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<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN"><html> <head> <meta http-equiv="Content-Type" content= "text/html; charset=iso-8859-1"> <meta name="KEYWORDS" lang="en-us" content= "missing values, incomplete data, EM algorithm, regularization, ridge regression, historic surface temperatures, climate data"> <meta name="DESCRIPTION" content="regularized EM algorithm"> <title>Analysis of incomplete datasets: Estimation of mean values and covariance matrices and imputation of missing values</title> <style> body{ background-color: #FFFFFF; color: #333333; } p.ref{ /* for bibliography */ margin-left: 4em; text-indent: -2.5em } code{ font-family: monospace; } p.code{ margin-left: 2.5em; text-align: left; font-family: monospace; } div.cent{ text-align: center } h1{ text-align: center; } h2{ text-align: left; margin-top: 2.5ex } a:link {color: #006699; text-decoration: none} a:active {color: #999933; text-decoration: none} a:visited {color: #006699; text-decoration: none} </style> </head> <body> <table border="0" cellspacing="0" cellpadding="0" width=550 align="center"> <tr><td> <h1 class="cent">Analysis of incomplete datasets: Estimation of mean values and covariance matrices and imputation of missing values<br></h1> <div class="cent"><font size="-1"> [<a href="#purpose">Purpose</a>] [<a href="#installation">Installation</a>] [<a href="#files">Module descriptions</a>] [<a href="#suggestions">Possible modifications</a>] </font></div> <h2><font size="+1"><a name="purpose">Purpose</a></font></h2> <p>What follows is a collection of <a href="http://www.mathworks.com/"> Matlab</a> modules for</p> <ul> <li>the estimation of mean values and covariance matrices from incomplete datasets, and </li> <li>the imputation of missing values in incomplete datasets.</li> </ul> <p>The modules implement the regularized EM algorithm described in</p> <p class="ref">T. Schneider, 2001: <a href= "../papers/imputation.pdf">Analysis of incomplete climate data: Estimation of mean values and covariance matrices and imputation of missing values</a>. <cite>Journal of Climate</cite>, <strong>14</strong>, 853–871.</p> <p>The EM algorithm for Gaussian data is based on iterated linear regression analyses. In the regularized EM algorithm, a regularized regression method replaces the conditional maximum likelihood estimation of regression parameters in the conventional EM algorithm for Gaussian data. The modules here provide truncated total least squares (with fixed truncation parameter) and ridge regression with generalized cross-validation as regularized regression methods.</p> <p>The implementation of the regularized EM algorithm is modular, so that the modules that perform the regularized regression (e.g., ridge regression and generalized cross-validation) can be exchanged for other regularization methods and other methods of determining a regularization parameter. Per-Christian Hansen's <a href="http://www.imm.dtu.dk/~pch/Regutools/regutools.html">Regularization Tools</a> contain Matlab modules implementing a collection of regularization methods that can be adapted to fit into the framework of the EM algorithm. The generalized cross-validation modules of the regularized EM algorithm are adapted from Hansen's generalized cross-validation modules.</p> <p>In the Matlab implementation of the regularized EM algorithm, more emphasis was placed on the modularity of the program code than on computational efficiency. Below are some <a href="#suggestions">suggestions</a> on how the regularized EM algorithm could be implemented more efficiently.</p> <h2><font size="+1"><a name="installation">Installation</a></font></h2> <p>The program package consists of several Matlab modules. To install the programs, copy the package (available as a <a href= "imputation.tar.gz">tar.gz-file</a>) into a directory that is accessible by Matlab. Unpack the package using</p> <p class="code"> gunzip imputation.tar.gz<br> tar -xvf imputation.tar </p> <p>Starting Matlab and invoking Matlab's online help function</p> <p class="code"> help <i>filename</i> </p> <p>displays information on the module <code><i>filename</i>.m</code>.</p> <h2><font size="+1"><a name="files">Module descriptions</a></font></h2> <dl> <dt><a href="CHANGES">CHANGES</a></dt> <dd>Recent significant changes of the programs.</dd> <dt><a href="center.m">center.m</a></dt> <dd>Centers data by subtracting the mean.</dd> <dt><a href="gcvfctn.m">gcvfctn.m</a> (auxiliary module to gcvridge.m)</dt> <dd>Evaluates generalized cross-validation function.</dd> <dt><a href="gcvridge.m">gcvridge.m</a></dt> <dd>Finds minimum of generalized cross-validation function for ridge regression.</dd> <dt><a href="iridge.m">iridge.m</a></dt> <dd>Computes regression parameters by individual ridge regressions.</dd> <dt><a href="mridge.m">mridge.m</a></dt> <dd>Computes regression parameters by a multiple ridge regression.</dd> <dt><a href="nancov.m">nancov.m</a></dt> <dd>Sample covariance matrix of available values in incomplete dataset.</dd> <dt><a href="nanmean.m">nanmean.m</a></dt> <dd>Sample mean of available values in incomplete dataset.</dd> <dt><a href="nanstd.m">nanstd.m</a></dt> <dd>Standard deviation of available values in incomplete dataset.</dd> <dt><a href="nansum.m">nansum.m</a></dt> <dd>Sum over available values in incomplete dataset.</dd> <dt><a href="peigs.m">peigs.m</a></dt> <dd>Computes positive eigenvalues and corresponding eigenvectors.</dd> <dt><a href="pttls.m">pttls.m</a></dt> <dd>Computes regression parameters by truncated total least squares.</dd> <dt><a href="regem.m">regem.m</a></dt> <dd>Driver module for regularized EM algorithm.</dd> <dt><a href="standardize.m">standardize.m</a></dt> <dd>Standardizes data by subtracting the mean and scaling with the standard deviation.</dd> </dl> <h2><font size="+1"><a name="suggestions">Possible modifications</a></font></h2> <p>More efficient implementations of the regularized EM algorithm are possible. For example, if the missing values in the dataset under consideration follow regular patterns, the algorithm might exploit the special structure of the dataset. Other possible modifications include the extensions mentioned in the <a href="../papers/imputation.pdf">above paper</a>:</p> <ul> <li>One could implement a regularized EM algorithm that exploits spatio-temporal covariability (cf. Section 4 of the <a href="../papers/imputation.pdf">above paper</a>).</li> <li>One could implement an adaptive method for the choice of truncation parameter if truncated total least squares (TTLS) is used as the regularized regression method in the regularized EM algorithm. Some criteria for the choice of truncation parameter in TTLS are discussed in Sima and van Huffel (2007), Level choice in truncated total least squares, <em>Comp. Stat. Data Anal.</em> (to appear). These methods require one additional eigendecomposition per record, in addition to the one eigendecomposition per iteration of the total covariance matrix required if TTLS is used. <li>One could find matching patterns of missing values in different records and compute a regression for each pattern of missing values instead of for each record.</li> <li>One could parallelize the algorithm, so that the computations for several records (or for several patterns of missing values) are carried out simultaneously.</li> <li>One could compute only one eigendecomposition per iteration, instead of one eigendecomposition per record (or per pattern of missing values), and compute the ridge regression via a singular value decomposition of a data matrix (cf. Section 3 of the <a href="../papers/imputation.pdf">above paper</a>). For datasets with many more variables than records, this procedure might be faster than computing one eigendecomposition per record and iteration.</li> </ul> </td></tr> </table> </body></html>⌨️ 快捷键说明
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