Trend Analysis - Download as PowerPoint by pub14358

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									  Main idea of the trend analysis
      forecasting method:
• a forecast is calculated by inserting a time
  value into the regression equation. The
  regression equation is determined from the
  time-series data using the “least squares
  method”




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     Prerequisite: Correlation

There should be a sufficient correlation
between the time parameter and the
values of the time-series data




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    The Correlation Coefficient
• The correlation coefficient, R, measure the
  strength and direction of linear relationships
  between two variables. It has a value
  between –1 and +1
• A correlation near zero indicates little linear
  relationship, and a correlation near one
  indicates a strong linear relationship
  between the two variables
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  Main idea of the trend analysis
             method
• Trend analysis uses a technique called least squares to fit a
  trend line to a set of time series data and then project the
  line into the future for a forecast.
• Trend analysis is a special case of regression analysis
  where the dependent variable is the variable to be
  forecasted and the independent variable is time.
• While moving average model limits the forecast to one
  period in the future, trend analysis is a technique for
  making forecasts further than one period into the future.


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The trend line is the “best-fit”
       line: an example
                         Municipal public libraries in Lithuania in 1991-2002
  Number of libraries




                        1700
                        1650
                        1600
                        1550
                        1500
                        1450
                        1400
                        1350
                               1990
                                      1991
                                              1992
                                                     1993
                                                            1994
                                                                   1995
                                                                          1996
                                                                                 1997
                                                                                        1998
                                                                                               1999
                                                                                                      2000
                                                                                                             2001
                                                                                                                    2002
                                                                                                                           2003
                                                                           Year


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Statistical measures of goodness
              of fit

 In trend analysis the following
 measures will be used:

• The Correlation Coefficient
• The Determination Coefficient



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The Coefficient of Determination R2

• The coefficient of determination, R2,
  measures the percentage of variation in the
  dependent variable that is explained by the
  regression or trend line. It has a value
  between zero and one, with a high value
  indicating a good fit.


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    Goodness of fit: Determination
          Coefficient R2
•   Range: [0, 1]
•   R2=1 means best fitting
•   R2=0 means worse fitting
•   In Excel R2 is denoted as RSQ (“R
    squared”)



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 Evaluation of the trend analysis
      forecasting method
• Advantages: Simple to use (if using
  appropriate software)
• Disadvantages: 1) not always applicable for
  the long-term time-series (because there
  exist several trends in such cases); 2) not
  applicable for seasonal and cyclic data
  patterns.

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  Using Excel to calculate linear
              trend

• Select a line on the diagram (left click on
  the line) 
• Right click and select Add Trend line 
• Select a type of the trend (Linear)



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                Non-linear trends
Excel provides easy calculation of the following trends

 •   Logarithmic
 •   Polynomial
 •   Power
 •   Exponential



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Examples of the non-linear trends




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                     Logarithmic trend

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         y = 4,6613Ln(x) + 1,0724
10
                R2 = 0,9963
 8
 6
 4
 2
 0
     0         2                       4                 6   8




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                          Trend (power)


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         y = 0,4826x1,5097
 8
           R2 = 0,9919
 6
 4
 2
 0
     0         2                       4                 6        8




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             Trend (exponential)

80
                   y = 0,0509e1,0055x
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                      R2 = 0,9808
40

20

0
     0   2                       4                 6   8




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                       Trend (polynomial)

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        y = -0,1142x3 + 1,6316x2 - 5,9775x + 7,7564
6
                        R2 = 0,9975
4

2

0
    0           2                      4                  6        8




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 Choosing the trend that fits best
• 1) Roughly: Visually, comparing the data
  pattern to the one of the 5 trends (linear,
  logarithmic, polynomial, power,
  exponential)
• 2) In a detailed way: By means of the
  determination coefficient


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