Nonlinear Regression
Nonlinear regression is a technique to analyze a nonlinear relationship between one or more independent variables and a dependent variable. The values of the independent variables are considered to be exact, while the values of the dependent variables are subject to error. The NonlinearRegressionModel class implements nonlinear regression in one variable.
Constructing Nonlinear Regression Models
The NonlinearRegressionModel class has three constructors.
The first constructor takes two arguments. The first is a numerical Vector<T> that represents the dependent variable. The second is a numerical Vector<T> that represent the independent variables.
NumericalVariable dependent = new NumericalVariable("y", yData);
NumericalVariable independent = new NumericalVariable("x1", x1Data);
NonlinearRegressionModel model1 = new NonlinearRegressionModel(dependent, independent);The third constructor takes two Vector<T> parameters. The first contains the data for the independent variable. The second contains the data for the dependent variable.
The fourth constructor takes 3 arguments. The first argument is a IDataFrame (a DataFrame<R, C> or Matrix<T>) that contains the variables to be used in the regression. The second argument is a string containing the name of the dependent variable. The third argument is a string containing the name of the independent. All the names must exist in the collection specified by the first parameter. All variables must be of type numerical Vector<T>.
VariableCollection variables = new VariableCollection();
variables.Add(dependent);
variables.Add(independent);
NonlinearRegressionModel model2 = new NonlinearRegressionModel(variables, "y", "x1");The fifth constructor also takes 3 arguments. The first argument is a DataTable object that contains the data for the regression analysis. The second argument is a string containing the name of the column that contains the data for the dependent variable. The third argument is a string containing the name of the column that contains the data for the independent variable. Both columns must be numerical or convertible to numerical values.
DataCollection table = new DataTable();
// Fill data table with data from some datasource.
NonlinearRegressionModel model3 = new NonlinearRegressionModel(table, "y", "x1");Defining the model
A nonlinear model is defined by a function that is nonlinear in the curve parameters and the independent variable. Such a function is represented by the NonlinearCurve class in the Extreme.Mathematics.Curves namespace
.To define the model, set the model's Curve to an instance of the curve you want to use. The Extreme.Mathematics.Curves.Nonlinear namespace defines a number of nonlinear curves. This includes exponential, rational and logistic curves. You can also define your own nonlinear curves. For details, see the chapter on Curve Fitting in the Mathematics Library for .NET User's Guide.
Computing the Regression
The Compute method performs the actual analysis. Most properties and methods throw an exception when they are accessed before the Compute method is called. You can verify that the model has been calculated by inspecting the Computed property.
model1.Compute();The Predictions property returns a Vector<T> that contains the values of the dependent variable as predicted by the model. The Residuals property returns a vector containing the difference between the actual and the predicted values of the dependent variable. Both vectors contain one element for each observation.
Regression Parameters
The NonlinearRegressionModel class' Parameters property returns a ParameterVector<T> object that contains the parameters of the regression model. The members of this collection are of type Parameter<T>. Regression parameters are created by the model. You cannot create them directly.
Parameters can be accessed by numerical index or by name. Parameters are automatically given the names A0, A1, and so on.
The Parameter class has four useful properties. The Value property returns the numerical value of the parameter, while the StandardError property returns the standard deviation of the parameter's distribution.
The Statistic property returns the value of the t-statistic corresponding to the hypothesis that the parameter equals zero. The PValue property returns the corresponding p-value. A high p-value indicates that the variable associated with the parameter does not make a significant contribution to explaining the data. The p-value always corresponds to a two-tailed test. The following example prints the properties of the slope parameter of our earlier example:
Parameter x1Parameter = model1.Parameters["x1"];
Console.WriteLine("Name: {0}", x1Parameter.Name);
Console.WriteLine("Value: {0}", x1Parameter.Value);
Console.WriteLine("St.Err.: {0}", x1Parameter.StandardError);
Console.WriteLine("t-statistic: {0}", x1Parameter.TStatistic);
Console.WriteLine("p-value: {0}", x1Parameter.PValue);The Parameter class has one method: GetConfidenceInterval. This method takes one argument: a confidence level between 0 and 1. A value of 0.95 corresponds to a confidence level of 95%. The method returns the confidence interval for the parameter at the specified confidence level as an Interval structure.
Verifying the Quality of the Regression
The ResidualSumOfSquares property gives the sum of the squares of the residuals. The regression line was found by minimizing this value. The StandardError property gives the standard deviation of the data.
The RSquared property returns the coefficient of determination. It is the ratio of the variation in the data that is explained by the model compared to the total variation in the data. Its value is always between 0 and 1, where 0 means the model explains nothing and 1 means the model explains the data perfectly.
An entirely different assessment is available through an analysis of variance. Here, the variation in the data is decomposed into a component explained by the model, and the variation in the residuals. The FStatistic property returns the F-statistic for the ratio of these two variances. The PValue property returns the corresponding p-value. A low p-value means that it is unlikely that the variation in the model is the same as the variation in the residuals. This means that the model is significant.
The results of the analysis of variance are also summarized in the regression model's ANOVA table, returned by the AdjustedRSquared property.