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Directionality can be seen in many stationary time series from various disciplines, but it is overlooked when fitting linear models with Gaussian errors. Moreover, we cannot rely on distinguishing directionality by comparing a plot of a time series in time order with a plot in reverse time order. In general, a statistical measure is required to detect and quantify directionality. There are several quite different qualitative forms of directionality, and we distinguish: rapid rises followed by slow recessions; rapid increases and rapid decreases from the mean followed by slow recovery toward the mean; directionality above or below some threshold; and intermittent directionality. The first objective is to develop a suite of statistical measures that will detect directionality and help classify its nature. The second objective is to demonstrate the potential benefits of detecting directionality. We consider applications from business, environmental science, finance, and medicine. Time series data are collected from many processes, both natural and anthropogenic, by a wide range of organizations, and directionality can easily be monitored as part of routine analysis. We suggest that doing so may provide new insights to the processes.

We provide empirical evidence of directionality in high‐frequency multivariate time series of the five largest U.S. banks between 1999 and 2017. The directionality is more apparent during crisis periods than during noncrisis periods, and it has only a low association with volatility. We use directionality and volatility as a regime‐switching criterion between two‐regime threshold vector autoregressive (TVAR) models for forecasting share prices. We compare the forecasting performances using mean relative error squared, and a weighted average of the forecasting error, with weights based on the estimated conditional variance, for individual model components and as a group. We have demonstrated that moving directionality can provide early warning of increased volatility and crisis periods, and has potential for improving one‐step ahead forecasts using TVAR(1) models. Refereed/Peer-reviewed

Daily rainfall records from seven stations in South Australia, with record lengths from 50 to 137 years and a common period of 36 years, are investigated for evidence of changes in the statistical distribution of annual total and annual average of monthly daily maxima. In addition, the monthly time series of monthly totals and monthly daily maxima are analysed for three stations for which records exceed 100 years. The monthly series show seasonality and provide evidence of a reduction in rainfall when the Southern Oscillation Index (SOI) is negative, which is modulated by the Pacific Decadal Oscillation (PDO). However, the monthly series do not provide any evidence of a consistent trend or of any changes in the seasonal pattern. Multivariate analyses, typically used in statistical quality control (SQC), are applied to time series of yearly totals and of averages of the 12 monthly daily maxima, during the common 36-year period. Although there are some outlying points in the charts, there is no evidence of any trend or step changes. However, some supplementary permutation tests do provide weak evidence of an increase of variability of rainfall measures. Furthermore, a factor analysis does provide some evidence of a change in the spatial structure of extremes. The variability of a factor which represents the difference between extremes in the Adelaide Hills and the plains increases in the second 18 years relative to the first 18 years. There is also some evidence that the mean of this factor has increased in absolute magnitude. Refereed/Peer-reviewed

This study compares time series models for stream flow that use lagged rainfall as an exogenous variable with models that use a small subset of discrete wavelet coefficients of lagged rainfall with cross product and quadratic terms of the wavelet coefficients. The models require the calculation of a moving discrete wavelet transform (MDWT) that is implemented as a multiscale transform. A comparison is made using data from three catchments in the Murray Darling Basin in Australia. For finite impulse response models, the adjusted coefficient of determination, (Radj2), increased with the MDWT from 0.36 to 0.46 for the Tooma River Basin (TRB), from 0.43 to 0.52 for the Jingellic Catchment (JC), and from 0.56 to 0.59 for the Ovens Catchment (OC). The autoregressive models with exogenous input (ARX) with the wavelet transform further improved these results. The MDWT is based on a Haar wavelet so the wavelet coefficients have a physical interpretation. Predictions are further improved by a two-stage pred...

Many time series show directionality as plots against time and against time-to-go are qualitatively different. A stationary linear model with Gaussian noise is non-directional (reversible). Directionality can be emulated by introducing non-Gaussian errors or by using a nonlinear model. Established measures of directionality are reviewed and modified for time series that are symmetrical about the time axis. The sunspot time series is shown to be directional with relatively sharp increases. A threshold autoregressive model of order 2, TAR(2) is fitted to the sunspot series by (nonlinear) least squares and is shown to give an improved fit on autoregressive models. However, this model does not model closely the directionality, so a penalized least squares procedure was implemented. The penalty function included a squared difference of the discrepancy between observed and simulated directionality. The TAR(2) fitted by penalized least squares gave improved out-of-sample forecasts and more realistic simulations of extreme values.

We compare time series of electroencephalograms (EEGs) from healthy volunteers with EEGs from subjects diagnosed with epilepsy. The EEG time series from the healthy group are recorded during awake state with their eyes open and eyes closed, and the records from subjects with epilepsy are taken from three different recording regions of pre-surgical diagnosis: hippocampal, epileptogenic and seizure zone. The comparisons for these 5 categories are in terms of deviations from linear time series models with constant variance Gaussian white noise error inputs. One feature investigated is directionality, and how this can be modelled by either non-linear threshold autoregressive models or non-Gaussian errors. A second feature is volatility, which is modelled by Generalized AutoRegressive Conditional Heteroskedasticity (GARCH) processes. Other features include the proportion of variability accounted for by time series models, and the skewness and the kurtosis of the residuals. The results suggest these comparisons may have diagnostic potential for epilepsy and provide early warning of seizures. Refereed/Peer-reviewed

A stationary time series model is directional if it has properties that are not symmetric with respect to time, for example if, after the expected value is adjusted to 0, the expected value of the product of the variable squared and the following variable differs from the expected value of the product of the variable and the following variable squared. Marked directionality in time series is apparent from a comparison of time series plots in chronological order and in reverse order (time-to-go), but formal quantitative tests are needed to determine the strength of evidence against an assumption of reversibility (no directionality). Linear time series models with Gaussian noise (LGN models) are reversible whereas linear models with other noise distributions and non-linear models are directional. Evidence of directionality in time series can be found in many disciplines including financial time series. Examples of directional financial time series are presented, together with values of directionality statistics. A method of calculating the p-values of the directional statistics, corresponding to a null hypothesis of reversibility, using simulation is presented. The time series are then modelled by threshold autoregressive models, with noise distributions, which can be non-Gaussian, fitted to match the empirical distribution of the residuals. The fitting criterion is a weighted sum of the error sum of squares and a measure of discrepancy between values of the directionality statistics in the time series and long simulations from the potential model. Strategies for determining suitable weights for the criterion will be investigated. The directional time series models will be assessed in terms of the distribution of extreme values of the time series, which is a feature that was not used in the fitting process. The assessment will be made by comparing the distribution of extremes from: the observed time series; a long simulation from the fitted directional model; and a long simulation from a LGN model fitted to the time series on the basis of the Akaike information criterion (AIC). Refereed/Peer-reviewed