FORECASTING Introduction Quantitative Models Time Series

Forecasting Models Forecasting QuantitativeQualitative Causal Model Time series Expert Judgment Trend Stationary Trend Trend + Seasonality Delphi Method Grassroots Market Research Jury Exec. Opinion

FORECASTING Introduction to Quantitative FORECASTING

Quantitative Forecasts Quantitative forecasting models possess two important and attractive features: They are expressed in mathematical notation. Thus, they establish an unambiguous record of how the forecast is made. With the use of spreadsheets and computers, quantitative models can be based on an amazing quantity of data.

Quantitative Forecasts Two types of quantitative forecasting models : Causal models Time-Series models

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

FORECASTING Time Series Forecasting FORECASTING

Forecasting Models Forecasting QuantitativeQualitative Causal Model Time series Expert Judgment Trend Stationary Trend Trend + Seasonality Delphi Method Grassroots Market Research Jury Exec. Opinion

Time Series Forecasting Models Time-series forecasting models produce forecasts by extrapolating the historical behavior of the values of a particular single variable of interest. Time-series data are historical data in chronological order, with only one value per time period.

Components of a Time Series

11 Time Series Components Time Trend Random movement Time Cycle Time Seasonal pattern Demand Time Trend with seasonal pattern

12 Time Series Forecasting Process Look at the data (Scatter Plot) Forecast using one or more techniques Evaluate the technique and pick the best one. Observations from the scatter Plot Techniques to tryWays to evaluate Data is reasonably stationary (no trend or seasonality) Heuristics - Averaging methods  Naive  Moving Averages  Simple Exponential Smoothing  MAD  MAPE  Standard Error  BIAS Data shows a consistent trend Regression  Linear  Non-linear Regressions (not covered in this course)  MAD  MAPE  Standard Error  BIAS  R-Squared Data shows both a trend and a seasonal pattern Classical decomposition  Find Seasonal Index  Use regression analyses to find the trend component  MAD  MAPE  Standard Error  BIAS  R-Squared

13 BIAS The arithmetic mean of the errors n is the number of forecast errors Mean Absolute Deviation - MAD Average of the absolute errors Evaluation of Forecasting Model

14 Mean Square Error - MSE Standard error Square Root of MSE Mean Absolute Percentage Error - MAPE Calculate the % error using the absolute error, then average the results Evaluation of Forecasting Model

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 The forecast is revised only when new data becomes available.  Stationary Model Types Naïve Forecast Moving Average/Weighted Moving Average Exponential Smoothing

16 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.

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

Naïve Forecast

20 Moving Average Model Simple n-Period Moving Average 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

Moving Averages

Moving Averages Forecast

Moving Average Forecast

Stability vs. Responsiveness Should I use a 3-period moving average or a 5-period moving average? The larger the “n” the more stable the forecast. A 3-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.

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

Weighted Moving Average Model Assumes data from some periods are more important than data from other periods (e.g. earlier periods). Uses weights to place more emphasis on some periods and less on others Historical values of the time series are assigned different weights when performing the forecast

27 Weighted Moving Average Model Weighted n-Period Moving Average Typically weights are decreasing: w 1 >w 2 >…>w n Sum of the weights =  w i = 1 Flexible weights reflect relative importance of each previous observation in forecasting

28 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

Weighted Moving Average

Operational Problems With Moving Averages The operational shortcoming of simple moving average models is that if n observations are to be included in the moving average, then (n-1) pieces of past data must be brought forward to be combined with the current (the nth) observation All this data must be stored in some way, in order to calculate the forecast. This may become a problem when a company needs to forecast the demand for thousands of individual products on an item-by-item basis. The next weighting scheme addresses this problem.

Exponential Smoothing Moving average technique that requires little record keeping of past data. Uses a smoothing constant α with a value between 0 and 1. (Usual range 0.1 to 0.3) Applies alpha to most recent period, and applies one minus alpha distributed to previous values α = The weight assigned to the latest period (smoothing constant) Forecast = α(Actual value in period t-1) + (1- α)(Forecast in period t-1) Can also be forecast for period t-1 plus α times the difference between the actual value and forecast in period t-1:

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

36 Simple Exponential Smoothing Properties of Simple Exponential Smoothing Widely used and successful model Requires very little data Larger , more responsive forecast Smaller , smoother forecast Suitable for relatively stable time series

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.

MAD for the moving average technique: MAD for the weighted moving average technique: Performance Measures – MAD for the Sample Example

MAPE for the moving average technique: MAPE for the weighted moving average technique: Performance Measures – MAPE for the Sample Example

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 (w i ). For the exponential smoothing: The exponential smoothing factor (  ). Excel Solver can be used to determine the values of the model parameters. Performance Measures – Selecting Model Parameters

Excel Solver EXCEL SOLVER EXAMPLE

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

44 Time Series Components Time Trend Random movement Time Cycle Time Seasonal pattern Demand Time Trend with seasonal pattern

Trend & Seasonality Trend 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!

Trend Analysis - Least Squares Method For Linear Regression 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: Y t = a + bt t is time index for each period, t = 1, 2, 3,…

Where = predicted value of the dependent variable (demand) a = Y- intercept b = Slope of the regression line t = independent variable (time period = 1, 2, 3, ….) Trend Analysis - Least Squares Method For Linear Regression

50 Curve Fitting: Simple Linear Regression One Independent Variable (X) is used to predict one Dependent Variable (Y): Y = a + b X 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 : Regression formula is an optional learning objective

51 Find the regression line with Excel Use Excel’s Tools | Data Analysis | Regression Curve Fitting: Multiple Regression 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 Use Excel’s Tools | Data Analysis | Regression Curve Fitting: Simple Linear Regression

Linear Trend Data & Error Analysis

Least Squares Graph

54 Pattern-based forecasting – Seasonal The methods we have learned (Heuristic methods and Regression) are not suitable for data that has pronounced fluctuations. If data seasonalized: Deseasonalize the data. Make forecast based on the deseasonalized data Reseasonalize the forecast Good forecast should mimic reality. Therefore, it is needed to give seasonality back.

55 Pattern-based forecasting – Seasonal Deseasonalize Forecast Reseasonalize Actual dataDeseasonalized data Example (SI + Regression)

56 Pattern-based forecasting – Seasonal Deseasonalization  Deaseasonalized Data = Reseasonalization  Reseasonalized Forecast = (Deseasonalized Forecast )* (Seasonal Index)

Forecasting Seasonal Data With Trend 1. Calculate the seasonal indices. 2. Calculate “deseasonalized” trend by divide the actual value (Y) by the seasonal index for that period. 3. Find the trend line, and extend the trend line into the desired forecast period. 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.

Forecasting Seasonal Data: Calculating Seasonal Indexes 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. ( )/2 = Ratio = Demand / Average Demand If Year 3 average monthly demand is expected to be 100 units. Forecast demand Year 3 January: 100 X = 96 units Forecast demand Year 3 May: 100 X = 131 units

Forecasting Seasonal Data: Calculating Seasonal Indexes 1.Take average of all Demand Values (Average Demand Column) 2.Get Ratio of “Actual Value: Average Value” for each period (Ratio Column) 3.Average the ratio for corresponding periods to get seasonal index

Seasonal Forecasting Seasonal Forecasting Example

61 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?