参考文章:http://www.cnblogs.com/Jason-MLiu/p/9243138.html?tdsourcetag=s_pctim_aiomsg
https://www.cnblogs.com/xingshansi/p/6545162.html
http://www.ilovematlab.cn/thread-201168-1-1.html
Function y=y0+(a/(w*sqrt(pi/2)))*exp(-2*((x-xc)/w)^2);
Data;
x=[-402.50,-397.50,-392.50,-387.50,-382.50,-377.50,-372.50,-367.50,-362.50,-357.50,-352.50,-347.50,-342.50,-337.50,-332.50,-327.50,-322.50,-317.50,-312.50,-307.50,-302.50,-297.50,-292.50,-287.50,-282.50,-277.50,-272.50,-267.50,-262.50,-257.50,-252.50,-247.50,-242.50,-237.50,-232.50,-227.50,-222.50,-217.50,-212.50,-207.50,-202.50,-197.50,-192.50,-187.50,-182.50,-177.50,-172.50,-167.50,-162.50];
y=[0,0,0,0,1,2,2,9,5,14,25,32,49,69,98,127,112,162,160,153,149,131,112,73,80,63,65,51,49,47,36,39,29,25,13,10,4,4,0,0,0,0,0,0,0,0,0,0,0];
均方差(RMSE): 11.7929171848004
残差平方和(SSE): 6814.57189065049
相关系数(R): 0.972279502902424
相关系数之平方(R^2): 0.945327431764185
决定系数(DC): 0.945327431764186
卡方系数(Chi-Square): 168.048569051438
F统计(F-Statistic): 274.360625154864
| 参数 | 最佳估算 |
|---|---|
| y0: | 4.35708080028293 |
| a: | 8932.86668800182 |
| w: | 49.2121917694601 |
| xc: | -308.013167059636 |
https://www.cnblogs.com/weizhen/p/5823877.html
y=y0+(a/(w * sqrt(pi/2))) * exp(-2 * ((x-xc)/w)^2)
formula
百度文库:https://wenku.baidu.com/view/cae4aefaf705cc175527093f.html
GAUSS
y0:偏移量
xc:峰中心位置
w:峰宽
A:峰面积
引用到了apache的common-math的jar包，主要用于矩阵运算(jar包找maven)
org.apache.commons
commons-math3
3.6.1
http://central.maven.org/maven2/org/apache/commons/commons-math3/3.6.1/commons-math3-3.6.1.jar
舍掉小数取整:Math.floor(3.5)=3
四舍五入取整:Math.rint(3.5)=4
进位取整:Math.ceil(3.1)=4
取绝对值：Math.abs(-3.5)=3.5
取余数：A%B = 余数
import org.apache.commons.math3.fitting.PolynomialCurveFitter;
import org.apache.commons.math3.fitting.WeightedObservedPoints;
import org.apache.commons.math3.linear.ArrayRealVector;
import org.apache.commons.math3.linear.DecompositionSolver;
import org.apache.commons.math3.linear.LUDecomposition;
import org.apache.commons.math3.linear.MatrixUtils;
import org.apache.commons.math3.linear.RealMatrix;
import org.apache.commons.math3.linear.RealVector;
public class MathUtil {
/**
* 一元线性拟合 y = a + b*x
*
* @param x
* @param y
* reuslt[0] = a result[1] = b result[2] 相关系数 result[3] 决定系数
* result[4] 点数量(长度) result[5] 自由度
*/
public static double[] lineFitting(double x[], double y[]) {
int size = x.length;
double xmean = 0.0;
double ymean = 0.0;
double xNum = 0.0;
double yNum = 0.0;
double xyNum = 0;
double xNum2 = 0;
double yNum2 = 0;
double rss = 0;
double tss = 0;
double result[] = new double[6];
for (int i = 0; i < size; i++) {
xmean += x[i];
ymean += y[i];
xNum2 += x[i] * x[i];
yNum2 += y[i] * y[i];
xyNum += x[i] * y[i];
}
xNum = xmean;
yNum = ymean;
xmean /= size;
ymean /= size;
double sumx2 = 0.0f;
double sumxy = 0.0f;
for (int i = 0; i < size; i++) {
sumx2 += (x[i] - xmean) * (x[i] - xmean);
sumxy += (y[i] - ymean) * (x[i] - xmean);
}
double b = sumxy / sumx2;
double a = ymean - b * xmean;
result[0] = a;
result[1] = b;
System.out.println("a = " + a + ", b=" + b);
double correlation = (xyNum - xNum * yNum / size)
/ Math.sqrt((xNum2 - xNum * xNum / size) * (yNum2 - yNum * yNum / size));
System.out.println("相关系数：" + correlation);
result[2] = correlation;
for (int i = 0; i < size; i++) {
rss += (y[i] - (a + b * x[i])) * (y[i] - (a + b * x[i]));
tss += (y[i] - ymean) * (y[i] - ymean);
}
double r2 = 1 - (rss / (size - 1 - 1)) / (tss / (size - 1));
result[3] = r2;
System.out.println("决定系数" + r2);
result[4] = x.length;
result[5] = x.length - 1 - 1;
return result;
}
/**
* 多元线性拟合 y = a + b*x1 + c*x2
*
* @param x
* @param y
* result[0] = a result[b] = b . . . result[len - 4] 点数 result[len -
* 3] 自由度 result[len - 2] 残差平方和 result[len - 1] 确定系数
*/
public static double[] lineFitting2(double x[][], double y[]) {
double[] a = new double[x.length + 1];
double[] v = new double[2]; // 这里的2为m
double[] dt = new double[4];
line2sqt(x, y, 2, 11, a, dt, v);
int i;
System.out.println("残差平方和：" + dt[0]);
double temp = a[a.length - 1];
// 更换输出位置，把常数放到第一位
for (i = a.length - 1; i > 0; i--) {
a[i] = a[i - 1];
}
a[0] = temp;
double[] result = new double[x.length + 5];
for (i = 0; i <= x.length; i++) {
result[i] = a[i];
}
result[x.length + 1] = y.length;
result[x.length + 2] = y.length - x.length;
result[x.length + 3] = dt[0];
result[x.length + 4] = getLine2R(x, y, a, x.length);
System.out.println("校正确定系数：" + result[x.length + 4]);
return result;
}
/**
* 多项式拟合 y = a + b*x1 + c*x1^2…… result[0] = a result[1] = b . . . result[n + 1]
* 点数 result[n + 2] 自由度 result[n + 3] 确定系数
*
* @param n
* 几级
* @return
*/
public static double[] dxsFitting(double x[], double y[], int n) {
double result[] = new double[n + 4];
WeightedObservedPoints obs = new WeightedObservedPoints();
for (int i = 0; i < x.length; i++) {
obs.add(x[i], y[i]);
}
// Instantiate a third-degree polynomial fitterm.
final PolynomialCurveFitter fitter = PolynomialCurveFitter.create(n);
// Retrieve fitted parameters (coefficients of the polynomial function).
final double[] coeff = fitter.fit(obs.toList());
for (int i = 0; i < coeff.length; i++) {
result[i] = coeff[i];
}
double s = getDxsR(x, y, coeff, 5);
System.out.println("确定系数:" + s);
result[n + 1] = x.length;
result[n + 2] = x.length - n - 1;
result[n + 3] = s;
return result;
}
/**
* 指数拟合 y = b*exp(ax) result[0] = a result[1] = b result[2] 数据点数 result[3] 自由度
* result[4] 确定系数
*
* @param x
* @param y
* @return
*/
public static double[] expFitting(double x[], double y[]) {
int size = x.length;
double xmean = 0.0;
double ymean = 0.0;
double rss = 0;
double tss = 0;
double result[] = new double[5];
for (int i = 0; i < size; i++) {
xmean += x[i];
y[i] = Math.log(y[i]);
ymean += y[i];
}
xmean /= size;
ymean /= size;
double sumx2 = 0.0f;
double sumxy = 0.0f;
for (int i = 0; i < size; i++) {
sumx2 += (x[i] - xmean) * (x[i] - xmean);
sumxy += (y[i] - ymean) * (x[i] - xmean);
}
double b = sumxy / sumx2;
double a = ymean - b * xmean;
for (int i = 0; i < size; i++) {
rss += (y[i] - (a + b * x[i])) * (y[i] - (a + b * x[i]));
tss += (y[i] - ymean) * (y[i] - ymean);
}
double r2 = 1 - (rss / (size - 1 - 1)) / (tss / (size - 1));
System.out.println("决定系数" + r2);
a = Math.exp(a);
System.out.println("a = " + a + ";b= " + b);
result[0] = a;
result[1] = b;
result[2] = x.length;
result[3] = x.length - 2;
result[4] = r2;
return result;
}
/**
* 对数拟合 y = ln(a * x + b) result[0] = a result[1] = b result[2] 数据点数 result[3]
* 自由度 result[4] 确定系数
*
* @param x
* @param y
*/
public static double[] logFitting(double x[], double y[]) {
int size = x.length;
double xmean = 0.0;
double ymean = 0.0;
double rss = 0;
double tss = 0;
double result[] = new double[5];
for (int i = 0; i < size; i++) {
xmean += x[i];
y[i] = Math.exp(y[i]);
ymean += y[i];
}
xmean /= size;
ymean /= size;
double sumx2 = 0.0f;
double sumxy = 0.0f;
for (int i = 0; i < size; i++) {
sumx2 += (x[i] - xmean) * (x[i] - xmean);
sumxy += (y[i] - ymean) * (x[i] - xmean);
}
double b = sumxy / sumx2;
double a = ymean - b * xmean;
for (int i = 0; i < size; i++) {
rss += (y[i] - (b + a * x[i])) * (y[i] - (b + a * x[i]));
tss += (y[i] - ymean) * (y[i] - ymean);
}
double r2 = 1 - (rss / (size - 1 - 1)) / (tss / (size - 1));
System.out.println("决定系数" + r2);
System.out.println("a = " + a + ";b= " + b);
result[0] = a;
result[1] = b;
result[2] = x.length;
result[3] = x.length - 2;
result[4] = r2;
return result;
}
/**
* 参考网址：http://wenku.baidu.com/link?url=T-xy9CUR_hVlCpA5o6xjqWb-Z1o5jraGEg-OHEtDgFLEynf6HedV68wMwC5u7RTs7PN5kiYq3qjyiddMveDFywLiuXai11MisNrNvv4KjAW&qq-pf-to=pcqq.c2c
* 峰拟合 y = y(max) * exp[-(x - x(max))^2/S]
*
* @param x
* @param y
* result[0] = x(max) result[1] = y(max) result[2] = S result[3] 点数
* result[4] 自由度 result[5] 确定系数
*/
public static double[] peakFitting(double x[], double y[]) {
int size = x.length;
double maxX = 0;
double maxY = 0;
double minY = Integer.MAX_VALUE;
double result[] = new double[6];
double[][] left = new double[x.length][3];
double[][] right = new double[x.length][1];
for (int i = 0; i < size; i++) {
if (y[i] > maxY) {
maxX = x[i];
maxY = y[i];
}
if (y[i] < minY) {
minY = y[i];
}
for (int j = 0; j < 3; j++) {
left[i][j] = getPeakValue(j, x[i]);
}
right[i][0] = Math.log(y[i]);
}
RealMatrix leftMatrix = MatrixUtils.createRealMatrix(left);
RealMatrix rightMatrix = MatrixUtils.createRealMatrix(right);
RealMatrix leftMatrix1 = leftMatrix.transpose();
RealMatrix m = leftMatrix1.multiply(leftMatrix);
RealMatrix tMatrix = new LUDecomposition(m).getSolver().getInverse().multiply(leftMatrix1)
.multiply(rightMatrix);
result[0] = maxX;
result[1] = maxY;
result[2] = -1.0 / tMatrix.getEntry(2, 0);
System.out.println(result[0] + " " + result[1] + " " + result[2]);
double aaaa = getPearR(x, y, result, 2);
System.out.println("确定系数:" + aaaa);
result[3] = x.length;
result[4] = x.length - 1 - 2;
result[5] = aaaa;
return result;
}
/**
* 用户自定义拟合
*
* @param x
* @param y
* @param sf
* 算法 常数
* x
* x^2
* x^3
* exp(x)
* ln(x)
* sin(x)
* cos(x)
*
* result[0] = 第一项系数 result[1] = 第二项系数 . . . result[n] = 第N项系数
* result[sf.length] 点数 result[sf.length+1] 自由度 result[sf.length+2]
* 确定系数
* @return
*/
public static double[] userDefineFitting(double x[], double y[], int sf[]) {
double[][] left = new double[sf.length][sf.length];
double[] right = new double[sf.length];
double[] result = new double[sf.length + 3];
result[sf.length] = x.length;
boolean containZero = false;
for (int i = 0; i < sf.length; i++) {
double yValue = 0;
// 数值中包括0
if (sf[i] == 0) {
containZero = true;
}
for (int j = 0; j < sf.length; j++) {
double xValue = 0;
// 计算X的的司格马
for (int k = 0; k < x.length; k++) {
xValue += getUserDefineValue(sf[i], x[k]) * getUserDefineValue(sf[j], x[k]);
}
left[i][j] = xValue;
}
// 计算Y的的司格马
for (int k = 0; k < x.length; k++) {
yValue += y[k] * getUserDefineValue(sf[i], x[k]);
}
right[i] = yValue;
}
// 计算自由度
result[sf.length + 1] = x.length - sf.length - 1;
// 计算自由度
if (containZero) {
result[sf.length + 1] = x.length - sf.length;
}
// 矩阵求解
RealMatrix leftMatrix = MatrixUtils.createRealMatrix(left);
DecompositionSolver solver = new LUDecomposition(leftMatrix).getSolver();
RealVector constants = new ArrayRealVector(right, false);
RealVector solution = solver.solve(constants);
System.out.println(solution);
// 获得系数值
for (int i = 0; i < solution.getDimension(); i++) {
result[i] = solution.getEntry(i);
}
double rss = 0, tss = 0, ymean = 0;
for (int i = 0; i < x.length; i++) {
ymean += y[i];
}
ymean /= x.length;
for (int i = 0; i < x.length; i++) {
rss += (y[i] - getUserDefineValueByX(sf, x[i], solution))
* (y[i] - getUserDefineValueByX(sf, x[i], solution));
tss += (y[i] - ymean) * (y[i] - ymean);
}
double r2 = 1 - (rss / result[sf.length + 1]) / (tss / (x.length - 1));
System.out.println("决定系数" + r2);
result[sf.length + 2] = r2;
return result;
}
/**
* 用户自定义函数，传入X值获得Y值
*
* @param n
* @param x
* @param solution
* @return
*/
private static double getUserDefineValueByX(int[] n, double x, RealVector solution) {
double value = 0;
for (int i = 0; i < n.length; i++) {
value += getUserDefineValue(n[i], x) * solution.getEntry(i);
}
return value;
}
/**
* 获得用户自定义的值， 常数
* x x^2
* x^3
* exp(x)
* ln(x)
* sin(x)
* cos(x)
*
* @param i
* @param x
* @return
*/
private static double getUserDefineValue(int i, double x) {
// 常数
if (i == 0) {
return 1;
} else if (i == 1) {
// x
return x;
} else if (i == 2) {
// x^2
return x * x;
} else if (i == 3) {
// x^3
return x * x * x;
} else if (i == 4) {
// exp(x)
return Math.pow(Math.E, x);
} else if (i == 5) {
// ln(x)
return Math.log(x);
} else if (i == 6) {
// sin(x)
return Math.sin(x);
} else if (i == 7) {
// cos(x)
return Math.cos(x);
}
return 0;
}
/**
* 获得决定系数
*
* @param coeff
* @return
*/
private static double getPearR(double x[], double y[], double coeff[], int n) {
int size = x.length;
double ymean = 0.0;
double rss = 0;
double tss = 0;
for (int i = 0; i < size; i++) {
ymean += y[i];
}
ymean /= size;
for (int i = 0; i < size; i++) {
rss += (y[i] - getPeakValueByX(x[i], coeff)) * (y[i] - getPeakValueByX(x[i], coeff));
tss += (y[i] - ymean) * (y[i] - ymean);
}
double r2 = 1 - (rss / (size - n - 1)) / (tss / (size - 1));
return r2;
}
/**
* 通过X获得Y的值 y = y(max) * exp[-(x - x(max))^2/S]
*
* @param x
* @param y
* result[0] = x(max) result[1] = y(max) result[2] = S
* @return
*/
private static double getPeakValueByX(double x, double result[]) {
double b = Math.exp(-Math.pow((x - result[0]), 2) / result[2]) * result[1];
return b;
}
private static double getPeakValue(int i, double x) {
// 常数
if (i == 0) {
return 1;
} else if (i == 1) {
return x;
} else if (i == 2) {
// x^2
return x * x;
}
return 0;
}
/**
* 获得决定系数
*
* @param coeff
* @return
*/
private static double getDxsR(double x[], double y[], double coeff[], int n) {
int size = x.length;
double ymean = 0.0;
double rss = 0;
double tss = 0;
for (int i = 0; i < size; i++) {
ymean += y[i];
}
ymean /= size;
for (int i = 0; i < size; i++) {
rss += (y[i] - getDxsValueByX(x[i], coeff)) * (y[i] - getDxsValueByX(x[i], coeff));
tss += (y[i] - ymean) * (y[i] - ymean);
}
double r2 = 1 - (rss / (size - n - 1)) / (tss / (size - 1));
return r2;
}
/**
* 通过X获得Y的值
*
* @param x
* @param coeff
* @return
*/
private static double getDxsValueByX(double x, double coeff[]) {
int size = coeff.length;
double result = coeff[0];
for (int i = 1; i < size; i++) {
result += coeff[i] * Math.pow(x, i);
}
return result;
}
/**
* 多元线性回归分析
*
* @param x[m][n]
* 每一列存放m个自变量的观察值
* @param y[n]
* 存放随即变量y的n个观察值
* @param m
* 自变量的个数
* @param n
* 观察数据的组数
* @param a
* 返回回归系数a0,...,am
* @param dt[4]
* dt[0]偏差平方和q,dt[1] 平均标准偏差s dt[2]返回复相关系数r dt[3]返回回归平方和u
* @param v[m]
* 返回m个自变量的偏相关系数
*/
private static void line2sqt(double[][] x, double[] y, int m, int n, double[] a, double[] dt, double[] v) {
int i, j, k, mm;
double q, e, u, p, yy, s, r, pp;
double[] b = new double[(m + 1) * (m + 1)];
mm = m + 1;
b[mm * mm - 1] = n;
for (j = 0; j <= m - 1; j++) {
p = 0.0;
for (i = 0; i <= n - 1; i++)
p = p + x[j][i];
b[m * mm + j] = p;
b[j * mm + m] = p;
}
for (i = 0; i <= m - 1; i++)
for (j = i; j <= m - 1; j++) {
p = 0.0;
for (k = 0; k <= n - 1; k++)
p = p + x[i][k] * x[j][k];
b[j * mm + i] = p;
b[i * mm + j] = p;
}
a[m] = 0.0;
for (i = 0; i <= n - 1; i++)
a[m] = a[m] + y[i];
for (i = 0; i <= m - 1; i++) {
a[i] = 0.0;
for (j = 0; j <= n - 1; j++)
a[i] = a[i] + x[i][j] * y[j];
}
chlk(b, mm, 1, a);
yy = 0.0;
for (i = 0; i <= n - 1; i++)
yy = yy + y[i] / n;
q = 0.0;
e = 0.0;
u = 0.0;
for (i = 0; i <= n - 1; i++) {
p = a[m];
for (j = 0; j <= m - 1; j++)
p = p + a[j] * x[j][i];
q = q + (y[i] - p) * (y[i] - p);
e = e + (y[i] - yy) * (y[i] - yy);
u = u + (yy - p) * (yy - p);
}
s = Math.sqrt(q / n);
r = Math.sqrt(1.0 - q / e);
for (j = 0; j <= m - 1; j++) {
p = 0.0;
for (i = 0; i <= n - 1; i++) {
pp = a[m];
for (k = 0; k <= m - 1; k++)
if (k != j)
pp = pp + a[k] * x[k][i];
p = p + (y[i] - pp) * (y[i] - pp);
}
v[j] = Math.sqrt(1.0 - q / p);
}
dt[0] = q;
dt[1] = s;
dt[2] = r;
dt[3] = u;
}
private static int chlk(double[] a, int n, int m, double[] d) {
int i, j, k, u, v;
if ((a[0] + 1.0 == 1.0) || (a[0] < 0.0)) {
System.out.println("fail\n");
return (-2);
}
a[0] = Math.sqrt(a[0]);
for (j = 1; j <= n - 1; j++)
a[j] = a[j] / a[0];
for (i = 1; i <= n - 1; i++) {
u = i * n + i;
for (j = 1; j <= i; j++) {
v = (j - 1) * n + i;
a[u] = a[u] - a[v] * a[v];
}
if ((a[u] + 1.0 == 1.0) || (a[u] < 0.0)) {
System.out.println("fail\n");
return (-2);
}
a[u] = Math.sqrt(a[u]);
if (i != (n - 1)) {
for (j = i + 1; j <= n - 1; j++) {
v = i * n + j;
for (k = 1; k <= i; k++)
a[v] = a[v] - a[(k - 1) * n + i] * a[(k - 1) * n + j];
a[v] = a[v] / a[u];
}
}
}
for (j = 0; j <= m - 1; j++) {
d[j] = d[j] / a[0];
for (i = 1; i <= n - 1; i++) {
u = i * n + i;
v = i * m + j;
for (k = 1; k <= i; k++)
d[v] = d[v] - a[(k - 1) * n + i] * d[(k - 1) * m + j];
d[v] = d[v] / a[u];
}
}
for (j = 0; j <= m - 1; j++) {
u = (n - 1) * m + j;
d[u] = d[u] / a[n * n - 1];
for (k = n - 1; k >= 1; k--) {
u = (k - 1) * m + j;
for (i = k; i <= n - 1; i++) {
v = (k - 1) * n + i;
d[u] = d[u] - a[v] * d[i * m + j];
}
v = (k - 1) * n + k - 1;
d[u] = d[u] / a[v];
}
}
return (2);
}
/**
* 获得相关系数
*
* @param coeff
* @return
*/
private static double getLine2R(double x[][], double y[], double a[], int n) {
int size = x[0].length;
double ymean = 0.0;
double rss = 0;
double tss = 0;
for (int i = 0; i < size; i++) {
ymean += y[i];
}
ymean /= size;
for (int i = 0; i < size; i++) {
rss += (y[i] - getLine2ValueByX(x, a, i)) * (y[i] - getLine2ValueByX(x, a, i));
tss += (y[i] - ymean) * (y[i] - ymean);
}
double r2 = 1 - (rss / (size - n - 1)) / (tss / (size - 1));
return r2;
}
/**
* 通过X获得Y的值
*
* @param x
* @return
*/
private static double getLine2ValueByX(double x[][], double a[], int n) {
int size = a.length;
double result = a[0];
for (int i = 1; i < size; i++) {
result += x[i - 1][n] * a[i];
}
return result;
}
}