时间序列分析：三重指数平滑

Author: nex3z 2019-07-27

Time Series, Triple Exponential Smoothing

　　双重指数平滑引入了趋势，但还不能处理序列中存在的季节性。为了体现序列中的趋势，对 $x_{n+1}$ 的预测 $\hat x_{n+1}$ 应由局部水平 $L_n$ 、趋势 $T_n$ 以及季节项 $S_{n+1-m}$ 组成，其中 $m$ 为季节周期。

$\begin{equation} \hat x_{n+1} = L_n + T_n + S_{n+1-m} \tag{1} \end{equation}$

　　其中局部水平 $L_n$ 需要移除季节项的干扰，是对 $x_n – S_{n-m}$ 的指数平滑，即

$\begin{equation} L_n = \alpha (x_n – S_{n-m}) + (1 – \alpha)(\hat x_n – S_{n-m}) \tag{2} \end{equation}$

其中

$\begin{equation} \hat x_{n} = L_{n-1} + T_{n-1} + S_{n-m} \end{equation}$

于是

$\begin{align} L_n = \alpha (x_n – S_{n-m}) + (1 – \alpha)(L_{n-1} + T_{n-1}) \tag{3} \end{align}$

　　趋势项 $T_n$ 的定义与双重指数平均相同，为

$\begin{equation} T_n = \beta \cdot (L_n – L_{n-1}) + (1 – \beta) \cdot T_{n-1} \tag{4} \end{equation}$

　　季节项 $S_n$ 是对季节的平滑，需要移除局部水平的干扰，即对 $x_n – L_n$ 进行指数平滑

$\begin{equation} S_n = \gamma (x_n – L_n) + (1 – \gamma )S_{n-m} \tag{5} \end{equation}$

　　式 $(1)$ 、 $(3)$ 、 $(4)$ 、 $(5)$ 即构成了三重滑动平均模型。当使用前 $n$ 个数据对 $n+h$ 时刻进行预测时，有

$\begin{equation} \hat x_{n+h} = L_n + h \cdot T_n + S_{n+h-m} \tag{6} \end{equation}$

　　前面引入的季节性都是加性的，如果存在乘性的季节性，则式 $(1)$ 变为

$\begin{equation} \hat x_{n+1} = (L_n + T_n) \cdot S_{n+1-m} \tag{7} \end{equation}$

式 $(6)$ 变为

$\begin{equation} \hat x_{n+h} = (L_n + h \cdot T_n) \cdot S_{n+h-m} \tag{8} \end{equation}$

2. 一个例子

　　载入国际航空旅客人数的月度数据（单位为千人），绘制图像如图 1。

plot(AirPassengers)

plot(AirPassengers)

图 1

可见该数据具有季节性，周期振幅（方差）随时间增长，具有乘性周期。

　　对数据进行 $\log$ 变换后，绘制图像如图 2。

data.ts.log <- log10(data.ts)

plot(data.ts.log)

data.ts.log <- log10(data.ts) plot(data.ts.log)

data.ts.log <- log10(data.ts)
plot(data.ts.log)

可见此时的季节性就变成了加性的了。

　　使用 HoltWinters 函数进行拟合

model <- HoltWinters(data.ts.log)

model

model <- HoltWinters(data.ts.log) model

model <- HoltWinters(data.ts.log)
model

得到的结果为

Holt-Winters exponential smoothing with trend and additive seasonal component.

Call:

HoltWinters(x = data.ts.log)

Smoothing parameters:

alpha: 0.326612

beta : 0.005744246

gamma: 0.8207255

Coefficients:

[,1]

a 2.680598830

b 0.003900787

s1 -0.031790733

s2 -0.061224237

s3 -0.015941495

s4 0.006307818

s5 0.014138008

s6 0.067260071

s7 0.127820295

s8 0.119893006

s9 0.038321663

s10 -0.014181699

s11 -0.085995400

s12 -0.044672707

Holt-Winters exponential smoothing with trend and additive seasonal component. Call: HoltWinters(x = data.ts.log) Smoothing parameters: alpha: 0.326612 beta : 0.005744246 gamma: 0.8207255 Coefficients: [,1] a 2.680598830 b 0.003900787 s1 -0.031790733 s2 -0.061224237 s3 -0.015941495 s4 0.006307818 s5 0.014138008 s6 0.067260071 s7 0.127820295 s8 0.119893006 s9 0.038321663 s10 -0.014181699 s11 -0.085995400 s12 -0.044672707

Holt-Winters exponential smoothing with trend and additive seasonal component.

Call:
HoltWinters(x = data.ts.log)

Smoothing parameters:
 alpha: 0.326612
 beta : 0.005744246
 gamma: 0.8207255

Coefficients:
            [,1]
a    2.680598830
b    0.003900787
s1  -0.031790733
s2  -0.061224237
s3  -0.015941495
s4   0.006307818
s5   0.014138008
s6   0.067260071
s7   0.127820295
s8   0.119893006
s9   0.038321663
s10 -0.014181699
s11 -0.085995400
s12 -0.044672707

手动计算 $h = 1$ （1961 年 1 月）和 $h = 13$ （1962 年 1 月）时的预测值

2.680598830 + 1 * 0.003900787 - 0.031790733

2.680598830 + 13 * 0.003900787 - 0.031790733

2.680598830 + 1 * 0.003900787 - 0.031790733 2.680598830 + 13 * 0.003900787 - 0.031790733

2.680598830 + 1 * 0.003900787 - 0.031790733
2.680598830 + 13 * 0.003900787 - 0.031790733

结果分别为 2.652708884 和 2.699518328。

　　使用模型进行预测

library(forecast)

model.forecast <- forecast(model)

model.forecast

library(forecast) model.forecast <- forecast(model) model.forecast

library(forecast)

model.forecast <- forecast(model)
model.forecast

结果为

Point Forecast Lo 80 Hi 80 Lo 95 Hi 95

Jan 1961 2.652709 2.630898 2.674520 2.619351 2.686066

Feb 1961 2.627176 2.604218 2.650134 2.592065 2.662287

Mar 1961 2.676360 2.652297 2.700422 2.639560 2.713160

Apr 1961 2.702510 2.677380 2.727640 2.664077 2.740942

May 1961 2.714241 2.688076 2.740406 2.674225 2.754257

Jun 1961 2.771264 2.744092 2.798436 2.729708 2.812820

Jul 1961 2.835725 2.807571 2.863878 2.792667 2.878782

Aug 1961 2.831698 2.802586 2.860811 2.787174 2.876222

Sep 1961 2.754028 2.723977 2.784079 2.708069 2.799987

Oct 1961 2.705425 2.674454 2.736396 2.658059 2.752791

Nov 1961 2.637512 2.605638 2.669386 2.588765 2.686259

Dec 1961 2.682736 2.649974 2.715497 2.632631 2.732840

Jan 1962 2.699518 2.661306 2.737731 2.641078 2.757959

Feb 1962 2.673986 2.635014 2.712957 2.614383 2.733588

Mar 1962 2.723169 2.683445 2.762894 2.662416 2.783923

Apr 1962 2.749319 2.708848 2.789790 2.687424 2.811214

May 1962 2.761050 2.719838 2.802262 2.698022 2.824078

Jun 1962 2.818073 2.776126 2.860020 2.753921 2.882226

Jul 1962 2.882534 2.839857 2.925211 2.817265 2.947803

Aug 1962 2.878508 2.835105 2.921910 2.812129 2.944886

Sep 1962 2.800837 2.756714 2.844960 2.733356 2.868318

Oct 1962 2.752234 2.707395 2.797074 2.683658 2.820811

Nov 1962 2.684322 2.638770 2.729873 2.614656 2.753987

Dec 1962 2.729545 2.683285 2.775805 2.658796 2.800294

Point Forecast Lo 80 Hi 80 Lo 95 Hi 95 Jan 1961 2.652709 2.630898 2.674520 2.619351 2.686066 Feb 1961 2.627176 2.604218 2.650134 2.592065 2.662287 Mar 1961 2.676360 2.652297 2.700422 2.639560 2.713160 Apr 1961 2.702510 2.677380 2.727640 2.664077 2.740942 May 1961 2.714241 2.688076 2.740406 2.674225 2.754257 Jun 1961 2.771264 2.744092 2.798436 2.729708 2.812820 Jul 1961 2.835725 2.807571 2.863878 2.792667 2.878782 Aug 1961 2.831698 2.802586 2.860811 2.787174 2.876222 Sep 1961 2.754028 2.723977 2.784079 2.708069 2.799987 Oct 1961 2.705425 2.674454 2.736396 2.658059 2.752791 Nov 1961 2.637512 2.605638 2.669386 2.588765 2.686259 Dec 1961 2.682736 2.649974 2.715497 2.632631 2.732840 Jan 1962 2.699518 2.661306 2.737731 2.641078 2.757959 Feb 1962 2.673986 2.635014 2.712957 2.614383 2.733588 Mar 1962 2.723169 2.683445 2.762894 2.662416 2.783923 Apr 1962 2.749319 2.708848 2.789790 2.687424 2.811214 May 1962 2.761050 2.719838 2.802262 2.698022 2.824078 Jun 1962 2.818073 2.776126 2.860020 2.753921 2.882226 Jul 1962 2.882534 2.839857 2.925211 2.817265 2.947803 Aug 1962 2.878508 2.835105 2.921910 2.812129 2.944886 Sep 1962 2.800837 2.756714 2.844960 2.733356 2.868318 Oct 1962 2.752234 2.707395 2.797074 2.683658 2.820811 Nov 1962 2.684322 2.638770 2.729873 2.614656 2.753987 Dec 1962 2.729545 2.683285 2.775805 2.658796 2.800294

         Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
Jan 1961       2.652709 2.630898 2.674520 2.619351 2.686066
Feb 1961       2.627176 2.604218 2.650134 2.592065 2.662287
Mar 1961       2.676360 2.652297 2.700422 2.639560 2.713160
Apr 1961       2.702510 2.677380 2.727640 2.664077 2.740942
May 1961       2.714241 2.688076 2.740406 2.674225 2.754257
Jun 1961       2.771264 2.744092 2.798436 2.729708 2.812820
Jul 1961       2.835725 2.807571 2.863878 2.792667 2.878782
Aug 1961       2.831698 2.802586 2.860811 2.787174 2.876222
Sep 1961       2.754028 2.723977 2.784079 2.708069 2.799987
Oct 1961       2.705425 2.674454 2.736396 2.658059 2.752791
Nov 1961       2.637512 2.605638 2.669386 2.588765 2.686259
Dec 1961       2.682736 2.649974 2.715497 2.632631 2.732840
Jan 1962       2.699518 2.661306 2.737731 2.641078 2.757959
Feb 1962       2.673986 2.635014 2.712957 2.614383 2.733588
Mar 1962       2.723169 2.683445 2.762894 2.662416 2.783923
Apr 1962       2.749319 2.708848 2.789790 2.687424 2.811214
May 1962       2.761050 2.719838 2.802262 2.698022 2.824078
Jun 1962       2.818073 2.776126 2.860020 2.753921 2.882226
Jul 1962       2.882534 2.839857 2.925211 2.817265 2.947803
Aug 1962       2.878508 2.835105 2.921910 2.812129 2.944886
Sep 1962       2.800837 2.756714 2.844960 2.733356 2.868318
Oct 1962       2.752234 2.707395 2.797074 2.683658 2.820811
Nov 1962       2.684322 2.638770 2.729873 2.614656 2.753987
Dec 1962       2.729545 2.683285 2.775805 2.658796 2.800294

可见 Jan 1961 和 Jan 1962 的预测值与前面手工计算的结果一致。

　　绘制预测结果如图 3 所示。

plot(model.forecast)

plot(model.forecast)

图 3

注意这里的预测值都是经过 $\log$ 变换的，需要进行反变换得到真正的预测值。

2019 年 7 月
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