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Multiple Linear Regression
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Multiple Linear Regression
Multiple linear regression is a technique to analyze a
linear 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
Extreme Optimization Numerical Libraries for .NET supports linear
regression through the LinearRegressionModel class.
Constructing Multiple Linear Regression Models
The LinearRegressionModel class has four constructors.
The first constructor takes two parameters. The first is a NumericalVariable that
represents the dependent variable. The second is an array of NumericalVariable objects that
represent the independent variables.
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NumericalVariable dependent = new NumericalVariable("y", yData);
NumericalVariable independent1 = new NumericalVariable("x1", x1Data);
NumericalVariable independent2 = new NumericalVariable("x2", x2Data);
LinearRegressionModel model1 = new LinearRegressionModel(dependent, independent); |
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Dim dependent As NumericalVariable = New NumericalVariable("y", yData)
Dim independent1 As NumericalVariable = New NumericalVariable("x1", xData)
Dim independent2 As NumericalVariable = New NumericalVariable("x2", xData)
Dim model1 As LinearRegressionModel = _
New LinearRegressionModel(dependent, independent) |
The second constructor takes 3 parameters. The first parameter is a
VariableCollection object that contains the variables to be used in the
regression. The second parameter is a string containing the name of the
dependent variable. The third parameter is an array of strings containing
the names of the independent variables. All the names must exist in
the collection specified by the first parameter. All variables must be of type NumericalVariable.
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VariableCollection variables = new VariableCollection();
variables.Add(dependent);
variables.Add(independent1);
variables.Add(independent2);
LinearRegressionModel model2 = new LinearRegressionModel(variables, "y", new string() {"x1", "x2"}); |
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Dim variables As VariableCollection = New VariableCollection()
variables.Add(dependent)
variables.Add(independent1)
variables.Add(independent2)
Dim model2 As LinearRegressionModel = _
New LinearRegressionModel(variables, "y", New String() {"x1", "x2"}) |
The third constructor also takes 3 parameters. The first parameter is
a DataTable object that contains the data for the regression analysis. The
second parameter is a string containing the name of the column that contains the
data for the dependent variable. The third parameter 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.
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// Fill data table with data from some datasource.
LinearRegressionModel model3 = new LinearRegressionModel(table, "y", new string() {"x1", "x2"}); |
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Dim table As DataTable = New DataTable()
' Fill data table with data from some datasource.
Dim model3 As LinearRegressionModel = _
New LinearRegressionModel(table, "y", New String() {"x1", "x2"}) |
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.
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model1.Compute(); |
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model1.Compute() |
The PredictedValues property returns a Vector
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
LinearRegressionModel class' Parameters
property returns a ParameterCollection
object that contains the parameters of the regression model. The members of
this collection are of type Parameter.
Regression parameters are created by the model. You cannot create them
directly.
Parameters can be accessed by numerical index or by name. The name of a parameter is usually the name of the variable
associated with it.
A multiple linear regression model has as many parameters as there are independent variables, plus one for
the intercept (constant term) when it is included. The intercept, if present, is the first parameter in the collection, with index 0.
The name of the intercept parameter can be retrieved or set through the InterceptParameterName property.
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:
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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); |
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Dim x1Parameter As = 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 parameter: 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.
When the model contains many independent variables, the additional variables
may be modeling the errors in the data rather than the data itself. This
causes the full model to be less reliable for making predictions. The AdjustedRSquared
property returns an adjusted R2 value that attempts to compensate for
this phenomenon.
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 AnovaTable
property.
Up: Regression Analysis Next: Polynomial Regression Previous: Simple Linear Regression Contents
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