FORECASTING DR. MOHAMMAD ABDUL MUKHYI, SE., MM
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Apa Arti Runtut Waktu? • Data runtut waktu (time series) merupakan data yang dikumpulkan, p , dicatat,, atau diobservasi sepanjang p j g waktu secara berurutan • Periode waktu dapat tahun, kuartal, bulan, minggu, dan dibeberapa kasus hari atau jam. jam • Runtut waktu dianalisis untuk menemukan pola yang g dapat p dipergunakan p g untuk: variasi masa lalu y (1) memprakirakan nilai masa depan dan membantu dalam manajemen operasi bisnis; (2) membuat perencanaan bahan baku, fasilitas produksi, dan jumlah staf guna memenuhi permintaan pe taa d dimasa asa mendatang. e data g 6/3/2008
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Mengapa g p Mempelajari p j Analisis Runtut Waktu? Karena dengan K d mengamati ti data d t runtut t t waktu kt akan k terlihat empat komponen yang mempengaruhi suatu pola data masa lalu dan sekarang, yang cenderung berulang dimasa mendatang
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Teknik Forecasting Pendekatan Basis
Teknik
Hasil
Peramalan ekstrapolatif
Ekstrapolasi trend
Analisis rangkaian-waktu Teknik benang-hitam Teknik OLS Pembobotan eksponensial Transformasi data Metode katastrofi
Projeksi
Peramalan Teoretis
Teori
Pemetaan teori Analisis jalur Analisis Input Input-Output Output Pemrograman linier Analisis regresi Estimasi interval Analisis hubungan
Prediksi
Peramalan intuitif
Penilaian subjektif
Delphi konvensional Delphi kebijakan Analisis dampak-silang Penilaian kelayakan
Konjektur
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Asumsi Peramalan Ekstrapolatif 1. Keajegan (persistence): Pola yang terjadi di masa lalu akan tetap terjadi di masa mendatang. Mis: jika konsumsi energi di masa lalu meningkat, ia akan selalu meningkat di masa depan. 2 Keteraturan (regularity): Variasi di masa lalu 2. akan secara teratur muncul di masa depan. Mis: jika banjir besar di Jakarta terjadi setiap 16 tah tahun n sekali, sekali pola yg g sama akan terjadi lagi lagi. 3. Keandalan (reliability) dan kesahihan (validity) data: Ketepatan p ramalan tergantung g g kepada p keandalan dan kesahihan data yg tersedia. Mis: data ttg laporan kejahatan seringkali tidak sesuai dg insiden kejahatan yg sesungguhnya, data ttg gaji bukan merupakan ukuran tepat 6/3/2008 5 dari pendapatan masyarakat.
Klasifikasi Metode Peramalan … Forecasting Method Subjective (Judgmental) Forecasting Methods
Objective Forecasting Methods Time Series M th d Methods
Causal Methods M th d
Analogies
Naïve Methods
Simple Regression
Moving Averages
Multiple Regression
Exponential Smoothing
Neural Networks
Delphi PERT
Simple Regression
Survey techniques
ARIMA Neural Networks
References : Combination of Time Series – Causal Methods
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Intervention Model Transfer Function (ARIMAX) VARIMA (VARIMAX) Neural Networks
Makridakis et al. Hanke and Reitsch Wei, W.W.S. B Box, JJenkins ki and d Reinsel R i l
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Ilustrasi M d l Matematis Model M i …
Klasifikasi Metode Peramalan :
Forecasting Method
Objective Subjective (Judgmental) Examples : sales l = f (price ( i , advert d t , …)) Forecasting Methods Forecasting Methods Examples : sales l (t) = f (sales ( l (t-1), sales l (t-2), …))
(t)
(t)
(t)
Combination of Time Series – Causal Methods Yt= f (Yt-j , j>0 ; Xt-i , i 0)
Examples : Time Series Methods Causal Methods , advert , advert , …) ¨ sales = f (sales (t)
Yt= f (Yt-1, Yt-2, … , Yt-k) 6/3/2008
(t-1)
(t)
(t-1)
Yt= f (X1t, X2t, … , Xkt) 7
Klasifikasi Model Time Series : Berdasarkan B MODELS Bentuk t k atau t Fungsi F i… TIME SERIES LINEAR Time Series Models
NONLINEAR Time Series Models
ARIMA Box-Jenkins
Models from time series theory nonlinear autoregressive, etc ...
Intervention Model
Flexible statistical parametric models neural network model, etc ...
Transfer Function (ARIMAX)
State-dependent, time-varying parameter and long-memory models
VARIMA (VARIMAX)
References : Timo Terasvirta, Dag Tjostheim and Clive W.J. Granger, (1994) “Aspects of Modelling Nonlinear Time Series”
6/3/2008 Handbook of Econometrics, Volume IV, Chapter 48. Edited by R.F. Engle and D.I. McFadden
Nonparametric models
Models from economic theory
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MODEL O PERAMALAN RUNTUT U U WAKTU U DENGAN ATAU TANPA TREN RUNTUT U U WKATU U EXPONENTIAL SMOOTING MENGANDUNG UNSUR TREND
TIDAK
MOVING AVERAGE
YA
TREND LINEAR
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TREND KUADRATIK
TREND EKSPONENSIAL
MODEL AUTOREGRESIF
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Empat komponen yang ditemukan dalam analisis runtut waktu adalah: 1. Trend, yaitu komponen jangka panjang yang mendasari p pertumbuhan ((atau p penurunan)) suatu data runtut waktu. 2. Siklikal (cyclical), yaitu suatu pola fluktuasi atau siklus dari data runtut waktu akibat perubahan kondisi ekonomi. 3. Musiman (seasonal), yaitu fluktuasi musiman yang sering i dijumpai dij i pada d data d t kuartalan, k t l b l bulanan atau t mingguan. 4. Tak beraturan (irregular), yaitu pola acak yang disebabkan oleh peristiwa yang tidak dapat diprediksi atau tidak beraturan, seperti perang, pemogokan, pemilu,, atau longsor p g maupun p bencana alam lainnya y 6/3/2008
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General of Time Series Patterns … Time Series S Patterns
Stationer
9 Nonseasonal Stationaryy models
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Trend Effect
Seasonal Effect
9 Nonseasonal Nonstationaryy models
9 Seasonal and p models Multiplicative
Cyclic Effect
9 Intervention models
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Time Series Analysis Deret berkala adalah suatu pengamatan atas suatu kumpulan variabel kuantitatif dari waktu ke waktu. waktu Contoh • angka indeks rata-rata industri Dow Jones • data historis penjualan, penjualan persediaan, persediaan jumlah pelanggan, pelanggan tingkat bunga, biaya-biaya, dan lain-lain Dunia Bisnis sangat tertarik akan peramalan dengan mengunakan variabel berkala Sering, bahwa variabel independen adalah tidak tersedia untuk membangun model regresi dari variabel deret berkala Dalam analisis deret berkala, kita meneliti perilaku dari suatu variabel masa lalu dalam rangka meramalkan perilakunya.masa depan 6/3/2008 17
Pola data General Time Series “PATTERN” Stationer St ti Trend (linear or nonlinear) Seasonal (additive or multiplicative) Cyclic Calendar Variation
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Pendekatan Analisis Deret Berkala Ada banyak teknik deret berkala. I i biasanya Ini bi mungkin ki untuk t k mengetahui t h i teknik t k ik mana yang terbaik untuk data tertentu. Biasanya mencoba beberapa teknik berbeda dan memilih salah satu terbaik. Untuk menjadi suatu model deret berkala yang efektif, ini harus menyediakan beberapa teknikderet berkala di dalam “ tool box.”
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Measuring Accuracy • We need a way to compare different time series techniques for a given data set. • Four common techniques are the:
∑
$ Yi − Y i n
n
Yi − Yˆi
n
– mean absolute deviation,
MAD =
i =1 =1
100 – mean absolute percent error, MAPE = ∑ n i =1
Yi
2 $ Yi − Yi ) ( MSE = ∑ n i =1 n
– the mean square error, – root mean square error. 6/3/2008
RMSE =
MSE
We will focus on the MSE.
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Extrapolation Models • Extrapolation models try to account for the past behavior of a time series variable in an effort to predict di t th the ffuture t behavior b h i off th the variable. i bl
$ = f (Y , Y , Y ,K) Y t +1 t t −1 t −2 We’ll ’ll first f talk lk about b severall extrapolation l techniques that are appropriate for stationary data.
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An Example • Hasil produksi padi Indonesia dari tahun 1970 sampai tahun 2008 sampai bulan Mei. • Hasil produksi ini berdasarkan musiman • Ada 39 tahun
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tahun 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 6/3/2008
produksi padi 18 693 649 18.693.649 20.483.687 19.393.933 21.490.578 22.476.073 22.339.455 23.300.939 23.347.132 25.771.570 26.282.663 29.651.905 32 774 176 32.774.176 33.583.677 35.303.106 38.136.446 39 032 945 39.032.945 39.726.761 40.078.195 41.676.170 44.725.582
tahun 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008
produksi padi 45 178 751 45.178.751 44.688.247 48.240.009 48.181.087 46.641.524 49.744.140 51.101.506 49.377.054 49.236.692 50.866.387 51.898.852 50 460 782 50.460.782 51.489.694 52.137.604 54.088.468 54 151 097 54.151.097 54.454.937 57.051.679 58.268.796 23
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Moving Averages Y + Y + Y t t-1 t- k +1 $ Yt +11 = k No general method exists for determining k. We must try out several k values to see what works best.
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Implementing the Model
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A Comment on Comparing MSE Values • Care should be taken when comparing MSE values of two different forecasting techniques. techniques • The lowest MSE may result from a technique that fits older values very well but fits recent values poorly. • It is sometimes wise to compute the MSE using only the most recent values.
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Forecasting With The Moving Average Model Forecasts for time periods 25 and 26 at time period 24:
Y24 + Y23 36 + 35 $ Y25 = = = 355 . 2 2 $ +Y Y 35 5 + 36 35.5 25 24 $ Y26 = = = 35.75 2 2
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Weighted Moving Average • The moving average technique assigns equal weight to all previous observations 1 1 1 $ Yt +1 = Yt + Yt-1 +L+ Yt- k -1 k k k The weighted moving average technique allows for different weights to be assigned to previous observations. b ti $ = w Y + w Y +L+ w Y Y t +1 1 t 2 t-1 k t- k -1 where h 0 ≤ wi ≤ 1 and d
∑w
i
=1
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Forecasting With The Weighted Moving Average Model Forecasts for time p periods 25 5 and 26 6 at time p period 24:
Yˆ25 = w1Y24 + w2 Y23 = 0.291× 36 + 0.709 × 35 = 35.29 Yˆ26 = w1Yˆ25 + w2 Y24 = 0.291× 35.29 + 0.709 × 36 = 35.79
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Exponential Smoothing $ =Y $ + α (Y − Y $ ) Y t +1 t t t where 0 ≤ α ≤ 1 It can be shown that the above equation is equivalent to:
$ = αY + α (1 − α ) Y + α (1 − α ) 2 Y +L+ α (1 − α ) n Y +L Y t +1 t t −1 t −2 t −n
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Examples of Two Exponential Smoothing Functions 42 40
Un nits Sold
38 36 34 32 Number of VCRs Sold Exp. Smoothing alpha=0.1
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Exp. Smoothing alpha=0.9
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Time Period 6/3/2008
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Implementing the Model
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Forecasting With The Exponential Smoothing Model Forecasts for time p periods 25 and 26 at time p period 24: ˆ =Y ˆ + α (Y − Y ˆ ) = 35.74 + 0.268(36 − 35.74) = 35.81 Y 25 24 24 24 ˆ =Y ˆ + α (Y − Y ˆ )≈Y ˆ + α (Y ˆ −Y ˆ )=Y ˆ = 35.81 Y 26 25 25 25 25 25 25 25
Note that,
$ = 35.81, for t = 25, 26, 27, K Y t
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Seasonality • Seasonality is a regular, repeating pattern in time series data. • May y be additive or multiplicative p in nature...
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Stationary Seasonal Effects Additive S easonal Effects
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Tim e Pe r iod
Multiplicative S easonal E ffects
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T im e Pe r io d
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Stationaryy Data With Additive Seasonal Effects where
ˆ Y t + n = E t + St + n − p
E t = α (Yt - St-p ) + (1- α )Et −1
St = β (Yt - E t ) + (1- β )St − p 0 ≤α ≤1 0 ≤ β ≤1 p represents the number of seasonal periods
• Et is the expected level at time period t. • St iis the th seasonall factor f t for f time ti period i d t. t 6/3/2008
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Implementing the Model
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Forecasting With The Additive Seasonal Effects Model Forecasts for time p periods 25 to 28 at time p period 24:
ˆ Y 24+ n = E 24 + S24+ n − 4 ˆ = E + S = 354.44 + 8.45 = 363.00 Y 25 24 21 ˆ = E + S = 354.44 − 17.82 = 336.73 Y 26
24
22
ˆ = E + S = 354.44 + 46.58 = 401.13 Y 27 24 23 ˆ = E + S = 354.44 − 31.73 = 322.81 Y 28 24 24 6/3/2008
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Stationaryy Data With Multiplicative Seasonal Effects where
ˆ Y t + n = E t × St + n − p
E t = α (Yt /Stt-pp ) + ((1- α ))Et −1
St = β (Yt /E t ) + (1- β )St − p 0 ≤α ≤1 0 ≤ β ≤1 p represents the number of seasonal periods
• Et is the expected level at time period t. • St is the seasonal factor for time period t. 6/3/2008
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Forecasting With The Multiplicative Seasonal Effects Model Forecasts for time periods 25 to 28 at time period 24:
ˆ Y 24+ n = E 24 × S 24+ n − 4 ˆ = E × S = 353.95 ×1.015 = 359.13 Y 25 24 21 ˆ = E × S = 354.44 × 0.946 = 334.94 Y 26 24 22 ˆ = E × S = 354.44 ×1.133 = 400.99 Y 27 24 23 ˆ = E × S = 354.44 × 0.912 = 322.95 Y 28 24 24 6/3/2008
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Trend Models • Trend is the long-term sweep or general direction of movement in a time series. • We’ll now consider some nonstationary time series techniques that are appropriate for data exhibiting upward or downward trends.
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An Example p • WaterCraft Inc. is a manufacturer of personal water crafts (also known as jet skis) skis). • The company has enjoyed a fairly steady growth in sales of its products products. • The officers of the company are preparing sales and a d manufacturing a u actu g p plans a s for o tthe e co coming g yea year. • Forecasts are needed of the level of sales that the company p y expects p to achieve each q quarter. • See file
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Double Moving g Average g ˆ Y t + n = E t + nTt where
E t = 2M t − D t Tt = 2(M t − D t ) /(k − 1) M t = (Yt + Yt −1 + L + Yt − k +1) / k D t = (Mt + Mt −1 + L + Mt − k +1) / k
• Et is the expected base level at time period t. • Tt is the expected trend at time period t. 6/3/2008
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Forecasting With The Double Moving Average Model Forecasts for time p periods 21 to 24 at time p period 20:
ˆ Y 20+ n = E 20 + nT20 ˆ = E + 1T = 2385.33 + 1×139.9 = 2525.23 Y 21 20 20 ˆ = E + 2T = 2385.33 + 2 × 139.9 = 2665.13 Y 22 20 20 ˆ = E + 3T = 2385.33 + 3 ×139.9 = 2805.03 Y 23 20 20 ˆ = E + 4T = 2385.33 + 4 × 139.9 = 2944.94 Y 24 20 20 6/3/2008
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Double Exponential Smoothing (Holt’s (Holt s Method) ˆ Y Tt t + n = E t + nT where Et = αYt + (1-α)(Et-1+ Tt-1) Tt = β(Et −Et-1) + (1-β) Tt-1 0 ≤ α ≤ 1 and 0 ≤ β ≤ 1
• Et is the expected base level at time period t. t • Tt is the expected trend at time period t. 6/3/2008
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Forecasting g With Holt’s Model Forecasts for time periods 21 to 24 at time period 20:
ˆ Y 20+ n = E 20 + nT20
$ = E + 1T = 2336.8 + 1 × 152.1 = 2488.9 Y 21 20 20 $ = E + 2T = 2336.8 + 2 × 152.1 = 2641.0 Y 22 20 20 $ = E + 3T = 2336.8 + 3 × 152.1 = 27931 Y . 23
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$ = E + 4T = 2336.8 + 4 × 152.1 = 2945.2 Y 24 20 20
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Holt-Winter’s Method For Additive Seasonal Effects Yˆt + n = E t + nTt + St + n − p where
(
)
E t = α Yt − St − p + (1- α )(Et −1 + Tt −1 )
Tt = β (E t − E t −1 ) + (1 - β )Tt −1 St = γ (Yt − E t ) + (1- γ )St − p 0 ≤α ≤1 0 ≤ β ≤1 0 ≤ γ ≤1
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Forecasting With Holt-Winter’s Additive Seasonal Effects Method Forecasts for time periods 21 to 24 at time period 20:
Yˆ20+ n = E 20 + nT20 + S20+ n −4 ˆ = E + 1× T + S = 2253.3 + 1× 154.3 + 262.66 = 2670.3 Y 21 20 20 17 ˆ = E + 2 × T + S = 2253.3 + 2 ×154.3 − 312.59 = 2249.3 Y 22 20 20 18 ˆ = E + 3 × T + S = 2253.3 + 3 ×154.3 + 205.40 = 2921.6 Y 23 20 20 19 ˆ = E + 4 × T + S = 2253.3 + 4 × 154.3 + 386.12 = 3256.6 Y 24 20 20 20
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Holt-Winter’s Method For M ltiplicati e Seasonal Effects Multiplicative Yˆt + n = (E t + nTt )St + n − p where
(
)
E t = α Yt / St − p + (1- α )(Et −1 + Tt −1 )
Tt = β (E t − E t −1 ) + ((1 - β ))Tt −1 St = γ (Yt / E t ) + (1- γ )St − p 0 ≤α ≤1 0 ≤ β ≤1 0 ≤ γ ≤1 6/3/2008
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Forecasting With Holt-Winter’s Multiplicative Seasonal Effects Method Forecasts for time p periods 21 to 24 at time period p 20:
Yˆ20+ n = (E 20 + nT20 ) S20+ n −4 ˆ = (E + 1T ) S = (2217.6 + 1×137.3)1.152 = 2713.7 Y 21 20 20 17 ˆ = (E + 2T ) S = (2217.6 + 2 ×137.3)0.849 = 2114.9 Y 22 20 20 18 ˆ = (E + 3T ) S = (2217.6 + 3 ×137.3)1.103 = 2900.5 Y 23 20 20 19 ˆ = (E + 4T ) S = (2217.6 + 4 ×137.3)1.190 = 3293.9 Y 24 20 20 20 6/3/2008
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The Linear Trend Model
Y$ t = b0 + b1X1t where X1t = t For example: X11 = 1, X12 = 2, X13 = 3, K
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Forecasting With Th Li The Linear T Trend dM Model d l Forecasts for time periods 21 to 24 at time period 20: $ = b + b X = 3751 Y . + 92.6255 × 21 = 2320.3 21 0 1 121 $ = b + b X = 3751 Y . + 92.6255 × 22 = 2412.9 22 0 1 122 $ = b + b X = 3751 Y . + 92.6255 × 23 = 2505.6 23 0 1 123 $ = b + b X = 3751 Y . + 92.6255 × 24 = 2598.2 24 0 1 124
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The TREND() Function TREND(Y-range, X-range, X-value for prediction) where: Y range is the spreadsheet range containing the dependent Y-range Y variable, X-range is the spreadsheet range containing the independent X variable(s), X-value for prediction is a cell (or cells) containing the values for the independent p X variable(s) ( ) for which we want an estimated value of Y. Note: The TREND( ) function is dynamically updated whenever any inputs to the function change. However, it does not provide the statistical information provided by the regression tool. It is best two use these two different approaches to doing regression in conjunction with one another. 6/3/2008
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The Quadratic Trend Model
$ = b +b X +b X Y t 0 1 1t 2 2t where X1t = t and X 2t = t 2
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Forecasting With The Quadratic Trend Model Forecasts for time periods 21 to 24 at time period 20: 2 ˆ = b +b X +b X Y 21 0 1 12 1 2 2 2 1 = 653 .67 + 16 .671 × 21 + 3.617 × 21 = 2598 .9 2 ˆ = b +b X +b X Y = 653 . 67 + 16 . 671 × 22 + 3 . 617 × 22 = 2771 .1 22 0 1 12 2 2 22 2 2 ˆ = b +b X +b X Y = 653 . 67 + 16 . 671 × 23 + 3 . 617 × 23 = 2950 .4 23 0 1 12 3 2 22 3 2 ˆ = b +b X +b X Y = 653 . 67 + 16 . 671 × 24 + 3 . 617 × 24 = 3137 .1 24 0 1 12 4 2 22 4
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Computing Multiplicative Seasonal Indices • We can compute multiplicative seasonal adjustment indices for period p as follows:
Yi ∑i Y$ i , for all i occuring Sp = g in season p np The final forecast for period i is then
$ adjusted = Y $ × S , for any i occuring in season p Y i i p 6/3/2008
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Forecasting With Seasonal Factors Applied To The Quadratic Trend Model Forecasts for time periods 21 to 24 at time period 20: ˆ = (b + b X + b X ) S = 2598 .9 × 105 .7% = 2747 .8 Y 21 0 1 12 1 2 22 1 1 ˆ = (b + b X + b X ) S = 2771 .1 × 80 .1% = 2219 .6 Y 22 0 1 12 2 2 22 2 2 ˆ = (b + b X + b X ) S = 2950 .5 × 103 .1% = 3041 .4 Y 23 0 1 12 3 2 22 3 3
ˆ = (b + b X + b X ) S = 3137 .2 × 111 .1% = 3486 .1 Y 24 0 1 12 4 2 22 4 4
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Summary of the Calculation and Use of Seasonal Indices 1. Create a trend model and calculate the estimated ˆ ) for each observation in the sample. value ( Y t 2. For each observation, calculate the ratio of the actual ˆ . value to the predicted trend value: Yt / Y t ˆ ). (For additive effects, compute the difference: Y − Y t
t
3. For each season, compute the average of the ratios calculated in step 2. These are the seasonal indices. 4. Multiply any forecast produced by the trend model by the appropriate seasonal index calculated in step 3. (For additive seasonal effects, add the appropriate factor to the forecast forecast.)) 6/3/2008
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Refining e g tthe e Seaso Seasonal a Indices d ces • Note that Solver can be used to simultaneously determine the optimal values of the seasonal indices and the parameters of the trend model being used. • There is no guarantee that this will produce a better forecast,, but it should produce p a model that fits the data better in terms of the MSE. See file Fig11-39.xls
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Seasonal Regression Models • Indicator variables may also be used in regression models to represent seasonal effects. • If there are p seasons, seasons we need p -1 1 indicator variables. Our example problem involves quarterly data, data so p=4 and we define the following 3 indicator variables: ⎧1, if Yt is an observation from quarter 1 X 3t = ⎨ ⎩0, otherwise ⎧1, if Yt is an observation from quarter 2 X 4t = ⎨ ⎩0, otherwise ⎧1, if Yt is an observation from quarter 3 X 5t = ⎨ th i ⎩0, otherwise 6/3/2008
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Implementing the Model • The regression function is: $ = b +b X +b X +b X +b X +b X Y t 0 1 1t 2 2t 3 3t 4 4t 5 5t where X1t = t and X 2t = t 2
See file Fig11-42.xls
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Forecasting With The S Seasonal lR Regression i M Model d l Forecasts for time periods 21 to 24 at time period 20: Yˆ21 = 824.471+ 17.319(21) + 3.485(21) 2 − 86.805(1) − 424.736(0) − 123.453(0) = 2638.5 Yˆ 22 = 824.471+ 17.319(22) + 3.485(22) 2 − 86.805(0) − 424.736(1) − 123.453(0) = 2467.7 Yˆ23 = 824.471+ 17.319(23) + 3.485(23) 2 − 86.805(0) − 424.736(0) − 123.453(1) = 2943.2 Yˆ24 = 824.471+ 17.319(24) + 3.485(24) 2 − 86.805(0) − 424.736(0) − 123.453(0) = 3247.8
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StatTools • StatTools is an add-in add in that simplifies the process of performing time series analysis in Excel. • A trial version of StatTools is available on p y g this book. the CD-ROM accompanying • For more information on StatTools see: http://www.palisade.com
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Combining Forecasts • It is also possible to combine forecasts to create a composite forecast. • Suppose we used three different forecasting methods on a given data set. Denote the predicted value of time period t using each method as follows:
F1t , F2t , and F3t We could create a composite forecast as follows:
$ = b +b F +b F +b F Y t 0 1 1t 2 2t 3 3t 6/3/2008
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ARIMA Model: Produksi padi ARIMA model for Produksi padi Estimates at each iteration Iteration SSE Parameters 0 451374984309793 0,100 0,100 0,100 0,100 1 368364447306473 0,021 0,199 0,250 0,152 2 325456920491419 -0,129 0,154 0,348 0,186 3 285571989081979 -0,279 0,118 0,475 0,229 4 245711224167492 -0,324 0,193 0,542 0,187 5 187899115792845 -0,474 0,260 0,657 0,041 6 155112700780088 -0,527 0,353 0,704 -0,109 7 141714161123773 -0,447 0 447 0 0,503 503 0 0,647 647 -0,201 0 201 8 129630495049884 -0,368 0,653 0,585 -0,287 9 117486920579732 -0,301 0,803 0,523 -0,376 10 101549593049237 -0,291 0,953 0,498 -0,503 11 87502739697009 -0,393 1,051 0,567 -0,653 , 1,076 , 0,614 , -0,708 , 12 81737678770148 -0,543 13 79314052359269 -0,619 1,074 0,596 -0,687 14 79233477547537 -0,622 1,073 0,587 -0,677 15 79162421478230 -0,620 1,072 0,581 -0,671 16 79124151604006 -0,619 1,071 0,577 -0,667 17 79104368397552 -0,619 1,071 0,576 -0,665 18 79092164140468 -0,619 1,070 0,575 -0,664 19 79085747120700 -0,620 1,070 0,575 -0,664 Relative change in each estimate less than 0,0010
6/3/2008
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Final Estimates of Parameters Type Coef SE Coef T AR 1 -0,6197 0,1611 -3,85 MA 1 1,0698 0,0298 35,85 MA 2 0,5753 0,1894 3,04 MA 3 -0,6641 0,6641 0,1882 -3,53 3,53
P 0,001 0,000 0,005 0,001
Differencing: 3 regular differences Number of observations: Original series 39, after differencing 36 Residuals: es dua s SS = 70758139198041 0 58 39 980 (bac (backforecasts o ecasts e excluded) c uded) MS = 2211191849939 DF = 32 Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag g 12 24 36 48 Chi-Square 20,9 38,4 * * DF 8 20 * * P-Value 0,008 0,008 * *
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Model-model Time Series Regression g
1. Model Regresi untuk LINEAR TREND Yt = a + b.t + error
Ö t = 1, 2, … (dummy waktu)
2. Model Regresi untuk Data SEASONAL (variasi konstan) Yt = a + b1 D1 + … + bS-1 DS-1 + error dengan : D1, D2, …, DS-1 S 1 adalah dummy waktu dalam satu periode seasonal.
3. Model Regresi untuk Data dengan LINEAR TREND dan SEASONAL (variasi konstan) Yt = a + b.t + c1 D1 + … + cS-1 DS-1 + error  Gabungan model 1 dan 2. 6/3/2008
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Naïve Model
³ The recent periods are the best predictors of the future. 1. The simplest model for stationary data is
Yˆt +1 = Yt 2. The simplest model for trend data is
Yˆt +1 = Yt + (Yt − Yt −1 ) or Yt ˆ Yt +1 = Yt Yt −1 3. The simplest model for seasonal data is
Yˆt +1 = Y(t +1) − s 6/3/2008
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Average Methods
1. Simple Averages ³ obtained by finding the mean for all the relevant values and then using g this mean to forecast the next period. p n
Yt ˆ Yt +1 = ∑ t =1 n
for stationary data
2. Moving Averages ³ obtained by finding the mean for a specified set of values and then using this mean to forecast the next period period.
(Y + Yt −1 + K + Yt − n +1 ) M t = Yˆt +1 = t n 6/3/2008
for stationary data
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Average Methods …
(continued)
3. Double Moving Averages ³ one set of moving averages is computed, and then a second set is computed as a moving average of the first set.
(Y + Yt −1 + K + Yt − n +1 ) M t = Yˆt +1 = t n ( M t + M t −1 + K + M t − n +1 ) (ii) M t′ = (ii). n (i).
(iii). at = 2M t − M t′ (iv). bt =
2 ( M t − M t′ ) n −1
Yˆt + p = at + bt p 6/3/2008
f for a linear trend data li t dd t 72
Exponential Smoothing Methods 9 Single Exponential Smoothing
Ö for stationary data
Yˆt +1 = αYt + (1 − α )Yˆt 9 Exponential Smoothing Adjusted for Trend : Holt’s Method 1. The exponentially smoothed series : At = α Yt + (1−α) (At-1+ Tt-1) 2. The trend estimate : Tt = β (At − At-1) + (1 − β) Tt-1 3. Forecast p periods into the future :
Yˆt + p = At + pTt 6/3/2008
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Exponential Smoothing Adjusted for Trend and Seasonal Variation : Winter’s Method
1. The exponentially smoothed series : Y At = α t + (1 − α ) ( At −1 + Tt −1) St − L 2. The trend estimate : Tt = β ( At − At −1) + (1 − β )Tt −1
3. The seasonality estimate :
Three p parameters models
Y St = γ t + (1 − γ ) St −1 At 4. Forecast p periods into the future : Yˆt + p = ( At − pTt ) St − L + p 6/3/2008
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