Lecture No. 9
Held on: Monday, August 28, 2000
Notes Prepared by: Nitin Navish Gupta & Rohit Govil
Additive
trend and Ratio seasonality model
We have
underlying trend with randomness around it. We assume that the demand follows
the following relation:
Dt = (m
+ct)dt +
et
m = initial
average demand (at t = 0)
c =
Increment (per unit time)
dt
=
Seasonality factor
et
= error
(randomly distributed)
At any time
t we need to keep,
St-1
= Best estimate of average at time t-1
Tt-1
= Best estimate of increment at time t-1
It-L,
, It-1 = Best estimates of seasonality
factors.
At time t
, demand Dt is observed. The parameters are then updated as
follows:
St
= a(
Dt/ It-L) + (1-a)(
St + St-1)
Tt
= b(
St - St-1) + (1-b)
Tt-1
It
= g(
Dt/ St) + (1-g)
It-L
where
a,b and
g are
appropriately chosen constants,
Now,
forecast at time t for time t+1 is given by:
Ft,t+1
= (St + Tt) It-L+1
Similarly,
forecast at time t for time t+m is given by:
Ft,t+m
= (St + mTt) It-L+m
Solved
Example:
1992
1993
1994
Q1
10
12
16
Q2
20
23
33
Q3
26
30
34
Q4
17
22
26
a=0.2,b=0.1 and
g=0.1
First we
need to find S0 and T0
D92
= 18.25 (avg. demand in 1992)
D93
= 21.75 (avg. demand in 1993)
T0
= (21.75-18.25)/4 = 0.875 = avg. increment per quarter
In the
middle of last quarter,
S0
= 21.75 + 1.5 T0 = 23.06
/* why 1.5 T0 is needed :
average demand in middle of last quarter = demand after second quarter + increment in third quarter + increment
in half of fourth quarter = 21.75 + T0 + .5 T0 = 23.06
*/
The estimate
of average in different time periods is as follows:
SQ,92
SQ,93
Q1
16.94
20.4
Q2
17.8
21.3
Q3
18.7
22.2
Q4
19.6
23.06
Seasonal
Indices
1992
1993
Avg.
Normalised average
Q1
0.5904
0.5872
0.5888
.5872
Q2
1.123
1.079
1.101
1.1038
Q3
1.391
1.352
1.38
1.3835
Q4
0.869
0.9539
0.92
.9223
We have
(1/L)ε
dt
=1.
F0,1
= (S0 + T0)I93 = (23.06
+0.875)x0.59
F0,2
= (23.06 + 2x0.875)x 1.1
and so
on
.
DQ1,94
= 16
SQ1,94
= 0.2(16/0.59) + 0.8(23.06 + 0.875) = 24.57
T1
= b(
SQ1,94 SQ4,93) + (1-b)
T0 = 0.1(24.57-23.06) + 0.9x0.875 = 0.9385
IQ1,94
= g(
DQ1,94/ SQ1,94) + (1-g)
IQ1,93 = 0.1(16/24.57) + 0.9x0.59 = 0.5961
Multiply
each Ii with (4/εIi)
to get normalized values.
Causal or
Explanatory methods
Here we try
to relate forecasts to factors in the economy that cause the trend, seasonal and
fluctuation . Factors used in causal models are of several types : disposable
income, new marriages, housing starts, inventories, action of competitors
etc.
Regrtession analysis is used to develop mathematical relationship between forecast and causal functions. If only one independent (causal) variable is used to estimate the dependent (forecast) variable, the relationship between the two is established using simple regression analysis
\
Yi
= f (X1, X2, X3,
.., Xn)
= a X1 + b
(say)
Yi
is the demand depending on various factors Xi
Aim: How to come
up with best values for a and b.
a and b
are selected to minimise the sum of squared error in the observed
data
For a given
set of data points (X1,
Y1), (X2, Y2),
, (Xn,
Yn)
We need to
minimize the error, E = ε
(Yi (a Xi + b))2 .
ή (dE/da) = 0
and (dE/db) = 0
Denoting,
Xn = (1/n)ε
Xi and Yn = (1/n) ε
Yi
we
get,
b = (ε
XiYi - n Xn
Yn)/(ε
Xi2 -
Xn2)
a = Yn - b Xn
Syx
= Standard error of estimate = Φ(ε(Yi
Yi)2)/(n-2) where Yi
is the forecasted value
And the
demand is assumed to be normally distributed with mean as the forecast and
standard deviation given by the standard error
In many
cases, error is normally distributed i.e. Y @ N (aX + b,
S2yx)
We have
probability of the forecast lying within a certain range
e.g. P(YΞ (aX + b
+ 2Syx)) @ 95%