Standard Time Series Methods Research Articles

Robust graphical methods for diagnosing trend in irregularly spaced water quality time series

Graphical methods can play an important role in the reliable assessment of trends in typically ill behaved river quality data series both as diagnostic tools and as visual corroborative evidence when assumptions required for formal statistical tests are not met. Robust, graphically-oriented trend diagnosis procedures are presented for data series characterized by nonnormal populations, uneven time spacing, nonmonotonic trend and other factors which can create serious problems for standard parametric time series methods. Cleveland's robust locally weighted regression (RLWR) developed for investigating nonlinearity in x-y scatterplots is adapted as a robust/resistant smoothing filter for the analysis of irregular time series comprising quantitative observations. Low powered RLWR trend lines reveal temporally local phenomena, e.g. abrupt jumps (often associated with point source impacts) and periodicities, while higher powered RLWR yields smooth lines characterizing medium and longer term trends. Simple variants of Tukey smoothing concepts are developed for series with censored observations. Applications to Ontario river quality series reveal that graphical evidence is frequently sufficient to obviate the need for formal trend testing. The methods are generally applicable to most time series.

Using Standardized Time Series to Estimate Confidence Intervals for the Difference Between Two Stationary Stochastic Processes

This paper discusses confidence interval estimation for the difference between the means of two independent, strictly stationary phi-mixing stochastic processes. The confidence intervals are based on “batched” subseries of the two time series and are obtained by using Schruben's standardized time series method. We compare these intervals with the classical result applied to the batch means, and consider different assumptions on the variances and the number of observations. The comparison results show that the four new estimators for each case have asymptotic properties that clearly dominate the classical estimator. Two applications of these intervals are (1) validating stationary discrete event simulation models, and (2) comparing two alternative policies or system designs via simulation.

A kalman filter model for single and two-stage repeated surveys

Scott and Smith (1974) have derived predictors for the mean of a time-dependent population by using standard time series methods when the variance is known. In this paper, adopting a normal superpopulation model, a Bayesian approach is applied to the analysis of single and two-stage repeated surveys under the assumption that a linear combination of the population means follows the Kalman filter model. A numerical example illustrates the performance of the Kalman predictor (K.P.) of the population total at time t for a single-stage, with the variances either known or unknown.

Cycle detection: a technique for estimating the frequency and amplitude of episodic fluctuations in blood hormone and substrate concentrations.

Investigation of episodic endocrine secretion has been hampered by inadequate analytical techniques for describing patterns of blood concentrations over time. Although standard time series methods, such as autocorrelation and power spectral analysis, are available, their use is limited to special cases in which rhythms are regular. To facilitate the analysis of our own episodic LH data, we have developed a process for determining the frequency and amplitude of both regular and irregular endocrine rhythms (signals) in the presence of high levels of random measurement errors (noise). This process, called cycle detection, engages an iterative, computerized procedure which scans data identifying sequential increases and decreases greater than an initial, preset threshold value. One complete cycle is defined as two increases greater than threshold separated by a decrease which is also greater than threshold. For an initial first pass estimate of frequency and amplitude, the threshold is set at 2.7 times the noise standard deviation. On the next pass, the threshold is readjusted, based on an empirically derived formula, and the data are scanned again. This process is repeated until the threshold reaches a stable value. We have tested the reliability of the cycle detection process by simulating irregular rhythm fluctuations of different frequencies, corrupted by various levels of noise and evaluating the signal characteristics with cycle detection analysis. These tests indicate that cycle detection provides excellent estimates of cycle frequency and amplitude, even with signal to noise ratios as low as 1.5. The ability of this process to analyze cyclic signals of almost any shape, with either regular or irregular rhythms, should make it a valuable tool in the hands of endocrine researchers.

Analysis of repeated surveys using time series methods.

Abstract Standard time series methods are applied to the analysis of repeated surveys under the assumption that the population parameters at each time period follow a stochastic model. Both overlapping and nonoverlapping surveys are considered in general terms, and specific results are obtained for the time series models assumed in the work of Patterson [10].