我想用不同的样本大小 n、不同的 SD 值和不同的 theta 值模拟一阶 MA(1) 的移动平均过程 [英] I want to simulate moving average process of order one MA(1) with varying sample size n, varying SD values and varying theta values
问题描述
我想用 mean = 0 但变化来模拟一些时间序列数据:在数学上，一阶移动平均过程，MA(1)表示为
I want to simulate some time series data with mean = 0 but varying: Mathematically, moving average process of order one, MA(1) is presented as
$$x_t=\mu+\varepsilon_{t}+\theta_{1}\varepsilon_{t-1}$$
$x_t$ 是 MA(1) 过程$\mu$ 是在我的情况下可以为零的平均值(就像回归方程中的截距一样)$\varepsilon_{t}$ 是误差项$\theta_{1}$ 是一个需要指定的常量(在我的例子中，一个介于 +-1 之间的不同数字).示例:在 $x=a+b*x_{i}$ 的简单回归方程中，$theta$ 就像 $b$
$x_t$ is the MA(1) process $\mu$ is the mean which can be zero in my case (just like intercept in regression equation) $\varepsilon_{t}$ is the error term $\theta_{1}$ is a constant which need be specified (in my case, a varying number in between +-1). Example: in simple regression equation of $x=a+b*x_{i}$, $theta$ is like the $b$
数字 N = 15、20、30、50、100、200.
标准偏差SD=1、4、9、16、25.
和 theta 值 \theta = +-0.2, +-0.4, +-0.6, +-0.8, +-0.9, +-0.95, +-0.99
set.seed(123)
# here I am only using first sample size 15
n <- 15
# white noise:
wnsd1<-ts(rnorm(n, mean=0, sd=1^2))
wnsd4<-ts(rnorm(n, mean=0, sd=2^2))
wnsd9<-ts(rnorm(n, mean=0, sd=3^2))
wnsd16<-ts(rnorm(n, mean=0, sd=4^2))
wnsd25<-ts(rnorm(n, mean=0, sd=5^2))
# initialise the first two values:
ma1 <- wnsd1[1:2]
# loop through and create the 3:15th values:
for(i in 3:n){
# here I only use when SD=1
ma1[i] <- wnsd1[i - 1] * 0.2 + wnsd1[i]
}
#turn them into time series, and for the last two, "integrate" them via cumulative sum
ma1 <- ts(ma1)
我想要一种成熟的方法来改变样本大小 N、标准差 SD 和 MA(1) \theta 的估计
I want a mature way of varying the sample size N, the standard deviation SD and the estimate of MA(1) \theta
推荐答案
这是一个好的方法.请注意，我不知道 phi 是如何使用的，因为它没有在代码中明确显示.如果你修改你的代码，我会尝试解决它.
Here's an OK way. Note, I do not know how phi is used as it wasn't explicitly in the code. If you modify your code, I would try to address it.
N <- c(15L, 20L)
SD = c(1, 2)^2
phi = c(0.2, 0.4)
set.seed(123)
res <- lapply(N,
function(n)
lapply(SD,
function(s.d.) {
wn <- ts(rnorm(n, 0, s.d.))
ar1 <- ma1 <- arma11 <- arma22 <- vector('numeric', n)
ar1 <- ma1 <- arma11 <- arma22 <- wn[1:2]
for (i in 3:n) {
ar1[i] <- ar1[i - 1] * 0.2 + wn[i]
ma1[i] <- wn[i - 1] * 2.8000 + wn[i]
arma11[i] <- arma11[i - 1] * 0.2 + wn[i - 1] * 2.80003769654 + wn[i]
arma22[i] <- arma22[i - 1] * 0.862537 + arma22[i - 2] * (-0.3) + 0.2 * wn[i - 1] - 0.3 * wn[i -
2] + wn[i]
}
#turn them into time series, and for the last two, "integrate" them via cumulative sum
return(data.frame(ar1 = ts(ar1),
ma1 = ts(ma1),
arma11 = ts(arma11),
arima111 = ts(cumsum(arma11)),
arima222 = ts(cumsum(cumsum(arma22)))
))
}))
res <- setNames(lapply(res, setNames, paste('SD', SD, sep = '_')), paste('n', N, sep = '_'))
res
结果 - 仅截断为一种组合:
Result - truncated to only one combination:
$n_15
$n_15$SD_1
ar1 ma1 arma11 arima111 arima222
1 -0.5604756 -0.56047565 -0.56047565 -0.56047565 -0.5604756
2 -0.2301775 -0.23017749 -0.23017749 -0.79065314 -1.3511288
3 1.5126728 0.91421134 0.86816717 0.07751403 -0.4913603
4 0.3730430 4.43489167 4.60858386 4.68609790 2.3123144
5 0.2038963 0.32671123 1.24843066 5.93452856 5.9733306
6 1.7558443 2.07707065 2.32676165 8.26129021 11.5104337
7 0.8120851 5.26309817 5.72851515 13.98980536 19.1736717
8 -1.1026442 0.02550414 1.17122455 15.16102991 26.4205560
9 -0.9073817 -4.22902431 -3.99482709 11.16620282 31.5923395
10 -0.6271383 -2.36884996 -3.16784126 7.99836155 34.8956636
11 1.0986541 -0.02377172 -0.65735677 7.34100478 38.5509080
12 0.5795447 3.78724286 3.65581765 10.99682243 43.8085632
13 0.5166804 1.40825017 2.13942726 13.13624969 50.4482906
14 0.2140188 1.23284278 1.66074334 14.79699303 57.8822760
15 -0.5130374 -0.24592953 0.08622331 14.88321634 64.9327807
编辑:此方法类似，但使用显式 for 循环而不是 lapply 并且仅返回 ma 变量:
Edit: This approach is similar but uses explicit for loops instead of lapply and only returns the ma variable:
N <- c(15L, 20L)
SD = c(1, 2) ^ 2
phi = c(0.2, 0.4)
res <- vector('list', length(N))
names(res) <- paste('N', N, sep = '_')
set.seed(123L)
for (i in seq_along(N)){
res[[i]] <- vector('list', length(SD))
names(res[[i]]) <- paste('SD', SD, sep = '_')
ma <- matrix(NA_real_, nrow = N[i], ncol = length(phi))
for (j in seq_along(SD)){
wn <- rnorm(N[i], mean = 0, sd = SD[j])
ma[1:2, ] <- wn[1:2]
for (k in 3:N[i]){
ma[k, ] <- wn[k - 1L] * phi + wn[k]
}
colnames(ma) <- paste('ma_theta', phi, sep = '_')
res[[i]][[j]] <- ma
}
}
res
$N_15
$N_15$SD_1
ma_theta_0.2 ma_theta_0.4
[1,] 0.68374552 0.68374552
[2,] -0.06082195 -0.06082195
[3,] 0.62079632 0.60863193
[4,] 1.46210976 1.58870190
[5,] 0.27439361 0.54149714
[6,] 1.01901666 1.02047467
[7,] -0.98492231 -0.78141058
[8,] -0.95929125 -1.19697805
[9,] 1.37489682 1.23057594
[10,] 0.68123152 0.98507506
[11,] -1.97674523 -1.90126763
[12,] -1.77448202 -2.18492658
[13,] -0.47358851 -0.74639600
[14,] 0.82562320 0.78546700
[15,] 0.07127263 0.24442851
$N_15$SD_4
ma_theta_0.2 ma_theta_0.4
[1,] 2.4967499 2.4967499
[2,] 3.8360215 3.8360215
[3,] 7.4514236 8.2186279
[4,] 1.5609108 2.8977547
[5,] -0.1631142 -0.1183009
[6,] -7.0545350 -7.0961205
[7,] -1.0052795 -2.4078694
[8,] -2.2079382 -2.1284761
[9,] -4.3535184 -4.8109984
[10,] -1.4988326 -2.2780403
[11,] 3.9158477 3.7719227
[12,] -7.1590394 -6.3470849
[13,] -3.3033159 -4.8975147
[14,] 0.1247257 -0.2170977
[15,] -3.4795205 -3.3862106
$N_20
$N_20$SD_1
ma_theta_0.2 ma_theta_0.4
[1,] 0.33390294 0.3339029
[2,] 0.41142992 0.4114299
[3,] 0.04924982 0.1315358
[4,] -2.47250543 -2.4791127
[5,] 2.07827851 1.5850989
[6,] 0.30899237 0.8232840
[7,] 0.61013343 0.5690736
[8,] 0.40400515 0.5342438
[9,] 1.07942653 1.1341798
[10,] 1.02259409 1.2275287
[11,] -0.04626128 0.1172706
[12,] 0.33620914 0.2942505
[13,] -0.86977528 -0.7941417
[14,] 0.66784124 0.4787595
[15,] -0.28965374 -0.1182691
[16,] 2.32456569 2.2323580
[17,] -1.16769422 -0.6843396
[18,] -0.79419702 -1.1244068
[19,] 0.73258241 0.6397850
[20,] 0.67520852 0.8402845
$N_20$SD_4
ma_theta_0.2 ma_theta_0.4
[1,] -2.35792415 -2.35792415
[2,] -3.98712297 -3.98712297
[3,] -0.21952177 -1.01694637
[4,] 0.05835091 0.17393147
[5,] -7.17257088 -7.18401681
[6,] -1.29402072 -2.72624571
[7,] 0.78856212 0.81620297
[8,] 0.85108984 1.00327409
[9,] -4.08028705 -3.94050594
[10,] 1.06051948 0.21650585
[11,] 5.89518717 6.27609379
[12,] 2.92780172 4.03065783
[13,] -4.17736476 -3.81237564
[14,] -2.65105266 -3.55952343
[15,] 1.03589810 0.68738173
[16,] -2.31129963 -2.03441673
[17,] -9.14822185 -9.66585835
[18,] 1.81088621 0.08476914
[19,] -2.61050979 -1.90310913
[20,] -2.95782317 -3.62140526
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