Forecasting Examples

Forecasting Examples:
Maintenance Demand
Dr Mehran Ullah

Quantitative Forecasting
1. Simple Moving Averages
• used for stationary time series
• It is composed of a constant term plus random fluctuation
• example could be the load exerted on an electronic component
• All previous observations are assumed to be equally important, i.e.,
equally weighted

• Since Ft (forecast for time periods t) is the average of the last N actual
observations, it is called a simple N-period moving average

2. Weighted Moving Average
• In simple moving average, an equal weight is given to all n data
points.
• Since individual weight is equal to 1/n, then sum of the weights is
n(1/n) = 1
• However, a more realistic view is that more recent data points have
more forecasting value than older data points
• the simple moving average method is modified by including weights
that decrease with the age of the data

• The values of wt must be non-decreasing with respect to t.
• These values can be empirically determined based on error analysis,
or subjectively estimated based on experience, hence combining
qualitative and quantitative forecasting approaches

3. Regression Analysis
• Regression analysis is used to develop a relationship between the
independent variable being forecasted and one or more independent
predictor variables
• For example if the maintenance cost for the current period m(t) is a
linear function of the number of operational hours in the same period
h(t), then the model is given by
• That is a straight-line regression relationship with a single
independent predictor variable, namely h(t)

• Parameters a and b are respectively called the intercept and the slope
of this line
• Regression analysis is the process of estimating these parameters
using the least-squares method
• This method finds the best values of a and b that minimize the sum of
the squared vertical distances (errors) from the line

• The general straight-line equation showing a linear trend of
maintenance work demand Di over time is
• For n historical data points: (t1, D1), (t2, D2), ..., (tn, Dn). The least-
squares method estimates a and b by minimizing the following sum of
squared errors:

• By taking partial derivatives with respect to a and b and setting them
equal to zero we obtain the following values of a and b
• Least-squares regression methodology can easily accommodate
multiple variables and also polynomial or nonlinear functional
relationships

• Example: Demand for a given spare part is given below for the last 4
years. Use linear regression to determine the best-fit straight line and
to forecast spare part demand in year 5.
• To find a and b, we need to find the following

• Utilizing the formulas of of a and b and the equation of linear
regression model;

4. Exponential Smoothing
a) Simple Exponential Smoothing (ES)
• Simple exponential smoothing (ES) is similar to weighted moving
average (WMA) in assigning higher weights to more recent data, but it
differs in two important aspects;
1. WMA is a weighted average of only the last N data points, while ES
is a weighted average of all past data
2. The weights in WMA are mostly arbitrary, while the weights in ES
are well structured
1. The weights in ES decrease exponentially with the age of the data

• Exponential smoothing is very easy to use, and very easy to update by
including new data as it becomes available
• The current forecast is a weighted average of the last forecast and the
last actual observation
• Given the value of smoothing constant α, the forecast is obtained by:

• The greater the value of α, the more weight of the last observation.
However, large values of α lead to highly variable, less stable,
forecasts.
• For forecast stability, a value of α between 0.1 and 0.3 is usually
recommended for smooth planning.
• The best value of α can be determined from experience or by trial and
error (choosing the value with minimum error)

4. Exponential Smoothing
b). Double Exponential Smoothing (Holt’s Method)
• The simple exponential smoothing can be used to estimate the
parameters for a constant (stationary) model.
• Double or triple exponential smoothing approaches can be used to
deal with linear, polynomial, and seasonal forecasting models
• Holt’s double exponential smoothing method is important tool to
handle linear models

• Holt’s double exponential smoothing method requires two smoothing
constants: α and β (β ≤ α)
• Two smoothing equations are applied: one for at, the intercept at
time t, and another for bt, the slope at time t

5. Seasonal Forecasting
• Demand for many products follows a seasonal or cyclic pattern, which
repeats itself every N periods
• For example, demand for electricity has a daily cycle, demand for
restaurants has a weekly cycle, while demand for clothes has a yearly
cycle
• Maintenance workload may show seasonal variation due to periodic
changes in demand, weather, or operational conditions

• If product demand for products is seasonal, then greater production
rates during the high-season intensify equipment utilization and
increase the probability of failure.
• If product demand is not seasonal, high temperatures during summer
months may cause overheating and more frequent equipment
failures
• Plotting the data is important to judge whether or not it has
seasonality trend, or both patterns

a). Forecasting for Stationary Seasonal Data
• The model representing this data is similar to the model presented for
Stationary Data, but it allows for seasonal variations
• Given data for at least two cycles (2N), following steps are used to obtain
the seasonal factor 𝑐𝑡:
1. Calculate the overall average μ;
2. Divide each data point by μ to obtain seasonal factor estimate;
3. Calculate seasonal factors 𝑐𝑡 by averaging all factors for similar periods;
• Note: See example 8.6 from the pdf.

b). Forecasting for Seasonal Data with a Trend
• It is possible for a time series to have both seasonal and trend
components.
• For example, the demand for airline travel increases during summer,
but it also keeps growing every year
• For such data the model is:
• The usual approach to forecast with seasonal-trend data is to
estimate each component by trying to remove the effect of the other
one

The same general approach is followed that consists of:
i. remove trend to estimate seasonality
ii. remove seasonality to estimate trend
iii. forecast using both seasonality and trend
The cycle average method is used for Seasonal Data with a Trend
1. Divide each cycle by its corresponding cycle average to remove trend.
2. Average the de-trended values for similar periods to determine seasonal
factors 𝑐1, …, 𝑐𝑁.
3. Use any appropriate trend-based method to forecast cycle averages.
4. Forecast by multiplying the trend-based cycle average by appropriate seasonal
factor.
Note: See example 8.6 from the pdf.

Error Analysis
• The forecasting error ε𝑡 in time period t is defined as the difference
between the actual and the forecasted value for the same period:
• Error analysis is used to evaluate performance of the forecasting model
and to check how closely it fits the given actual data
• Error analysis is also used to compare objectively and systematically the
alternative models (different available methods) in order to choose the
best one

a). Sum of the errors (SOE)
• SOE can be deceiving as large positive errors may cancel out with large
negative errors
• This measure is good for checking bias, if the forecast is unbiased, SOE
should be close to zero

b). Mean absolute deviation (MAD)
• Neutralizes the opposite signs of errors by taking their absolute values
c). Mean squared error (MSE)
• Neutralizes the opposite signs of errors by squaring them

d). Mean absolute percent error (MAPE)
• Independent measure for evaluating the “goodness” of an individual
forecast
• All the other measures only compare different forecasting models relative
to each other

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