计算样本数据的方差, 标准方差与协方差

计算样本数据的方差, 标准方差与协方差

在图像处理中有时候会涉及计算图像像素数据的方差，标准方差与协方差等统计学属性作为中间数据。因此知道什么是方差、标准方差、协方差很重要。

二：代码实现

Java代码实现计算数据的方差，标准方差、协方差

package com.gloomyfish.image.gmm;

public class CalculateVariance {

    public double mean(double[] data) {
        double sum = 0;
        int len = data.length;
        for(int i=0; i<len; i++) {
            sum += data[i];
        }
        return sum / len;
    }

    public double variance(double[] data) {
        double mean = mean(data);
        double sum = 0;
        int len = data.length;
        double delta = 0;
        for(int i=0; i<len; i++) {
            delta = data[i] - mean;
            sum += (delta*delta);
        }
        return sum / len;
    }

    public double sd(double[] data) {
        return Math.sqrt(variance(data));
    }

    public double covariance(double[] X, double[] Y) {
        double mx = mean(X);
        double my = mean(Y);
        // 理论上应该通过插值保证长度一致
        int len = X.length == Y.length ? X.length : Math.min(X.length, Y.length);
        double sum = 0;
        for(int i=0; i<len; i++) {
            sum += ((X[i]-mx)*(Y[i]-my));
        }
        return sum / len;
    }

    public static void main(String[] args) {
        int len = 20;
        double[] X = new double[len];
        double[] Y = new double[len];
        for(int i=0; i<len; i++) {
            X[i] = Math.random()*100;
            Y[i] = Math.random()*100;
        }
        CalculateVariance cv = new CalculateVariance();
        System.out.println("X Mean: " + cv.mean(X));
        System.out.println("Y Mean: " + cv.mean(Y));
        System.out.println();

        System.out.println("X Variance: " + cv.variance(X));
        System.out.println("Y Variance: " + cv.variance(Y));
        System.out.println();

        System.out.println("X Standard Deviation: " + cv.sd(X));
        System.out.println("Y Standard Deviation: " + cv.sd(Y));
        System.out.println();

        System.out.println("XY Covariance: " + cv.covariance(X, Y));
        System.out.println();
    }

}

三：意义表述

协方差可以分析样本数据之间的线性相关性，协方差为正数时候，一般情况表示相关，协方差为负数的时候则表示不相关，常见的相关性计算就是基于协方差实现。在图像的直方图数据比较中是常规的手段之一。OpenCV与ImageJ中均有代码实现。