Practical statistical methods for call centres with a case study addressing urgent medical care delivery

Our interest is in forecasting for call centres, and in particular out-of-hours call centres (OOHCC) which deal with patient requests for medical advice outside normal working hours. Planning needs accurate forecasts of incoming call volumes. These vary by hour, day, and season, and must account for calendar effects such as Christmas. Using historical data, we explain how to use simple regression models to forecast call volumes arriving on specified days, taking into account calendar effects. We then show how we forecast the pattern of arrivals of calls during a specified day. These result in predictions for volumes of calls arriving for each day of the year, and their pattern of arrival during the day. We show how simulation models may then be used for resource allocation, uncertainty analysis, and staff scheduling. The data are details of call numbers and queue lengths from all parts of the patient-advice process for around five years, for a call centre based in Newcastle-upon-Tyne. There are around 350,000 complete cases in total. The methods are easily extended to other kinds of call centre. We describe the impact Swine flu had on call volumes in the summer of 2009, and our reactions to amend models in order to maintain forecast quality

Dimension reduction via principal variables

For many large-scale datasets it is necessary to reduce dimensionality to the point where further exploration and analysis can take place. Principal variables are a subset of the original variables and preserve, to some extent, the structure and information carried by the original variables. Dimension reduction using principal variables is considered and a novel algorithm for determining such principal variables is proposed. This method is tested and compared with 11 other variable selection methods from the literature in a simulation study and is shown to be highly effective. Extensions to this procedure are also developed, including a method to determine longitudinal principal variables for repeated measures data, and a technique for incorporating utilities in order to modify the selection process. The method is further illustrated with real datasets, including some larger UK data relating to patient outcome after total knee replacement

Time-weighted multi-touch attribution and channel relevance in the customer journey to online purchase

We address statistical issues in attributing revenue to marketing channels and inferring the importance of individual channels in customer journeys towards an online purchase. We describe the relevant data structures and introduce an example. We suggest an asymmetric bathtub shape as appropriate for time-weighted revenue attribution to the customer journey, provide an algorithm, and illustrate the method. We suggest a modification to this method when there is independent information available on the relative values of the channels. To infer channel importance, we employ sequential data analysis ideas and restrict to data which ends in a purchase. We propose metrics for source, intermediary, and destination channels based on twoand three-step transitions in fragments of the customer journey. We comment on the practicalities of formal hypothesis testing. We illustrate the ideas and computations using data from a major UK online retailer. Finally, we compare the revenue attributions suggested by the methods in this paper with several common attribution methods

Bayesian Graphical Models for Software Testing

This paper describes a new approach to the problem of software testing. The approach is based on Bayesian graphical models and presents formal mechanisms for the logical structuring of the software testing problem, the probabilistic and statistical treatment of the uncertainties to be addressed, the test design and analysis process, and the incorporation and implication of test results. Once constructed, the models produced are dynamic representations of the software testing problem. They may be used to drive test design, answer what-if questions, and provide decision support to managers and testers. The models capture the knowledge of the software tester for further use. Experiences of the approach in case studies are briefly discusse

Bayes linear sufficiency in non-exchangeable multivariate multiple regressions

We consider sufficiency for Bayes linear revision for multivariate multiple regression problems, and in particular where we have a sequence of multivariate observations at different matrix design points, but with common parameter vector. Such sequences are not usually exchangeable. However, we show that there is a sequence of transformed observations which is exchangeable and we demonstrate that their mean is sufficient both for Bayes linear revision of the parameter vector and for prediction of future observations. We link these ideas to making revisions of belief over replicated structure such as graphical templates of model relationships. We show that the sufficiencies lead to natural residual collections and thence to sequential diagnostic assessments. We show how each finite regression problem corresponds to a parallel implied infinite exchangeable sequence which may be exploited to solve the sample-size design problem. Bayes linear methods are based on limited specifications of belief, usually means, variances, and covariances. As such, the methodology is well suited to highdimensional regression problems where a full Bayesian analysis is difficult or impossible, but where a linear Bayes approach offers a pragmatic way to combine judgements with data in order to produce posterior summaries

Inferring marketing channel relevance in the customer journey to online purchase

In this paper we address the problem of inferring marketing channel importance for the customer journey to online purchase, using sequential data analysis ideas. We suggest a method for inferring the relative value of channels using historical data. We propose metrics for source, intermediary, and destination channels based on two- and three-step transitions in fragments of the customer journey. We comment on the difficulties of formal hypothesis testing. We illustrate the ideas and computations using data from a major UK online retailer