Non-Linear Regression

This method is used to create multifactorial regression models and to forecast the analyzed process using these models.

A time series regression model is the following relation:

Xt = F(Y1, …, Ykt, A) + errt

Where:

Xt. Explained variable.
Y1, …, Ykt. Associated independent (explanatory) variables.
A. Vector of parameters of the selected function F.
errt. Disturbance with zero mean of distribution and the variance δ2, which values are independent at different moments in time and uniformly distributed (otherwise, white noise).

This model has unknown parameters A and δ2 that should be estimated based on available data on the studied process.

Time series regression models may have linear dependency between explanatory variables, which results in multicollinearity and poor conditionality of calculation procedures, as well as in poor statistical properties of model estimated coefficients. Multicollinearity is caused by high correlation between explanatory variables. To eliminate this problem, the user can use linear combinations of explanatory variables with poor or no correlation between them as new variables. This is the basis for the principal component analysis used in this system to estimate regression parameters; the method is considered to be a robust (stable) method of estimation.

After the unknown parameters have been estimated, the user can move to forecasting frequency of the analyzed series, which requires future or forecasted values of factorial series.

Xt = F(Y1t + T, …, Ykt + T, A)

Where T – lead period.

One of the characteristics of the regression model quality is the determination coefficient or the adjusted determination coefficient, which is an alternative version of the former. The latter is an unbiased estimate of the real determination coefficient.

Model estimated coefficients follow the Student's distribution in the asymptotic form. This is used as the basis for checking the hypothesis of the coefficients equality to zero and for calculating confidence limits of the coefficients. The Durbin-Watson coefficient is commonly used for checking residuals autocorrelation.