Erratic/Irregular Fluctuations, Business Mathematics and Statistics Video Lecture | SSC CGL Tier 2 - Study Material, Online Tests, Previous Year

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well this session I shall be introducing
the topic of time series analysis and
today we shall see its introduction
we'll see what is the definition of time
series what are its objectives its
components and the analysis of time
series so here we go let's begin with
the definition a series of observation
on a variable recorded after successive
intervals of time is called a time
series the success of intervals are
usually time intervals of any unit which
could be say 10 years a year a month a
week a day and R or so on it should be
noted here that the time series data are
bivariate data in which one of the
variable is the time which is
represented in the x-axis and it's
denoted by the letter T the other
variable could be anything else which is
denoted by the letter Y T Y with a
subscript T so now let's see what are
the objectives of time series
study the analysis of Time series
implies its decomposition into various
factors that affect the value of its
variable in a given period of time and
we shall be seeing very shortly what are
these components
the main objectives of time-series study
is to study the past behavior of data
and to make forecasts for future well we
can understand the present in a more
efficient way by analyzing the time
series data and this helps us further in
planning of the business and also helps
in comparative study of two different
series of data so now let's look at the
components of time series basically we
have four components of time series
which are the secular trend or simply
trend then we have the periodic or
oscillatory variations which are of two
types the long-term variation is called
the cyclic variation denoted by the
letter C and the short-term variation is
called the seasonal variation which is
denoted by the letter S and then lastly
we have the random or irregular
variation denoted by the letter R so
let's look at what all of these
components one by one let's begin with
the first one
that's the secular trend secular trend
or simply trend is the general tendency
of the data to increase or decrease or
stagnate over a long period of time most
of the business and economic time series
would reveal a tendency to increase all
the degrees over a number of years for
example the data relating to industrial
production or agricultural production or
say population bank deposit financial
deficits etc show an increase in trend
in general that is the magnitude of
these data have been raising over a
fairly long period of time as opposed to
this our time series may also reveal a
declining trend for example with the
improved medical facilities the death
rate is likely to show a declining trend
the time series data are usually taken
for a longer period of time say 10 years
or above and the longer the period the
better or more smooth the trend value
obtained would be it shows that the
variable has increased or decreased over
a long period of time okay so what are
the objectives of studying the trend or
measuring the trend first is to study
the past growth or decline of the series
and we know that this will help us in
our business decision making assuming
same behavior to continue in future this
could be used in forecasting influence
of other factors could be determined by
identifying and eliminating the trend
once we eliminate the trend the residual
factors will have the seasonal cyclic or
irregular variations and further the
trend of two or more time series data
could be used for their comparison in
the sense we can check which of the data
is increasing more rapidly which has
higher rate of increase or lower rate of
increase or decrease now if we look at
how the trend is we can demonstrate it
using a graph here let's say we have
here a graph with x-axis having the time
which is say in years 1 2 3 4 5 6 7 8 9
10 and the y-axis say for example has
the rare the in the population of a city
let's say the population is in say
Millions at certain point of time say
the population was here then increased
it was increasing say very gradually -
some values like this and then suddenly
it had a rapid increase then it
stagnated almost for certain period and
so on so when we join these dots we'll
get a smooth curve like this now if I
have to get the trend of this data I can
simply take the draw a straight line let
me use a different color here and we see
that it's gradually increasing in this
manner so we see that it is having a
rising trend this is the linear trend of
this data say the data is irregular
let's say we have we get a curve
something like this say which is related
to the production of certain products
then we can get the trend in this manner
trend is the line which would represent
or which will pass through the whole of
data so this is your trend line and here
this is the trend line we can have
linear trend we can also have nonlinear
trend we will see the analysis of that
in the upcoming videos this is just to
understand how the trend is alright so
next we have is the cyclic variation
now cyclic variation repeats after
regular interval of time the time is
called the period of oscillation it does
not follow any regular pattern however
it repeats after certain interval of
time regularly and the time interval is
usually long which could be say between
3 years to 10 years or so on we have a
cyclic variation something like this
let's say you have data and this the say
this is the the x axis has years as its
unit when we have data say let's say
it's increasing in a manner like this
and then gradually it comes down again
it goes up and again it comes down and
so on this is actually the cyclic manner
in which if we look at the data here we
see that if we look at this pattern it
has gradually increased which we can say
and then it reaches a maximum point
let's say that the business is at boom
and then there is a recession finally
there is a depression here the lowest
point and again there is recovery okay
so the cyclic manner in which the data
tends to increase and decrease after
regular interval of time is called the
cyclic variation okay now this time
interval need not necessarily be the
same and same as the Mackay in the case
of magnitude we need not have the same
magnitude every time each cycle could be
of a different length that's why we say
that it's not very regular but it
repeats after a certain interval of time
for example we have seen in the recent
past that the IT industry had its own
boom and recession
every certain years certain number of
years there is a boom in the industry
and then there is a recession all right
so now let's move on to the next one and
that is the seasonal variation or short
term variation the short term variation
is defined as the repetitive and
predictable movement around the
trendline
in one year or less so we see that this
is or a year or less it's detected by
measuring the quantity of interest for
small time intervals such as days weeks
months or quarter of a year and this
variation occurs over a short period of
time within a year due to seasonal
factors which could be the climatic
conditions of an area the customs and
the traditions etc so here the short
term variation let me write it first
this is the trend this is the cyclic
variation and then the short term
variation is something like this you
have the time and the other variable
here within a year if there is a
fluctuation increase in the sales and
then decrease and so on the short-term
fluctuation deme when we you know when
we look at these data say for example
and when we enlarge this we will see
that there will be some ups and downs
here that's what is denoted here an
example could be the sale of say warm
clothes woolen clothes which rises in
the month of December when it's very
cold and it decreases when it is hot of
course if you are in the northern
hemisphere we know that the month of
December and January is very cold and
then the demand for these kind of
dresses or it could be the demand of an
air-conditioner or any other product
which are seasonal or maybe a product
which is related to a certain festival
like for example the sale
crackers during the festival of
Deepavali okay and finally we have the
random variation denoted by the letter R
these random variations are
unpredictable in nature okay they are
unpredictable variations occurring due
to some random factors there is nothing
fixed and there's nothing regular it
does not follow any regular pattern and
can be obtained as a residue after the
elimination of all the other factors we
do not have a specific way of finding
the random factor variation the only way
is to eliminate all the remaining
factors and as a residue if there is any
random variation there in the in the
data it will be obtained although it
does not occur regularly it's in fact
are always very intense what an example
let's say the death rate if we look at
the death rate of the population in a
city let's say we denote the death rate
here okay
we may see that it is usually following
a pattern something like this maybe
which is no almost gradual in different
months say D we have the months here
okay
now suddenly what happens at a certain
point may be due to some say natural
disaster or something or maybe and it
may increase it may rise to a value a
very high value okay this will happen
may be due to a natural disaster or it
could happen due to any other say
climatic conditions or any other reason
okay maybe there is a thing an attack or
a war or say for example a disease which
has spread around in an area during
which suddenly in a certain month or
maybe between a certain time period the
death rate may increase and go to a very
high level in that city so now this is
due to a random factor it's not seasonal
it's not regular and
when I mean if it has occurred here when
will it occur next nobody knows it may
not it may never occur or it may occur
maybe again after a sudden maybe a
certain small period of time say there
is an earthquake or there is a flood or
something so that way you know the there
is no way by which we can predict these
factors and also we do not know what
would be the magnitude of these they do
not have anything regular okay so that
is the random variation and then let's
move on now to the analysis of time
series the purpose of analysis of time
series is to decompose it into its
various components which we have seen
just now and there are basically two
models based on the assumption regarding
the manner in which these components
have combined themselves these models
are additive model and the
multiplicative model so let's look at
both of these one by one
the additive model is based on the
assumption that the value of the
variables of a time series at a point of
time T is some of the four components
and symbolically this can be written as
Y T which is the the factor which we
have yester study is equal to the sum of
DT which is the trend at that point of
time plus the cyclic variation plus the
seasonal variation plus the random
variation we have the symbols t t c TS t
and RT as trend cyclic seasonal and
random components of time series add the
time T now this model assumes that all
the four components act independently
that is one component has no effect on
other irrespective of their magnitudes
whereas the multiplicative model we see
that it is based on the assumption that
the value of the variable of a time
series at a point of time T is the
product of the four components
symbolically it can be written as Y T
equals T T multiplied by CT multiplies
by st x r t where t t c TS t NR t we
know are the trend cyclic seasonal and
random components of time series at the
time T this model implies that although
the four components may be due to
different causes they are these are
strictly speaking not independent of
each other now which of these models are
commonly in use well there is very
little agreement about the validity of
the models which are assumed above okay
and it is not certain as to how the
components combine themselves we do not
know and consequently various mixed
types of models have also been suggested
where say two components are added and
the remaining two are multiplied or
three are added and one is multiplied to
it and so on but these combined or mixed
type of models are not used very often
the most commonly used models are either
the additive or the multiplicative
models though the additive model may be
appropriate in certain cases it is the
multiplicative model which characterizes
the majority of time series in economic
and business fields so we shall be
taking the multiplicative model for our
study okay so that was a brief
introduction of time series analysis
further we will see how to measure the
various components one by one we'll
start with secular trend then we'll move
on to the cyclic seasonal and random
variations gradually after one after the
other
so in the next video we will have we
will take up the measurement of secular
trend thanks for watching edgy PD of all
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