Example Polynomial Fitting

In this example it is shown how EJML can be used to fit a polynomial of arbitrary degree to a set of data. The key concepts shown here are; 1) how to create a linear using LinearSolverFactory, 2) use an adjustable linear solver, 3) and effective matrix reshaping. This is all done using the procedural interface.

First a best fit polynomial is fit to a set of data and then a outliers are removed from the observation set and the coefficients recomputed. Outliers are removed efficiently using an adjustable solver that does not resolve the whole system again.

External Resources:

• PolynomialFit.java source code
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PolynomialFit Example Code

/**
 * <p>
 * This example demonstrates how a polynomial can be fit to a set of data.  This is done by
 * using a least squares solver that is adjustable.  By using an adjustable solver elements
 * can be inexpensively removed and the coefficients recomputed.  This is much less expensive
 * than resolving the whole system from scratch.
 * </p>
 * <p>
 * The following is demonstrated:<br>
 * <ol>
 *  <li>Creating a solver using LinearSolverFactory</li>
 *  <li>Using an adjustable solver</li>
 *  <li>reshaping</li>
 * </ol>
 * @author Peter Abeles
 */
public class PolynomialFit {

    // Vandermonde matrix
    DenseMatrix64F A;
    // matrix containing computed polynomial coefficients
    DenseMatrix64F coef;
    // observation matrix
    DenseMatrix64F y;

    // solver used to compute
    AdjustableLinearSolver solver;

    /**
     * Constructor.
     *
     * @param degree The polynomial's degree which is to be fit to the observations.
     */
    public PolynomialFit( int degree ) {
        coef = new DenseMatrix64F(degree+1,1);
        A = new DenseMatrix64F(1,degree+1);
        y = new DenseMatrix64F(1,1);

        // create a solver that allows elements to be added or removed efficiently
        solver = LinearSolverFactory.adjustable();
    }

    /**
     * Returns the computed coefficients
     *
     * @return polynomial coefficients that best fit the data.
     */
    public double[] getCoef() {
        return coef.data;
    }

    /**
     * Computes the best fit set of polynomial coefficients to the provided observations.
     *
     * @param samplePoints where the observations were sampled.
     * @param observations A set of observations.
     */
    public void fit( double samplePoints[] , double[] observations ) {
        // Create a copy of the observations and put it into a matrix
        y.reshape(observations.length,1,false);
        System.arraycopy(observations,0, y.data,0,observations.length);

        // reshape the matrix to avoid unnecessarily declaring new memory
        // save values is set to false since its old values don't matter
        A.reshape(y.numRows, coef.numRows,false);

        // set up the A matrix
        for( int i = 0; i < observations.length; i++ ) {

            double obs = 1;

            for( int j = 0; j < coef.numRows; j++ ) {
                A.set(i,j,obs);
                obs *= samplePoints[i];
            }
        }

        // process the A matrix and see if it failed
        if( !solver.setA(A) )
            throw new RuntimeException("Solver failed");

        // solver the the coefficients
        solver.solve(y,coef);
    }

    /**
     * Removes the observation that fits the model the worst and recomputes the coefficients.
     * This is done efficiently by using an adjustable solver.  Often times the elements with
     * the largest errors are outliers and not part of the system being modeled.  By removing them
     * a more accurate set of coefficients can be computed.
     */
    public void removeWorstFit() {
        // find the observation with the most error
        int worstIndex=-1;
        double worstError = -1;

        for( int i = 0; i < y.numRows; i++ ) {
            double predictedObs = 0;

            for( int j = 0; j < coef.numRows; j++ ) {
                predictedObs += A.get(i,j)*coef.get(j,0);
            }

            double error = Math.abs(predictedObs- y.get(i,0));

            if( error > worstError ) {
                worstError = error;
                worstIndex = i;
            }
        }

        // nothing left to remove, so just return
        if( worstIndex == -1 )
            return;

        // remove that observation
        removeObservation(worstIndex);

        // update A
        solver.removeRowFromA(worstIndex);

        // solve for the parameters again
        solver.solve(y,coef);
    }

    /**
     * Removes an element from the observation matrix.
     *
     * @param index which element is to be removed
     */
    private void removeObservation( int index ) {
        final int N = y.numRows-1;
        final double d[] = y.data;

        // shift
        for( int i = index; i < N; i++ ) {
            d[i] = d[i+1];
        }
        y.numRows--;
    }
}