I sought to demonstrate the motivation, construction and application of Linear Least Square Regression.
see: sections - Mapping: Point to Point, Mapping: Point to Function
The sections Mapping: Point to Point and Mapping: Point to Function were constructed to familiarize the reader with the move from considering functions x -f(parameters, form)-> y as simply belonging to the plane A(x,y) to belonging to the collection A(x,y) X B(parameters).
We demonstrate the form of coefficents belonging to B(parameters) which when mapped to A(x,y) always contain the points Q = {(x0, y0), (x1, y1), (x2, y2)} in plane A(x,y). Considering all forms in B(parameters) containing Combinations_Choose_2(Q) we show by example that those forms intersect in plane B(parameters). We then demonstrate that Combinations_Choose_3(Q), there exist forms which do not simulteneously intersect showing that there exists no such form which describes all three points. This then begs the question, does there exist a form in B(parameters) that reasonabally represents the points Q belong to A(x,y)?
see: Least Square Regression, Least Square Regression and Matrix Calculus, The Matrix Derivitive, The Weighted Matrix
In sections Least Square Regression, Least Square Regression and Matrix Calculus and The Matrix Derivitive we describe and formulate the method of Least Square Regression as a means to construct a model of the given data.
In section The Weighted Matrix, I describe the action of the introduction of a weight term to our matrix form and propose a form for the elements of such a matrix.
see: Experiments
I apply our method to forecast NYSE and NIST datasets. We use RSS to measure the goodness of fit for our model and demonstrate the effectiveness of the Weighted Matrix term.
Model building is an incredibly useful tool. It allows one to understand processes outside a first principle approach. I hope to be able to apply and extend this method in the months and years to come.