This document defines time series and provides examples of time series data. It discusses the components of time series, including trends, seasonal variations, cyclical variations, and random variations. Graphs of time series are presented, along with uses of time series analysis such as prediction and comparison. Mathematical models for time series analysis, including additive and multiplicative models, are also introduced.

1. NADEEM UDDIN
ASSOCIATE PROFESSOR
OF STATISTICS
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2. Definition of Time Series:
Time series refers to an arrangement and
presentation of statistical data in chronological
order. The statistical data is collected over a period
of time. According to Spiegel, “A time series is a
set of observations taken at specified times, usually
at equal intervals.” There exist various forces that
affect the values of the phenomenon in a time
series. These are also the components of the time
series analysis.

3. Examples of time series:
1- The monthly sales of mobiles in Pakistan.
2- The daily consumption of electricity in a sugar mill.
3- The annual rainfall recorded in Karachi.
4- Pakistan’s Imports from 1980 to 2000.
5- Pakistan’s Exports from 1980 to 2000.
6- The hourly temperature recorded in a city.
7- Yearly wheat production in Punjab.
8- The festival sale.
9- The number of patients suffering from flue in winter.
10- The increase in oil price due to Gulf war.

4. Graph of time series(Historigram):
Time series graphs help to show trends or patterns.
Time taking along x-axis and the values of time series along y-axis
and then join the plotted points by a straight line.
Example-1
An ice cream company, shows its sales over the past three years, taken
every three months, on the time series graph below. (Quarter 1 is for
January, February and March)

5. Uses of Time Series:
a) The most important use of studying time series is that it
helps us to predict the future behaviour of the variable based
on past experience.
b) It is helpful for business planning as it helps in comparing the
actual current performance with the expected one.
c) From time series, we get to study the past behaviour of the
phenomenon or the variable under consideration.
d) We can compare the changes in the values of different
variables at different times or places, etc.

6. Components(Variations) of Time Series:
You may have heard people saying that the price of
a particular commodity has increased or decreased
with time. This commodity can be anything like
gold, silver, any eatables, petrol, diesel etc. These
types of data are the time series of data.
fluctuation in a time series is data called the
components of the time series.

7. Following are the various components of the time series:
Secular Trend or Simple trend or Long term movement:
Secular trend refers to the general tendency of data to increase or
decrease or stagnate over a long period of time. Time series relating to
Economic, Business, and Commerce may show an upward or increasing
tendency. Whereas, the time series relating to death rates, birth rates,
share prices, etc. may show a downward or decreasing tendency.

8. Seasonal variations:
Seasonal variations refer to the changes that take place due
to the rhythmic forces which operate in a regular and
periodic manner. These forces usually have the same or
most similar pattern year after year. When we record data
weekly, monthly or quarterly, we can see and calculate
seasonal variations. Thus, when a time series consists of
data only based on annual figures, there will be seen no
seasonal variations. These variations may be due to seasons,
weather conditions, habits, customs or traditions. For
example, in summers the sale of ice-cream increases.

10. Cyclical variations:
Cyclical variations are due to the ups and downs recurring after a
period from time to time. These are due to the business cycle and
every organization has to phase all the four phases of a business cycle
some time or the other. Prosperity or boom, recession, depression, and
recovery are the four phases of a business cycle.

11. Random or irregular variations:
Random variations are fluctuations which are a result of unforeseen
and unpredictable forces. These forces operate in an absolutely
random or erratic manner and do not have any definite pattern. Thus,
these variations may be due to floods, famines, earthquakes, strikes,
etc.

12. Signal and noise
The systematic fluctuation or regular pattern in time series is
called signal.
The unsystematic or irregular pattern is called noise.
Linear and Non-Linear Trend:
If we plot the time series values on a graph in accordance with
time t. The pattern of the data clustering shows the type of trend. If
the set of data cluster more or less round a straight line, then the
trend is linear otherwise it is non-linear (Curvilinear).

13. Time Series Analysis:
There are certain phenomena that define our society such
as population, birth or death rates, incomes, etc. And the
one parameter which shows a clear variation in all of these
phenomena is time. So, when we plot a collection of
readings with respect to a phenomenon against time we
call it a series in time. Furthermore, we can also categorize
the nature of the trend (upward tendency or downward
tendency) which a particular series is displaying. The study
of these various series is what we call time series analysis.

14. The advantages of time series analysis are as follows:
Reliability:
Time series analysis uses historical data to represent
conditions along with a progressive linear chart. The
information or data used is collected over a period of time
say, weekly, monthly, quarterly or annually. This makes the
data and forecasts reliable.
Seasonal Patterns:
As the data related to a series of periods, it helps us to
understand and predict the seasonal pattern. For example,
the time series may reveal that the demand of clothes not
only increases during festival but also during the wedding
season.

15. Estimation of trends:
The time series analysis helps in the identification of
trends. The data tendencies are useful to managers as they
show an increase or decrease in sales, production, share
prices, etc.
Growth:
Time series analysis helps in the measurement of financial
growth. It also helps in measuring the internal growth of an
organization that leads to economic growth.

16. Mathematical Model for Time Series Analysis:
Mathematically, a time series is given as
yt = f (t)
Additive Model for Time Series Analysis:
If yt is the time series value at time t. Tt, St, Ct, and Rt are the trend
value, seasonal, cyclic and random fluctuations at time t respectively.
According to the Additive Model, a time series can be expressed as
yt = Tt + St + Ct + Rt.
This model assumes that all four components of the time series act
independently of each other.

17. Multiplicative Model for Time Series Analysis:
The multiplicative model assumes that the various components in a
time series operate proportionately to each other. According to this
model
yt = Tt × St × Ct × Rt