单次计算
μ = ∑ i = 1 n x i n \mu = \frac{\sum_{i=1}^{n} x_i}{n} μ=n∑i=1nxi
σ 2 = ∑ i = 1 n ( x i − μ ) 2 n = ∑ i = 1 n x i 2 − 2 ∑ i = 1 n x i μ + n μ 2 n = ∑ i = 1 n x i 2 − n μ 2 n = ∑ i = 1 n x i 2 n − μ 2 \begin{array}{ll} \sigma^2 &= \frac{\sum_{i=1}^{n}(x_i - \mu)^2}{n} \\ &= \frac{\sum_{i=1}^{n} x_i^2 -2\sum_{i=1}^{n} x_i\mu + n\mu^2}{n} \\ &= \frac{\sum_{i=1}^{n} x_i^2 - n\mu^2}{n} \\ &= \frac{\sum_{i=1}^{n} x_i^2}{n} - \mu^2 \end{array} σ2=n∑i=1n(xi−μ)2=n∑i=1nxi2−2∑i=1nxiμ+nμ2=n∑i=1nxi2−nμ2=n∑i=1nxi2−μ2
增量计算
| 指标 | 第一批次 | 第二批次 | 合并 |
|---|---|---|---|
| 总数 | n 1 n_1 n1 | n 2 n_2 n2 | n 1 + n 2 n_1+n_2 n1+n2 |
| 均值 | μ 1 \mu_1 μ1 | μ 2 \mu_2 μ2 | n 1 μ 1 + n 2 μ 2 n 1 + n 2 \frac{n_1 \mu_1 + n_2\mu_2}{n_1 + n_2} n1+n2n1μ1+n2μ2 |
| 方差 | σ 1 \sigma_1 σ1 | σ 2 \sigma_2 σ2 | ? |
| ∑ x i 2 \sum x_i^2 ∑xi2 | n 1 σ 1 2 + n 1 μ 1 2 n_1 \sigma_1^2 + n_1 \mu_1^2 n1σ12+n1μ12 | n 2 σ 2 2 + n 2 μ 2 2 n_2 \sigma_2^2 + n_2 \mu_2^2 n2σ22+n2μ22 | n 1 σ 1 2 + n 1 μ 1 2 + n 2 σ 2 2 + n 2 μ 2 2 n_1 \sigma_1^2 + n_1 \mu_1^2 + n_2 \sigma_2^2 + n_2 \mu_2^2 n1σ12+n1μ12+n2σ22+n2μ22 |
σ 2 = ∑ i = 1 n x i 2 n − μ 2 = n 1 σ 1 2 + n 1 μ 1 2 + n 2 σ 2 2 + n 2 μ 2 2 n 1 + n 2 − ( n 1 μ 1 + n 2 μ 2 n 1 + n 2 ) 2 = ( n 1 + n 2 ) ( n 1 σ 1 2 + n 1 μ 1 2 + n 2 σ 2 2 + n 2 μ 2 2 ) − ( n 1 μ 1 + n 2 μ 2 ) 2 ( n 1 + n 2 ) 2 = n 1 σ 1 2 + n 2 σ 2 2 n 1 + n 2 + n 1 n 2 μ 1 2 + n 1 n 2 μ 2 2 − 2 n 1 n 2 μ 1 μ 2 ( n 1 + n 2 ) 2 = n 1 σ 1 2 + n 2 σ 2 2 n 1 + n 2 + n 1 n 2 ( μ 1 − μ 2 ) 2 ( n 1 + n 2 ) 2 \begin{array}{ll} \sigma^2 &= \frac{\sum_{i=1}^{n} x_i^2}{n} - \mu^2 \\ &= \frac{n_1 \sigma_1^2 + n_1 \mu_1^2 + n_2 \sigma_2^2 + n_2 \mu_2^2}{n_1+n_2} - (\frac{n_1 \mu_1 + n_2\mu_2}{n_1 + n_2})^2 \\ &= \frac{(n_1 + n_2)(n_1 \sigma_1^2 + n_1 \mu_1^2 + n_2 \sigma_2^2 + n_2 \mu_2^2) - (n_1 \mu_1 + n_2\mu_2)^2}{(n_1 + n_2)^2} \\ &= \frac{n_1 \sigma_1^2 + n_2 \sigma_2^2}{n_1 + n_2} + \frac{ n_1n_2\mu_1^2 + n_1n_2\mu_2^2 - 2n_1n_2\mu_1\mu_2}{(n_1 +n_2)^2} \\ &= \frac{n_1 \sigma_1^2 + n_2 \sigma_2^2}{n_1 + n_2} + \frac{ n_1n_2(\mu_1 - \mu_2)^2 }{(n_1 +n_2)^2} \end{array} σ2=n∑i=1nxi2−μ2=n1+n2n1σ12+n1μ12+n2σ22+n2μ22−(n1+n2n1μ1+n2μ2)2=(n1+n2)2(n1+n2)(n1σ12+n1μ12+n2σ22+n2μ22)−(n1μ1+n2μ2)2=n1+n2n1σ12+n2σ22+(n1+n2)2n1n2μ12+n1n2μ22−2n1n2μ1μ2=n1+n2n1σ12+n2σ22+(n1+n2)2n1n2(μ1−μ2)2
方差的增量来自均值漂移