Introduction to Forecasting
Forecasting Methods:
FORECASTING - a method for translating past experience into estimates of the future
Read: The University Bookstore Student Computer Purchase Program page 497 in the text.
Key questions which must be answered:
forecasting horizons:
forecasting methods:
qualitative methods
quantitative methods
- causal methods
- time series methods
qualitative forecasting methods are based on educated opinions of appropriate persons
1. delphi method: forecast is developed by a panel of experts who anonymously answer a series of questions; responses are fed back to panel members who then may change their original responses
- very time consuming and expensive
- new groupware makes this process much more feasible
2. market research: panels, questionnaires, test markets, surveys, etc.
3. product life-cycle analogy: forecasts based on life-cycles of similar products, services, or processes
4. expert judgement by management, sales force, or other knowledgeable persons
time series forecasting methods are based on analysis of historical data (time series: a set of observations measured at successive times or over successive periods). They make the assumption that past patterns in data can be used to forecast future data points.
1. moving averages (simple moving average, weighted moving average): forecast is based on arithmetic average of a given number of past data points
2. exponential smoothing (single exponential smoothing, double exponential smoothing): a type of weighted moving average that allows inclusion of trends, etc.
3. mathematical models (trend lines, log-linear models, Fourier series, etc.): linear or non-linear models fitted to time-series data, usually by regression methods
4. Box-Jenkins methods: autocorrelation methods used to identify underlying time series and to fit the "best" model
COMPONENTS OF TIME SERIES DEMAND
1. average: the mean of the observations over time
2. trend: a gradual increase or decrease in the average over time
3. seasonal influence: predictable short-term cycling behaviour due to time of day, week, month, season, year, etc.
4. cyclical movement: unpredictable long-term cycling behaviour due to business cycle or product/service life cycle
5. random error: remaining variation that cannot be explained by the other four components
moving average techniques forecast demand by calculating an average of actual demands from a specified number of prior periods
each new forecast drops the demand in the oldest period and replaces it with the demand in the most recent period; thus, the data in the calculation "moves" over time
simple moving average: A_{t} = D_{t} + D_{t-1} + D_{t-2} + ... + D_{t-N+1}
N
where N = total number of periods in the average
forecast for period t+1: F_{t+1} = A_{t}
Key Decision: N - How many periods should be considered in the forecast
Tradeoff: Higher value of N - greater smoothing, lower responsiveness
Lower value of N - less smoothing, more responsiveness
- the more periods (N) over which the moving average is calculated, the less susceptible the forecast is to random variations, but the less responsive it is to changes
- a large value of N is appropriate if the underlying pattern of demand is stable
- a smaller value of N is appropriate if the underlying pattern is changing or if it is important to identify short-term fluctuations
a weighted moving average is a moving average where each historical demand may be weighted differently
average: A_{t} = W_{1} D_{t} + W_{2} D_{t-1} + W_{3} D_{t-2} + ... + W_{N} D_{t-N+1}
where:
N = total number of periods in the average
W_{t} = weight applied to period t's demand
_{Sum of all the weights = 1}
forecast: F_{t+1} = A_{t} = forecast for period t+1
exponential smoothing gives greater weight to demand in more recent periods, and less weight to demand in earlier periods
average: A_{t} = a D_{t} + (1 - a) A_{t-1} = a D_{t} + (1 - a) F_{t}
forecast for period t+1: F_{t+1} = A_{t}
where:
A_{t-1} = "series average" calculated by the exponential smoothing model to period t-1
a = smoothing parameter between 0 and 1
the larger the smoothing parameter , the greater the weight given to the most recent demand
when a trend exists, the forecasting technique must consider the trend as well as the series average ignoring the trend will cause the forecast to always be below (with an increasing trend) or above (with a decreasing trend) actual demand
double exponential smoothing smooths (averages) both the series average and the trend
forecast for period t+1: F_{t+1} = A_{t} + T_{t}
average: A_{t} = aD_{t} + (1 - a) (A_{t-1} + T_{t-1}) = aD_{t} + (1 - a) F_{t}
average trend: T_{t} = B CT_{t} + (1 - B) T_{t-1}
current trend: CT_{t} = A_{t} - A_{t-1}
forecast for p periods into the future: F_{t+p} = A_{t} + p T_{t}
where:
A_{t} = exponentially smoothed average of the series in period t
T_{t} = exponentially smoothed average of the trend in period t
CT_{t} = current estimate of the trend in period t
a = smoothing parameter between 0 and 1 for smoothing the averages
B = smoothing parameter between 0 and 1 for smoothing the trend
What happens when the patterns you are trying to predict display seasonal effects?
What is seasonality? - It can range from true variation between seasons, to variation between months, weeks, days in the week and even variation during a single day or hour.
To deal with seasonal effects in forecasting two tasks must be completed:
the multiplicative seasonal method adjusts a given forecast by multiplying the forecast by a seasonal factor
Step 1: calculate the average demand _{y} per period for each year (y) of past data by dividing total demand for the year by the number of periods in the year
Step 2: divide the actual demand D_{y,t} for each period (t) by the average demand _{y} per period (calculated in Step 1) to get a seasonal factor f_{y,t} for each period; repeat for each year of data
Step 3: calculate the average seasonal factor _{t} for each period by summing all the seasonal factors f_{y,t} for that period and dividing by the number of seasonal factors
Step 4: determine the forecast for a given period in a future year by multiplying the average seasonal factor _{t} by the forecasted demand in that future year
Seasonal Forecasting (multiplicative method)
Actual Demand
Year | Q1 | Q2 | Q3 | Q4 | Total | Avg |
1 | 100 | 70 | 60 | 90 | 320 | 80 |
2 | 120 | 80 | 70 | 110 | 380 | 95 |
3 | 134 | 80 | 70 | 100 | 381 | 96 |
Seasonal Factor
Year | Q1 | Q2 | Q3 | Q4 |
1 | 1.25 | .875 | .75 | 1.125 |
2 | 1.26 | .84 | .74 | 1.16 |
3 | 1.4 | .83 | .73 | 1.04 |
Avg. Seasonal Factor | 1.30 | .85 | .74 | 1.083 |
Seasonal Factor - the percentage of average quarterly demand that occurs in each quarter.
Annual Forecast for year 4 is predicted to be 400 units.
Average forecast per quarter is 400/4 = 100 units.
Quarterly Forecast = avg. forecast × seasonal factor.
causal forecasting methods are based on a known or perceived relationship between the factor to be forecast and other external or internal factors
1. regression: mathematical equation relates a dependent variable to one or more independent variables that are believed to influence the dependent variable
2. econometric models: system of interdependent regression equations that describe some sector of economic activity
3. input-output models: describes the flows from one sector of the economy to another, and so predicts the inputs required to produce outputs in another sector
4. simulation modelling
MEASURING FORECAST ERRORS
There are two aspects of forecasting errors to be concerned about - Bias and Accuracy
Bias - A forecast is biased if it errs more in one direction than in the other
- The method tends to under-forecasts or over-forecasts.
Accuracy - Forecast accuracy refers to the distance of the forecasts from actual demand ignore the direction of that error.
Example: For six periods forecasts and actual demand have been tracked The following table gives actual demand D_{t} and forecast demand F_{t} for six periods:
t | D_{t} | F_{t} | E_{t} | (E_{t})^{2} | |E_{t}| | | E_{t}|/D_{t} |
1 | 170 | 200 | -30 | 900 | 30 | 17.6% |
2 | 230 | 195 | 35 | 1225 | 35 | 15.2% |
3 | 250 | 210 | 40 | 1600 | 40 | 16.0% |
4 | 200 | 220 | -20 | 400 | 20 | 10.0% |
5 | 185 | 210 | -25 | 625 | 25 | 13.5% |
6 | 180 | 200 | -20 | 400 | 20 | 11.1% |
Total | -20 | 5150 | 170 | 83.5% |
Forecast Measure
What information does each give?
conclusions:
forecast has a tendency to over-estimate demand
average error per forecast was 28.33 units, or 13.9% of actual demand
sampling distribution of forecast errors has standard deviation of 29.3 units.
CRITERIA FOR SELECTING A FORECASTING METHOD
Objectives: 1. Maximize Accuracy and 2. Minimize Bias
Potential Rules for selecting a time series forecasting method. Select the method that
or others. It appears obvious that some measure of both accuracy and bias should be used together. How?
What about the number of periods to be sampled?
"focus forecasting" refers to an approach to forecasting that develops forecasts by various techniques, then picks the forecast that was produced by the "best" of these techniques, where "best" is determined by some measure of forecast error.
FOCUS FORECASTING: EXAMPLE
For the first six months of the year, the demand for a retail item has been 15, 14, 15, 17, 19, and 18 units.
A retailer uses a focus forecasting system based on two forecasting techniques: a two-period moving average, and a trend-adjusted exponential smoothing model with = 0.1 and = 0.1. With the exponential model, the forecast for January was 15 and the trend average at the end of December was 1.
The retailer uses the mean absolute deviation (MAD) for the last three months as the criterion for choosing which model will be used to forecast for the next month.
a. What will be the forecast for July and which model will be used?
b. Would you answer to Part a. be different if the demand for May had been 14 instead of 19?
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