如何在R中通过roll::roll_sd计算指数移动标准差(EMA_SD)？

观察图片Difference_EMA_SMA_Standard_Deviation可知，指数移动标准差(EMA_SD)与标准移动标准差(SMA_SD)有时存在显著差异。目前可通过以下代码用roll::roll_sd计算SMA_SD：

# 数据集
dataset <- structure(list(Time = structure(c(1702391700, 1702392000, 1702392300, 
1702392600, 1702392900, 1702393200, 1702393500, 1702393800, 1702394100, 
1702394400, 1702394700, 1702395000, 1702395300, 1702395600, 1702395900, 
1702396200, 1702396500, 1702396800, 1702397100, 1702397400, 1702397700, 
1702398000, 1702398300, 1702398600, 1702398900, 1702399200, 1702399500, 
1702399800, 1702400100, 1702400400, 1702400700, 1702401000, 1702401300, 
1702401600, 1702401900, 1702402200, 1702402500, 1702402800, 1702403100, 
1702403400, 1702403700, 1702404000, 1702404300, 1702404600, 1702404900, 
1702405200, 1702405500, 1702405800, 1702406100, 1702406400), tzone = "", class = c("POSIXct", 
"POSIXt")), Open = c(132.95, 133.93, 133.32, 133.62, 133.8, 134.7, 
135.14, 134.91, 135.49, 135.87, 135.88, 135.96, 136.2, 136.97, 
137.74, 137.34, 137.93, 138.47, 138.47, 138.71, 138.94, 139.31, 
139.85, 138.82, 138.75, 139.34, 138.94, 138.72, 138.42, 138.25, 
138.24, 137.6, 137.93, 138.01, 137.77, 137.52, 137.67, 137.21, 
137.33, 137.1, 137.35, 137.3, 137.78, 137.14, 137.01, 136.7, 
136.19, 136.46, 136.22, 135.92), High = c(134.14, 134.28, 133.64, 
134.26, 134.89, 135.2, 135.15, 135.57, 136.28, 136.02, 136.06, 
136.54, 137.03, 137.74, 137.74, 137.93, 138.58, 138.87, 138.87, 
139, 139.54, 139.89, 139.86, 139.2, 139.47, 139.38, 139.14, 138.83, 
138.59, 138.5, 138.26, 137.99, 138.09, 138.14, 137.84, 137.89, 
137.68, 137.51, 137.59, 137.42, 137.54, 137.79, 137.79, 137.26, 
137.34, 136.74, 136.48, 136.68, 136.41, 136.48), Low = c(132.71, 
132.52, 132.82, 133.5, 133.79, 134.4, 134.68, 134.65, 135.46, 
135.56, 135.71, 135.96, 136.1, 136.95, 137.18, 137.29, 137.9, 
138.27, 138.36, 138.62, 138.94, 139.3, 138.8, 138.71, 138.43, 
138.65, 138.61, 138.24, 138.16, 138.02, 137.51, 137.37, 137.63, 
137.7, 137.43, 137.4, 137.16, 137.19, 137.04, 137.09, 137.19, 
136.99, 137.12, 136.67, 136.7, 136.05, 136.1, 136.06, 135.71, 
135.91), Close = c(133.94, 133.32, 133.61, 133.82, 134.69, 135.16, 
134.92, 135.49, 135.87, 135.87, 135.96, 136.19, 136.97, 137.73, 
137.34, 137.92, 138.47, 138.47, 138.71, 138.94, 139.31, 139.85, 
138.82, 138.75, 139.32, 138.94, 138.71, 138.42, 138.25, 138.22, 
137.6, 137.94, 138.02, 137.77, 137.52, 137.68, 137.21, 137.33, 
137.09, 137.36, 137.31, 137.77, 137.13, 137.01, 136.71, 136.2, 
136.46, 136.22, 135.91, 136.45), Volume = c(2439400, 2089200, 
1507400, 1755500, 2029600, 1939100, 1169100, 1687600, 2181900, 
1316700, 1126100, 1588700, 1875000, 2122800, 1916900, 1825100, 
2206300, 2179400, 1473300, 1780200, 2012100, 1921700, 2295300, 
1868100, 1954600, 1183300, 1067900, 1247700, 929600, 1407500, 
1552400, 1386500, 1162000, 910900, 753700, 770100, 922700, 664900, 
903500, 568700, 697400, 1127800, 785700, 1210500, 969900, 1297000, 
973400, 1228400, 1176400, 940800)), row.names = c(NA, -50L), class = c("data.table", 
"data.frame"))

# 计算SMA_SD
library(roll)
dataset$Roll_SD <- roll::roll_sd(x = dataset$Close, width = 20, center = TRUE, weights = rep(1, 20))

之前尝试用EMA权重直接传入roll::roll_sd的方式无法得到正确结果，代码如下：

dataset$Roll_SD_EMA  <- roll::roll_sd(x = dataset$Close, width = 20, center = TRUE, weights = ( 1 - (2/(20 +1)))^(0:19), online = FALSE)

正确计算EMA_SD的方法 #

EMA是递归计算的指数加权平均，因此EMA标准差不能直接用窗口加权的roll::roll_sd计算，正确步骤是：

1. 计算收盘价的EMA均值（平滑系数alpha=2/(N+1)，N=20）
2. 计算每个收盘价与对应EMA均值的平方偏差
3. 对平方偏差计算EMA（使用相同的alpha）
4. 对平方偏差的EMA结果开根号，得到EMA_SD

使用TTR包的EMA函数可以便捷实现：

library(TTR)

# 设置参数
N <- 20
alpha <- 2/(N+1)

# 步骤1：计算收盘价的EMA均值
dataset$EMA_mean <- EMA(dataset$Close, n = N, wilder = FALSE)

# 步骤2：计算平方偏差
dataset$sq_dev <- (dataset$Close - dataset$EMA_mean)^2

# 步骤3：计算平方偏差的EMA
dataset$EMA_sq_dev <- EMA(dataset$sq_dev, n = N, wilder = FALSE)

# 步骤4：开根号得到EMA_SD
dataset$EMA_SD <- sqrt(dataset$EMA_sq_dev)

结果验证 #

此时对比Roll_SD（SMA_SD）和EMA_SD，就能复现图片中两者的显著差异。

内容的提问来源于stack exchange，提问作者Humba Bumba