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

Improving the health and nutrition of dairy cows by investigating the farmer and stakeholder attitudes and behaviours that influence health in the transition period

During the transition from the dry period to lactation the dairy cow undergoes a period of physiological, metabolic and immunological change, and is at greater risk of developing disease, to the detriment of health, welfare and production. Many studies have been undertaken to determine appropriate management strategies to improve health and welfare during the transition period, however the incidence of disease during this period remains high. In this study, 22 dairy farmers calving all year round (AYR), 10 dairy farmers block calving herds, 12 veterinary advisors and 12 non-veterinary advisors were interviewed. A farm audit of the 22 AYR herds showed that 11 of the herds had more than 15% lame cows in the pre-calving or early lactation groups. Most dietary minerals were oversupplied in early lactation and pre-calving diets, although 12 out of 22 farms did not supply enough magnesium pre-calving as judged by NRC requirements. The qualitative data showed a lack of awareness of metabolic disease and potential risk factors in farmers with AYR calving herds. A key theme arising from the advisor interviews was a perceived lack of focused transition management advice provided by advisors, and a lack of cooperation between veterinarians and nutritionists. A nationwide questionnaire was also conducted, finding the majority of farmers (52%) were actively seeking advice to improve their transition management. The questionnaire demonstrated that many of the themes derived from the interviews can be applied to other dairy farmers in England, such as farmers having positive relationships with their veterinarians and nutritionists. Heterogeneity in farmer attitudes, management systems and infrastructure highlighted the difficulty in delivering a one-size-fits-all approach to metabolic disease control. Future initiatives should focus on a tailored approach, understanding the main priorities of the farmer, and acknowledging the farm-specific infrastructure and layout

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

Application of Outlier Detection and Missing Value Estimation Techniques to Various Forms of Traffic Count Data

This paper reports on the application of suitable techniques for detecting outliers and suggesting estimates for missing values in various forrns of traffic count data. The data used in this study came from three sources. The first set was provided by the Department of Transport's (DOT) regional office in Leeds and consists of automatic hourly traffic counts at four sites. The second set was part of a larger database provided by West Yorkshire Highways, Engineering and Technical Services (HETS). This set consists of automatic half hourly traffic counts on a single site. The third and final set was provided by Nottinghan University and consists of automatic five minute traffic counts at 40 locations, in close proximity to each other, from Leicester. Three suitable techniques emerged from pilot studies of such series conducted by Watson et a1 (1992a) and Redfern et a1 (1992). The three techniques are: a) Maintaining an average and variability measure over time; b) ARIMA modelling with detection of large residuals; C) A point's influence on the correlation structure of the series. A fourth technique, by-eye detection and estimation, provides an intuitive comparison for the first three techniques

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

Using Administrative Data to Count Local Populations

There is growing evidence that official population statistics based on the decennial census are inaccurate at the local authority level—the fundamental administrative unit of the UK. This paper investigates the use of locally available administrative data sets for counting populations. The method uses truth tables for combining different data sources with different population coverage according to a defined and therefore replicable set of rules. The result is timelier and geographically more flexible data which is more cost-effective to produce than a survey-based census. Associated techniques for linking diverse data sources at individual and household level are briefly discussed. The methodology is then applied to administrative data from a London borough with about 170,000 people. The results are evaluated and compared with other population sources. The paper concludes by discussing potential improvements including scaling up the work to cover multiple local authorities. The practicalities of using alternative central government data sets are briefly considered. A sequel paper in this journal provides examples of key applications of this approach at local level