Step by Step solution about How to resolve the algorithm Multiple regression step by step in the Ruby programming language

The Ruby code snippet you provided demonstrates the implementation of linear regression to determine the coefficients of a linear equation that best fits a given set of data points. It takes in two arrays: y representing the dependent variable and x representing the independent variables. Then, the code uses the Matrix library to perform matrix operations and compute the regression coefficients.

A breakdown of the code:

1. require 'matrix': This line includes the 'Matrix' library, which provides classes and methods for working with matrices in Ruby.
2. def regression_coefficients y, x: This line defines a method called regression_coefficients that takes two parameters: y (an array of dependent variable values) and x (an array of independent variable values).
3. y = Matrix.column_vector y.map { |i| i.to_f }: Here, the dependent variable values (y) are converted into a column vector of type Matrix::ColumnVector using the column_vector method. This is done by mapping each element of the y array to its floating-point representation using to_f.
4. x = Matrix.columns x.map { |xi| xi.map { |i| i.to_f }}: The independent variable values (x) are converted into a matrix of type Matrix using the columns method. This is achieved by first mapping each row of the x array (xi represents each row) to a row vector, and then mapping each element of the row vector to its floating-point representation using to_f.
5. (x.t * x).inverse * x.t * y: This line calculates the regression coefficients using matrix operations.
   • x.t: Transposes the x matrix.
   • (x.t * x): Multiplies the transposed x matrix by itself, effectively calculating the covariance matrix.
   • (x.t * x).inverse: Inverts the covariance matrix.
   • (x.t * x).inverse * x.t: Multiplies the inverse of the covariance matrix by the transposed x matrix.
   • (x.t * x).inverse * x.t * y: Finally, multiplies the result by the column vector y, which contains the dependent variable values. The resulting matrix represents the regression coefficients.
6. The code demonstrates the usage of the regression_coefficients method by providing two examples:
   • The first example computes regression coefficients for a set of data points with one independent variable.
   • The second example computes regression coefficients for a set of data points with two independent variables.

In summary, this code defines a method for linear regression that calculates regression coefficients based on a given set of data points. It utilizes the 'Matrix' library for matrix operations, and demonstrates the method's usage with two examples of linear regression.

require 'matrix'

def regression_coefficients y, x
  y = Matrix.column_vector y.map { |i| i.to_f }
  x = Matrix.columns x.map { |xi| xi.map { |i| i.to_f }}

  (x.t * x).inverse * x.t * y
end


puts regression_coefficients([1, 2, 3, 4, 5], [ [2, 1, 3, 4, 5] ])


puts regression_coefficients([3,4,5], [ [1,2,1], [1,1,2] ])