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PubMed2000

Nonparametric analysis of recurrent events and death.

This article is concerned with the analysis of recurrent events in the presence of a terminal event such as death. We consider the mean frequency function, defined as the marginal mean of the cumulative number of recurrent events over time. A simple nonparametric estimator for this quantity is presented. It is shown that the estimator, properly normalized, converges weakly to a zero-mean Gaussian process with an easily estimable covariance function. Nonparametric statistics for comparing two mean frequency functions and for combining data on recurrent events and death are also developed. The asymptotic null distributions of these statistics, together with consistent variance estimators, are derived. The small-sample properties of the proposed estimators and test statistics are examined through simulation studies. An application to a cancer clinical trial is provided.

Antineoplastic Agents, Alkylating↗
PubMed2001

On the use of the variogram in checking for independence in spatial data.

The variogram is a standard tool in the analysis of spatial data, and its shape provides useful information on the form of spatial correlation that may be present. However, it is also useful to be able to assess the evidence for the presence of any spatial correlation. A method of doing this, based on an assessment of whether the true function underlying the variogram is constant, is proposed. Nonparametric smoothing of the squared differences of the observed variables, on a suitably transformed scale, is used to estimate variogram shape. A statistic based on a ratio of quadratic forms is proposed and the test is constructed by investigating the distributional properties of this statistic under the assumption of an independent Gaussian process. The power of the test is investigated. Reference bands are proposed as a graphical follow-up. An example is discussed.

Biometry↗
PubMed2000

Intersample fluctuations in phosphocreatine concentration determined by 31P-magnetic resonance spectroscopy and parameter estimation of metabolic responses to exercise in humans.

The ATP turnover rate during constant-load exercise is often estimated from the initial rate of change of phosphocreatine concentration ([PCr]) using 31P-magnetic resonance spectroscopy (MRS). However, the phase and amplitude characteristics of the sample-to-sample fluctuations can markedly influence this estimation (as well as that for the time constant (tau) of the [PCr] change) and confound its physiological interpretation especially for small amplitude responses. This influence was investigated in six healthy males who performed repeated constant-load quadriceps exercise of a moderate intensity in a whole-body MRS system. A transmit- receive surface coil was placed under the right quadriceps, allowing determination of intramuscular [PCr]; pulmonary oxygen uptake (VO2) was simultaneously determined, breath-by-breath, using a mass spectrometer and a turbine volume measuring module. The probability density functions (PDF) of [PCr] and VO2 fluctuations were determined for each test during the steady states of rest and exercise and the PDF was then fitted to a Gaussian function. The standard deviation of the [PCr] and VO2 fluctuations at rest and during exercise (sr and sw, respectively) and the peak centres of the distributions (xc(r) and xc(w)) were determined, as were the skewness (gamma1) and kurtosis (gamma2) coefficients. There was no difference between sr and sw for [PCr] relative to the resting control baseline (s(r) = 1.554 %delta (s.d. = 0.44), s(w) = 1.514 %delta (s.d. = 0.35)) or the PDF peak centres (xc(r) = -0.013 %delta (s.d. = 0.09), xc(w) -0.197 %delta (s.d. = 0.18)). The standard deviation and peak centre of the 'noise' in VO2 also did not vary between rest and exercise (sr = 0.0427 l min(-1) (s.d. = 0.0104), s(w) = 0.0640 l min(-1) (s.d. = 0.0292); xc(r) = -0.0051 l min(-1) (s.d. = 0.0069), xc(w) 0.0022 l min(-1) (s.d. = 0.0034)). Our results demonstrate that the intersample 'noise' associated with [PCr] determination by 31P-MRS may be characterised as a stochastic Gaussian process that is uncorrelated with work rate, as previously described for VO2. This 'noise' can significantly affect the estimation of tau[PCr] and especially the initial rate of change of [PCr], i.e. the fluctuations can lead to variations in estimation of the initial rate of change of [PCr] of more than twofold, if the inherent 'noise' is not accounted for. This 'error' may be significantly reduced in such cases if the initial rate of change is estimated from the time constant and amplitude of the response.

Adenosine Triphosphate↗
PubMed2006

Coregionalized single- and multiresolution spatially varying growth curve modeling with application to weed growth.

Modeling of longitudinal data from agricultural experiments using growth curves helps understand conditions conducive or unconducive to crop growth. Recent advances in Geographical Information Systems (GIS) now allow geocoding of agricultural data that help understand spatial patterns. A particularly common problem is capturing spatial variation in growth patterns over the entire experimental domain. Statistical modeling in these settings can be challenging because agricultural designs are often spatially replicated, with arrays of subplots, and interest lies in capturing spatial variation at possibly different resolutions. In this article, we develop a framework for modeling spatially varying growth curves as Gaussian processes that capture associations at single and multiple resolutions. We provide Bayesian hierarchical models for this setting, where flexible parameterization enables spatial estimation and prediction of growth curves. We illustrate using data from weed growth experiments conducted in Waseca, Minnesota, that recorded growth of the weed Setaria spp. in a spatially replicated design.

Bayes Theorem↗
PubMed1967

The (+) reference: accuracy of estimated mean components in average response studies.

The (+/-) reference is defined as the result of alternate addition and subtraction and division by N (the number of sample functions). Under suitable conditions both the (+/-) reference and the variable component (noise) of the usual average tend to be derived from the same Gaussian process, and the former can be used as a measure of the latter. This property is most easily applied when the noise is derived from a stationary process. Application of the (+/-) reference and the average of the square of the voltage in studies of evoked response is discussed.

Electrophysiology↗
PubMed2001

On a class of support vector kernels based on frames in function Hilbert spaces.

There has been an increasing interest in kernel-based techniques, such as support vector techniques, regularization networks, and gaussian processes. There are inner relationships among those techniques, with the kernel function playing a central role. This article discusses a new class of kernel functions derived from the so-called frames in a function Hilbert space.

Algorithms↗
PubMed2002

Bayesian model assessment and comparison using cross-validation predictive densities.

In this work, we discuss practical methods for the assessment, comparison, and selection of complex hierarchical Bayesian models. A natural way to assess the goodness of the model is to estimate its future predictive capability by estimating expected utilities. Instead of just making a point estimate, it is important to obtain the distribution of the expected utility estimate because it describes the uncertainty in the estimate. The distributions of the expected utility estimates can also be used to compare models, for example, by computing the probability of one model having a better expected utility than some other model. We propose an approach using cross-validation predictive densities to obtain expected utility estimates and Bayesian bootstrap to obtain samples from their distributions. We also discuss the probabilistic assumptions made and properties of two practical cross-validation methods, importance sampling and k-fold cross-validation. As illustrative examples, we use multilayer perceptron neural networks and gaussian processes with Markov chain Monte Carlo sampling in one toy problem and two challenging real-world problems.

Arm↗
PubMed2005

A statistical approach for array CGH data analysis.

BACKGROUND: Microarray-CGH experiments are used to detect and map chromosomal imbalances, by hybridizing targets of genomic DNA from a test and a reference sample to sequences immobilized on a slide. These probes are genomic DNA sequences (BACs) that are mapped on the genome. The signal has a spatial coherence that can be handled by specific statistical tools. Segmentation methods seem to be a natural framework for this purpose. A CGH profile can be viewed as a succession of segments that represent homogeneous regions in the genome whose BACs share the same relative copy number on average. We model a CGH profile by a random Gaussian process whose distribution parameters are affected by abrupt changes at unknown coordinates. Two major problems arise: to determine which parameters are affected by the abrupt changes (the mean and the variance, or the mean only), and the selection of the number of segments in the profile. RESULTS: We demonstrate that existing methods for estimating the number of segments are not well adapted in the case of array CGH data, and we propose an adaptive criterion that detects previously mapped chromosomal aberrations. The performances of this method are discussed based on simulations and publicly available data sets. Then we discuss the choice of modeling for array CGH data and show that the model with a homogeneous variance is adapted to this context. CONCLUSIONS: Array CGH data analysis is an emerging field that needs appropriate statistical tools. Process segmentation and model selection provide a theoretical framework that allows precise biological interpretations. Adaptive methods for model selection give promising results concerning the estimation of the number of altered regions on the genome.

Algorithms↗
PubMed2000

Nonlinearity in human resting, eyes-closed EEG: an in-depth case study.

The question of nonlinearity in the human electroencephalogram (EEG) is important, since linear methods of EEG analysis are more well-developed and computationally faster than nonlinear methods. Furthermore, the presence or absence of nonlinearity has important theoretical implications for understanding the nature of the brain's oscillatory activity. Using a linear summary measure as a control, we report a failure to reject the null hypothesis of a (largely) stationary linear-Gaussian process for normal, resting, eyes-closed EEG from a single participant. We found significant evidence of nonlinearity at two occipital sites (O1 and O2) where the 8-12.5 Hz alpha rhythm was prominent. However, this element of nonlinear structure appeared trivial, as (1) we found no evidence of time irreversibility at these loci, and (2) best-fitting linear models accounted on-average for over 94% of the variance in the data with nonlinear modeling doing no better. Half of the remaining variance could be accounted for by nonstationarity. While our findings technically apply only to the one individual tested, his EEG was typical of those seen under the conditions that we employed.

Alpha Rhythm↗
PubMed1988

Maximum entropy methods in dark field electron micrographs and elemental maps.

A maximum entropy algorithm is described which not only fits a model to the data consistent with the size of the noise and the maximum entropy principle but also distributes the residuals between the data and the model in a way consistent with the noise in the data having been generated by a random gaussian process. The results of applying the algorithm to profiles of electron micrographs, electron micrographs and model data is presented. The algorithm is found to achieve various degrees of signal to noise ratio enhancement. Preliminary results show that the spatial resolution is not suppressed. A biassing artifact is described.

Algorithms↗
PubMedDate not supplied

Waking and sleeping states in the rat from an EEG data analysis point of view.

This article presents the characteristics of ECoGs of arousal, slow wave sleep and paradoxical sleep in the rat, in terms of analysis of data. In a first part, we have applied four different methods of analysis to the three tracings: the instantaneous amplitude histograms computation, the integrative method of Drohocki, the spectral analysis and the normalized slope descriptor method of Hjorth. Each method provides, after data reduction, characteristic parameters of the tracings. A graph which displays peak spectral frequency versus mean integrated value is enough to discriminate between the 3 quantified tracings. Multivariate discriminant analysis reveals that 3 coefficients altogether allow a good discrimination. In the second part we ask the question: which kind of signal is the paradoxical sleep tracing? After the impossibility to choose between a narrow-band Gaussian process or a sinusoidal wave burried in noise, we propose a third kind of signal found after modulation analysis. This signal is modulated both in amplitude and frequency around a carrier frequency beeing the dominant theta rhythm.

Animals↗
PubMed1995

Tests for genetic linkage and homogeneity.

This report concerns likelihood ratio tests in a heterogeneous model for linkage, where the recombination fraction has a binomial mixture distribution with an unknown proportion of unlinked families. We consider families of unequal sizes with known or unknown phase data. In both cases, the limit distributions of the linkage test statistics are a mixture of a mass at ) and of a X2/1 distribution in equal proportions and homogeneity test statistics tend to the supremum of Gaussian processes. The critical values of the homogeneity tests are simulated, and the power functions of the linkage and homogeneity tests are compared in a simulation study.

Biometry↗
PubMed1993

Gaussian models for genetic linkage analysis using complete high-resolution maps of identity by descent.

Gaussian-process models are developed to detect genetic linkage using complete high-resolution maps of identity by descent between affected relative pairs. Approximations are given for the significance level and power of the likelihood-ratio test of no linkage and for likelihood-ratio confidence regions for trait loci. The sample sizes required to detect linkage by using different classes of affected relative pairs are compared, and the problem of combining data from different classes of relatives is discussed.

Genetic Linkage↗
PubMed2002

Large-deviation functions for nonlinear functionals of a Gaussian stationary Markov process.

We introduce a general method, based on a mapping onto quantum mechanics, for investigating the large-T limit of the distribution P(r,T) of the nonlinear functional r[V]=(1/T)integral(T)(0)dT' V[X(T')], where V(X) is an arbitrary function of the stationary Gaussian Markov process X(T). For T-->infinity at fixed r we obtain P(r,T) approximately exp[-theta(r)T], where theta(r) is a large-deviation function. We present explicit results for a number of special cases including V(X)=XH(X) [where H(X) is the Heaviside function], which is related to the cooling and the heating degree days relevant to weather derivatives.

Journal Article↗
PubMed2004

Non-Markovian rotating unstable processes driven by Gaussian colored noise.

In this paper the statistical properties of the mean passage time distribution are used to characterize the decay process of non-Markovian rotating unstable processes driven by Gaussian colored noise and subjected to the influence of a constant external force. The time characterization will be linear and studied in two limiting cases: large and intermediate times. General systems of two variables are studied. In both schemes we show that, for small correlation time of the noise, the non-Markovian effects are taken into account by an effective noise intensity. To compare qualitatively the non-Markovian time scale with respect to the Markovian case, we apply those results to determine the detection bandwidth of a large external signal in a laser system.

Journal Article↗
PubMed1995

Gaussian models for degradation processes-Part I: Methods for the analysis of biomarker data.

We present two stochastic models that describe the relationship between biomarker process values at random time points, event times, and a vector of covariates. In both models the biomarker processes are degradation processes that represent the decay of systems over time. In the first model the biomarker process is a Wiener process whose drift is a function of the covariate vector. In the second model the biomarker process is taken to be the difference between a stationary Gaussian process and a time drift whose drift parameter is a function of the covariates. For both models we present statistical methods for estimation of the regression coefficients. The first model is useful for predicting the residual time from study entry to the time a critical boundary is reached while the second model is useful for predicting the latency time from the infection until the time the presence of the infection is detected. We present our methods principally in the context of conducting inference in a population of HIV infected individuals.

Biomarkers↗