam", to = "list")

this is essentially equivalent to triplet(object).

+
signature(from = "spam", to = "vector")

this is essentially equivalent to object@entries + (structurebased=TRUE) or c(object).

+
signature(from = "spam", to = "logical")

the entries are forced to logicals (nonzeros only in case of + structurebased=TRUE).

+
signature(from = "spam", to = "integer")

the entries are forced to integers (nonzeros only in case of + structurebased=TRUE).

+
+ + +

Examples

+
ifelse( diag.spam(2)*c(0,1), TRUE, FALSE)
#> [1] FALSE TRUE
+
+ +
+ + +
+ + + + + + diff --git a/public/reference/coercion.html b/public/reference/coercion.html new file mode 100644 index 0000000000000000000000000000000000000000..54597b6f9fc992d279abab9c2a747452f85f65a2 --- /dev/null +++ b/public/reference/coercion.html @@ -0,0 +1,228 @@ + + + + + + + + +Coercion to a Vector — Coercion • spam + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + + +
+ +
+
+ + +
+ +

Coercion of spam matrices to proper vector objects

+ +
+ +
# S4 method for spam
+as.vector(x, mode = "any")
+ +

Arguments

+ + + + + + + + + + +
x

spam object.

mode

character string naming an atomic mode or "any"/"list"/"expression".

+ +

Value

+ +

If structurebased=TRUE, the vector x@entries.
+ Conversely, if structurebased=FALSE, the result is identical to + one with as.vector(as.matrix(x)).

+ +

Details

+ +

This coercion allows smooth transitions between different + matrix formats, see example below.
+The Cholesky factors are first transformed to a spam object.

+ +

See also

+ + + + +

Examples

+
x <- diag(2) +ifelse( x, x, 1-x)
#> [,1] [,2] +#> [1,] 1 1 +#> [2,] 1 1
ifelse( x, as.vector(x), 1-as.vector(x))
#> [,1] [,2] +#> [1,] 1 1 +#> [2,] 1 1
+x <- diag.spam(2) +options(spam.structurebased=FALSE) +ifelse( x, as.vector(x), 1-as.vector(x))
#> [1] 1 1 1 1
options(spam.structurebased=TRUE) +ifelse( x, as.vector(x), 1-as.vector(x))
#> [1] 1 1
+
+ +
+ + +
+ + + + + + diff --git a/public/reference/complexity.html b/public/reference/complexity.html new file mode 100644 index 0000000000000000000000000000000000000000..2975d43bb616bba239a954c430bf272acf7acb12 --- /dev/null +++ b/public/reference/complexity.html @@ -0,0 +1,218 @@ + + + + + + + + +Complexity for Sparse Matrices — complexity • spam + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + + +
+ +
+
+ + +
+ +

A few results of computational complexities for selected + sparse algoritms in spam

+ +
+ + +

Details

+ +

A Cholesky factorization of an n-matrix requires n^3/3 flops. In case of + banded matrices (bandwidth p, p<<n) a factorization requires about 2np^2 + flops. Forward- and backsolves for banded matrices require essentially + 2np flops.

+

George and Liu (1981) proves that any reordering would require at + least O(n^3/2) flops for the factorization and produce at least O(n + log(n)) fill-ins for square lattices with a local neighbor hood.
+ They also show that algorithms based on nested dissection are optimal + in the order of magnitude sense.

+

More to follow.

+

+ + + +

+

+

+ +

References

+ +

+ + + +George, A. and Liu, J. (1981) +Computer Solution of Large Sparse Positive Definite Systems, +Prentice Hall.

+ +

See also

+ + + + +
+ +
+ + +
+ + + + + + diff --git a/public/reference/constructors.html b/public/reference/constructors.html new file mode 100644 index 0000000000000000000000000000000000000000..9f7c0bad3bd0ac55cf5a87d37995cab6859fb5b2 --- /dev/null +++ b/public/reference/constructors.html @@ -0,0 +1,216 @@ + + + + + + + + +Slot modification — constructors • spam + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + + +
+ +
+
+ + +
+ +

Modify slots of spam objects

+ +
+ +
rowpointers( x) <- value
+colindices( x) <- value
+entries( x) <- value
+ +

Arguments

+ + + + + + + + + + +
x

a spam matrix

value

vector of appropriate length.

+ +

Value

+ +

Modified spam object.

+ +

Details

+ +

Various tests are performed. Thus much slower than direct + assignment.
+ Slot dimension should be changed through pad or dim

+ + +

Examples

+
x <- diag.spam( 2) +rowpointers( x) <- c(1,1,3) + +# The last line is equivalent to +x@rowpointers <- as.integer( c(1,1,3))
+
+ +
+ + +
+ + + + + + diff --git a/public/reference/cov.html b/public/reference/cov.html new file mode 100644 index 0000000000000000000000000000000000000000..ee32e0736acac9813a8e33fc9d41b99c133cf286 --- /dev/null +++ b/public/reference/cov.html @@ -0,0 +1,266 @@ + + + + + + + + +Covariance functions — covmat • spam + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + + +
+ +
+
+ + +
+ +

Evaluate a covariance function.

+ +
+ +
covmat(h, theta, ... , type="sph")
+
+cov.exp(h, theta, ... , eps= getOption("spam.eps"))
+cov.sph(h, theta, ... , eps= getOption("spam.eps"))
+cov.nug(h, theta, ... , eps= getOption("spam.eps"))
+cov.wu1(h, theta, ... , eps= getOption("spam.eps"))
+cov.wu2(h, theta, ... , eps= getOption("spam.eps"))
+cov.wu3(h, theta, ... , eps= getOption("spam.eps"))
+cov.wend1(h, theta, ... , eps= getOption("spam.eps"))
+cov.wend2(h, theta, ... , eps= getOption("spam.eps"))
+cov.mat(h, theta, ... , eps= getOption("spam.eps"))
+ +

Arguments

+ + + + + + + + + + + + + + + + + + + + + + +
h

object containing the lags.

theta

parameter of the covariance function, see + ‘Details’.

type

covariance function specification.

arguments passed from other methods.

eps

tolerance level.

+ +

Details

+ +

covmat is a wrapper that calls the other functions + according to the argument type. The nomenclature is similar to + premat +
+ The parametrization is (range, partial- sill, [smoothness = 1], [nugget = 0]), where + only the range needs to be specified. + In case of negative parameter values, a warning is issued and the + absolute value is retained. + Although more cryptic, having all arguments as a single vector + simplifies optimization with optim. +
+ Currently, the functions distinguish between a sparse spam + object h and any other numeric type. In the future, this might + change and appropriate methods will be implemented.

+ +

Value

+ +

Covariance function evaluated on h.

+ +

References

+ +

Any classical book about geostatistics.

+ +

See also

+ + + + +

Examples

+
locs <- cbind(runif(10),runif(10)) +h <- nearest.dist(locs, delta=.3) +Sigma <- cov.sph(h, c(.3, 1, .1))
# NOT RUN { +h <- seq(0, to=1, length.out=100) +plot( h, cov.exp(h, c(1/3,1)), type='l', ylim=c(0,1)) +type <- c("sph","wendland1","wendland2","wu1","wu2","wu3") +for (i in 1:6) + lines( h, covmat(h, 1, type=type[i]), col=i+1) +legend('topright',legend=type, col=2:7, lty=1) + +# }
+
+ +
+ + +
+ + + + + + diff --git a/public/reference/crossprod.html b/public/reference/crossprod.html new file mode 100644 index 0000000000000000000000000000000000000000..57caa5d61f099cc36f8f085a2823b49e29f40311 --- /dev/null +++ b/public/reference/crossprod.html @@ -0,0 +1,218 @@ + + + + + + + + +Spam Matrix Crossproduct — crossprod • spam + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + + +
+ +
+
+ + +
+ +

Given matrices x and y as arguments, return a matrix + cross-product. This is formally equivalent to (but usually + slightly faster than) the call t(x) %*% y (crossprod.spam) or x + %*% t(y) (tcrossprod.spam).

+ +
+ +
crossprod.spam(x, y = NULL)
+
+     tcrossprod.spam(x, y = NULL)
+ +

Arguments

+ + + + + + +
x, y

matrices: y = NULL is taken to be the + same matrix as x. Vectors are promoted to single-column or + single-row matrices, depending on the context.

+ +

Value

+ +

A double matrix

+ +

Note

+ +

When x or y are not matrices, they are treated as column or + row matrices.

+ + +

Examples

+
crossprod.spam(diag.spam(2),1:2)
#> [,1] +#> [1,] 1 +#> [2,] 2
+
+
+ +
+ + +
+ + + + + + diff --git a/public/reference/defunct.html b/public/reference/defunct.html new file mode 100644 index 0000000000000000000000000000000000000000..1c47de538537f6ec4cea4d881bf537a9c3a811f6 --- /dev/null +++ b/public/reference/defunct.html @@ -0,0 +1,193 @@ + + + + + + + + +Defunct Objects in Package <span class="pkg">spam</span> — spam-defunct • spam + + + + + + + + + + + + + + + + + + + + + + + +spam — spam-defunct" /> + + + + + + + + + + + + + + + + + + + + +
+
+ + + +
+ +
+
+ + +
+ +

The functions or variables listed here are defunct, i.e. thorw an error when used.

+ +
+ +
validspamobject(...)
+ +

Arguments

+ + + + + + +
...

some arguments

+ +

See also

+ + + + +
+ +
+ + +
+ + + + + + diff --git a/public/reference/det.html b/public/reference/det.html new file mode 100644 index 0000000000000000000000000000000000000000..9d9780e3607a12d5abbc1374f2251a8ef48ca731 --- /dev/null +++ b/public/reference/det.html @@ -0,0 +1,264 @@ + + + + + + + + +Calculate the determinant of a positive definite Sparse Matrix — det • spam + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + + +
+ +
+
+ + +
+ +

det and determinant calculate the determinant of a + positive definite sparse matrix. determinant returns separately the + modulus of the determinant, optionally on the logarithm scale, and + the sign of the determinant.

+ +
+ +
det(x, ...)
+determinant(x, logarithm = TRUE, ...)
+ +

Arguments

+ + + + + + + + + + + + + + +
x

sparse matrix of class spam or a Cholesky factor of + class spam.chol.NgPeyton.

logarithm

logical; if TRUE (default) return the logarithm of the + modulus of the determinant.

...

Optional arguments. Examples include method argument + and additional parameters used by the method.

+ +

Value

+ +

For det, the determinant of x. For determinant, a list with + components

+

+
modulus

a numeric value. The modulus (absolute value) of the + determinant if logarithm is FALSE; otherwise the + logarithm of the modulus.

+ +
sign

integer; either +1 or -1 according to whether the determinant + is positive or negative.

+ + + +

Details

+ +

If the matrix is not positive definite, the function issues a + warning and returns NA.

+

The determinant is based on the product of the diagonal entries of a + Cholesky factor, i.e. internally, a Cholesky decomposition is + performed. By default, the NgPeyton algorithm with minimal degree + ordering us used. To change the methods or supply additonal parameters + to the Cholesky factorization function, see the help for chol.

+

The determinant of a Cholesky factor is also defined.

+ +

References

+ +

Ng, E. G. and B. W. Peyton (1993) Block sparse Cholesky algorithms +on advanced uniprocessor computers, SIAM J. Sci. Comput., 14, +1034--1056.

+ +

See also

+ + + + +

Examples

+
x <- spam( c(4,3,0,3,5,1,0,1,4),3) +det( x)
#> [1] 40
determinant( x)
#> $modulus +#> [1] 3.688879 +#> attr(,"logarithm") +#> [1] TRUE +#> +#> $sign +#> [1] 1 +#> +#> attr(,"class") +#> [1] "det"
+det( chol( x))
#> [1] 6.324555
+
+ +
+ + +
+ + + + + + diff --git a/public/reference/diag.html b/public/reference/diag.html new file mode 100644 index 0000000000000000000000000000000000000000..29c1259c236993b7d84554ad232025383c4ce0e8 --- /dev/null +++ b/public/reference/diag.html @@ -0,0 +1,256 @@ + + + + + + + + +Sparse Matrix diagonals — diag • spam + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + + +
+ +
+
+ + +
+ +

Extract or replace the diagonal of a matrix, or construct a + diagonal matrix.

+ +
+ +
## diag(x)
+## diag(x=1, nrow, ncol, names = TRUE)
+diag(x) <- value
+
+diag.spam(x=1, nrow, ncol)
+spam_diag(x=1, nrow, ncol)
+diag.spam(x) <- value
+ +

Arguments

+ + + + + + + + + + + + + + +
x

a spam matrix, a vector or a scalar.

nrow, ncol

Optional dimensions for the result.

value

either a single value or a vector of length equal to that of + the current diagonal.

+ +

Value

+ +

If x is a spam matrix then diag(x) returns the diagonal of x.

+

The assignment form sets the diagonal of the sparse matrix x to the + given value(s).

+

diag.spam works as diag for spam matrices: + If x is a vector (or 1D array) of length two or more, then + diag.spam(x) returns a diagonal matrix whose diagonal is + x. + spam_diag is an alias for diag.spam and in the spirit + of the result of diag is a spam object.

+

If x is a vector of length one then diag.spam(x) returns an + identity matrix of order the nearest integer to x. The + dimension of the returned matrix can be specified by nrow and + ncol (the default is square).

+

The assignment form sets the diagonal of the matrix x to the + given value(s).

+ + + +

Details

+ +

Using diag(x) can have unexpected effects if x is a vector + that could be of length one. Use diag(x, nrow = length(x)) for + consistent behaviour.

+ +

See also

+ + + + +

Examples

+
diag.spam(2, 4) # 2*I4
#> [,1] [,2] [,3] [,4] +#> [1,] 2 0 0 0 +#> [2,] 0 2 0 0 +#> [3,] 0 0 2 0 +#> [4,] 0 0 0 2 +#> Class 'spam' (32-bit)
smat <- diag.spam(1:5) +diag( smat)
#> [1] 1 2 3 4 5
diag( smat) <- 5:1 + +# The last line is equivalent to +diag.spam( smat) <- 5:1 + +# Note that diag.spam( 1:5) <- 5:1 not work of course.
+
+ +
+ + +
+ + + + + + diff --git a/public/reference/diff.html b/public/reference/diff.html new file mode 100644 index 0000000000000000000000000000000000000000..6f11efbbbe3ac1e4a7682c86fe2cd0a0ce04d208 --- /dev/null +++ b/public/reference/diff.html @@ -0,0 +1,236 @@ + + + + + + + + +Lagged Differences — diff • spam + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + + +
+ +
+
+ + +
+ +

Returns suitably lagged and iterated differences.

+ +
+ +
# diff.spam(x, lag = 1, differences = 1, ...)
+# S4 method for spam
+diff(x, lag = 1, differences = 1, ...)
+ +

Arguments

+ + + + + + + + + + + + + + + + + + +
x

a spam matrix containing the values to be + differenced.

lag

an integer indicating which lag to use.

differences

an integer indicating the order of the difference.

...

further arguments to be passed to or from methods.

+ +

Value

+ +

A spam matrix with elements similar to + as.spam(diff(as.matrix(x), ...)).

+ +

See also

+ +

diff in base, precmat.

+ + +

Examples

+
# incidence matrix for a RW(3) model +D <- diff.spam(diag.spam(10), lag=1, differences=3) + t(D)
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7] +#> [1,] -1 0 0 0 0 0 0 +#> [2,] 3 -1 0 0 0 0 0 +#> [3,] -3 3 -1 0 0 0 0 +#> [4,] 1 -3 3 -1 0 0 0 +#> [5,] 0 1 -3 3 -1 0 0 +#> [6,] 0 0 1 -3 3 -1 0 +#> [7,] 0 0 0 1 -3 3 -1 +#> [8,] 0 0 0 0 1 -3 3 +#> [9,] 0 0 0 0 0 1 -3 +#> [10,] 0 0 0 0 0 0 1 +#> Class 'spam' (32-bit)
# NOT RUN { + +# } +
+
+ +
+ + +
+ + + + + + diff --git a/public/reference/dim.html b/public/reference/dim.html new file mode 100644 index 0000000000000000000000000000000000000000..ab6f3efffdd5d7a152b167c3230f78a9127543bc --- /dev/null +++ b/public/reference/dim.html @@ -0,0 +1,229 @@ + + + + + + + + +Dimensions of an Object — dim • spam + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + + +
+ +
+
+ + +
+ +

Retrieve or set the dimension of an spam object.

+ +
+ +
# dim(x)
+# dim(x) <- value
+
+ +

Arguments

+ + + + + + + + + + +
x

a spam matrix

value

A numeric two-vector, which is coerced to integer (by truncation).

+ +

Value

+ +

dim retrieves + the dimension slot of the object. It is a vector + of mode integer.

+

The replacemnt method changes the dimension of the object by rearranging.

+ +

Details

+ +

In older version of spam, the behavior of the replacement + method was different and is now implemented in pad.spam.

+ +

See also

+ + + + +

Examples

+
x <- diag(4) +dim(x)<-c(2,8) +x
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] +#> [1,] 1 0 0 0 0 1 0 0 +#> [2,] 0 0 1 0 0 0 0 1
+s <- diag.spam(4) +dim(s) <- c(2,8) # result is different than x + +s <- diag.spam(4) +pad(s) <- c(7,3) # any positive value can be used
+
+ +
+ + +
+ + + + + + diff --git a/public/reference/display-1.png b/public/reference/display-1.png new file mode 100644 index 0000000000000000000000000000000000000000..eca8b944f02993e769cfa3890e9389587f9c56fd Binary files /dev/null and b/public/reference/display-1.png differ diff --git a/public/reference/display-2.png b/public/reference/display-2.png new file mode 100644 index 0000000000000000000000000000000000000000..71a57203369290e3d5f88d676d674e049f0d4a10 Binary files /dev/null and b/public/reference/display-2.png differ diff --git a/public/reference/display.html b/public/reference/display.html new file mode 100644 index 0000000000000000000000000000000000000000..37756aff8203f0cf22d5d12e78c0e65267b3d0e6 --- /dev/null +++ b/public/reference/display.html @@ -0,0 +1,228 @@ + + + + + + + + +Graphially Represent the Nonzero Entries — display • spam + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + + +
+ +
+
+ + +
+ +

The function represents the nonzero entries in a simple + bicolor plot.

+ +
+ +
display(x, ...)
+ +

Arguments

+ + + + + + + + + + +
x

matrix of class spam or spam.chol.NgPeyton.

...

any other arguments passed + to image.default/plot.

+ +

Details

+ +

spam.getOption("imagesize") determines if the sparse matrix is + coerced into a matrix and the plotted with image.default or if + the matrix is simply represented as a scatterplot with + pch=".". The points are scaled according to + cex*getOption("spam.cex")/(nrow + ncol). + For some devices or for non-square matrices, + cex needs probably some adjustment.

+ +

See also

+ + + + +

Examples

+
set.seed(13) + +smat <- spam_random(8) +par(mfcol=c(1,2), pty='s') +options(spam.imagesize = 1000) +display(smat) +options(spam.imagesize = 10) +display(smat, cex=.25)
+ +# very large but very sparse matrix +smat <- spam_random(2^14, distribution=rnorm, density=1e-5, verbose=TRUE)
#> Density is 1.04e-05, specified is 1e-05 (nnz=2798).
par(mfcol=c(1, 1), mai=c(.4,.4,.1,.1), pty='s') +display(smat)
#> Warning: default value for 'cex' in 'display' might not be the optimal choice
+
+ +
+ + +
+ + + + + + diff --git a/public/reference/eigen.html b/public/reference/eigen.html new file mode 100644 index 0000000000000000000000000000000000000000..3469dd69bd3da7a32c1103c6b4032503ed87b22d --- /dev/null +++ b/public/reference/eigen.html @@ -0,0 +1,20610 @@ + + + + + + + + +Eigenvalues for Sparse Matrices — eigen • spam + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + + +
+ +
+
+ + +
+ +

Functions to calculate eigenvalues and eigenvectors of sparse matrices. + It uses the value of spam.options("inefficiencywarning") to dispatch between base::eigen() or the Implicitly Restarted Arnoldi Process, using 'ARPACK'.

+

eigen.spam is a wrapper function of eigen_approx and transforms its output to base::eigen like.

+ +
+ +
eigen.spam(x, nev = 10, symmetric, only.values = FALSE, control = list())
+eigen_approx(x, nev, ncv, nitr, mode, only.values = FALSE, verbose = FALSE, f_routine)
+ +

Arguments

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
x

a matrix of class spam whose nev eigenvalues and eigenvectors are to be computed.

nev

number of eigenvalues to calculate.

symmetric

if TRUE, the matrix is assumed to be symmetric.

only.values

if TRUE, only nev eigenvalues are computed and returned, otherwise nev eigenvalues and eigenvectors are returned.

control

additional options, see ‘Details’.

ncv

see ‘Details’, use the control option for eigen.spam.

nitr

see ‘Details’, use the control option for eigen.spam.

mode

see ‘Details’, use the control option for eigen.spam.

verbose

see ‘Details’, use the control option for eigen.spam.

f_routine

only for eigen_approx, to call the Fortran routine for symmetric matrices set this option to "ds_eigen_f" and for non symmetric to "dn_eigen_f".

+ +

Note

+ +

For non symmetric matrices it can not be guaranteed, that the Fortran code is stable. +The user is advised to choose the control options carefully, see ‘Details’ for more information.

+ +

Value

+ +

A vector of the length corresponding to the dimension of the input matrix. + Containing the required nev eigenvalues. + If requested also the corresponding eigenvectors. + In the non symmetric case, the eigenvalues are returned in a matrix with a column containing the real parts and a column containing the imaginary parts of the eigenvalues. + The eigenvectors are then returned in two matrices.

+ +

Details

+ +

+
mode = "LM":

there are different modes available for this function, each mode returns a different range of eigenvalues. + Also the available modes are dependent, whether the input matrix is symmetric or not: +

+
"LM":

Eigenvalues with largest magnitude (sym, non sym), that is, largest eigenvalues in the Euclidean norm of complex numbers.

+
"SM":

Eigenvalues with smallest magnitude (sym, non sym), that is, smallest eigenvalues in the Euclidean norm of complex numbers.

+
"LR":

Eigenvalues with largest real part (non sym).

+
"SR":

Eigenvalues with smallest real part (non sym).

+
"LI":

Eigenvalues with largest imaginary part (non sym).

+
"SI":

Eigenvalues with smallest imaginary part (non sym).

+
"LA":

Eigenvalues with largest algebraic value (sym), that is, largest eigenvalues inclusive of any negative sign.

+
"SA":

Eigenvalues with smallest algebraic value (syn), that is, smallest eigenvalues inclusive of any negative sign.

+

% item +
ncv:

the largest number of basis vectors that will be used in the Implicitly Restarted Arnoldi Process. + Work per major iteration is proportional to x@dimension[1]*ncv*ncv. + The default is set if symmetric to min(x@dimension[1] + 1, max(2 * nev + 1, 200)) or else to min(x@dimension[1] - 1, max(2 * nev + 1, 100)). + Note, this value should not be chosen arbitrary large, but slightly larger than nev. + Otherwise it could lead to memory allocation problems.

+
nitr:

the maximum number of iterations. + The default is set to ncv + 1000

+
spamflag = FALSE:

if TRUE, the Implicitly Restarted Arnoldi Process is used, independent of the dimension of the respective matrix.

+
verbose = FALSE:

print additional information.

+
cmplxeps:

threshold to determine whether a double value is zero, while transforming the ARPACK output to R class complex. + The default is set to .Machine$double.eps.

+

+ +

References

+ +

Lehoucq, R. B. and Sorensen, D. C. and Yang, C. (1997) ARPACK Users Guide: Solution of Large Scale Eigenvalue Problems by Implicitly Restarted Arnoldi Methods.

+ +

See also

+ +

Option "inefficiencywarning" in spam.options.

+ + +

Examples

+
gmrf <- precmat.GMRFreglat(50, 50, c(.4,.3,.2,.1),'m2p4') +eigen.spam(gmrf, nev = 50, control = list(mode = "LM"))
#> $values +#> [1] 1.828108 1.828108 1.827761 1.827760 1.827192 1.827191 1.826419 1.826419 +#> [9] 1.825474 1.825473 1.825004 1.825004 1.824671 1.824671 1.824315 1.824315 +#> [17] 1.824010 1.824009 1.823175 1.823174 1.823030 1.823029 1.822035 1.822034 +#> [25] 1.821686 1.821686 1.820735 1.820735 1.820207 1.820207 1.820156 1.820155 +#> [33] 1.819748 1.819746 1.819323 1.819323 1.818960 1.818959 1.818583 1.818583 +#> [41] 1.818113 1.818113 1.817626 1.817623 1.816941 1.816940 1.816714 1.816709 +#> [49] 1.816072 1.816068 NA NA NA NA NA NA +#> [57] NA NA NA NA NA NA NA NA +#> [65] NA NA NA NA NA NA NA NA +#> [73] NA NA NA NA NA NA NA NA +#> [81] NA NA NA NA NA NA NA NA +#> [89] NA NA NA NA NA NA NA NA +#> [97] NA NA NA NA NA NA NA NA +#> [105] NA NA NA NA NA NA NA NA +#> [113] NA NA NA NA NA NA NA NA +#> [121] NA NA NA NA NA NA NA NA +#> [129] NA NA NA NA NA NA NA NA +#> [137] NA NA NA NA NA NA NA NA +#> [145] NA NA NA NA NA NA NA NA +#> [153] NA NA NA NA NA NA NA NA +#> [161] NA NA NA NA NA NA NA NA +#> [169] NA NA NA NA NA NA NA NA +#> [177] NA NA NA NA NA NA NA NA +#> [185] NA NA NA NA NA NA NA NA +#> [193] NA NA NA NA NA NA NA NA +#> [201] NA NA NA NA NA NA NA NA +#> [209] NA NA NA NA NA NA NA NA +#> [217] NA NA NA NA NA NA NA NA +#> [225] NA NA NA NA NA NA NA NA +#> [233] NA NA NA NA NA NA NA NA +#> [241] NA NA NA NA NA NA NA NA +#> [249] NA NA NA NA NA NA NA NA +#> [257] NA NA NA NA NA NA NA NA +#> [265] NA NA NA NA NA NA NA NA +#> [273] NA NA NA NA NA NA NA NA +#> [281] NA NA NA NA NA NA NA NA +#> [289] NA NA NA NA NA NA NA NA +#> [297] NA NA NA NA NA NA NA NA +#> [305] NA NA NA NA NA NA NA NA +#> [313] NA NA NA NA NA NA NA NA +#> [321] NA NA NA NA NA NA NA NA +#> [329] NA NA NA NA NA NA NA NA +#> [337] NA NA NA NA NA NA NA NA +#> [345] NA NA NA NA NA NA NA NA +#> [353] NA NA NA NA NA NA NA NA +#> [361] NA NA NA NA NA NA NA NA +#> [369] NA NA NA NA NA NA NA NA +#> [377] NA NA NA NA NA NA NA NA +#> [385] NA NA NA NA NA NA NA NA +#> [393] NA NA NA NA NA NA NA NA +#> [401] NA NA NA NA NA NA NA NA +#> [409] NA NA NA NA NA NA NA NA +#> [417] NA NA NA NA NA NA NA NA +#> [425] NA NA NA NA NA NA NA NA +#> [433] NA NA NA NA NA NA NA NA +#> [441] NA NA NA NA NA NA NA NA +#> [449] NA NA NA NA NA NA NA NA +#> [457] NA NA NA NA NA NA NA NA +#> [465] NA NA NA NA NA NA NA NA +#> [473] NA NA NA NA NA NA NA NA +#> [481] NA NA NA NA NA NA NA NA +#> [489] NA NA NA NA NA NA NA NA +#> [497] NA NA NA NA NA NA NA NA +#> [505] NA NA NA NA NA NA NA NA +#> [513] NA NA NA NA NA NA NA NA +#> [521] NA NA NA NA NA NA NA NA +#> [529] NA NA NA NA NA NA NA NA +#> [537] NA NA NA NA NA NA NA NA +#> [545] NA NA NA NA NA NA NA NA +#> [553] NA NA NA NA NA NA NA NA +#> [561] NA NA NA NA NA NA NA NA +#> [569] NA NA NA NA NA NA NA NA +#> [577] NA NA NA NA NA NA NA NA +#> [585] NA NA NA NA NA NA NA NA +#> [593] NA NA NA NA NA NA NA NA +#> [601] NA NA NA NA NA NA NA NA +#> [609] NA NA NA NA NA NA NA NA +#> [617] NA NA NA NA NA NA NA NA +#> [625] NA NA NA NA NA NA NA NA +#> [633] NA NA NA 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3.875059e-02 -8.640516e-03 -5.477438e-02 -1.116518e-02 -3.806770e-02 +#> [1877,] -3.969913e-02 -4.206616e-03 5.523495e-02 -6.775971e-03 3.713823e-02 +#> [1878,] 3.633366e-02 1.668505e-02 -4.979202e-02 2.375917e-02 -3.245832e-02 +#> [1879,] -2.903152e-02 -2.734040e-02 3.916401e-02 -3.785190e-02 2.483458e-02 +#> [1880,] 1.865765e-02 3.494164e-02 -2.469041e-02 4.750828e-02 -1.537595e-02 +#> [1881,] -6.452623e-03 -3.864173e-02 8.148567e-03 -5.176152e-02 5.327043e-03 +#> [1882,] -6.130579e-03 3.808649e-02 8.485534e-03 5.033552e-02 4.106127e-03 +#> [1883,] 1.761842e-02 -3.345632e-02 -2.329835e-02 -4.366121e-02 -1.191258e-02 +#> [1884,] -2.671041e-02 2.543486e-02 3.468504e-02 3.279693e-02 1.739333e-02 +#> [1885,] 3.244714e-02 -1.510892e-02 -4.154689e-02 -1.926602e-02 -2.022452e-02 +#> [1886,] -3.432926e-02 3.814278e-03 4.341322e-02 4.836302e-03 2.046161e-02 +#> [1887,] 3.237052e-02 7.050470e-03 -4.047281e-02 8.725952e-03 -1.848857e-02 +#> [1888,] -2.707828e-02 -1.621507e-02 3.351275e-02 -1.989201e-02 1.492437e-02 +#> [1889,] 1.936472e-02 2.270630e-02 -2.377667e-02 2.756325e-02 -1.050477e-02 +#> [1890,] -1.040277e-02 -2.596992e-02 1.276508e-02 -3.119424e-02 5.959921e-03 +#> [1891,] 1.448509e-03 2.592887e-02 -2.009106e-03 3.083075e-02 -1.907426e-03 +#> [1892,] 6.343209e-03 -2.296995e-02 -7.148559e-03 -2.706066e-02 -1.223667e-03 +#> [1893,] -1.208237e-02 1.786231e-02 1.373324e-02 2.088801e-02 3.228633e-03 +#> [1894,] 1.526231e-02 -1.162097e-02 -1.725332e-02 -1.355236e-02 -4.120950e-03 +#> [1895,] -1.581579e-02 5.337259e-03 1.773595e-02 6.322629e-03 4.089852e-03 +#> [1896,] 1.410115e-02 -2.482275e-06 -1.568108e-02 -2.980198e-04 -3.433061e-03 +#> [1897,] -1.082145e-02 -3.647346e-03 1.194433e-02 -3.753571e-03 2.479726e-03 +#> [1898,] 6.890344e-03 5.241780e-03 -7.569662e-03 5.495973e-03 -1.519257e-03 +#> [1899,] -3.266446e-03 -4.824056e-03 3.600102e-03 -5.050447e-03 7.494117e-04 +#> [1900,] 7.829620e-04 2.829802e-03 -8.975592e-04 2.945293e-03 -2.522267e-04 +#> [1901,] 9.523437e-04 -1.350462e-03 -1.854938e-03 -2.300289e-03 -2.552534e-03 +#> [1902,] -2.706234e-03 1.979375e-03 5.023478e-03 3.210170e-03 6.442475e-03 +#> [1903,] 4.856676e-03 -1.536904e-03 -8.735520e-03 -2.185342e-03 -1.065339e-02 +#> [1904,] -6.867283e-03 -1.267032e-04 1.203204e-02 -9.439934e-04 1.403907e-02 +#> [1905,] 8.162059e-03 2.917564e-03 -1.393378e-02 5.918573e-03 -1.553241e-02 +#> [1906,] -8.229560e-03 -6.494606e-03 1.362101e-02 -1.206751e-02 1.434433e-02 +#> [1907,] 6.722664e-03 1.030859e-02 -1.059493e-02 1.839733e-02 -1.011976e-02 +#> [1908,] -3.537199e-03 -1.367943e-02 4.793524e-03 -2.374072e-02 3.024366e-03 +#> [1909,] -1.144658e-03 1.590133e-02 3.359092e-03 2.694306e-02 6.253163e-03 +#> [1910,] 6.855272e-03 -1.635997e-02 -1.298043e-02 -2.705808e-02 -1.658481e-02 +#> [1911,] -1.288918e-02 1.464392e-02 2.283360e-02 2.352195e-02 2.656096e-02 +#> [1912,] 1.839262e-02 -1.063178e-02 -3.149422e-02 -1.627683e-02 -3.469830e-02 +#> [1913,] -2.248335e-02 4.540169e-03 3.755709e-02 5.822671e-03 3.966575e-02 +#> [1914,] 2.438376e-02 3.077310e-03 -3.985207e-02 6.814251e-03 -4.049377e-02 +#> [1915,] -2.354772e-02 -1.138293e-02 3.763671e-02 -2.019459e-02 3.673434e-02 +#> [1916,] 1.976112e-02 1.935419e-02 -3.073525e-02 3.264734e-02 -2.854668e-02 +#> [1917,] -1.319989e-02 -2.591737e-02 1.960104e-02 -4.249482e-02 1.669504e-02 +#> [1918,] 4.434207e-03 3.009703e-02 -5.290261e-03 4.828947e-02 -2.458141e-03 +#> [1919,] 5.624216e-03 -3.116012e-02 -1.065230e-02 -4.902792e-02 -1.253652e-02 +#> [1920,] -1.582681e-02 2.873338e-02 2.638177e-02 4.431058e-02 2.652952e-02 +#> [1921,] 2.493194e-02 -2.287560e-02 -3.999072e-02 -3.442180e-02 -3.786560e-02 +#> [1922,] -3.176855e-02 1.409248e-02 4.976306e-02 2.031714e-02 4.521484e-02 +#> [1923,] 3.539711e-02 -3.289706e-03 -5.440853e-02 -3.517326e-03 -4.774731e-02 +#> [1924,] -3.524466e-02 -8.331282e-03 5.324395e-02 -1.407793e-02 4.523677e-02 +#> [1925,] 3.119398e-02 1.942192e-02 -4.629504e-02 3.043147e-02 -3.807992e-02 +#> [1926,] -2.361298e-02 -2.865982e-02 3.430334e-02 -4.363228e-02 2.723032e-02 +#> [1927,] 1.331857e-02 3.492296e-02 -1.863627e-02 5.214467e-02 -1.405842e-02 +#> [1928,] -1.478588e-03 -3.744041e-02 1.112285e-03 -5.500819e-02 1.596164e-04 +#> [1929,] -1.053563e-02 3.589786e-02 1.623478e-02 5.196027e-02 1.286094e-02 +#> [1930,] 2.132622e-02 -3.048262e-02 -3.141965e-02 -4.346468e-02 -2.359567e-02 +#> [1931,] -2.965087e-02 2.186095e-02 4.276084e-02 3.064217e-02 3.100511e-02 +#> [1932,] 3.458470e-02 -1.109013e-02 -4.909209e-02 -1.511353e-02 -3.452961e-02 +#> [1933,] -3.563893e-02 -5.226166e-04 4.990083e-02 -1.222123e-03 3.412908e-02 +#> [1934,] 3.281964e-02 1.159487e-02 -4.537559e-02 1.644212e-02 -3.024938e-02 +#> [1935,] -2.661825e-02 -2.085280e-02 3.635702e-02 -2.885203e-02 2.372430e-02 +#> [1936,] 1.793525e-02 2.729792e-02 -2.420145e-02 3.719914e-02 -1.563116e-02 +#> [1937,] -7.948176e-03 -3.033319e-02 1.057800e-02 -4.081845e-02 7.124016e-03 +#> [1938,] -2.056635e-03 2.983074e-02 2.770300e-03 3.969218e-02 7.299196e-04 +#> [1939,] 1.086424e-02 -2.613191e-02 -1.426643e-02 -3.441661e-02 -7.086817e-03 +#> [1940,] -1.749673e-02 1.998040e-02 2.270576e-02 2.608374e-02 1.141778e-02 +#> [1941,] 2.134040e-02 -1.239906e-02 -2.740006e-02 -1.609762e-02 -1.354533e-02 +#> [1942,] -2.221603e-02 4.529359e-03 2.824017e-02 5.954163e-03 1.362396e-02 +#> [1943,] 2.038271e-02 2.542108e-03 -2.567072e-02 2.981644e-03 -1.207265e-02 +#> [1944,] -1.647546e-02 -7.943618e-03 2.058373e-02 -9.667905e-03 9.474350e-03 +#> [1945,] 1.138716e-02 1.114066e-02 -1.415000e-02 1.351937e-02 -6.461299e-03 +#> [1946,] -6.113684e-03 -1.200224e-02 7.616124e-03 -1.446402e-02 3.606464e-03 +#> [1947,] 1.586807e-03 1.080145e-02 -2.099779e-03 1.291661e-02 -1.338387e-03 +#> [1948,] 1.478502e-03 -8.150840e-03 -1.584735e-03 -9.675938e-03 -1.081908e-04 +#> [1949,] -2.698308e-03 4.883509e-03 3.043976e-03 5.764735e-03 7.008484e-04 +#> [1950,] 2.077198e-03 -1.899120e-03 -2.348929e-03 -2.239217e-03 -5.798308e-04 +#> [1951,] -1.456247e-03 -2.156133e-04 2.659582e-03 -5.804949e-04 3.359242e-03 +#> [1952,] 2.666204e-03 1.360654e-03 -4.728291e-03 2.823962e-03 -5.668096e-03 +#> [1953,] -3.185344e-03 -3.258380e-03 5.430443e-03 -6.331285e-03 6.035855e-03 +#> [1954,] 2.672798e-03 5.544416e-03 -4.215143e-03 1.038589e-02 -3.928667e-03 +#> [1955,] -9.646826e-04 -7.721095e-03 8.753564e-04 -1.406785e-02 -7.156339e-04 +#> [1956,] -1.879029e-03 9.238674e-03 4.387012e-03 1.640916e-02 7.451394e-03 +#> [1957,] 5.563335e-03 -9.591234e-03 -1.096303e-02 -1.656612e-02 -1.537510e-02 +#> [1958,] -9.586582e-03 8.412099e-03 1.790822e-02 1.398075e-02 2.325487e-02 +#> [1959,] 1.330602e-02 -5.553019e-03 -2.407495e-02 -8.504547e-03 -2.971998e-02 +#> [1960,] -1.603206e-02 1.133185e-03 2.828720e-02 4.623945e-04 3.348101e-02 +#> [1961,] 1.713773e-02 4.451700e-03 -2.953155e-02 9.356268e-03 -3.354595e-02 +#> [1962,] -1.616680e-02 -1.056423e-02 2.713471e-02 -1.978135e-02 2.939765e-02 +#> [1963,] 1.292338e-02 1.640118e-02 -2.089926e-02 2.941146e-02 -2.110400e-02 +#> [1964,] -7.527802e-03 -2.110122e-02 1.117422e-02 -3.680865e-02 9.341788e-03 +#> [1965,] 4.281480e-04 2.386954e-02 1.153365e-03 4.071070e-02 4.670941e-03 +#> [1966,] 7.637015e-03 -2.410160e-02 -1.474812e-02 -4.022957e-02 -1.932586e-02 +#> [1967,] -1.572181e-02 2.148691e-02 2.798744e-02 3.500540e-02 3.283717e-02 +#> [1968,] 2.279224e-02 -1.607648e-02 -3.917292e-02 -2.529146e-02 -4.348562e-02 +#> [1969,] -2.786561e-02 8.301791e-03 4.676292e-02 1.195462e-02 4.985798e-02 +#> [1970,] 3.015064e-02 1.059840e-03 -4.959310e-02 3.611581e-03 -5.104697e-02 +#> [1971,] -2.916797e-02 -1.097100e-02 4.705252e-02 -1.965174e-02 4.678241e-02 +#> [1972,] 2.483261e-02 2.026087e-02 -3.918860e-02 3.426993e-02 -3.747453e-02 +#> [1973,] -1.748500e-02 -2.777641e-02 2.672414e-02 -4.567743e-02 2.416344e-02 +#> [1974,] 7.864221e-03 3.253729e-02 -1.098185e-02 5.243243e-02 -8.382404e-03 +#> [1975,] 2.974759e-03 -3.387262e-02 -6.274993e-03 -5.363769e-02 -8.045093e-03 +#> [1976,] -1.378999e-02 3.151979e-02 2.306460e-02 4.906762e-02 2.324494e-02 +#> [1977,] 2.331060e-02 -2.567010e-02 -3.744142e-02 -3.920626e-02 -3.553461e-02 +#> [1978,] -3.040260e-02 1.695364e-02 4.774836e-02 2.518961e-02 4.363414e-02 +#> [1979,] 3.421928e-02 -6.364231e-03 -5.283132e-02 -8.660002e-03 -4.681716e-02 +#> [1980,] -3.431485e-02 -4.866245e-03 5.218617e-02 -8.447759e-03 4.498285e-02 +#> [1981,] 3.070490e-02 1.542725e-02 -4.601780e-02 2.415871e-02 -3.864129e-02 +#> [1982,] -2.386452e-02 -2.410071e-02 3.520319e-02 -3.671342e-02 2.881750e-02 +#> [1983,] 1.466380e-02 2.992007e-02 -2.116447e-02 4.479141e-02 -1.689089e-02 +#> [1984,] -4.249488e-03 -3.229476e-02 5.670233e-03 -4.767163e-02 4.395187e-03 +#> [1985,] -6.110768e-03 3.108239e-02 9.406028e-03 4.530820e-02 7.190955e-03 +#> [1986,] 1.519944e-02 -2.659850e-02 -2.233593e-02 -3.831222e-02 -1.663303e-02 +#> [1987,] -2.201107e-02 1.956264e-02 3.175931e-02 2.784391e-02 2.308367e-02 +#> [1988,] 2.588204e-02 -1.098863e-02 -3.684936e-02 -1.543215e-02 -2.616208e-02 +#> [1989,] -2.656953e-02 2.035395e-03 3.739753e-02 2.748733e-03 2.596236e-02 +#> [1990,] 2.426876e-02 6.159797e-03 -3.380808e-02 8.628884e-03 -2.299431e-02 +#> [1991,] -1.956652e-02 -1.264097e-02 2.700528e-02 -1.743195e-02 1.806970e-02 +#> [1992,] 1.333850e-02 1.676245e-02 -1.826944e-02 2.286097e-02 -1.215488e-02 +#> [1993,] -6.606434e-03 -1.826839e-02 9.027907e-03 -2.467040e-02 6.213931e-03 +#> [1994,] 3.771246e-04 1.731115e-02 -6.335351e-04 2.316698e-02 -1.066875e-03 +#> [1995,] 4.511923e-03 -1.440667e-02 -5.835442e-03 -1.912454e-02 -2.716401e-03 +#> [1996,] -7.516384e-03 1.033410e-02 9.724206e-03 1.363062e-02 4.872781e-03 +#> [1997,] 8.459425e-03 -5.996104e-03 -1.087717e-02 -7.890693e-03 -5.447912e-03 +#> [1998,] -7.544972e-03 2.262046e-03 9.630852e-03 3.021772e-03 4.747542e-03 +#> [1999,] 5.308105e-03 1.806674e-04 -6.727073e-03 1.316318e-04 -3.250354e-03 +#> [,6] [,7] [,8] [,9] [,10] +#> [1,] 1.340582e-03 -1.322555e-04 1.473409e-03 5.433327e-05 1.251948e-03 +#> [2,] -2.547822e-03 -4.537218e-04 -2.818091e-03 -6.598295e-04 -2.344136e-03 +#> [3,] 3.362017e-03 1.660536e-03 3.761123e-03 1.736357e-03 3.087435e-03 +#> [4,] -3.582683e-03 -3.294511e-03 -4.080324e-03 -3.143075e-03 -3.322491e-03 +#> [5,] 3.098948e-03 5.088644e-03 3.636945e-03 4.687435e-03 2.939229e-03 +#> [6,] -1.907581e-03 -6.736392e-03 -2.398656e-03 -6.140801e-03 -1.894329e-03 +#> [7,] 1.159683e-04 7.932354e-03 4.492743e-04 7.260536e-03 2.241438e-04 +#> [8,] 2.070571e-03 -8.414739e-03 2.016765e-03 -7.817262e-03 1.949665e-03 +#> [9,] -4.375568e-03 8.003987e-03 -4.715206e-03 7.624828e-03 -4.424068e-03 +#> [10,] 6.488702e-03 -6.632111e-03 7.305991e-03 -6.569421e-03 6.929588e-03 +#> [11,] -8.108559e-03 4.358317e-03 -9.436522e-03 4.633562e-03 -9.156152e-03 +#> [12,] 8.984923e-03 -1.368126e-03 1.078869e-02 -1.910720e-03 1.078805e-02 +#> [13,] -8.954727e-03 -2.044634e-03 -1.112346e-02 -1.393087e-03 -1.154440e-02 +#> [14,] 7.966589e-03 5.511266e-03 1.031674e-02 4.972824e-03 1.122011e-02 +#> [15,] -6.090442e-03 -8.633133e-03 -8.381413e-03 -8.453195e-03 -9.721335e-03 +#> [16,] 3.510795e-03 1.103239e-02 5.471784e-03 1.143034e-02 7.089680e-03 +#> [17,] -5.044698e-04 -1.240179e-02 -1.869388e-03 -1.352183e-02 -3.510229e-03 +#> [18,] -2.594207e-03 1.254676e-02 -2.048627e-03 1.441863e-02 -6.994420e-04 +#> [19,] 5.436013e-03 -1.141376e-02 5.854709e-03 -1.393152e-02 5.119508e-03 +#> [20,] -7.701416e-03 9.100673e-03 -9.123899e-03 1.202480e-02 -9.274696e-03 +#> [21,] 9.141082e-03 -5.847447e-03 1.148758e-02 -8.831330e-03 1.269214e-02 +#> [22,] -9.606815e-03 2.007905e-03 -1.268063e-02 4.645381e-03 -1.496270e-02 +#> [23,] 9.068840e-03 1.993780e-03 1.257555e-02 1.071422e-04 1.579702e-02 +#> [24,] -7.617277e-03 -5.716102e-03 -1.119859e-02 -4.918002e-03 -1.506794e-02 +#> [25,] 5.447866e-03 8.755591e-03 8.725839e-03 9.259336e-03 1.283243e-02 +#> [26,] -2.834208e-03 -1.079696e-02 -5.459473e-03 -1.265122e-02 -9.328943e-03 +#> [27,] 9.061550e-05 1.165117e-02 1.787687e-03 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2.572896e-03 1.405393e-02 +#> [1042,] -7.381029e-03 -7.066991e-03 -2.756950e-02 -6.388906e-03 -1.237115e-02 +#> [1043,] 5.330898e-03 1.365704e-02 2.140372e-02 9.302034e-03 9.512410e-03 +#> [1044,] -3.319565e-03 -1.720522e-02 -1.408317e-02 -1.092404e-02 -5.936087e-03 +#> [1045,] 1.663730e-03 1.773851e-02 6.862378e-03 1.109052e-02 2.225080e-03 +#> [1046,] -5.275590e-04 -1.575122e-02 -8.258795e-04 -9.896122e-03 1.017864e-03 +#> [1047,] -7.730027e-05 1.208624e-02 -3.271290e-03 7.682810e-03 -3.281375e-03 +#> [1048,] 2.599147e-04 -7.767074e-03 5.100242e-03 -4.982032e-03 4.243119e-03 +#> [1049,] -1.962044e-04 3.810156e-03 -4.779046e-03 2.420151e-03 -3.835815e-03 +#> [1050,] 6.662591e-05 -1.049163e-03 2.817741e-03 -6.030628e-04 2.268817e-03 +#> [1051,] -2.354556e-03 -6.305202e-05 1.959589e-04 2.980248e-03 1.699208e-03 +#> [1052,] 2.899620e-03 2.275533e-04 -3.180080e-04 -6.681716e-03 -1.400014e-03 +#> [1053,] -1.198200e-03 -5.457643e-04 5.326094e-04 1.009994e-02 -1.062264e-03 +#> [1054,] 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1.980682e-02 -2.927047e-02 +#> [1067,] -4.143347e-02 5.917796e-03 -2.495213e-02 -1.012231e-02 3.386763e-02 +#> [1068,] 4.488528e-02 2.213529e-03 2.791813e-02 -4.589902e-04 -3.506398e-02 +#> [1069,] -4.355170e-02 -1.158055e-02 -2.801572e-02 1.070686e-02 3.280805e-02 +#> [1070,] 3.747091e-02 2.110558e-02 2.477864e-02 -1.947865e-02 -2.749192e-02 +#> [1071,] -2.728966e-02 -2.954764e-02 -1.816476e-02 2.587498e-02 1.987708e-02 +#> [1072,] 1.417853e-02 3.567438e-02 8.607562e-03 -2.935190e-02 -1.096880e-02 +#> [1073,] 3.261086e-04 -3.844754e-02 3.003134e-03 2.977416e-02 1.861030e-03 +#> [1074,] -1.452981e-02 3.719256e-02 -1.541293e-02 -2.740429e-02 6.422426e-03 +#> [1075,] 2.680673e-02 -3.172443e-02 2.715694e-02 2.283293e-02 -1.306535e-02 +#> [1076,] -3.581225e-02 2.240729e-02 -3.675555e-02 -1.686492e-02 1.755492e-02 +#> [1077,] 4.065142e-02 -1.013496e-02 4.292323e-02 1.038235e-02 -1.972353e-02 +#> [1078,] -4.098004e-02 -3.768132e-03 -4.475868e-02 -4.207903e-03 1.973372e-02 +#> [1079,] 3.702621e-02 1.771365e-02 4.188624e-02 -1.009353e-03 -1.801316e-02 +#> [1080,] -2.953204e-02 -3.006040e-02 -3.452519e-02 4.868857e-03 1.515420e-02 +#> [1081,] 1.962714e-02 3.933903e-02 2.347432e-02 -7.245061e-03 -1.179768e-02 +#> [1082,] -8.655270e-03 -4.445653e-02 -1.001206e-02 8.265246e-03 8.521955e-03 +#> [1083,] -2.018580e-03 4.484926e-02 -4.274288e-03 -8.246823e-03 -5.755699e-03 +#> [1084,] 1.119144e-02 -4.056076e-02 1.770545e-02 7.608760e-03 3.727663e-03 +#> [1085,] -1.796996e-02 3.223192e-02 -2.874094e-02 -6.775342e-03 -2.458968e-03 +#> [1086,] 2.186280e-02 -2.100468e-02 3.618884e-02 6.090617e-03 1.795575e-03 +#> [1087,] -2.280808e-02 8.353807e-03 -3.936313e-02 -5.758664e-03 -1.471348e-03 +#> [1088,] 2.113638e-02 4.127410e-03 3.816488e-02 5.819080e-03 1.187039e-03 +#> [1089,] -1.747988e-02 -1.495463e-02 -3.307573e-02 -6.159876e-03 -6.884015e-04 +#> [1090,] 1.264601e-02 2.295823e-02 2.506581e-02 6.562676e-03 -1.722743e-04 +#> [1091,] -7.478000e-03 -2.743631e-02 -1.543160e-02 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1.301882e-02 1.176259e-03 -1.981538e-03 2.134604e-04 -1.339330e-02 +#> [1105,] -1.673383e-02 -6.871615e-04 2.198170e-03 4.835298e-03 1.599702e-02 +#> [1106,] 1.855093e-02 2.621195e-04 -2.089540e-03 -1.131331e-02 -1.643451e-02 +#> [1107,] -1.764145e-02 -1.062787e-04 1.778830e-03 1.825595e-02 1.413121e-02 +#> [1108,] 1.357780e-02 3.229185e-04 -1.492195e-03 -2.450584e-02 -8.939539e-03 +#> [1109,] -6.436990e-03 -8.707829e-04 1.494186e-03 2.889742e-02 1.190509e-03 +#> [1110,] -3.174316e-03 1.556931e-03 -2.008908e-03 -3.044916e-02 8.332006e-03 +#> [1111,] 1.418004e-02 -2.068387e-03 3.144580e-03 2.853224e-02 -1.849142e-02 +#> [1112,] -2.517304e-02 2.037710e-03 -4.839034e-03 -2.298799e-02 2.795804e-02 +#> [1113,] 3.461225e-02 -1.130180e-03 6.840028e-03 1.417475e-02 -3.540341e-02 +#> [1114,] -4.104925e-02 -8.648350e-04 -8.727676e-03 -2.935709e-03 3.969903e-02 +#> [1115,] 4.335017e-02 3.957692e-03 9.978015e-03 -9.507561e-03 -4.008846e-02 +#> [1116,] -4.087843e-02 -7.914981e-03 -1.005831e-02 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1.039934e-02 2.296905e-02 +#> [1817,] 3.116316e-02 -5.068068e-04 -3.765766e-03 -1.926747e-02 -2.069383e-02 +#> [1818,] -2.394902e-02 2.730849e-05 2.512063e-03 2.751009e-02 1.494378e-02 +#> [1819,] 1.414614e-02 1.138592e-03 -1.985429e-03 -3.374030e-02 -5.966409e-03 +#> [1820,] -2.924464e-03 -2.528541e-03 2.472219e-03 3.670238e-02 -5.433262e-03 +#> [1821,] -8.381035e-03 3.581853e-03 -4.015844e-03 -3.549578e-02 1.796772e-02 +#> [1822,] 1.845570e-02 -3.757122e-03 6.391422e-03 2.975640e-02 -3.003073e-02 +#> [1823,] -2.618227e-02 2.651422e-03 -9.133952e-03 -1.976102e-02 3.992665e-02 +#> [1824,] 3.078389e-02 -9.793848e-05 1.161627e-02 6.432559e-03 -4.612957e-02 +#> [1825,] -3.191324e-02 -3.777565e-03 -1.316354e-02 8.759371e-03 4.752978e-02 +#> [1826,] 2.967685e-02 8.553752e-03 1.318390e-02 -2.399066e-02 -4.362571e-02 +#> [1827,] -2.459410e-02 -1.356690e-02 -1.129125e-02 3.733910e-02 3.462769e-02 +#> [1828,] 1.750012e-02 1.801266e-02 7.397084e-03 -4.706504e-02 -2.145362e-02 +#> [1829,] -9.410129e-03 -2.107967e-02 -1.753319e-03 5.187121e-02 5.614642e-03 +#> [1830,] 1.367495e-03 2.209223e-02 -5.063727e-03 -5.109958e-02 1.099275e-02 +#> [1831,] 5.700651e-03 -2.063670e-02 1.222748e-02 4.483270e-02 -2.635618e-02 +#> [1832,] -1.109925e-02 1.664887e-02 -1.878252e-02 -3.388292e-02 3.863581e-02 +#> [1833,] 1.443715e-02 -1.044564e-02 2.379367e-02 1.967100e-02 -4.642596e-02 +#> [1834,] -1.564738e-02 2.693746e-03 -2.649780e-02 -4.013903e-03 4.894869e-02 +#> [1835,] 1.495708e-02 5.680994e-03 2.643267e-02 -1.114376e-02 -4.614946e-02 +#> [1836,] -1.281607e-02 -1.362999e-02 -2.351995e-02 2.401134e-02 3.868131e-02 +#> [1837,] 9.799123e-03 2.014188e-02 1.808765e-02 -3.320210e-02 -2.778251e-02 +#> [1838,] -6.500326e-03 -2.439987e-02 -1.082684e-02 3.791288e-02 1.506991e-02 +#> [1839,] 3.437447e-03 2.591006e-02 2.688594e-03 -3.800719e-02 -2.283317e-03 +#> [1840,] -9.815296e-04 -2.457894e-02 5.262329e-03 3.399225e-02 -8.976622e-03 +#> [1841,] -6.789006e-04 2.072656e-02 -1.201354e-02 -2.689789e-02 1.747486e-02 +#> [1842,] 1.536304e-03 -1.503250e-02 1.676323e-02 1.808135e-02 -2.249809e-02 +#> [1843,] -1.729527e-03 8.422926e-03 -1.904431e-02 -8.993107e-03 2.391876e-02 +#> [1844,] 1.486108e-03 -1.917143e-03 1.879332e-02 9.440532e-04 -2.216155e-02 +#> [1845,] -1.055123e-03 -3.541076e-03 -1.635243e-02 5.088046e-03 1.808327e-02 +#> [1846,] 6.450828e-04 7.239214e-03 1.240336e-02 -8.582930e-03 -1.279116e-02 +#> [1847,] -3.795381e-04 -8.812473e-03 -7.844155e-03 9.516586e-03 7.433877e-03 +#> [1848,] 2.787503e-04 8.300904e-03 3.628743e-03 -8.313162e-03 -3.002087e-03 +#> [1849,] -2.699849e-04 -6.136826e-03 -5.964493e-04 5.723775e-03 1.722831e-04 +#> [1850,] 2.227592e-04 3.063775e-03 -6.797757e-04 -2.657892e-03 7.819661e-04 +#> [1851,] 3.375628e-03 5.579110e-04 -2.893928e-03 9.505584e-05 1.782812e-03 +#> [1852,] -7.084572e-03 7.188856e-04 5.654345e-03 -1.188501e-03 -3.007621e-03 +#> [1853,] 1.002132e-02 -3.529398e-03 -7.439912e-03 2.788659e-03 3.306602e-03 +#> [1854,] -1.119351e-02 7.247745e-03 7.639109e-03 -4.318590e-03 -2.609649e-03 +#> [1855,] 9.908175e-03 -1.105605e-02 -5.978693e-03 5.266730e-03 1.139879e-03 +#> [1856,] -5.907919e-03 1.410671e-02 2.565695e-03 -5.315865e-03 6.586755e-04 +#> [1857,] -5.654558e-04 -1.568380e-02 2.141723e-03 4.422723e-03 -2.243710e-03 +#> [1858,] 8.790729e-03 1.533468e-02 -7.426511e-03 -2.831193e-03 3.119830e-03 +#> [1859,] -1.766756e-02 -1.294947e-02 1.244909e-02 1.016276e-03 -2.973875e-03 +#> [1860,] 2.588156e-02 8.775748e-03 -1.640175e-02 4.294711e-04 1.769998e-03 +#> [1861,] -3.210809e-02 -3.368339e-03 1.865265e-02 -9.429329e-04 2.176483e-04 +#> [1862,] 3.522243e-02 -2.514679e-03 -1.885392e-02 1.393041e-04 -2.443973e-03 +#> [1863,] -3.448335e-02 8.049258e-03 1.699562e-02 2.069901e-03 4.198211e-03 +#> [1864,] 2.966098e-02 -1.249010e-02 -1.339755e-02 -5.416981e-03 -4.755808e-03 +#> [1865,] -2.108900e-02 1.529652e-02 8.642490e-03 9.292504e-03 3.546979e-03 +#> [1866,] 9.632737e-03 -1.621733e-02 -3.464850e-03 -1.283567e-02 -3.073525e-04 +#> [1867,] 3.421982e-03 1.532071e-02 -1.383780e-03 1.508676e-02 -4.821760e-03 +#> [1868,] -1.654250e-02 -1.296549e-02 5.266936e-03 -1.517488e-02 1.126728e-02 +#> [1869,] 2.815929e-02 9.720833e-03 -7.769452e-03 1.250577e-02 -1.808273e-02 +#> [1870,] -3.688094e-02 -6.250740e-03 8.758890e-03 -6.912928e-03 2.408535e-02 +#> [1871,] 4.168235e-02 3.185638e-03 -8.392264e-03 -1.259394e-03 -2.805069e-02 +#> [1872,] -4.203920e-02 -1.005205e-03 7.068427e-03 1.116150e-02 2.893164e-02 +#> [1873,] 3.799008e-02 -4.606250e-05 -5.336425e-03 -2.153606e-02 -2.606274e-02 +#> [1874,] -3.011886e-02 -4.468016e-06 3.778071e-03 3.089600e-02 1.931159e-02 +#> [1875,] 1.946211e-02 8.745217e-04 -2.887392e-03 -3.775463e-02 -9.147335e-03 +#> [1876,] -7.357017e-03 -2.093350e-03 2.969807e-03 4.087005e-02 -3.389843e-03 +#> [1877,] -4.746298e-03 3.104341e-03 -4.079903e-03 -3.946160e-02 1.681734e-02 +#> [1878,] 1.547971e-02 -3.383941e-03 6.009265e-03 3.336021e-02 -2.941847e-02 +#> [1879,] -2.373117e-02 2.554131e-03 -8.326407e-03 -2.306466e-02 3.949781e-02 +#> [1880,] 2.877396e-02 -4.670926e-04 1.046113e-02 9.691717e-03 -4.564180e-02 +#> [1881,] -3.033498e-02 -2.754126e-03 -1.181738e-02 5.173946e-03 4.694129e-02 +#> [1882,] 2.859568e-02 6.726884e-03 1.189344e-02 -1.970708e-02 -4.313858e-02 +#> [1883,] -2.412979e-02 -1.087056e-02 -1.038693e-02 3.210993e-02 3.467369e-02 +#> [1884,] 1.779088e-02 1.450756e-02 7.264557e-03 -4.087752e-02 -2.262193e-02 +#> [1885,] -1.057004e-02 -1.698551e-02 -2.783451e-03 4.501714e-02 8.532455e-03 +#> [1886,] 3.446580e-03 1.779840e-02 -2.540968e-03 -4.418655e-02 5.805387e-03 +#> [1887,] 2.745249e-03 -1.668444e-02 8.015397e-03 3.872946e-02 -1.862961e-02 +#> [1888,] -7.414927e-03 1.368229e-02 -1.288364e-02 -2.960396e-02 2.846340e-02 +#> [1889,] 1.026776e-02 -9.134884e-03 1.645596e-02 1.821757e-02 -3.432168e-02 +#> [1890,] -1.130662e-02 3.639050e-03 -1.822975e-02 -6.197773e-03 3.583447e-02 +#> [1891,] 1.078795e-02 2.051040e-03 1.797973e-02 -4.863105e-03 -3.326927e-02 +#> [1892,] -9.143587e-03 -7.150179e-03 -1.580104e-02 1.364640e-02 2.745221e-02 +#> [1893,] 6.884699e-03 1.097637e-02 1.209654e-02 -1.929085e-02 -1.960447e-02 +#> [1894,] -4.505825e-03 -1.306928e-02 -7.509562e-03 2.149370e-02 1.112389e-02 +#> [1895,] 2.405397e-03 1.327163e-02 2.812837e-03 -2.051609e-02 -3.350161e-03 +#> [1896,] -8.349959e-04 -1.175827e-02 1.227691e-03 1.709602e-02 -2.646970e-03 +#> [1897,] -1.166015e-04 9.006401e-03 -3.990610e-03 -1.228778e-02 6.222387e-03 +#> [1898,] 5.054371e-04 -5.710824e-03 5.113970e-03 7.258008e-03 -7.229862e-03 +#> [1899,] -4.876638e-04 2.657446e-03 -4.566297e-03 -3.074941e-03 6.007300e-03 +#> [1900,] 2.600775e-04 -5.763305e-04 2.659600e-03 5.268938e-04 -3.279662e-03 +#> [1901,] -2.556030e-03 2.787892e-03 2.067751e-03 -2.587003e-03 -1.147726e-03 +#> [1902,] 3.233056e-03 -6.443542e-03 -2.139191e-03 5.320059e-03 5.268783e-04 +#> [1903,] -1.546573e-03 9.928581e-03 2.904403e-05 -7.339582e-03 1.645993e-03 +#> [1904,] -2.507691e-03 -1.223448e-02 3.981678e-03 8.008231e-03 -4.803001e-03 +#> [1905,] 8.437562e-03 1.257211e-02 -9.205588e-03 -7.034006e-03 8.168429e-03 +#> [1906,] -1.532365e-02 -1.051776e-02 1.467974e-02 4.516055e-03 -1.093779e-02 +#> [1907,] 2.196334e-02 6.091420e-03 -1.934266e-02 -9.069781e-04 1.245513e-02 +#> [1908,] -2.706658e-02 2.443638e-04 2.222581e-02 -3.097957e-03 -1.235275e-02 +#> [1909,] 2.947233e-02 -7.662985e-03 -2.262597e-02 6.720034e-03 1.062575e-02 +#> [1910,] -2.835059e-02 1.511419e-02 2.022944e-02 -9.270920e-03 -7.627548e-03 +#> [1911,] 2.335748e-02 -2.150094e-02 -1.516777e-02 1.030490e-02 3.988446e-03 +#> [1912,] -1.471787e-02 2.586167e-02 7.997370e-03 -9.718064e-03 -4.749167e-04 +#> [1913,] 3.221282e-03 -2.752689e-02 3.919608e-04 7.773983e-03 -2.180924e-03 +#> [1914,] 9.869484e-03 2.622432e-02 -8.922050e-03 -5.050185e-03 3.446229e-03 +#> [1915,] -2.298431e-02 -2.211625e-02 1.649924e-02 2.315927e-03 -3.112324e-03 +#> [1916,] 3.446172e-02 1.576493e-02 -2.218525e-02 -3.655380e-04 1.353063e-03 +#> [1917,] -4.278289e-02 -8.034003e-03 2.533762e-02 -1.590972e-04 1.296760e-03 +#> [1918,] 4.678871e-02 -5.541428e-05 -2.569721e-02 -1.079276e-03 -4.040546e-03 +#> [1919,] -4.584781e-02 7.488236e-03 2.341039e-02 4.018226e-03 5.978608e-03 +#> [1920,] 3.995065e-02 -1.341053e-02 -1.898455e-02 -8.178809e-03 -6.296376e-03 +#> [1921,] -2.971628e-02 1.725069e-02 1.318752e-02 1.273317e-02 4.445463e-03 +#> [1922,] 1.631111e-02 -1.878792e-02 -6.910185e-03 -1.664643e-02 -2.814799e-04 +#> [1923,] -1.292414e-03 1.815910e-02 1.017173e-03 1.886804e-02 -5.870063e-03 +#> [1924,] -1.360039e-02 -1.580606e-02 3.788319e-03 -1.853773e-02 1.323540e-02 +#> [1925,] 2.667577e-02 1.237535e-02 -7.063456e-03 1.516730e-02 -2.070111e-02 +#> [1926,] -3.650865e-02 -8.590555e-03 8.673334e-03 -8.763210e-03 2.699263e-02 +#> [1927,] 4.211678e-02 5.120741e-03 -8.781896e-03 -1.350306e-04 -3.089234e-02 +#> [1928,] -4.307150e-02 -2.468420e-03 7.791003e-03 1.051071e-02 3.145848e-02 +#> [1929,] 3.952936e-02 8.958354e-04 -6.241505e-03 -2.101307e-02 -2.820563e-02 +#> [1930,] -3.218320e-02 -4.006571e-04 4.696227e-03 3.016473e-02 2.121308e-02 +#> [1931,] 2.214310e-02 7.427998e-04 -3.627222e-03 -3.659861e-02 -1.114028e-02 +#> [1932,] -1.076702e-02 -1.514759e-03 3.327848e-03 3.928349e-02 -8.556659e-04 +#> [1933,] -5.323773e-04 2.240195e-03 -3.864917e-03 -3.769912e-02 1.328782e-02 +#> [1934,] 1.047593e-02 -2.480878e-03 5.078140e-03 3.193001e-02 -2.456513e-02 +#> [1935,] -1.807500e-02 1.931259e-03 -6.625125e-03 -2.266129e-02 3.323814e-02 +#> [1936,] 2.273664e-02 -4.828669e-04 8.061689e-03 1.107664e-02 -3.822327e-02 +#> [1937,] -2.430533e-02 -1.753107e-03 -8.940923e-03 1.324455e-03 3.896796e-02 +#> [1938,] 2.303968e-02 4.468527e-03 8.911270e-03 -1.295934e-02 -3.552913e-02 +#> [1939,] -1.953241e-02 -7.221217e-03 -7.794431e-03 2.240553e-02 2.855273e-02 +#> [1940,] 1.458977e-02 9.525356e-03 5.627986e-03 -2.861401e-02 -1.915875e-02 +#> [1941,] -9.091882e-03 -1.095158e-02 -2.664434e-03 3.105756e-02 8.752485e-03 +#> [1942,] 3.856569e-03 1.121720e-02 -6.733430e-04 -2.979086e-02 1.205102e-03 +#> [1943,] 4.724789e-04 -1.024887e-02 3.871917e-03 2.541412e-02 -9.428872e-03 +#> [1944,] -3.498069e-03 8.204895e-03 -6.427475e-03 -1.894923e-02 1.500412e-02 +#> [1945,] 5.094500e-03 -5.452074e-03 7.947257e-03 1.165125e-02 -1.751289e-02 +#> [1946,] -5.385656e-03 2.500240e-03 -8.232677e-03 -4.788912e-03 1.707302e-02 +#> [1947,] 4.684419e-03 9.440108e-05 7.327456e-03 -5.680311e-04 -1.428598e-02 +#> [1948,] -3.407575e-03 -1.839806e-03 -5.520096e-03 3.719260e-03 1.010545e-02 +#> [1949,] 1.983623e-03 2.412018e-03 3.298089e-03 -4.434642e-03 -5.651156e-03 +#> [1950,] -7.708715e-04 -1.732190e-03 -1.260036e-03 2.970388e-03 2.001408e-03 +#> [1951,] -1.139809e-03 -3.376431e-03 1.773699e-03 2.841545e-03 -2.387549e-03 +#> [1952,] 4.294077e-03 5.235585e-03 -5.412589e-03 -3.773254e-03 6.094245e-03 +#> [1953,] -8.797217e-03 -4.844311e-03 1.001015e-02 2.427802e-03 -1.004343e-02 +#> [1954,] 1.364396e-02 1.930200e-03 -1.442795e-02 1.046305e-03 1.312919e-02 +#> [1955,] -1.766275e-02 3.265319e-03 1.750786e-02 -6.033239e-03 -1.443924e-02 +#> [1956,] 1.972265e-02 -1.002479e-02 -1.828621e-02 1.158650e-02 1.343224e-02 +#> [1957,] -1.893999e-02 1.727574e-02 1.617310e-02 -1.662487e-02 -1.003763e-02 +#> [1958,] 1.485044e-02 -2.377312e-02 -1.106698e-02 2.014867e-02 4.661495e-03 +#> [1959,] -7.517631e-03 2.831171e-02 3.385554e-03 -2.143435e-02 1.900034e-03 +#> [1960,] -2.440919e-03 -2.993227e-02 5.991073e-03 2.017093e-02 -8.620583e-03 +#> [1961,] 1.391665e-02 2.808713e-02 -1.585463e-02 -1.651360e-02 1.444421e-02 +#> [1962,] -2.545379e-02 -2.273824e-02 2.485720e-02 1.104607e-02 -1.848735e-02 +#> [1963,] 3.545343e-02 1.437228e-02 -3.172232e-02 -4.661016e-03 2.020324e-02 +#> [1964,] -4.240746e-02 -3.931657e-03 3.544388e-02 -1.616769e-03 -1.947777e-02 +#> [1965,] 4.512714e-02 -7.326148e-03 -3.544085e-02 6.829132e-03 1.663970e-02 +#> [1966,] -4.293112e-02 1.801699e-02 3.164378e-02 -1.026649e-02 -1.238453e-02 +#> [1967,] 3.576420e-02 -2.683501e-02 -2.450072e-02 1.159784e-02 7.628415e-03 +#> [1968,] -2.422787e-02 3.274126e-02 1.490376e-02 -1.092426e-02 -3.321244e-03 +#> [1969,] 9.517092e-03 -3.510677e-02 -4.049881e-03 8.746466e-03 2.559439e-04 +#> [1970,] 6.728187e-03 3.378892e-02 -6.739671e-03 -5.853401e-03 1.088790e-03 +#> [1971,] -2.263590e-02 -2.913178e-02 1.621181e-02 3.154354e-03 -6.418015e-04 +#> [1972,] 3.634849e-02 2.189348e-02 -2.335522e-02 -1.487568e-03 -1.249175e-03 +#> [1973,] -4.626806e-02 -1.311524e-02 2.752889e-02 1.442860e-03 3.890238e-03 +#> [1974,] 5.126529e-02 3.955437e-03 -2.852788e-02 -3.232936e-03 -6.384031e-03 +#> [1975,] -5.082284e-02 4.483218e-03 2.657926e-02 6.638720e-03 7.814942e-03 +#> [1976,] 4.509401e-02 -1.130608e-02 -2.227328e-02 -1.103984e-02 -7.441081e-03 +#> [1977,] -3.486994e-02 1.593567e-02 1.644460e-02 1.552522e-02 4.855987e-03 +#> [1978,] 2.146162e-02 -1.816310e-02 -1.002589e-02 -1.906334e-02 -8.860356e-05 +#> [1979,] -6.515501e-03 1.813792e-02 3.899389e-03 2.070062e-02 -6.378366e-03 +#> [1980,] -8.209824e-03 -1.630256e-02 1.229668e-03 -1.975057e-02 1.367634e-02 +#> [1981,] 2.107362e-02 1.328673e-02 -4.918595e-03 1.593806e-02 -2.069343e-02 +#> [1982,] -3.075736e-02 -9.782469e-03 7.021142e-03 -9.471418e-03 2.626631e-02 +#> [1983,] 3.641604e-02 6.422557e-03 -7.668449e-03 1.027837e-03 -2.938646e-02 +#> [1984,] -3.775776e-02 -3.682850e-03 7.204994e-03 8.346033e-03 2.938342e-02 +#> [1985,] 3.504387e-02 1.823294e-03 -6.094719e-03 -1.740129e-02 -2.605156e-02 +#> [1986,] -2.901374e-02 -8.746734e-04 4.816666e-03 2.489218e-02 1.969701e-02 +#> [1987,] 2.074911e-02 6.694301e-04 -3.769999e-03 -2.978336e-02 -1.109621e-02 +#> [1988,] -1.150061e-02 -9.070110e-04 3.205177e-03 3.142219e-02 1.372921e-03 +#> [1989,] 2.503953e-03 1.238517e-03 -3.192179e-03 -2.964591e-02 8.184441e-03 +#> [1990,] 5.187265e-03 -1.352757e-03 3.629065e-03 2.480508e-02 -1.633306e-02 +#> [1991,] -1.082878e-02 1.046567e-03 -4.286354e-03 -1.769968e-02 2.207672e-02 +#> [1992,] 1.405697e-02 -2.661438e-04 4.876077e-03 9.439934e-03 -2.482264e-02 +#> [1993,] -1.489779e-02 -8.876805e-04 -5.130029e-03 -1.256674e-03 2.446766e-02 +#> [1994,] 1.371650e-02 2.189997e-03 4.870525e-03 -5.705184e-03 -2.139965e-02 +#> [1995,] -1.112060e-02 -3.350946e-03 -4.058802e-03 1.057418e-02 1.641541e-02 +#> [1996,] 7.834701e-03 4.090605e-03 2.810892e-03 -1.288896e-02 -1.057060e-02 +#> [1997,] -4.569345e-03 -4.213750e-03 -1.377234e-03 1.266146e-02 4.988770e-03 +#> [1998,] 1.905751e-03 3.668459e-03 8.945067e-05 -1.035729e-02 -6.629778e-04 +#> [1999,] -2.141746e-04 -2.575287e-03 7.157107e-04 6.798034e-03 -1.715770e-03 +#> [,11] [,12] [,13] [,14] [,15] +#> [1,] -9.777349e-04 -6.374138e-04 -1.108604e-03 -1.191302e-03 -5.757383e-05 +#> [2,] 2.120014e-03 6.793418e-04 2.600441e-03 1.596665e-03 -6.360326e-04 +#> [3,] -3.152103e-03 -1.182500e-04 -4.079241e-03 -1.157546e-03 1.987419e-03 +#> [4,] 3.823999e-03 -9.377213e-04 5.175260e-03 -7.441697e-06 -3.806110e-03 +#> [5,] -3.956005e-03 2.287147e-03 -5.617842e-03 1.636060e-03 5.824812e-03 +#> [6,] 3.470127e-03 -3.674213e-03 5.284819e-03 -3.383055e-03 -7.730848e-03 +#> [7,] -2.401966e-03 4.836380e-03 -4.219760e-03 4.894704e-03 9.204846e-03 +#> [8,] 8.915601e-04 -5.551726e-03 2.614798e-03 -5.881795e-03 -9.961694e-03 +#> [9,] 8.443326e-04 5.677252e-03 -7.637341e-04 6.176290e-03 9.788488e-03 +#> [10,] -2.552677e-03 -5.171491e-03 -1.004011e-03 -5.760667e-03 -8.574603e-03 +#> [11,] 3.989366e-03 4.097969e-03 2.394435e-03 4.764755e-03 6.330053e-03 +#> [12,] -4.960700e-03 -2.609786e-03 -3.210140e-03 -3.431655e-03 -3.189863e-03 +#> [13,] 5.353752e-03 9.191153e-04 3.389196e-03 2.060615e-03 -5.961864e-04 +#> [14,] -5.150708e-03 7.419921e-04 -3.014045e-03 -9.393323e-04 4.687308e-03 +#> [15,] 4.425508e-03 -2.161818e-03 2.289127e-03 2.800749e-04 -8.686406e-03 +#> [16,] -3.324524e-03 3.182788e-03 -1.492373e-03 -1.730461e-04 1.218198e-02 +#> [17,] 2.035926e-03 -3.723123e-03 9.110733e-04 5.666547e-04 -1.479282e-02 +#> [18,] -7.541595e-04 3.783239e-03 -7.757296e-04 -1.278707e-03 1.621077e-02 +#> [19,] -3.535013e-04 -3.436883e-03 1.205846e-03 2.036114e-03 -1.623702e-02 +#> [20,] 1.172292e-03 2.809900e-03 -2.179063e-03 -2.534837e-03 1.480794e-02 +#> [21,] -1.653124e-03 -2.051687e-03 3.530221e-03 2.507663e-03 -1.200726e-02 +#> [22,] 1.812186e-03 1.305377e-03 -4.980815e-03 -1.785779e-03 8.062946e-03 +#> [23,] -1.718316e-03 -6.826064e-04 6.193100e-03 3.414021e-04 -3.328065e-03 +#> [24,] 1.471768e-03 2.473063e-04 -6.838107e-03 1.697499e-03 -1.752854e-03 +#> [25,] -1.179404e-03 -1.083818e-05 6.663998e-03 -4.063735e-03 6.689176e-03 +#> [26,] 9.316158e-04 -6.169217e-05 -5.551110e-03 6.395795e-03 -1.099671e-02 +#> [27,] -7.854854e-04 3.633426e-05 3.542613e-03 -8.302600e-03 1.425168e-02 +#> [28,] 7.570551e-04 9.602186e-06 -8.445526e-04 9.435644e-03 -1.613983e-02 +#> [29,] -8.234209e-04 -7.593835e-06 -2.205025e-03 -9.554663e-03 1.649425e-02 +#> [30,] 9.332827e-04 -8.618674e-05 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3.553659e-02 1.074515e-02 +#> [1895,] 2.193625e-02 1.962695e-02 -2.066208e-02 -2.563677e-02 -5.434352e-03 +#> [1896,] -2.349897e-02 -9.889261e-03 2.367514e-02 1.462114e-02 6.149599e-04 +#> [1897,] 2.102660e-02 1.399815e-03 -2.194871e-02 -4.770784e-03 2.996969e-03 +#> [1898,] -1.566814e-02 4.264617e-03 1.678005e-02 -2.111236e-03 -4.882092e-03 +#> [1899,] 9.141886e-03 -6.230965e-03 -1.006938e-02 5.043648e-03 4.836866e-03 +#> [1900,] -3.358430e-03 4.511780e-03 3.883551e-03 -4.028182e-03 -3.032733e-03 +#> [1901,] -5.927396e-04 2.713305e-03 7.667816e-04 -5.188497e-03 -1.316961e-03 +#> [1902,] -6.803742e-04 -5.611247e-03 1.919914e-03 1.020314e-02 3.227115e-03 +#> [1903,] 3.772389e-03 7.700734e-03 -7.600895e-03 -1.330714e-02 -5.094768e-03 +#> [1904,] -8.160079e-03 -8.088374e-03 1.504605e-02 1.315468e-02 6.110686e-03 +#> [1905,] 1.293063e-02 6.195165e-03 -2.250974e-02 -9.104977e-03 -5.494646e-03 +#> [1906,] -1.695944e-02 -1.912960e-03 2.810288e-02 1.390474e-03 2.708686e-03 +#> [1907,] 1.914436e-02 -4.333815e-03 -3.020477e-02 8.899346e-03 2.368306e-03 +#> [1908,] -1.864804e-02 1.163864e-02 2.782810e-02 -2.001853e-02 -9.346745e-03 +#> [1909,] 1.509683e-02 -1.876954e-02 -2.085880e-02 2.991104e-02 1.733882e-02 +#> [1910,] -8.691973e-03 2.440301e-02 1.011734e-02 -3.663274e-02 -2.508044e-02 +#> [1911,] 2.058956e-04 -2.738875e-02 2.775404e-03 3.876183e-02 3.114253e-02 +#> [1912,] 9.140053e-03 2.699080e-02 -1.570386e-02 -3.571096e-02 -3.419190e-02 +#> [1913,] -1.789522e-02 -2.305364e-02 2.647485e-02 2.787353e-02 3.325093e-02 +#> [1914,] 2.465424e-02 1.605628e-02 -3.326118e-02 -1.656980e-02 -2.790388e-02 +#> [1915,] -2.833148e-02 -7.039287e-03 3.497990e-02 3.799739e-03 1.840800e-02 +#> [1916,] 2.837843e-02 -2.583580e-03 -3.152491e-02 8.150075e-03 -5.685583e-03 +#> [1917,] -2.489784e-02 1.129347e-02 2.380264e-02 -1.716681e-02 -8.803279e-03 +#> [1918,] 1.862837e-02 -1.776078e-02 -1.355929e-02 2.174482e-02 2.328461e-02 +#> [1919,] -1.080032e-02 2.110562e-02 3.031113e-03 -2.130172e-02 -3.594959e-02 +#> [1920,] 2.889448e-03 -2.106977e-02 5.513170e-03 1.630932e-02 4.523746e-02 +#> [1921,] 3.681757e-03 1.806296e-02 -1.024555e-02 -8.208420e-03 -5.007090e-02 +#> [1922,] -7.834316e-03 -1.307162e-02 1.016989e-02 -8.790333e-04 5.000329e-02 +#> [1923,] 9.053760e-03 7.446872e-03 -5.348662e-03 8.585435e-03 -4.525470e-02 +#> [1924,] -7.494465e-03 -2.614101e-03 -3.084841e-03 -1.279378e-02 3.663595e-02 +#> [1925,] 3.943401e-03 -2.361774e-04 1.313585e-02 1.209402e-02 -2.538079e-02 +#> [1926,] 3.516029e-04 4.112355e-04 -2.234083e-02 -6.105032e-03 1.292291e-02 +#> [1927,] -3.955351e-03 2.123621e-03 2.826717e-02 -4.406453e-03 -6.623324e-04 +#> [1928,] 5.564639e-03 -6.723618e-03 -2.902041e-02 1.763846e-02 -1.023525e-02 +#> [1929,] -4.306715e-03 1.218951e-02 2.365629e-02 -3.107995e-02 1.897272e-02 +#> [1930,] -4.704617e-05 -1.701600e-02 -1.241268e-02 4.198900e-02 -2.517170e-02 +#> [1931,] 6.999755e-03 1.971388e-02 -3.288734e-03 -4.792987e-02 2.883991e-02 +#> [1932,] -1.540066e-02 -1.913977e-02 2.107525e-02 4.726278e-02 -3.027354e-02 +#> [1933,] 2.365889e-02 1.476378e-02 -3.804799e-02 -3.948821e-02 2.992461e-02 +#> [1934,] -3.005833e-02 -6.817092e-03 5.131742e-02 2.537585e-02 -2.826756e-02 +#> [1935,] 3.311016e-02 -3.714985e-03 -5.854532e-02 -6.850229e-03 2.569556e-02 +#> [1936,] -3.187071e-02 1.525723e-02 5.838722e-02 -1.334655e-02 -2.246717e-02 +#> [1937,] 2.615712e-02 -2.593613e-02 -5.074820e-02 3.215030e-02 1.870906e-02 +#> [1938,] -1.661369e-02 3.394154e-02 3.680192e-02 -4.673235e-02 -1.446629e-02 +#> [1939,] 4.611504e-03 -3.788537e-02 -1.876937e-02 5.501088e-02 9.779201e-03 +#> [1940,] 8.001832e-03 3.708309e-02 -4.986985e-04 -5.601661e-02 -4.760485e-03 +#> [1941,] -1.925364e-02 -3.169844e-02 1.805294e-02 5.004311e-02 -3.536838e-04 +#> [1942,] 2.742850e-02 2.271912e-02 -3.138565e-02 -3.855453e-02 5.196573e-03 +#> [1943,] -3.140316e-02 -1.176541e-02 3.887440e-02 2.387229e-02 -9.313955e-03 +#> [1944,] 3.086418e-02 7.687793e-04 -4.004389e-02 -8.705567e-03 1.224936e-02 +#> [1945,] -2.636183e-02 8.414875e-03 3.559778e-02 -4.378929e-03 -1.365137e-02 +#> [1946,] 1.918686e-02 -1.437809e-02 -2.721908e-02 1.344172e-02 1.337549e-02 +#> [1947,] -1.109383e-02 1.644290e-02 1.718415e-02 -1.751427e-02 -1.155237e-02 +#> [1948,] 3.926922e-03 -1.479042e-02 -7.872017e-03 1.673594e-02 8.600688e-03 +#> [1949,] 7.742554e-04 1.040995e-02 1.270892e-03 -1.226161e-02 -5.175212e-03 +#> [1950,] -2.106258e-03 -4.876210e-03 1.417878e-03 5.964469e-03 2.056052e-03 +#> [1951,] -1.690335e-03 -1.595956e-03 4.197362e-03 3.282741e-03 1.977232e-04 +#> [1952,] 4.320190e-03 1.925553e-03 -9.973856e-03 -3.459359e-03 -7.074228e-04 +#> [1953,] -7.147570e-03 -5.842828e-04 1.559545e-02 5.150465e-05 1.000641e-03 +#> [1954,] 9.293123e-03 -2.438782e-03 -1.927624e-02 6.559968e-03 -5.842659e-04 +#> [1955,] -9.937033e-03 6.728339e-03 1.957445e-02 -1.519404e-02 -8.355945e-04 +#> [1956,] 8.513768e-03 -1.150436e-02 -1.572362e-02 2.410114e-02 3.239567e-03 +#> [1957,] -4.861636e-03 1.576849e-02 7.828017e-03 -3.132193e-02 -6.259413e-03 +#> [1958,] -7.077303e-04 -1.850729e-02 3.121743e-03 3.510051e-02 9.231303e-03 +#> [1959,] 7.438830e-03 1.891066e-02 -1.542005e-02 -3.426622e-02 -1.132790e-02 +#> [1960,] -1.425890e-02 -1.655895e-02 2.697299e-02 2.850348e-02 1.174112e-02 +#> [1961,] 1.997889e-02 1.153733e-02 -3.571544e-02 -1.845034e-02 -9.875487e-03 +#> [1962,] -2.352912e-02 -4.449587e-03 4.003370e-02 5.601564e-03 5.509075e-03 +#> [1963,] 2.418120e-02 -3.675443e-03 -3.910564e-02 7.968238e-03 1.114773e-03 +#> [1964,] -2.170877e-02 1.156954e-02 3.308587e-02 -2.000776e-02 -9.287285e-03 +#> [1965,] 1.645101e-02 -1.796712e-02 -2.309417e-02 2.853753e-02 1.792894e-02 +#> [1966,] -9.262309e-03 2.185493e-02 1.100536e-02 -3.225192e-02 -2.576232e-02 +#> [1967,] 1.354761e-03 -2.267745e-02 9.200771e-04 3.079032e-02 3.153030e-02 +#> [1968,] 5.936738e-03 2.045372e-02 -1.049130e-02 -2.481892e-02 -3.421712e-02 +#> [1969,] -1.142004e-02 -1.577874e-02 1.603621e-02 1.590218e-02 3.322957e-02 +#> [1970,] 1.428720e-02 9.705903e-03 -1.675111e-02 -6.186466e-03 -2.850361e-02 +#> [1971,] -1.428075e-02 -3.532145e-03 1.287697e-02 -2.042535e-03 2.051708e-02 +#> [1972,] 1.174791e-02 -1.471621e-03 -5.662011e-03 6.839877e-03 -1.020781e-02 +#> [1973,] -7.568572e-03 4.329454e-03 -2.885794e-03 -7.020433e-03 -1.184928e-03 +#> [1974,] 2.969195e-03 -4.571954e-03 1.041006e-02 2.429306e-03 1.232365e-02 +#> [1975,] 7.411221e-04 2.344297e-03 -1.469990e-02 5.995387e-03 -2.198105e-02 +#> [1976,] -2.458453e-03 1.614859e-03 1.416683e-02 -1.638498e-02 2.921519e-02 +#> [1977,] 1.535409e-03 -6.122663e-03 -8.206664e-03 2.629980e-02 -3.348197e-02 +#> [1978,] 2.058303e-03 9.801516e-03 -2.630260e-03 -3.321704e-02 3.466920e-02 +#> [1979,] -7.702901e-03 -1.137875e-02 1.669951e-02 3.505062e-02 -3.305379e-02 +#> [1980,] 1.423402e-02 9.974582e-03 -3.154666e-02 -3.059867e-02 2.919833e-02 +#> [1981,] -2.016881e-02 -5.317084e-03 4.436284e-02 1.982689e-02 -2.381434e-02 +#> [1982,] 2.400881e-02 -2.163536e-03 -5.252404e-02 -3.927571e-03 1.762374e-02 +#> [1983,] -2.455854e-02 1.138511e-02 5.410845e-02 -1.486175e-02 -1.124680e-02 +#> [1984,] 2.119351e-02 -2.080233e-02 -4.828894e-02 3.364834e-02 5.136196e-03 +#> [1985,] -1.402029e-02 2.869925e-02 3.552131e-02 -4.940927e-02 4.361446e-04 +#> [1986,] 3.891853e-03 -3.352804e-02 -1.748908e-02 5.955003e-02 -5.345935e-03 +#> [1987,] 7.729041e-03 3.422506e-02 -3.187172e-03 -6.238795e-02 9.554280e-03 +#> [1988,] -1.903473e-02 -3.043874e-02 2.342040e-02 5.746556e-02 -1.303929e-02 +#> [1989,] 2.821452e-02 2.261981e-02 -4.021712e-02 -4.563253e-02 1.574407e-02 +#> [1990,] -3.380540e-02 -1.195228e-02 5.121786e-02 2.888009e-02 -1.755735e-02 +#> [1991,] 3.497970e-02 1.356057e-04 -5.511322e-02 -9.960815e-03 1.833276e-02 +#> [1992,] -3.170965e-02 1.094008e-02 5.185541e-02 -8.130852e-03 -1.794084e-02 +#> [1993,] 2.477279e-02 -1.956109e-02 -4.262584e-02 2.271504e-02 1.633772e-02 +#> [1994,] -1.559455e-02 2.452475e-02 2.956801e-02 -3.191041e-02 -1.362840e-02 +#> [1995,] 5.958096e-03 -2.537204e-02 -1.534148e-02 3.495170e-02 1.010237e-02 +#> [1996,] 2.360304e-03 2.246864e-02 2.588042e-03 -3.228080e-02 -6.225008e-03 +#> [1997,] -7.955456e-03 -1.691556e-02 6.582483e-03 2.539724e-02 2.579141e-03 +#> [1998,] 1.008087e-02 1.030600e-02 -1.098398e-02 -1.650259e-02 2.360466e-04 +#> [1999,] -8.798390e-03 -4.377379e-03 1.054709e-02 8.014675e-03 -1.729120e-03 +#> [,16] [,17] [,18] [,19] [,20] +#> [1,] 1.627092e-03 -2.118607e-03 1.609317e-03 -2.182989e-03 2.148938e-03 +#> [2,] -3.109242e-03 4.701752e-03 -1.896545e-03 5.059618e-03 -2.989979e-03 +#> [3,] 4.169005e-03 -7.126147e-03 8.013791e-04 -8.049960e-03 2.364934e-03 +#> [4,] -4.573341e-03 8.806163e-03 1.467671e-03 1.056269e-02 -3.521510e-04 +#> [5,] 4.167139e-03 -9.302991e-03 -4.475757e-03 -1.208837e-02 -2.749053e-03 +#> [6,] -2.897906e-03 8.405070e-03 7.645767e-03 1.227825e-02 6.469573e-03 +#> [7,] 8.279571e-04 -6.165072e-03 -1.036779e-02 -1.099549e-02 -1.024838e-02 +#> 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3.437051e-03 1.279424e-02 +#> [1848,] 7.230093e-03 -1.208476e-04 8.380045e-04 9.110263e-04 -9.932280e-03 +#> [1849,] -6.216351e-03 5.432394e-04 -6.939915e-04 -2.861817e-03 5.987486e-03 +#> [1850,] 3.548653e-03 -5.324751e-04 2.832909e-04 2.364094e-03 -2.290368e-03 +#> [1851,] 2.857939e-03 5.184721e-03 2.533286e-03 -1.303982e-03 -4.412574e-03 +#> [1852,] -5.250915e-03 -8.105299e-03 -8.013060e-03 -5.215662e-04 9.177180e-03 +#> [1853,] 6.119920e-03 7.542557e-03 1.498380e-02 5.340087e-03 -1.260372e-02 +#> [1854,] -4.602439e-03 -3.080326e-03 -2.158127e-02 -1.221395e-02 1.324631e-02 +#> [1855,] 2.651361e-04 -4.797115e-03 2.593995e-02 1.960414e-02 -1.025163e-02 +#> [1856,] 6.740524e-03 1.479966e-02 -2.659915e-02 -2.570711e-02 3.575899e-03 +#> [1857,] -1.563769e-02 -2.509625e-02 2.282221e-02 2.885329e-02 5.976899e-03 +#> [1858,] 2.511335e-02 3.369272e-02 -1.476258e-02 -2.787915e-02 -1.690318e-02 +#> [1859,] -3.352538e-02 -3.884224e-02 3.443519e-03 2.239162e-02 2.730035e-02 +#> [1860,] 3.919055e-02 3.940235e-02 9.442007e-03 -1.286866e-02 -3.525333e-02 +#> [1861,] -4.069908e-02 -3.506509e-02 -2.186741e-02 5.762843e-04 3.922274e-02 +#> [1862,] 3.719462e-02 2.641456e-02 3.187269e-02 1.267701e-02 -3.835179e-02 +#> [1863,] -2.856411e-02 -1.480297e-02 -3.794502e-02 -2.489310e-02 3.262855e-02 +#> [1864,] 1.549898e-02 2.074423e-03 3.930166e-02 3.426700e-02 -2.287305e-02 +#> [1865,] 5.864135e-04 9.803279e-03 -3.601916e-02 -3.951818e-02 1.055762e-02 +#> [1866,] -1.776542e-02 -1.912073e-02 2.898625e-02 4.010298e-02 2.495692e-03 +#> [1867,] 3.390064e-02 2.474393e-02 -1.969500e-02 -3.626877e-02 -1.447150e-02 +#> [1868,] -4.698111e-02 -2.630288e-02 9.918460e-03 2.894656e-02 2.389365e-02 +#> [1869,] 5.543369e-02 2.421719e-02 -1.345044e-03 -1.951415e-02 -2.986490e-02 +#> [1870,] -5.835352e-02 -1.955912e-02 -4.752898e-03 9.488519e-03 3.216842e-02 +#> [1871,] 5.561488e-02 1.378882e-02 7.748562e-03 -2.189711e-04 -3.121868e-02 +#> [1872,] -4.784744e-02 -8.421947e-03 -7.744131e-03 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