Bayesian error bars for regression

Regression problems are concerned with predicting the values of one or more continuous quantities, given the values of a number of input variables. For virtually every application of regression, however, it is also important to have an indication of the uncertainty in the predictions. Such uncertainties are expressed in terms of the error bars, which specify the standard deviation of the distribution of predictions about the mean. Accurate estimate of error bars is of practical importance especially when safety and reliability is an issue. The Bayesian view of regression leads naturally to two contributions to the error bars. The first arises from the intrinsic noise on the target data, while the second comes from the uncertainty in the values of the model parameters which manifests itself in the finite width of the posterior distribution over the space of these parameters. The Hessian matrix which involves the second derivatives of the error function with respect to the weights is needed for implementing the Bayesian formalism in general and estimating the error bars in particular. A study of different methods for evaluating this matrix is given with special emphasis on the outer product approximation method. The contribution of the uncertainty in model parameters to the error bars is a finite data size effect, which becomes negligible as the number of data points in the training set increases. A study of this contribution is given in relation to the distribution of data in input space. It is shown that the addition of data points to the training set can only reduce the local magnitude of the error bars or leave it unchanged. Using the asymptotic limit of an infinite data set, it is shown that the error bars have an approximate relation to the density of data in input space.

Statnote 25: Stepwise multiple regression

An investigator may also wish to select a small subset of the X variables which give the best prediction of the Y variable. In this case, the question is how many variables should the regression equation include? One method would be to calculate the regression of Y on every subset of the X variables and choose the subset that gives the smallest mean square deviation from the regression. Most investigators, however, prefer to use a ‘stepwise multiple regression’ procedure. There are two forms of this analysis called the ‘step-up’ (or ‘forward’) method and the ‘step-down’ (or ‘backward’) method. This Statnote illustrates the use of stepwise multiple regression with reference to the scenario introduced in Statnote 24, viz., the influence of climatic variables on the growth of the crustose lichen Rhizocarpon geographicum (L.)DC.

Impact of RFID in reverse logistics

This paper aims to discuss the recent literature on Radio Frequency Identification (RFID) and reverse logistics (RL). Particular attention is applied to the bullwhip effect and its increase as RL activities are integrated into the supply chain. RFID is investigated as a tool to assist with integrating reverse and forward logistics into a seamless supply chain and reduce the bullwhip effect. However, further research is required within this area and in particular the return on investment for RFID-enabled integrated systems.

Mathematical modelling of patient risks in a hospital environment, using multiple regression analysis

The aim of this research work was primarily to examine the relevance of patient parameters, ward structures, procedures and practices, in respect of the potential hazards of wound cross-infection and nasal colonisation with multiple resistant strains of Staphylococcus aureus, which it is thought might provide a useful indication of a patient's general susceptibility to wound infection. Information from a large cross-sectional survey involving 12,000 patients from some 41 hospitals and 375 wards was collected over a five-year period from 1967-72, and its validity checked before any subsequent analysis was carried out. Many environmental factors and procedures which had previously been thought (but never conclusively proved) to have an influence on wound infection or nasal colonisation rates, were assessed, and subsequently dismissed as not being significant, provided that the standard of the current range of practices and procedures is maintained and not allowed to deteriorate. Retrospective analysis revealed that the probability of wound infection was influenced by the patient's age, duration of pre-operative hospitalisation, sex, type of wound, presence and type of drain, number of patients in ward, and other special risk factors, whilst nasal colonisation was found to be influenced by the patient's age, total duration of hospitalisation, sex, antibiotics, proportion of occupied beds in the ward, average distance between bed centres and special risk factors. A multi-variate regression analysis technique was used to develop statistical models, consisting of variable patient and environmental factors which were found to have a significant influence on the risks pertaining to wound infection and nasal colonisation. A relationship between wound infection and nasal colonisation was then established and this led to the development of a more advanced model for predicting wound infections, taking advantage of the additional knowledge of the patient's state of nasal colonisation prior to operation.

Statnote 24: Multiple regression

In previous statnotes, the application of correlation and regression methods to the analysis of two variables (X,Y) was described. These methods can be used to determine whether there is a linear relationship between the two variables, whether the relationship is positive or negative, to test the degree of significance of the linear relationship, and to obtain an equation relating Y to X. This Statnote extends the methods of linear correlation and regression to situations where there are two or more X variables, i.e., 'multiple linear regression’.

Multi-criteria logistics distribution network design using SAS/OR

This paper explores the use of the optimization procedures in SAS/OR software with application to the contemporary logistics distribution network design using an integrated multiple criteria decision making approach. Unlike the traditional optimization techniques, the proposed approach, combining analytic hierarchy process (AHP) and goal programming (GP), considers both quantitative and qualitative factors. In the integrated approach, AHP is used to determine the relative importance weightings or priorities of alternative warehouses with respect to both deliverer oriented and customer oriented criteria. Then, a GP model incorporating the constraints of system, resource, and AHP priority is formulated to select the best set of warehouses without exceeding the limited available resources. To facilitate the use of integrated multiple criteria decision making approach by SAS users, an ORMCDM code was implemented in the SAS programming language. The SAS macro developed in this paper selects the chosen variables from a SAS data file and constructs sets of linear programming models based on the selected GP model. An example is given to illustrate how one could use the code to design the logistics distribution network.

Strategic logistics outsourcing:an integrated QFD and fuzzy AHP approach

This paper develops an integratedapproach, combining quality function deployment (QFD), fuzzy set theory, and analytic hierarchy process (AHP) approach, to evaluate and select the optimal third-party logistics service providers (3PLs). In the approach, multiple evaluating criteria are derived from the requirements of company stakeholders using a series of house of quality (HOQ). The importance of evaluating criteria is prioritized with respect to the degree of achieving the stakeholder requirements using fuzzyAHP. Based on the ranked criteria, alternative 3PLs are evaluated and compared with each other using fuzzyAHP again to make an optimal selection. The effectiveness of proposed approach is demonstrated by applying it to a Hong Kong based enterprise that supplies hard disk components. The proposed integratedapproach outperforms the existing approaches because the outsourcing strategy and 3PLs selection are derived from the corporate/business strategy.

Gaussian process quantile regression using expectation propagation

Direct quantile regression involves estimating a given quantile of a response variable as a function of input variables. We present a new framework for direct quantile regression where a Gaussian process model is learned, minimising the expected tilted loss function. The integration required in learning is not analytically tractable so to speed up the learning we employ the Expectation Propagation algorithm. We describe how this work relates to other quantile regression methods and apply the method on both synthetic and real data sets. The method is shown to be competitive with state of the art methods whilst allowing for the leverage of the full Gaussian process probabilistic framework.

Strategic logistics outsourcing: an integrated QFD and AHP approach

Third-party logistics service providers (3PLs) play a vital role in contemporary supply chain management. Evaluation and selection of the right 3PLs depends on a wide range of quantitative and qualitative criteria rather than cost-based factors. Although various multi-criteria decision making approaches have been proposed, they have not considered the impact of business objectives and requirements of company stakeholders on the evaluating criteria. To enable the "voice" of company stakeholders is considered, this paper develops an integrated approach for selecting 3PL strategically. In the approach, multiple evaluating criteria are derived from the requirements of company stakeholders using a series of house of quality (HOQ). The importance of evaluating criteria is prioritized with respect to the degree of achieving the stakeholder requirements using analytic hierarchy process (AHP). Based on the ranked criteria, alternative 3PLs are evaluated and compared with each other using AHP again to make an optimal selection.

Investigating local delivery by road to retail and non-retail customers in the UK logistics sector

Over recent years, hub-and-spoke distribution techniques have attracted widespread research attention. Despite there being a growing body of literature in this area there is less focus on the spoke-terminal element of the hub-and-spoke system as being a key component in the overall service received by the end-user. Current literature is highly geared towards discussing bulk optimization of freight units rather than to the more discrete and individualistic profile characteristics of shared-user Less-than-truckload (LTL) freight. In this paper, a literature review is presented to review the role hub-and-spoke systems play in meeting multi-profile customer demands, particularly in developing sectors with more sophisticated needs, such as retail. The paper also looks at the use of simulation technology as a suitable tool for analyzing spoke-terminal operations within developing hub-and spoke systems.

The multiple pheromone ant clustering algorithm and its application to real world domains

The Multiple Pheromone Ant Clustering Algorithm (MPACA) models the collective behaviour of ants to find clusters in data and to assign objects to the most appropriate class. It is an ant colony optimisation approach that uses pheromones to mark paths linking objects that are similar and potentially members of the same cluster or class. Its novelty is in the way it uses separate pheromones for each descriptive attribute of the object rather than a single pheromone representing the whole object. Ants that encounter other ants frequently enough can combine the attribute values they are detecting, which enables the MPACA to learn influential variable interactions. This paper applies the model to real-world data from two domains. One is logistics, focusing on resource allocation rather than the more traditional vehicle-routing problem. The other is mental-health risk assessment. The task for the MPACA in each domain was to predict class membership where the classes for the logistics domain were the levels of demand on haulage company resources and the mental-health classes were levels of suicide risk. Results on these noisy real-world data were promising, demonstrating the ability of the MPACA to find patterns in the data with accuracy comparable to more traditional linear regression models. © 2013 Polish Information Processing Society.

The evolution of logistics service providers and the role of Internet-based applications in facilitating global operations

The need for global logistics services has increased dramatically and become extremely complex and dynamic as a result of a number of changes in manufacturing and in industrial production. In response, the logistics industry is changing in a variety of ways, including mergers to form integrated transportation service providers, outsourcing and increased use of information technology. The aim of this chapter is to provide an overview of the evolution and the most important trends in the logistics services provider (LSP) industry. Specific emphasis will be given to the role of Internet-based applications. Within this context, the chapter will also present the role of logistics e-marketplaces. In particular, based on the secondary research of currently existing logistics on-line marketplaces, an analysis and classification of them is provided with the aim of identifying service gaps. The analysis reveals that logistics electronic marketplaces, despite the increased range of services currently offered, still face limitations with reference to integrated customs links or translation services, which both reduce the efficiency of global operations.

Logistics by rail:is this the key to more sustainable traffic policies in Northern Germany?

Despite the fact that Germany has a well- expanded traffic infrastructure, the country con-fronts a strong growth in freight volumes and it is very likely that in the forthcoming yearsit will not be able to reasonably cope with the increasing demand. Against this back-ground, the aim of this paper is to provide an in-depth analysis of the transport sector in North Germany in an effort to identify, if any, the possibilities of collapse as a result of the continuous increase in the demand of freight traffic. The research based on the DelphiTechnique, collects, analyses and summarizes the opinions of a group of experts in theaforementioned issues. Results indicate that railways could represent the solution to theforecasted growing freight volumes in the next years, not only in Germany, but also in thewhole European transport sector. In spite of continuous efforts undertaken by the politicsand the economy, the existing logistics and freight traffic concepts are not sufficient. Fi-nancing is too scarce; traffic concepts take issues like sustainability, environment protec-tion and working conditions into little consideration.

Factors determining disease duration in Alzheimer's disease:a postmortem study of 103 cases using the Kaplan-Meier estimator and Cox regression

Factors associated with duration of dementia in a consecutive series of 103 Alzheimer's disease (AD) cases were studied using the Kaplan-Meier estimator and Cox regression analysis (proportional hazard model). Mean disease duration was 7.1 years (range: 6 weeks-30 years, standard deviation = 5.18); 25% of cases died within four years, 50% within 6.9 years, and 75% within 10 years. Familial AD cases (FAD) had a longer duration than sporadic cases (SAD), especially cases linked to presenilin (PSEN) genes. No significant differences in duration were associated with age, sex, or apolipoprotein E (Apo E) genotype. Duration was reduced in cases with arterial hypertension. Cox regression analysis suggested longer duration was associated with an earlier disease onset and increased senile plaque (SP) and neurofibrillary tangle (NFT) pathology in the orbital gyrus (OrG), CA1 sector of the hippocampus, and nucleus basalis of Meynert (NBM). The data suggest shorter disease duration in SAD and in cases with hypertensive comorbidity. In addition, degree of neuropathology did not influence survival, but spread of SP/NFT pathology into the frontal lobe, hippocampus, and basal forebrain was associated with longer disease duration. © 2014 R. A. Armstrong.