Time_series_analysis1a

8/4/2019 Time_series_analysis1a

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Time series analysis


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Sales

2000-01 Apr 6.6

May 6.7

Jun 5.9

Jul 4.9

Aug 5.8Sep 6.4

Oct 6.2

Nov 6.3

Dec 9.6

Jan 7.5

Feb 7.6Mar 8.1

2001-02 Apr 6.6

May 6.7

Jun 5.9

Jul 4.9

Aug 5.8

Sep 6.4Oct 6.2

Nov 6.3

Dec 9.6

Jan 7.5

Feb 7.6

Mar 8.1

Copy and paste monthly

sales year on year onebelow the other


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Sales MA-12

2000-01 Apr 6.6

May 6.7

Jun 5.9

Jul 4.9

Aug 5.8Sep 6.4 6.80

Oct 6.2 6.80

Nov 6.3 6.80

Dec 9.6 6.80

Jan 7.5 6.80

Feb 7.6 6.80

Mar 8.1 6.80

2001-02 Apr 6.6 6.80

May 6.7 6.80

Jun 5.9 6.80

Jul 4.9 6.80

Aug 5.8 6.80

Sep 6.4 6.80Oct 6.2 6.89

Nov 6.3 7.01

Dec 9.6 7.24

Jan 7.5 7.52

Feb 7.6 7.60

Mar 8.1 7.61

Formula for Moving

average = Enter@average(c2..c13) in d2

Copy and paste the

formula to d3

D3 will read as@average(c3..c14)

Now copy till D.., leaving

the last 6

Next center Movingaverage


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Canceling random variation

Inherent in the collection of data taken overtime is some form of random variation.

There exist methods for reducing of canceling

the effect due to random variation. An often-used technique in industry is

"smoothing".

This technique, when properly applied, revealsmore clearly the underlying trend, seasonaland cyclic components.


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Moving Average Method

It is one of the most popular method for

calculating Long Term Trend.

This method is also used for Seasonal

fluctuation, cyclical fluctuation & irregular

fluctuation.

In this method we calculate the Moving

Average for certain periods.


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Sales and Moving Averages


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Seasonal variation: Seasonal variation are short-term

fluctuation in a time series which occurperiodically in a year. This continues torepeat year after year.

The major factors that are weather conditionsand customs of people.

More woolen clothes are sold in winter than inthe season of summer .

each year more ice creams are sold in summer

and very little in Winter season.

The sales in the departmental stores are moreduring festive seasons that in the normal days.


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Seasonal Variation in one year or less

= ratio actual to Moving averageMonth Sales MA MAC Act/Mac

2000-01 Apr 1 6.6

May 2 6.7

Jun 3 5.9

Jul 4 4.9

Aug 5 5.8

Sep 6 6.4 6.800

Oct 7 6.2 6.825 6.813 0.910

Nov 8 6.3 6.875 6.850 0.920

Dec 9 9.6 6.983 6.929 1.385Jan 10 7.5 7.175 7.079 1.059

Feb 11 7.6 7.292 7.233 1.051

Mar 12 8.1 7.383 7.338 1.104

2001-02 Apr 13 6.9 7.467 7.425 0.929

May 14 7.3 7.558 7.513 0.972

Jun 15 7.2 7.350 7.454 0.966

Jul 16 7.2 7.375 7.363 0.978Aug 17 7.2 7.417 7.396 0.974

Sep 18 7.5 7.467 7.442 1.008

Oct 19 7.2 7.533 7.500 0.960

Nov 20 7.4 7.600 7.567 0.978

Dec 21 7.1 7.725 7.663 0.927

Jan 22 7.8 7.808 7.767 1.004

Feb 23 8.1 7.775 7.792 1.040Mar 24 8.7 7.692 7.733 1.125


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Averaging Moving averages of different years

and rectification

Copy and paste seasonality ratios in year wise columns,calculate averages and adjust for total =12

2000-

01

2001-

02

2002-

03

2003-

04

2004-

05

2005-

06

2006-

07

2007-

08

2008-

09 Mean Index

Apr 0.929 0.999 1.130 0.884 1.115 1.040 1.028 1.012 1.017 1.019

May 0.972 1.047 1.060 0.988 1.120 1.073 1.151 0.932 1.043 1.045Jun 0.966 1.119 1.079 1.038 1.010 1.044 1.100 0.949 1.038 1.040

Jul 0.978 1.052 0.887 1.186 1.107 0.952 0.987 0.965 1.014 1.016

Aug 0.974 0.877 0.990 0.886 0.924 0.898 1.030 0.940 0.942

Sep 1.008 0.846 0.930 0.909 0.844 0.893 0.859 1.040 0.916 0.918

Oct 0.910 0.960 0.985 1.019 0.967 0.815 0.853 0.828 1.030 0.930 0.931

Nov 0.920 0.978 0.989 0.977 1.012 0.858 0.909 0.876 1.037 0.951 0.953

Dec 0.927 0.987 0.988 0.999 0.974 0.962 0.988 0.974 0.975 0.977

Jan 1.059 1.004 0.980 0.981 0.999 1.038 1.068 1.050 1.017 1.022 1.024

Feb 1.051 1.040 0.939 1.004 0.936 1.119 1.106 1.082 1.023 1.033 1.035

Mar 1.104 1.125 1.016 1.143 0.889 1.276 1.128 1.110 1.099 1.101

Sum 11.977 12.000


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De-seasoning data

Actual sales

De-seasonalised sales = _______________

Monthly seasonal Index


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Month Sales MAC Sales/Mac Index Salesd

2000-01 Apr 1 6.6 1.019 6.477

May 2 6.7 1.045 6.413

Jun 3 5.9 1.040 5.674

Jul 4 4.9 1.016 4.821

Aug 5 5.8 0.942 6.160Sep 6 6.4 0.918 6.973

Oct 7 6.2 6.81 0.910 0.931 6.656

Nov 8 6.3 6.85 0.920 0.953 6.614

Dec 9 9.6 6.93 1.385 0.977 9.829

Jan 10 7.5 7.08 1.059 1.024 7.327

Feb 11 7.6 7.23 1.051 1.035 7.342Mar 12 8.1 7.34 1.104 1.101 7.357

2001-02 Apr 13 6.9 7.43 0.929 1.019 6.772

May 14 7.3 7.51 0.972 1.045 6.987

Jun 15 7.2 7.45 0.966 1.040 6.924

Jul 16 7.2 7.36 0.978 1.016 7.084

Aug 17 7.2 7.40 0.974 0.942 7.646

Sep 18 7.5 7.44 1.008 0.918 8.171

Oct 19 7.2 7.50 0.960 0.931 7.729

Nov 20 7.4 7.57 0.978 0.953 7.769

Dec 21 7.1 7.66 0.927 0.977 7.270

Jan 22 7.8 7.77 1.004 1.024 7.620

Feb 23 8.1 7.79 1.040 1.035 7.825

Mar 24 8.7 7.73 1.125 1.101 7.902


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Linear Trend: Least Squares

Slope of the best fitting line

sum XY- n*X mean*Y mean

b =_____________________________

Sum X^2 n * Xmean^2

Easy to remember variance xy/variance x^2

Since X mean and Y mean is on the regression

line

A = Y mean b* X mean


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Simplify X in Time series

X = 2005, 2006, 2007, 2008, 2009,2010, 2011 X = -3, -2,-1, 0, 1, 2, 3 . X mean = 0

X = 2005, 2006, 2007, 2007.5, 2008,

2009,2010 X= 2*(-2.5), 2*(-1.5), 2*(-0.5), 2*(.5), 2*(1.5),

2*(2.5)

X = -5, -3, -1,1, 3,5 X mean = 0 N*X mean * Y mean = 0*Y mean =0

b = sum XY/sum X^2, a = Y mean


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SUMMARY OUTPUT

Regression Statistics

Multiple R 0.799894

R Square 0.639831

Adjusted R

Square 0.636586

Standard

Error 0.771784

Observatio

ns 113

ANOVA

df SS MS F

Significanc

e F

Regression 1 117.4554 117.4554 197.1884 2.29E-26

Residual 111 66.11723 0.595651

Total 112 183.5726

Coefficient

s

Standard

Error t Stat P-value Lower 95% Upper 95%

Lower

95.0%

Upper

95.0%

Intercept 6.761015 0.146176 46.25262 2.27E-74 6.471358 7.050672 6.471358 7.050672

Month 0.031256 0.002226 14.04238 2.29E-26 0.026845 0.035666 0.026845 0.035666

Intercept and coefficients are

different as periods (x) is

numbered from 1 to 113


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Cyclical Variations:Cyclical variations are recurrent upward or downward

movements in a time series but the period of cycle is

greater than a year. Also these variations are not regularas seasonal variation.

A business cycle showing these oscillatory movements hasto pass through four phases-prosperity, recession,depression and recovery. In a business, these four phasesare completed by passing one to another in this order.


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Sales MA-12 MAC Trend MAC/trend

2000-01 Apr 6.6 6.57

May 6.7 6.61

Jun 5.9 6.64

Jul 4.9 6.67

Aug 5.8 6.71Sep 6.4 6.80 6.80 6.74 1.01

Oct 6.2 6.80 6.80 6.78 1.00

Nov 6.3 6.80 6.80 6.81 1.00

Dec 9.6 6.80 6.80 6.85 0.99

Jan 7.5 6.80 6.80 6.88 0.99

Feb 7.6 6.80 6.80 6.91 0.98Mar 8.1 6.80 6.80 6.95 0.98

2001-02 Apr 6.6 6.80 6.80 6.98 0.97

May 6.7 6.80 6.80 7.02 0.97

Jun 5.9 6.80 6.80 7.05 0.96

Jul 4.9 6.80 6.80 7.08 0.96

Aug 5.8 6.80 6.80 7.12 0.96

Sep 6.4 6.80 6.81 7.15 0.96

Oct 6.2 6.82 6.82 7.19 0.97

Nov 6.3 6.83 6.88 7.22 0.99

Dec 9.6 6.93 7.06 7.26 1.02

Jan 7.5 7.19 7.29 7.29 1.04

Feb 7.6 7.39 7.47 7.32 1.04

Mar 8.1 7.55 7.55 7.36 1.04

Cyclical

Factors


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Time Series Model Addition Model:

Y = T + S + C + IWhere:- Y = Original Data

T = Trend Value

S = Seasonal FluctuationC = Cyclical Fluctuation

I =

I = IrregularFluctuation

Multiplication Model:Y = T x S x C x I

or

Y = TSCI


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Irregular variation:Irregular variations are fluctuations in time series that are

short in duration, erratic in nature and follow no

regularity in the occurrence pattern. These variationsare also referred to as residual variations since by

definition they represent what is left out in a time series

after trend ,cyclical and seasonal variations. Irregular

fluctuations results due to the occurrence of unforeseen

events like :

Floods,

Earthquakes,

Wars,

Famines


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Ad d T i


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Advanced Topic

Parabolic Curve for Trend

Many times the line which draw by Least SquareMethod is not prove Line of best fit because it is

not present actual long term trend So we distributedTime Series in sub- part and make following equation:-

Yc = a + bx + cx2 If this equation is increase up to second degree then it is

Parabola of second degree and if it is increase up to thirddegree then it Parabola of third degree. There are three

constant a, b and c.

Its are calculated by following three equation:-


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If we take the deviation from Mean year then the allthree equation are presented like this:

422

2

2

XcXaYX

XbXY

XCNaY

4322

32

2

XcXbXaYX

XcXbXaXY

XcXbNaY

Parabola of second degree:-


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Year Production Dev. From MiddleYear(x)xY x2 x2Y x3 x4 Trend ValueY = a + bx + cx2

1992

1993

1994

1995

1996

5

7

4

9

10

-2

-1

0

1

2

-10

-7

0

9

20

4

1

0

1

4

20

7

0

9

40

-8

-1

0

1

8

16

1

0

1

16

5.7

5.6

6.3

8.0

10.5

= 35 = 0 =12 = 10 = 76 = 0 = 34Y X

XY

2X

YX2

3

X 4

X

Example:Draw a parabola of second degree from the following data:-

Year 1992 1993 1994 1995 1996

Production (000) 5 7 4 9 10


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422

2

2

XcXaYX

XbXY

XNaY

Now we put the value of

35 = 5a + 10c (i)

12 = 10b (ii)76 = 10a + 34c .. (iii)

From equation (ii) we get b = = 1.2

NXXXXYYX ,&,,,,,432

10

12

We take deviation from middle year so the

equations are as below:


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Equation (ii) is multiply by 2 and subtracted from (iii):

10a + 34c = 76 .. (iv)10a + 20c = 70 .. (v)

14c = 6 or c = = 0.43

Now we put the value of c in equation (i)

5a + 10 (0.43) = 35

5a = 35-4.3 = 5a = 30.7

a = 6.14

Now after putting the value of a, b and c, Parabola of second

degree is made that is:

Y = 6.34 + 1.2x + 0.43x2

14

6


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Equation (ii) is multiply by 2 and subtracted from (iii):

10a + 34c = 76 .. (iv)10a + 20c = 70 .. (v)

14c = 6 or c = = 0.43

Now we put the value of c in equation (i)

5a + 10 (0.43) = 35

5a = 35-4.3 = 5a = 30.7

a = 6.14

Now after putting the value of a, b and c, Parabola of second

degree is made that is:

2

14

6

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Time_series_analysis1a