Measurement of Trend Component of a Time Series and How to Remove (or Correct) it from Time Series?

Measurement of Trend Component of a Time Series and How to Remove (or Correct) it from Time Series? 

Any time series may contain some or all of the following components:

1.    Trend Component (T)

2.    Cyclical Component (C)

3.    Seasonal Component (S)

4.    Irregular Component (I)

These components may get combined usually as below:

If time series is represented as y_t, then 

y_t =  T * C * S * I

or

y_t =  T + C + S + I

The trend component is the long-term pattern of a time series.  It can be positive or negative, as it depends on whether the series exhibits an increasing or decreasing long-term pattern. 

If a time series does not have either increasing or decreasing pattern, then the series is stationary as per its mean.

How to Measure Trend Component of a Time Series

The simplest and more commonly employed methods to measure Trend Component of a Time Series are: 

1.      Graphic or Free hand Method:  First of all the data is plotted on a graph paper and the trend line is fitted by a line or a freehand curve by just inspecting and following the graph of the series.  The curve needs to be smooth and with almost equal number of points above and below it.  By eye estimate, the sum of the vertical deviations of the given points above the trend line should approximately equal to the sum of the vertical deviations of the given points below the trend line. Also the sum of the squares of the vertical deviations of the given points from the trend line should be minimum possible. 

2.      Moving Average Method:  This method comprise of taking arithmetic means of the data/values for a certain span and then placing the value so calculated against the middle of the time span.  The time span should be equal to the average fluctuation period.  If this span is of period k, then the moving averages obtained by averaging k at a time are called Moving Averages of period of extent k.   If k is even, the successive values of moving averages are placed in the center/middle of the period/span of time. 

How to remove trend component from a time series?

In order to correct/remove the trend component from the first multiplicative model, one has to divide this expression by the trend component (T).  In order to correct/remove the trend component from the second additive model, one has to subtract this expression by the trend component (T).