Lineární regrese

Linear regression

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Extensive Linear regression module offers unusually rich commented text and graphical diagnostics for many types of linear regression models including polynomials, multivariate Taylor models, general transformed regreesion. The module goes beyond least squares regression, it offers several robust methods like Lp, M-estimates, Quantile regression, Bounded Influence regression, All possible subsets regression, and much more. Diagnostics includes diagnostics of model, data and regression method, vhich will make results as reliable as possible.

Linear regression - Pdf manual

Regression model coefficients
Curve fitting
Variable prediction
Find most suitable model (Stepwise, All possible subsets)
Calibration, validation
Response surface optimization function
Robust methods for "real world" data
Extensive diagnostic tools will find any problems and pecularities in data or model

Models:

Plain linear model
Polynomial models
Taylor quadratic hypersurfaces
User-defined models
Weighted models
Quasilinear response correction
Implicit models
All posible subset, find best model

Methods:

Least squares
Rank correction/regularization
Quantile regression
Lp-norm regression
Least median of Squares (LMS)
IRWLS
M-estimates
Robust/resistant BIR method
Stepwise/All model selection

Text output:

Basic analysis
Correlations in X
Multicollinearity
Eigenvalues analysis
Analysis of variance (ANOVA)
Regression coefficients statistics
Confidence intervals
Statistical residual analysis
R, RSS, AIC, MEP, etc.
Classical residuals
Dependences in residuals
Model/Data testing
Tests for data and residuals
Tests of model
Predicted statistics
Influential points analysis
Jackknife residuals
Hat Matrix
Cook distance
Atkinson distance
Andrews-Pregibon statistics
Likelihood distances
Prediction

Graphical output:

Regression curve
Y-prediction
Residuals vs. prediction
Absolute residuals
Squared residuals
Residual QQ-plot
Autocorrelation plot
Heteroscedasticity plot
Jackknife residuals
Predicted residuals
Partial regression plots
Partial residual plots
Hat matrix diagonal plot
Predicted residuals QQ-plot
Pregibon statsitic plot
Williams statistic plot
McCulloh statistic plot
L-R plot
Cook distance
Atkinson distance
Studentized residuals
Andrews plot
Jackknife residual QQ-plot

Main panel of Linear regression

Output specification panel

User-defined model panel

Example outputs:

Regression line with identified cases (numbers instead of points) and confidence band (red):

Comparison of two models - model building example (how important is the regression diagniostics):

First model consiered was derived theoretically, has 5 parameters and fits the data well.
Model: [Rate] ~ Abs + [pH-value] + [pH-value]^2 + Ln([pH-value]) + Exp([pH-value])
Its prediction capability is low however, and moreover three out of the five parameters proved insignificant, which makes this model unusable.

Variable	Estimate	Std dev	Conclusion	P-value	Lower C.I.	Upper C.I.
Abs	2.1789	0.47815	Significant	2.00E-005	1.226	3.131
[pH-value]	-0.251	0.64328	Insignificant	0.69727	-1.532	1.03
[pH-value]^2	0.00641	0.03973	Insignificant	0.872108	-0.072	0.085
Ln([pH-value])	1.5023	1.1092	Insignificant	0.17974	-0.707	3.712
Exp([pH-value])	0.000261	3.1980627E-005	Significant	6.29E-012	0.00019	0.0003

Residual variability	12.21360278
F-statistic	359.6264707
Multiple correlation R	0.9752305217
Determination coefficient R^2	0.9510745705
Predicted correlation coef. Rp	0.9418693616
Mean error of prediction MEP	0.1836906899
Akaike information criterion	-137.4849057

In the Second model we dropped one of the insignificant parameters. Model still fits the data still very well.

Model: [Rate] ~ Abs + [pH-value] + [pH-value]^2 + Exp([pH-value])

Its prediction capability outside the interval of measured data has improved. Morover all the model parameters are now significant, so we can compute physical constants from the data. Notice much higher F-statistic, which means overall significance of the model.

Variable	Estimate	Std Dev	Conclusion	p-value	Lower C.I.	Upper C.I.
Abs	1.61910	0.241700	Significant	3.3881E-009	1.13761	2.1005
[pH-value]	0.605230	0.118673	Significant	2.4763E-006	0.36881	0.8416
[pH-value]^2	-0.04453	0.0128498	Significant	0.000877	-0.07013	-0.018
Exp([pH-value])	0.000290	2.356E-005	Significant	0	0.000243	0.00033

Residual variability	12.51634119
F-statistic	473.622371
Multiple correlation R	0.9746085658
Determination coefficient R^2	0.9498618565
Predicted correlation coef. Rp	0.9417364687
Mean error of prediction MEP	0.1841106266
Akaike information criterion	-137.5506086

All possible subsets regression

This method can search up to 8000 regression submodels to select the best one to describe the given data. Criteria for the selection are F-statistic, Mean error of prediction (MEP) and Akaike infiormation statistic (AIC)

Quantile regression

This methods finds the regression quantile curve with a given probability ot data below it. This is very important for example when modelling reliability.