* y = b dot x + e = [b0, b1] dot [1, x1] + e = b0 + b1 * x1 + e *
* where 'e' represents the residuals (the part not explained by the model). * @param x the input/design matrix augmented with a first column of ones * @param y the response vector */ class SimpleRegression (x: MatriD, y: VectoD) extends PredictorMat (x, y) { if (x.dim2 != 2) flaw ("constructor", "design matrix must have 2 columns") //:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: /** Train the predictor by fitting the parameter vector (b-vector) in the * simple regression equation *
* y = b dot x + e = [b0, b1] dot [1, x1] + e *
* using the least squares method. * @see www.analyzemath.com/statistics/linear_regression.html * @param yy the response vector */ def train (yy: VectoD = y): SimpleRegression = { val x1 = x.col(1) // get column 1 of x = [1.0, x1] val sx = x1.sum // sum of x values val sy = y.sum // sum of y values val ssx = x1 dot x1 // sum of squares x val ssy = y dot y // sum of squares y val sxy = x1 dot y // sum of cross products b = new VectorD (2) // parameter vector [b0, b1] b(1) = (m * sxy - sx * sy) / (m * ssx - sx*sx) // slope b(0) = (sy - b(1) * sx) / m // intercept this } // train //:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: /** Perform 'k'-fold cross-validation. * @param k the number of folds */ def crossVal (k: Int = 10) { crossValidate ((x: MatriD, y: VectoD) => new SimpleRegression (x, y), k) } // crossVal } // SimpleRegression class //::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: /** The `SimpleRegression` companion object provides a simple factory method * for building simple regression linear regression models. */ object SimpleRegression { //::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: /** Create a Simple Linear Regression model, automatically prepending the * column of ones (form matrix from two column vectors [1 x]). * @param x the input/design m-by-1 vector * @param y the response m-vector */ def apply (x: VectoD, y: VectoD): SimpleRegression = { new SimpleRegression (MatrixD.form_cw (1.0, x), y) } // apply } // SimpleRegression object //::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: /** The `SimpleRegressionTest` object to test the `SimpleRegression` class: *
* y = b0 + b1 * x *
* > runMain scalation.analytics.SimpleRegressionTest */ object SimpleRegressionTest extends App { // 4 data points: val x = VectorD (1, 2, 3, 4) val y = VectorD (1, 3, 3, 4) // val y = VectorD (1, 3, 2, 4) println ("x = " + x) println ("y = " + y) val rg = SimpleRegression (x, y) rg.train ().eval () println ("coefficient = " + rg.coefficient) println ("fitMap = " + rg.fitMap) } // SimpleRegressionTest object //::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: /** The `SimpleRegressionTest2` object is used to test the `SimpleRegression` class. *
* y = b dot x = [b0, b1] dot [1, x1] *
* @see http://www.analyzemath.com/statistics/linear_regression.html * > runMain scalation.analytics.SimpleRegressionTest2 */ object SimpleRegressionTest2 extends App { // 5 data points: constant x1 val x = new MatrixD ((5, 2), 1.0, 0.0, // x 5-by-2 matrix 1.0, 1.0, 1.0, 2.0, 1.0, 3.0, 1.0, 4.0) val y = VectorD (2.0, 3.0, 5.0, 4.0, 6.0) // y vector println ("x = " + x) println ("y = " + y) val rg = new SimpleRegression (x, y) rg.train ().eval () println ("coefficient = " + rg.coefficient) println ("fitMap = " + rg.fitMap) val z = VectorD (1.0, 5.0) // predict y for one point val yp = rg.predict (z) println ("predict (" + z + ") = " + yp) val yyp = VectorD (for (i <- x.range1) yield rg.predict (x(i))) // predict y for several points println ("predict (" + x + ") = " + yyp) new Plot (x.col(1), y, yyp) } // SimpleRegressionTest2 object //::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: /** The `SimpleRegressionTest3` object is used to test the `SimpleRegression` class *
* y = b dot x = b0 + b1 * x1 *
* @see http://mathbits.com/mathbits/tisection/Statistics2/linear.htm * > runMain scalation.analytics.SimpleRegressionTest3 */ object SimpleRegressionTest3 extends App { // 20 data points: just x1 coordinate val x1 = VectorD ( 4.0, 9.0, 10.0, 14.0, 4.0, 7.0, 12.0, 22.0, 1.0, 3.0, 8.0, 11.0, 5.0, 6.0, 10.0, 11.0, 16.0, 13.0, 13.0, 10.0) val y = VectorD (390.0, 580.0, 650.0, 730.0, 410.0, 530.0, 600.0, 790.0, 350.0, 400.0, 590.0, 640.0, 450.0, 520.0, 690.0, 690.0, 770.0, 700.0, 730.0, 640.0) println ("x1 = " + x1) println ("y = " + y) val x = MatrixD.form_cw (1.0, x1) // form matrix x from vector x1 val rg = new SimpleRegression (x, y) rg.train ().eval () println ("coefficient = " + rg.coefficient) println ("fitMap = " + rg.fitMap) val z = VectorD (1.0, 15.0) // predict y for one point val yp = rg.predict (z) println ("predict (" + z + ") = " + yp) val yyp = VectorD (for (i <- x.range1) yield rg.predict (x(i))) // predict y for several points println ("predict (" + x + ") = " + yyp) new Plot (x1, y, yyp) } // SimpleRegressionTest3 object