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Simple Linear Regression
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Simple Linear Regression
Simple linear regression is a technique to analyze a
linear relationship between two variables. The
Extreme Optimization Numerical Libraries for .NET supports simple linear regression
through the SimpleRegressionModel class.
Constructing Simple Regression Models
The SimpleRegressionModel class has five constructors.
The first constructor takes two arrays of double values. The first array
contains the data for the dependent variable. The second array contains the data
for the independent variable. The dependent variable is named 'Y,' while the
independent variable is named 'X.'
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double[] xData = {0.2, 337.4, 118.2, 884.6, 10.1,
226.5, 666.3, 996.3, 448.6, 777.0, 558.2, 0.4, 0.6,
775.5, 666.9, 338.0, 447.5, 11.6, 556.0, 228.1, 995.8,
887.6, 120.2, 0.3, 0.3, 556.8, 339.1, 887.2, 999.0,
779.0, 11.1, 118.3, 229.2, 669.1, 448.9, 0.5};
double[] yData = {0.1, 338.8, 118.1, 888.0, 9.2,
228.1, 668.5, 998.5, 449.1, 778.9, 559.2, 0.3, 0.1,
778.1, 668.8, 339.3, 448.9, 10.8, 557.7, 228.3, 998.0,
888.8, 119.6, 0.3, 0.6, 557.6, 339.3, 888.0, 998.5,
778.9, 10.2 , 117.6, 228.9, 668.4, 449.2, 0.2};
SimpleRegressionModel model1 = new SimpleRegressionModel(yData, xData); |
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Dim xData As Double() = {0.2, 337.4, 118.2, 884.6, 10.1, _
226.5, 666.3, 996.3, 448.6, 777.0, 558.2, 0.4, 0.6, _
775.5, 666.9, 338.0, 447.5, 11.6, 556.0, 228.1, 995.8, _
887.6, 120.2, 0.3, 0.3, 556.8, 339.1, 887.2, 999.0, _
779.0, 11.1, 118.3, 229.2, 669.1, 448.9, 0.5}
Dim yData As Double() = {0.1, 338.8, 118.1, 888.0, 9.2, _
228.1, 668.5, 998.5, 449.1, 778.9, 559.2, 0.3, 0.1, _
778.1, 668.8, 339.3, 448.9, 10.8, 557.7, 228.3, 998.0, _
888.8, 119.6, 0.3, 0.6, 557.6, 339.3, 888.0, 998.5, _
778.9, 10.2, 117.6, 228.9, 668.4, 449.2, 0.2}
Dim model1 As SimpleRegressionModel = _
New SimpleRegressionModel(yData, xData) |
The second constructor takes two Vector objects. Once again, the
first vector contains the data for the dependent variable. The
second vector contains the data for the independent variable. The dependent
variable is named 'Y,' while the independent variable is named 'X.'
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Vector independentVector = new GeneralVector(xData);
SimpleRegressionModel model2 = new SimpleRegressionModel(dependentVector, independentVector); |
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Dim dependentVector As Vector = New GeneralVector(yData)
Dim independentVector As Vector = New GeneralVector(xData)
Dim model2 As SimpleRegressionModel = _
New SimpleRegressionModel(dependentVector, independentVector) |
The third constructor takes two NumericalVariable objects as its only two
parameters. The first variable represents the dependent variable. The second
variable represents the independent variable.
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NumericalVariable independent = new NumericalVariable("x", xData);
SimpleRegressionModel model3 = new SimpleRegressionModel(dependent, independent); |
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Dim dependent As NumericalVariable = New NumericalVariable("y", yData)
Dim independent As NumericalVariable = New NumericalVariable("x", xData)
Dim model3 As SimpleRegressionModel = _
New SimpleRegressionModel(dependent, independent) |
The fourth 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 a string containing the name of the
independent variable. The two names must exist in the collection specified
by the first parameter, and the variables must be of type NumericalVariable.
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variables.Add(dependent);
variables.Add(independent);
SimpleRegressionModel model4 = new SimpleRegressionModel(variables, "y", "x"); |
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Dim variables As VariableCollection = New VariableCollection()
variables.Add(dependent)
variables.Add(independent)
Dim model4 As SimpleRegressionModel = New SimpleRegressionModel(variables, "y", "x") |
The fifth 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.
SimpleRegressionModel model5 = new SimpleRegressionModel(table, "y", "x"); |
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Dim table As DataTable = New DataTable()
' Fill data table with data from some datasource.
Dim model5 As SimpleRegressionModel = New SimpleRegressionModel(table, "y", "x") |
Computing the Regression Line
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.
The GetRegressionLine
method returns a Line
object that represents the regression line.
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model1.Compute();
Extreme.Mathematics.Curves.Line regressionLine = model1.GetRegressionLine(); |
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model1.Compute()
Dim regressionLine As Extreme.Mathematics.Curves.Line = _
model1.GetRegressionLine() |
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
SimpleRegressionModel 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.
A simple linear regression model has two parameters. The first, with index 0,
is the intercept: the Y-value where the regression line crosses the Y-axis. The
second parameter, with index 1, is the slope of the regression line.
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 slope = model1.Parameters[1];
Console.WriteLine("Name: {0}", slope.Name);
Console.WriteLine("Value: {0}", slope.Value);
Console.WriteLine("St.Err.: {0}", slope.StandardError);
Console.WriteLine("t-statistic: {0}", slope.TStatistic);
Console.WriteLine("p-value: {0}", slope.PValue); |
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Dim slope As = model1.Parameters(1)
Console.WriteLine("Name: {0}", slope.Name)
Console.WriteLine("Value: {0}", slope.Value)
Console.WriteLine("St.Err.: {0}", slope.StandardError)
Console.WriteLine("t-statistic: {0}", slope.TStatistic)
Console.WriteLine("p-value: {0}", slope.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 Line
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: Multiple Linear Regression Previous: Regression Analysis Contents
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