Simple Linear Regression
Simple linear regression is a technique to analyze a linear relationship between two variables. A common generalization is to study relationships between two variables that can be transformed into a linear relationship, which we will call linearized. Simple linear regression is implemented by the SimpleRegressionModel class, and supports both linear and linearized regression.
Constructing Simple Regression Models
The SimpleRegressionModel class has six constructors that come in pairs. The first constructor takes two vectors and creates a linear regression model. 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.
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};
var y = Vector.Create(yData);
var x = Vector.Create(xData);
var model1 = new SimpleRegressionModel(y, x);
var model2 = new SimpleRegressionModel(yData, xData);
Note that, because arrays can be implicitly converted to vectors, it is also possible to pass arrays instead of vectors.
The second constructor takes 3 arguments. The first argument is an 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 variable. The two names must exist in the data frame and their element type must be convertible to Double.
var dataFrame = DataFrame.FromColumns(new Dictionary<string, object>()
{ { "y", y }, {"x", x } });
var model3 = new SimpleRegressionModel(dataFrame, "y", "x");
The third and fourth constructors are similar to the first two, but add a third argument that specifies the kind of relationship between the two variables. This argument is of type SimpleRegressionKind, which can take on the following values:
Value | Description |
---|---|
Exponential | The logarithm of the dependent variable depends linearly on the independent variable: Y = a * b^{X}. |
Inverse | The dependent variable depends linearly on the inverse (reciprocal) of the independent variable: Y = a / x + b. |
Linear | The dependent variable depends linearly on the independent variable: Y = a X + b. This is the default. |
Logarithmic | The dependent variable depends linearly on the logarithm of the independent variable: Y = a log X + b |
Power | The logarithm of the dependent variable depends linearly on the logarithm of the independent variable: Y = a * X^{b}. |
var model4 = new SimpleRegressionModel(y, x,
SimpleRegressionKind.Exponential);
var model5 = new SimpleRegressionModel(dataFrame, "y", "x",
SimpleRegressionKind.Exponential);
The fifth constructor is like the first but has 5 arguments in total: The third argument is a vector containing weights for the observations. The fourth argument is the kind of relationship between the variables. The fifth argument is a boolean that indicates whether the intercept (constant term) should be excluded from the model. The sixth constructor is like the second and takes the same 3 additional arguments:
var weights = Vector.Reciprocal(x);
var model6 = new SimpleRegressionModel(y, x, weights,
kind: SimpleRegressionKind.Exponential, noIntercept: false);
Computing the Regression Curve
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 Polynomial that represents the regression line. For linearized regression, the GetRegressionCurve method returns a Curve object that represents the fitted curve:
model1.Fit();
var line = model1.GetRegressionLine();
model4.Fit();
var curve = model4.GetRegressionCurve();
The Predictions 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 ParameterVector<T> 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.
A simple linear regression model takes two arguments. 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. For linearized
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:
var 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.Statistic);
Console.WriteLine("p-value: {0}", slope.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 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.
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.