Multiple regression

Multiple regression
You are encouraged to solve this task according to the task description, using any language you may know.

Given a set of data vectors in the following format:

$y=\{y_{1},y_{2},...,y_{n}\}\,$

$x_{i}=\{x_{i1},x_{i2},...,x_{in}\},i\in 1..k\,$

Compute the vector $\beta =\{\beta _{1},\beta _{2},...,\beta _{n}\}$ using ordinary least squares regression using the following equation:

$y_{i}=\beta \cdot x_{i}\,$

You can assume y is given to you as an array, and x is given to you as a two-dimensional array.

Wikipedia: http://en.wikipedia.org/wiki/Ordinary_least_squares

Note: This is more general than Polynomial Fitting, which only deals with 2 datasets and only deals with polynomial equations. Ordinary least squares can deal with an arbitrary number of datasets (limited by the processing power of the machine) and can have more advanced equations such as: $y=\beta _{1}log(x_{1})+\beta _{2}2^{x_{1}}+\beta _{3}sin(x_{2})$

Ruby

Using the standard library Matrix class:

<lang ruby>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 </lang>

Testing: <lang ruby> regression_coefficients([1, 2, 3, 4, 5], [ [2, 1, 3, 4, 5] ]) # => Matrix0.981818181818182 </lang>