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.
Autocorrelation (left) and Partial Correlation (right) Graphs
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 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.
For partial autocorrelation, the blue lines represent ± 2 standard errors for approximate 95% confidence limits, where the standard error is computed as follows:
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.
Variogram Graph (left) and AR Coefficient Graph (right)
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.
Spectral Density Plots
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.
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 whether the differenced series is stationary.
The Differencing Specification dialog appears in the report window when you select the Difference command. It enables you to specify the differencing operation that you want to apply to the time series. Click Estimate to see the results of the differencing operation. The Specify Differencing dialog enables 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.
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.
Removes the linear trend from the data, where Trend = β0 + β1 time. A Time Series report for the detrended series is added to the report window. Also, a column containing the most recently detrended series is added to the data table.
Note: When you select the Remove Linear Trend option, JMP adds a column to the data table that contains the detrended data. If this column is already present in the data table when the option is selected, JMP overwrites the existing column.
Note: When you select the Remove Cycle option, JMP adds a column to the data table that contains the decycled data. If this column is already present in the data table when the option is selected, JMP overwrites the existing column.
C is the (optional) Constant.
A is the Amplitude of the cosine wave.
U is the number of Units per Cycle.
P is the Phase of the cosine wave.
t is one less than the row number of a given observation.
Ot is the original time series
Ct is the trend cycle component
St is the seasonal component
It is the irregular component
Overlays the X11-adjusted time series on the original time series, Ot. Original and Adjusted Plot for X11 Analysis for Monthly shows an example of the original and adjusted time series plot. The X11-adjusted values are for the multiplicative adjustment and for the additive adjustment.
Original and Adjusted Plot for X11 Analysis for Monthly
Saves four columns to the data table: the seasonally adjusted time series, the seasonal components (St), the trend cycle components (Ct), and the irregular series components (It).
For details about the contents of the X11 output tables, see Shisken et. al. (1967) or SAS/ETS 13.1 User’s Guide (search for “The output from PROC X11”).
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.
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.