Smoothing

Basic Smoothing Models Moving average, weighted moving average, exponential smoothing Single and Double Smoothing First order exponential smoothing with trends and/or seasonal patterns

Simple M-Period Moving Average adalah rata-rata dari M titik data terakhir Pada dasarnya mengasumsikan data merupakan series stabil (tidak ada trend)

Moving Average hari Nilai tukar USD/IDR HariNilai Tukar F 4 = ( ) / 3 = F 4 = F 5 = ( ) / 3 = HariNilai Tukar F 4 = ( ) / 3 = F 4 = F 5 = ( ) / 3 =

Moving Average hari Nilai tukar USD/IDR Nilai aktual Moving average dengan rentang 3 hari Moving average dengan rentang 6 hari

Weighted Moving Averages W i adalah bobot dari setiap titik data historis. Pada prinsipnya, semua teknik forecasting data time series univariate merupakan weighted moving averages

Weighted Moving Average hari Nilai tukar USD/IDR Bobot yang ditentukan t0.7 t – 10.2 t – 20.1 F 4 = 0.7* * *10319 F 4 = F 5 = 0.7* * *10325 F 5 = Bobot yang ditentukan t0.7 t – 10.2 t – 20.1 F 4 = 0.7* * *10319 F 4 = F 5 = 0.7* * *10325 F 5 =

Simple Exponential Smoothing F t+1 (t) = Forecast for time t+1 made at time t A t = Actual outcome at time t 0<  <1 is the “smoothing parameter” F t+1 (t) = F t (t-1) +  [A t – F t (t-1)] –Adjust forecast based on last forecast error OR F t+1 (t) = (1-  )F t (t-1) +  A t –Weighted average of last forecast and last Actual

Exponential Smoothing Include all past observations Weight recent observations much more heavily than very old observations Bobot Sekarang Decreasing weight given to older observations   (1-  )  (1-  ) 2

Simple Exponential Smoothing Is appropriate when the underlying time series behaves like a constant + Noise –X t =  + N t –That is, for a quite stable process Not appropriate when trends or seasonality present

Simple Exponential Smoothing hari Nilai tukar USD/IDR F t+1 (t) = (1-  )F t (t-1) +  A t Initial F 3 = ( )/2 = A3 = F 4 = 0.3 (10319) (10287) = D 4 = F 5 = 0.3 (10325) ( ) = F t+1 (t) = (1-  )F t (t-1) +  A t Initial F 3 = ( )/2 = A3 = F 4 = 0.3 (10319) (10287) = D 4 = F 5 = 0.3 (10325) ( ) =

Simple Exponential Smoothing Thus, new forecast is weighted sum of old forecast and actual data Notes: –Only 2 values (A t and F t ) are required, comparedwith n for moving average –Parameter α determined empirically (whatever works best) –Rule of thumb: α < 0.5 –Typically, α = 0.2 or α = 0.3 work well Large α value emphasizes recent data levels and a forecast responsive to trends Small α value takes past demands into account and provide a more stable forecast

Single Exponential Smoothing

Double Smoothing Terdiri atas dua tahap –Melakukan smoothing terhadap series asal –Melakukan smoothing terhadap series hasil smoothing Cocok dikunakan untuk series yang memiliki pola trend naik/turun Dapat dilakukan pada moving average maupun exponential smoothing

Smoothing untuk Data yang mengandung trend dan seasonal Holt-Winter Method Memasukkan term trend (fungsi linear data terhadap waktu) serta pengaruh data pada periode ke-(t-s) sebagai cara menangkap seasonal. Bentuk model komponen: aditif atau multiplikatif

Error Measures Errors: The difference between actual and predicted (one period earlier) e t = V t – P t (t-1) –e t =can be positive or negative Absolute error |e t | –Always positive Squared Error e t 2 –Always positive The percentage error PE t = 100e t / V t –Can be positive or negative

Bias and error magnitude Forecasts can be: –Consistently too high or too low (bias) –Right on average, but with large deviations both positive and negative (error magnitude) Should monitor both for changes

Error Measures Look at errors over time Cumulative measures summed or averaged over all data –Error Total (ET) –Mean Percentage Error (MPE) –Mean Absolute Percentage Error (MAPE) –Mean Squared Error (MSE) –Root Mean Squared Error (RMSE) –Mean Absolute Deviation (MAD) Measure Bias Measure error magnitude

Error Total Sum of all errors Uses raw (positive or negative) errors ET can be positive or negative Measures bias in the forecast Should stay close to zero as we saw in last presentation Average of percent errors Can be positive or negative Measures bias, should stay close to zero MPE

MSE and RMSE Average of squared errors Always positive Measures “magnitude” of errors Units are “data units squared” Square root of MSE Always positive Measures “magnitude” of errors Units are “data units” Standard deviation of forecast errors

MAPE and MAD Average of absolute percentage errors Always positive Measures magnitude of errors Units are “percentage” Smoothed absolute errors Always positive Measures magnitude of errors Looks at the recent past

Percentage or Actual units Often errors naturally increase as the level of the series increases Natural, thus no reason for alarm If true, percentage based measured preferred Actual units are more intuitive

Pelatihan Time Series Analysis Squared or Absolute Errors Absolute errors are more intuitive Standard deviation units less so –66% within  1 S.D. –95% within  2 S.D. When using measures for automatic model selection, there are statistical reasons for preferring measures based on squared errors

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