4b014time series analysis

8/6/2019 4b014Time Series Analysis

1/18

Name of Institution

1

TIME-SERIES ANALYSIS


2/18

Name of Institution

2

TIME-SERIES ANALYSIS When data is collected, observed or recorded at successive intervals

of time, such data are referred to as Time Series i.e a Time Series

consists of statistical data in chronological order (in accordance with

time).

When we observe numerical data at different points of time, the set

of observations are known as Time Series.

Ex. Data of production, sales, imports etc. at different points oftime.


3/18

Name of Institution

3

COMPONENTS OF

TIME SERIES1. Secular Trend

The general movement persisting over a long period of time

represented by the diagonal line drawn trough the irregular

curve is called Secular Trend.

The general tendency of the data to grow or decline over a

long period of time.

Sudden, Erratic and short term movements in either direction

have nothing to do with trend.

Example: GDP growth, population growth, prices, literacy rateetc.


4/18

Name of Institution

4

2. Seasonal Variations SV are the fluctuations which completes the whole sequence of

change within the span of a year and has about the same pattern

year after year.

It includes any kind of variation which is of periodic natures &

whose repeating cycles are of relatively short durations.

SV can be because of:-

- Climate and weather conditions.

- Customs, traditions & habits.


5/18

Name of Institution

5

3. Cyclical Variations These refers to the recurrent variations in time series that usually

last longer than a year and are regular.

Cyclical fluctuations are long term movements that represent

consistently recurring rises and declines in activity.

Example: Business cycles.

4. Irregular Variations

Refers to variations in business activities which do not repeat in a

definite pattern. These are variations caused by unpredictable factors like sudden

political instability, earthquakes, strikes, wars etc.


6/18

Name of InstitutionUSES OF A TIME SERIES

It enables us to study the past behaviour of the

phenomenon under consideration.

It helps to study the components which are of paramount

importance to a businessman in the planning of futureoperations and in the formulation of executive and policy

decisions.

It helps to compare the actual current performance or

accomplishments with the expected ones and analyze

the causes of such variations.

It helps us to compare the changes in the values of

different phenomenon at different places.

6


7/18

Name of Institution

7

METHODS OF

MEASUREMENTS1. Freehand or Graphical Method.

2. Semi-Average Method.

3. Method of Moving Averages, and

4. Least Squares Methods.


8/18

Name of Institution

8

FITTING

OFSTRAIGHT LINE TRENDBY METHOD OF LEAST SQUARES

Let Yc=a+bX represents equation of a straight line(trend line),where:--

Yc : represents calculated values of Y.

a : designates the Y-intercept.b : represents the slope of the line, i.e., rate of change of

Y per unit change in X.

X : The X variable in time series analysis representstime.

In order to determine the values of constants a and b, the followingtwo normal equations are to be solved:-

2XbXaXY

XbNaY

77!7

7!7


9/18

Name of Institution

9

Ques 1. Determine the trend line which best fits the following data and also find thetrend values for the given years.

Year Sales

(in Rs. 000)

2000 35

2001 56

2002 79

2003 80

2004 40


10/18

Name of Institution

10

Sol.

Year Sales

(in Rs. 000):

(Y)

Deviations from

middle year:

(X), i.e. 2002

X2 XY Trend

values

Yc

2000 35 -2

2001 56 -1

2002 79 0

2003 80 1

2004 40 2

Y = X= 0 X2= XY=


11/18

Name of Institution

11

Ques 2. Given below are the figures of production (in lakh kg.) of a sugar factory.Fit a straight line trend by the least square method and tabulate the trend. Also

estimate the trend for the year2006.

Year Production

1999 40

2000 45

2001 46

2002 42

2003 47

2004 502005 46


12/18

Name of Institution

12

Sol.

Year Production

(Lac Kgs.):

(Y)

Deviations from

middle year:

(X), i.e. 2002

X2 XY Trend

values

Yc

1999 40 -3

2000 45 -2

2001 46 -1

2002 42 0

2003 47 1

2004 50 2

2005 46 3

Y = X= 0 X2= XY=


13/18

Name of Institution

13

Ques 3. Fit a straight line trend by the method of least squares to the followingdata. Assuming that the same rate of change continues what would be the

predicted earnings for the year 1992?

Year Earnings

(Rs. cr.)

1981 38

1982 40

1983 65

1984 72

1985 69

1986 60

1987 87

1988 95


14/18

Name of Institution

14

Sol.

Year Earnings

(Rs. cr.):

(Y)

Deviations

from middle

year:

i.e. 1984.5

X2 XY Trend

values

Yc

1981 38 -3.5

1982 40 -2.5

1983 65 -1.5

1984 72 -0.5

1985 69 0.5

1986 60 1.5

1987 87 2.5

1988 95 3.5

N = 8 Y = X= 0 X2

= XY=


15/18

Name of Institution

15

Qs 4. From the data given below fit a straight line trend by the method of leastsquares and find the trend values. Calculate the estimated milk consumption for

the year 1997, assuming same trend continues.

Year Milk

consumption

(million litres)1988 102.3

1989 101.9

1990 105.8

1991 112.0

1992 114.8

1993 118.7

1994 124.5

1995 129.9


16/18

Name of Institution

16

Sol.

Year Milk

consumption

(mln. Lts) :

(Y)

Deviations

from middle

year:

i.e. 1991.5

X2 XY Trend

values

Yc

1988 102.3 -3.5

1989 101.9 -2.5

1990 105.8 -1.5

1991 112.0 -0.5

1992 114.8 0.5

1993 118.7 1.5

1994 124.5 2.5

1995 129.9 3.5

N = 8 Y = X= 0 X2

= XY=


17/18

Name of Institution

17

Qs 5. The following data show the experience of machine operators and theirperformance ratings as given by the number of good parts turned out per 100

pieces.

Develop a linear trend for this data and estimate the probable performance if an

operator has 10 years experience.

Operator

experience

Performance

Rating

16 8712 88

18 89

4 68

3 78

10 80

5 75

12 83


18/18

Name of Institution

18

Sol.

Operator

experience

(X)

Performance

Rating

(Y)

X2 XY

16 87

12 88

18 89

4 68

3 78

10 80

5 75

12 83

X= Y = X2= XY=

4b014time series analysis

Documents