Self modeling with flexible, random time transformations

Methods for modeling sets of complex curves where the curves must be aligned in time (or in another continuous predictor) fall into the general class of functional data analysis and include self-modeling regression and time-warping procedures. Self-modeling regression (SEMOR), also known as a shape invariant model (SIM), assumes the curves have a common shape, modeled nonparametrically, and curve-specific differences in amplitude and timing, traditionally modeled by linear transformations. When curves contain multiple features that need to be aligned in time, SEMOR may be inadequate since a linear time transformation generally cannot align more than one feature. Time warping procedures focus on timing variability and on finding flexible time warps to align multiple data features. We draw on these methods to develop a SIM that models the time transformations as random, flexible, monotone functions. The model is motivated by speech movement data from the University of Wisconsin X-ray microbeam speech production project and is applied to these data to test the effect of different speaking conditions on the shape and relative timing of movement profiles.     

Predicting germination response to temperature. II. Three-dimensional regression, statistical gridding and iterative-probit optimization using measured and interpolated-subpopulation data

BACKGROUND AND AIMS: Most current thermal-germination models are parameterized with subpopulation-specific rate data, interpolated from cumulative-germination-response curves. The purpose of this study was to evaluate the relative accuracy of three-dimensional models for predicting cumulative germination response to temperature. Three-dimensional models are relatively more efficient to implement than two-dimensional models and can be parameterized directly with measured data. METHODS: Seeds of four rangeland grass species were germinated over the constant-temperature range of 3 to 38 degrees C and monitored for subpopulation variability in germination-rate response. Models for estimating subpopulation germination rate were generated as a function of temperature using three-dimensional regression, statistical gridding and iterative-probit optimization using both measured and interpolated-subpopulation data as model inputs. KEY RESULTS: Statistical gridding is more accurate than three-dimensional regression and iterative-probit optimization for modelling germination rate and germination time as a function of temperature and subpopulation. Optimization of the iterative-probit model lowers base-temperature estimates, relative to two-dimensional cardinal-temperature models, and results in an inability to resolve optimal-temperature coefficients as a function of subpopulation. Residual model error for the three-dimensional model was extremely high when parameterized with measured-subpopulation data. Use of measured data for model evaluation provided a more realistic estimate of predictive error than did evaluation of the larger set of interpolated-subpopulation data. CONCLUSIONS: Statistical-gridding techniques may provide a relatively efficient method for estimating germination response in situations where the primary objective is to estimate germination time. This methodology allows for direct use of germination data for model parameterization and automates the significant computational requirements of a two-dimensional piece-wise-linear model, previously shown to produce the most accurate estimates of germination time.     

A Symmetric Extended Logistic Model with Applications to Experimental Toxicity Data

The fit of the logit and probit models for quantal response data can be improved by embedding these classical models within a richer parametric family indexed by one or two shape parameters. In this paper, a symmetric extended logistic model indexed by a shape parameter λ is discussed with application to dose response curves. The usual maximum likelihood method is employed to estimate the parameters of the model. The need to include the shape parameter λ is illustrated by analyzing a set of real experimental data and comparing the fit of the extended logistic model to those obtained by the standard logit and probit models.     

Estimating Subject‐Specific Dependent Competing Risk Profile with Censored Event Time Observations

Summary In a longitudinal study, suppose that the primary endpoint is the time to a specific event. This response variable, however, may be censored by an independent censoring variable or by the occurrence of one of several dependent competing events. For each study subject, a set of baseline covariates is collected. The question is how to construct a reliable prediction rule for the future subject's profile of all competing risks of interest at a specific time point for risk‐benefit decision making. In this article, we propose a two‐stage procedure to make inferences about such subject‐specific profiles. For the first step, we use a parametric model to obtain a univariate risk index score system. We then estimate consistently the average competing risks for subjects who have the same parametric index score via a nonparametric function estimation procedure. We illustrate this new proposal with the data from a randomized clinical trial for evaluating the efficacy of a treatment for prostate cancer. The primary endpoint for this study was the time to prostate cancer death, but had two types of dependent competing events, one from cardiovascular death and the other from death of other causes.     

A new non-orthogonal decomposition method to determine effective torques for three-dimensional joint rotation

This paper describes a new non-orthogonal decomposition method to determine effective torques for three-dimensional (3D) joint rotation. A rotation about a joint coordinate axis (e.g. shoulder internal/external rotation) cannot be explained only by the torque about the joint coordinate axis because the joint coordinate axes usually deviate from the principal axes of inertia of the entire kinematic chain distal to the joint. Instead of decomposing torques into three orthogonal joint coordinate axes, our new method decomposes torques into three "non-orthogonal effective axes" that are determined in such a way that a torque about each effective axis produces a joint rotation only about one of the joint coordinate axes. To demonstrate the validity of this new method, a simple internal/external rotation of the upper arm with the elbow flexed at 90 degrees was analyzed by both orthogonal and non-orthogonal decomposition methods. The results showed that only the non-orthogonal decomposition method could explain the cause-effect mechanism whereby three angular accelerations at the shoulder joint are produced by the gravity torque, resultant joint torque, and interaction torque. The proposed method would be helpful for biomechanics and motor control researchers to investigate the manner in which the central nervous system coordinates the gravity torque, resultant joint torque, and interaction torque to control 3D joint rotations.     

Determining alternative models for vegetation response analysis: a non-parametric approach

Abstract. Vegetation models based on multiple logistic regression are of growing interest in environmental studies and decision making. The relatively simple sigmoid Gaussian optimum curves are most common in current vegetation models, although several different other response shapes are known. However, improvements in the technical means for handling statistical data now facilitate fast and interactive calculation of alternative complex, more data-related, non-parametric models. The aim in this study was to determine whether, and if so how often, a complex response shape could be more adequate than a linear or quadratic one. Using the framework of Generalized Additive Models, both parametric (linear and quadratic) and non-parametric (smoothed) stepwise multiple logistic regression techniques were applied to a large data set on wetlands and water plants and to six environmental variables: pH, chloride, orthophosphate, inorganic nitrogen, thickness of the sapropelium layer and depth of the water-body. All models were tested for their goodness-of-fit and significance. Of all 156 generalized additive models calculated, 77 % were found to contain at least one smoothed predictor variable, i.e. an environmental variable with a response better fitted by a complex, non-parametric, than by a linear or quadratic parametric curve. Chloride was the variable with the highest incidence of smoothed responses (48 %). Generally, a smoothed curve was preferable in 23 % of all species-variable correlations calculated, compared to 25 % and 18 % for sigmoid and Gaussian shaped curves, respectively. Regression models of two plant species are presented in detail to illustrate the potential of smoothers to produce good fitting and biologically sound response models in comparison to linear and polynomial regression models. We found Generalized Additive Modelling a useful and practical technique for improving current regression-based vegetation models by allowing for alternative, complex response shapes.     

Linear mixed models for longitudinal shape data with applications to facial modeling

We present a novel application of methods for analysis of high-dimensional longitudinal data to a comparison of facial shape over time between babies with cleft lip and palate and similarly aged controls. A pairwise methodology is used that was introduced in Fieuws and Verbeke (2006) in order to apply a linear mixed-effects model to data of high dimensions, such as describe facial shape. The approach involves fitting bivariate linear mixed-effects models to all the pairwise combinations of responses, where the latter result from the individual coordinate positions, and aggregating the results across repeated parameter estimates (such as the random-effects variance for a particular coordinate). We describe one example using landmarks and another using facial curves from the cleft lip study, the latter using B-splines to provide an efficient parameterization. The results are presented in 2 dimensions, both in the profile and in the frontal views, with bivariate confidence intervals for the mean position of each landmark or curve, allowing objective assessment of significant differences in particular areas of the face between the 2 groups. Model comparison is performed using Wald and pseudolikelihood ratio tests.     

Prediction of protein-protein interactions based on surface patch comparison

A method to predict if two proteins interact, based on their three-dimensional structures, is presented. It consists of five steps: (i) the surface of each protein, represented by the solvent accessible atoms, is divided into small patches; (ii) the shape of each patch is described by the atom distributions along its principal axes; (iii) the shape complementarity between two patches is estimated by comparing, through contingency table analysis, their atom distributions along their principal axes; (iv) given protein A, with nA surface patches, and protein B, with nB surface patches, nA x nB shape complementarity values are obtained; and (v) the distribution of the latter allows one to discriminate pairs of interacting and of noninteracting proteins. Only a few seconds are necessary to predict if two proteins interact, with accuracy close to 80%, sensitivity over 70% and specificity close to 50%.     

Propagation of excitation in idealized anisotropic two-dimensional tissue

This paper reports on a simulation of propagation for anisotropic two-dimensional cardiac tissue. The tissue structure assumed was that of a Hodgin-Huxley membrane separating inside and outside anisotropic media, obeying Ohm's law in each case. Membrane current was found by an integral expression involving partial spatial derivatives of Vm weighted by a function of distance. Numerical solutions for transmembrane voltage as a function of time following excitation at a single central site were computed using an algorithm that examined only the portion of the tissue undergoing excitation at each moment; thereby, the number of calculations required was reduced to a large but achievable number. Results are shown for several combinations of the four conductivity values: With isotropic tissue, excitation spread in circles, as expected. With tissue having nominally normal ventricular conductivities, excitation spread in patterns close to ellipses. With reciprocal conductivities, isochrones approximated a diamond shape, and were in conflict with the theoretical predictions of Muler and Markin; the time constant of the foot of the action potentials, as computed, varied between sites along axes as compared with sites along the diagonals, even though membrane properties were identical everywhere. Velocity of propagation changed for several milliseconds following the stimulus. Patterns that would have been expected from well-known studies in one dimension did not always occur in two dimensions, with the magnitude of the difference varying from nil for isotropic conductivities to quite large for reciprocal conductivities.     

Species response curves along environmental gradients. A case study from SE Norwegian swamp forests

Abstract. Vegetation science has relied on untested paradigms relating to the shape of species response curves along environmental gradients. To advance in this field, we used the HOF approach to model response curves for 112 plant species along six environmental gradients and three ecoclines (as represented by DCA ordination axes) in SE Norwegian swamp forests. Response curve properties were summarized in three binary response variables: (1) model unimodal or monotonous (determinate) vs. indeterminate; (2) for determinate models, unimodal vs. monotonous and (3) for unimodal models, skewed vs. symmetric. We used logistic regression to test the influence, singly and jointly, of seven predictor variables on each of three response variables. Predictor variables included gradient type (environmental or ecocline) and length (compositional turnover); species category (vascular plant, moss, Sphagnum or hepatic), species frequency and richness, tolerance (the fraction of the gradient along which the species occurs) and position of species along each gradient. The probability for fitting a determinate model increased as the main occurrence of species approached gradient extremes and with increasing species tolerance and frequency and gradient length. Appearance of unimodal models was favoured by low species tolerance and disfavoured by closeness of species to gradient extremes. Appearance of skewed models was weakly related to predictors but was slightly favoured by species optima near gradient extremes. Contrary to the results of previous studies, species category, gradient type and variation in species richness along gradients did not contribute independently to model prediction. The overall best predictors of response curve shape were position along the gradient (relative to extremes) and tolerance; the latter also expressing gradient length in units of compositional turnover. This helps predicting species responses to gradients from gradient specific species properties. The low proportion of skewed response curves and the large variation of species response curves along all gradients indicate that skewed response curves is a smaller problem for the performance of ordination methods than often claimed. We find no evidence that DCA ordination increases the unimodality, or symmetry, of species response curves more than expected from the higher compositional turnover along ordination axes. Thus ordination axes may be appropriate proxies for ecoclines, applicable for use in species response modelling.     

Association models for a multivariate binary response

Models for a multivariate binary response are parameterized by univariate marginal probabilities and dependence ratios of all orders. The w-order dependence ratio is the joint success probability of w binary responses divided by the joint success probability assuming independence. This parameterization supports likelihood-based inference for both regression parameters, relating marginal probabilities to explanatory variables, and association model parameters, relating dependence ratios to simple and meaningful mechanisms. Five types of association models are proposed, where responses are (1) independent given a necessary factor for the possibility of a success, (2) independent given a latent binary factor, (3) independent given a latent beta distributed variable, (4) follow a Markov chain, and (5) follow one of two first-order Markov chains depending on the realization of a binary latent factor. These models are illustrated by reanalyzing three data sets, foremost a set of binary time series on auranofin therapy against arthritis. Likelihood-based approaches are contrasted with approaches based on generalized estimating equations. Association models specified by dependence ratios are contrasted with other models for a multivariate binary response that are specified by odds ratios or correlation coefficients.     

Isomap transform for segmenting human body shapes

Segmentation of the 3D human body is a very challenging problem in applications exploiting volume capture data. Direct clustering in the Euclidean space is usually complex or even unsolvable. This paper presents an original method based on the Isomap (isometric feature mapping) transform of the volume data-set. The 3D articulated posture is mapped by Isomap in the pose of Da Vinci's Vitruvian man. The limbs are unrolled from each other and separated from the trunk and pelvis, and the topology of the human body shape is recovered. In such a configuration, Hoshen–Kopelman clustering applied to concentric spherical shells is used to automatically group points into the labelled principal curves. Shepard interpolation is utilised to back-map points of the principal curves into the original volume space. The experimental results performed on many different postures have proved the validity of the proposed method. Reliability of less than 2 cm and 3° in the location of the joint centres and direction axes of rotations has been obtained, respectively, which qualifies this procedure as a potential tool for markerless motion analysis.     

Bayesian regularization for flexible baseline hazard functions in Cox survival models

Fully Bayesian methods for Cox models specify a model for the baseline hazard function. Parametric approaches generally provide monotone estimations. Semi‐parametric choices allow for more flexible patterns but they can suffer from overfitting and instability. Regularization methods through prior distributions with correlated structures usually give reasonable answers to these types of situations. We discuss Bayesian regularization for Cox survival models defined via flexible baseline hazards specified by a mixture of piecewise constant functions and by a cubic B‐spline function. For those “semi‐parametric” proposals, different prior scenarios ranging from prior independence to particular correlated structures are discussed in a real study with microvirulence data and in an extensive simulation scenario that includes different data sample and time axis partition sizes in order to capture risk variations. The posterior distribution of the parameters was approximated using Markov chain Monte Carlo methods. Model selection was performed in accordance with the deviance information criteria and the log pseudo‐marginal likelihood. The results obtained reveal that, in general, Cox models present great robustness in covariate effects and survival estimates independent of the baseline hazard specification. In relation to the “semi‐parametric” baseline hazard specification, the B‐splines hazard function is less dependent on the regularization process than the piecewise specification because it demands a smaller time axis partition to estimate a similar behavior of the risk.     

Isomap transform for segmenting human body shapes

Segmentation of the 3D human body is a very challenging problem in applications exploiting volume capture data. Direct clustering in the Euclidean space is usually complex or even unsolvable. This paper presents an original method based on the Isomap (isometric feature mapping) transform of the volume data-set. The 3D articulated posture is mapped by Isomap in the pose of Da Vinci's Vitruvian man. The limbs are unrolled from each other and separated from the trunk and pelvis, and the topology of the human body shape is recovered. In such a configuration, Hoshen-Kopelman clustering applied to concentric spherical shells is used to automatically group points into the labelled principal curves. Shepard interpolation is utilised to back-map points of the principal curves into the original volume space. The experimental results performed on many different postures have proved the validity of the proposed method. Reliability of less than 2?cm and 3° in the location of the joint centres and direction axes of rotations has been obtained, respectively, which qualifies this procedure as a potential tool for markerless motion analysis.     

The sophisticated visual system of a tiny Cambrian crustacean: analysis of a stalked fossil compound eye

Fossilized compound eyes from the Cambrian, isolated and three-dimensionally preserved, provide remarkable insights into the lifestyle and habitat of their owners. The tiny stalked compound eyes described here probably possessed too few facets to form a proper image, but they represent a sophisticated system for detecting moving objects. The eyes are preserved as almost solid, mace-shaped blocks of phosphate, in which the original positions of the rhabdoms in one specimen are retained as deep cavities. Analysis of the optical axes reveals four visual areas, each with different properties in acuity of vision. They are surveyed by lenses directed forwards, laterally, backwards and inwards, respectively. The most intriguing of these is the putatively inwardly orientated zone, where the optical axes, like those orientated to the front, interfere with axes of the other eye of the contralateral side. The result is a three-dimensional visual net that covers not only the front, but extends also far laterally to either side. Thus, a moving object could be perceived by a two-dimensional coordinate (which is formed by two axes of those facets, one of the left and one of the right eye, which are orientated towards the moving object) in a wide three-dimensional space. This compound eye system enables small arthropods equipped with an eye of low acuity to estimate velocity, size or distance of possible food items efficiently. The eyes are interpreted as having been derived from individuals of the early crustacean Henningsmoenicaris scutula pointing to the existence of highly efficiently developed eyes in the early evolutionary lineage leading towards the modern Crustacea.     

Comparing Two Samples of Response Curves by Nonparamètric Tests of Predicted Order

Starting from MOSTELLER'S (1955) one-sample test of predicted order, some two-sample versions are proposed for comparing nonparametrically a treatment group (m response curves) with a control group (n response curves) as to a predicted shape represented by a specified rank order. The tests are applied to two groups of learning curves.     

Direction ofE for maximum response of a retinula cell

Summary Except for very special fused rhabdoms, e. g. those with orthogonal microvilli like the worker bee, the direction of the electric vector E of linear polarized light necessary for a maximum response from a retinula cell is not parallel (or perpendicular) to the microvilli of the recorded cell. This is because the rhabdomeres of a fused rhabdom are optically coupled, i. e. the properties of each rhabdomere influence the manner in which light is transmitted down the composite rhabdom structure. A rhabdom is analogous to a non-uniform absorbing optical crystal. Such a crystal has two coordinate (optical) axes along which E remains linear polarized as it propagates. Only when the microvilli of the recorded cell are parallel to one of these axes will the direction ofE for maximum retinula cell response be parallel to the microvilli. The locust-type of rhabdom is used as an example.     

Reirradiation at long time intervals in mouse kidney: a comparison between experimental results and the predictions of the F-type tissue model

The tolerance of a late-responding tissue to reirradiation after long time intervals has been analysed using the F-type tissue model. In this model the tissue is composed of identical cells, each of which is capable of extensive proliferation and of tissue-specific function. The model was adapted to calculate the response to two fractions of radiation given in a variable overall time. For two equal doses of radiation the repair of tissue damage after the first fraction could be detected theoretically by a change in the rate of cell depletion after retreatment and by an increase in the minimum cell number attained. For an 'experimental set-up', in which a constant first dose was followed by a range of retreatment doses in a variable overall time, the repair of tissue damage theoretically could be detected most sensitively by a shift of the dose-response curves to higher retreatment doses as the time interval between the two doses was increased. A prerequisite for a proper comparison of these dose-response curves was that the responses were evaluated at times after the first dose determined by the minimal latency times after high retreatment doses. From a comparison of these theoretical results with experimental findings for mouse kidneys it was concluded that no recovery of tissue function took place over a 6-month period. Instead it appeared that the kidneys had become more sensitive to irradiation over this period.