Using Non-Parametric Tests to Evaluate Traffic Forecasting Performance.

This paper proposes the use of a number of nonparametric comparison methods for evaluating traffic flow forecasting techniques. The advantage to these methods is that they are free of any distributional assumptions and can be legitimately used on small datasets. To demonstrate the applicability of these tests, a number of models for the forecasting of traffic flows are developed. The one-step-ahead forecasts produced are then assessed using nonparametric methods. Consideration is given as to whether a method is universally good or good at reproducing a particular aspect of the original series. That choice will be dictated, to a degree, by the user’s purpose for assessing traffic flow

Outlier Detection and Missing Value Estimation in Time Series Traffic Count Data: Final Report of SERC Project GR/G23180.

A serious problem in analysing traffic count data is what to do when missing or extreme values occur, perhaps as a result of a breakdown in automatic counting equipment. The objectives of this current work were to attempt to look at ways of solving this problem by: 1)establishing the applicability of time series and influence function techniques for estimating missing values and detecting outliers in time series traffic data; 2)making a comparative assessment of new techniques with those used by traffic engineers in practice for local, regional or national traffic count systems Two alternative approaches were identified as being potentially useful and these were evaluated and compared with methods currently employed for `cleaning' traffic count series. These were based on evaluating the effect of individual or groups of observations on the estimate of the auto-correlation structure and events influencing a parametric model (ARIMA). These were compared with the existing methods which included visual inspection and smoothing techniques such as the exponentially weighted moving average in which means and variances are updated using observations from the same time and day of week. The results showed advantages and disadvantages for each of the methods. The exponentially weighted moving average method tended to detect unreasonable outliers and also suggested replacements which were consistently larger than could reasonably be expected. Methods based on the autocorrelation structure were reasonably successful in detecting events but the replacement values were suspect particularly when there were groups of values needing replacement. The methods also had problems in the presence of non-stationarity, often detecting outliers which were really a result of the changing level of the data rather than extreme values. In the presence of other events, such as a change in level or seasonality, both the influence function and change in autocorrelation present problems of interpretation since there is no way of distinguishing these events from outliers. It is clear that the outlier problem cannot be separated from that of identifying structural changes as many of the statistics used to identify outliers also respond to structural changes. The ARIMA (1,0,0)(0,1,1)7 was found to describe the vast majority of traffic count series which means that the problem of identifying a starting model can largely be avoided with a high degree of assurance. Unfortunately it is clear that a black-box approach to data validation is prone to error but methods such as those described above lend themselves to an interactive graphics data-validation technique in which outliers and other events are highlighted requiring acceptance or otherwise manually. An adaptive approach to fitting the model may result in something which can be more automatic and this would allow for changes in the underlying model to be accommodated. In conclusion it was found that methods based on the autocorrelation structure are the most computationally efficient but lead to problems of interpretation both between different types of event and in the presence of non-stationarity. Using the residuals from a fitted ARIMA model is the most successful method at finding outliers and distinguishing them from other events, being less expensive than case deletion. The replacement values derived from the ARIMA model were found to be the most accurate

Setar Modelling of Traffic Count Data.

As part of a SERC funded project investigating outlier detection and replacement with transport data, univariate Box-Jenkins (1976) models have already been successfully applied to traffic count series (see Redfern et al, 1992). However, the underlying assumption of normality for ARIMA models implies they are not ideally suited for time series exhibiting certain behavioural characteristics. The limitations of ARIMA models are discussed in some detail by Tong (1983), including problems with time irreversibility, non-normality, cyclicity and asymmetry. Data with irregularly spaced extreme values are unlikely to be modelled well by ARIMA models, which are better suited to data where the probability of a very high value is small. Tong (1983) argues that one way of modelling such non-normal behaviour might be to retain the general ARIMA framework and allow the white noise element to be non-gaussian. As an alternative he proposes abandoning the linearity assumption and defines a group of non linear structures, one of which is the Self-Exciting Threshold Autoregressive (SETAR) model. The model form is described in more detail below but basically consists of two (or more) piecewise linear models, with the time series "tripping" between each model according to its value with respect to a threshold point. The model is called "Self-Exciting" because the indicator variable determining the appropriate linear model for each piece of data is itself a function of the data series. Intuitively this means the mechanism driving the alternation between each model form is not an external input such as a related time series (other models can be defined where this exists), but is actually contained within the series itself. The series is thus Self-Exciting. The three concepts embedded within the SETAR model structure are those of the threshold, limit cycle and time delay, each of which can be illustrated by the diverse applications such models can take. The threshold can be defined as some point beyond which, if the data falls, the series structure changes inherently and so an alternative linear model form would be appropriate. In hydrology this is seen as the non-linearity of soil infiltration, where at the soil saturation point (threshold) a new model for infiltration would become appropriate. Limit cycles describe the stable cyclical phenomena which we sometimes observe within time series. The cyclical behaviour is stationary, ie consists of regular, sustained oscillations and is an intrinsic property of the data. The limit cycle phenomena is physically observable in the field of radio-engineering where a triode valve is used to generate oscillations (see Tong, 1983 for a full description). Essentially the triode value produces self-sustaining oscillations between emitting and collecting electrons, according to the voltage value of a grid placed between the anode and cathode (thereby acting as the threshold indicator). The third essential concept within the SETAR structure is that of the time delay and is perhaps intuitively the easiest to grasp. It can be seen within the field of population biology where many types of non-linear model may apply. For example within the cyclical oscillations of blowfly population data there is an inbuilt "feedback" mechanism given by the hatching period for eggs, which would give rise to a time delay parameter within the model. For some processes this inherent delay may be so small as to be virtually instantaneous and so the delay parameter could be omitted. In general time series Tong (1983) found the SETAR model well suited to the cyclical nature of the Canadian Lynx trapping series and for modelling riverflow systems (Tong, Thanoon & Gudmundsson, 1984). Here we investigate their applicability with time series traffic counts, some of which have exhibited the type of non-linear and cyclical characteristics which could undermine a straightforward linear modelling process

An Influence Method for Outliers Detection Applied to Time Series Traffic Data

The applicability of an outlier detection statistic developed for standard time series is assessed in estimating missing values and detecting outliers in traffic count data. The work of Chernick, Downing and Pike (1982) is extended to form a quantitive outlier detection statistic for use with time series data. The statistic is formed from the squared elements of the Influence Function Matrix, where each element of the matrix gives the influence on pk, of a pair of observations at time lag k. Approximate first four moments for the statistic are derived and by fitting Johnson curves to those theoretical moments, critical points are also produced. The statistic is also used to form the basis of an adjustment procedure to treat outliers or estimate missing values in the time series. Chernick et al's (1982) nuclear power data and the Department of Transport's traffic count data are used for practical illustration

Development of an Influence Statistic for Outlier Detection With Time Series Traffic Data.

As part of a SERC funded project investigating the detection and treatment of outlying time series transport data, the practical applicability of the Influence Statistic described by Watson et al(1991) is assessed here. Missing or outlying data occur in a variety of transport time series such as traflic counts or journey times for many reasons including broken machinery and recording errors. In practice such data is patched largely by subjective opinion or using simple aggregate methods. In the analysis of non-transport time series several methods have been recently developed to both detect and treat outliers, including work by Kohn and Ansley (1986), Hau and Tong (1984) and Bruce and Martin (1989). These methods use either an intervention modelling approach (where the outlier is modelled as part of an ARIMA structure) or look at the influence an observation exerts on a particular parameter associated with the model. An alternative is the Influence Statistic proposed by Watson (1987) and Watson et al (1992) which examines the influence of an observation on the sample autocorrelation function. Initial research showed the statistic has practical application in a transport context, and a replacement procedure based on the method was found to be effective in treating maverick data. Here we report the results from a wider application of the statistic using traffic count data fmm. the Department of Transport. Further developments are suggested and investigated for the replacement procedure and a comparison is made between possible variations in the method

Detection of Outliers in Time Series.

As part of a SERC funded project this study aims to summarise the most relevant and recent literature with respect to outlier detection for time series and missing value estimation in traffic count data. Many types of transport data are collected over time and are potentlally suited to the application of time series analysis techniques. including accident data, ticket sales and traffic counts. Missing data or outliers in traffic counts can cause problems when analysing the data, for example in order to produce forecasts. At present it seems that little work has been undertaken to assess the merits of alternative methods to treat such data or develop a more analytic approach. Here we intend to review current practices in the transport field and summarise more general time series techniques for handling outlying or missing data. The literature study forms the fist stage of a research project aiming to establish the applicability of time series and other techniques in estimating missing values and outlier detection/replacement in a variety of transport data. Missing data and outliers can occur for a variety of reasons, for example the breakdown of automatic counters. Initial enquiries suggest that methods for patching such data can be crude. Local authorities are to be approached individually usinga short questionnaire enquiry form in order to attempt to ascertain their current practices. Having reviewed current practices the project aims to transfer recently developed methods for dealing with outliers in general time series into a transport context. It is anticipated that comparisons between possible methods could highlight an alternative and more analytical approach to current practices. A description of the main methods ior detecting outliers in time series is given within the first section. In the second section practical applications of Box-Jenkins methods within a transport context are given. current practices for dealing with outlying and missing data within transport are discussed in section three. Recommendations for methods to be used in our current research are followed by the appendices containing most of the mathematical detail

Approximate solution of the Duffin-Kemmer-Petiau equation for a vector Yukawa potential with arbitrary total angular momenta

The usual approximation scheme is used to study the solution of the Duffin-Kemmer-Petiau (DKP) equation for a vector Yukawa potential in the framework of the parametric Nikiforov-Uvarov (NU) method. The approximate energy eigenvalue equation and the corresponding wave function spinor components are calculated for arbitrary total angular momentum in closed form. Further, the approximate energy equation and wave function spinor components are also given for case. A set of parameter values is used to obtain the numerical values for the energy states with various values of quantum levelsComment: 17 pages; Communications in Theoretical Physics (2012). arXiv admin note: substantial text overlap with arXiv:1205.0938, and with arXiv:quant-ph/0410159 by other author

Pyrrolo- and pyridomorphinans:Non-selective opioid antagonists and delta opioid agonists/mu opioid partial agonists

Opioid ligands have found use in a number of therapeutic areas, including for the treatment of pain and opiate addiction (using agonists) and alcohol addiction (using antagonists such as naltrexone and nalmefene). The reaction of imines, derived from the opioid ligands oxymorphone and naltrexone, with Michael acceptors leads to pyridomorphinans with structures similar to known pyrrolo- and indolomorphinans. One of the synthesized compounds, 5e, derived from oxymorphone had substantial agonist activity at delta opioid receptors but not at mu and/or kappa opioid receptors and in that sense profiled as a selective delta opioid receptor agonist. The pyridomorphinans derived from naltrexone and naloxone were all found to be non-selective potent antagonists and as such could have utility as treatments for alcohol abuse

Modelling Outliers and Missing Values in traffic Count Data Using the ARIMA Model.

This paper considers the application of the methodology to traffic count time series in which both missing values and outliers are present. Intervention analysis and detection using large residuals are shown to he reasonably effective but possible problems that result from non- stationarity in the data are identified. It is shown that despite considerable variabilty in the types of series the model selected from the ARIMA family is surprisingly homogeneous

Post-training inactivation of the anterior thalamic nuclei impairs spatial performance on the radial arm maze

The limbic thalamus, specifically the anterior thalamic nuclei (ATN), contains brain signals including that of head direction cells, which fire as a function of an animal\u27s directional orientation in an environment. Recent work has suggested that this directional orientation information stemming from the ATN contributes to the generation of hippocampal and parahippocampal spatial representations, and may contribute to the establishment of unique spatial representations in radially oriented tasks such as the radial arm maze. While previous studies have shown that ATN lesions can impair spatial working memory performance in the radial maze, little work has been done to investigate spatial reference memory in a discrimination task variant. Further, while previous studies have shown that ATN lesions can impair performance in the radial maze, these studies produced the ATN lesions prior to training. It is therefore unclear whether the ATN lesions disrupted acquisition or retention of radial maze performance. Here, we tested the role of ATN signaling in a previously learned spatial discrimination task on a radial arm maze. Rats were first trained to asymptotic levels in a task in which two maze arms were consistently baited across training. After 24 h, animals received muscimol inactivation of the ATN before a 4 trial probe test. We report impairments in post-inactivation trials, suggesting that signals from the ATN modulate the use of a previously acquired spatial discrimination in the radial-arm maze. The results are discussed in relation to the thalamo-cortical limbic circuits involved in spatial information processing, with an emphasis on the head direction signal. © 2017 Harvey, Thompson, Sanchez, Yoder and Clark