Moving Average (MA):
A moving average, also called rolling average, rolling mean or running average, is a
type of finite impulse response filter used to analyze a set of data points by creating a
series of averages of different subsets of the full data set.
A moving average is commonly used with time series data to smooth out short-term
fluctuations and highlight longer-term trends or cycles. The threshold between short-term
and long-term depends on the application, and the parameters of the moving average will
be set accordingly.
For example, it is often used in technical analysis of financial data, like stock prices,
returns or trading volumes. It is also used in economics to examine gross domestic
product, employment or other macroeconomic time series.
Examples:
Ex No.1:
The demand for a product in each of the last five months is shown below.
Months
Demand ('00s)
1
2
3
4
5
13 17 19 23 24
Use a two month moving average to generate a forecast for demand in month 6.
Solution:
The two month moving average for month’s two to five is given by:
m2 = (13 + 17)/2 = 15.0
m3 = (17 + 19)/2 = 18.0
m4 = (19 + 23)/2 = 21.0
m5 = (23 + 24)/2 = 23.5
The forecast for month six is just the moving average for the month before that i.e. the
moving average for month 5= m5 = 2350.
Ex No.2:
The table below shows the demand for a particular brand of razor in a shop for each of
the last nine months.
Month
Demand
1 2 3 4 5 6 7 8 9
10 12 13 17 15 19 20 21 20
Calculate a three month moving average for months three to nine. What would be your
forecast for the demand in month ten?
Solution
The three month moving average for months 3 to 9 is given by:
m3 = (10 + 12 + 13)/3 = 11.67
m4 = (12 + 13 + 17)/3 = 14.00
m5 = (13 + 17 + 15)/3 = 15.00
m6 = (17 + 15 + 19)/3 = 17.00
m7 = (15 + 19 + 20)/3 = 18.00
m8 = (19 + 20 + 21)/3 = 20.00
m9 = (20 + 21 + 20)/3 = 20.33
The forecast for month 10 is just the moving average for the month before that i.e. the
moving average for month 9 = m9 = 20.33.
Hence (as we cannot have fractional demand) the forecast for month 10 is 20.
WEIGHTED MOVING AVERAGE (WMA):
A Weighted Moving Average is an average of data calculated over a
period of time, where greater weight is attached to the most recent
data.A weighted moving average is a moving average where each historical demand
may be weighted differently
average: At = W1 Dt + W2 Dt-1 + W3 Dt-2 + ... + WN Dt-N+1
where:
N = total number of periods in the average
Wt = weight applied to period t's demand
Sum of all the weights = 1
forecast: Ft+1 = At = forecast for period t+1
Example:
Question: Given the weekly demand information and weights, what is the weighted moving
average forecast of the 5th period or week?
Week
1
2
3
4
Demand
820
775
680
655
Weights:
t-1
0.7
t-2
0.2
t-3
0.1
Solution
Week
1
2
3
4
5
Demand
820
775
680
655
Forecast
672
F5 = (0.1)(755)+(0.2)(680)+(0.7)(655)= 672
Note: More weight age would be given to recent most values.
EXPONENTIAL SMOOTHING
Exponential smoothing is a technique that can be applied to time series data, either to
produce smoothed data for presentation, or to make forecasts. The time series data
themselves are a sequence of observations. The observed phenomenon may be an
essentially random process, or it may be an orderly, but noisy, process. Whereas in the
simple moving average the past observations are weighted equally, exponential
smoothing assigns exponentially decreasing weights over time.
Exponential smoothing gives greater weight to demand in more recent periods, and less
weight to demand in earlier periods
average: At = a Dt + (1 - a) At-1 = a Dt + (1 - a) Ft
forecast for period t+1: Ft+1 = At
where:
At-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
Example:
The demand for a product in each of the last five months is shown below.
Month
1 2 3 4 5
Demand ('00s) 13 17 19 23 24
Apply exponential smoothing with a smoothing constant of 0.9 to generate a forecast for
demand for demand in month 6.
Solution
Applying exponential smoothing with a smoothing constant of 0.9 we get:
M1 = Y1 = 13
M2 = 0.9Y2 + 0.1M1 = 0.9(17) + 0.1(13) = 16.60
M3 = 0.9Y3 + 0.1M2 = 0.9(19) + 0.1(16.60) = 18.76
M4 = 0.9Y4 + 0.1M3 = 0.9(23) + 0.1(18.76) = 22.58
M5 = 0.9Y5 + 0.1M4 = 0.9(24) + 0.1(22.58) = 23.86
As before the forecast for month six is just the average for month 5= M5 = 2386