Statnote 20: non-linear regression: fitting a general polynomial curve

In some circumstances, there may be no scientific model of the relationship between X and Y that can be specified in advance and indeed the objective of the investigation may be to provide a ‘curve of best fit’ for predictive purposes. In such an example, the fitting of successive polynomials may be the best approach. There are various strategies to decide on the polynomial of best fit depending on the objectives of the investigation.

Data methods in optometry. Part 10: non-linear regression analysis

1. The techniques associated with regression, whether linear or non-linear, are some of the most useful statistical procedures that can be applied in clinical studies in optometry. 2. In some cases, there may be no scientific model of the relationship between X and Y that can be specified in advance and the objective may be to provide a ‘curve of best fit’ for predictive purposes. In such cases, the fitting of a general polynomial type curve may be the best approach. 3. An investigator may have a specific model in mind that relates Y to X and the data may provide a test of this hypothesis. Some of these curves can be reduced to a linear regression by transformation, e.g., the exponential and negative exponential decay curves. 4. In some circumstances, e.g., the asymptotic curve or logistic growth law, a more complex process of curve fitting involving non-linear estimation will be required.

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.

A computational model of prosody for Yorùbá text-to-speech synthesis

This work examines prosody modelling for the Standard Yorùbá (SY) language in the context of computer text-to-speech synthesis applications. The thesis of this research is that it is possible to develop a practical prosody model by using appropriate computational tools and techniques which combines acoustic data with an encoding of the phonological and phonetic knowledge provided by experts. Our prosody model is conceptualised around a modular holistic framework. The framework is implemented using the Relational Tree (R-Tree) techniques (Ehrich and Foith, 1976). R-Tree is a sophisticated data structure that provides a multi-dimensional description of a waveform. A Skeletal Tree (S-Tree) is first generated using algorithms based on the tone phonological rules of SY. Subsequent steps update the S-Tree by computing the numerical values of the prosody dimensions. To implement the intonation dimension, fuzzy control rules where developed based on data from native speakers of Yorùbá. The Classification And Regression Tree (CART) and the Fuzzy Decision Tree (FDT) techniques were tested in modelling the duration dimension. The FDT was selected based on its better performance. An important feature of our R-Tree framework is its flexibility in that it facilitates the independent implementation of the different dimensions of prosody, i.e. duration and intonation, using different techniques and their subsequent integration. Our approach provides us with a flexible and extendible model that can also be used to implement, study and explain the theory behind aspects of the phenomena observed in speech prosody.

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.

Development of a conceptual model of the soil-moisture-plant sub-system of the hydrological cycle

The soil-plant-moisture subsystem is an important component of the hydrological cycle. Over the last 20 or so years a number of computer models of varying complexity have represented this subsystem with differing degrees of success. The aim of this present work has been to improve and extend an existing model. The new model is less site specific thus allowing for the simulation of a wide range of soil types and profiles. Several processes, not included in the original model, are simulated by the inclusion of new algorithms, including: macropore flow; hysteresis and plant growth. Changes have also been made to the infiltration, water uptake and water flow algorithms. Using field data from various sources, regression equations have been derived which relate parameters in the suction-conductivity-moisture content relationships to easily measured soil properties such as particle-size distribution data. Independent tests have been performed on laboratory data produced by Hedges (1989). The parameters found by regression for the suction relationships were then used in equations describing the infiltration and macropore processes. An extensive literature review produced a new model for calculating plant growth from actual transpiration, which was itself partly determined by the root densities and leaf area indices derived by the plant growth model. The new infiltration model uses intensity/duration curves to disaggregate daily rainfall inputs into hourly amounts. The final model has been calibrated and tested against field data, and its performance compared to that of the original model. Simulations have also been carried out to investigate the effects of various parameters on infiltration, macropore flow, actual transpiration and plant growth. Qualitatively comparisons have been made between these results and data given in the literature.

Compositional semantics of dataflow networks with query-driven communication of exact values

We develop and study the concept of dataflow process networks as used for exampleby Kahn to suit exact computation over data types related to real numbers, such as continuous functions and geometrical solids. Furthermore, we consider communicating these exact objectsamong processes using protocols of a query-answer nature as introduced in our earlier work. This enables processes to provide valid approximations with certain accuracy and focusing on certainlocality as demanded by the receiving processes through queries. We define domain-theoretical denotational semantics of our networks in two ways: (1) directly, i. e. by viewing the whole network as a composite process and applying the process semantics introduced in our earlier work; and (2) compositionally, i. e. by a fixed-point construction similarto that used by Kahn from the denotational semantics of individual processes in the network. The direct semantics closely corresponds to the operational semantics of the network (i. e. it iscorrect) but very difficult to study for concrete networks. The compositional semantics enablescompositional analysis of concrete networks, assuming it is correct. We prove that the compositional semantics is a safe approximation of the direct semantics. Wealso provide a method that can be used in many cases to establish that the two semantics fully coincide, i. e. safety is not achieved through inactivity or meaningless answers. The results are extended to cover recursively-defined infinite networks as well as nested finitenetworks. A robust prototype implementation of our model is available.

An X-ray micro-fluorescence study to investigate the distribution of Al, Si, P and Ca ions in the surrounding soft tissue after implantation of a calcium phosphate-mullite ceramic composite in a rabbit animal model

Synthetic calcium phosphates, despite their bioactivity, are brittle. Calcium phosphate-mullite composites have been suggested as potential dental and bone replacement materials which exhibit increased toughness. Aluminium, present in mullite, has however been linked to bone demineralisation and neurotoxicity: it is therefore important to characterise the materials fully in order to understand their in vivo behaviour. The present work reports the compositional mapping of the interfacial region of a calcium phosphate-20 wt% mullite biocomposite/soft tissue interface, obtained from the samples implanted into the long bones of healthy rabbits according to standard protocols (ISO-10993) for up to 12 weeks. X-ray micro-fluorescence was used to map simultaneously the distribution of Al, P, Si and Ca across the ceramic-soft tissue interface. A well defined and sharp interface region was present between the ceramic and the surrounding soft tissue for each time period examined. The concentration of Al in the surrounding tissue was found to fall by two orders of magnitude, to the background level, within similar to 35 mu m of the implanted ceramic.

Feature LDA:a supervised topic model for automatic detection of web API documentations from the web

Web APIs have gained increasing popularity in recent Web service technology development owing to its simplicity of technology stack and the proliferation of mashups. However, efficiently discovering Web APIs and the relevant documentations on the Web is still a challenging task even with the best resources available on the Web. In this paper we cast the problem of detecting the Web API documentations as a text classification problem of classifying a given Web page as Web API associated or not. We propose a supervised generative topic model called feature latent Dirichlet allocation (feaLDA) which offers a generic probabilistic framework for automatic detection of Web APIs. feaLDA not only captures the correspondence between data and the associated class labels, but also provides a mechanism for incorporating side information such as labelled features automatically learned from data that can effectively help improving classification performance. Extensive experiments on our Web APIs documentation dataset shows that the feaLDA model outperforms three strong supervised baselines including naive Bayes, support vector machines, and the maximum entropy model, by over 3% in classification accuracy. In addition, feaLDA also gives superior performance when compared against other existing supervised topic models.

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.

Joint sentiment/topic model for sentiment analysis

Sentiment analysis or opinion mining aims to use automated tools to detect subjective information such as opinions, attitudes, and feelings expressed in text. This paper proposes a novel probabilistic modeling framework based on Latent Dirichlet Allocation (LDA), called joint sentiment/topic model (JST), which detects sentiment and topic simultaneously from text. Unlike other machine learning approaches to sentiment classification which often require labeled corpora for classifier training, the proposed JST model is fully unsupervised. The model has been evaluated on the movie review dataset to classify the review sentiment polarity and minimum prior information have also been explored to further improve the sentiment classification accuracy. Preliminary experiments have shown promising results achieved by JST.

Semiparametric smooth-coefficient stochastic frontier model

This paper proposes a semiparametric smooth-coefficient (SPSC) stochastic production frontier model where regression coefficients are unknown smooth functions of environmental factors (ZZ). Technical inefficiency is specified in the form of a parametric scaling function which also depends on the ZZ variables. Thus, in our SPSC model the ZZ variables affect productivity directly via the technology parameters as well as through inefficiency. A residual-based bootstrap test of the relevance of the environmental factors in the SPSC model is suggested. An empirical application is also used to illustrate the technique.

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.

Impact of high-performance work systems on individual- and branch-level performance:test of a multilevel model of intermediate linkages

We proposed and tested a multilevel model, underpinned by empowerment theory, that examines the processes linking high-performance work systems (HPWS) and performance outcomes at the individual and organizational levels of analyses. Data were obtained from 37 branches of 2 banking institutions in Ghana. Results of hierarchical regression analysis revealed that branch-level HPWS relates to empowerment climate. Additionally, results of hierarchical linear modeling that examined the hypothesized cross-level relationships revealed 3 salient findings. First, experienced HPWS and empowerment climate partially mediate the influence of branch-level HPWS on psychological empowerment. Second, psychological empowerment partially mediates the influence of empowerment climate and experienced HPWS on service performance. Third, service orientation moderates the psychological empowerment-service performance relationship such that the relationship is stronger for those high rather than low in service orientation. Last, ordinary least squares regression results revealed that branch-level HPWS influences branch-level market performance through cross-level and individual-level influences on service performance that emerges at the branch level as aggregated service performance. © 2011 American Psychological Association.

A fuzzy decision tree-based duration model for Standard Yorùbá text-to-speech synthesis

In this paper, we present syllable-based duration modelling in the context of a prosody model for Standard Yorùbá (SY) text-to-speech (TTS) synthesis applications. Our prosody model is conceptualised around a modular holistic framework. This framework is implemented using the Relational Tree (R-Tree) techniques. An important feature of our R-Tree framework is its flexibility in that it facilitates the independent implementation of the different dimensions of prosody, i.e. duration, intonation, and intensity, using different techniques and their subsequent integration. We applied the Fuzzy Decision Tree (FDT) technique to model the duration dimension. In order to evaluate the effectiveness of FDT in duration modelling, we have also developed a Classification And Regression Tree (CART) based duration model using the same speech data. Each of these models was integrated into our R-Tree based prosody model. We performed both quantitative (i.e. Root Mean Square Error (RMSE) and Correlation (Corr)) and qualitative (i.e. intelligibility and naturalness) evaluations on the two duration models. The results show that CART models the training data more accurately than FDT. The FDT model, however, shows a better ability to extrapolate from the training data since it achieved a better accuracy for the test data set. Our qualitative evaluation results show that our FDT model produces synthesised speech that is perceived to be more natural than our CART model. In addition, we also observed that the expressiveness of FDT is much better than that of CART. That is because the representation in FDT is not restricted to a set of piece-wise or discrete constant approximation. We, therefore, conclude that the FDT approach is a practical approach for duration modelling in SY TTS applications. © 2006 Elsevier Ltd. All rights reserved.