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In class independent PCA we generate a single covariance using the points from all data sets. After computing the covariance we then use it to determine the transform in the manner described below. Points from the current space are mapped to the new feature space my multiplying it with the transpose of the transform. The point are also normalized by dividing them by their corresponding eigen value to obtain the Whitening transformation which is represented by the equation below.
  • The linear transformation T which is used to transform points from the current space to a new feature space is determined using:


    The main objective of transformation is to make the covariance of Y an identity matrix.
    We desire T to be an orthonormal transformation.

  • The transform is defined by the eigen vectors and eigen values of the covariance. The eigen vectors describe the coordinate system of the new feature space and the eigen values describe the variance of the data set in the new feature space. Note that the reason we use the covariance to obtain the transform is because we need to maintain the structure of the distribution, and the covariance of the data set gives us the structure of the distribution. The transform is obtained using the following formula:
    Where are the eigenvalues and Phi are the eigenvectors of the covariance.

  • Here is a brief example of how the class independent PCA scheme works:

    First select the Two Gaussian data set from the Patterns menu. Following that select the Class Independent PCA option under the Algorithms menu. Initialize this algorithm by selecting Initialize from the Go menu.In order to compute the line of discrimination select the Next option under the Go menu. This will display the first step of the process, data sets in both the input plot (top left) and the output plot (bottom left) of the applet. Also, the process description box indicates which step you are currently on and the algorithm that is currently being used to compute the line of discrimination.

    Class Independent PCA

  • The second step of the process computes the mean of the each data set. The mean of each data sets is displayed on the output plot as black dots near the corresponding data sets. The value of the mean for each data set, which corresponds to the current scale, is displayed on the process description box. The covariance and transformation matrices used to compute the line of discrimination for the data set are also displayed in the process description box.

    Class Independent PCA

  • The third step of the process displays the line of discrimination of the given data sets as determined by the class independent PCA algorithm. Also, the classification error for each data set along with the total classification error is displayed on the process description box.

    Class Independent PCA

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