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Module 4. Forecasting MGS3100 1 Forecasting Forecasting Quantitative Qualitative Causal Model Expert Judgment Trend Delphi Method Time series Grassroots Stationary Trend Trend + Seasonality 2 Quantitative Forecasting --Forecasting based on data and models • Casual Models: Price Population Causal Year 2000 Advertising Model Sales …… • Time Series Models: Sales1999 Sales1998 Time Series Year 2000 Sales1997 Model Sales …… 3 Causal forecasting • Regression Find a straight line that fits the data best. 12 Best line! 10 8 6 4 2 0 Intercept 10 11 12 13 14 15 16 17 18 19 20 y = Intercept + slope * x (= b0 + b1x) Slope = change in y / change in x 4 Causal Forecasting Models • Curve Fitting: Simple Linear Regression – One Independent Variable (X) is used to predict one Dependent Variable (Y): Y = a + b X – Given n observations (Xi, Yi), 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 (Yi - a - bXi)2: b XiYi Xi Yi / Xi 2 ( Xi ) 2 n n a Yi b Xi n n Regression formula is an optional learning objective 5 • Curve Fitting: Simple Linear Regression – Find the regression line with Excel • Use Function: a = INTERCEPT(Y range; X range) b = SLOPE(Y range; X range) • Use Solver • Use Excel’s Tools | Data Analysis | Regression • Curve Fitting: Multiple Regression – Two or more independent variables are used to predict the dependent variable: Y = b0 + b1X1 + b2X2 + … + bpXp – Use Excel’s Tools | Data Analysis | Regression 6 Time Series Forecasting Process Look at the data Forecast using one or Evaluate the technique (Scatter Plot) more techniques and pick the best one. Observations from the Techniques to try Ways to evaluate scatter Plot Heuristics - Averaging methods MAD Data is reasonably Naive MAPE stationary Moving Averages Standard Error (no trend or seasonality) Simple Exponential Smoothing BIAS MAD Regression MAPE Data shows a consistent Linear Standard Error trend Non-linear Regressions (not BIAS covered in this course) R-Squared MAD Classical decomposition MAPE Data shows both a trend and Find Seasonal Index Standard Error a seasonal pattern Use regression analyses to find BIAS the trend component R-Squared 7 Evaluation of Forecasting Model • BIAS - The arithmetic mean of the errors BIAS (Actual - Forecast) Error n n – n is the number of forecast errors – Excel: =AVERAGE(error range) • Mean Absolute Deviation - MAD MAD | Actual - Forecast | | Error | n n – No direct Excel function to calculate MAD 8 Evaluation of Forecasting Model • Mean Square Error - MSE MSE (Actual - Forecast) (Error) 2 2 n n – Excel: =SUMSQ(error range)/COUNT(error range) – Standard error is square root of MSE • Mean Absolute Percentage Error - MAPE | Actual - Forecast| Actual *100% M APE n • R2 - only for curve fitting model such as regression • In general, the lower the error measure (BIAS, MAD, MSE) or the higher the R2, the better the forecasting model 9 Stationary data forecasting • Naïve I sold 10 units yesterday, so I think I will sell 10 units today. • n-period moving average For the past n days, I sold 12 units on average. Therefore, I think I will sell 12 units today. • Exponential smoothing I predicted to sell 10 units at the beginning of yesterday; At the end of yesterday, I found out I sold in fact 8 units. So, I will adjust the forecast of 10 (yesterday’s forecast) by adding adjusted error (α * error). This will compensate over (under) forecast of yesterday. 10 Naïve Model • The simplest time series forecasting model • Idea: “what happened last time (last year, last month, yesterday) will happen again this time” • Naïve Model: – Algebraic: Ft = Yt-1 • Yt-1 : actual value in period t-1 • Ft : forecast for period t – Spreadsheet: B3: = A2; Copy down 11 Moving Average Model • Simple n-Period Moving Average Sum of actual values in previous n periods F t n Y Y Y = t 1 t2 tn n • Issues of MA Model – Naïve model is a special case of MA with n = 1 – Idea is to reduce random variation or smooth data – All previous n observations are treated equally (equal weights) – Suitable for relatively stable time series with no trend or seasonal pattern 12 Smoothing Effect of MA Model Longer-period moving averages (larger n) react to actual changes more slowly 13 Moving Average Model • Weighted n-Period Moving Average F =w Y w Y w Y t 1 t 1 2 t 2 n t n – Typically weights are decreasing: w1>w2>…>wn – Sum of the weights = wi = 1 – Flexible weights reflect relative importance of each previous observation in forecasting – Optimal weights can be found via Solver 14 Weighted MA: An Illustration Month Weight Data August 17% 130 September 33% 110 October 50% 90 November forecast: FNov = (0.50)(90)+(0.33)(110)+(0.17)(130) = 103.4 15 Exponential Smoothing • Concept is simple! – Make a forecast, any forecast – Compare it to the actual – Next forecast is • Previous forecast plus an adjustment • Adjustment is fraction of previous forecast error – Essentially • Not really forecast as a function of time • Instead, forecast as a function of previous actual and forecasted value 16 Simple Exponential Smoothing • A special type of weighted moving average – Include all past observations – Use a unique set of weights that weight recent observations much more heavily than very old observations: weight 0 1 Decreasing weights given to older observations (1 ) (1 ) 2 (1 ) 3 Today 17 Simple ES: The Model Ft Yt 1 (1 )Yt 2 (1 ) 2 Yt 3 Ft Yt 1 (1 )Yt 2 (1 a)Yt 3 Ft Yt 1 (1 ) Ft 1 New forecast = weighted sum of last period actual value and last period forecast – : Smoothing constant – Ft : Forecast for period t – Ft-1: Last period forecast – Yt-1: Last period actual value 18 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 19 Time Series Components • Trend – persistent upward or downward pattern in a time series • Seasonal – Variation dependent on the time of year – Each year shows same pattern • Cyclical – up & down movement repeating over long time frame – Each year does not show same pattern • Noise or random fluctuations – follow no specific pattern – short duration and non-repeating 20 Time Series Components Cycle Trend Random movement Time Time Seasonal Trend with Demand pattern seasonal pattern 21 Time Time Trend Model • Curve fitting method used for time series data (also called time series regression model) • Useful when the time series has a clear trend • Can not capture seasonal patterns • Linear Trend Model: Yt = a + bt – t is time index for each period, t = 1, 2, 3,… 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 22 Pattern-based forecasting - Trend • Regression – Recall Independent Variable X, which is now time variable – e.g., days, months, quarters, years etc. Find a straight line that fits the data best. 12 Best line! 10 8 6 4 2 0 Intercept 10 11 12 13 14 15 16 17 18 19 20 y = Intercept + slope * x (= b0 + b1x) Slope = change in y / change in x 23 Pattern-based forecasting – Seasonal • Once data turn out to be seasonal, deseasonalize the data. – The methods we have learned (Heuristic methods and Regression) is not suitable for data that has pronounced fluctuations. • Make forecast based on the deseasonalized data • Reseasonalize the forecast – Good forecast should mimic reality. Therefore, it is needed to give seasonality back. 24 Pattern-based forecasting – Seasonal Example (SI + Regression) Actual data Deseasonalized data Deseasonalize Forecast Reseasonalize 25 Pattern-based forecasting – Seasonal • Deseasonalization Deseasonalized data = Actual / SI • Reseasonalization Reseasonalized forecast = deseasonalized forecast * SI 26 Seasonal Index • What’s an index? – Ratio – SI = ratio between actual and average demand • Suppose – SI for quarter demand is 1.20 • What’s that mean? • Use it to forecast demand for next fall – So, where did the 1.20 come from?! 27 Calculating Seasonal Indices • Quick and dirty method of calculating SI – For each year, calculate average demand – Divide each demand by its yearly average • This creates a ratio and hence a raw index • For each quarter, there will be as many raw indices as there are years – Average the raw indices for each of the quarters – The result will be four values, one SI per quarter 28 Classical decomposition • Start by calculating seasonal indices • Then, deseasonalize the demand – Divide actual demand values by their SI values y ’ = y / SI – Results in transformed data (new time series) – Seasonal effect removed • Forecast – Regression if deseasonalized data is trendy – Heuristics methods if deseasonalized data is stationary • Reseasonalize with SI 29 Causal or Time series? • What are the difference? • Which one to use? 30 Can you… • describe general forecasting process? • compare and contrast trend, seasonality and cyclicality? • describe the forecasting method when data is stationary? • describe the forecasting method when data shows trend? • describe the forecasting method when data shows seasonality? 31