org.hipparchus.stat.regression

## Class SimpleRegression

• All Implemented Interfaces:
Serializable, UpdatingMultipleLinearRegression

public class SimpleRegression
extends Object
implements Serializable, UpdatingMultipleLinearRegression
Estimates an ordinary least squares regression model with one independent variable.

 y = intercept + slope * x

Standard errors for intercept and slope are available as well as ANOVA, r-square and Pearson's r statistics.

Observations (x,y pairs) can be added to the model one at a time or they can be provided in a 2-dimensional array. The observations are not stored in memory, so there is no limit to the number of observations that can be added to the model.

Usage Notes:

• When there are fewer than two observations in the model, or when there is no variation in the x values (i.e. all x values are the same) all statistics return NaN. At least two observations with different x coordinates are required to estimate a bivariate regression model.
• Getters for the statistics always compute values based on the current set of observations -- i.e., you can get statistics, then add more data and get updated statistics without using a new instance. There is no "compute" method that updates all statistics. Each of the getters performs the necessary computations to return the requested statistic.
• The intercept term may be suppressed by passing false to the SimpleRegression(boolean) constructor. When the hasIntercept property is false, the model is estimated without a constant term and getIntercept() returns 0.

Serialized Form
• ### Constructor Summary

Constructors
Constructor and Description
SimpleRegression()
Create an empty SimpleRegression instance
SimpleRegression(boolean includeIntercept)
Create a SimpleRegression instance, specifying whether or not to estimate an intercept.
• ### Method Summary

All Methods
Modifier and Type Method and Description
void addData(double[][] data)
Adds the observations represented by the elements in data.
void addData(double x, double y)
Adds the observation (x,y) to the regression data set.
void addObservation(double[] x, double y)
Adds one observation to the regression model.
void addObservations(double[][] x, double[] y)
Adds a series of observations to the regression model.
void append(SimpleRegression reg)
Appends data from another regression calculation to this one.
void clear()
Clears all data from the model.
double getIntercept()
Returns the intercept of the estimated regression line, if hasIntercept() is true; otherwise 0.
double getInterceptStdErr()
Returns the standard error of the intercept estimate, usually denoted s(b0).
double getMeanSquareError()
Returns the sum of squared errors divided by the degrees of freedom, usually abbreviated MSE.
long getN()
Returns the number of observations that have been added to the model.
double getR()
double getRegressionSumSquares()
Returns the sum of squared deviations of the predicted y values about their mean (which equals the mean of y).
double getRSquare()
Returns the coefficient of determination, usually denoted r-square.
double getSignificance()
Returns the significance level of the slope (equiv) correlation.
double getSlope()
Returns the slope of the estimated regression line.
double getSlopeConfidenceInterval()
Returns the half-width of a 95% confidence interval for the slope estimate.
double getSlopeConfidenceInterval(double alpha)
Returns the half-width of a (100-100*alpha)% confidence interval for the slope estimate.
double getSlopeStdErr()
Returns the standard error of the slope estimate, usually denoted s(b1).
double getSumOfCrossProducts()
Returns the sum of crossproducts, xi*yi.
double getSumSquaredErrors()
Returns the sum of squared errors (SSE) associated with the regression model.
double getTotalSumSquares()
Returns the sum of squared deviations of the y values about their mean.
double getXSumSquares()
Returns the sum of squared deviations of the x values about their mean.
boolean hasIntercept()
Returns true if the model includes an intercept term.
double predict(double x)
Returns the "predicted" y value associated with the supplied x value, based on the data that has been added to the model when this method is activated.
RegressionResults regress()
Performs a regression on data present in buffers and outputs a RegressionResults object.
RegressionResults regress(int[] variablesToInclude)
Performs a regression on data present in buffers including only regressors indexed in variablesToInclude and outputs a RegressionResults object
void removeData(double[][] data)
Removes observations represented by the elements in data.
void removeData(double x, double y)
Removes the observation (x,y) from the regression data set.
• ### Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
• ### Constructor Detail

• #### SimpleRegression

public SimpleRegression()
Create an empty SimpleRegression instance
• #### SimpleRegression

public SimpleRegression(boolean includeIntercept)
Create a SimpleRegression instance, specifying whether or not to estimate an intercept.

Use false to estimate a model with no intercept. When the hasIntercept property is false, the model is estimated without a constant term and getIntercept() returns 0.

Parameters:
includeIntercept - whether or not to include an intercept term in the regression model
• ### Method Detail

public void addData(double x,
double y)
Adds the observation (x,y) to the regression data set.

Uses updating formulas for means and sums of squares defined in "Algorithms for Computing the Sample Variance: Analysis and Recommendations", Chan, T.F., Golub, G.H., and LeVeque, R.J. 1983, American Statistician, vol. 37, pp. 242-247, referenced in Weisberg, S. "Applied Linear Regression". 2nd Ed. 1985.

Parameters:
x - independent variable value
y - dependent variable value
• #### removeData

public void removeData(double x,
double y)
Removes the observation (x,y) from the regression data set.

Mirrors the addData method. This method permits the use of SimpleRegression instances in streaming mode where the regression is applied to a sliding "window" of observations, however the caller is responsible for maintaining the set of observations in the window.

The method has no effect if there are no points of data (i.e. n=0)
Parameters:
x - independent variable value
y - dependent variable value

public void addData(double[][] data)
throws MathIllegalArgumentException
Adds the observations represented by the elements in data.

(data,data) will be the first observation, then (data,data), etc.

This method does not replace data that has already been added. The observations represented by data are added to the existing dataset.

To replace all data, use clear() before adding the new data.

Parameters:
data - array of observations to be added
Throws:
MathIllegalArgumentException - if the length of data[i] is not greater than or equal to 2

public void addObservation(double[] x,
double y)
throws MathIllegalArgumentException
Adds one observation to the regression model.
Specified by:
addObservation in interface UpdatingMultipleLinearRegression
Parameters:
x - the independent variables which form the design matrix
y - the dependent or response variable
Throws:
MathIllegalArgumentException - if the length of x does not equal the number of independent variables in the model

public void addObservations(double[][] x,
double[] y)
throws MathIllegalArgumentException
Adds a series of observations to the regression model. The lengths of x and y must be the same and x must be rectangular.
Specified by:
addObservations in interface UpdatingMultipleLinearRegression
Parameters:
x - a series of observations on the independent variables
y - a series of observations on the dependent variable The length of x and y must be the same
Throws:
MathIllegalArgumentException - if x is not rectangular, does not match the length of y or does not contain sufficient data to estimate the model
• #### removeData

public void removeData(double[][] data)
Removes observations represented by the elements in data.

If the array is larger than the current n, only the first n elements are processed. This method permits the use of SimpleRegression instances in streaming mode where the regression is applied to a sliding "window" of observations, however the caller is responsible for maintaining the set of observations in the window.

To remove all data, use clear().

Parameters:
data - array of observations to be removed
• #### clear

public void clear()
Clears all data from the model.
Specified by:
clear in interface UpdatingMultipleLinearRegression
• #### getN

public long getN()
Returns the number of observations that have been added to the model.
Specified by:
getN in interface UpdatingMultipleLinearRegression
Returns:
n number of observations that have been added.
• #### predict

public double predict(double x)
Returns the "predicted" y value associated with the supplied x value, based on the data that has been added to the model when this method is activated.

 predict(x) = intercept + slope * x

Preconditions:

• At least two observations (with at least two different x values) must have been added before invoking this method. If this method is invoked before a model can be estimated, Double,NaN is returned.

Parameters:
x - input x value
Returns:
predicted y value
• #### getIntercept

public double getIntercept()
Returns the intercept of the estimated regression line, if hasIntercept() is true; otherwise 0.

The least squares estimate of the intercept is computed using the normal equations. The intercept is sometimes denoted b0.

Preconditions:

• At least two observations (with at least two different x values) must have been added before invoking this method. If this method is invoked before a model can be estimated, Double,NaN is returned.

Returns:
the intercept of the regression line if the model includes an intercept; 0 otherwise
SimpleRegression(boolean)
• #### hasIntercept

public boolean hasIntercept()
Returns true if the model includes an intercept term.
Specified by:
hasIntercept in interface UpdatingMultipleLinearRegression
Returns:
true if the regression includes an intercept; false otherwise
SimpleRegression(boolean)
• #### getSlope

public double getSlope()
Returns the slope of the estimated regression line.

The least squares estimate of the slope is computed using the normal equations. The slope is sometimes denoted b1.

Preconditions:

• At least two observations (with at least two different x values) must have been added before invoking this method. If this method is invoked before a model can be estimated, Double.NaN is returned.

Returns:
the slope of the regression line
• #### getSumSquaredErrors

public double getSumSquaredErrors()
Returns the sum of squared errors (SSE) associated with the regression model.

The sum is computed using the computational formula

SSE = SYY - (SXY * SXY / SXX)

where SYY is the sum of the squared deviations of the y values about their mean, SXX is similarly defined and SXY is the sum of the products of x and y mean deviations.

The sums are accumulated using the updating algorithm referenced in addData(double, double).

The return value is constrained to be non-negative - i.e., if due to rounding errors the computational formula returns a negative result, 0 is returned.

Preconditions:

• At least two observations (with at least two different x values) must have been added before invoking this method. If this method is invoked before a model can be estimated, Double,NaN is returned.

Returns:
sum of squared errors associated with the regression model
• #### getTotalSumSquares

public double getTotalSumSquares()
Returns the sum of squared deviations of the y values about their mean.

This is defined as SSTO here.

If n < 2, this returns Double.NaN.

Returns:
sum of squared deviations of y values
• #### getXSumSquares

public double getXSumSquares()
Returns the sum of squared deviations of the x values about their mean. If n < 2, this returns Double.NaN.

Returns:
sum of squared deviations of x values
• #### getSumOfCrossProducts

public double getSumOfCrossProducts()
Returns the sum of crossproducts, xi*yi.
Returns:
sum of cross products
• #### getRegressionSumSquares

public double getRegressionSumSquares()
Returns the sum of squared deviations of the predicted y values about their mean (which equals the mean of y).

This is usually abbreviated SSR or SSM. It is defined as SSM here

Preconditions:

• At least two observations (with at least two different x values) must have been added before invoking this method. If this method is invoked before a model can be estimated, Double.NaN is returned.

Returns:
sum of squared deviations of predicted y values
• #### getMeanSquareError

public double getMeanSquareError()
Returns the sum of squared errors divided by the degrees of freedom, usually abbreviated MSE.

If there are fewer than three data pairs in the model, or if there is no variation in x, this returns Double.NaN.

Returns:
sum of squared deviations of y values
• #### getR

public double getR()
Returns Pearson's product moment correlation coefficient, usually denoted r.

Preconditions:

• At least two observations (with at least two different x values) must have been added before invoking this method. If this method is invoked before a model can be estimated, Double,NaN is returned.

Returns:
Pearson's r
• #### getRSquare

public double getRSquare()
Returns the coefficient of determination, usually denoted r-square.

Preconditions:

• At least two observations (with at least two different x values) must have been added before invoking this method. If this method is invoked before a model can be estimated, Double,NaN is returned.

Returns:
r-square
• #### getInterceptStdErr

public double getInterceptStdErr()
Returns the standard error of the intercept estimate, usually denoted s(b0).

If there are fewer that three observations in the model, or if there is no variation in x, this returns Double.NaN.

Additionally, a Double.NaN is returned when the intercept is constrained to be zero
Returns:
standard error associated with intercept estimate
• #### getSlopeStdErr

public double getSlopeStdErr()
Returns the standard error of the slope estimate, usually denoted s(b1).

If there are fewer that three data pairs in the model, or if there is no variation in x, this returns Double.NaN.

Returns:
standard error associated with slope estimate
• #### getSlopeConfidenceInterval

public double getSlopeConfidenceInterval()
throws MathIllegalArgumentException
Returns the half-width of a 95% confidence interval for the slope estimate.

The 95% confidence interval is

(getSlope() - getSlopeConfidenceInterval(), getSlope() + getSlopeConfidenceInterval())

If there are fewer that three observations in the model, or if there is no variation in x, this returns Double.NaN.

Usage Note:
The validity of this statistic depends on the assumption that the observations included in the model are drawn from a Bivariate Normal Distribution.

Returns:
half-width of 95% confidence interval for the slope estimate
Throws:
MathIllegalArgumentException - if the confidence interval can not be computed.
• #### getSlopeConfidenceInterval

public double getSlopeConfidenceInterval(double alpha)
throws MathIllegalArgumentException
Returns the half-width of a (100-100*alpha)% confidence interval for the slope estimate.

The (100-100*alpha)% confidence interval is

(getSlope() - getSlopeConfidenceInterval(), getSlope() + getSlopeConfidenceInterval())

To request, for example, a 99% confidence interval, use alpha = .01

Usage Note:
The validity of this statistic depends on the assumption that the observations included in the model are drawn from a Bivariate Normal Distribution.

Preconditions:

• If there are fewer that three observations in the model, or if there is no variation in x, this returns Double.NaN.
• (0 < alpha < 1); otherwise an MathIllegalArgumentException is thrown.

Parameters:
alpha - the desired significance level
Returns:
half-width of 95% confidence interval for the slope estimate
Throws:
MathIllegalArgumentException - if the confidence interval can not be computed.
• #### getSignificance

public double getSignificance()
Returns the significance level of the slope (equiv) correlation.

Specifically, the returned value is the smallest alpha such that the slope confidence interval with significance level equal to alpha does not include 0. On regression output, this is often denoted Prob(|t| > 0)

Usage Note:
The validity of this statistic depends on the assumption that the observations included in the model are drawn from a Bivariate Normal Distribution.

If there are fewer that three observations in the model, or if there is no variation in x, this returns Double.NaN.

Returns:
significance level for slope/correlation
Throws:
MathIllegalStateException - if the significance level can not be computed.
• #### regress

public RegressionResults regress()
throws MathIllegalArgumentException
Performs a regression on data present in buffers and outputs a RegressionResults object.

If there are fewer than 3 observations in the model and hasIntercept is true a MathIllegalArgumentException is thrown. If there is no intercept term, the model must contain at least 2 observations.

Specified by:
regress in interface UpdatingMultipleLinearRegression
Returns:
RegressionResults acts as a container of regression output
Throws:
MathIllegalArgumentException - if the model is not correctly specified
MathIllegalArgumentException - if there is not sufficient data in the model to estimate the regression parameters
• #### regress

public RegressionResults regress(int[] variablesToInclude)
throws MathIllegalArgumentException
Performs a regression on data present in buffers including only regressors indexed in variablesToInclude and outputs a RegressionResults object
Specified by:
regress in interface UpdatingMultipleLinearRegression
Parameters:
variablesToInclude - an array of indices of regressors to include
Returns:
RegressionResults acts as a container of regression output
Throws:
MathIllegalArgumentException - if the variablesToInclude array is null or zero length
MathIllegalArgumentException - if a requested variable is not present in model