The Time Series platform begins by showing a time series plot, like the one shown previously in Time Series Plot of Seriesg (Airline Passenger) Data. The Graph command on the platform popup menu has a submenu of controls for the time series plot with the following commands.
hides or displays a horizontal line in the time series graph that depicts the mean of the time series.
The autocorrelation graph describes the correlation between all the pairs of points in the time series for a given separation in time (lag). Autocorrelation and partial autocorrelation graphs can help you determine whether the time series is stationary (meaning it has a fixed mean and standard deviation over time) and what model might be appropriate to fit the time series.
Tip: The autocorrelation graph of the sample is often called the sample autocorrelation function.
For autocorrelation, the Ljung-Box Q and p-values appear for each lag. The Q-statistic can be used to test whether a group of autocorrelations is significantly different from zero, or to test that the residuals from a model can be distinguished from white noise.
The number of lags begins with 0 to provide a broader picture of the analysis. To compute correlations beginning with lag 1, modify the JMP preferences before generating the graph. Select File > Preferences > Platforms > Time Series and then select Suppress Lag 0 in ACF and PACF.
The autocorrelation for the kth lag is computed as follows:
where is the mean of the N non-missing points in the time series. The bars graphically depict the autocorrelations.
By definition, the first autocorrelation (lag 0) always has length 1. The curves show twice the large-lag standard error (± 2 standard errors), computed as
For partial autocorrelation, the blue lines represent ± 2 standard errors for approximate 95% confidence limits, where the standard error is computed as follows:
for all k
The Variogram command alternately displays or hides the graph of the variogram. The variogram measures the variance of the differences of points k lags apart and compares it to that for points one lag apart. The variogram is computed from the autocorrelations as
where rk is the autocorrelation at lag k. The plot on the left in Variogram Graph (left) and AR Coefficient Graph (right) shows the Variogram graph for the Seriesg data.
The AR Coefficients command alternately displays or hides the graph of the least squares estimates of the autoregressive (AR) coefficients. The definition of these coefficients is given below. These coefficients approximate those that you would obtain from fitting a high-order, purely autoregressive model. The right-hand graph in Variogram Graph (left) and AR Coefficient Graph (right) shows the AR coefficients for the Seriesg data.
The Spectral Density command alternately displays or hides the graphs of the spectral density as a function of period and frequency (Spectral Density Plots).
The periodogram is smoothed and scaled by 1/(4π) to form the spectral density.
The Fisher’s Kappa statistic tests the null hypothesis that the values in the series are drawn from a normal distribution with variance 1 against the alternative hypothesis that the series has some periodic component. Kappa is the ratio of the maximum value of the periodogram, I(fi), and its average value. The probability of observing a larger Kappa if the null hypothesis is true is given by
where q = N / 2 if N is even, q = (N - 1) / 2 if N is odd, and κ is the observed value of Kappa. The null hypothesis is rejected if this probability is less than the significance level α.
For q - 1 > 100, Bartlett’s Kolmogorov-Smirnov compares the normalized cumulative periodogram to the cumulative distribution function of the uniform distribution on the interval (0, 1). The test statistic equals the maximum absolute difference of the cumulative periodogram and the uniform CDF. If it exceeds , then reject the hypothesis that the series comes from a normal distribution. The values a = 1.36 and a = 1.63 correspond to significance levels 5% and 1% respectively.
Save Spectral Density creates a new table containing the spectral density and periodogram where the (i+1)th row corresponds to the frequency fi = i / N (that is, the ith harmonic of 1 / N).
is the period of the ith harmonic, 1 / fi.
is the periodogram, I(fi).
The Number of Forecast Periods command displays a dialog for you to reset the number of periods into the future that the fitted models will forecast. The initial value is set in the Time Series Launch dialog. All existing and future forecast results will show the new number of periods with this command.
Many time series do not exhibit a fixed mean. Such nonstationary series are not suitable for description by some time series models such as those with only autoregressive and moving average terms (ARMA models). However, these series can often be made stationary by differencing the values in the series. The differenced series is given by
where t is the time index and B is the backshift operator defined by Byt = yt-1.
The Difference command computes the differenced series and produces graphs of the autocorrelations and partial autocorrelations of the differenced series. These graphs can be used to determine if the differenced series is stationary.
Several of the time series models described in the next sections accommodate a differencing operation (the ARIMA, Seasonal ARIMA models, and some of the smoothing models). The Difference command is useful for determining the order of differencing that should be specified in these models.
The Differencing Specification dialog appears in the report window when you select the Difference command. It allows you to specify the differencing operation you want to apply to the time series. Click Estimate to see the results of the differencing operation. The Specify Differencing dialog allows you to specify the Nonseasonal Differencing Order, d, the Seasonal Differencing Order, D, and the number of Periods Per Season, s. Selecting zero for the value of the differencing order is equivalent to no differencing of that kind.
controls the plot of the differenced series and behaves the same as those under the Time Series Graph menu.
appends the differenced series to the original data table. The leading d + sD elements are lost in the differencing process. They are represented as missing values in the saved series.
The plot show how an observation at time t is related to another observation at time t +/- p. +/- p is the lag, because the plot allows negative values and positive values.