Example1.java

package applications.statistics;


import algorithms.optimizers.BatchGradientDescent;
import algorithms.optimizers.GDInput;
import algorithms.utils.DefaultIterativeAlgorithmController;
import algorithms.utils.IterativeAlgorithmResult;
import datastructs.maths.DenseMatrixSet;
import datastructs.maths.RowBuilder;
import datastructs.maths.Vector;
import datastructs.utils.RowType;
import maths.errorfunctions.MSEVectorFunction;
import maths.errorfunctions.SSEVectorFunction;
import maths.functions.LinearVectorPolynomial;
import ml.regression.LinearRegressor;
import plotting.PlotOptions;
import plotting.PlotScatter;
import tech.tablesaw.api.Table;
import utils.ListMaths;
import utils.TableDataSetLoader;

import java.io.File;
import java.io.IOException;

/** Category: Statistics
 * ID: Example1
 * Description: Goodness of fit of regression line
 * Taken From:
 * Details:
 * TODO
 */
public class Example1 {

    public static void main(String[] args)throws IOException {

        // load the data
        Table dataSet = TableDataSetLoader.loadDataSet(new File("src/main/resources/datasets/car_plant.csv"));

        // let's plot the dataset
        PlotOptions options = new PlotOptions();
        options.plotTitle = "Production vs Electricity Usage";
        PlotScatter.plot(options, dataSet, "Production", "Electricity Usage");

        Vector labels = new Vector(dataSet, "Electricity Usage");
        Table reducedDataSet = dataSet.removeColumns("Electricity Usage").first(dataSet.rowCount());

        DenseMatrixSet<Double> denseMatrixSet = new DenseMatrixSet(RowType.Type.DOUBLE_VECTOR, new RowBuilder(), reducedDataSet.rowCount(), 2, 1.0);
        denseMatrixSet.setColumn(1, reducedDataSet.doubleColumn(0));

        LinearVectorPolynomial hypothesis = new LinearVectorPolynomial(1);
        LinearRegressor regressor = new LinearRegressor(hypothesis);

        GDInput gdInput = new GDInput();
        gdInput.showIterations = false;
        gdInput.eta=0.01;
        gdInput.errF = new MSEVectorFunction(hypothesis);
        gdInput.iterationContorller = new DefaultIterativeAlgorithmController(10000,1.0e-8);

        BatchGradientDescent gdSolver = new BatchGradientDescent(gdInput);

        IterativeAlgorithmResult result = (IterativeAlgorithmResult) regressor.train(denseMatrixSet, labels, gdSolver);

        System.out.println(result);
        System.out.println("Intercept: "+hypothesis.getCoeff(0)+" slope: "+hypothesis.getCoeff(1));

        // let's see the max error over the dateset
        Vector errors = regressor.getErrors(denseMatrixSet, labels);
        double maxError = ListMaths.max(errors.getRawData());

        System.out.println("Maximum error over dataset: "+maxError);

        // let's get an estimate of the error variance.
        //The error variance sigma^2 can be estimated by considering the deviations between the observed
        //data values y_i and their fitted values \hat(y)_i . Specifically, the sum of squares for error SSE is defined
        //to be the sum of the squares of these deviations
        Vector yhat = regressor.predict(denseMatrixSet);

        double sseError = SSEVectorFunction.error(labels, yhat);
        double sigma2_hat = sseError/ (yhat.size()-2);
        System.out.println("Estimate of error variance: "+ sigma2_hat);

        // interval estimation
        double Sxx = ListMaths.sxx(denseMatrixSet.getColumn(1).getRawData());
        System.out.println("Estimate of Sxx: "+Sxx);

        // standard error for the slope
        double se_slope = Math.sqrt(sigma2_hat)/Math.sqrt(Sxx);
        System.out.println("Standard error for the slope: "+se_slope);

        // t-statistic
        double t = hypothesis.getCoeff(1)/se_slope;
        System.out.println("t-statistic: "+t);

        //The two-sided p-value is calculated as
        //p-value = 2 × P(X > 6.37) approx 0
        //where the random variable X has a t-distribution with 10 degrees of freedom. This low p-value
        //indicates that the null hypothesis is not plausible and so the slope parameter is known to be
        //nonzero. In other words, it has been established that the distribution of electricity usage does
        //depend on the level of production.

        // The proportion of the total variability in the dependent variable y that is accounted for by
        // the regression line is given by the coefficient of determination.
        // This coefficient takes a value between 0 and 1, and the closer it is to one the smaller is the
        // sum of squares for error SSE in relation to
        // the sum of squares for regression SSR. Thus, larger values of R^2 tend to indicate that the data
        // points are closer to the fitted regression line. Nevertheless, a low
        // value of R^2 should not necessarily be interpreted as implying that the fitted regression line is
        // not appropriate or is not useful. A fitted regression line may be accurate and informative even
        // though a small value of R^2 is obtained because of a large error variance sigma62.
        double sst = ListMaths.sse(labels.getRawData());
        double r_sqr =  1.0- sseError/sst;

        System.out.println("Coefficient of determination: "+r_sqr);


    }
}