Analisis Deret Waktu

(1)

Analisis Deret Waktu

Pertemuan 2

FMA, PKS. Dept. Statistika IPB

Jenis Data

• Cross section

¾ Beberapa pengamatan diamati bersama‐sama pada periode waktu tertentup p g p p ¾ Harga saham semua perusahaan yang tercatat di BEJ pada hari Rabu 27 Februari 2008

• Time Series

¾ Satu pengamatan diamati selama sekian periode secara teratur ¾ Harga saham P.T. TELKOM di BEJ dari 2 Januari 2008 hingga 27 Februari 2008

• Longitudinal/panel

¾ Beberapa pengamatan diamati bersama‐sama selama kurun waktu tertentu p p g (gabungan cross section dan time series) ¾ Harga saham P.T. TELKOM, P.T. INDOSAT, dan P.T. Mobile8 di BEJ dari 2 Januari 2008 hingga 27 Februari 2008

(2)

Pola Data Time Series

5 6 7 8 9 30 35 40 45 50 0 1 2 3 4 5 12345678910 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 0 5 10 15 20 25 12345678910 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 14 16 18 20 25 Konstan Trend

0 2 4 6 8 10 12 12345678910 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 0 5 10 15 12345678910 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 Seasonal Cyclic

Metode Forecasting

Metode forecasting dapat dibedakan menjadi

dua kelompok:

•Smoothing

¾Moving average, Single Exponential Smoothing,

Double Exponential Smoothing, Metode Winter

•Modeling

¾

/

¾ARIMA, ARCH/GARCH

(3)

Year 2000 ??

Smoothing

(4)

Sekilas Tentang Smoothing

• Prinsip dasar: pengenalan pola data dengan

h l k

i i l k l

menghaluskan variasi lokal.

• Prinsip penghalusan umumnya berupa rata‐

rata.

• Beberapa metode penghalusan hanya cocok

untuk pola data tertentu

untuk pola data tertentu.

Metode Yang Dibahas

• Single Moving Average

• Double Moving Average

• Single Exponential Smoothing

• Double Exponential Smoothing

• Metode Winter untuk musiman aditif

• Metode Winter untuk musiman multiplikatif

(5)

Ilustrasi

• All these methods will be illustrated with the following example: Suppose that a hospital would like to forecast the number of patients arrival from the following historical p g data:

Week Patients Arrival

1 400

2 380

3 411

4 415

• Note: Although week 4 data is given, some methods require that forecast for period 4 is first computed before computing forecast for period 5.

web4.uwindsor.ca/users/b/.../73.../Lecture_5_Forecasting_f04_331.ppt

Time Series Methods Simple Moving Average

450

A moving average of order N is simply the arithmetic average of the most recent N

observations For 3 week moving averages N=3; 450 —

430 —

410 —

390 —

ent arrivals

observations. For 3-week moving averages N=3; for 6-week moving averages N=6; etc.

(6)

450 Patient

450 — 430 — 410 — 390 — ent arrivals Patient Week Arrivals 1 400 2 380 3 411

Given 3-week data, one-step-ahead forecast for week 4 or two step ahead forecast for 370 —

Pati

Week

| | | | | |

0 5 10 15 20 25 30

for week 4 or two-step-ahead forecast for week 5 is simply the arithmetic average of the first 3-week data

450 Patient

4 f k forecast ahead -step -One 450 — 430 — 410 — 390 — ent arrivals at e t Week Arrivals 1 400 2 380 3 411 = 4 F 4 for week 370 — Pati Week | | | | | | 0 5 10 15 20 25 30

(7)

450

Patient 450 — 430 — 410 — 390 — ent arrivals Patient Week Arrivals 1 400 2 380 3 411 5 f k forecast ahead -step -Two 370 — Pati | | | | | | 0 5 10 15 20 25 30 Week 5 for week = 5 F 450 Patient

One-step-ahead forecast for week 5 is computed from the arithmetic average of weeks 2, 3 and 4 data 450 — 430 — 410 — 390 — ent arrivals at e t Week Arrivals 2 380 3 411 4 415 5 f k forecast ahead -step -One data 370 — Pati Week | | | | | | 0 5 10 15 20 25 30 5 for week = 5 F

(8)

450 3 k MA

450 — 430 — 410 — 390 — ent arrivals 3-week MA forecast 370 — Pati Week | | | | | | 0 5 10 15 20 25 30 Actual patient arrivals 450 3 k MA 6-week MA

450 — 430 — 410 — 390 — ent arrivals 3-week MA forecast 6-week MA forecast Week 370 — Pati | | | | | | 0 5 10 15 20 25 30 Actual patient arrivals

(9)

Single Moving Average

y Ide: data pada suatu periode dipengaruhi oleh data

beberapa periode sebelumnya

y Cocok untuk pola data konstan/stasioner

y Prinsip dasar:

¾ Data smoothing pada periode ke‐t merupakan rata‐rata dari m buah data dari data periode ke‐t hingga ke‐(t‐

m+1) Î 1 t

t i

S =

∑

X

¾ Data smoothing pada periode ke‐t berperan sebagai nilai

forecasting pada periode ke‐t+1 F_t= S_t‐1dan F_n,h= S_n

1

t i

i t m m = − +

∑

(10)

Ilustrasi MA dengan m=3

Periode (t) Data (X_t) Smoothing (S_t) Forecasting (F_t)

1 5 - -2 7 - -2 7 - -3 6 6 -4 4 5.6 6 5 5 5 5.6 6 6 5 5 7 8 6.3 5 8 7 7 6 3 8 7 7 6.3 9 8 7.6 7 10 7 7.3 7.6 11 7.3 12 7.3

Pengaruh Pemilihan Nilai m

8.00 9.00 2.00 3.00 4.00 5.00 6.00 7.00 Semula MA (m=3) MA (m=6)

0.00 1.00

1 23 4 5 67 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

Waktu

MA dengan m yang lebih besar menghasilkan pola data yang lebih halus.

(11)

Example: Weekly Department Store Sales

• The weekly sales figures (in

millions of dollars)

d i

h f ll

i

Period (t) Sales (y)

1 5,3

2 4,4

3 5,4

4 5,8

5 5,6

presented in the following

table are used by a major

department store to

determine the need for

temporary sales personnel.

5 5,6 6 4,8 7 5,6 8 5,6 9 5,4 10 6,5 11 5,1 12 5,8 13 5 14 6,2 15 5,6 16 6,7 17 5,2 18 5,5 19 5,8 20 5,1 21 5,8 22 6,7 23 5,2 24 6 25 5,8

Example: Weekly Department Store Sales

Weekly Sales 8 2 3 4 5 6 7 8 Sa le s Sales (y) 0 1 2 0 5 10 15 20 25 30 Weeks

(12)

Example: Weekly Department Store Sales

• Use a three-week moving average (k=3) for

th d

t

t t

l

t f

th

the department store sales to forecast for the

week 24 and 26.

• The forecast error is

9 . 5 3 8 . 5 7 . 6 2 . 5 3 ) ( ˆ 23 22 21 24 = + + = + + = y y y y 1 . 9 . 5 6 ˆ₂₄ 24 24=y −y = − = e

Example: Weekly Department Store Sales

• The forecast for the week 26 is

7 . 5 3 2 . 5 6 8 . 5 3 ˆ 25 24 23 26 = + + = + + =y y y y

(13)

Latihan: Weekly Department Store Sales

Period (t) Sales (y) forecast 1 5.3 • RMSE = 0.63 2 4.4 3 5.4 4 5.8 5.033333 5 5.6 5.2 6 4.8 5.6 7 5.6 5.4 8 5.6 5.333333 9 5.4 5.333333 10 6.5 5.533333 11 5.1 5.833333 12 5.8 5.666667 13 5 5.8 14 6.2 5.3 15 5.6 5.666667 16 6.7 5.6 RMSE 0.63

Weekly Sales Forecasts

3 4 5 6 7 8 Sal e s Sales (y) forecast 17 5.2 6.166667 18 5.5 5.833333 19 5.8 5.8 20 5.1 5.5 21 5.8 5.466667 22 6.7 5.566667 23 5.2 5.866667 24 6 5.9 25 5.8 5.966667 5.666667 0 1 2 3 0 5 10 15 20 25 30 Weeks faculty.wiu.edu/F-Dehkordi/DS-533/.../Moving-average-methods.ppt

Double Moving Average

• Mirip dengan single moving average

• Cocok untuk data yang berpola tren

• Proses penghalusan dengan rata‐rata

dilakukan dua kali

– Tahap I: 1 t S =

∑

X p – Tahap II: 1, 1 t i i t m S X m = − + =

∑

2, 1, 1 1 t t i i t m S S m= − + =

∑

(14)

Double Moving Average

(lanjutan)

• Forecasting dilakukan dengan formula

dengan

2, ,t t h t t( ) F ₊ = A +B h 1, 2,

2

t t t

A

=

S

−

S

(

1, 2,

)

2 1 t t t B S S m = − −

Ilustrasi DMA dengan m=3

t Xt S1,t S2,t At Bt F2,t 1 12.50 2 11.80 3 12.85 12.38 4 13.95 12.87 5 13.30 13.37 12.87 13.87 0.50 6 13.95 13.73 13.32 14.14 0.41 14.37 7 15.00 14.08 13.73 14.43 0.35 14.55 8 16.20 15.05 14.29 15.81 0.76 14.78 8 16.20 15.05 14.29 15.81 0.76 14.78 9 16.10 15.77 14.97 16.57 0.80 16.57 10 17.37 11 18.17 12 18.97

(15)

Pemilihan Model

(lanjutan)

8 9 3 4 5 6 7 8 Semula MA(m=3) MA(m=6) SES(0.3) SES(0.4)

0 1 2 3 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 Waktu