DSc 3120 Generalized Modeling Techniques with Applications Part II. Forecasting

Forecasting Models 2 Forecasting ß Why Forecasting?  Characteristics of Forecasts k Forecasts are usually wrong or seldom correct k Aggregate forecasts are usually more accurate k Less accurate further into the future ß Assumptions of Forecasting Models k Information (data) about the past is available k The pattern of the past will continue into the future.

Forecasting Models 3 Qualitative Forecasting  Sales force composites (field sales force) ß Consumer market survey (users’ expectations) ß Jury of executive ß The Delphi method -- Forecasting based on experience, judgement, and knowledge

Forecasting Models 4 Quantitative Forecasting  Casual Models: Causal Model Year 2000 Sales Price Population Advertising ……  Time Series Models: Time Series Model Year 2000 Sales Sales 1999 Sales 1998 Sales 1997 …… -- Forecasting based on data and models

Forecasting Models 5 Overview of Forecasting Models

Forecasting Models 6 Causal Forecasting Models ß Curve Fitting: Simple Linear Regression k One Independent Variable (X) is used to predict one Dependent Variable (Y): Y = a + b X k Given n observations (X i, Y i ), we can fit a line to the overall pattern of these data points. The Least Squares Method in statistics can give us the best a and b in the sense of minimizing  (Y i - a - bX i ) 2 :

Forecasting Models 7 ß Curve Fitting: Simple Linear Regression k Find the regression line with Excel l Use Function: a = INTERCEPT(Y range; X range) b = SLOPE(Y range; X range) l Use Solver l Use Excel’s Tools | Data Analysis | Regression ß Curve Fitting: Multiple Regression k Two or more independent variables are used to predict the dependent variable: Y = b 0 + b 1 X 1 + b 2 X 2 + … + b p X p k Use Excel’s Tools | Data Analysis | Regression

Forecasting Models 8 ß BIAS - The arithmetic mean of the errors k n is the number of forecast errors k Excel: =AVERAGE(error range) ß Mean Absolute Deviation - MAD k No direct Excel function to calculate MAD Evaluation of Forecasting Model

Forecasting Models 9 ß Mean Square Error - MSE k Excel: =SUMSQ(error range)/COUNT(error range) ßMean Absolute Percentage Error - MAPE ßR 2 - only for curve fitting model such as regression ß In general, the lower the error measure (BIAS, MAD, MSE) or the higher the R 2, the better the forecasting model Evaluation of Forecasting Model

Forecasting Models 10 Time Series Model Building ßHistorical data collection ßData plotting (time series plot) ßForecasting model building ßEvaluation and selection of model ßForecasting with the final selected model

Forecasting Models 11 Components of A Time Series  Trend: long term overall up or down movement  Seasonality: periodic pattern repeating every year  Cycles: up & down movement repeating over long time frame ßRandom Variations: random movements follow no pattern

Forecasting Models 12 Components of A Time Series Time Trend Random movement Time Cycle Time Seasonal pattern Demand Time Trend with seasonal pattern

Forecasting Models 13 Types of Time Series Models ß Nonseasonal Model k Trend k Naïve k Moving average k Exponential ß Seasonal Model k Time Series Decomposition

Forecasting Models 14 Trend Model ß Curve fitting method used for time series data (also called time series regression model) ß Useful when the time series has a clear trend ß Can not capture seasonal patterns ß Linear Trend Model: Y t = a + bt k t is time index for each period, t = 1, 2, 3,…

Forecasting Models 15 Trend Model (Cont.) ßNonlinear Trend Models k Power: Y t = at b k Quadratic: Y t = a + bt + ct 2 Power Quadratic

Forecasting Models 16 Naïve Model ß The simplest time series forecasting model ß Idea: “what happened last time (last year, last month, yesterday) will happen again this time” ß Naïve Model: k Algebraic: F t = Y t-1 l Y t-1 : actual value in period t-1 l F t : forecast for period t k Spreadsheet: B3: = A2; Copy down

Forecasting Models 17 Moving Average Model ß Simple n-Period Moving Average ß Issues of MA Model k Naïve model is a special case of MA with n = 1 k Idea is to reduce random variation or smooth data k All previous n observation are treated equally (equal weights) k Suitable for relatively stable time series with no trend or seasonal pattern

Forecasting Models 18 Smoothing Effect of MA Model Longer-period moving averages (larger n) react to actual changes more slowly

Forecasting Models 19 Moving Average Model ß Weighted n-Period Moving Average k Typically weights are decreasing: w 1 >w 2 >…>w n k Sum of the weights =  w i = 1 k Flexible weights reflect relative importance of each previous observation in forecasting k Optimal weights can be found via Solver

Forecasting Models 20 Weighted MA: An Illustration Month Weight Data August 17%130 September 33%110 October 50%90 November forecast: F Nov = (0.50)(90)+(0.33)(110)+(0.17)(130) = 103.4

Forecasting Models 21 Simple Exponential Smoothing ßA special type of weighted moving average k Include all past observations k Use a unique set of weights that weight recent observations much more heavily than very old observations: weight Decreasing weights given to older observations Today

Forecasting Models 22 Simple ES: The Model New forecast = weighted sum of last period actual value and last period forecast   : Smoothing constant  F t :Forecast for period t k F t-1 :Last period forecast k Y t-1 :Last period actual value

Forecasting Models 23 Simple Exponential Smoothing  Properties of Simple Exponential Smoothing  Widely used and successful model  Requires very little data  Larger , more responsive forecast; Smaller , smoother forecast (See Table 13.2)  “best”  can be found by Solver  Suitable for relatively stable time series

Forecasting Models 24 Holt’s Model – Exponential Smoothing with Trend k F t Forecast for period t k L t Level term (intercept) k T t Trend term (slope) k Y t-1 Last period actual value   Smoothing constant for Level L k  Smoothing constant for Trend T

Forecasting Models 25 ß Basic Idea: a time series is composed of several basic components: Trend, Seasonality, Cycle, and Random Error ß The multiplicative decomposition model:  These components contribute to time series value in a m ultiplicative way Time Series Decomposition Model

Forecasting Models 26 Time Series Decomposition ß The basic model is: k Y = Trend  Cyclical  Seasonal  Error ß Since we cannot easily extract or predict cycles, we will assume that the trend component will capture cycles during the forecast period ß Since we have to live with error (cannot predict it), our model is simplified to: k Y = Trend  Seasonal

Forecasting Models 27 I. Estimate Seasonal Component - Seasonal Index ß Step 1: Calculate 1-Year Moving Averages k For quarterly data, use 4-period MA k For monthly data, use 12-period MA ß Step 2: Calculate Centered Moving Averages k Simple average of two adjacent MAs ß Step 3: Calculate Seasonal Ratio (SR) k SR = Y / CMA ß Step 4: Calculate Seasonal Index (SI)

Forecasting Models 28 Calculate Seasonal Index (Steps 1-3)

Forecasting Models 29

Forecasting Models 30 Calculate Seasonal Index (Step 4) n ASR = Average Seasonal Ratio n GA = Grand Average = average of all ASR’s n The average of Seasonal Indices (SI) must be 1

Forecasting Models 31 II. Estimate Trend Component ß Step 1: Remove seasonal effect k Deseasonalized data t = Y t / SI t ß Step 2: Fit a trend line to deseasonalized data using least squares method ß Step 3: Calculate the trend value for each period s Note: If the deseasonalized data look stable (no apparent trend), simple exponential smoothing may be used in Steps 2 and 3 to calculate the forecast (rather than trend) for each period.

Forecasting Models 32 III. Forecast ß Combine seasonal and trend components k F t = Trend Value t  Seasonal Index t n This final step is also called reseasonalizing n Trend Value t is the trend estimate for the period t, based on the trend model fitted to the deseasonalized data