A brief intro to Time Series Components

1. Time Series Analysis

2. Time Series Analysis
A Time Series is a collection of observations made
sequentially in time.
According to Ya-lun Chou, “A Time Series may be
defined as a collection of readings belonging to
different time periods, of some economic variables
or composite of variables”
Examples: Financial time series, scientific time series,
Demographic time series, Meteorological time series

3. Time series data Vs. Cross Sectional data
Time series data Cross Sectional data
Time-series data is a set of observations
collected at usually equally spaced time
intervals.
Cross-sectional data are observations
that coming from different individuals or
groups at a single point in time
Time series data usually follows one
subject's changes over the course of time.
Cross-sectional data refers to data
collected by observing many subjects
(such as individuals, firms or
countries/regions) at the same point of
time.
It focuses on results gained over an
extended period of time, often within a
small area
It focuses on the information received
from surveys and opinions at a particular
time, in various locations, depending on
the information sought.
Example: The daily closing price of a
certain stock recorded over the last six
weeks is an example of time-series data
Example: The closing prices of a group of
20 different stocks on December 15, 1986
this would be an example of cross-
sectional data

4. Cont…
A study on random sample of 4000 graphics from 15 of the
world’s news papers published between 1974 and 1989
found that more than 75% of all graphics were time series.
Sales figures jan 98 - dec 01
0
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jun-97
jan-98
jul-98
feb-99
aug-99
mar-00
okt-00
apr-01
nov-01
maj-02

5. Cont…
Tot-P ug/l, Råån, Helsingborg 1980-2001
0
100
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1980-01-15
1981-01-15
1982-01-15
1983-01-15
1984-01-15
1985-01-15
1986-01-15
1987-01-15
1988-01-15
1989-01-15
1990-01-15
1991-01-15
1992-01-15
1993-01-15
1994-01-15
1995-01-15
1996-01-15
1997-01-15
1998-01-15
1999-01-15
2000-01-15
2001-01-15

6. Cont…
Mathematically,
Ut = f(t)
Ut : Value of the phenomenon or variable under
consideration at time t.
For example, (i) population of a country or region (Ut) in
different year (t)
(ii) Number of births and deaths (Ut) in different months
(t)
(iii) Sales of a store (Ut) in different months (t)
(iv) Temperatures (Ut) of a place in different days (t) etc.

7. Cont…
Time series gives a bi-variate distribution, one
of the variables being time (t) and the other
being the value (Ut)
 Time t may be yearly, monthly, weekly,
daily or even hourly
Usually equal interval

8. Components of a time series
 The pattern or behavior of the data in a time series
has several components.
 Theoretically, any time series can be decomposed
into:
Secular Trend or Long term movement
Periodic change or short term movement
(i) Seasonal (ii) Cyclical
Irregular or random components
 However, this decomposition is often not straight-
forward because these factors interact.

9. Trend component
 The trend component accounts for the gradual shifting of the
time series to relatively higher or lower values over a long
period of time.
 Trend is usually the result of long-term factors such as
changes in the population, demographics, technology, or
consumer preferences.

10. Cont…
 Downward trend: Declining birth or death rate
 Upward trend: Population growth, agricultural
production
 Mathematically trend may be Linear or non-linear
(curvi-linear)
 The term “long time period” is a relative term.

11. Periodic movements
Forces which prevent the smooth flow of
the series in a particular direction and
tend to repeat themselves over a period
of time
Seasonal variations or fluctuations
Cyclical variations or fluctuations

12. Seasonal Variations
The component responsible for the regular rise
or fall (fluctuations) in the time series during a
period not more than 1 year.
Fluctuations occur in regular sequence
(periodical)
The period being a month, a week, a day, or
even a fraction of the day, an hour etc.

13. Cont…

14. Cont…
Time series data depicted annually do not
represent seasonal variations. Seasonal
variations may be attributed to the following
reasons:
1. Natural forces : Weather or seasons
2. Man-made conventions: Habits, Fashions,
Customs or rituals etc.

15. Cyclical Variations
Cycle refers to recurrent variations/oscillatory
movements in time series
Cyclical variations usually last longer than a
year
One complete period is called “Cycle”

16. Cont…
Business Cycle (Four phase Cycle)
ProsPerity (Period of Boom)
recovery recession
dePression

17. Cont…

18. Irregular or Random Variations
Random or irregular or residual fluctuations
Beyond the control of human (unpredictable)
Earthquakes, Wars, Floods, Revolutions etc.
Short duration and non-repeating

19. Cont…

20. Purpose of Time series
 To identify the components, the net effects of
whose interaction is exhibited by the
movement of a time series
 To isolate, study, analyze and measure them
independently i.e; holding the other things
constant

21. Uses of Time Series
To study the past behavior of the variable
To formulate policy decisions and planning of
future operations.
To predict or estimate or forecast the behavior
of the phenomenon in future which is very
essential for business planning
To compare the changes in the values of
different phenomenon at different times

22. Decomposition of Time series
 Decomposition by Additive hypothesis
Ut= Tt + St + Ct + Rt
Ut= Time Series value at time t
Tt = Trend component
St = Seasonal component
Ct = Cyclical component
Rt= Random component

23. Cont…
 Decomposition by Multiplicative hypothesis
Ut= Tt x St x Ct x Rt
= logU˃ t= logTt + logSt + logCt + logRt

24. Measurement of Trend
The following methods are used to measure
“Trend”:
1. Graphic method
2. Method of Semi-Averages
3. Method of Curve fitting by principles of least
squares
4. Method of Moving average