/******************************************************************************
* Compilation: javac LinearRegression.java
* Execution: java LinearRegression
* Dependencies: none
*
* Compute least squares solution to y = beta * x + alpha.
* Simple linear regression.
*
******************************************************************************/
/**
* The {@code LinearRegression} class performs a simple linear regression
* on an set of n data points (yi, xi).
* That is, it fits a straight line y = α + β x,
* (where y is the response variable, x is the predictor variable,
* α is the y-intercept, and β is the slope)
* that minimizes the sum of squared residuals of the linear regression model.
* It also computes associated statistics, including the coefficient of
* determination R2 and the standard deviation of the
* estimates for the slope and y-intercept.
*
* @author Robert Sedgewick
* @author Kevin Wayne
*/
public class LinearRegression {
private final double intercept, slope;
private final double r2;
private final double svar0, svar1;
/**
* Performs a linear regression on the data points {@code (y[i], x[i])}.
*
* @param x the values of the predictor variable
* @param y the corresponding values of the response variable
* @throws IllegalArgumentException if the lengths of the two arrays are not equal
*/
public LinearRegression(double[] x, double[] y) {
if (x.length != y.length) {
throw new IllegalArgumentException("array lengths are not equal");
}
int n = x.length;
// first pass
double sumx = 0.0, sumy = 0.0, sumx2 = 0.0;
for (int i = 0; i < n; i++) {
sumx += x[i];
sumx2 += x[i]*x[i];
sumy += y[i];
}
double xbar = sumx / n;
double ybar = sumy / n;
// second pass: compute summary statistics
double xxbar = 0.0, yybar = 0.0, xybar = 0.0;
for (int i = 0; i < n; i++) {
xxbar += (x[i] - xbar) * (x[i] - xbar);
yybar += (y[i] - ybar) * (y[i] - ybar);
xybar += (x[i] - xbar) * (y[i] - ybar);
}
slope = xybar / xxbar;
intercept = ybar - slope * xbar;
// more statistical analysis
double rss = 0.0; // residual sum of squares
double ssr = 0.0; // regression sum of squares
for (int i = 0; i < n; i++) {
double fit = slope*x[i] + intercept;
rss += (fit - y[i]) * (fit - y[i]);
ssr += (fit - ybar) * (fit - ybar);
}
int degreesOfFreedom = n-2;
r2 = ssr / yybar;
double svar = rss / degreesOfFreedom;
svar1 = svar / xxbar;
svar0 = svar/n + xbar*xbar*svar1;
}
/**
* Returns the y-intercept α of the best of the best-fit line y = α + β x.
*
* @return the y-intercept α of the best-fit line y = α + β x
*/
public double intercept() {
return intercept;
}
/**
* Returns the slope β of the best of the best-fit line y = α + β x.
*
* @return the slope β of the best-fit line y = α + β x
*/
public double slope() {
return slope;
}
/**
* Returns the coefficient of determination R2.
*
* @return the coefficient of determination R2,
* which is a real number between 0 and 1
*/
public double R2() {
return r2;
}
/**
* Returns the standard error of the estimate for the intercept.
*
* @return the standard error of the estimate for the intercept
*/
public double interceptStdErr() {
return Math.sqrt(svar0);
}
/**
* Returns the standard error of the estimate for the slope.
*
* @return the standard error of the estimate for the slope
*/
public double slopeStdErr() {
return Math.sqrt(svar1);
}
/**
* Returns the expected response {@code y} given the value of the predictor
* variable {@code x}.
*
* @param x the value of the predictor variable
* @return the expected response {@code y} given the value of the predictor
* variable {@code x}
*/
public double predict(double x) {
return slope*x + intercept;
}
/**
* Returns a string representation of the simple linear regression model.
*
* @return a string representation of the simple linear regression model,
* including the best-fit line and the coefficient of determination
* R2
*/
public String toString() {
StringBuilder s = new StringBuilder();
s.append(String.format("%.2f n + %.2f", slope(), intercept()));
s.append(" (R^2 = " + String.format("%.3f", R2()) + ")");
return s.toString();
}
}