Detrending time series is a process of eliminating the trend component from a time series, where a trend refers to a change in the mean over time (a continuous decrease or increase over time). It means that when data is detrended, an aspect from that data has been removed that you think is causing some kind of distortion.
Assuming the multiplicative model:
$$Detrended\, value = \frac{Y}{T} = \frac{TSCI}{T}=SCI $$
Assuming additive model:
$$Detrended\, value = Y-T=T+S+C+I-T = S+C+I$$
Component of Time Series Data
Detrending Time Series (Stationary Time Series)
The detrending time series is a process of removing the trend from a non-stationary time series. A detrended time series is known as a stationary time series, while a time series with a trend is a non-stationary time series. A stationary time series oscillates about the horizontal line. If a series does not have a trend or we remove the trend successfully, the series is said to be trend stationary.
Eliminating the trend component may be thought of as rotating the trend line to a horizontal position. The trend component can be eliminated from the observed time series by computing either the ratios to the trend if the multiplicative model is assumed or the deviations from the trend if the additive model is assumed.
Note that the best detrending method depends on the nature of your trend:
Use differencing for stationary trends (constant increase/decrease).
Use model fitting for more complex trends (curves, changing slopes).
Detrending is often a preparatory step for further analysis such as forecasting and identifying seasonal patterns. On the other hand, detrending might not be necessary if the trend is already incorporated into your analysis. Some methods, like deseasonalizing, can involve both detrending and removing seasonal effects.
When a straight line does not describe accurately the long-term movement of a time series, then one might detect some curvature and decide to fit a curve instead of a straight line.
Table of Contents
The most commonly used curve, to describe the nonlinear secular trend in a time series, are:
Exponential curve, and
Second-degree parabola
Exponential (Nonlinear) Curve
The exponential curve describes the trend (nonlinear) in a time series that changes by a constant percentage rate. The equation of the curve is $\hat{y} = ab^x$
Taking logarithm, we get the linear form $log\, \hat{y}=log\, a + (log\,b)x$
The method of least squares gives the normal equations:
\begin{align*} \sum log\, y & = n\, log\, a + log\, b \sum x\\ \sum log\, y & = n\, log\, a \sum x + log\, b \sum x^2 \end{align*}
However, if $\sum x=0$ the normal equations becomes
\begin{align*} \sum log\,y & = n\, log a\\ \sum x log\, y &= log\, b \sum x^2 \end{align*}
The values of $log\, a$ and $log\, b$ are
\begin{align*} log\, a &=\frac{\sum log\, y}{n}\\ log\, b&= \frac{\sum x log\, y}{\sum x^2} \end{align*}
Taking $antilog$ of of $log\, a$ and $log\, b$, we get the values of $a$ and $b$.
Question: The population of a country for the years 1911 to 1971 in ten yearly intervals in millions is 5.38, 7.22, 9.64, 12.70, 17.80, 24.02, and 31.34. (i) Fit a curve of the type $\hat{y}=ab^x$ to this time series and find the trend values, (ii) Forecast the population for the year 1991.
solution
(i) We have $\overline{t}=\frac{(1991+1971)}{2}=1941$. Let $x=\frac{t-\overline{t}}{10}=\frac{5-1941}{10}$ so that coded year number $x$ is measured in a unit of 10 years.
Year $t$
Population $y$
Coded Year $x=\frac{x-1941}{10}$
$log y$
$x log\, y$
$x^2$
$\hat{y}=13.029(1.345)^x$
1911
5.38
-3
0.73078
-2.19234
9
5.355
1921
7.22
-2
0.85854
-1.71708
4
7.202
1931
9.64
-1
0.98408
-0.98498
1
9.687
1941
12.70
0
1.10380
0
0
13.029
1951
17.808
1
1.25042
1.25042
1
17.524
1961
24.02
2
1.38057
2.76114
4
23.570
1971
31.34
3
1.49610
4.48830
9
31.701
The least squares exponential curve is $\hat{y} = ab^x$
Taking logarithm, $log\, \hat{y} = log a + (log\, b)x$
since $\sum x=0$, therefore
\begin{align*} log\, a &= \frac{\sum log\, y}{n} = \frac{7.80429}{7}=1.1149\\ log\, b &= \frac{\sum x log\, y}{\sum x^2} = \frac{3.60636}{28}=0.12880\\ a &= antilog(1.1149)=13.029\\ b &= antilog(0.1288)=1.345\\ \hat{y} &=13.029 (1.345)^x,\quad \text{with origin at 1941 and unit of $x$ as 10 years} \end{align*}
(ii) For $t=1941$ we have $x=\frac{t-1941}{10}= \frac{1991-1994}{10}=5$. Putting $x=5$, in the least squares exponential curve, we have $\hat{y} = 13.029 (1.345)^5 = 57.348$ millions
Second Degree Parabola (Nonlinear Trend)
It describes the trend (nonlinear) in a time series where a change in the amount of change is constant per unit of time. The quadratic (parabolic) trend can be described by the equation
\begin{align*} \hat{y} = a + bx + cx^2 \end{align*}
The method of least squares gives the normal equations as
\begin{align*} \sum y &= na + b\sum x + c \sum x^2\\ \sum xy &= a\sum x + b\sum x^2 + c \sum x^3\\ \sum x^2y &= a \sum x^2 + b\sum x^3 + c\sum x^4 \end{align*}
However if $\sum x = 0 \sum x^3$ then the normal equation reduces to
\begin{align*} \sum y &= na + c\sum x^2\\ \sum xy &= b\sum x^2\\ \sum x^2 y &= a \sum x^2 + c \sum x^4\\ & \text{the values of $a$, $b$, and $c$ can be found as}\\ c &= \frac{n \sum x^2 y – (\sum x^2)(\sum y)}{n \sum x^2 -(\sum x^2)^2}\\ a&=\frac{\sum y – c\sum x^2}{n}\\ b&= \frac{\sum xy}{\sum x^2} \end{align*}
Question: Given the following time series
Year
1931
1933
1935
1937
1939
1941
1943
1945
Price Index
96
87
91
102
108
139
307
289
Fit a second-degree parabola taking the origin in 1938.
Find the trend values
What would have been the equation of the parabola if the origin were in 1933
Solution
(i)
Year $t$
Price index $y$
Coded Year $x=t-1938$
$x^2$
$x^4$
$xy$
$x^2y$
Trend values $y=110.2+15.48x+2.01 x^2$
1931
96
-7
49
2401
-672
4704
100.33
1933
87
-5
25
625
-435
2175
83.05
1935
91
-3
9
81
-273
819
81.85
1937
102
-1
1
1
-102
102
96.73
1939
108
1
1
1
108
108
127.69
1941
139
3
9
81
417
1251
174.73
1943
307
5
25
625
1535
7675
237.85
1945
289
7
49
2401
2023
14161
317.05
Total
1219
0
168
6216
2601
30995
(ii) Different trend values are already computed in the above table.
\begin{align*} \hat{y} &= a + b x + c x^2\\ c &= \frac{n\sum x^2 y-(\sum x^2)(\sum y)}{n \sum x^4 -(\sum x^2)^2} =\frac{8(30995)-(168)(1219)}{8(6126)-(168)^2}=2.01\\ a &= \frac{\sum y – a \sum x^2}{n}=\frac{1219-(2.01)(168)}{8}=119.2\\ b &= \frac{\sum xy}{\sum x^2}=\frac{2601}{168} = 15.48\\ \hat{y} &= 110.2 + 15.48x + 2.01^2,\quad \text{with origin at the year 1938} \end{align*}
For different values of $x$, the trend values are obtained in the table.
For shifting the origin at 1933, replace $x$ by $(x-5)$
The method of least squares gives the most satisfactory measurement of the secular trend in a time series when the distribution of the deviations is approximately normal.
The least-squares estimates are unbiased estimates of the parameters.
The method can be used when the trend is linear, exponential, or quadratic.
Demerits of Least Squares
The method of least squares method gives too much weight to extremely large deviations from the trend
The least-squares line is the best only for the period to which it has reference.
The elimination or addition for a few or more periods may change its position.
The least-squares principle (Method of Least Squares) says that “the sum of squares of the deviations of the observed values from the corresponding expected values should be least”. Among all the trend lines, the trend line is called a least-squares fit for which the sum of the squares of the deviations of the observed values from their corresponding expected values is the least.
Note that the usual probabilistic assumptions made in regression and correlation analysis are not met in the case of time series data.
Secular Trend — Linear Trend
It is useful to describe the trend in a time series where the amount of change is constant per unit of time.
Let $(x_1, y_1), (x_2, y_2), \cdots, (x_n,y_n)$ be the $n$ pairs of observed sample values of a time series variable $y$, with $x$ representing the coded time value. We can plot these $n$ points on a graph.
Let us suppose that we want to fit a straight line expressed in slope-intercept form as:
\begin{align} \hat{y} = a + bx, \quad \quad (eq1) \end{align}
The line (eq-1) will be called the least squares line if it makes $\sum(y-a-bx)^2$ minimum. The method of least squares yields the following normal equations:
\begin{align*} \sum y &= na + b \sum x\\ \sum xy &= a \sum x + b \sum x^2 \end{align*}
The normal equations give the value of $a$ and $b$ as:
\begin{align*} b &= \frac{n \sum xy – (\sum x \sum y )}{n \sum x^2 -(\sum x)^2}\\ a & = \overline{y}-b\overline{x} \end{align*}
However, if $\sum x=0$ the usual normal equations reduces to
The trend values $\hat{y}$ are computed from the least-squares line $\hat{y}=a+bx$ by substituting the values of $x$ corresponding to the different time periods.
Properties of the Method of Least Squares
The least-squares line always passes through the point ($\overline{x}, \overline{y}$) called the center of gravity of the data.
The sum of deviations $\sum(y-\hat{y})$ of the observed values $y$ from their corresponding expected values $\hat{y}$ is zero, that is, $\sum(y-\hat{y})=0$, hence $\sum y= \sum \hat{y}$
The sum of squares of the deviations $\sum (y-\hat{y})^2$ measures how well the trend line fits the data. A smaller $\sum (y-\hat{y})^2$ means the better fit.
Moving Averages and Least Squares Linear Trend: The least-squares linear trend values corresponding to the central time period in each group of $k$ observations are equal to the $k$-period moving averages.
Question: Determine the trend line by the least-squares method from the following data. Plot the actual values and the linear trend on the same graph.
Year
1945
1946
1947
1948
1949
1950
1951
1952
1953
Price
3
6
2
10
7
9
14
12
18
Solution
The equation of the trend line is
\begin{align*} \hat{Y} = a + b\, X \end{align*}
Normal Equations are:
\begin{align*} \Sigma Y & = n\, a + b \, \Sigma X \tag{i}\\ \Sigma XY& = a\, \Sigma X+ b\, \Sigma X^2 \tag{ii} \end{align*}
From (1) $a=\frac{81}{9}=9$, and from (2) $b=\frac{101}{60}=1.7$.
Fitted trend line is $\hat{Y}=9 + 1.7\,X$.
The method of least squares is a valuable tool for analyzing trends in time series data. By understanding the strengths and limitations of the methods, you can effectively use them to gain insights, make predictions, and compare trends across different time series datasets.
