Chapter 3 Lecture 4 Forecasting

Time Series is a sequence of measurements over time, usually obtained at equally spaced intervals – Daily – Monthly – Quarterly – Yearly Time Series Forecasts

Time ordered sequence of observations taken at regular observations taken at regular intervals. Statistical techniques that make use of historical data collected over a long period of time. Methods assume that what has occurred in the past will continue to occur in the future. Forecasts based on only one factor - time.

Time Series Patterns

Time Series Forecasts Naive Forecasts Techniques for Averaging Moving Average Weighted Moving Average Exponential Smoothing Techniques for Trend Trend Techniques for Seasonality Techniques for Cycles

Naive Forecasts Uh, give me a minute.... We sold 250 wheels last week.... Now, next week we should sell.... The forecast for any period equals the previous period’s actual value. The recent periods are the best predictors of the future

Simple to use Virtually no cost Quick and easy to prepare Data analysis is nonexistent Easily understandable Cannot provide high accuracy Naïve Forecasts

F(t) = A(t-1)F(t) = A(t-1) Stable time series data (stationer) F(t) = A(t-1) + (A(t-1) – A(t-2))F(t) = A(t-1) + (A(t-1) – A(t-2)) Data with trends trends F(t) = A(t-n)F(t) = A(t-n) Seasonal variations Uses for Naïve Forecasts at different data patterns

Naïve Forecasts DemandYear Stationer F(t) = A(t-1) Ex.

Naïve Forecasts DemandYear Trend F(t) = A(t-1) + (A(t-1) – A(t-2)) Ex.

Naïve Forecasts DemandYear 101/ / / / / / /2011 Seasonal F(t) = A(t-1) + (A(t-1) – A(t-2)) Ex.

Naïve Forecast Graph

A technique that averages a number of recent actual values, updated as new values become available. F t = forecast for time period t MA n = n period moving average A t-1 = actual value in period t-1 n = number of periods ( data points ) F t = MA n = n A t-n + … A t-2 + A t-1 Moving Averages, no pattern ( random variation )

Compute a 3-period moving average forecast given demand for shopping carts for the last five periods as shown: t=6 Compute a 3-period moving average forecast given demand for shopping carts for the last five periods as shown: t=6 F t = MA n = n A t-n + … A t-2 + A t period ??? demand F 6 = MA 3 = 3 A 3 + A 4 + A 5 = Moving Averages Ex.

t=7 t= period ??? demand F 7 = MA 3 = 3 A 4 + A 5 + A 6 = forecast period ??? demand actual period demand forecast Moving Averages Ex. (cont.)

Moving Averages Ex.

Moving Averages Ex. (cont.)

Moving Averages Ex.

Moving Averages Ex. (cont.)

Moving Averages Ex.

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 must balance stability with responsiveness If responsiveness is required, average with few data points should be used,

As data points in an moving average technique increased, the sensitivity ( responsiveness ) of the average to new values decreased. If responsiveness is required, average with few data points should be used, Decreasing the number of data points in an moving average technique, increase the weight of more recent values Moving Averages

It is easy to compute It is easy to understand It is easy to compute It is easy to understand All values in the average are weighted equally, the oldest value has the same weight as the most recent value But Moving Averages most recent observations must be better indicators of the future than older observations Idea

Weighted Moving Averages Historical values of the time series are assigned different weights when performing the forecast

More recent values in a series are given more weight in computing the forecast. Trial and error used to find the suitable weighting scheme The sum of all weights must be = 1 Weighted average technique is more reflective of the most recent occurrences. Weighted Moving Averages

In a weighted moving average, weights are assigned to the most recent data. Formula: Weighted Moving Averages

Compute a 4-period weighted moving average forecast given demand for shopping carts for the last periods values and weights as shown: F6 = W5*A5 + W4*A4 + W3*A3 + W2*A2 F6 = 0.4(41) + 0.3(40) + 0.2(43) + 0.1(40) = 41 If the actual value of F6 is 39 then F7 = 0.4(39) + 0.3(41) + 0.2(40) + 0.1(43) = period ??? demand weight Weighted Moving Averages Ex.

Market Mixer, Inc. sells can openers. Monthly sales for an eight-month period were as follows: MonthSalesMonthSales Forecast next month’s sales using a 3-month weighted moving average, where the weight for the most recent data value is 0.60; the next most recent, 0.30; and the earliest, Solution: PeriodSales Weighted Moving Average Forecast (450*.10) + (425*.30) + (445*.60) = (425*.10) + (445*.30) + (435*.60) = (445*.10) + (435*.30) + (460*.60) = (435*.10) + (460*.30) + (445*.60) = (460*.10) + (455*.30) + (430*.60) = (455*.10) + (430*.30) + (420*.60) = Comments: 1. Any forecasts beyond Period 9 will have the same value as the Period 9 forecast, i.e., WMA gives greater weight to more recent values in the moving average and is more responsive to recent changes in the data. 427 Weighted Moving Averages

The most recent observations might have the highest predictive value.The most recent observations might have the highest predictive value. Therefore, we should give more weight to the more recent time periods when forecasting. Therefore, we should give more weight to the more recent time periods when forecasting. Exponential Smoothing

F t = F t-1 +  ( A t-1 - F t-1 ) Weighted averaging method based on previous forecast plus a percentage of the forecast error Exponential Smoothing Determination of  is usually judgmental and subjective and often based on trial-and -error experimentation. The most commonly used values of  are between.10 and.50.

Exponential Smoothing Ex.

.1 .4 Actual Selecting a smoothing constant α is a matter of judgment or trial and error Picking a Smoothing Constant

Parabolic Exponential Growth Techniques for Trend Nonlinear Trends

Linear Trends Techniques for Trend

F t = Forecast for period t t = Specified number of time periods a = Value of F t at t = 0 b = Slope of the line F t = a + b t t FtFt Techniques for Trend

b = n(ty)- ty nt 2 - (t) 2 a = y- bt n    Calculating a and b

Linear Trend Equation Ex.

y = t a= (15) 5 = b= 5 (2499)- 15(812) 5(55)- 225 = = Linear Trend Equation

Moving average Weighted moving average Exponential smoothing Techniques for Averaging