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Java Library for Statistical Analysis.
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# Example 6: Goodness of fit of regression line
 
 ## Contents
 * [Overview](#overview) 
     * [Goodness of fit of regression line](#goodness_of_fit)
 * [Import files](#include_files)
 * [The main function](#m_func)
 * [Results](#results)
 * [Source Code](#source_code)
 
 ##  Overview
 
 ###  Goodness of fit of regression line
 

 ##  Import files
 
 ```
package examples.stats.example3;

import optimization.GradientDescent;
import optimization.GDInput;
import utils.DefaultIterativeAlgorithmController;
import utils.IterativeAlgorithmResult;
import datasets.DenseMatrixSet;
import datastructs.RowBuilder;
import datasets.VectorDouble;
import datastructs.RowType;
import maths.errorfunctions.MSEVectorFunction;
import maths.errorfunctions.SSEVectorFunction;
import maths.functions.LinearVectorPolynomial;
import ml.regression.LinearRegressor;

import tech.tablesaw.api.Table;
import utils.ListMaths;
import utils.TableDataSetLoader;

import java.io.File;
import java.io.IOException;
 ```
 
 ##  The main function
 
 ```
 public class Example3 {
 
     public static void main(String[] args)throws IOException {
 
         // load the data
         Table dataSet = TableDataSetLoader.loadDataSet(new File("src/main/resources/datasets/car_plant.csv"));
 
         Vector labels = new Vector(dataSet, "Electricity Usage");
         Table reducedDataSet = dataSet.removeColumns("Electricity Usage").first(dataSet.rowCount());
 
         DenseMatrixSet 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);
  
     }
 }
     
 ```
 
 ##  Results
 
 ```
Converged: true
Tolerance: 9.995007266283551E-9
# Threads: 1
Iterations: 7076

Intercept: 0.37857734128519877 slope: 0.5049674670001678
Maximum error over dataset: 0.2780755638140122
Estimate of error variance: 0.02992964035512725
Estimate of Sxx: 4.8723000000000525
Standard error for the slope: 0.07837611613854814
t-statistic: 6.442874333138931
Coefficient of determination: 0.8019860710106865
 
 ```
 
 ##  Source Code
 
 Example3.java