Problem 2043. Six Steps to PCA - Step 1: Centre and Standardize

Created by Neil Caithness

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Invented in 1901 by Karl Pearson the method is mostly used today as a tool in exploratory data analysis and dimension reduction, but also for making predictive models in machine learning.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>Step 1: Centre and Standardize</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>A first step for many multivariate methods begins by removing the influence of location and scale from variables in the raw data. Also commonly known as the z-scores of</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>X</w:t></w:r><w:r><w:t>,</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>Z</w:t></w:r><w:r><w:t> is a transformation of</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>X</w:t></w:r><w:r><w:t> such that the columns are centered to have mean 0 and scaled to have standard deviation 1 (unless a column of</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>X</w:t></w:r><w:r><w:t> is constant, in which case that column of</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>Z</w:t></w:r><w:r><w:t> is constant at 0). Strictly speaking, z-scores are based on population parameters, whereas the analogous calculation based on sample mean and standard deviation is the Student's t-statistic.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>Task</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Write a function to centre and standardize the input matrix</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>X</w:t></w:r><w:r><w:t>, returning as the output a structure with the following fields:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr></w:pPr><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>Z</w:t></w:r><w:r><w:t>: the centred and standardized matrix corresponding to the input</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>X</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr></w:pPr><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>Mu</w:t></w:r><w:r><w:t>: a vector of the original means of columns of</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>X</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr></w:pPr><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>Sigma</w:t></w:r><w:r><w:t>: a vector of the original standard deviations of columns of</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>X</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>Tips</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr></w:pPr><w:r><w:t>Matlab's</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>zscore</w:t></w:r><w:r><w:t> function is part of the Stats Toolbox which is not available in Cody. You'll have to write your own.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr></w:pPr><w:r><w:t>You should take care to avoid division by zero when a column is invariant.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>Following problems in the series</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr></w:pPr><w:hyperlink w:docLocation=\"http://www.mathworks.co.uk/matlabcentral/cody/problems/2049-six-steps-to-pca-step-2-covariance\"><w:r><w:t>Step 2: Covariance</w:t></w:r></w:hyperlink></w:p></w:body></w:document>"},{"partUri":"/matlab/output.xml","contentType":"text/xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\" ?><embeddedOutputs><metaData><evaluationState>manual</evaluationState><layoutState>code</layoutState><outputStatus>ready</outputStatus></metaData><outputArray type=\"array\"/><regionArray type=\"array\"/></embeddedOutputs>"}]}

IntroductionPrincipal Component Analysis (PCA) is a classic among the many methods of multivariate data analysis. Invented in 1901 by Karl Pearson the method is mostly used today as a tool in exploratory data analysis and dimension reduction, but also for making predictive models in machine learning.Step 1: Centre and StandardizeA first step for many multivariate methods begins by removing the influence of location and scale from variables in the raw data. Also commonly known as the z-scores of <tt>X</tt>, <tt>Z</tt> is a transformation of <tt>X</tt> such that the columns are centered to have mean 0 and scaled to have standard deviation 1 (unless a column of <tt>X</tt> is constant, in which case that column of <tt>Z</tt> is constant at 0). Strictly speaking, z-scores are based on population parameters, whereas the analogous calculation based on sample mean and standard deviation is the Student's t-statistic.TaskWrite a function to centre and standardize the input matrix <tt>X</tt>, returning as the output a structure with the following fields:<ul><li><tt>Z</tt>: the centred and standardized matrix corresponding to the input <tt>X</tt></li><li><tt>Mu</tt>: a vector of the original means of columns of <tt>X</tt></li><li><tt>Sigma</tt>: a vector of the original standard deviations of columns of <tt>X</tt></li></ul>Tips<ul><li>Matlab's <tt>zscore</tt> function is part of the Stats Toolbox which is not available in Cody. You'll have to write your own.</li><li>You should take care to avoid division by zero when a column is invariant.</li></ul>Following problems in the series<ul><li><a href = "http://www.mathworks.co.uk/matlabcentral/cody/problems/2049-six-steps-to-pca-step-2-covariance">Step 2: Covariance</a></li></ul>

*Introduction*

Principal Component Analysis (PCA) is a classic among the many methods of multivariate data analysis. Invented in 1901 by Karl Pearson the method is mostly used today as a tool in exploratory data analysis and dimension reduction, but also for making predictive models in machine learning.

*Step 1: Centre and Standardize*

A first step for many multivariate methods begins by removing the influence of location and scale from variables in the raw data. Also commonly known as the z-scores of |X|, |Z| is a transformation of |X| such that the columns are centered to have mean 0 and scaled to have standard deviation 1 (unless a column of |X| is constant, in which case that column of |Z| is constant at 0). Strictly speaking, z-scores are based on population parameters, whereas the analogous calculation based on sample mean and standard deviation is the Student's t-statistic.

*Task*

Write a function to centre and standardize the input matrix |X|, returning as the output a structure with the following fields:

* |Z|: the centred and standardized matrix corresponding to the input |X|
* |Mu|: a vector of the original means of columns of |X|
* |Sigma|: a vector of the original standard deviations of columns of |X|

*Tips*

* Matlab's |zscore| function is part of the Stats Toolbox which is not available in Cody. You'll have to write your own.
* You should take care to avoid division by zero when a column is invariant.

*Following problems in the series*

* <http://www.mathworks.co.uk/matlabcentral/cody/problems/2049-six-steps-to-pca-step-2-covariance Step 2: Covariance>

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Last Solution submitted on Jul 28, 2023

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Jean-Marie Sainthillier on 10 Feb 2014

Your definition of a constant (or invariant) data with rand is problematic. If you increase the size of your data (n=1000, n=10000...), you can always increase the deviations (so what threshold for sigma ?). I think that with real data, this artifact isn't possible. No ?

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