![Page 1: Pemodelan Data Deret Waktu - WordPress.com · 2019-09-02 · Pemodelan Data Deret Waktu (AR dan MA) 2 The forecasting methods based on the smoothing may be inefficient and sometimes](https://reader031.vdocuments.us/reader031/viewer/2022013008/5e44a0d4b29961000b66f59b/html5/thumbnails/1.jpg)
Dr. Kusman Sadik, M.Si
Sekolah Pascasarjana Departemen Statistika IPB
Semester Ganjil 2019/2020
Analisis Deret Waktu (STK 651)
IPB University─ Bogor Indonesia ─ Inspiring Innovation with Integrity
Pemodelan Data Deret Waktu
(AR dan MA)
![Page 2: Pemodelan Data Deret Waktu - WordPress.com · 2019-09-02 · Pemodelan Data Deret Waktu (AR dan MA) 2 The forecasting methods based on the smoothing may be inefficient and sometimes](https://reader031.vdocuments.us/reader031/viewer/2022013008/5e44a0d4b29961000b66f59b/html5/thumbnails/2.jpg)
2
The forecasting methods based on the smoothing
may be inefficient and sometimes inappropriate
because they do not take advantage of the serial
dependence in the observations in the most effective
way.
To formally incorporate this dependent structure, in
this course we will explore a general class of models
called autoregressive integrated moving average
models or ARIMA models (also known as Box-Jenkinsmodels).
![Page 3: Pemodelan Data Deret Waktu - WordPress.com · 2019-09-02 · Pemodelan Data Deret Waktu (AR dan MA) 2 The forecasting methods based on the smoothing may be inefficient and sometimes](https://reader031.vdocuments.us/reader031/viewer/2022013008/5e44a0d4b29961000b66f59b/html5/thumbnails/3.jpg)
3
AR(p) : Autoregressive ber-ordo p
I(d) : Integrated ber-ordo d
MA(q) : Moving Average ber-ordo q
![Page 4: Pemodelan Data Deret Waktu - WordPress.com · 2019-09-02 · Pemodelan Data Deret Waktu (AR dan MA) 2 The forecasting methods based on the smoothing may be inefficient and sometimes](https://reader031.vdocuments.us/reader031/viewer/2022013008/5e44a0d4b29961000b66f59b/html5/thumbnails/4.jpg)
4
Model Stasioner:
AR(p), MA(q), dan ARMA(p, q)
Model Tidak-Stasioner:
ARI(p, d), IMA(d, q), dan ARIMA(p, d, q)
![Page 5: Pemodelan Data Deret Waktu - WordPress.com · 2019-09-02 · Pemodelan Data Deret Waktu (AR dan MA) 2 The forecasting methods based on the smoothing may be inefficient and sometimes](https://reader031.vdocuments.us/reader031/viewer/2022013008/5e44a0d4b29961000b66f59b/html5/thumbnails/5.jpg)
5
E(Yt) = E(Yt-1 ) = … = E(Yt-k ) Konstan
V(Yt) = V(Yt-1 ) = … = V(Yt-k ) Konstan
![Page 6: Pemodelan Data Deret Waktu - WordPress.com · 2019-09-02 · Pemodelan Data Deret Waktu (AR dan MA) 2 The forecasting methods based on the smoothing may be inefficient and sometimes](https://reader031.vdocuments.us/reader031/viewer/2022013008/5e44a0d4b29961000b66f59b/html5/thumbnails/6.jpg)
6
Peristilahan
Autocovariance : Koragam Diri
Autocorrelation : Korelasi Diri
![Page 7: Pemodelan Data Deret Waktu - WordPress.com · 2019-09-02 · Pemodelan Data Deret Waktu (AR dan MA) 2 The forecasting methods based on the smoothing may be inefficient and sometimes](https://reader031.vdocuments.us/reader031/viewer/2022013008/5e44a0d4b29961000b66f59b/html5/thumbnails/7.jpg)
7
![Page 8: Pemodelan Data Deret Waktu - WordPress.com · 2019-09-02 · Pemodelan Data Deret Waktu (AR dan MA) 2 The forecasting methods based on the smoothing may be inefficient and sometimes](https://reader031.vdocuments.us/reader031/viewer/2022013008/5e44a0d4b29961000b66f59b/html5/thumbnails/8.jpg)
8
![Page 9: Pemodelan Data Deret Waktu - WordPress.com · 2019-09-02 · Pemodelan Data Deret Waktu (AR dan MA) 2 The forecasting methods based on the smoothing may be inefficient and sometimes](https://reader031.vdocuments.us/reader031/viewer/2022013008/5e44a0d4b29961000b66f59b/html5/thumbnails/9.jpg)
9
![Page 10: Pemodelan Data Deret Waktu - WordPress.com · 2019-09-02 · Pemodelan Data Deret Waktu (AR dan MA) 2 The forecasting methods based on the smoothing may be inefficient and sometimes](https://reader031.vdocuments.us/reader031/viewer/2022013008/5e44a0d4b29961000b66f59b/html5/thumbnails/10.jpg)
10
![Page 11: Pemodelan Data Deret Waktu - WordPress.com · 2019-09-02 · Pemodelan Data Deret Waktu (AR dan MA) 2 The forecasting methods based on the smoothing may be inefficient and sometimes](https://reader031.vdocuments.us/reader031/viewer/2022013008/5e44a0d4b29961000b66f59b/html5/thumbnails/11.jpg)
11
Misalkan diketahui data deret waktu sebagai berikut: 2, 3, 2, 5.
Hitung secara manual penduga fungsi autokorelasi (ACF) untuk k = 1
dan 2:
r1 = {(3-3)(2-3)+(2-3)(3-3)+(5-3)(2-3)}/{(-1)2+(0)2+(-1)2+(2)2} = - 0.333
r2 = {(2-3)(2-3)+(5-3)(3-3)}/{(-1)2+(0)2+(-1)2+(2)2} = 0.167
= 3
> data <- c(2, 3, 2, 5)
> acf(data, lag.max = 3, plot = FALSE)
Autocorrelations of series ‘data’, by lag
0 1 2 3
1.000 -0.333 0.167 -0.333
Program R
![Page 12: Pemodelan Data Deret Waktu - WordPress.com · 2019-09-02 · Pemodelan Data Deret Waktu (AR dan MA) 2 The forecasting methods based on the smoothing may be inefficient and sometimes](https://reader031.vdocuments.us/reader031/viewer/2022013008/5e44a0d4b29961000b66f59b/html5/thumbnails/12.jpg)
12
(Langkah Acak)
![Page 13: Pemodelan Data Deret Waktu - WordPress.com · 2019-09-02 · Pemodelan Data Deret Waktu (AR dan MA) 2 The forecasting methods based on the smoothing may be inefficient and sometimes](https://reader031.vdocuments.us/reader031/viewer/2022013008/5e44a0d4b29961000b66f59b/html5/thumbnails/13.jpg)
13
![Page 14: Pemodelan Data Deret Waktu - WordPress.com · 2019-09-02 · Pemodelan Data Deret Waktu (AR dan MA) 2 The forecasting methods based on the smoothing may be inefficient and sometimes](https://reader031.vdocuments.us/reader031/viewer/2022013008/5e44a0d4b29961000b66f59b/html5/thumbnails/14.jpg)
14
et ~ Normal(0, σe2)
(4.1.4)
![Page 15: Pemodelan Data Deret Waktu - WordPress.com · 2019-09-02 · Pemodelan Data Deret Waktu (AR dan MA) 2 The forecasting methods based on the smoothing may be inefficient and sometimes](https://reader031.vdocuments.us/reader031/viewer/2022013008/5e44a0d4b29961000b66f59b/html5/thumbnails/15.jpg)
15
Pada beberapa software, termasuk R, untuk pertimbangan
komputasi penulisan model MA(q) menggunakan tanda plus
pada parameter θ, yaitu:
Hal ini harus diperhatikan pada saat pendugaan parameter,
karena Program R akan menghasilkan penduga θ yang
berlawanan tanda (plus / minus) dengan model MA(q)
pada persamaan (4.1.4) di atas.
Untuk pembangkitan data MA di R bisa menggunakan
persamaan (4.1.4), yang berbeda hanya saat pendugaan.
![Page 16: Pemodelan Data Deret Waktu - WordPress.com · 2019-09-02 · Pemodelan Data Deret Waktu (AR dan MA) 2 The forecasting methods based on the smoothing may be inefficient and sometimes](https://reader031.vdocuments.us/reader031/viewer/2022013008/5e44a0d4b29961000b66f59b/html5/thumbnails/16.jpg)
16
Karena et ~ Normal(0, σe2) maka:
E(et) = E(et-1 ) = … = E(et-k ) = 0
V(et) = V(et-1 ) = … = V(et-k ) = (σe)2
E(et.ek) = 0 untuk semua t ≠ k
![Page 17: Pemodelan Data Deret Waktu - WordPress.com · 2019-09-02 · Pemodelan Data Deret Waktu (AR dan MA) 2 The forecasting methods based on the smoothing may be inefficient and sometimes](https://reader031.vdocuments.us/reader031/viewer/2022013008/5e44a0d4b29961000b66f59b/html5/thumbnails/17.jpg)
17
V(Yt) = V(et - θet-1)
= V(et) + θ2V(et-1)
= σe2 + θ2σe
2
Cov(-θet-1 , et-1)
= -θCov(et-1 , et-1)
= -θVar(et-1)
= -θσe2
![Page 18: Pemodelan Data Deret Waktu - WordPress.com · 2019-09-02 · Pemodelan Data Deret Waktu (AR dan MA) 2 The forecasting methods based on the smoothing may be inefficient and sometimes](https://reader031.vdocuments.us/reader031/viewer/2022013008/5e44a0d4b29961000b66f59b/html5/thumbnails/18.jpg)
18
![Page 19: Pemodelan Data Deret Waktu - WordPress.com · 2019-09-02 · Pemodelan Data Deret Waktu (AR dan MA) 2 The forecasting methods based on the smoothing may be inefficient and sometimes](https://reader031.vdocuments.us/reader031/viewer/2022013008/5e44a0d4b29961000b66f59b/html5/thumbnails/19.jpg)
19
ACF / Autocorrelation Function : ρk = (γk/γ0)
![Page 20: Pemodelan Data Deret Waktu - WordPress.com · 2019-09-02 · Pemodelan Data Deret Waktu (AR dan MA) 2 The forecasting methods based on the smoothing may be inefficient and sometimes](https://reader031.vdocuments.us/reader031/viewer/2022013008/5e44a0d4b29961000b66f59b/html5/thumbnails/20.jpg)
20
t = k Cov(αet , θek) = (αθ)σe2
t ≠ k Cov(αet , θek) = 0
![Page 21: Pemodelan Data Deret Waktu - WordPress.com · 2019-09-02 · Pemodelan Data Deret Waktu (AR dan MA) 2 The forecasting methods based on the smoothing may be inefficient and sometimes](https://reader031.vdocuments.us/reader031/viewer/2022013008/5e44a0d4b29961000b66f59b/html5/thumbnails/21.jpg)
21
![Page 22: Pemodelan Data Deret Waktu - WordPress.com · 2019-09-02 · Pemodelan Data Deret Waktu (AR dan MA) 2 The forecasting methods based on the smoothing may be inefficient and sometimes](https://reader031.vdocuments.us/reader031/viewer/2022013008/5e44a0d4b29961000b66f59b/html5/thumbnails/22.jpg)
22
![Page 23: Pemodelan Data Deret Waktu - WordPress.com · 2019-09-02 · Pemodelan Data Deret Waktu (AR dan MA) 2 The forecasting methods based on the smoothing may be inefficient and sometimes](https://reader031.vdocuments.us/reader031/viewer/2022013008/5e44a0d4b29961000b66f59b/html5/thumbnails/23.jpg)
23
![Page 24: Pemodelan Data Deret Waktu - WordPress.com · 2019-09-02 · Pemodelan Data Deret Waktu (AR dan MA) 2 The forecasting methods based on the smoothing may be inefficient and sometimes](https://reader031.vdocuments.us/reader031/viewer/2022013008/5e44a0d4b29961000b66f59b/html5/thumbnails/24.jpg)
24
Var(Yt) = Var(ϕYt-1 + et) = ϕ2Var(Yt-1) + Var(et)
![Page 25: Pemodelan Data Deret Waktu - WordPress.com · 2019-09-02 · Pemodelan Data Deret Waktu (AR dan MA) 2 The forecasting methods based on the smoothing may be inefficient and sometimes](https://reader031.vdocuments.us/reader031/viewer/2022013008/5e44a0d4b29961000b66f59b/html5/thumbnails/25.jpg)
25
![Page 26: Pemodelan Data Deret Waktu - WordPress.com · 2019-09-02 · Pemodelan Data Deret Waktu (AR dan MA) 2 The forecasting methods based on the smoothing may be inefficient and sometimes](https://reader031.vdocuments.us/reader031/viewer/2022013008/5e44a0d4b29961000b66f59b/html5/thumbnails/26.jpg)
26
![Page 27: Pemodelan Data Deret Waktu - WordPress.com · 2019-09-02 · Pemodelan Data Deret Waktu (AR dan MA) 2 The forecasting methods based on the smoothing may be inefficient and sometimes](https://reader031.vdocuments.us/reader031/viewer/2022013008/5e44a0d4b29961000b66f59b/html5/thumbnails/27.jpg)
27
![Page 28: Pemodelan Data Deret Waktu - WordPress.com · 2019-09-02 · Pemodelan Data Deret Waktu (AR dan MA) 2 The forecasting methods based on the smoothing may be inefficient and sometimes](https://reader031.vdocuments.us/reader031/viewer/2022013008/5e44a0d4b29961000b66f59b/html5/thumbnails/28.jpg)
28
![Page 29: Pemodelan Data Deret Waktu - WordPress.com · 2019-09-02 · Pemodelan Data Deret Waktu (AR dan MA) 2 The forecasting methods based on the smoothing may be inefficient and sometimes](https://reader031.vdocuments.us/reader031/viewer/2022013008/5e44a0d4b29961000b66f59b/html5/thumbnails/29.jpg)
29
![Page 30: Pemodelan Data Deret Waktu - WordPress.com · 2019-09-02 · Pemodelan Data Deret Waktu (AR dan MA) 2 The forecasting methods based on the smoothing may be inefficient and sometimes](https://reader031.vdocuments.us/reader031/viewer/2022013008/5e44a0d4b29961000b66f59b/html5/thumbnails/30.jpg)
30
Ordo (nilai q) pada model MA(q) dapat diidentifikasi
dari plot ACF-nya.
Ordo (nilai p) pada model AR(p) TIDAK dapat
diidentifikasi dari plot ACF-nya karena polanya
berbentuk eksponensial.
Ordo pada model AR(p) dapat diidentifikasi dari plot yt
dengan yt-k, untuk k = 1, 2, 3, …
![Page 31: Pemodelan Data Deret Waktu - WordPress.com · 2019-09-02 · Pemodelan Data Deret Waktu (AR dan MA) 2 The forecasting methods based on the smoothing may be inefficient and sometimes](https://reader031.vdocuments.us/reader031/viewer/2022013008/5e44a0d4b29961000b66f59b/html5/thumbnails/31.jpg)
31
![Page 32: Pemodelan Data Deret Waktu - WordPress.com · 2019-09-02 · Pemodelan Data Deret Waktu (AR dan MA) 2 The forecasting methods based on the smoothing may be inefficient and sometimes](https://reader031.vdocuments.us/reader031/viewer/2022013008/5e44a0d4b29961000b66f59b/html5/thumbnails/32.jpg)
32
![Page 33: Pemodelan Data Deret Waktu - WordPress.com · 2019-09-02 · Pemodelan Data Deret Waktu (AR dan MA) 2 The forecasting methods based on the smoothing may be inefficient and sometimes](https://reader031.vdocuments.us/reader031/viewer/2022013008/5e44a0d4b29961000b66f59b/html5/thumbnails/33.jpg)
33
![Page 34: Pemodelan Data Deret Waktu - WordPress.com · 2019-09-02 · Pemodelan Data Deret Waktu (AR dan MA) 2 The forecasting methods based on the smoothing may be inefficient and sometimes](https://reader031.vdocuments.us/reader031/viewer/2022013008/5e44a0d4b29961000b66f59b/html5/thumbnails/34.jpg)
34
![Page 35: Pemodelan Data Deret Waktu - WordPress.com · 2019-09-02 · Pemodelan Data Deret Waktu (AR dan MA) 2 The forecasting methods based on the smoothing may be inefficient and sometimes](https://reader031.vdocuments.us/reader031/viewer/2022013008/5e44a0d4b29961000b66f59b/html5/thumbnails/35.jpg)
35
![Page 36: Pemodelan Data Deret Waktu - WordPress.com · 2019-09-02 · Pemodelan Data Deret Waktu (AR dan MA) 2 The forecasting methods based on the smoothing may be inefficient and sometimes](https://reader031.vdocuments.us/reader031/viewer/2022013008/5e44a0d4b29961000b66f59b/html5/thumbnails/36.jpg)
36
# Simulasi Data MA(1) dan AR(1)
# Install packages : "forecast", "TTR", "TSA", "graphics"
library("forecast")
library("TTR")
library("TSA")
library("graphics")
set.seed(1001)
e <- rnorm(175,0,1)
n <- length(e)
# Membangkitkan X, MA(1) dengan tetha = 0.95
tetha <- 0.95
x <- c(1:n)
for (i in 2:n) { x[i] <- e[i] - tetha*e[i-1] } # Model MA Cryer 4.1.4
x.ma1 <- x[-c(1:50)] # membuang 50 data pertama
plot.ts(x.ma1, lty=1)
points(x.ma1)
acf(x.ma1, lag.max=20) # menampilkan plot acf
acf(x.ma1, lag.max=20, plot=FALSE) # menampilkan nilai acf
![Page 37: Pemodelan Data Deret Waktu - WordPress.com · 2019-09-02 · Pemodelan Data Deret Waktu (AR dan MA) 2 The forecasting methods based on the smoothing may be inefficient and sometimes](https://reader031.vdocuments.us/reader031/viewer/2022013008/5e44a0d4b29961000b66f59b/html5/thumbnails/37.jpg)
37
# Membangkitkan y, AR(1) dengan Phi = 0.85
y <- c(1:n)
for (i in 2:n) { y[i] <- 0.85*y[i-1] + e[i] }
y.ar1 <- y[-c(1:50)] # membuang 50 data pertama
plot.ts(y.ar1, lty=1)
points(y.ar1)
plot(x=zlag(y.ar1,1),y=y.ar1,xlab=expression(Y[t-1]),
ylab=expression(Y[t]),type='p')
plot(x=zlag(y.ar1,2),y=y.ar1,xlab=expression(Y[t-2]),
ylab=expression(Y[t]),type='p')
acf(y.ar1, lag.max=20) # menampilkan plot acf
acf(y.ar1, lag.max=20, plot=FALSE) # menampilkan nilai acf
![Page 38: Pemodelan Data Deret Waktu - WordPress.com · 2019-09-02 · Pemodelan Data Deret Waktu (AR dan MA) 2 The forecasting methods based on the smoothing may be inefficient and sometimes](https://reader031.vdocuments.us/reader031/viewer/2022013008/5e44a0d4b29961000b66f59b/html5/thumbnails/38.jpg)
38
![Page 39: Pemodelan Data Deret Waktu - WordPress.com · 2019-09-02 · Pemodelan Data Deret Waktu (AR dan MA) 2 The forecasting methods based on the smoothing may be inefficient and sometimes](https://reader031.vdocuments.us/reader031/viewer/2022013008/5e44a0d4b29961000b66f59b/html5/thumbnails/39.jpg)
39
![Page 40: Pemodelan Data Deret Waktu - WordPress.com · 2019-09-02 · Pemodelan Data Deret Waktu (AR dan MA) 2 The forecasting methods based on the smoothing may be inefficient and sometimes](https://reader031.vdocuments.us/reader031/viewer/2022013008/5e44a0d4b29961000b66f59b/html5/thumbnails/40.jpg)
40
> acf(x.ma1, lag.max=20, plot=FALSE) # menampilkan nilai acf
Autocorrelations of series ‘x.ma1’, by lag
1 2 3 4 5 6 7 8 9 10 11
-0.390 -0.055 -0.029 0.009 -0.001 0.026 -0.113 0.105 -0.113 0.195 -0.045
12 13 14 15 16 17 18 19 20
-0.149 0.137 -0.051 -0.029 -0.060 0.058 -0.007 0.116 -0.157
![Page 41: Pemodelan Data Deret Waktu - WordPress.com · 2019-09-02 · Pemodelan Data Deret Waktu (AR dan MA) 2 The forecasting methods based on the smoothing may be inefficient and sometimes](https://reader031.vdocuments.us/reader031/viewer/2022013008/5e44a0d4b29961000b66f59b/html5/thumbnails/41.jpg)
41
![Page 42: Pemodelan Data Deret Waktu - WordPress.com · 2019-09-02 · Pemodelan Data Deret Waktu (AR dan MA) 2 The forecasting methods based on the smoothing may be inefficient and sometimes](https://reader031.vdocuments.us/reader031/viewer/2022013008/5e44a0d4b29961000b66f59b/html5/thumbnails/42.jpg)
42
![Page 43: Pemodelan Data Deret Waktu - WordPress.com · 2019-09-02 · Pemodelan Data Deret Waktu (AR dan MA) 2 The forecasting methods based on the smoothing may be inefficient and sometimes](https://reader031.vdocuments.us/reader031/viewer/2022013008/5e44a0d4b29961000b66f59b/html5/thumbnails/43.jpg)
43
![Page 44: Pemodelan Data Deret Waktu - WordPress.com · 2019-09-02 · Pemodelan Data Deret Waktu (AR dan MA) 2 The forecasting methods based on the smoothing may be inefficient and sometimes](https://reader031.vdocuments.us/reader031/viewer/2022013008/5e44a0d4b29961000b66f59b/html5/thumbnails/44.jpg)
44
![Page 45: Pemodelan Data Deret Waktu - WordPress.com · 2019-09-02 · Pemodelan Data Deret Waktu (AR dan MA) 2 The forecasting methods based on the smoothing may be inefficient and sometimes](https://reader031.vdocuments.us/reader031/viewer/2022013008/5e44a0d4b29961000b66f59b/html5/thumbnails/45.jpg)
45
![Page 46: Pemodelan Data Deret Waktu - WordPress.com · 2019-09-02 · Pemodelan Data Deret Waktu (AR dan MA) 2 The forecasting methods based on the smoothing may be inefficient and sometimes](https://reader031.vdocuments.us/reader031/viewer/2022013008/5e44a0d4b29961000b66f59b/html5/thumbnails/46.jpg)
46
1. Misalkan diketahui data deret waktu sebagai berikut: 3, 8, 5, 9, 12, 20.
Hitung secara manual penduga fungsi autokorelasi (ACF) untuk k = 1, 2,
dan 3, kemudian bandingkan hasilnya dengan keluaran Program R.
2. Melalui Program R, bangkitkan data yt, (n = 165), berupa MA(2) dengan
θ1 = 0.65 dan θ2 = - 0.85 serta et ~ Normal(0,1). Gunakan 150 data terakhir,
kemudian buat correlogramnya. Apa yang dapat disimpulkan dari
correlogram tersebut?
3. Melalui Program R, bangkitkan data yt, (n = 165), berupa AR(2) dengan
Φ1 = 0.75 dan Φ2 = - 0.65 dan et ~ Normal(0,1). Gunakan 150 data terakhir:
a. Buatlah correlogramnya. Apa yang dapat disimpulkan dari correlogram
tersebut?
b. Buatlah plot antara yt dengan yt-1. Apa kesimpulan Anda?
c. Buatlah plot antara yt dengan yt-2. Apa kesimpulan Anda?
d. Buatlah plot antara yt dengan yt-3. Apa kesimpulan Anda?
![Page 47: Pemodelan Data Deret Waktu - WordPress.com · 2019-09-02 · Pemodelan Data Deret Waktu (AR dan MA) 2 The forecasting methods based on the smoothing may be inefficient and sometimes](https://reader031.vdocuments.us/reader031/viewer/2022013008/5e44a0d4b29961000b66f59b/html5/thumbnails/47.jpg)
47
![Page 48: Pemodelan Data Deret Waktu - WordPress.com · 2019-09-02 · Pemodelan Data Deret Waktu (AR dan MA) 2 The forecasting methods based on the smoothing may be inefficient and sometimes](https://reader031.vdocuments.us/reader031/viewer/2022013008/5e44a0d4b29961000b66f59b/html5/thumbnails/48.jpg)
48
Montgomery, D.C., et.al. 2008. Forecasting Time Series Analysis
2nd. John Wiley.
Cryer, J.D. and Chan, K.S. 2008. Time Series Analysis with
Application in R. Springer.
Cowpertwait, P.S.P. and Metcalfe, A.V. 2009. Introductory Time
Series with R. Springer New York.
Wei, William, W.S. 1990. Time Series Analysis, Univariate and
Multivariate Methods. Adison-Wesley Publishing Company Inc,
Canada.
![Page 49: Pemodelan Data Deret Waktu - WordPress.com · 2019-09-02 · Pemodelan Data Deret Waktu (AR dan MA) 2 The forecasting methods based on the smoothing may be inefficient and sometimes](https://reader031.vdocuments.us/reader031/viewer/2022013008/5e44a0d4b29961000b66f59b/html5/thumbnails/49.jpg)
49
Bisa di-download di
kusmansadik.wordpress.com
![Page 50: Pemodelan Data Deret Waktu - WordPress.com · 2019-09-02 · Pemodelan Data Deret Waktu (AR dan MA) 2 The forecasting methods based on the smoothing may be inefficient and sometimes](https://reader031.vdocuments.us/reader031/viewer/2022013008/5e44a0d4b29961000b66f59b/html5/thumbnails/50.jpg)
50