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/*
* Licensed to the Apache Software Foundation (ASF) under one or more
* contributor license agreements. See the NOTICE file distributed with
* this work for additional information regarding copyright ownership.
* The ASF licenses this file to You under the Apache License, Version 2.0
* (the "License"); you may not use this file except in compliance with
* the License. You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
package org.apache.commons.math3.stat.regression;
import java.io.Serializable;
import org.apache.commons.math3.exception.OutOfRangeException;
import org.apache.commons.math3.distribution.TDistribution;
import org.apache.commons.math3.exception.MathIllegalArgumentException;
import org.apache.commons.math3.exception.NoDataException;
import org.apache.commons.math3.exception.util.LocalizedFormats;
import org.apache.commons.math3.util.FastMath;
import org.apache.commons.math3.util.Precision;
/**
* 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 {@code false} to
* the {@link #SimpleRegression(boolean)} constructor. When the
* {@code hasIntercept} property is false, the model is estimated without a
* constant term and {@link #getIntercept()} returns {@code 0}.
*
*
* @version $Id: SimpleRegression.java 1244703 2012-02-15 20:46:30Z luc $
*/
public class SimpleRegression implements Serializable, UpdatingMultipleLinearRegression {
/** Serializable version identifier */
private static final long serialVersionUID = -3004689053607543335L;
/** sum of x values */
private double sumX = 0d;
/** total variation in x (sum of squared deviations from xbar) */
private double sumXX = 0d;
/** sum of y values */
private double sumY = 0d;
/** total variation in y (sum of squared deviations from ybar) */
private double sumYY = 0d;
/** sum of products */
private double sumXY = 0d;
/** number of observations */
private long n = 0;
/** mean of accumulated x values, used in updating formulas */
private double xbar = 0;
/** mean of accumulated y values, used in updating formulas */
private double ybar = 0;
/** include an intercept or not */
private final boolean hasIntercept;
// ---------------------Public methods--------------------------------------
/**
* Create an empty SimpleRegression instance
*/
public SimpleRegression() {
this(true);
}
/**
* Create a SimpleRegression instance, specifying whether or not to estimate
* an intercept.
*
* Use {@code false} to estimate a model with no intercept. When the
* {@code hasIntercept} property is false, the model is estimated without a
* constant term and {@link #getIntercept()} returns {@code 0}.
*
* @param includeIntercept whether or not to include an intercept term in
* the regression model
*/
public SimpleRegression(boolean includeIntercept) {
super();
hasIntercept = includeIntercept;
}
/**
* 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.
*
*
* @param x independent variable value
* @param y dependent variable value
*/
public void addData(final double x,final double y) {
if (n == 0) {
xbar = x;
ybar = y;
} else {
if( hasIntercept ){
final double fact1 = 1.0 + n;
final double fact2 = n / (1.0 + n);
final double dx = x - xbar;
final double dy = y - ybar;
sumXX += dx * dx * fact2;
sumYY += dy * dy * fact2;
sumXY += dx * dy * fact2;
xbar += dx / fact1;
ybar += dy / fact1;
}
}
if( !hasIntercept ){
sumXX += x * x ;
sumYY += y * y ;
sumXY += x * y ;
}
sumX += x;
sumY += y;
n++;
return;
}
/**
* 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)
*
* @param x independent variable value
* @param y dependent variable value
*/
public void removeData(final double x,final double y) {
if (n > 0) {
if (hasIntercept) {
final double fact1 = n - 1.0;
final double fact2 = n / (n - 1.0);
final double dx = x - xbar;
final double dy = y - ybar;
sumXX -= dx * dx * fact2;
sumYY -= dy * dy * fact2;
sumXY -= dx * dy * fact2;
xbar -= dx / fact1;
ybar -= dy / fact1;
} else {
final double fact1 = n - 1.0;
sumXX -= x * x;
sumYY -= y * y;
sumXY -= x * y;
xbar -= x / fact1;
ybar -= y / fact1;
}
sumX -= x;
sumY -= y;
n--;
}
return;
}
/**
* Adds the observations represented by the elements in
* data.
*
* (data[0][0],data[0][1]) will be the first observation, then
* (data[1][0],data[1][1]), 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.
*
* @param data array of observations to be added
* @throws ModelSpecificationException if the length of {@code data[i]} is not
* greater than or equal to 2
*/
public void addData(final double[][] data) {
for (int i = 0; i < data.length; i++) {
if( data[i].length < 2 ){
throw new ModelSpecificationException(LocalizedFormats.INVALID_REGRESSION_OBSERVATION,
data[i].length, 2);
}
addData(data[i][0], data[i][1]);
}
return;
}
/**
* Adds one observation to the regression model.
*
* @param x the independent variables which form the design matrix
* @param y the dependent or response variable
* @throws ModelSpecificationException if the length of {@code x} does not equal
* the number of independent variables in the model
*/
public void addObservation(final double[] x,final double y) throws ModelSpecificationException{
if( x == null || x.length == 0 ){
throw new ModelSpecificationException(LocalizedFormats.INVALID_REGRESSION_OBSERVATION,x!=null?x.length:0, 1);
}
addData( x[0], y );
return;
}
/**
* Adds a series of observations to the regression model. The lengths of
* x and y must be the same and x must be rectangular.
*
* @param x a series of observations on the independent variables
* @param y a series of observations on the dependent variable
* The length of x and y must be the same
* @throws ModelSpecificationException if {@code x} is not rectangular, does not match
* the length of {@code y} or does not contain sufficient data to estimate the model
*/
public void addObservations(final double[][] x,final double[] y) {
if ((x == null) || (y == null) || (x.length != y.length)) {
throw new ModelSpecificationException(
LocalizedFormats.DIMENSIONS_MISMATCH_SIMPLE,
(x == null) ? 0 : x.length,
(y == null) ? 0 : y.length);
}
boolean obsOk=true;
for( int i = 0 ; i < x.length; i++){
if( x[i] == null || x[i].length == 0 ){
obsOk = false;
}
}
if( !obsOk ){
throw new ModelSpecificationException(
LocalizedFormats.NOT_ENOUGH_DATA_FOR_NUMBER_OF_PREDICTORS,
0, 1);
}
for( int i = 0 ; i < x.length ; i++){
addData( x[i][0], y[i] );
}
return;
}
/**
* 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().
*
* @param data array of observations to be removed
*/
public void removeData(double[][] data) {
for (int i = 0; i < data.length && n > 0; i++) {
removeData(data[i][0], data[i][1]);
}
}
/**
* Clears all data from the model.
*/
public void clear() {
sumX = 0d;
sumXX = 0d;
sumY = 0d;
sumYY = 0d;
sumXY = 0d;
n = 0;
}
/**
* Returns the number of observations that have been added to the model.
*
* @return n number of observations that have been added.
*/
public long getN() {
return n;
}
/**
* 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.
*
*
* @param x input x value
* @return predicted y value
*/
public double predict(final double x) {
final double b1 = getSlope();
if (hasIntercept) {
return getIntercept(b1) + b1 * x;
}
return b1 * x;
}
/**
* Returns the intercept of the estimated regression line, if
* {@link #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.
*
*
* @return the intercept of the regression line if the model includes an
* intercept; 0 otherwise
* @see #SimpleRegression(boolean)
*/
public double getIntercept() {
return hasIntercept ? getIntercept(getSlope()) : 0.0;
}
/**
* Returns true if the model includes an intercept term.
*
* @return true if the regression includes an intercept; false otherwise
* @see #SimpleRegression(boolean)
*/
public boolean hasIntercept() {
return hasIntercept;
}
/**
* 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.
*
*
* @return the slope of the regression line
*/
public double getSlope() {
if (n < 2) {
return Double.NaN; //not enough data
}
if (FastMath.abs(sumXX) < 10 * Double.MIN_VALUE) {
return Double.NaN; //not enough variation in x
}
return sumXY / sumXX;
}
/**
* 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
* {@link #addData}.
*
* 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.
*
*
* @return sum of squared errors associated with the regression model
*/
public double getSumSquaredErrors() {
return FastMath.max(0d, sumYY - sumXY * sumXY / sumXX);
}
/**
* 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.
*
* @return sum of squared deviations of y values
*/
public double getTotalSumSquares() {
if (n < 2) {
return Double.NaN;
}
return sumYY;
}
/**
* Returns the sum of squared deviations of the x values about their mean.
*
* If n < 2, this returns Double.NaN.
*
* @return sum of squared deviations of x values
*/
public double getXSumSquares() {
if (n < 2) {
return Double.NaN;
}
return sumXX;
}
/**
* Returns the sum of crossproducts, xi*yi.
*
* @return sum of cross products
*/
public double getSumOfCrossProducts() {
return sumXY;
}
/**
* 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.
*
*
* @return sum of squared deviations of predicted y values
*/
public double getRegressionSumSquares() {
return getRegressionSumSquares(getSlope());
}
/**
* 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.
*
* @return sum of squared deviations of y values
*/
public double getMeanSquareError() {
if (n < 3) {
return Double.NaN;
}
return hasIntercept ? (getSumSquaredErrors() / (n - 2)) : (getSumSquaredErrors() / (n - 1));
}
/**
* 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.
*
*
* @return Pearson's r
*/
public double getR() {
double b1 = getSlope();
double result = FastMath.sqrt(getRSquare());
if (b1 < 0) {
result = -result;
}
return result;
}
/**
* 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.
*
*
* @return r-square
*/
public double getRSquare() {
double ssto = getTotalSumSquares();
return (ssto - getSumSquaredErrors()) / ssto;
}
/**
* 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
*
* @return standard error associated with intercept estimate
*/
public double getInterceptStdErr() {
if( !hasIntercept ){
return Double.NaN;
}
return FastMath.sqrt(
getMeanSquareError() * ((1d / n) + (xbar * xbar) / sumXX));
}
/**
* 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.
*
*
* @return standard error associated with slope estimate
*/
public double getSlopeStdErr() {
return FastMath.sqrt(getMeanSquareError() / sumXX);
}
/**
* 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.
*
* @return half-width of 95% confidence interval for the slope estimate
* @throws OutOfRangeException if the confidence interval can not be computed.
*/
public double getSlopeConfidenceInterval() {
return getSlopeConfidenceInterval(0.05d);
}
/**
* 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
* OutOfRangeException is thrown.
*
*
* @param alpha the desired significance level
* @return half-width of 95% confidence interval for the slope estimate
* @throws OutOfRangeException if the confidence interval can not be computed.
*/
public double getSlopeConfidenceInterval(final double alpha) {
if (alpha >= 1 || alpha <= 0) {
throw new OutOfRangeException(LocalizedFormats.SIGNIFICANCE_LEVEL,
alpha, 0, 1);
}
TDistribution distribution = new TDistribution(n - 2);
return getSlopeStdErr() *
distribution.inverseCumulativeProbability(1d - alpha / 2d);
}
/**
* 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.
*
* @return significance level for slope/correlation
* @throws org.apache.commons.math3.exception.MaxCountExceededException
* if the significance level can not be computed.
*/
public double getSignificance() {
TDistribution distribution = new TDistribution(n - 2);
return 2d * (1.0 - distribution.cumulativeProbability(
FastMath.abs(getSlope()) / getSlopeStdErr()));
}
// ---------------------Private methods-----------------------------------
/**
* Returns the intercept of the estimated regression line, given the slope.
*
* Will return NaN if slope is NaN.
*
* @param slope current slope
* @return the intercept of the regression line
*/
private double getIntercept(final double slope) {
if( hasIntercept){
return (sumY - slope * sumX) / n;
}
return 0.0;
}
/**
* Computes SSR from b1.
*
* @param slope regression slope estimate
* @return sum of squared deviations of predicted y values
*/
private double getRegressionSumSquares(final double slope) {
return slope * slope * sumXX;
}
/**
* Performs a regression on data present in buffers and outputs a RegressionResults object
* @return RegressionResults acts as a container of regression output
* @throws ModelSpecificationException if the model is not correctly specified
*/
public RegressionResults regress() throws ModelSpecificationException{
if( hasIntercept ){
if( n < 3 ){
throw new NoDataException( LocalizedFormats.NOT_ENOUGH_DATA_REGRESSION );
}
if( FastMath.abs( sumXX ) > Precision.SAFE_MIN ){
final double[] params = new double[]{ getIntercept(), getSlope() };
final double mse = getMeanSquareError();
final double _syy = sumYY + sumY * sumY / n;
final double[] vcv = new double[]{
mse * (xbar *xbar /sumXX + 1.0 / n),
-xbar*mse/sumXX,
mse/sumXX };
return new RegressionResults(
params, new double[][]{vcv}, true, n, 2,
sumY, _syy, getSumSquaredErrors(),true,false);
}else{
final double[] params = new double[]{ sumY / n, Double.NaN };
//final double mse = getMeanSquareError();
final double[] vcv = new double[]{
ybar / (n - 1.0),
Double.NaN,
Double.NaN };
return new RegressionResults(
params, new double[][]{vcv}, true, n, 1,
sumY, sumYY, getSumSquaredErrors(),true,false);
}
}else{
if( n < 2 ){
throw new NoDataException( LocalizedFormats.NOT_ENOUGH_DATA_REGRESSION );
}
if( !Double.isNaN(sumXX) ){
final double[] vcv = new double[]{ getMeanSquareError() / sumXX };
final double[] params = new double[]{ sumXY/sumXX };
return new RegressionResults(
params, new double[][]{vcv}, true, n, 1,
sumY, sumYY, getSumSquaredErrors(),false,false);
}else{
final double[] vcv = new double[]{Double.NaN };
final double[] params = new double[]{ Double.NaN };
return new RegressionResults(
params, new double[][]{vcv}, true, n, 1,
Double.NaN, Double.NaN, Double.NaN,false,false);
}
}
}
/**
* Performs a regression on data present in buffers including only regressors
* indexed in variablesToInclude and outputs a RegressionResults object
* @param variablesToInclude an array of indices of regressors to include
* @return RegressionResults acts as a container of regression output
* @throws ModelSpecificationException if the model is not correctly specified
* @throws MathIllegalArgumentException if the variablesToInclude array is null or zero length
* @throws OutOfRangeException if a requested variable is not present in model
*/
public RegressionResults regress(int[] variablesToInclude) throws ModelSpecificationException{
if( variablesToInclude == null || variablesToInclude.length == 0){
throw new MathIllegalArgumentException(LocalizedFormats.ARRAY_ZERO_LENGTH_OR_NULL_NOT_ALLOWED);
}
if( variablesToInclude.length > 2 || (variablesToInclude.length > 1 && !hasIntercept) ){
throw new ModelSpecificationException(
LocalizedFormats.ARRAY_SIZE_EXCEEDS_MAX_VARIABLES,
(variablesToInclude.length > 1 && !hasIntercept) ? 1 : 2);
}
if( hasIntercept ){
if( variablesToInclude.length == 2 ){
if( variablesToInclude[0] == 1 ){
throw new ModelSpecificationException(LocalizedFormats.NOT_INCREASING_SEQUENCE);
}else if( variablesToInclude[0] != 0 ){
throw new OutOfRangeException( variablesToInclude[0], 0,1 );
}
if( variablesToInclude[1] != 1){
throw new OutOfRangeException( variablesToInclude[0], 0,1 );
}
return regress();
}else{
if( variablesToInclude[0] != 1 && variablesToInclude[0] != 0 ){
throw new OutOfRangeException( variablesToInclude[0],0,1 );
}
final double _mean = sumY * sumY / n;
final double _syy = sumYY + _mean;
if( variablesToInclude[0] == 0 ){
//just the mean
final double[] vcv = new double[]{ sumYY/(((n-1)*n)) };
final double[] params = new double[]{ ybar };
return new RegressionResults(
params, new double[][]{vcv}, true, n, 1,
sumY, _syy+_mean, sumYY,true,false);
}else if( variablesToInclude[0] == 1){
//final double _syy = sumYY + sumY * sumY / ((double) n);
final double _sxx = sumXX + sumX * sumX / n;
final double _sxy = sumXY + sumX * sumY / n;
final double _sse = FastMath.max(0d, _syy - _sxy * _sxy / _sxx);
final double _mse = _sse/((n-1));
if( !Double.isNaN(_sxx) ){
final double[] vcv = new double[]{ _mse / _sxx };
final double[] params = new double[]{ _sxy/_sxx };
return new RegressionResults(
params, new double[][]{vcv}, true, n, 1,
sumY, _syy, _sse,false,false);
}else{
final double[] vcv = new double[]{Double.NaN };
final double[] params = new double[]{ Double.NaN };
return new RegressionResults(
params, new double[][]{vcv}, true, n, 1,
Double.NaN, Double.NaN, Double.NaN,false,false);
}
}
}
}else{
if( variablesToInclude[0] != 0 ){
throw new OutOfRangeException(variablesToInclude[0],0,0);
}
return regress();
}
return null;
}
}