# VB Least Squares Example

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```Imports System
Imports System.Globalization

Imports CenterSpace.NMath.Core

Namespace CenterSpace.NMath.Core.Examples.VisualBasic

' A .NET example in Visual Basic showing how to use the least squares classes to solve
' linear least squares problems.
Module LeastSquaresExample

Sub Main()

' This example uses strings representing numbers in the US locale
' so change the current culture info.  For example, "\$4.30"

' Calculate the slope and intercept of the linear least squares fit
' through the five points:
' (20, .446) (30, .601), (40, .786), (50, .928), (60, .950)
Dim A As New DoubleMatrix("5x1[20.0  30.0  40.0  50.0  60.0]")
Dim Y As New DoubleVector("[.446 .601 .786 .928 .950]")

' Back to original culture

' We want our straight line to be of the form y = mx + b, where b is
' not necessarily equal to zero. Thus we will set the third
' constructor argument to true so that we calculate the intercept
' parameter.
Dim Lsq As New DoubleLeastSquares(A, Y, True)

Console.WriteLine()
Console.WriteLine()
Console.WriteLine("Y-intercept = {0}", Lsq.X(0))
Console.WriteLine("Slope = {0}", Lsq.X(1))

' We can look at the residuals which are the difference between the
' actual value of y at a point x, and the corresponding point y on
' line for the same x.
Console.WriteLine("Residuals = {0}", Lsq.Residuals.ToString("F3"))

' Finally, we can look at the residual sum of squares, which is the
' sum of the squares of the elements in the residual vector.
Console.WriteLine("Residual Sum of Squares (RSS) = {0}", Lsq.ResidualSumOfSquares.ToString("F3"))

' The least squares class can also be used to solve "rank-deficient" least
' square problems:
A = New DoubleMatrix("6x4 [0 9 -6 3  -3 0 -3 0  1 3 -1 1  1 3 -1 1  -2 0 -2 0  3 6 -1 2]")
Y = New DoubleVector("[-3 5 -2 2 1 -2]")

' For this problem we will specify a tolerance for computing the effective rank
' of the matrix A, and we will not have the class add an intercept parameter
' for us.
Lsq = New DoubleLeastSquares(A, Y, 0.0000000001)
Console.WriteLine("Least squares solution = {0}", Lsq.X.ToString("F3"))
Console.Write("Rank computed using a tolerance of ")
Console.Write(Lsq.Tolerance.ToString() + ", = ")
Console.WriteLine(Lsq.Rank.ToString())

' You can even use the least squares class to solve under-determined systems
' (the case where A has more columns than rows).
A = New DoubleMatrix("6x4 [-3 -1 6 -5  5 4 -6 8  7 5 0 -4  -7 4 0 3  -7 7 -8 2  3 4 2 -4]")
Y = New DoubleVector("[-3 1 8 -2]")
Lsq = New DoubleLeastSquares(A.Transpose(), Y, 0.00000001)

Console.Write("Solution to under-determined system = ")
Console.WriteLine(Lsq.X.ToString("F3"))

Console.Write("Rank computed using a tolerance of ")
Console.Write(Lsq.Tolerance.ToString() + " is ")
Console.WriteLine(Lsq.Rank.ToString())

Console.WriteLine()
Console.WriteLine("Press Enter Key")