﻿ SimpleLinearRegression Class   # SimpleLinearRegression Class

Simple Linear Regression of the form y = Ax + B. Inheritance Hierarchy
SystemObject
Accord.MachineLearningTransformBaseDouble, Double
Accord.Statistics.Models.Regression.LinearSimpleLinearRegression

Namespace:  Accord.Statistics.Models.Regression.Linear
Assembly:  Accord.Statistics (in Accord.Statistics.dll) Version: 3.8.0 Syntax
```[SerializableAttribute]
public class SimpleLinearRegression : TransformBase<double, double>,
ILinearRegression```

The SimpleLinearRegression type exposes the following members. Constructors
NameDescription SimpleLinearRegression
Creates a new Simple Linear Regression of the form y = Ax + B.
Top Properties
NameDescription Intercept
Linear coefficient (Intercept). NumberOfInputs
Gets the number of inputs accepted by the model.
(Inherited from TransformBaseTInput, TOutput.) NumberOfOutputs
Gets the number of outputs generated by the model.
(Inherited from TransformBaseTInput, TOutput.) NumberOfParameters
Gets the number of parameters in the model (returns 2). Slope
Angular coefficient (Slope).
Top Methods
NameDescription CoefficientOfDetermination(Double, Double, Double)
Gets the coefficient of determination, or R² (r-squared). CoefficientOfDetermination(Double, Double, Boolean, Double)
Gets the coefficient of determination, as known as R² (r-squared). Compute(Double) Obsolete.
Computes the regression for a single input. Compute(Double) Obsolete.
Computes the regression output for a given input. Equals
Determines whether the specified object is equal to the current object.
(Inherited from Object.) Finalize
Allows an object to try to free resources and perform other cleanup operations before it is reclaimed by garbage collection.
(Inherited from Object.)  FromData
Creates a new linear regression directly from data points. GetConfidenceInterval
Gets the confidence interval for an input point. GetDegreesOfFreedom
Gets the degrees of freedom when fitting the regression. GetHashCode
Serves as the default hash function.
(Inherited from Object.) GetPredictionInterval
Gets the prediction interval for an input point. GetPredictionStandardError
Gets the standard error of the prediction for a particular input point. GetStandardError(Double, Double)
Gets the overall regression standard error. GetStandardError(Double, Double, Double)
Gets the standard error of the fit for a particular input point. GetType
Gets the Type of the current instance.
(Inherited from Object.) MemberwiseClone
Creates a shallow copy of the current Object.
(Inherited from Object.) Regress Obsolete.
Performs the regression using the input and output data, returning the sum of squared errors of the fit. ToString
Returns a System.String representing the regression.
(Overrides ObjectToString.) ToString(IFormatProvider)
Returns a System.String representing the regression. ToString(String)
Returns a System.String representing the regression. ToString(String, IFormatProvider)
Returns a System.String representing the regression. Transform(TInput)
Applies the transformation to a set of input vectors, producing an associated set of output vectors.
(Inherited from TransformBaseTInput, TOutput.) Transform(Double)
Applies the transformation to an input, producing an associated output.
(Overrides TransformBaseTInput, TOutputTransform(TInput).) Transform(TInput, TOutput)
Applies the transformation to an input, producing an associated output.
(Inherited from TransformBaseTInput, TOutput.)
Top Extension Methods
NameDescription HasMethod
Checks whether an object implements a method with the given name.
(Defined by ExtensionMethods.) IsEqual
Compares two objects for equality, performing an elementwise comparison if the elements are vectors or matrices.
(Defined by Matrix.) To(Type)Overloaded.
Converts an object into another type, irrespective of whether the conversion can be done at compile time or not. This can be used to convert generic types to numeric types during runtime.
(Defined by ExtensionMethods.) ToTOverloaded.
Converts an object into another type, irrespective of whether the conversion can be done at compile time or not. This can be used to convert generic types to numeric types during runtime.
(Defined by ExtensionMethods.)
Top Remarks
In linear regression, the model specification is that the dependent variable, y is a linear combination of the parameters (but need not be linear in the independent variables). As the linear regression has a closed form solution, the regression coefficients can be efficiently computed using the Regress method of this class. Examples

Let's say we have some univariate, continuous sets of input data, and a corresponding univariate, continuous set of output data, such as a set of points in R². A simple linear regression is able to fit a line relating the input variables to the output variables in which the minimum-squared-error of the line and the actual output points is minimum.

```// Let's say we have some univariate, continuous sets of input data,
// and a corresponding univariate, continuous set of output data, such
// as a set of points in R². A simple linear regression is able to fit
// a line relating the input variables to the output variables in which
// the minimum-squared-error of the line and the actual output points
// is minimum.

// Declare some sample test data.
double[] inputs = { 80, 60, 10, 20, 30 };
double[] outputs = { 20, 40, 30, 50, 60 };

// Use Ordinary Least Squares to learn the regression
OrdinaryLeastSquares ols = new OrdinaryLeastSquares();

// Use OLS to learn the simple linear regression
SimpleLinearRegression regression = ols.Learn(inputs, outputs);

// Compute the output for a given input:
double y = regression.Transform(85); // The answer will be 28.088

// We can also extract the slope and the intercept term
// for the line. Those will be -0.26 and 50.5, respectively.
double s = regression.Slope;     // -0.264706
double c = regression.Intercept; // 50.588235```

Now, let's say we would like to perform a regression using an intermediary transformation, such as for example logarithmic regression. In this case, all we have to do is to first transform the input variables into the desired domain, then apply the regression as normal:

```// This is the same data from the example available at
// http://mathbits.com/MathBits/TISection/Statistics2/logarithmic.htm

// Declare your inputs and output data
double[] inputs = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 };
double[] outputs = { 6, 9.5, 13, 15, 16.5, 17.5, 18.5, 19, 19.5, 19.7, 19.8 };

// Transform inputs to logarithms
double[] logx = Matrix.Log(inputs);

// Use Ordinary Least Squares to learn the regression
OrdinaryLeastSquares ols = new OrdinaryLeastSquares();

// Use OLS to learn the simple linear regression
SimpleLinearRegression lr = ols.Learn(logx, outputs);

// Compute predicted values for inputs
double[] predicted = lr.Transform(logx);

// Get an expression representing the learned regression model
// We just have to remember that 'x' will actually mean 'log(x)'
string result = lr.ToString("N4", CultureInfo.InvariantCulture);

// Result will be "y(x) = 6.1082x + 6.0993"

// The mean squared error between the expected and the predicted is
double error = new SquareLoss(outputs).Loss(predicted); // 0.261454``` See Also