Example: Covariance and Principal Component Analysis

Functions > Vector and Matrix > Array Characteristics > Example: Covariance and Principal Component Analysis

Calculate the covariance of a data set, and use the eigenvals and eigenvec functions to perform principal component analysis.

1. Define the following matrix D.

Each row of D represents one observation, each column a measured characteristic.

2. Calculate the covariance matrix of the sample.

3. Compute and sort the eigenvalues.

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    <define xmlns="http://schemas.mathsoft.com/math50">
      <id label="None" xml:space="preserve">V</id>
      <apply>
        <id label="Function" xml:space="preserve" label-is-contextual="true">reverse</id>
        <apply>
          <id label="Function" xml:space="preserve" label-is-contextual="true">sort</id>
          <apply>
            <id label="Function" xml:space="preserve" label-is-contextual="true">eigenvals</id>
            <id label="Variable" xml:space="preserve" label-is-contextual="true">S1</id>
          </apply>
        </apply>
      </apply>
    </define>
    <resultFormat>
      <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
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4. Compute the transform matrix.

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        <id label="Function" xml:space="preserve" label-is-contextual="true">eigenvec</id>
        <sequence>
          <id label="Variable" xml:space="preserve" label-is-contextual="true">S1</id>
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            <id label="Variable" xml:space="preserve" label-is-contextual="true">V</id>
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      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
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            <real>0</real>
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      <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
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5. Transform the original data.

6. Calculate the covariance matrix of the transformed data.

The principal components of the data are the diagonal elements of matrix S2.