最近在寫一個熒光圖像分析軟件，需要自己擬合方程。一元回歸線公式的算法參考了《Java數值方法》，擬合度R^2（絕對系數）是自己寫的，歡迎討論。計算結果和Excel完全一致。
總共三個文件：
DataPoint.java
/**
* A data point for interpolation and regression.
*/
public class DataPoint
{
/** the x value */ public float x;
/** the y value */ public float y;
/**
* Constructor.
* @param x the x value
* @param y the y value
*/
public DataPoint(float x, float y)
{
this.x = x;
this.y = y;
}
}
/**
* A least-squares regression line function.
*/
import java.util.*;
import java.math.BigDecimal;
public class RegressionLine
//implements Evaluatable
{
/** sum of x */ private double sumX;
/** sum of y */ private double sumY;
/** sum of x*x */ private double sumXX;
/** sum of x*y */ private double sumXY;
/** sum of y*y */ private double sumYY;
/** sum of yi-y */ private double sumDeltaY;
/** sum of sumDeltaY^2 */ private double sumDeltaY2;
/**誤差 */
private double sse;
private double sst;
private double E;
private String[] xy ;
private ArrayList listX ;
private ArrayList listY ;
private int XMin,XMax,YMin,YMax;
/** line coefficient a0 */ private float a0;
/** line coefficient a1 */ private float a1;
/** number of data points */ private int pn ;
/** true if coefficients valid */ private boolean coefsValid;
/**
* Constructor.
*/
public RegressionLine() {
XMax = 0;
YMax = 0;
pn = 0;
xy =new String[2];
listX = new ArrayList();
listY = new ArrayList();
}
/**
* Constructor.
* @param data the array of data points
*/
public RegressionLine(DataPoint data[])
{
pn = 0;
xy =new String[2];
listX = new ArrayList();
listY = new ArrayList();
for (int i = 0; i < data.length; ++i) {
addDataPoint(data[i]);
}
}
/**
* Return the current number of data points.
* @return the count
*/
public int getDataPointCount() { return pn; }
/**
* Return the coefficient a0.
* @return the value of a0
*/
public float getA0()
{
validateCoefficients();
return a0;
}
/**
* Return the coefficient a1.
* @return the value of a1
*/
public float getA1()
{
validateCoefficients();
return a1;
}
/**
* Return the sum of the x values.
* @return the sum
*/
public double getSumX() { return sumX; }
/**
* Return the sum of the y values.
* @return the sum
*/
public double getSumY() { return sumY; }
/**
* Return the sum of the x*x values.
* @return the sum
*/
public double getSumXX() { return sumXX; }
/**
* Return the sum of the x*y values.
* @return the sum
*/
public double getSumXY() { return sumXY; }
public double getSumYY() { return sumYY; }
public int getXMin() {
return XMin;
}
public int getXMax() {
return XMax;
}
public int getYMin() {
return YMin;
}
public int getYMax() {
return YMax;
}
/**
* Add a new data point: Update the sums.
* @param dataPoint the new data point
*/
public void addDataPoint(DataPoint dataPoint)
{
sumX += dataPoint.x;
sumY += dataPoint.y;
sumXX += dataPoint.x*dataPoint.x;
sumXY += dataPoint.x*dataPoint.y;
sumYY += dataPoint.y*dataPoint.y;
if(dataPoint.x > XMax){
XMax = (int)dataPoint.x;
}
if(dataPoint.y > YMax){
YMax = (int)dataPoint.y;
}
//把每個點的具體坐標存入ArrayList中，備用
xy[0] = (int)dataPoint.x+ "";
xy[1] = (int)dataPoint.y+ "";
if(dataPoint.x!=0 && dataPoint.y != 0){
System.out.print(xy[0]+",");
System.out.println(xy[1]);
try{
//System.out.println("n:"+n);
listX.add(pn,xy[0]);
listY.add(pn,xy[1]);
}
catch(Exception e){
e.printStackTrace();
}
/*
System.out.println("N:" + n);
System.out.println("ArrayList listX:"+ listX.get(n));
System.out.println("ArrayList listY:"+ listY.get(n));
*/
}
++pn;
coefsValid = false;
}
/**
* Return the value of the regression line function at x.
* (Implementation of Evaluatable.)
* @param x the value of x
* @return the value of the function at x
*/
public float at(int x)
{
if (pn < 2) return Float.NaN;
validateCoefficients();
return a0 + a1*x;
}
public float at(float x)
{
if (pn < 2) return Float.NaN;
validateCoefficients();
return a0 + a1*x;
}
/**
* Reset.
*/
public void reset()
{
pn = 0;
sumX = sumY = sumXX = sumXY = 0;
coefsValid = false;
}
/**
* Validate the coefficients.
* 計算方程系數 y=ax+b 中的a
*/
private void validateCoefficients()
{
if (coefsValid) return;
if (pn >= 2) {
float xBar = (float) sumX/pn;
float yBar = (float) sumY/pn;
a1 = (float) ((pn*sumXY - sumX*sumY)
/(pn*sumXX - sumX*sumX));
a0 = (float) (yBar - a1*xBar);
}
else {
a0 = a1 = Float.NaN;
}
coefsValid = true;
}
/**
* 返回誤差
*/
public double getR(){
//遍歷這個list并計算分母
for(int i = 0; i < pn -1; i++) {
float Yi= (float)Integer.parseInt(listY.get(i).toString());
float Y = at(Integer.parseInt(listX.get(i).toString()));
float deltaY = Yi - Y;
float deltaY2 = deltaY*deltaY;
/*
System.out.println("Yi:" + Yi);
System.out.println("Y:" + Y);
System.out.println("deltaY:" + deltaY);
System.out.println("deltaY2:" + deltaY2);
*/
sumDeltaY2 += deltaY2;
//System.out.println("sumDeltaY2:" + sumDeltaY2);
}
sst = sumYY - (sumY*sumY)/pn;
//System.out.println("sst:" + sst);
E =1- sumDeltaY2/sst;
return round(E,4) ;
}
//用于實現精確的四舍五入
public double round(double v,int scale){
if(scale<0){
throw new IllegalArgumentException(
"The scale must be a positive integer or zero");
}
BigDecimal b = new BigDecimal(Double.toString(v));
BigDecimal one = new BigDecimal("1");
return b.divide(one,scale,BigDecimal.ROUND_HALF_UP).doubleValue();
}
public float round(float v,int scale){
if(scale<0){
throw new IllegalArgumentException(
"The scale must be a positive integer or zero");
}
BigDecimal b = new BigDecimal(Double.toString(v));
BigDecimal one = new BigDecimal("1");
return b.divide(one,scale,BigDecimal.ROUND_HALF_UP).floatValue();
}
}
演示程序：
LinearRegression.java
/**
* <p><b>Linear Regression</b>
* <br>
* Demonstrate linear regression by constructing the regression line for a set
* of data points.
*
* <p>require DataPoint.java,RegressionLine.java
*
* <p>為了計算對于給定數據點的最小方差回線，需要計算SumX,SumY,SumXX,SumXY; (注：SumXX = Sum (X^2))
* <p><b>回歸直線方程如下： f(x)=a1x+a0 </b>
* <p><b>斜率和截距的計算公式如下：</b>
* <br>n: 數據點個數
* <p>a1=(n(SumXY)-SumX*SumY)/(n*SumXX-(SumX)^2)
* <br>a0=(SumY - SumY * a1)/n
* <br>(也可表達為a0=averageY-a1*averageX)
*
* <p><b>畫線的原理：兩點成一直線，只要能確定兩個點即可</b><br>
* 第一點：(0,a0) 再隨意取一個x1值代入方程，取得y1，連結(0,a0)和(x1,y1)兩點即可。
* 為了讓線穿過整個圖,x1可以取橫坐標的最大值Xmax，即兩點為(0,a0),(Xmax,Y)。如果y=a1*Xmax+a0,y大于
* 縱坐標最大值Ymax，則不用這個點。改用y取最大值Ymax，算得此時x的值，使用(X,Ymax)， 即兩點為(0,a0),(X,Ymax)
*
* <p><b>擬合度計算：(即Excel中的R^2)</b>
* <p> *R2 = 1 - E
* <p>誤差E的計算：E = SSE/SST
* <p>SSE=sum((Yi-Y)^2) SST=sumYY - (sumY*sumY)/n;
* <p>
*/
public class LinearRegression
{
private static final int MAX_POINTS = 10;
private double E;
/**
* Main program.
*
* @param args
* the array of runtime arguments
*/
public static void main(String args[])
{
RegressionLine line = new RegressionLine();
line.addDataPoint(new DataPoint(20, 136));
line.addDataPoint(new DataPoint(40, 143));
line.addDataPoint(new DataPoint(60, 152));
line.addDataPoint(new DataPoint(80, 162));
line.addDataPoint(new DataPoint(100, 167));
printSums(line);
printLine(line);
}
/**
* Print the computed sums.
*
* @param line
* the regression line
*/
private static void printSums(RegressionLine line)
{
System.out.println("\n數據點個數 n = " + line.getDataPointCount());
System.out.println("\nSum x = " + line.getSumX());
System.out.println("Sum y = " + line.getSumY());
System.out.println("Sum xx = " + line.getSumXX());
System.out.println("Sum xy = " + line.getSumXY());
System.out.println("Sum yy = " + line.getSumYY());
}
/**
* Print the regression line function.
*
* @param line
* the regression line
*/
private static void printLine(RegressionLine line)
{
System.out.println("\n回歸線公式: y = " +
line.getA1() +
"x + " + line.getA0());
System.out.println("擬合度： R^2 = " + line.getR());
}
}