TIME SERIES ANALYSIS COMPONENTS

TIME SERIES
ANALYSIS
BirinderSingh,AssistantProfessor,PCTE
Baddowal
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WHAT IS A TIME SERIES?
 Essentially, Time Series is a sequence of numerical
data obtained at regular time intervals.
 Occurs in many areas: economics, finance,
environment, medicine
 The aims of time series analysis are
 to describe and summarize time series data,
 fit models, and make forecasts
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WHY ARE TIME SERIES DATA DIFFERENT
FROM OTHER DATA?
 Data are not independent
 Much of the statistical theory relies on the
data being independent and identically
distributed
 Large samples sizes are good, but long
time series are not always the best
 Series often change with time, so bigger isn’t
always better
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BirinderSingh,AssistantProfessor,
PCTEBaddowal

WHAT ARE USERS LOOKING FOR IN AN
ECONOMIC TIME SERIES?
 Important features of economic indicator series
include
 Direction
 Turning points
 In addition, we want to see if the
series is increasing/decreasing
more slowly/faster than it was
before
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WHEN SHOULD TIME SERIES ANALYSIS
BEST BE USED?
 We do not assume the existence of deterministic model
governing the behaviour of the system considered.
 Instances where deterministic factors are not readily
available and the accuracy of the estimate can be
compromised on the need..(be careful!)
 We will only consider univariate time series
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FORECASTING HORIZONS
 Long Term
 5+ years into the future
 R&D, plant location, product planning
 Principally judgement-based
 Medium Term
 1 season to 2 years
 Aggregate planning, capacity planning, sales forecasts
 Mixture of quantitative methods and judgement
 Short Term
 1 day to 1 year, less than 1 season
 Demand forecasting, staffing levels, purchasing,
inventory levels
 Quantitative methods
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EXAMPLES OF TIME SERIES DATA
 Number of babies born in each hour
 Daily closing price of a stock.
 The monthly trade balance of Japan for each year.
 GDP of the country, measured each year.
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TIME SERIES
 Coordinates (t,x) is established in the 2
axis
 (1, 44,320)
 (2, 52,865)
 (3, 53,092)
 etc..
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Exports
30,000
35,000
40,000
45,000
50,000
55,000
60,000
65,000
70,000
75,000
80,000
1988 1990 1992 1994 1996 1998 2000

TIME SERIES
 A graphical representation of time series.
 We use x as a function of t: x= f(t)
 Data points connected by a curve
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Exports
30,000
35,000
40,000
45,000
50,000
55,000
60,000
65,000
70,000
75,000
80,000
1988 1990 1992 1994 1996 1998 2000

IMPORTANCE OF TIME SERIES ANALYSIS
 Understand the past.
What happened over the last years, months?
 Forecast the future.
Government wants to know future of unemployment
rate, percentage increase in cost of living etc.
For companies to predict the demand for their
product etc.
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TIME-SERIES COMPONENTS
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Time-Series
Cyclical
Random
Trend
Seasonal

COMPONENTS OF TIME SERIES
 Secular Trend / Trend (T)
 Seasonal variation (S)
 Cyclical variation (C)
 Random variation (I)
or irregular
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COMPONENTS OF TIME SERIES
SECULAR TREND (T)
 Trend: the long-term patterns or movements in
the data.
 Anytime series shows various fluctuations from
time to time, but in a long period of time, that
series has the increasing or declining trend in
one direction.
 Overall or persistent, long-term upward or
downward pattern of movement.
 Secular Trend is usually of two types:
 Linear Trend Y = a + bX
 Parabolic Trend Y = a + bX + cX2
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SEASONAL VARIATION (S)
 Regular periodic fluctuations that occur within
year.
Examples:
 Consumption of heating oil, which is high in winter,
and low in other seasons of year.
 Demand of cold drinks, juices etc. in summers tends
to be greater in comparison to other months
 Gasoline consumption, which is high in summer
when most people go on vacation.
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Components of Time Series

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-10
-5
0
5
10
15
20
25
30
Seasonal variation (S)
Summer
Winter Winter
Summer

EXAMPLE
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Quarterly with Seasonal Components
0
5
10
15
20
25
0 5 10 15 20 25 30 35
Time
Sales

SEASONAL COMPONENTS REMOVED
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Quarterly without Seasonal Components
0
5
10
15
20
25
0 5 10 15 20 25 30 35
Time
Sales
Y(t)

WHY DO USERS WANT SEASONALLY
ADJUSTED DATA?
Seasonal movements can make features difficult
or impossible to see
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CAUSES OF SEASONAL EFFECTS
 Possible causes are
 Natural factors
 Administrative or legal measures
 Social/cultural/religious traditions (e.g., fixed
holidays, timing of vacations)
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Cyclical variation ( Ct )
• Cyclical variations are similar to seasonal
variations. Cycles are often irregular both in
height of peak and duration.
• These refer to oscillatory variations in a
time series having duration of 2-10 years.
• Examples:
• Long-term product demand cycles.
• Cycles in the monetary and financial
sectors. (Important for economists!)

CYCLICAL COMPONENT
 Long-term wave-like patterns
 Regularly occur but may vary in length
 Often measured peak to peak or trough to trough
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Sales
1 Cycle
Year

IRREGULAR VARIATIONS (I)
 Unpredictable, random, “residual” fluctuations
 Generally short term variations
 Due to random variations of
 Nature
 Accidents or unusual events
 “Noise” in the time series
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CAUSES OF IRREGULAR EFFECTS
 Possible causes
 Unseasonable weather/natural disasters
 Strikes
 Sampling error
 Nonsampling error
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ANALYSIS OR DECOMPOSITION OF
TIME SERIES
 Decompose the series into various components
 Trend – long term movements in the level of the series
 Seasonal effects – cyclical fluctuations reasonably stable
in terms of annual timing (including moving holidays
and working day effects)
 Cycles – cyclical fluctuations longer than a year
 Irregular – other random or short-term unpredictable
fluctuations
 Models of Time Series:
 Additive Model : O = T + S + C + I
 Multiplicative Model : O = TSCI
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Not easy to understand the pattern!

OUR AIM
 is to understand and identify different variations
so that we can easily predict the future variations
separately and combine together
 Look how the above complicated series could be
understood as follows separately
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FEW VARIATIONS SEPARATELY
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Can you imagine how all components aggregate
together to form this?

MULTIPLICATIVE TIME-SERIES
MODEL FOR ANNUAL DATA
 Used primarily for forecasting
 Observed value in time series is the product of
components
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where Ti = Trend value at year i
Ci = Cyclical value at year i
Ii = Irregular (random) value at year i
iiii ICTY 

MULTIPLICATIVE TIME-SERIES MODEL
WITH A SEASONAL COMPONENT
 Used primarily for forecasting
 Allows consideration of seasonal variation
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where Ti = Trend value at time i
Si = Seasonal value at time i
Ci = Cyclical value at time i
Ii = Irregular (random) value at time i
iiiii ICSTY 

METHODS OF MEASURING TREND
Free Hand Curve Method
Semi Average Method
Moving Average Method
Least Square Method
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LEAST SQUARE METHOD
 Best Method of Trend Fitting
 Trend Line is fitted in such a way that following
two conditions are fulfilled:
 Σ 𝑌 − 𝑌𝑐 = 0, i.e. the sum of the deviations of the
actual values of Y and computed trend values (Yc) is
zero.
 Σ 𝑌 − 𝑌𝑐 2 is least.
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DIRECT METHOD
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SHORTCUT METHOD
 Middle Year is taken as the year of origin with
value 0 & deviations from other years are computed.
 Sum of the deviations will always be zero i.e. ƩX = 0
 Compute ƩY, ƩXY, ƩX2.
 Calculate a & b where a =
ƩY
𝑁
, b =
ƩXY
ƩX2
 Put values of a & b in Y = a + bX
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PRACTICE PROBLEMS
Q: Fit a straight line by method of least squares:
Also show on graph paper. Ans: Y = 90 + 2X
Estimate the sales for 2002. Ans: Y = 60 + 5X, 85
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Year 1993 1994 1995 1996 1997 1998 1999
Prod. 80 90 92 83 94 99 92
Year 1995 1996 1997 1998 1999
Sales 45 56 78 46 75

PRACTICE PROBLEMS
Estimate the sales for 2001. Ans: Y = 35.67 + 2X, 49.7
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Year 1995 1996 1997 1998 1999 2000
Sales 28 32 29 35 40 50

SEMI AVERAGE METHOD
 First of all, time series is divided into two equal
parts and thereafter, separate arithmetic mean is
calculated for each part.
 The two values for Arithmetic Means is plotted
on graph corresponding to the time periods.
 Therefore, a straight line is formed, & is called a
Trend Line.
 Two Cases:
 No. of years is even
 No. of years is odd
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PRACTICE PROBLEMS
Q: Fit a straight line by the method of semi average
to the data given below:
Q: Fit a trend line by the method of semi average to
the data given below:
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Year 2000 2001 2002 2003 2004 2005 2006 2007
Sales 412 438 444 454 470 482 490 500
Year 1991 1992 1993 1994 1995 1996 1997
Profit 20 22 27 26 30 29 40

MOVING AVERAGE METHOD
 In this, one has to decide what moving average
should be taken up for consideration i.e. 3 year, 4
year, 5 year, 7 year etc.
 Moving average method is studied in two
different situations:
 Odd period moving average
 Even period moving average
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PRACTICE PROBLEMS
Q: From the following data, calculate trend values
using 3 yearly, 5 yearly & 7 yearly moving average:
Q: Calculate Trend Values using 4 yearly moving
average from the following data:
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Year 1991 1992 1993 1994 1995 1996 1997
Profit 412 438 446 454 470 483 490
Year 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979
Sales 7 8 9 11 10 12 8 6 5 10

TIME SERIES ANALYSIS COMPONENTS

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