Regression in statistics refers to the capability of modeling and relating a series of one or more independent variables with one target dependent variable. The regression equation can be thought of as a function of variables X (the independent variables) and β (the parameters of the regression):
where Y is the dependent variable.
Subject of the Regression Analysis is then finding the parameters β so that the sum of squared error residuals
is minimized. This procedure is sometimes referred to as least squares estimation.
Linear Multiple Regression
The linear regression assumes that the relationships between independent variables are all linear that is all the β coefficients appear with power one. The following equations:
are both linear regressions even though the independent variables are combined using non-linear functions or powers because both expressions are linear in the parameters β of the regression.
The general linear data model:
can be solved with respect to parameters β by solving the normal matrix equation:
You can download the following example of Linear Regression built using the methods available in Ipredict’s library. The example builds a very simple one week ahead predictor of the Forex EUR/USD.
The logistic regression is used when the dependent variable assumes only two values partitioning in effect the output like in life/death, male/female or buy/sell decisions. Then:
is the regression equation that will be minimized using a least squares approach as in the linear regression case.
You can download the following example of Logistic Regression. The example builds a simple one week ahead predictor of the Forex EUR/USD using a Logistic Regression.
Tikhonov regularization, also known as Ridge Regression, is a common method used to regularize ill-posed problems.
The linear matrix equation
requires the inversion of the matrix
that can be ill-conditioned or singular (does not have an inverse). In this case a solution can be found by solving an alternative problem:
where I is the identity matrix. Of course for this reduces to the standard unregularized least squares regression. The optimal determination of the parameter is a very complex problem and this parameter is normally determined using manual or ad-hoc methods.
Optimal Linear Predictor
The Optimal Linear Predictor is a digital filter that extrapolates linearly from past values the future values of a time-series. This is related to the Autoregressive Model (or AR Model) that is the model that is supposed to be underlying the time-series:
Where Y is the time-series, the are the autoregressive parameters and is white noise. The Optimal Linear Predictor computes all the parameters and then extrapolates the time-series to future values.
You can download the following example of Linear Predictor. The example shows how effective the Linear Predictor is in several useful time-series.
Optimal Detrended Linear Predictor
The Optimal Detrended Linear Predictor is a digital filter that borrows from the Optimal Linear Predictor the basics. This filter is particularly designed to forecast data that has a linear trend. The method essentially detrends the data before applying the Optimal Linear Predictor operator. The forecast is then re-trended using the original trend parameters.