Introduction - Objectives Of Studying Time Series Analysis - Variations In Time Series
- Methods Of Estimating Trend: Freehand Method - Moving Average Method - Semi-Average Method - Least
Square Method
2. Introduction
• Forecasting or predicting is an essential tool in any decision-making
process
• Used for Inventory management to annual sales
• Quality depends on the quantity of the past data
• Pattern is used to arrive at an estimate in the future
• This analysis helps us cope with uncertainty about the future
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3. Variations in Time Series
4 different variation involved in time series:
1. Secular Trend
2. Cyclical Fluctuation
3. Seasonal Variation
4. Irregular Variation
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4. Secular Trend
• The value of the variable tends to increase or decrease over a long
period
• Steady increase in cost of living recorded by the Consumer Price Index
is an example of secular trend
• In terms of long term period, complete cost of living varies a great
deal
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5. Cyclical Trend
• Business cycle is the most common example
• Peak at sometimes and likely to slump at the other
• The cycle may extend up to 1 year to 15 to 20 years
• There is no regular pattern but will move in little unpredictable
manner
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6. Seasonal Variation
• Involves patterns of change within a year
• They tend to have a cycle from year to year
• Substantial peak and irregular through at particular periods in a year
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7. Irregular Variations
• Value of a variable may be completely unpredictable
• Effect of one situation ripples to the impact of any other inter related
commodity
• White Revolution and effect on Gas
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8. Reasons for Studying Trends
• The study of secular trends allows us to describe a historical pattern
• Evaluating the eating lifecycle resulted in creation of Maggie
• Studying secular trends permits us to project past patterns, or trends,
into the future
• Growth trend of population helps predict the population projections
• In many situations, studying the secular trend of a time series allows
us to eliminate the trend component from the series
• Make in India Campaign
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9. Fitting the Linear Trend
by the least-squares method
• Assuming the trend is in the straight line
• The general equation of a straight line:
y = mx + c
• y is the dependent axis
• X is the independent axis
• c is the intercept of the line
• m is the slope of the trend line
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10. Slope and Intercept
Slope of the best fitting Regression Line
m =
𝑋𝑌 −(𝑛∗𝑚𝑒𝑎𝑛 𝑋 ∗𝑚𝑒𝑎𝑛 𝑌 )
𝑋2−(𝑛∗𝑚𝑒𝑎𝑛(𝑋)2)
Y-intercept of the Best-Fitting Regression Line
c=mean(Y)-m*mean(X)
• X = values of dependent axis
• Y = values of independent axis
• n = number of data points in the time series
• m= slope
• c= Y-Intercept
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11. Translating or coding time
• It is tedious to calculate in the equation to find the slope
• We can convert the traditional measures of time into the following
• If there are 3 points of time 1992, 1993, 1994
• They can be represented as -1, 0, 1
• Can be achieved by subtracting the mean from all the 3 points
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12. Time Coding
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S No X X-Mean(X) Coded Time
1 1989 1989-1992 -3
2 1990 1990-1992 -2
3 1991 1991-1992 -1
4 1992 1992-1992 0
5 1993 1993-1992 1
6 1994 1994-1992 2
7 1995 1995-1992 3
S No X X-Mean(X) X-Mean(X)
Coded Time
(X-Mean(X))*2
1 1990 1990-1992.5 -2.5 -5
2 1991 1991-1992.5 -1.5 -3
3 1992 1992-1992.5 -0.5 -1
4 1993 1993-1992.5 0.5 1
5 1994 1994-1992.5 1.5 3
6 1995 1995-1992.5 2.5 5
13. Slope and intercept of coded time
• Slope of the trend line for coded time values
m=
𝑥𝑌
𝑥2
• Intercept of the trend line for coded time values
a= mean (Y)
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14. Problem 1
• Calculate the slope and y intercept for the following trend
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X Y
(1) (2)
1988 98
1989 105
1990 116
1991 119
1992 135
1993 156
1994 177
1995 208
15. Problem 1
• Calculate the slope and y intercept for the following trend
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X Y X-mean(X) (X-mean(X))*2 XY X^2
(1) (2) (3) (3)*2=(4) (4)*(2) (4)^2
1988 98 -3.5 -7 -686 49
1989 105 -2.5 -5 -525 25
1990 116 -1.5 -3 -348 9
1991 119 -0.5 -1 -119 1
1992 135 0.5 1 135 1
1993 156 1.5 3 468 9
1994 177 2.5 5 885 25
1995 208 3.5 7 1456 49
Mean: 1991.50 Sum: 1266 168
Slope: 1266/168 = 7.536
Y-intercept = 139.25
Trend line is
y= 7.536 * x + 139.25
16. Problem 2
Jim is a part time plumber. Now he wanted to hire some staff for
himself and wanted to forecast the next 3 years trends
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Year
Avg Client per
month
2001 6.4
2002 11.3
2003 14.7
2004 18.4
2005 19.6
2006 25.7
2007 32.5
2008 48.7
2009 55.4
2010 75.7
2011 94.3
17. Problem 2
Jim is a part time plumber. Now he wanted to hire some staff for
himself and wanted to forecast the next 3 years trends
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Year
Avg Client per
month
Time
Code
X * Y X^2
2001 6.4 -5 -32 25
2002 11.3 -4 -45.2 16
2003 14.7 -3 -44.1 9
2004 18.4 -2 -36.8 4
2005 19.6 -1 -19.6 1
2006 25.7 0 0 0
2007 32.5 1 32.5 1
2008 48.7 2 97.4 4
2009 55.4 3 166.2 9
2010 75.7 4 302.8 16
2011 94.3 5 471.5 25
Intercept: 36.60909091 Sum: 892.7 110
Slope: 892.7/110 = 8.11
Y-intercept = 36.6091
Trend line is
y= 8.11 * x + 36.6091
2012:
y= 8.11 * 6 + 36.6091 = 85.3
2013:
y= 8.11 * 7 + 36.6091 = 93.4
2014:
y= 8.11 * 8 + 36.6091 = 101.5
18. Methods of estimating Trend
• Freehand Method
• Moving Average Method
• Semi-Average Method
• Least Square Method
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19. Freehand method
• Briefly described for drawing frequency curves
• Observations is plotted against time on the horizontal axis and a freehand
smooth curve is drawn through the plotted points
• Smoothness should not be scarified in trying to let the points fall exactly on
the curve
• Eliminates the short term and long term oscillations and the irregular
movements from the time series, and elevates the general trend
Disadvantages:
• Different individuals draw curves or lines that differ in slope and intercept
• Used only in situations where the scatter diagram of the original data
conforms to some well define trends
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20. Problem 1
Measure the trend using the method of the freehand curve from the
given data of production of wheat in a particular area of the world.
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Years
Production Million
Metric Tons
1981 6.6
1982 6.9
1983 5.6
1984 6.3
1985 8.4
1986 7.2
1987 7.2
1988 8.5
1989 8.5
21. Problem 1
Measure the trend using the method of the freehand curve from the
given data of production of wheat in a particular area of the world.
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Years
Production Million
Metric Tons
1981 6.6
1982 6.9
1983 5.6
1984 6.3
1985 8.4
1986 7.2
1987 7.2
1988 8.5
1989 8.5
22. Moving Average Method
• A n-period moving average for time period t is the arithmetic average of the time series
values for the n most recent time periods
• For example: A 3-period moving average at period (t+1) is calculated by (yt-2 + yt-1 +
yt)/3
• Advantages of Moving Average Method
• Easily understood
• Easily computed
• Provides stable forecasts
• Disadvantages of Moving Average Method
• Requires saving all past n data points
• Lags behind a trend
• Ignores complex relationships in data
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23. Example 1
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Period Actual MA (3) MA (5)
1 42
2 40
3 43
4 40 41.67
5 41 41.00
6 39 41.33 41.2
7 46 40.00 40.6
8 44 42.00 41.8
9 45 43.00 42
10 38 45.00 43
11 40 42.33 42.4
12 41.00 42.6
34
36
38
40
42
44
46
48
1 2 3 4 5 6 7 8 9 10 11 12
Weighted Moving Average
Actual MA (3) MA (5)
24. Semi Average Method
• This method is as simple and relatively objective as the free hand
method
• Data is divided in two equal halves and the arithmetic mean is
calculated
• If the number of observations is even the division into halves
• If the number of observations is odd, then the middle most item is
dropped
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25. Advantages and Disadvantages of the Semi-
Averages Method
• Advantages
• This method is very simple and easy to understand, and also it does
not require many calculations.
• Disadvantages
• For non-linear trends this method is not applicable.
• averages are affected by extreme values
• extreme value should either be omitted or this method should not be
applied
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28. Example 1 Solution
• Trend of 1 year is called the
slope = m = 3.656
• Trend of 1st year in the
series is the y –intercept = c
= 25.008
• Then the trend line is
y= 3.656x + 25.008
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