Trend estimation
The seasonal trend model is the traditional approach to time series analysis. The modeling is performed using a mathematical model comprises the following components:
- A trend component,
- A seasonal component and
- A noise component.
Missing, for example, the seasonal component, one speaks only of a trend model.
- 2.1 Linear or polynomial trend model
- 2.2 Exponential model
- 2.3 Moving Averages
- 3.1 Additive seasonal variation
- 3.2 Multiplicative seasonal variation
Model structure
If the observed time-series, then a trend is initially estimated. Possible are linear, polynomial or exponential trends, but also moving averages.
From the residuals one can estimate an additive or multiplicative seasonal component. It is expected that by the deviations of the trend function of the observed values are subject to a seasonal pattern.
Example
The graph below shows the unemployment rates in the Federal Republic of Germany from January 2005 to December 2008 (top left) and a linear trend function. Top right, the deviation between the observed unemployment figures and the estimates of the trend is shown. One can see that in the spring of each year, the trend function underestimates the unemployment figures and autumn overestimated ( same color = same month). The graph on the left below shows the average deviation over the years for each month. This deviation is added for the corresponding month for trend function. This results in the graph on the right below the seasonal trend model ( red line).
Trend estimate
The trend of a time series are the global history of a time series again. For various regression approaches are used:
- A linear or polynomial model:
- An exponential model: or
- Also moving averages with a correspondingly high order.
Linear or polynomial trend model
In the linear or polynomial trend model is simply a linear or polynomial regression with respect to the time variable performed to estimate the trend:
While the estimated values , ... depend on how the time is parameterized, the estimated trend values are independent of the parameterization.
The following table shows two parameterizations of time for a linear trend model:
- Corresponds to the first trend model January 2005 and the same
- February 2005 the same,
- January 2005 and the same
- February 2005 the same.
Then the values for and for all subsequent months are fixed.
Since coming out the same estimates for the two parameterizations, one can select any:
- The first parametrization allows easy interpretation of the trend function. Starting from an unemployment rate of 4.825 million in December 2004 (), the unemployment rate falls by an average of about 26,150 people per month until December 2011.
- The second parameterization would be useful if you had to calculate the regression coefficients by hand. Among other things, the arithmetic mean is needed to the results here. Furthermore, we see that 3.71363 million people in the period January 2005 to December 2011 were unemployed on average.
In the present data, a linear trend function would be unsuitable because they only poorly reproduces the global history of the series. This also shows the previous graphic. It also shows that a quadratic function be better trend:
Exponential model
An exponential trend model is used when the data suggest it. In the right graph, we see the number of phones ( in thousands) in the United States from 1891 to 1979 as well as an exponential and a linear trend function. Obviously, the exponential trend describes the data better than the linear trend.
Furthermore, the exponential trend model
The advantage that in the back-calculation results
The estimated value of each.
The estimate of the regression coefficients is done by reduction to the linear model, that is, both the logarithms and then and appreciated.
In contrast to the linear or polynomial trend function of both the estimated values of the regression coefficients and the estimated values thereof depend on how time is parameterized. In the graph corresponds to the year 1891 and 1892 equal to equal
Moving Averages
Another alternative for trend estimation are moving averages of sufficiently high order. The value is calculated as the average of the observed values in one place. Distinction must be made the calculation for even and odd orders:
For an even order the edge points, and each flow with the weight 1/2 and all points between them first with the weight
However, this is only one way to calculate moving averages; for more see the main article: Moving average.
However, the moving averages throw three problems:
However, the advantage of the moving averages is the better fit to a non- linear trend in the data.
Season estimate
In the season estimate, it is assumed that there is a structure in the time series that repeats seasonally. The length of a season is known in advance. With the unemployment figures, we know that due to the weather conditions increase the unemployment figures for the winter back regularly as they fall back down to summer. So there is an annual patterns in the data.
Essentially, seasonal variations are modeled either additive or multiplicative:
With the value of a trend estimate and an index that is repeated in every season.
The following table shows the values of the unemployment rate in Germany from January 2005 to December 2011 (), a trend estimate () with a moving average of order 13, and the deviations between the observed values and the estimated trend for an additive or multiplicative seasonal model.
Additives seasonal variation
Each time a season having a predetermined length is assigned to a season index. Then, the difference between the observed value and the estimated trend value is calculated
After all values are averaged for a fixed
In the unemployment example ( ) first so all January deviations are averaged ():
This is repeated for all months to December ():
This can be calculated from the estimated trend and the averaged seasonal variations the final time series estimation.
Multiplicative seasonal variation
Each time a season having a predetermined length is assigned to a season index. Then the quotient between the observed value and the estimated trend value is calculated
After all values are averaged for a fixed.
In the unemployment example ( ) first so all January deviations are averaged ():
This is repeated for all months to December ():
This can be calculated from the estimated trend and the averaged seasonal variations the final time series estimation.
Quality of a seasonal trend model
Because there are several options for both the trend estimate and estimate for the season, there is the question of which model is the best. Since both models can be non-linear, you can not necessarily proceed in two stages, that is, only take "best" trend model and then choose the best seasonal model that; only a combination of trend and seasonal estimation should be examined.
Based on the linear regression coefficient of determination for a seasonal trend model is defined:
With the average of all, for which a prediction is made. In general, the determination of a seasonal trend model is significantly larger than in the linear regression.
The following table shows data for the unemployed in Germany from January 2005 to December 2011 the Bestimmheitsmaße for different trend or seasonal trend models.
The graph shows the nine seasonal trend models. One can see that both the blue ( linear trend ) and the green models ( exponential trend ) is not a good fit to the data. The red models ( moving averages ) best fit to the data.