/*
* 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.math.stat.regression;
import java.io.Serializable;
import org.apache.commons.math.MathException;
import org.apache.commons.math.exception.OutOfRangeException;
import org.apache.commons.math.distribution.TDistribution;
import org.apache.commons.math.distribution.TDistributionImpl;
import org.apache.commons.math.exception.util.LocalizedFormats;
import org.apache.commons.math.util.FastMath;
/**
* Estimates an ordinary least squares regression model
* with one independent variable.
* <p>
* <code> y = intercept + slope * x </code></p>
* <p>
* Standard errors for <code>intercept</code> and <code>slope</code> are
* available as well as ANOVA, r-square and Pearson's r statistics.</p>
* <p>
* 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.</p>
* <p>
* <strong>Usage Notes</strong>: <ul>
* <li> 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 <code>NaN</code>. At least two observations with
* different x coordinates are requred to estimate a bivariate regression
* model.
* </li>
* <li> 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.</li>
* </ul></p>
*
* @version $Id: SimpleRegression.java 1131229 2011-06-03 20:49:25Z luc $
*/
public class SimpleRegression implements Serializable {
/** Serializable version identifier */
private static final long serialVersionUID = -3004689053607543335L;
/** the distribution used to compute inference statistics. */
private TDistribution distribution;
/** 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;
// ---------------------Public methods--------------------------------------
/**
* Create an empty SimpleRegression instance
*/
public SimpleRegression() {
this(1);
}
/**
* Create an empty SimpleRegression using the given distribution object to
* compute inference statistics.
*
* @param degrees Number of degrees of freedom of the distribution used
* to compute inference statistics.
* @since 2.2
*/
public SimpleRegression(int degrees) {
distribution = new TDistributionImpl(degrees);
}
/**
* Adds the observation (x,y) to the regression data set.
* <p>
* 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.</p>
*
*
* @param x independent variable value
* @param y dependent variable value
*/
public void addData(double x, double y) {
if (n == 0) {
xbar = x;
ybar = y;
} else {
double dx = x - xbar;
double dy = y - ybar;
sumXX += dx * dx * (double) n / (n + 1d);
sumYY += dy * dy * (double) n / (n + 1d);
sumXY += dx * dy * (double) n / (n + 1d);
xbar += dx / (n + 1.0);
ybar += dy / (n + 1.0);
}
sumX += x;
sumY += y;
n++;
if (n > 2) {
distribution = new TDistributionImpl(n - 2);
}
}
/**
* Removes the observation (x,y) from the regression data set.
* <p>
* 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.</p>
*
* 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(double x, double y) {
if (n > 0) {
double dx = x - xbar;
double dy = y - ybar;
sumXX -= dx * dx * (double) n / (n - 1d);
sumYY -= dy * dy * (double) n / (n - 1d);
sumXY -= dx * dy * (double) n / (n - 1d);
xbar -= dx / (n - 1.0);
ybar -= dy / (n - 1.0);
sumX -= x;
sumY -= y;
n--;
if (n > 2) {
distribution = new TDistributionImpl(n - 2);
}
}
}
/**
* Adds the observations represented by the elements in
* <code>data</code>.
* <p>
* <code>(data[0][0],data[0][1])</code> will be the first observation, then
* <code>(data[1][0],data[1][1])</code>, etc.</p>
* <p>
* This method does not replace data that has already been added. The
* observations represented by <code>data</code> are added to the existing
* dataset.</p>
* <p>
* To replace all data, use <code>clear()</code> before adding the new
* data.</p>
*
* @param data array of observations to be added
*/
public void addData(double[][] data) {
for (int i = 0; i < data.length; i++) {
addData(data[i][0], data[i][1]);
}
}
/**
* Removes observations represented by the elements in <code>data</code>.
* <p>
* 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.</p>
* <p>
* To remove all data, use <code>clear()</code>.</p>
*
* @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" <code>y</code> value associated with the
* supplied <code>x</code> value, based on the data that has been
* added to the model when this method is activated.
* <p>
* <code> predict(x) = intercept + slope * x </code></p>
* <p>
* <strong>Preconditions</strong>: <ul>
* <li>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, <code>Double,NaN</code> is
* returned.
* </li></ul></p>
*
* @param x input <code>x</code> value
* @return predicted <code>y</code> value
*/
public double predict(double x) {
double b1 = getSlope();
return getIntercept(b1) + b1 * x;
}
/**
* Returns the intercept of the estimated regression line.
* <p>
* The least squares estimate of the intercept is computed using the
* <a href="http://www.xycoon.com/estimation4.htm">normal equations</a>.
* The intercept is sometimes denoted b0.</p>
* <p>
* <strong>Preconditions</strong>: <ul>
* <li>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, <code>Double,NaN</code> is
* returned.
* </li></ul></p>
*
* @return the intercept of the regression line
*/
public double getIntercept() {
return getIntercept(getSlope());
}
/**
* Returns the slope of the estimated regression line.
* <p>
* The least squares estimate of the slope is computed using the
* <a href="http://www.xycoon.com/estimation4.htm">normal equations</a>.
* The slope is sometimes denoted b1.</p>
* <p>
* <strong>Preconditions</strong>: <ul>
* <li>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, <code>Double.NaN</code> is
* returned.
* </li></ul></p>
*
* @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 <a href="http://www.xycoon.com/SumOfSquares.htm">
* sum of squared errors</a> (SSE) associated with the regression
* model.
* <p>
* The sum is computed using the computational formula</p>
* <p>
* <code>SSE = SYY - (SXY * SXY / SXX)</code></p>
* <p>
* where <code>SYY</code> is the sum of the squared deviations of the y
* values about their mean, <code>SXX</code> is similarly defined and
* <code>SXY</code> is the sum of the products of x and y mean deviations.
* </p><p>
* The sums are accumulated using the updating algorithm referenced in
* {@link #addData}.</p>
* <p>
* 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.</p>
* <p>
* <strong>Preconditions</strong>: <ul>
* <li>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, <code>Double,NaN</code> is
* returned.
* </li></ul></p>
*
* @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.
* <p>
* This is defined as SSTO
* <a href="http://www.xycoon.com/SumOfSquares.htm">here</a>.</p>
* <p>
* If <code>n < 2</code>, this returns <code>Double.NaN</code>.</p>
*
* @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 <code>n < 2</code>, this returns <code>Double.NaN</code>.</p>
*
* @return sum of squared deviations of x values
*/
public double getXSumSquares() {
if (n < 2) {
return Double.NaN;
}
return sumXX;
}
/**
* Returns the sum of crossproducts, x<sub>i</sub>*y<sub>i</sub>.
*
* @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).
* <p>
* This is usually abbreviated SSR or SSM. It is defined as SSM
* <a href="http://www.xycoon.com/SumOfSquares.htm">here</a></p>
* <p>
* <strong>Preconditions</strong>: <ul>
* <li>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, <code>Double.NaN</code> is
* returned.
* </li></ul></p>
*
* @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.
* <p>
* If there are fewer than <strong>three</strong> data pairs in the model,
* or if there is no variation in <code>x</code>, this returns
* <code>Double.NaN</code>.</p>
*
* @return sum of squared deviations of y values
*/
public double getMeanSquareError() {
if (n < 3) {
return Double.NaN;
}
return getSumSquaredErrors() / (n - 2);
}
/**
* Returns <a href="http://mathworld.wolfram.com/CorrelationCoefficient.html">
* Pearson's product moment correlation coefficient</a>,
* usually denoted r.
* <p>
* <strong>Preconditions</strong>: <ul>
* <li>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, <code>Double,NaN</code> is
* returned.
* </li></ul></p>
*
* @return Pearson's r
*/
public double getR() {
double b1 = getSlope();
double result = FastMath.sqrt(getRSquare());
if (b1 < 0) {
result = -result;
}
return result;
}
/**
* Returns the <a href="http://www.xycoon.com/coefficient1.htm">
* coefficient of determination</a>,
* usually denoted r-square.
* <p>
* <strong>Preconditions</strong>: <ul>
* <li>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, <code>Double,NaN</code> is
* returned.
* </li></ul></p>
*
* @return r-square
*/
public double getRSquare() {
double ssto = getTotalSumSquares();
return (ssto - getSumSquaredErrors()) / ssto;
}
/**
* Returns the <a href="http://www.xycoon.com/standarderrorb0.htm">
* standard error of the intercept estimate</a>,
* usually denoted s(b0).
* <p>
* If there are fewer that <strong>three</strong> observations in the
* model, or if there is no variation in x, this returns
* <code>Double.NaN</code>.</p>
*
* @return standard error associated with intercept estimate
*/
public double getInterceptStdErr() {
return FastMath.sqrt(
getMeanSquareError() * ((1d / (double) n) + (xbar * xbar) / sumXX));
}
/**
* Returns the <a href="http://www.xycoon.com/standerrorb(1).htm">standard
* error of the slope estimate</a>,
* usually denoted s(b1).
* <p>
* If there are fewer that <strong>three</strong> data pairs in the model,
* or if there is no variation in x, this returns <code>Double.NaN</code>.
* </p>
*
* @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.
* <p>
* The 95% confidence interval is</p>
* <p>
* <code>(getSlope() - getSlopeConfidenceInterval(),
* getSlope() + getSlopeConfidenceInterval())</code></p>
* <p>
* If there are fewer that <strong>three</strong> observations in the
* model, or if there is no variation in x, this returns
* <code>Double.NaN</code>.</p>
* <p>
* <strong>Usage Note</strong>:<br>
* The validity of this statistic depends on the assumption that the
* observations included in the model are drawn from a
* <a href="http://mathworld.wolfram.com/BivariateNormalDistribution.html">
* Bivariate Normal Distribution</a>.</p>
*
* @return half-width of 95% confidence interval for the slope estimate
* @throws MathException if the confidence interval can not be computed.
*/
public double getSlopeConfidenceInterval() throws MathException {
return getSlopeConfidenceInterval(0.05d);
}
/**
* Returns the half-width of a (100-100*alpha)% confidence interval for
* the slope estimate.
* <p>
* The (100-100*alpha)% confidence interval is </p>
* <p>
* <code>(getSlope() - getSlopeConfidenceInterval(),
* getSlope() + getSlopeConfidenceInterval())</code></p>
* <p>
* To request, for example, a 99% confidence interval, use
* <code>alpha = .01</code></p>
* <p>
* <strong>Usage Note</strong>:<br>
* The validity of this statistic depends on the assumption that the
* observations included in the model are drawn from a
* <a href="http://mathworld.wolfram.com/BivariateNormalDistribution.html">
* Bivariate Normal Distribution</a>.</p>
* <p>
* <strong> Preconditions:</strong><ul>
* <li>If there are fewer that <strong>three</strong> observations in the
* model, or if there is no variation in x, this returns
* <code>Double.NaN</code>.
* </li>
* <li><code>(0 < alpha < 1)</code>; otherwise an
* <code>IllegalArgumentException</code> is thrown.
* </li></ul></p>
*
* @param alpha the desired significance level
* @return half-width of 95% confidence interval for the slope estimate
* @throws MathException if the confidence interval can not be computed.
*/
public double getSlopeConfidenceInterval(double alpha)
throws MathException {
if (alpha >= 1 || alpha <= 0) {
throw new OutOfRangeException(LocalizedFormats.SIGNIFICANCE_LEVEL,
alpha, 0, 1);
}
return getSlopeStdErr() *
distribution.inverseCumulativeProbability(1d - alpha / 2d);
}
/**
* Returns the significance level of the slope (equiv) correlation.
* <p>
* Specifically, the returned value is the smallest <code>alpha</code>
* such that the slope confidence interval with significance level
* equal to <code>alpha</code> does not include <code>0</code>.
* On regression output, this is often denoted <code>Prob(|t| > 0)</code>
* </p><p>
* <strong>Usage Note</strong>:<br>
* The validity of this statistic depends on the assumption that the
* observations included in the model are drawn from a
* <a href="http://mathworld.wolfram.com/BivariateNormalDistribution.html">
* Bivariate Normal Distribution</a>.</p>
* <p>
* If there are fewer that <strong>three</strong> observations in the
* model, or if there is no variation in x, this returns
* <code>Double.NaN</code>.</p>
*
* @return significance level for slope/correlation
* @throws MathException if the significance level can not be computed.
*/
public double getSignificance() throws MathException {
return 2d * (1.0 - distribution.cumulativeProbability(
FastMath.abs(getSlope()) / getSlopeStdErr()));
}
// ---------------------Private methods-----------------------------------
/**
* Returns the intercept of the estimated regression line, given the slope.
* <p>
* Will return <code>NaN</code> if slope is <code>NaN</code>.</p>
*
* @param slope current slope
* @return the intercept of the regression line
*/
private double getIntercept(double slope) {
return (sumY - slope * sumX) / n;
}
/**
* Computes SSR from b1.
*
* @param slope regression slope estimate
* @return sum of squared deviations of predicted y values
*/
private double getRegressionSumSquares(double slope) {
return slope * slope * sumXX;
}
}