Slides 13b: Time-Series Models; Measuring Forecast Error MGS3100 Chapter 13 Forecasting Slides 13b: Time-Series Models; Measuring Forecast Error

Forecasting Models

Time Series Models General Form: Y = T * C * S ± ε, where T = Trend - long term movement of mean C = (Business) Cycle - an upturn or downturn not caused by seasonal variation; effect of the economy S = Seasonal Variation - repetitive pattern observed over a specific time period ε = Error (random variation) Practical Forecast Form: Ŷ = T * S C is important, but difficult to forecast Don’t forecast an error!

Components of a Time Series value Linear trend and seasonality time series Future Linear trend time series A stationary time series Time

Time Series: Stationary Models Stationary Model Assumptions Assumes item forecasted will stay steady over time (constant mean; random variation only) Techniques will smooth out short-term irregularities Forecast for period t+1 is equal to forecast for period t+k; the forecast is revised only when new data becomes available. Stationary Model Types Naïve Forecast Moving Average Weighted Moving Average Exponential Smoothing

Stationary Time Series Models: The Naïve Model Whatever happened last period will happen again this time The model is simple and flexible Provides a baseline to measure other models Attempts to capture seasonal factors at the expense of ignoring trend or

Measures of Forecast Error Bias - The arithmetic sum of the errors MAD - Mean Absolute Deviation MAPE – Mean Absolute Percentage Error Mean Square Error (MSE) - Similar to simple sample variance Standard Error - Standard deviation of the sampling distribution (the square root of the MSE) Bias, MAD, and MAPE - typically used for time series Bias is difference between the actual value and the forecasted value.

Naïve Forecast

Naïve Forecast Graph

Stationary Time Series Models: Moving Averages The Moving Average Method The forecast is the average of the last n observations of the time series.

Moving Averages

Moving Averages Forecast

Moving Averages Graph

Stability vs. Responsiveness Should I use a 2-period moving average or a 3-period moving average? The larger the “n” the more stable the forecast. A 2-period model will be more responsive to change. We don’t want to chase outliers. But we don’t want to take forever to correct for a real change. We must balance stability with responsiveness.

Stationary Time Series Models: Weighted Moving Averages The Weighted Moving Average Method Historical values of the time series are assigned different weights when performing the forecast = w1Yt + w2Yt-1 +w3Yt-2 + …+ wnYt-n+1 Swi = 1

Weighted Moving Average

Weighted Moving Average

Stationary Time Series Models: Exponential Smoothing Moving average technique that requires a minimum amount of past data Uses a smoothing constant α with a value between 0 and 1 (Usual range 0.1 to 0.3) Forecast for period t = Forecast for period t-1 plus α times the difference between the actual value and forecast in period t-1: Ŷt = Ŷt-1 + α(Yt-1 - Ŷt-1), or Can also be expressed as: Ŷt = α(Yt-1) + (1- α)(Ŷt-1) = α(Actual value in period t-1) + (1- α)(Forecast in period t-1) Both moving averages and weighted moving averages are effective in smoothing out sudden fluctuations in the demand pattern in order to provide stable estimates. Increasing the size of k (number of periods averaged) smoothes out fluctuations even better. This requires keeping extensive historical records.

Exponential Smoothing Data Class Exercise: What is the forecast for January of the following year? How about March? Find the Bias, Mad & MAPE. (Note: α equals 0.1.)

Exponential Smoothing (Alpha = .419)

Exponential Smoothing

Evaluating the Performance of Forecasting Techniques Several forecasting methods have been presented. Which one of these forecasting methods gives the “best” forecast?

Performance Measures – Sample Example Find the forecasts and the errors for each forecasting technique applied to the following stationary time series. Time 1 2 3 4 5 6 Time series: 100 110 90 80 105 115 3-Period Moving average: 100 93.33 91.6 Error for the 3-Period MA: - 20 11.67 23.4 3-Period Weighted MA(.5, .3, .2) 98 89 85.5 Error for the 3-Period WMA - 18 16 29.5

Performance Measures – MAD for the Sample Example MAD for the moving average technique: MAD for the weighted moving average technique: |-20| + |11.67| + |23.4| 3 MAD = = S |D t| n = 18.35 |-18| + |116| + |29.5| 3 MAD = = S |D t| n = 21.17

Performance Measures – MAPE for the Sample Example MAPE for the moving average technique: MAPE for the weighted moving average technique: |-20|/80 + |11.67|/105+ |23.4|/115 3 MAPE= = S |D t| n |-18|/80 + |16|/105 + |29.5|/115 = .188 = .211

Performance Measures – Selecting Model Parameters Use the performance measures to select a good set of values for each model parameter. For the moving average: the number of periods (n). For the weighted moving average: The number of periods (n), The weights (wi). For the exponential smoothing: The exponential smoothing factor (a). Excel Solver can be used to determine the values of the model parameters.

Trend & Seasonality Trend analysis Seasonality analysis Technique that fits a trend equation (or curve) to a series of historical data points Projects the equation into the future for medium and long term forecasts. Typically do not want to forecast into the future more than half the number of time periods used to generate the forecast Seasonality analysis Adjustment to time series data due to variations at certain periods. Adjust with seasonal index - ratio of average value of the item in a season to the overall annual average value. Examples: demand for coal in winter months; demand for soft drinks in the summer and over major holidays

Linear Trend Analysis Midwestern Manufacturing Sales

Least Squares for Linear Regression Midwestern Manufacturing Objective: Minimize the squared deviations!

Least Squares Method X = value of the independent variable (time) Where = predicted value of the dependent variable (demand) X = value of the independent variable (time) a = Y-axis intercept = - b* b = Slope of the regression line =

Linear Trend Data & Error Analysis

Least Squares Graph

Another way to Determine Trend: Use the Excel Regression Function Run linear regression to test b1 in the model Yt=b0+b1t+et Excel results: 0.71601 This large P-value indicates that there is little evidence that trend exists Conclusion: A stationary model is appropriate.

Forecasting Seasonal Data: Quick Method Ratio = Demand / Average Demand Seasonal Index – ratio of the average value of the item in a season to the overall average annual value. Example: average of year 1 January ratio to year 2 January ratio. (0.851 + 1.064)/2 = 0.957 A seasonal index with value below 1 indicates demand below average that month, and an index above 1 indicates demand above average that month. Using these seasonal indices, the future demand for any future month can be adjusted. For example, if the average demand for answering machines in year three is expected to be 100 units, then the forecast for January’s demand is 100 X 0.957 = 96 units, which is below average. May’s forecast is 100 X 1.309 = 131 units, which is above average. If Year 3 average monthly demand is expected to be 100 units. Forecast demand Year 3 January: 100 X 0.957 = 96 units Forecast demand Year 3 May: 100 X 1.309 = 131 units

Forecasting Seasonal Data With Trend Calculate the seasonal indices (as shown on the previous slide) Calculate “deseasonalized” treand by dividing the actual value (Y) by the seasonal index for that period: Deseasonalized Trend = Y / Seasonal index (e.g., 80 units/ 0.957 = 83.595) Find the trend line, and extend the trend line into the desired forecast period.

Forecasting Seasonal Data With Trend: Calculating the Seasonal Forecast 4. Now that we have the Seasonal Indices and Trend line, we can reseasonalize the data and generate the “seasonalized” forecast by multiplying the trend line values in the forecast period by the appropriate seasonal indices for each time period as follows: Ŷ = Trend x Seasonal Index