An example multivariate regression over the fuel consumption for a set of cars given a set of attributes from the R dataset mtcars

  mtcars - A data frame with 32 observations on 11 variables.
  *
  [, 1]	mpg	Miles/(US) gallon
  [, 2]	cyl	Number of cylinders
  [, 3]	disp	Displacement (cu.in.)
  [, 4]	hp	Gross horsepower
  [, 5]	drat	Rear axle ratio
  [, 6]	wt	Weight (1000 lbs)
  [, 7]	qsec	1/4 mile time
  [, 8]	vs	V/S
  [, 9]	am	Transmission (0 = automatic, 1 = manual)
  [,10]	gear	Number of forward gears
  [,11]	carb	Number of carburetors
  

The use example is taken from the example class ExampleMtCarsConsumption. The data is read as CSV and the target column is separated from the predictor variables.

  val inputFile = "data/cars/mtcars.csv"

  def readCars() = {
    val reader = new MatrixReader {}
    val M = reader.read(new File(inputFile))
    val X = M(::, 1 until M.cols)
    val Y = M(::, 0)
    (X, Y.toDenseVector)
  }
        

The OrdLeastSquares class can be trained on the predictor and target variable, the example below trains 3 versions of the model and applies a polynomial transformation by specifying the degree of the polynomial as the third parameter to the apply function.

    val (xMat, yVec) = readCars
    val ols1 = OrdLeastSquares(xMat, yVec, 1)
    val (est1, error1) = ols1.train()
    val Y1 = ols1.predict(xMat)

    val ols2 = OrdLeastSquares(xMat, yVec, 2)
    val (est2, error2) = ols2.train()
    val Y2 = ols2.predict(xMat)

    val ols3 = OrdLeastSquares(xMat, yVec, 3)
    val (est3, error3) = ols3.train()
    val Y3 = ols3.predict(xMat)
        

Note for the sake of the example, the same input data xMat is supplied to the predict function the output of the prediction and set of residuals is demonstrated below.

The Beta parameter z-scores and p-values in the example are accessed using the following:

    println(s"Model 1: P-Values\n")
    printValues(ols1.betaPValue.toDenseVector)
    println("\n")
    println("Model 1: Z-Values\n")
    printValues(ols1.betaZScore.toDenseVector)

    println(s"Model 1: Critical Beta Value: ${ols1.criticalBetaZValue}")
    println(s"Model 1: Critical PValue: ${ols1.criticalPValue}")

    

The example program prints out these values as:

            Model 1: P-Values

0.0
0.3652613211066145
0.3984387352974198
0.3976368502242137
0.0048969511992071005
1.0568802173001641E-43
0.0033237033130584254
0.19474123424914283
1.0222461526756827E-20
0.018876079706805125
0.30077756098169905

            Model 1: Z-Values

46.37027436574356
-0.42000880973224936
0.05025928055824771
-0.08096411105282786
2.966548141714111
-14.002635401119749
3.0944263212321235
1.197618530512598
9.498501040691133
2.470190781809722
-0.7515926480275557

            Model 1: Critical Beta Value: 1.6000000000000003

            Model 1: Critical PValue: 0.11092083467945553
        

Those z values whose absolute values are larger than the critical beta value are considered not to have a mean value of 0.0 (hence contribute to the linear model).