Multivariate Nonparametric Regression Research Articles

Depression and Neurocognitive Performance in Portuguese Patients Infected with HIV

This study focused on neurocognitive disorders associated with HIV infection, characterizes the most affected neuropsychological domains and their potential as factors related with depression. 130 HIV-positive individuals with CD4 >200 cells/mm(3), undetectable viral load, treated with HAART and with all kinds of risk behaviors were included. A structured interview composed by seven sections was used and relevant clinical and laboratory data was assessed. For the neuropsychological and depression assessment the HIV Neurobehavioral Research Center (HNRC) Battery and Hamilton Rating Scale for Depression (HAM-D) were chosen. Univariate nonparametric statistics and multivariate regression model were performed. Among 34% of depressed subjects, 20.7% had attempted suicide, 45.4% were under psychiatric medication and only 19% had no diagnostic criteria for HIV Associated Neurocognitive Disorders (HAND). The effect of depression on cognitive function revealed a significant deterioration in five of the eight measures. In the multivariate analysis, five variables significantly determined depression: sex, social class, antidepressant medication, history of suicide attempt and Dexterity and motor speed (neuropsychological test battery). Despite good patient adherence to treatment (HAART and antidepressive) and good immune status, the prevalence of HAND and depression are high and suggest a significant relation between severity of depression and cognitive functioning.

Surface Estimation, Variable Selection, and the Nonparametric Oracle Property.

Variable selection for multivariate nonparametric regression is an important, yet challenging, problem due, in part, to the infinite dimensionality of the function space. An ideal selection procedure should be automatic, stable, easy to use, and have desirable asymptotic properties. In particular, we define a selection procedure to be nonparametric oracle (np-oracle) if it consistently selects the correct subset of predictors and at the same time estimates the smooth surface at the optimal nonparametric rate, as the sample size goes to infinity. In this paper, we propose a model selection procedure for nonparametric models, and explore the conditions under which the new method enjoys the aforementioned properties. Developed in the framework of smoothing spline ANOVA, our estimator is obtained via solving a regularization problem with a novel adaptive penalty on the sum of functional component norms. Theoretical properties of the new estimator are established. Additionally, numerous simulated and real examples further demonstrate that the new approach substantially outperforms other existing methods in the finite sample setting.

Multivariate Adaptive Regression Splines as a Tool to Improve the Accuracy of Parts Produced by FSPIF

Feature Assisted Single Point Incremental Forming (FSPIF) is a technique to increase the accuracy of the SPIF process. FSPIF generates an optimized toolpath based on the features detected in the workpiece geometry and using knowledge of the behavior of these features during incremental forming. Using this optimized toolpath, parts can be formed with higher accuracy. The prediction of the dimensional deviations occurring in different features during forming as a function of their type (e.g. planar, ruled, freeform or ribs ) and various process parameters, such as sheet thickness, wall angle, tool diameter, rolling direction, etc., is an important step in the FSPIF method. Due to the great number of parameters and combinations that are possible, a mathematical tool should be used in order to automate the prediction process. One such tool is MARS or Multivariate Adaptive Regression Splines, a fast, non-parametric multivariate regression technique with automatic variable selection, which generates continuous surfaces as a response function. In this paper, the authors describe and validate the use of MARS as a tool to predict deviations in uncompensated tests by training the MARS model using only a limited number of experiments. Using this validated model, compensation strategies are developed and implemented, which have shown significant improvements in accuracy in new test cases.

Low ratio of 4-(methylnitrosamino)-1-(3-pyridyl)-1-butanol-glucuronides (NNAL-Gluc)/free NNAL increases urothelial carcinoma risk

Cigarette smoking is the most important risk factor for bladder cancer. The compound 4-(methylnitrosamino)-1-(3-pyridyl)-1-butanone (NNK) and its metabolite 4-(methylnitrosamino)-1-(3-pyridyl)-1-butanol (NNAL) are viewed as biomarkers for cigarette smoking exposure. Therefore, we wanted to explore the effects of these urinary metabolites on urothelial carcinoma (UC) risk. We recruited 127 pairs of UC cases and matched healthy participants for a hospital-based case–control study. Participants completed questionnaires of medical and social information, including smoking history, and provided 50 mL urine samples. Urine samples were analyzed for free NNAL and NNAL-Gluc using the liquid chromatography-tandem mass spectrometry method. Nonparametric analysis and multivariate logistic regression were applied to compare the differences in NNK-related metabolites between UC cases and controls, and to estimate the UC risk associated with certain risk factors. Overall, controls with higher cumulative cigarette smoking exposure had higher total NNAL, free NNAL and NNAL-Gluc. In addition, a decreased NNAL-Gluc/free NNAL ratio corresponded to a significantly increased UC risk. The association between the NNAL-Gluc/free NNAL ratio and UC risk was significant in a dose–response manner. Furthermore, cumulative cigarette smoking exposure was found to interact significantly with low NNAL-Gluc/free NNAL ratio to affect UC risk in this study. This is the first study to conclude that the metabolic products of total NNAL, free NNAL and NNAL-Gluc might be measured as biomarkers of cigarette smoking exposure. Furthermore, the NNAL-Gluc/free NNAL ratio was a better biomarker to evaluate UC risk than total NNAL.

Cerebral extracellular chemistry and outcome following traumatic brain injury: a microdialysis study of 223 patients.

Secondary insults can adversely influence outcome following severe traumatic brain injury. Monitoring of cerebral extracellular chemistry with microdialysis has the potential for early detection of metabolic derangements associated with such events. The objective of this study was to determine the relationship between the fundamental biochemical markers and neurological outcome in a large cohort of patients with traumatic brain injury. Prospectively collected observational neuromonitoring data from 223 patients were analysed. Monitoring modalities included digitally recorded intracranial pressure, cerebral perfusion pressure, cerebrovascular pressure reactivity index and microdialysis markers glucose, lactate, pyruvate, glutamate, glycerol and the lactate/pyruvate ratio. Outcome was assessed using the Glasgow Outcome Scale at 6 months post-injury. Patient-averaged values of parameters were used in statistical analysis, which included univariate non-parametric methods and multivariate logistic regression. Monitoring with microdialysis commenced on median (interquartile range) Day 1 (1-2) from injury and median (interquartile range) duration of monitoring was 4 (2-7) days. Averaged over the total monitoring period levels of glutamate (P = 0.048), lactate/pyruvate ratio (P = 0.044), intracranial pressure (P = 0.006) and cerebrovascular pressure reactivity index (P = 0.01) were significantly higher in patients who died. During the initial 72 h of monitoring, median glycerol levels were also higher in the mortality group (P = 0.014) and median lactate/pyruvate ratio (P = 0.026) and lactate (P = 0.033) levels were significantly lower in patients with favourable outcome. In a multivariate logistic regression model (P < 0.0001), which employed data averaged over the whole monitoring period, significant independent positive predictors of mortality were glucose (P = 0.024), lactate/pyruvate ratio (P = 0.016), intracranial pressure (P = 0.029), cerebrovascular pressure reactivity index (P = 0.036) and age (P = 0.003), while pyruvate was a significant independent negative predictor of mortality (P = 0.004). The results of this study suggest that extracellular metabolic markers are independently associated with outcome following traumatic brain injury. Whether treatment-related improvement in biochemistry translates into better outcome remains to be established.

Daily peak electricity load forecasting in South Africa using a multivariate non-parametric regression approach

Accurate prediction of daily peak load demand is very important for decision makers in the energy sector. This helps in the determination of consistent and reliable supply schedules during peak periods. Accurate short term load forecasts enable effective load shifting between transmission substations, scheduling of startup times of peak stations, load flow analysis and power system security studies. A multivariate adaptive regression splines (MARS) modelling approach towards daily peak electricity load forecasting in South Africa is presented in this paper for the period 2000 to 2009. MARS is a non-parametric multivariate regression method which is used in high-dimensional problems with complex model structures, such as nonlinearities, interactions and missing data, in a straight forward manner and produces results which may easily be explained to management. The models developed in this paper consist of components that represent calendar and meteorological data. The performances of the models are evaluated by comparing them to a piecewise linear regression model. The results from the study show that the MARS models achieve better forecast accuracy.

Impact of reclamation of surface-mined boreal forest soils on microbial community composition and function

Large-scale mining for oil extraction in the Canadian boreal forest has prompted the need to assess the effectiveness of reclamation strategies on key biotic components of reconstructed soil profiles. Capitalizing on a network of long-term monitoring plots, we evaluated how enzyme activities and microbial community composition responded to reclamation of surface-mined soil in the Athabasca oil sands region of northern Alberta. Bulk soil (organic and mineral horizons) was sampled from undisturbed boreal forest sites and from sites representing seven reclamation prescriptions applied 30 years ago. Microbial community composition of both bacteria and fungi was determined with denaturing gradient gel electrophoresis (DGGE) and phospholipid fatty acid (PLFA) profiling. Microbial function was assessed by measuring the activities of β-glucosidase, β-xylosidase, phenoloxidase, peroxidase, phosphatase, chitinase, and urease. We found a general decrease in enzyme activities in reclaimed sites. Overburden (low-C) based prescriptions affected enzyme activities the most, causing a decrease in phenoloxidase activity (mineral horizon) in comparison to productive natural sites and an increase in β-glucosidase activity (organic horizon) in relation to nutrient-poor natural sites. Tailings-sand-based prescription enhanced microbial community dissimilarities with natural productive sites, although these were only evident with PLFA profiles. Effects of time-since-reclamation were not apparent, probably because they were masked by the larger influence of reclamation treatment. According to a principal components analysis of PLFA profiles, biomarker 18:1ω7c (Gram-negative) dominated in reclaimed sites and marker 18:1ω9c (fungi) was more abundant in natural sites. An increased fungal component in natural sites was presumably dominated by ectomycorrhizae, as suggested by sequencing of ITS regions. Non-parametric multivariate multiple regression indicated that the fungal-to-bacterial-biomass ratio largely explained the overall variation in PLFA profiles of reclaimed sites, whereas in natural sites soil nitrogen explained most of the community structure variability. Soil pH and woody debris accumulation emerged as significant explanatory variables when all sites were analyzed together. In contrast to PLFA profiles, DGGE profiles revealed significant regional-scale spatial structuring within reclaimed sites (organic horizon) and in natural sites (mineral horizon). Our results indicate that the response of abundant microbial populations to reclamation is likely governed by soil abiotic properties and, indirectly, by the effects of reclamation on plant growth.

Partial Generalized Additive Models: An Information-Theoretic Approach for Dealing With Concurvity and Selecting Variables

Scientists are often interested in which covariates are important, and how these covariates affect the response variable, rather than just making predictions. This requires inputs from both statistical modeling and background knowledge. Generalized additive models (GAMs) are a class of interpretable, multivariate nonparametric regression models which are very useful data exploration tools for these purposes, but concurvity among covariates (the nonlinear analogue of collinearity for linear regression) can lead GAMs to produce unstable or even wrong estimates of the covariates’ functional effects. We develop a new procedure called partial generalized additive models (pGAM), based on mutual information (MI), a measure of nonlinear dependence between variables. Our procedure is similar in spirit to the Gram–Schmidt method for linear least squares. By building a GAM on a selected set of transformed variables, pGAM produces more stable models, selects variables parsimoniously, and provides insight into the nature of concurvity between the covariates by calculating functional dependencies among them. With simulation experiments and real-data examples, we show that pGAM produces much better estimates of the covariates’ functional effects, and also incorporates a reasonable and meaningful variable selection method. R code for fitting pGAMs is available online (see Supplemental Materials Section).

Self-injurious behavior among Greek male prisoners: Prevalence and risk factors

Background Self-harm among prisoners is a common phenomenon. This study aims to estimate the prevalence of self-injurious behavior (SIB) among Greek male prisoners, record their motives and determine independent risk factors. Methods A self-administered, anonymous questionnaire was administered to 173 male prisoners in the Chalkida prison, Greece. The questionnaire included items on self-harm/SIB, demographic parameters, childhood history, family history, physical and mental disease, lifestyle and smoking habits, alcohol dependence (CAGE questionnaire), illicit substance use, aggression (Buss–Perry Aggression Questionnaire [BPAQ] and Lifetime History of Aggression [LTHA]), impulsivity (Barrat Impulsivity Scale-11) and suicidal ideation (Spectrum of Suicidal Behavior Scale). Univariate nonparametric statistics and multivariate ordinal logistic regression were performed. Results Of all the participants, 49.4% (95% CI: 41.5–57.3%) disclosed self-harm (direct or indirect). The prevalence of SIB was equal to 34.8% (95% CI: 27.5–42.6%). Most frequently, SIB coexisted with indirect self-harm (80.7%). The most common underlying motives were to obtain emotional release (31.6%) and to release anger (21.1%). At the univariate analysis, SIB was positively associated with a host of closely related factors: low education, physical/sexual abuse in childhood, parental neglect, parental divorce, alcoholism in family, psychiatric condition in family, recidivism, age, sentence already served, impulsivity, aggression, alcohol dependence, self-reported diagnosed psychiatric condition and illicit substance use. Childhood variables were particularly associated with the presence of diagnosed psychiatric condition. At the multivariate analysis, however, only three parameters were proven independent risk factors: self-reported diagnosed psychiatric condition, illicit substance use and aggression (BPAQ scale). Conclusion The prevalence of SIB is particularly high. Psychiatric condition, illicit substance use and aggression seem to be the most meaningful risk factors; childhood events seem only to act indirectly.

Nonparametric multivariate regression methods to determine dexamethasone concentration using ELISA measurements

Dexamethasone is a synthetic steroid used in veterinary medicine, but its presence in food must remain below certain limits to avoid risk to human health. HPLC methods determine dexamethasone concentration with a high level of accuracy but these methods are expensive and the field instrumentation is difficult to obtain. On the other hand, the classical ELISA method can be used to estimate dexamethasone concentration but it suffers from natural steroid interferences which result in false negatives and positives. In this article we propose the concept of multi-detection in order to estimate dexamethasone concentration from a vector of absorbances. The function which maps the absorbance vector into dexamethasone concentration is calculated by means of non-linear regression algorithms. Results demonstrate that the proposed methods perform much better than the classical ELISA method when interfering steroids are present. This implies that results of immunoassays with cross-reactivity could be treated with nonparametric multivariate regression methods to improve accuracy.

Relationship Between Soil Properties and Patterns of Bacterial β-diversity Across Reclaimed and Natural Boreal Forest Soils

Productivity gradients in the boreal forest are largely determined by regional-scale changes in soil conditions, and bacterial communities are likely to respond to these changes. Few studies, however, have examined how variation in specific edaphic properties influences the composition of soil bacterial communities along environmental gradients. We quantified bacterial compositional diversity patterns in ten boreal forest sites of contrasting fertility. Bulk soil (organic and mineral horizons) was sampled from sites representing two extremes of a natural moisture-nutrient gradient and two distinct disturbance types, one barren and the other vegetation-rich. We constructed 16S rRNA gene clone libraries to characterize the bacterial communities under phylogenetic- and species-based frameworks. Using a nucleotide analog to label DNA-synthesizing bacteria, we also assessed the composition of active taxa in disturbed sites. Most sites were dominated by sequences related to the alpha-Proteobacteria, followed by acidobacterial and betaproteobacterial sequences. Non-parametric multivariate regression indicated that pH, which was lowest in the natural sites, explained 34% and 16% of the variability in community structure as determined by phylogenetic-based (UniFrac distances) and species-based (Jaccard similarities) metrics, respectively. Soil pH was also a significant predictor of richness (Chao1) and diversity (Shannon) measures. Within the natural edaphic gradient, soil moisture accounted for 32% of the variance in phylogenetic (but not species) community structure. In the boreal system we studied, bacterial beta-diversity patterns appear to be largely related to "master" variables (e.g., pH, moisture) rather than to observable attributes (e.g., plant cover) leading to regional-scale fertility gradients.

Regression analysis of the structure function for reliability evaluation of continuous-state system

Technical systems are designed to perform an intended task with an admissible range of efficiency. According to this idea, it is permissible that the system runs among different levels of performance, in addition to complete failure and the perfect functioning one. As a consequence, reliability theory has evolved from binary-state systems to the most general case of continuous-state system, in which the state of the system changes over time through some interval on the real number line. In this context, obtaining an expression for the structure function becomes difficult, compared to the discrete case, with difficulty increasing as the number of components of the system increases. In this work, we propose a method to build a structure function for a continuum system by using multivariate nonparametric regression techniques, in which certain analytical restrictions on the variable of interest must be taken into account. Once the structure function is obtained, some reliability indices of the system are estimated. We illustrate our method via several numerical examples.

Genetic background comparison using distance-based regression, with applications in population stratification evaluation and adjustment.

Population stratification (PS) can lead to an inflated rate of false-positive findings in genome-wide association studies (GWAS). The commonly used approach of adjustment for a fixed number of principal components (PCs) could have a deleterious impact on power when selected PCs are equally distributed in cases and controls, or the adjustment of certain covariates, such as self-identified ethnicity or recruitment center, already included in the association analyses, correctly maps to major axes of genetic heterogeneity. We propose a computationally efficient procedure, PC-Finder, to identify a minimal set of PCs while permitting an effective correction for PS. A general pseudo F statistic, derived from a non-parametric multivariate regression model, can be used to assess whether PS exists or has been adequately corrected by a set of selected PCs. Empirical data from two GWAS conducted as part of the Cancer Genetic Markers of Susceptibility (CGEMS) project demonstrate the application of the procedure. Furthermore, simulation studies show the power advantage of the proposed procedure in GWAS over currently used PS correction strategies, particularly when the PCs with substantial genetic variation are distributed similarly in cases and controls and therefore do not induce PS.

Testing for multivariate volatility functions using minimum volume sets and inverse regression

We propose two new types of nonparametric tests for investigating multivariate regression functions. The tests are based on cumulative sums coupled with either minimum volume sets or inverse regression ideas; involving no multivariate nonparametric regression estimation. The methods proposed facilitate the investigation for different features such as if a multivariate regression function is (i) constant, (ii) of a bathtub shape, and (iii) in a given parametric form. The inference based on those tests may be further enhanced through associated diagnostic plots. Although the potential use of those ideas is much wider, we focus on the inference for multivariate volatility functions in this paper, i.e. we test for (i) heteroscedasticity, (ii) the so-called ‘smiling effect’, and (iii) some parametric volatility models. The asymptotic behavior of the proposed tests is investigated, and practical feasibility is shown via simulation studies. We further illustrate our methods with real financial data.

Nonparametric Multivariate Conditional Distribution and Quantile Regression

In nonparametric multivariate regression analysis, one usually seeks methods to reduce the dimensionality of the regression function to bypass the difficulty caused by the curse of dimensionality. We study nonparametric estimation of multivariate conditional distribution and quantile regression via local univariate quadratic estimation of partial derivatives of bivariate copulas. Without restricting the form of underlying regression function or using dimensional reduction, we show that a d-dimensional multivariate conditional distribution and quantile regression could be estimated by d(d 1)/2 times of univariate smoothers. The asymptotic bias and variance as well as smoothing parameter selection method are derived. Simulations show that the method works quite well. The techniques are illustrated by application to exchange rate data.

Penalized Spline Estimation for Varying-Coefficient Models

Varying-coefficient models are useful extensions of classical linear models. They arise from multivariate nonparametric regression, nonlinear time series modeling and forecasting, longitudinal data analysis, and others. This article proposes the penalized spline estimation for the varying-coefficient models. Assuming a fixed but potentially large number of knots, the penalized spline estimators are shown to be strong consistency and asymptotic normality. A systematic optimization algorithm for the selection of multiple smoothing parameters is developed. One of the advantages of the penalized spline estimation is that it can accommodate varying degrees of smoothness among coefficient functions due to multiple smoothing parameters being used. Some simulation studies are presented to illustrate the proposed methods.

Variance function estimation in multivariate nonparametric regression with fixed design

Variance function estimation in multivariate nonparametric regression is considered and the minimax rate of convergence is established in the iid Gaussian case. Our work uses the approach that generalizes the one used in [A. Munk, Bissantz, T. Wagner, G. Freitag, On difference based variance estimation in nonparametric regression when the covariate is high dimensional, J. R. Stat. Soc. B 67 (Part 1) (2005) 19–41] for the constant variance case. As is the case when the number of dimensions d = 1 , and very much contrary to standard thinking, it is often not desirable to base the estimator of the variance function on the residuals from an optimal estimator of the mean. Instead it is desirable to use estimators of the mean with minimal bias. Another important conclusion is that the first order difference based estimator that achieves minimax rate of convergence in the one-dimensional case does not do the same in the high dimensional case. Instead, the optimal order of differences depends on the number of dimensions.

Smooth backfitting in generalized additive models

Generalized additive models have been popular among statisticians and data analysts in multivariate nonparametric regression with non-Gaussian responses including binary and count data. In this paper, a new likelihood approach for fitting generalized additive models is proposed. It aims to maximize a smoothed likelihood. The additive functions are estimated by solving a system of nonlinear integral equations. An iterative algorithm based on smooth backfitting is developed from the Newton-Kantorovich theorem. Asymptotic properties of the estimator and convergence of the algorithm are discussed. It is shown that our proposal based on local linear fit achieves the same bias and variance as the oracle estimator that uses knowledge of the other components. Numerical comparison with the recently proposed two-stage estimator [Ann. Statist. 32 (2004) 2412-2443] is also made.

Local influence analysis for penalized Gaussian likelihood estimation in partially linear single-index models

Single-index model is a potentially tool for multivariate nonparametric regression, generalizes both the generalized linear models(GLM) and the missing-link function problem in GLM. In this paper, we extend Cook’s local influence analysis to the penalized Gaussian likelihood estimator based on P-spline for the partially linear single-index model. Some influence measures, based on the minor perturbation of the model, are derived for the penalized least squares estimation. An illustrative example is also presented.

Generalized likelihood ratio tests for the structure of semiparametric additive models

AbstractSemiparametric additive models (SAMs) are very useful in multivariate nonparametric regression. In this paper, the authors study nonparametric testing problems for the nonparametric components of SAMs. Using the backfitting algorithm and the local polynomial smoothing technique, they extend to SAMs the generalized likelihood ratio tests of Fan &amp;Jiang (2005). The authors show that the proposed tests possess the Wilks‐type property and that they can detect alternatives nearing the null hypothesis with a rate arbitrarily close to root‐n while error distributions are unspecified. They report simulations which demonstrate the Wilks phenomenon and the powers of their tests. They illustrate the performance of their approach by simulation and using the Boston housing data set.