ppt on maths

1. By:-
HEEMA
SUMANT
& ABHISHEK

2. Moving average method
 A quantitative method of forecasting or smoothing a
time series by averaging each successive group (no. of
observations) of data values.
 term MOVING is used because it is obtained by summing
and averaging the values from a given no of periods, each
time deleting the oldest value and adding a new value.

3.  For applying the method of moving averages the
period of moving averages has to be selected
 This period can be 3- yearly moving averages 5yr
moving averages 4yr moving averages etc.
 For ex:- 3-yearly moving averages can be calculated
from the data : a, b, c, d, e, f can be computed as :

4.  If the moving average is an odd no of values e.g., 3 years,
there is no problem of centring it. Because the moving
total for 3 years average will be centred besides the 2nd
year and for 5 years average be centred besides 3rd year.
 But if the moving average is an even no, e.g., 4 years
moving average, then the average of 1st 4 figures will be
placed between 2nd and 3rd year.
 This process is called centering of the averages. In
case of even period of moving averages, the trend
values are obtained after centering the averages a
second time.

5. Goals : –
 Smooth out the short-term fluctuations.
 Identify the long-term trend.

6. MERITS Of Moving average
method
 simple method.
 flexible method.
OBJECTIVE :-
 If the period of moving averages coincides with the
period of cyclic fluctuations in the data , such
fluctuations are automatically eliminated
 This method is used for determining seasonal, cyclic and
irregular variations beside the trend values.

7. LIMITATIONS Of Moving average
method
 No trend values for some year.
 M.A is not represented by mathematical function - not
helpful in forecasting and predicting.
 The selection of the period of moving average is a difficult
task.
 In case of non-linear trend the values obtained by this
method are biased in one or the other direction.

8. Moving Average Example
Year Units Moving
1994 2
1995 5 3
1996 2 3
1997 2 3.67
1998 7 5
1999 6
John is a building contractor with a record of a total
of 24 single family homes constructed over a 6-year
period. Provide John with a 3-year moving average
graph.
Avg.

9. Moving Average Example
Solution
Year Response Moving
Avg.
1994 2
1995 5 3
1996 2 3
1997 2 3.67
1998 7 5
1999 6
94 95 96 97 98 99
8
6
4
2
0
Sales
L = 3
No MA for 2 years

10. Calculation of moving average
based on period
 When period is odd-
example:-
Calculate the 3-yearly moving averages of the data
given below:
yrs 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989
Sales
(million
of
rupees)
3 4 8 6 7 11 9 10 14 12

11. year Sales,(millions
of rupees)
3-yearly totals 3-yearly moving
averages(trends)
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
3
4
8
6
7
1 1
9
1 0
1 4
1 2
1 5
1 8
2 1
2 4
2 7
3 0
3 3
3 6
5=(1 5/3)
6=(1 8/3)
7=(2 1/3)
8=(2 4/3)
9=(2 7/3)
1 0=(3 0/3)
1 1=(3 3/3)
1 2=(3 6/3)

12. In Figure, 3-yrs MA plotted on graph fall on a straight line, and the cyclic
f luctuation have been smoothed out. The straight Line is the required trend
line.
1980 1981 1982 1983 1984 1985 1986 1987 1988 1989
1
2
3
years
sales
4
6
8
10
12
Actual line

13. Calculation of moving average
based on period
When period is even:-
Example :-
Compute 4-yearly moving averages from the
following data:
year 1991 1992 1993 1994 1995 1996 1997 1998
Annual sale(Rs
in crores)
36 43 43 34 44 54 34 24

14. Year
(1)
Annual
sales (Rs in
crores) (2)
4-yearly
moving
total (T)
(3)
4-yearly
moving
averages (A)
(3)/4 {4}
4-yearly centred moving averages
OR (trend values) (5)
1991 36
1992 43
156 39
1993 43 (39+41)/2=80/2=40
164 41
1994 34 (41+43.75)/2=84.75/2=
42.375
175 43.75
1995 44 (43.75+41.50)/2=42.625
166 41.50
1996 54 (41.50+39)/2=40.25
156 39
1997 34
1998 24

15. sales
year

16. Method of least squares
 This is the best method for obtaining trend values.
 It provides a convenient basis for obtaining the line of best
fit in a series.
 Line of the best fit is a line from which the sum of the
deviations of various points on its either side is zero.
 The sum of the squares of the deviation of various
points from the line of best fit is the least. – That is
why this method is known as method of least squares.

17. Method of least squares
 Least squares, also used in regression analysis,
determines the unique trend line forecast which
minimizes the mean squares of deviations. The
independent variable is the time period and the
dependent variable is the actual observed value in
the time series
 equation of straight line trend:
Ŷ=a+bX

18. b = ∑XY -
∑X2
-n 2
a= -b
Where,
a = Y-intercept
b = slope of the best-fitting estimating line.
X = value of independent variable
Y = value of dependent variable
= mean of the values of the independent
variable
= mean of the values of the dependent
variable
x
x
y
xn
y x
y

19.  When =0
then b= ∑XY
∑X2
a= =∑Y/N
x
y

20. MERITS
 This method gives the trend values for the entire time
period.
 This method can be used to forecast future trend
because trend line establishes a functional relationship
between the values and the time.
 This is a completely objective method.

21. LIMITATIONS
 It requires some amount of calculations and may
appear tedious and complicated for some.
 Future forecasts made by this method are based only
on trend values; seasonal, cyclical or irregular
variations are ignored.
 If even a single item is added to the series a new
equation has to be formed.

22. Example:-
Determine trend line:-
Year: 2000 2001 2002 2003 2004
Sales(
in Rs
‘000):
35 56 79 80 40

23. Calculation
Year (x) Sales (Y) X
X2 XY
2000
2001
2002
2003
2004
35
56
79
80
40
-2
-1
0
1
2
4
1
0
1
4
-70
-56
0
80
80
TOTAL ∑Y=290 ∑X=0 ∑X2
=10 ∑XY=34

24. Calculation
 Now a = =∑ Y/ N =290/ 5=58
and b = ∑XY/∑X =34/10 =3.4
Substituting these values in equation of trend
line which is
Y=58+3.4X ,with 2002=0
y
Year
(x)
X
=x-2002
Trend values
(Y=58+3.4X)
2000
2001
2002
2003
2004
-2
-1
0
1
2
58+3.4×(-2)=51.2
58+3.4×(-1)=54.6
58+3.4×(0)=58.0
58+3.4×(1)=61.4
58+3.4×(2)=64.8

26. A problem involving all four
component of a time series
 firm that specializes in producing recreational
equipment . To forecast future sale firm has collected
the information.
 Given time series -1)trend 2)cyclic 3)seasonal
Quarterly
sales
Sales per quarter(× $10,000)
Year I II III IV
1991 16 21 9 18
1992 15 20 10 18
1993 17 24 13 22
1994 17 25 11 21
1995 18 26 14 25

27. Solution
 Procedure for describing information in time series
consist of four stages:-
1. finding seasonal indices- using moving average
method
2. Deseasonalized the given data.
3. Developing the trend line.
4.Finding the cyclical variation around the trend line.

28. Calculating the Seasonal Indexes
1. Compute a series of n -period centered moving
averages, where n is the number of seasons in the
time series.
2. If n is an even number, compute a series of 2-
period centered moving averages.
3. Divide each time series observation by the
corresponding centered moving average to identify
the seasonal-irregular effect in the time series.
4. For each of the n seasons, average all the
computed seasonal-irregular values for that season
to eliminate the irregular influence and obtain an
estimate of the seasonal influence, called the
seasonal index, for that season.

29. Deseasonalizing the Time Series
 The purpose of finding seasonal indexes is to
remove the seasonal effects from the time series.
 This process is called deseasonalizing the time
series.
 By dividing each time series observation by the
corresponding seasonal index, the result is a
deseasonalized time series.
 With deseasonalized data, relevant comparisons
can be made between observations in successive
periods.

30. Calculation of 4-Qr centered moving average :
Year
(1)
Quarter
(2)
Actual
sales
(3)
Step 1:
4-Qr moving
total
(4)
Step 3:
4-Qr centered
moving average
(6)
Step:4
% of actual to
moving averages
(7)={(3)×100}÷(6)
1991 I.
II.
III.
IV.
16
21
9
18
64
63
15.875
15.625
56.7
115.2
1992
I.
II.
III.
IV.
15
20
10
18
62
63
63
65
15.625
15.750
16.000
16.750
96.0
127.0
62.5
107.5
1993
I.
II.
III.
IV.
17
24
13
22
69
72
76
76
17.625
18.500
19.000
19.125
96.5
129.7
68.4
115.0

31. year
(1)
quarter
(2)
Actual sales
(3)
Step 1:
4-Qr
moving
total
(4)
Step 2:
4-Qr
moving
average
(5)=(4)÷4
Step 3:
4-Qr
centered
moving
average
(6)
Step:4
% of actual to
moving
averages
(7)={(3)×100}÷(
6)
1994
I.
II.
III.
IV.
17
25
11
21
77
75
74
75
19.25
18.75
18.50
18.75
19.000
18.625
18.625
18.875
89.5
134.2
59.1
111.3
1995
I.
II.
III.
IV.
18
26
14
25
76
79
83
19.00
19.75
20.75
19.375
20.250
92.9
128.4

32. Computing the seasonal index
Year I II III IV
1991 - - 56.7 115.2
1992 96.0 127.0 62.5 107.5
1993 96.5 129.7 68.4 115.0
1994 89.5 134.2 59.1 111.3
1995 92.9 128.4 - -
modified sum=188.9 258.1 121.6 226.3
modified mean: Qr I: 188÷2=94.45
II: 258.1÷2=129.05
III: 121.6÷2=60.80
IV: 226.3÷2=113.15
397.45
Quarter indices × Adjusting factor = seasonal indices
=400/397.45=1.0064
I 94.45 1.0064 = 95.1
II 129.05 1.0064 = 129.9
III 60.80 1.0064 = 61.2
IV 113.15 1.0064 = 113.9
sum of seasonal indices = 400.1

33. Calculation 0f deaseasonalised
time series valuesYear
(1)
Quarter
(2)
Actual sales
(3)
Seasonal index/100
(4)
Deseasonalized sales
(5)=(3)÷(4)
1991 I
II
III
IV
16
21
9
18
0.951
1.299
0.612
1.139
16.8
16.2
14.7
15.8
1992 I
II
III
IV
15
20
10
18
0.951
1.299
0.612
1.139
15.8
15.4
16.3
15.8
1993 I
II
III
IV
17
24
13
22
0.951
1.299
0.612
1.139
17.9
18.5
21.2
19.3
1994 I
II
III
IV
17
25
11
21
0.951
1.299
0.612
1.139
17.9
19.2
18.0
18.4
1995 I
II
III
IV
18
26
14
25
0.951
1.299
0.612
1.139
18.9
20.0
22.9
21.9

34. Identifying the trend component
Year
(1)
Qr
(2)
Deseasonalized sales
(3)
Translating or
coding the time
variable(4)
x
(5)=(4)×2
xY
(6)=(5)×(3)
x²
(7)=(5)²
1991 I
II
III
IV
16.8
16.2
14.7
15.8
-9.5
-8.5
-7.5
-6.5
-19
-17
-15
-13
-319.2
-275.4
-220.5
-205.4
361
289
225
169
1992 I
II
III
IV
15.8
15.4
16.3
15.8
-5.5
-4.5
-3.5
-2.5
-11
-9
-7
-5
-173.8
-138.6
-114.1
-79.0
121
81
49
25
1993
Mean
I
II
III
IV
17.9
18.5
21.2
19.3
-1.5
-0.5
0*
0.5
1.5
-3
-1
1
3
-53.7
-18.5
21.2
57.9
9
1
1
9
1994 I
II
III
IV
17.9
19.2
18.0
18.4
2.5
3.5
4.5
5.5
5
7
9
11
89.5
134.4
162.0
202.4
25
49
81
121
1995 I
II
III
IV
18.9
20.0
22.9
21.9
∑Y=360.9
6.5
7.5
8.5
9.5
13
15
17
19
245.7
300.0
389.3
416.1
169
225
189
361
We assign mean=0 ,b/w II&III(1993) & measure translated time,x,by 0.5 because
periods is even

35. b= ∑XY
∑X2
a= =∑Y/N

36. Identifying the cyclical variation
Year
(1)
Quarter
(2)
Deseasonalised sales
(3)
Ŷ=a+bx
(4)
(Y×100)/Ŷ
percent of trend
(5)
1991 I
II
III
IV
16.8
16.2
14.7
15.8
18+0.16(-19)=14.96
18+0.16(-17)=15.28
18+0.16(-15)=15.60
18+0.16(-13)=15.92
112.3
106.0
94.2
99.2
1992 I
II
III
IV
15.8
15.4
16.3
15.8
18+0.16(-11)=16.24
18+0.16(-9)=16.56
18+0.16(-7)=16.88
18+0.16(-5)=17.20
97.3
93.0
96.6
91.9
1993 I
II
III
IV
17.9
18.5
21.2
19.3
18+0.16(-3)=17.52
18+0.16(-1)=17.84
18+0.16(1)=18.16
18+0.16(3)=18.48
102.2
103.7
116.7
104.4
1994 I
II
III
IV
19.9
19.2
18.0
18.4
18+0.16(5)=18.80
18+0.16(7)=19.12
18+0.16(9)=19.44
18+0.16(11)=19.76
95.2
100.4
92.6
93.1
1995 I
II
III
IV
18.9
20.0
22.9
21.9
18+0.16(13)=20.08
18+0.16(15)=20.40
18+0.16(17)=20.72
18+0.16(19)=21.04
94.1
98.0
110.5
104.1

37. Graph of time series , trend line and 4-Qr centered moving
average for sales data
4-Qr centered moving average
(both trend & cyclical component)
sales
X=0