Autocorrelation - Revision history

https://csml-wiki.northwestern.edu/index.php?action=history&feed=atom&title=Autocorrelation Autocorrelation - Revision history 2026-04-09T11:49:48Z Revision history for this page on the wiki MediaWiki 1.39.2 https://csml-wiki.northwestern.edu/index.php?title=Autocorrelation&diff=861&oldid=prev Administrator at 15:00, 11 May 2025 2025-05-11T15:00:56Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 10:00, 11 May 2025</td> </tr><tr> <td colspan="2" class="diff-lineno">Line 9:</td> <td colspan="2" class="diff-lineno">Line 9:</td> </tr> <tr> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><tt>filename</tt> is a plain text file. Lines starting with a '#' will be ignored (use this feature to insert column descriptions and other information into your simulation data). If <tt>filename</tt> ends with '<tt>.gz</tt>' the file is assumed to be compressed with [http://en.wikipedia.org/wiki/Gzip gzip] and will be decompressed on the fly. Note that this happens in memory; no decompressed version of the file is written to disk. This has the advantage that no additional disk space is required and that no additional time is required to compress the data again after the analysis.</div></td> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><tt>filename</tt> is a plain text file. Lines starting with a '#' will be ignored (use this feature to insert column descriptions and other information into your simulation data). If <tt>filename</tt> ends with '<tt>.gz</tt>' the file is assumed to be compressed with [http://en.wikipedia.org/wiki/Gzip gzip] and will be decompressed on the fly. Note that this happens in memory; no decompressed version of the file is written to disk. This has the advantage that no additional disk space is required and that no additional time is required to compress the data again after the analysis.</div></td> </tr> <tr> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>To read from standard input instead of a file, specify ''STDIN'' as the filename.</div></td> <td class="diff-marker" data-marker="+"></td> <td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>To read from standard input instead of a file, specify ''STDIN'' as the filename<ins style="font-weight: bold; text-decoration: none;">. ''STDIN.gz'' is also permitted to indicate a compressed data stream</ins>.</div></td> </tr> <tr> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>When redirecting the output, note that the '''autocorrelation function''' for each column is written to standard output, whereas the '''autocorelation time''' (and all other information) is written to standard error. See [[#Special usage notes|usage notes]] below.</div></td> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>When redirecting the output, note that the '''autocorrelation function''' for each column is written to standard output, whereas the '''autocorelation time''' (and all other information) is written to standard error. See [[#Special usage notes|usage notes]] below.</div></td> </tr> </table>

Administrator https://csml-wiki.northwestern.edu/index.php?title=Autocorrelation&diff=859&oldid=prev Administrator at 18:22, 10 May 2025 2025-05-10T18:22:49Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 13:22, 10 May 2025</td> </tr><tr> <td colspan="2" class="diff-lineno">Line 19:</td> <td colspan="2" class="diff-lineno">Line 19:</td> </tr> <tr> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="a-option">-a</tt><br>Compute the autocorrelation time by integrating the absolute autocorrelation function. This is useful to eliminate cancellations due to anticorelations (which would result in an underestimation of the correlation time), but also will enhance noise in the tail of the autocorrelation function (where the function fluctuates around zero; normally such fluctuations cancel out).</div></td> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="a-option">-a</tt><br>Compute the autocorrelation time by integrating the absolute autocorrelation function. This is useful to eliminate cancellations due to anticorelations (which would result in an underestimation of the correlation time), but also will enhance noise in the tail of the autocorrelation function (where the function fluctuates around zero; normally such fluctuations cancel out).</div></td> </tr> <tr> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="c-option">-c n:m</tt><br>Compute the cross correlation of columns <tt>n</tt> and <tt>m</tt>. Note: columns are numbered from 0.</div></td> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="c-option">-c n:m</tt><br>Compute the cross correlation of columns <tt>n</tt> and <tt>m</tt>. Note: columns are numbered from 0.</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="e-option">-d</tt><br>Only print the function (autocorrelation, cross correlation, or mean squared displacement) up to the lagtime specified, even when using an FFT-based calculation (which in turn is enabled via the <del style="font-weight: bold; text-decoration: none;">-f</del> option). Since the FFT-based calculation yields these functions over the entire domain at no additional computational cost, normally the maximum lagtime is ignored for printing purposes; this option overrides this.</div></td> <td class="diff-marker" data-marker="+"></td> <td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="e-option">-d</tt><br>Only print the function (autocorrelation, cross correlation, or mean squared displacement) up to the lagtime specified, even when using an FFT-based calculation (which in turn is enabled via the <ins style="font-weight: bold; text-decoration: none;">–f</ins> option). Since the FFT-based calculation yields these functions over the entire domain at no additional computational cost, normally the maximum lagtime is ignored for printing purposes; this option overrides this.</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="e-option">-e</tt><br>Disable truncating the data set when using an FFT-based calculation (which in turn is enabled via the <del style="font-weight: bold; text-decoration: none;">-f</del> option). Normally, use of <del style="font-weight: bold; text-decoration: none;">-f</del> truncates the data set to a multiple of 256 (for more than 10240 samples) or a multiple of 16384 (for more than 10<sup>6</sup> samples). Note that disabling this truncation carries a significant speed penalty (and is primarily used for debugging purposes). However, it makes the results identical to those obtained without using the FFT. (If the size of the original data set is already a multiple of 256 (or 16384), then the results with and without FFT are identical even without this option.)</div></td> <td class="diff-marker" data-marker="+"></td> <td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="e-option">-e</tt><br>Disable truncating the data set when using an FFT-based calculation (which in turn is enabled via the <ins style="font-weight: bold; text-decoration: none;">–f</ins> option). Normally, use of <ins style="font-weight: bold; text-decoration: none;">–f</ins> truncates the data set to a multiple of 256 (for more than 10240 samples) or a multiple of 16384 (for more than 10<sup>6</sup> samples). Note that disabling this truncation carries a significant speed penalty (and is primarily used for debugging purposes). However, it makes the results identical to those obtained without using the FFT. (If the size of the original data set is already a multiple of 256 (or 16384), then the results with and without FFT are identical even without this option.)</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="f-option">-f</tt><br>Employ a Fast Fourier Transform to greatly reduce the computational effort, by analyzing the data as a convolution in the frequency domain. This works for the autocorrelation function ([http://mathworld.wolfram.com/Wiener-KhinchinTheorem.html Wiener-Khinchin Theorem]) as well as for the cross correlation (option <tt><del style="font-weight: bold; text-decoration: none;">-c</del></tt>) ([http://mathworld.wolfram.com/Cross-CorrelationTheorem.html Cross-Correlation Theorem]) and the mean squared displacement (option <tt><del style="font-weight: bold; text-decoration: none;">-m</del></tt>). Note that this requires computing the function over the entire time domain; therefore, for very rapidly decaying functions, where ''maxtime'' can be kept small, conventional evaluation might be faster.</div></td> <td class="diff-marker" data-marker="+"></td> <td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="f-option">-f</tt><br>Employ a Fast Fourier Transform to greatly reduce the computational effort, by analyzing the data as a convolution in the frequency domain. This works for the autocorrelation function ([http://mathworld.wolfram.com/Wiener-KhinchinTheorem.html Wiener-Khinchin Theorem]) as well as for the cross correlation (option <tt><ins style="font-weight: bold; text-decoration: none;">–c</ins></tt>) ([http://mathworld.wolfram.com/Cross-CorrelationTheorem.html Cross-Correlation Theorem]) and the mean squared displacement (option <tt><ins style="font-weight: bold; text-decoration: none;">–m</ins></tt>). Note that this requires computing the function over the entire time domain; therefore, for very rapidly decaying functions, where ''maxtime'' can be kept small, conventional evaluation might be faster.</div></td> </tr> <tr> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="m-option">-m</tt><br>Calculate the mean squared displacement (MSD). See [[#Algorithm|algorithm]] below for the relation between the MSD and the autocorrelation function.</div></td> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="m-option">-m</tt><br>Calculate the mean squared displacement (MSD). See [[#Algorithm|algorithm]] below for the relation between the MSD and the autocorrelation function.</div></td> </tr> <tr> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="o-option">-o</tt> ''filename''<br>Write autocorrelation function to ''filename''.</div></td> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="o-option">-o</tt> ''filename''<br>Write autocorrelation function to ''filename''.</div></td> </tr> </table>

Administrator https://csml-wiki.northwestern.edu/index.php?title=Autocorrelation&diff=858&oldid=prev Administrator at 18:21, 10 May 2025 2025-05-10T18:21:56Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 13:21, 10 May 2025</td> </tr><tr> <td colspan="2" class="diff-lineno">Line 19:</td> <td colspan="2" class="diff-lineno">Line 19:</td> </tr> <tr> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="a-option">-a</tt><br>Compute the autocorrelation time by integrating the absolute autocorrelation function. This is useful to eliminate cancellations due to anticorelations (which would result in an underestimation of the correlation time), but also will enhance noise in the tail of the autocorrelation function (where the function fluctuates around zero; normally such fluctuations cancel out).</div></td> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="a-option">-a</tt><br>Compute the autocorrelation time by integrating the absolute autocorrelation function. This is useful to eliminate cancellations due to anticorelations (which would result in an underestimation of the correlation time), but also will enhance noise in the tail of the autocorrelation function (where the function fluctuates around zero; normally such fluctuations cancel out).</div></td> </tr> <tr> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="c-option">-c n:m</tt><br>Compute the cross correlation of columns <tt>n</tt> and <tt>m</tt>. Note: columns are numbered from 0.</div></td> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="c-option">-c n:m</tt><br>Compute the cross correlation of columns <tt>n</tt> and <tt>m</tt>. Note: columns are numbered from 0.</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="e-option">-<del style="font-weight: bold; text-decoration: none;">e</del></tt><br><del style="font-weight: bold; text-decoration: none;">Disable</del> <del style="font-weight: bold; text-decoration: none;">truncating</del> the <del style="font-weight: bold; text-decoration: none;">data</del> <del style="font-weight: bold; text-decoration: none;">set</del> when using FFT-based calculation (which in turn is enabled via the -f option). <del style="font-weight: bold; text-decoration: none;">Normally,</del> <del style="font-weight: bold; text-decoration: none;">use of</del> -<del style="font-weight: bold; text-decoration: none;">f</del> <del style="font-weight: bold; text-decoration: none;">truncates</del> <del style="font-weight: bold; text-decoration: none;">the</del> <del style="font-weight: bold; text-decoration: none;">data</del> <del style="font-weight: bold; text-decoration: none;">set</del> <del style="font-weight: bold; text-decoration: none;">to</del> <del style="font-weight: bold; text-decoration: none;">a</del> <del style="font-weight: bold; text-decoration: none;">multiple</del> <del style="font-weight: bold; text-decoration: none;">of</del> <del style="font-weight: bold; text-decoration: none;">256</del> <del style="font-weight: bold; text-decoration: none;">(for</del> <del style="font-weight: bold; text-decoration: none;">more</del> <del style="font-weight: bold; text-decoration: none;">than</del> <del style="font-weight: bold; text-decoration: none;">10240</del> <del style="font-weight: bold; text-decoration: none;">samples)</del> <del style="font-weight: bold; text-decoration: none;">or</del> <del style="font-weight: bold; text-decoration: none;">a</del> <del style="font-weight: bold; text-decoration: none;">multiple of 16384 (for more than 10<sup>6</sup> samples). Note that disabling this truncation carries a significant speed penalty (and</del> is <del style="font-weight: bold; text-decoration: none;">primarily used</del> for <del style="font-weight: bold; text-decoration: none;">debugging</del> purposes<del style="font-weight: bold; text-decoration: none;">).</del> <del style="font-weight: bold; text-decoration: none;">However,</del> <del style="font-weight: bold; text-decoration: none;">it makes the results identical to those obtained without using the FFT. (If the size of the original data set is already a multiple of 256 (or 16384), then the results with and without FFT are identical even without</del> this<del style="font-weight: bold; text-decoration: none;"> option</del>.<del style="font-weight: bold; text-decoration: none;">)</del></div></td> <td class="diff-marker" data-marker="+"></td> <td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="e-option">-<ins style="font-weight: bold; text-decoration: none;">d</ins></tt><br><ins style="font-weight: bold; text-decoration: none;">Only</ins> <ins style="font-weight: bold; text-decoration: none;">print</ins> the <ins style="font-weight: bold; text-decoration: none;">function</ins> <ins style="font-weight: bold; text-decoration: none;">(autocorrelation, cross correlation, or mean squared displacement) up to the lagtime specified, even</ins> when using<ins style="font-weight: bold; text-decoration: none;"> an</ins> FFT-based calculation (which in turn is enabled via the -f option). <ins style="font-weight: bold; text-decoration: none;">Since</ins> <ins style="font-weight: bold; text-decoration: none;">the</ins> <ins style="font-weight: bold; text-decoration: none;">FFT</ins>-<ins style="font-weight: bold; text-decoration: none;">based</ins> <ins style="font-weight: bold; text-decoration: none;">calculation</ins> <ins style="font-weight: bold; text-decoration: none;">yields</ins> <ins style="font-weight: bold; text-decoration: none;">these</ins> <ins style="font-weight: bold; text-decoration: none;">functions</ins> <ins style="font-weight: bold; text-decoration: none;">over</ins> <ins style="font-weight: bold; text-decoration: none;">the</ins> <ins style="font-weight: bold; text-decoration: none;">entire</ins> <ins style="font-weight: bold; text-decoration: none;">domain</ins> <ins style="font-weight: bold; text-decoration: none;">at</ins> <ins style="font-weight: bold; text-decoration: none;">no</ins> <ins style="font-weight: bold; text-decoration: none;">additional</ins> <ins style="font-weight: bold; text-decoration: none;">computational</ins> <ins style="font-weight: bold; text-decoration: none;">cost,</ins> <ins style="font-weight: bold; text-decoration: none;">normally</ins> <ins style="font-weight: bold; text-decoration: none;">the</ins> <ins style="font-weight: bold; text-decoration: none;">maximum</ins> <ins style="font-weight: bold; text-decoration: none;">lagtime</ins> is <ins style="font-weight: bold; text-decoration: none;">ignored</ins> for <ins style="font-weight: bold; text-decoration: none;">printing</ins> purposes<ins style="font-weight: bold; text-decoration: none;">;</ins> <ins style="font-weight: bold; text-decoration: none;">this</ins> <ins style="font-weight: bold; text-decoration: none;">option</ins> <ins style="font-weight: bold; text-decoration: none;">overrides</ins> this.</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="e-option">-e</tt><br>Disable truncating the data set when using an FFT-based calculation (which in turn is enabled via the -f option). Normally, use of -f truncates the data set to a multiple of 256 (for more than 10240 samples) or a multiple of 16384 (for more than 10<sup>6</sup> samples). Note that disabling this truncation carries a significant speed penalty (and is primarily used for debugging purposes). However, it makes the results identical to those obtained without using the FFT. (If the size of the original data set is already a multiple of 256 (or 16384), then the results with and without FFT are identical even without this option.)</div></td> </tr> <tr> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="f-option">-f</tt><br>Employ a Fast Fourier Transform to greatly reduce the computational effort, by analyzing the data as a convolution in the frequency domain. This works for the autocorrelation function ([http://mathworld.wolfram.com/Wiener-KhinchinTheorem.html Wiener-Khinchin Theorem]) as well as for the cross correlation (option <tt>-c</tt>) ([http://mathworld.wolfram.com/Cross-CorrelationTheorem.html Cross-Correlation Theorem]) and the mean squared displacement (option <tt>-m</tt>). Note that this requires computing the function over the entire time domain; therefore, for very rapidly decaying functions, where ''maxtime'' can be kept small, conventional evaluation might be faster.</div></td> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="f-option">-f</tt><br>Employ a Fast Fourier Transform to greatly reduce the computational effort, by analyzing the data as a convolution in the frequency domain. This works for the autocorrelation function ([http://mathworld.wolfram.com/Wiener-KhinchinTheorem.html Wiener-Khinchin Theorem]) as well as for the cross correlation (option <tt>-c</tt>) ([http://mathworld.wolfram.com/Cross-CorrelationTheorem.html Cross-Correlation Theorem]) and the mean squared displacement (option <tt>-m</tt>). Note that this requires computing the function over the entire time domain; therefore, for very rapidly decaying functions, where ''maxtime'' can be kept small, conventional evaluation might be faster.</div></td> </tr> <tr> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="m-option">-m</tt><br>Calculate the mean squared displacement (MSD). See [[#Algorithm|algorithm]] below for the relation between the MSD and the autocorrelation function.</div></td> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="m-option">-m</tt><br>Calculate the mean squared displacement (MSD). See [[#Algorithm|algorithm]] below for the relation between the MSD and the autocorrelation function.</div></td> </tr> </table>

Administrator https://csml-wiki.northwestern.edu/index.php?title=Autocorrelation&diff=857&oldid=prev Administrator at 16:03, 10 May 2025 2025-05-10T16:03:10Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 11:03, 10 May 2025</td> </tr><tr> <td colspan="2" class="diff-lineno">Line 23:</td> <td colspan="2" class="diff-lineno">Line 23:</td> </tr> <tr> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="m-option">-m</tt><br>Calculate the mean squared displacement (MSD). See [[#Algorithm|algorithm]] below for the relation between the MSD and the autocorrelation function.</div></td> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="m-option">-m</tt><br>Calculate the mean squared displacement (MSD). See [[#Algorithm|algorithm]] below for the relation between the MSD and the autocorrelation function.</div></td> </tr> <tr> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="o-option">-o</tt> ''filename''<br>Write autocorrelation function to ''filename''.</div></td> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="o-option">-o</tt> ''filename''<br>Write autocorrelation function to ''filename''.</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="u-option">-u</tt><br>Do not normalize the correlation function. Without this option, the <del style="font-weight: bold; text-decoration: none;">autocorelation</del> function C(t) or cross<del style="font-weight: bold; text-decoration: none;">-</del>correlation function X(t) is normalized by C(0) or X(0), respectively. Note that this option disables calculation of the autocorrelation time (see [[#Algorithm|algorithm]] below). This option has no meaning in conjunction with the –m option (MSD calculation).</div></td> <td class="diff-marker" data-marker="+"></td> <td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="u-option">-u</tt><br>Do not normalize the correlation function. Without this option, the <ins style="font-weight: bold; text-decoration: none;">autocorrelation</ins> function C(t) or cross<ins style="font-weight: bold; text-decoration: none;"> </ins>correlation function X(t) is normalized by C(0) or X(0), respectively. Note that this option disables calculation of the autocorrelation time (see [[#Algorithm|algorithm]] below). This option has no meaning in conjunction with the –m option (MSD calculation).</div></td> </tr> <tr> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="x-option">-x</tt><br>Only compute the cross term in the correlation function, i.e., the first term in C(t) or X(t) (see [[#Algorithm|algorithm]] below). This option has no meaning in conjunction with the –m option (MSD calculation).</div></td> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="x-option">-x</tt><br>Only compute the cross term in the correlation function, i.e., the first term in C(t) or X(t) (see [[#Algorithm|algorithm]] below). This option has no meaning in conjunction with the –m option (MSD calculation).</div></td> </tr> <tr> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td> </tr> </table>

Administrator https://csml-wiki.northwestern.edu/index.php?title=Autocorrelation&diff=856&oldid=prev Administrator at 16:01, 10 May 2025 2025-05-10T16:01:41Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 11:01, 10 May 2025</td> </tr><tr> <td colspan="2" class="diff-lineno">Line 92:</td> <td colspan="2" class="diff-lineno">Line 92:</td> </tr> <tr> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>=== Download binary versions (Linux and OS X) ===</div></td> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>=== Download binary versions (Linux and OS X) ===</div></td> </tr> <tr> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The current version is 3.<del style="font-weight: bold; text-decoration: none;">93</del>, dated <del style="font-weight: bold; text-decoration: none;">December</del> <del style="font-weight: bold; text-decoration: none;">2013</del>. It is strongly recommend that you upgrade from any earlier version. You can download binary versions of this program here, but note that this was created for internal lab use - we cannot provide any support.</div></td> <td class="diff-marker" data-marker="+"></td> <td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The current version is 3.<ins style="font-weight: bold; text-decoration: none;">96</ins>, dated <ins style="font-weight: bold; text-decoration: none;">May</ins> <ins style="font-weight: bold; text-decoration: none;">2025</ins>. It is strongly recommend that you upgrade from any earlier version. You can download binary versions of this program here, but note that this was created for internal lab use - we cannot provide any support.</div></td> </tr> <tr> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>(binaries coming soon, pending minor improvements)</div></td> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>(binaries coming soon, pending minor improvements)</div></td> </tr> </table>

Administrator https://csml-wiki.northwestern.edu/index.php?title=Autocorrelation&diff=855&oldid=prev Administrator at 16:00, 10 May 2025 2025-05-10T16:00:34Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 11:00, 10 May 2025</td> </tr><tr> <td colspan="2" class="diff-lineno">Line 23:</td> <td colspan="2" class="diff-lineno">Line 23:</td> </tr> <tr> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="m-option">-m</tt><br>Calculate the mean squared displacement (MSD). See [[#Algorithm|algorithm]] below for the relation between the MSD and the autocorrelation function.</div></td> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="m-option">-m</tt><br>Calculate the mean squared displacement (MSD). See [[#Algorithm|algorithm]] below for the relation between the MSD and the autocorrelation function.</div></td> </tr> <tr> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="o-option">-o</tt> ''filename''<br>Write autocorrelation function to ''filename''.</div></td> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="o-option">-o</tt> ''filename''<br>Write autocorrelation function to ''filename''.</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="u-option">-u</tt><br>Do not normalize the <del style="font-weight: bold; text-decoration: none;">autocorrelation</del> function. Without this option, the autocorelation function C(t) is normalized by C(0). Note that this option disables calculation of the autocorrelation time (see [[#Algorithm|algorithm]] below).</div></td> <td class="diff-marker" data-marker="+"></td> <td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="u-option">-u</tt><br>Do not normalize the <ins style="font-weight: bold; text-decoration: none;">correlation</ins> function. Without this option, the autocorelation function C<ins style="font-weight: bold; text-decoration: none;">(t) or cross-correlation function X</ins>(t) is normalized by C(0)<ins style="font-weight: bold; text-decoration: none;"> or X(0), respectively</ins>. Note that this option disables calculation of the autocorrelation time (see [[#Algorithm|algorithm]] below<ins style="font-weight: bold; text-decoration: none;">). This option has no meaning in conjunction with the –m option (MSD calculation</ins>).</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="x-option">-x</tt><br>Only compute the cross term in the <del style="font-weight: bold; text-decoration: none;">autocorrelation</del> function, i.e., the first term in C(t) (see [[#Algorithm|algorithm]] below).</div></td> <td class="diff-marker" data-marker="+"></td> <td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="x-option">-x</tt><br>Only compute the cross term in the <ins style="font-weight: bold; text-decoration: none;">correlation</ins> function, i.e., the first term in C<ins style="font-weight: bold; text-decoration: none;">(t) or X</ins>(t) (see [[#Algorithm|algorithm]] below<ins style="font-weight: bold; text-decoration: none;">). This option has no meaning in conjunction with the –m option (MSD calculation</ins>).</div></td> </tr> <tr> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>=== Interpreting output ===</div></td> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>=== Interpreting output ===</div></td> </tr> </table>

Administrator https://csml-wiki.northwestern.edu/index.php?title=Autocorrelation&diff=595&oldid=prev Administrator: /* Special usage notes */ 2015-10-31T16:11:28Z

<p><span dir="auto"><span class="autocomment">Special usage notes</span></span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 11:11, 31 October 2015</td> </tr><tr> <td colspan="2" class="diff-lineno">Line 41:</td> <td colspan="2" class="diff-lineno">Line 41:</td> </tr> <tr> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><li>Note that, per standard GNU style, options can be combined. Thus, for example, '<tt>-a -e -f</tt>' can be specified as '<tt>-aef</tt>'.</li></div></td> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><li>Note that, per standard GNU style, options can be combined. Thus, for example, '<tt>-a -e -f</tt>' can be specified as '<tt>-aef</tt>'.</li></div></td> </tr> <tr> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><li>(coming soon)</li></div></td> <td class="diff-marker" data-marker="+"></td> <td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><li>(<ins style="font-weight: bold; text-decoration: none;">more tips </ins>coming soon)</li></div></td> </tr> <tr> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></ul></div></td> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></ul></div></td> </tr> </table>

Administrator https://csml-wiki.northwestern.edu/index.php?title=Autocorrelation&diff=594&oldid=prev Administrator: /* Options */ 2015-10-31T16:10:06Z

<p><span dir="auto"><span class="autocomment">Options</span></span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 11:10, 31 October 2015</td> </tr><tr> <td colspan="2" class="diff-lineno">Line 19:</td> <td colspan="2" class="diff-lineno">Line 19:</td> </tr> <tr> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="a-option">-a</tt><br>Compute the autocorrelation time by integrating the absolute autocorrelation function. This is useful to eliminate cancellations due to anticorelations (which would result in an underestimation of the correlation time), but also will enhance noise in the tail of the autocorrelation function (where the function fluctuates around zero; normally such fluctuations cancel out).</div></td> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="a-option">-a</tt><br>Compute the autocorrelation time by integrating the absolute autocorrelation function. This is useful to eliminate cancellations due to anticorelations (which would result in an underestimation of the correlation time), but also will enhance noise in the tail of the autocorrelation function (where the function fluctuates around zero; normally such fluctuations cancel out).</div></td> </tr> <tr> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="c-option">-c n:m</tt><br>Compute the cross correlation of columns <tt>n</tt> and <tt>m</tt>. Note: columns are numbered from 0.</div></td> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="c-option">-c n:m</tt><br>Compute the cross correlation of columns <tt>n</tt> and <tt>m</tt>. Note: columns are numbered from 0.</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="e-option">-e</tt><br>Disable truncating the data set when using FFT-based calculation (which in turn is enabled via the -f option). Normally, use of -f truncates the data set to a multiple of 256 (for more than 10240 samples) or a multiple of 16384 (for more than 10<sup>6</sup> samples). Note that disabling this truncation carries a significant speed penalty (and primarily used for debugging purposes). However, it makes the results identical to those obtained without using the FFT. (If the size of the original data set is already a multiple of 256 (or 16384), then the results with and without FFT are identical even without this option.)</div></td> <td class="diff-marker" data-marker="+"></td> <td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="e-option">-e</tt><br>Disable truncating the data set when using FFT-based calculation (which in turn is enabled via the -f option). Normally, use of -f truncates the data set to a multiple of 256 (for more than 10240 samples) or a multiple of 16384 (for more than 10<sup>6</sup> samples). Note that disabling this truncation carries a significant speed penalty (and<ins style="font-weight: bold; text-decoration: none;"> is</ins> primarily used for debugging purposes). However, it makes the results identical to those obtained without using the FFT. (If the size of the original data set is already a multiple of 256 (or 16384), then the results with and without FFT are identical even without this option.)</div></td> </tr> <tr> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="f-option">-f</tt><br>Employ a Fast Fourier Transform to greatly reduce the computational effort, by analyzing the data as a convolution in the frequency domain. This works for the autocorrelation function ([http://mathworld.wolfram.com/Wiener-KhinchinTheorem.html Wiener-Khinchin Theorem]) as well as for the cross correlation (option <tt>-c</tt>) ([http://mathworld.wolfram.com/Cross-CorrelationTheorem.html Cross-Correlation Theorem]) and the mean squared displacement (option <tt>-m</tt>). Note that this requires computing the function over the entire time domain; therefore, for very rapidly decaying functions, where ''maxtime'' can be kept small, conventional evaluation might be faster.</div></td> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="f-option">-f</tt><br>Employ a Fast Fourier Transform to greatly reduce the computational effort, by analyzing the data as a convolution in the frequency domain. This works for the autocorrelation function ([http://mathworld.wolfram.com/Wiener-KhinchinTheorem.html Wiener-Khinchin Theorem]) as well as for the cross correlation (option <tt>-c</tt>) ([http://mathworld.wolfram.com/Cross-CorrelationTheorem.html Cross-Correlation Theorem]) and the mean squared displacement (option <tt>-m</tt>). Note that this requires computing the function over the entire time domain; therefore, for very rapidly decaying functions, where ''maxtime'' can be kept small, conventional evaluation might be faster.</div></td> </tr> <tr> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="m-option">-m</tt><br>Calculate the mean squared displacement (MSD). See [[#Algorithm|algorithm]] below for the relation between the MSD and the autocorrelation function.</div></td> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="m-option">-m</tt><br>Calculate the mean squared displacement (MSD). See [[#Algorithm|algorithm]] below for the relation between the MSD and the autocorrelation function.</div></td> </tr> </table>

Administrator https://csml-wiki.northwestern.edu/index.php?title=Autocorrelation&diff=445&oldid=prev Administrator: /* Options */ 2014-11-24T18:15:10Z

<p><span dir="auto"><span class="autocomment">Options</span></span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 13:15, 24 November 2014</td> </tr><tr> <td colspan="2" class="diff-lineno">Line 19:</td> <td colspan="2" class="diff-lineno">Line 19:</td> </tr> <tr> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="a-option">-a</tt><br>Compute the autocorrelation time by integrating the absolute autocorrelation function. This is useful to eliminate cancellations due to anticorelations (which would result in an underestimation of the correlation time), but also will enhance noise in the tail of the autocorrelation function (where the function fluctuates around zero; normally such fluctuations cancel out).</div></td> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="a-option">-a</tt><br>Compute the autocorrelation time by integrating the absolute autocorrelation function. This is useful to eliminate cancellations due to anticorelations (which would result in an underestimation of the correlation time), but also will enhance noise in the tail of the autocorrelation function (where the function fluctuates around zero; normally such fluctuations cancel out).</div></td> </tr> <tr> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="c-option">-c n:m</tt><br>Compute the cross correlation of columns <tt>n</tt> and <tt>m</tt>. Note: columns are numbered from 0.</div></td> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="c-option">-c n:m</tt><br>Compute the cross correlation of columns <tt>n</tt> and <tt>m</tt>. Note: columns are numbered from 0.</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="e-option">-e</tt><br>Disable truncating the data set when using FFT-based calculation (which in turn is enabled via the -f option). Normally, use of -f truncates the data set to a multiple of 256 (for more than 10240 samples) or a multiple of 16384 (for more than 10<sup>6</sup> samples). Note that disabling this truncation carries a significant speed penalty. However, it makes the results identical to those obtained without using the FFT.</div></td> <td class="diff-marker" data-marker="+"></td> <td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="e-option">-e</tt><br>Disable truncating the data set when using FFT-based calculation (which in turn is enabled via the -f option). Normally, use of -f truncates the data set to a multiple of 256 (for more than 10240 samples) or a multiple of 16384 (for more than 10<sup>6</sup> samples). Note that disabling this truncation carries a significant speed penalty<ins style="font-weight: bold; text-decoration: none;"> (and primarily used for debugging purposes)</ins>. However, it makes the results identical to those obtained without using the FFT.<ins style="font-weight: bold; text-decoration: none;"> (If the size of the original data set is already a multiple of 256 (or 16384), then the results with and without FFT are identical even without this option.)</ins></div></td> </tr> <tr> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="f-option">-f</tt><br>Employ a Fast Fourier Transform to greatly reduce the computational effort, by analyzing the data as a convolution in the frequency domain. This works for the autocorrelation function ([http://mathworld.wolfram.com/Wiener-KhinchinTheorem.html Wiener-Khinchin Theorem]) as well as for the cross correlation (option <tt>-c</tt>) ([http://mathworld.wolfram.com/Cross-CorrelationTheorem.html Cross-Correlation Theorem]) and the mean squared displacement (option <tt>-m</tt>). Note that this requires computing the function over the entire time domain; therefore, for very rapidly decaying functions, where ''maxtime'' can be kept small, conventional evaluation might be faster.</div></td> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="f-option">-f</tt><br>Employ a Fast Fourier Transform to greatly reduce the computational effort, by analyzing the data as a convolution in the frequency domain. This works for the autocorrelation function ([http://mathworld.wolfram.com/Wiener-KhinchinTheorem.html Wiener-Khinchin Theorem]) as well as for the cross correlation (option <tt>-c</tt>) ([http://mathworld.wolfram.com/Cross-CorrelationTheorem.html Cross-Correlation Theorem]) and the mean squared displacement (option <tt>-m</tt>). Note that this requires computing the function over the entire time domain; therefore, for very rapidly decaying functions, where ''maxtime'' can be kept small, conventional evaluation might be faster.</div></td> </tr> <tr> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="m-option">-m</tt><br>Calculate the mean squared displacement (MSD). See [[#Algorithm|algorithm]] below for the relation between the MSD and the autocorrelation function.</div></td> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <tt id="m-option">-m</tt><br>Calculate the mean squared displacement (MSD). See [[#Algorithm|algorithm]] below for the relation between the MSD and the autocorrelation function.</div></td> </tr> </table>

Administrator https://csml-wiki.northwestern.edu/index.php?title=Autocorrelation&diff=402&oldid=prev Administrator: /* Algorithm */ 2014-07-17T15:24:30Z

<p><span dir="auto"><span class="autocomment">Algorithm</span></span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 10:24, 17 July 2014</td> </tr><tr> <td colspan="2" class="diff-lineno">Line 64:</td> <td colspan="2" class="diff-lineno">Line 64:</td> </tr> <tr> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>\tau = \sum_{i=0}^{\mathrm{maxtime}} [C(i)/C(0)]</math>.</div></td> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>\tau = \sum_{i=0}^{\mathrm{maxtime}} [C(i)/C(0)]</math>.</div></td> </tr> <tr> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Some more nomenclature: Sometimes the autocorrelation refers to the ''cross term'' <math>\langle a_j a_{j+t}\rangle</math> (where the brackets indicate averaging over <math>j</math>), which is the first term in <math>C(t)</math>. In that case, <math>C(t)/C(0)</math> is referred to as the '''autocorrelation coefficient''' or autocorrelation <del style="font-weight: bold; text-decoration: none;">coeffient</del> ''function.'' To compute the cross term in '''AC''', specify the [[#x-option|<tt>-x</tt>]] option (or [[#u-option|<tt>-u</tt>]] [[#x-option|<tt>-x</tt>]] to suppress normalization).</div></td> <td class="diff-marker" data-marker="+"></td> <td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Some more nomenclature: Sometimes the autocorrelation refers to the ''cross term'' <math>\langle a_j a_{j+t}\rangle</math> (where the brackets indicate averaging over <math>j</math>), which is the first term in <math>C(t)</math>. In that case, <math>C(t)/C(0)</math> is referred to as the '''autocorrelation coefficient''' or autocorrelation <ins style="font-weight: bold; text-decoration: none;">coefficient</ins> ''function.'' To compute the cross term in '''AC''', specify the [[#x-option|<tt>-x</tt>]] option (or [[#u-option|<tt>-u</tt>]] [[#x-option|<tt>-x</tt>]] to suppress normalization).</div></td> </tr> <tr> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>For two time series <math>a_1, \ldots, a_N</math> and <math>b_1, \ldots, b_N</math>, the '''covariance''' is defined as</div></td> <td class="diff-marker"></td> <td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>For two time series <math>a_1, \ldots, a_N</math> and <math>b_1, \ldots, b_N</math>, the '''covariance''' is defined as</div></td> </tr> </table>

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