Metode Pemulusan Eksponensial Sederhana (Single Exponential Smoothing) KULIAH 3 METODE PERAMALAN DERET WAKTU rahmaanisa@apps.ipb.ac.id
Review Untuk apa metode pemulusan (smoothing) dilakukan terhadap data deret waktu? Kapan metode rataan bergerak sederhana digunakan? Kapan metode rataan bergerak ganda digunakan?
Review: The Process of Smoothing Data Set
Outline Konsep dasar pemulusan eksponensial Pemulusan eksponensial sederhana Peramalan melalui pemulusan eksponensial sederhana Contoh aplikasi pada data
Moving Average Vs Exponential Smoothing
Moving Average Vs Exponential Smoothing n Period Moving Average hanya menggunakan n data Exponential Smoothing: menggunakan semua data bobot yg lebih besar diberikan pada data yg lebih up to date
What do you think about the MA forecasts?
Introduction Hasil smoothing tidak sesuai dgn pola data MENGAPA? karena data tidak lagi konstan Perubahan hasil smoothing terlalu lambat Bagaimana solusinya?
Konsep Dasar Pemulusan Eksponensial
Exponential Smoothing It uses weighted averages of the past data The effect of recent observations is expected to decline exponentially over time The further back along the historical time path one travels, the less influence each observation has on the forecasts
Pemulusan Eksponensial Sederhana (SINGLE EXPONENTIAL SMOOTHING)
Single Exponential Smoothing give geometrically decreasing weights to the past observations. an exponentially weighted smoother is obtained by introducing a discount factor
Procedures of Single Exponential Smoothing Step 1: Compute the initial estimate of the mean (or level) of the series at time period t = 0 Step 2: Compute the updated estimate by using the smoothing equation l y (1 ) l T T T 1 where is a smoothing constant between 0 and 1. Slide 14
Procedures of Single Exponential Smoothing Note that l y (1 ) l T T T y (1 )[ y (1 ) l ] 1 T T 1 T 2 y (1 ) y (1 ) l 2 T T 1 T 2 y (1 ) y (1 ) y... (1 ) y (1 ) l 2 T 1 T T T 1 T 2 1 0 The coefficients measuring the contributions of the observations decrease exponentially over time. Slide 15
Single Exponential Smoothing l y (1 ) l T T T 1 This can also be seen as the linear combination of the current observation and the smoothed observation at the previous time unit.
Comparison of Weights: MA Vs SES
Intial Value 1. Set l 0 = y 1, if the changes in the process are expected to occur early and fast 2. Take a subset of the avalaible data. Set l 0 = y, if the process is at least at the beginning locally constant.
Ilustrasi Data Dow Jones 12000 11000 10000 9000 8000 7000 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 Actual SES(0.1) SES(0.9)
The Value of α α 0 maka hasil smoothing semakin smooth α 1 maka hasil smoothing semakin mendekati pola data aktual
The Value of α Perhatikan bahwa: α α α α α α α α α α Bagaimana pengaruh besarnya α terhadap Var y T?
The Value of α Thus the question will be how much smoothing is needed. In the literature, α values between 0.1 and 0.4 are often recommended and do indeed perform well in practice.
Peramalan
Single Exponential Smoothing Point forecast made at time T for y T+p yˆ ( T) l ( p 1,2,3,...) T p T Slide 24
Aplikasi pada Data
Example: Cod Catch The Bay City Seafood Company recorded the monthly cod catch for the previous two years, as follow: Cod Catch (In Tons) Month Year 1 Year 2 January 362 276 February 381 334 March 317 394 April 297 334 May 399 384 June 402 314 July 375 344 August 349 337 September 386 345 October 328 362 November 389 314 December 343 365 Slide 26
Time Series Plot 415 395 375 355 335 315 295 Cod Catch 275 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Period
Time Series Plot 415 395 375 355 335 315 295 Cod Catch 275 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Period
Single Exponential Smoothing Cod Period Catch 0 1 362 2 381 3 317 4 297 5 399 6 402 7 375 8 349 9 386 10 328 11 389 12 343 Lt Forecast Period Cod Catch 13 276 14 334 15 394 16 334 17 384 18 314 19 344 20 337 21 345 22 362 23 314 24 365 25 Lt Forecast
Single Exponential Smoothing Period Cod Catch Lt Forecast 0 362.0 1 362 362.0 362.0 2 381 369.6 362.0 3 317 348.6 369.6 4 297 327.9 348.6 5 399 356.4 327.9 6 402 374.6 356.4 7 375 374.8 374.6 8 349 364.5 374.8 9 386 373.1 364.5 10 328 355.0 373.1 11 389 368.6 355.0 12 343 358.4 368.6 Period Cod Catch Lt Forecast 13 276 325.4 358.4 14 334 328.9 325.4 15 394 354.9 328.9 16 334 346.5 354.9 17 384 361.5 346.5 18 314 342.5 361.5 19 344 343.1 342.5 20 337 340.7 343.1 21 345 342.4 340.7 22 362 350.2 342.4 23 314 335.7 350.2 24 365 347.4 335.7 25 347.4
The Forecasts 415.0 395.0 375.0 355.0 335.0 315.0 295.0 275.0 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 Aktual Forecast
Evaluasi
Latihan 1) Exercise 4.1 in Montgomery et al. (2015) 2) Exercise 4.2 in Montgomery et al. (2015)
Referensi Montgomery, D.C., Jennings, C.L., Kulahci, M. 2015. Introduction to Time Series Analysis and Forecasting, 2 nd ed. New Jersey: John Wiley & Sons. 34
Materi perkuliahan dapat diakses pada: stat.ipb.ac.id/en 35