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移动平均线与EMA

10-25 7万+

【炼丹技巧】指数移动平均(EMA)的原理及PyTorch实现

Nicolas ​

EMA的定义

指数移动平均(Exponential Moving Average)也叫权重移动平均(Weighted Moving Average),是一种给予近期数据更高权重的平均方法。

假设我们有n个数据: [\theta_1, \theta_2, . \theta_n]

  • 普通的平均数: \overline=\frac\sum_^n \theta_i
  • EMA: v_t = \beta\cdot v_ + (1-\beta)\cdot \theta_t ,其中, v_t 表示前 t 条的平均值 ( v_0=0 ), \beta 是加权权重值 (一般设为0.9-0.999)。

Andrew Ng在Course 2 Improving Deep Neural Networks中讲到,EMA可以近似看成过去 1/(移动平均线与EMA 1-\beta)​ 个时刻 v​ 值的平均。

普通的过去 n 时刻的平均是这样的:

类比EMA,可以发现当 \beta=\frac 时,两式形式上相等。需要注意的是,两个平均并不是严格相等的,这里只是为了帮助理解。

实际上,EMA计算时,过去 1/(1-\beta)​ 个时刻之前的数值平均会decay到 \frac​ 的加权比例,证明如下。

如果将这里的 v_t 展开,可以得到:

v_t = \alpha^n v_ + (1-\alpha)(\alpha^\theta_+ . +\alpha^0\theta_t)

在深度学习的优化中的EMA

上面讲的是广义的ema定义和计算方法,特别的,在深度学习的优化过程中, \theta_t 是t时刻的模型权重weights, v_t 是t时刻的影子权重(shadow weights)。在梯度下降的过程中,会一直维护着这个影子权重,但是这个影子权重并不会参与训练。基本的假设是,模型权重在最后的n步内,会在实际的最优点处抖动,所以我们取最后n步的平均,能使得模型更加的鲁棒。

EMA的偏差修正

实际使用中,如果令 v_0=0 ,且步数较少,ema的计算结果会有一定偏差。

EMA为什么有效

令第n时刻的模型权重(weights)为 v_n ,梯度为 g_n ,可得:

令第n时刻EMA的影子权重为 v_n ,可得:

\begin v_n &= \alpha v_+(1-\alpha)\theta_n \\ &= \alpha (\alpha v_+(1-\alpha)\theta_)+(1-\alpha)\theta_n \\ &= . \\ &= \alpha^n v_0+(1-\alpha)(\theta_n+\alpha\theta_+\alpha^2\theta_+. +\alpha^\theta_) \end

代入上面 \theta_n 的表达,令 v_0=\theta_1 展开上面的公式,可得:

\begin v_n &= \alpha^n v_0+(1-\alpha)(\theta_n+\alpha\theta_+\alpha^2\theta_+. +\alpha^\theta_)\\ &= \alpha^n v_0+(1-\alpha)(\theta_1-\sum_^g_i+\alpha(\theta_1-\sum_^g_i)+. + \alpha^(\theta_1-\sum_^g_i)+\alpha^\theta_)\\ &= \alpha^n v_0+(1-\alpha)(\frac\theta_1-\sum_^\frac>g_i) \\ &= \alpha^n v_0+(1-\alpha^n)\theta_1 -\sum_^(1-\alpha^)g_i\\ &= \theta_1 -\sum_^(1-\alpha^)g_i \end

\begin \theta_n &= \theta_1-\sum_^g_i \\ v_n &= \theta_1 -\sum_^(1-\alpha^)g_i \end

多种移动平均计算总结(MA,EMA,SMA,DMA,TMA,WMA)

diaohuaidi6355 于 2017-06-23 14:43:00 发布 1050 收藏 3

当前函数值= 当前权重 X 当前价格 + 当前权重的互补值 X 移动平均线与EMA 上一次函数值;

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