For example, we can transform the time series by calculating the natural log of the original data:. Here we can see that the size of the seasonal fluctuations and random fluctuations in the log-transformed time series seem to be roughly constant over time, and do not depend on the level of the time series. Thus, the log-transformed time series can probably be described using an additive model. Decomposing a time series means separating it into its constituent components, which are usually a trend component and an irregular component, and if it is a seasonal time series, a seasonal component.
A non-seasonal time series consists of a trend component and an irregular component. Decomposing the time series involves trying to separate the time series into these components, that is, estimating the the trend component and the irregular component. To estimate the trend component of a non-seasonal time series that can be described using an additive model, it is common to use a smoothing method, such as calculating the simple moving average of the time series.
For example, as discussed above, the time series of the age of death of 42 successive kings of England appears is non-seasonal, and can probably be described using an additive model, since the random fluctuations in the data are roughly constant in size over time:. Thus, we can try to estimate the trend component of this time series by smoothing using a simple moving average.
To smooth the time series using a simple moving average of order 3, and plot the smoothed time series data, we type:. There still appears to be quite a lot of random fluctuations in the time series smoothed using a simple moving average of order 3.
Thus, to estimate the trend component more accurately, we might want to try smoothing the data with a simple moving average of a higher order. This takes a little bit of trial-and-error, to find the right amount of smoothing. For example, we can try using a simple moving average of order The data smoothed with a simple moving average of order 8 gives a clearer picture of the trend component, and we can see that the age of death of the English kings seems to have decreased from about 55 years old to about 38 years old during the reign of the first 20 kings, and then increased after that to about 73 years old by the end of the reign of the 40th king in the time series.
A seasonal time series consists of a trend component, a seasonal component and an irregular component. Decomposing the time series means separating the time series into these three components: that is, estimating these three components.
This function estimates the trend, seasonal, and irregular components of a time series that can be described using an additive model. For example, as discussed above, the time series of the number of births per month in New York city is seasonal with a peak every summer and trough every winter, and can probably be described using an additive model since the seasonal and random fluctuations seem to be roughly constant in size over time:. For example, we can print out the estimated values of the seasonal component by typing:.
The estimated seasonal factors are given for the months January-December, and are the same for each year. The largest seasonal factor is for July about 1. The plot above shows the original time series top , the estimated trend component second from top , the estimated seasonal component third from top , and the estimated irregular component bottom.
We see that the estimated trend component shows a small decrease from about 24 in to about 22 in , followed by a steady increase from then on to about 27 in If you have a seasonal time series that can be described using an additive model, you can seasonally adjust the time series by estimating the seasonal component, and subtracting the estimated seasonal component from the original time series.
You can see that the seasonal variation has been removed from the seasonally adjusted time series. The seasonally adjusted time series now just contains the trend component and an irregular component.
If you have a time series that can be described using an additive model with constant level and no seasonality, you can use simple exponential smoothing to make short-term forecasts. The simple exponential smoothing method provides a way of estimating the level at the current time point. Smoothing is controlled by the parameter alpha; for the estimate of the level at the current time point.
The value of alpha; lies between 0 and 1. Values of alpha that are close to 0 mean that little weight is placed on the most recent observations when making forecasts of future values. We can read the data into R and plot it by typing:. You can see from the plot that there is roughly constant level the mean stays constant at about 25 inches. The random fluctuations in the time series seem to be roughly constant in size over time, so it is probably appropriate to describe the data using an additive model.
Thus, we can make forecasts using simple exponential smoothing. For example, to use simple exponential smoothing to make forecasts for the time series of annual rainfall in London, we type:.
The output of HoltWinters tells us that the estimated value of the alpha parameter is about 0. This is very close to zero, telling us that the forecasts are based on both recent and less recent observations although somewhat more weight is placed on recent observations. By default, HoltWinters just makes forecasts for the same time period covered by our original time series. In this case, our original time series included rainfall for London from , so the forecasts are also for The plot shows the original time series in black, and the forecasts as a red line.
The time series of forecasts is much smoother than the time series of the original data here. As a measure of the accuracy of the forecasts, we can calculate the sum of squared errors for the in-sample forecast errors, that is, the forecast errors for the time period covered by our original time series. It is common in simple exponential smoothing to use the first value in the time series as the initial value for the level.
For example, in the time series for rainfall in London, the first value is For example, to make forecasts with the initial value of the level set to As explained above, by default HoltWinters just makes forecasts for the time period covered by the original data, which is for the rainfall time series.
To use the forecast. When using the forecast. HoltWinters function, as its first argument input , you pass it the predictive model that you have already fitted using the HoltWinters function. For example, to make a forecast of rainfall for the years 8 more years using forecast. HoltWinters , we type:.
The forecast. For example, the forecasted rainfall for is about To plot the predictions made by forecast. We can only calculate the forecast errors for the time period covered by our original time series, which is for the rainfall data.
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Neumann" The null distribution of the serial correlation coefficient may be badly affected by departures from normality in the underlying process Cox, ; Bartels, It is therefore a good idea to consider using a nonparametric test for randomness if the normality of the underlying process is in doubt Bartels, Wald and Wolfowitz introduced the rank serial correlation coefficient, which for lag-1 autocorrelation is simply the Yule-Walker estimate Equation 5 above with the actual observations replaced with their ranks.
This statistic is bounded between 0 and 4, and for a purely random process is symmetric about 2. Small values of this statistic indicate possible positive autocorrelation, and large values of this statistics indicate possible negative autocorrelation.
Durbin and Watson , , proposed using this statistic in the context of checking the independence of residuals from a linear regression model and provided tables for the distribution of this statistic.
This test statistic does not allow for missing values. Bartels shows that asymptotically, the rank von Neumann ratio statistic is a linear transformation of the rank serial correlation coefficient, so any asymptotic results apply to both statistics.
Based on this exact distribution, Bartels presents a table of critical values for the numerator of the RVN statistic for sample sizes between 4 and Note : The definition of the beta distribution assumes the random variable ranges from 0 to 1. This definition can be generalized as follows. Neumann" , the function serialCorrelationTest does the following:. If the number of ties is small, however, they may not grossly affect the assumed p-value.
When ties are present, the function serialCorrelationTest issues a warning. Bartels, R.
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