有数列$\{x_i\}$，数列前$ t$个的均值为$M_t$，则 $$M_t=\frac{\sum_{i=1}^{t}x_i}{t}$$ $M_t$还有一种迭代的求取方法， $$M_t=\frac{t-1}{t}M_{t-1}+\frac{1}{t}x_t$$
推导过程
有 $$M_t={(\sum_{i=1}^{t}x_i)}/{t}$$ 将$\sum_{i=1}^{t}x_i$拆成$\sum_{i=1}^{t-1}x_i$和$x_t$的和，即 $$\begin{aligned} &M_t &= & (\sum_{i=1}^{t}x_i)/{t} \\ \\ & &=& (\sum_{i=1}^{t-1}x_i+x_t)/t \\ \\ & &=& \frac{1}{t}\sum_{i=1}^{t-1}x_i+\frac{1}{t}x_t\end{aligned}$$ 由$M_{t-1}=(\sum_{i=1}^{t-1}x_i)/(t-1)$可得$\sum_{i=1}^{t-1}x_i=M_{t-1}(t-1)$，带入上式， $$\begin{aligned} &M_t &= & \frac{1}{t}\sum_{i=1}^{t-1}x_i+\frac{1}{t}x_t \\ \\ & &=& \frac{t-1}{t}M_{t-1}+\frac{1}{t}x_t\end{aligned}$$
代码实现
迭代函数
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void iterative_mean(float& average, const float num, const int index)
{
float index_1 = static_cast<float>(index + 1);
average = average * static_cast<float>(index) / index_1 + num / index_1;
}
示例
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void iterative_mean(float& average, const float num, const int index)
{
float index_1 = static_cast<float>(index + 1);
average = average * static_cast<float>(index) / index_1 + num / index_1;
}
int main()
{
float a[10] = {0};
const int count = 10;
for (int i = 0; i < count; ++i)
{
a[i] = i+1;
std::cout << a[i] << " ";
}
std::cout << std::endl;
float iter_mean = 0;
float acc_mean = 0;
for (int i = 0; i < count; ++i)
{
// 迭代求平均值
iterative_mean(iter_mean, a[i], i);
// 累加
acc_mean += a[i];
}
// 累加计算平均值
acc_mean /= count;
std::cout << "accumulative mean: " << acc_mean << std::endl;
std::cout << "iterative mean: " << iter_mean << std::endl;
return 0;
}
输出：
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accumulative mean: 5.5
iterative mean: 5.5