计算样本数据的方差, 标准方差与协方差

小咪咪 2022-02-04 15:09 270阅读 0赞

## 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。  
![这里写图片描述][SouthEast]  
![这里写图片描述][SouthEast 1]

## 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。

[SouthEast]: /images/20220204/8bac81981d244054b8cdbbdc0e8030ac.png
[SouthEast 1]: /images/20220204/a60a5a8c861d4c3898b2a25287375e6b.png