Ecological implications of a latitudinal gradient in inter-annual climatic variability: a test using fractal and chaos theories

Although measures of seasonality are common, ecologists have seldom measured inter-annual variation or tested for the presence of a latitudinal gradient. Yet, species diversity and range size may be influenced by latitudinal gradients in climatic variability. Fractal and chaos theories offer new techniques to measure climatic unpredictability by analyzing the dynamics of nonlinear time-series data. We analyzed data on the number of freezing degree-days and date of first permanent ice from 12 weather stations (43–83°N) in Canada to describe latitudinal gradients in the timing of seasonal events. Time-series (ca 30 yr) of climatic variables were found to be nonlinear dynamic systems that were neither stochastic nor cyclic. The analyses reveal a latitudinal gradient in system memory length, with high latitude locations displaying shorter memory length (R/S fractal analysis and Lyapunov exponent). This latitudinal gradient is also characterized by more random and irregular fractal patterns (Hurst exponent) associated with higher latitudes. Both of these patterns may be due to the simpler climatic systems in high latitudes as indicated by fewer dynamic variables (lower correlation dimension). Application of fractal and chaos theories show that the timing of seasonal events (i.e. inter-annual variation) comprises a chaotic-dynamic system of comparatively low dimension.     

Stochastic modelling of environmental variation for biological populations

We examine stochastic effects, in particular environmental variability, in population models of biological systems. Some simple models of environmental stochasticity are suggested, and we demonstrate a number of analytic approximations and simulation-based approaches that can usefully be applied to them. Initially, these techniques, including moment-closure approximations and local linearization, are explored in the context of a simple and relatively tractable process. Our presentation seeks to introduce these techniques to a broad-based audience of applied modellers. Therefore, as a test case, we study a natural stochastic formulation of a non-linear deterministic model for nematode infections in ruminants, proposed by Roberts and Grenfell (1991). This system is particularly suitable for our purposes, since it captures the essence of more complicated formulations of parasite demography and herd immunity found in the literature. We explore two modes of behaviour. In the endemic regime the stochastic dynamic fluctuates widely around the non-zero fixed points of the deterministic model. Enhancement of these fluctuations in the presence of environmental stochasticity can lead to extinction events. Using a simple model of environmental fluctuations we show that the magnitude of this system response reflects not only the variance of environmental noise, but also its autocorrelation structure. In the managed regime host-replacement is modelled via periodic perturbation of the population variables. In the absence of environmental variation stochastic effects are negligible, and we examine the system response to a realistic environmental perturbation based on the effect of micro-climatic fluctuations on the contact rate. The resultant stochastic effects and the relevance of analytic approximations based on simple models of environmental stochasticity are discussed.     

Effective dimension reduction methods for tumor classification using gene expression data

MOTIVATION: One particular application of microarray data, is to uncover the molecular variation among cancers. One feature of microarray studies is the fact that the number n of samples collected is relatively small compared to the number p of genes per sample which are usually in the thousands. In statistical terms this very large number of predictors compared to a small number of samples or observations makes the classification problem difficult. An efficient way to solve this problem is by using dimension reduction statistical techniques in conjunction with nonparametric discriminant procedures. RESULTS: We view the classification problem as a regression problem with few observations and many predictor variables. We use an adaptive dimension reduction method for generalized semi-parametric regression models that allows us to solve the 'curse of dimensionality problem' arising in the context of expression data. The predictive performance of the resulting classification rule is illustrated on two well know data sets in the microarray literature: the leukemia data that is known to contain classes that are easy 'separable' and the colon data set.     

Reduction of stochastic conductance-based neuron models with time-scales separation

We introduce a method for systematically reducing the dimension of biophysically realistic neuron models with stochastic ion channels exploiting time-scales separation. Based on a combination of singular perturbation methods for kinetic Markov schemes with some recent mathematical developments of the averaging method, the techniques are general and applicable to a large class of models. As an example, we derive and analyze reductions of different stochastic versions of the Hodgkin Huxley (HH) model, leading to distinct reduced models. The bifurcation analysis of one of the reduced models with the number of channels as a parameter provides new insights into some features of noisy discharge patterns, such as the bimodality of interspike intervals distribution. Our analysis of the stochastic HH model shows that, besides being a method to reduce the number of variables of neuronal models, our reduction scheme is a powerful method for gaining understanding on the impact of fluctuations due to finite size effects on the dynamics of slow fast systems. Our analysis of the reduced model reveals that decreasing the number of sodium channels in the HH model leads to a transition in the dynamics reminiscent of the Hopf bifurcation and that this transition accounts for changes in characteristics of the spike train generated by the model. Finally, we also examine the impact of these results on neuronal coding, notably, reliability of discharge times and spike latency, showing that reducing the number of channels can enhance discharge time reliability in response to weak inputs and that this phenomenon can be accounted for through the analysis of the reduced model.     

Defining gait patterns using Parallel Factor 2 (PARAFAC2): A new analysis of previously published data

Three-dimensional gait analysis (3D–GA) is commonly used to answer clinical questions of the form “which joints and what variables are most affected during when”. When studying high-dimensional datasets, traditional dimension reduction methods (e.g. principal components analysis) require “data flattening”, which may make the ensuing solutions difficult to interpret. The aim of the present study is to present a case study of how a multi-dimensional dimension reduction technique, Parallel Factor 2 (PARAFAC2), provides a clinically interpretable set of solutions to typical biomechanical datasets where different variables are collected during walking and running. Three-dimensional kinematic and kinetic data used for the present analyses came from two publicly available datasets on walking (n = 33) and running (n = 28). For each dataset, a four-dimensional array was constructed as follows: Mode A was time normalized cycle points; mode B was the number of participants multiplied by the number of speed conditions tested; mode C was the number of joint degrees of freedom, and mode D was variable (angle, velocity, moment, power). Five factors for walking and four factors for running were extracted which explained 79.23% and 84.64% of their dataset’s variance. The factor which explains the greatest variance was swing-phase sagittal plane knee kinematics (walking), and kinematics and kinetics (running). Qualitatively, all extracted factors increased in magnitude with greater speed in both walking and running. This study is a proof of concept that PARAFAC2 is useful for performing dimension reduction and producing clinically interpretable solutions to guide clinical decision making.     

Low-dimensional,morphologically accurate models of subthreshold membrane potential

The accurate simulation of a neuron’s ability to integrate distributed synaptic input typically requires the simultaneous solution of tens of thousands of ordinary differential equations. For, in order to understand how a cell distinguishes between input patterns we apparently need a model that is biophysically accurate down to the space scale of a single spine, i.e., 1 μm. We argue here that one can retain this highly detailed input structure while dramatically reducing the overall system dimension if one is content to accurately reproduce the associated membrane potential at a small number of places, e.g., at the site of action potential initiation, under subthreshold stimulation. The latter hypothesis permits us to approximate the active cell model with an associated quasi-active model, which in turn we reduce by both time-domain (Balanced Truncation) and frequency-domain (H₂{\cal H}_2 approximation of the transfer function) methods. We apply and contrast these methods on a suite of typical cells, achieving up to four orders of magnitude in dimension reduction and an associated speed-up in the simulation of dendritic democratization and resonance. We also append a threshold mechanism and indicate that this reduction has the potential to deliver an accurate quasi-integrate and fire model.     

Dominant ionic mechanisms explored in spiking and bursting using local low-dimensional reductions of a biophysically realistic model neuron

The large number of variables involved in many biophysical models can conceal potentially simple dynamical mechanisms governing the properties of its solutions and the transitions between them as parameters are varied. To address this issue, we extend a novel model reduction method, based on “scales of dominance,” to multi-compartment models. We use this method to systematically reduce the dimension of a two-compartment conductance-based model of a crustacean pyloric dilator (PD) neuron that exhibits distinct modes of oscillation—tonic spiking, intermediate bursting and strong bursting. We divide trajectories into intervals dominated by a smaller number of variables, resulting in a locally reduced hybrid model whose dimension varies between two and six in different temporal regimes. The reduced model exhibits the same modes of oscillation as the 16 dimensional model over a comparable parameter range, and requires fewer ad hoc simplifications than a more traditional reduction to a single, globally valid model. The hybrid model highlights low-dimensional organizing structure in the dynamics of the PD neuron, and the dependence of its oscillations on parameters such as the maximal conductances of calcium currents. Our technique could be used to build hybrid low-dimensional models from any large multi-compartment conductance-based model in order to analyze the interactions between different modes of activity.     

New deep learning-based methods for visualizing ecosystem properties using environmental DNA metabarcoding data

Environmental DNA (eDNA) metabarcoding provides an efficient approach for documenting biodiversity patterns in marine and terrestrial ecosystems. The complexity of these data prevents current methods from extracting and analyzing all the relevant ecological information they contain, and new methods may provide better dimensionality reduction and clustering. Here we present two new deep learning-based methods that combine different types of neural networks (NNs) to ordinate eDNA samples and visualize ecosystem properties in a two-dimensional space: the first is based on variational autoencoders and the second on deep metric learning. The strength of our new methods lies in the combination of two inputs: the number of sequences found for each molecular operational taxonomic unit (MOTU) detected and their corresponding nucleotide sequence. Using three different datasets, we show that our methods accurately represent several biodiversity indicators in a two-dimensional latent space: MOTU richness per sample, sequence α-diversity per sample, Jaccard's and sequence β-diversity between samples. We show that our nonlinear methods are better at extracting features from eDNA datasets while avoiding the major biases associated with eDNA. Our methods outperform traditional dimension reduction methods such as Principal Component Analysis, t-distributed Stochastic Neighbour Embedding, Nonmetric Multidimensional Scaling and Uniform Manifold Approximation and Projection for dimension reduction. Our results suggest that NNs provide a more efficient way of extracting structure from eDNA metabarcoding data, thereby improving their ecological interpretation and thus biodiversity monitoring.     

Approximate reduction of multiregional models with environmental stochasticity

In this work we extend previous results regarding the use of approximate aggregation techniques to simplify the study of discrete time models for populations that live in an environment that changes randomly with time. Approximate aggregation techniques allow one to transform a complex system involving many coupled variables and in which there are processes with different time scales, by a simpler reduced model with a fewer number of 'global' variables, in such a way that the dynamics of the former can be approximated by that of the latter. We present the reduction of a stochastic multiregional model in which the population, structured by age and spatial location, lives in a random environment and in which migration is fast with respect to demography. However, the technique works in much more general settings as, for example, those of stage-structured populations living in a multipatch environment. By manipulating the original system and appropriately defining the global variables we obtain a simpler system. The paper concentrates on establishing relationships between the original and the reduced systems for a given separation of time scales between the two processes. In particular, we relate the original state variables and the global variables and, in the case the pattern of temporal variation is Markovian, we relate the presence of strong stochastic ergodicity for the original and reduced systems. Moreover, we relate different measures of asymptotic population growth for these systems.     

Novel method for the display of multivariate data using neural networks

A neural network has been used to reduce the dimensionality of multivariate data sets to produce two-dimensional (2D) displays of these sets. The data consisted of physicochemical properties for sets of biologically active molecules calculated by computational chemistry methods. Previous work has demonstrated that these data contain sufficient relevant information to classify the compounds according to their biological activity. The plots produced by the neural network are compared with results from two other techniques for linear and nonlinear dimension reduction, and are shown to give comparable and, in one case, superior results. Advantages of this technique are discussed.     

An Ant Colony Optimization Based Dimension Reduction Method for High-Dimensional Datasets

In this paper, a bionic optimization algorithm based dimension reduction method named Ant Colony Optimization -Selection (ACO-S) is proposed for high-dimensional datasets. Because microarray datasets comprise tens of thousands of features (genes), they are usually used to test the dimension reduction techniques. ACO-S consists of two stages in which two well-known ACO algorithms, namely ant system and ant colony system, are utilized to seek for genes, respectively. In the first stage, a modified ant system is used to filter the nonsignificant genes from high-dimensional space, and a number of promising genes are reserved in the next step. In the second stage, an improved ant colony system is applied to gene selection. In order to enhance the search ability of ACOs, we propose a method for calculating priori available heuristic information and design a fuzzy logic controller to dynamically adjust the number of ants in ant colony system. Furthermore, we devise another fuzzy logic controller to tune the parameter (q₀) in ant colony system. We evaluate the performance of ACO-S on five microarray datasets, which have dimensions varying from 7129 to 12000. We also compare the performance of ACO-S with the results obtained from four existing well-known bionic optimization algorithms. The comparison results show that ACO-S has a notable ability to generate a gene subset with the smallest size and salient features while yielding high classification accuracy. The comparative results generated by ACO-S adopting different classifiers are also given. The proposed method is shown to be a promising and effective tool for mining high-dimension data and mobile robot navigation.     

Iterative partial least squares with right-censored data analysis: a comparison to other dimension reduction techniques

In the linear model with right-censored responses and many potential explanatory variables, regression parameter estimates may be unstable or, when the covariates outnumber the uncensored observations, not estimable. We propose an iterative algorithm for partial least squares, based on the Buckley-James estimating equation, to estimate the covariate effect and predict the response for a future subject with a given set of covariates. We use a leave-two-out cross-validation method for empirically selecting the number of components in the partial least-squares fit that approximately minimizes the error in estimating the covariate effect of a future observation. Simulation studies compare the methods discussed here with other dimension reduction techniques. Data from the AIDS Clinical Trials Group protocol 333 are used to motivate the methodology.     

Classification using partial least squares with penalized logistic regression

MOTIVATION: One important aspect of data-mining of microarray data is to discover the molecular variation among cancers. In microarray studies, the number n of samples is relatively small compared to the number p of genes per sample (usually in thousands). It is known that standard statistical methods in classification are efficient (i.e. in the present case, yield successful classifiers) particularly when n is (far) larger than p. This naturally calls for the use of a dimension reduction procedure together with the classification one. RESULTS: In this paper, the question of classification in such a high-dimensional setting is addressed. We view the classification problem as a regression one with few observations and many predictor variables. We propose a new method combining partial least squares (PLS) and Ridge penalized logistic regression. We review the existing methods based on PLS and/or penalized likelihood techniques, outline their interest in some cases and theoretically explain their sometimes poor behavior. Our procedure is compared with these other classifiers. The predictive performance of the resulting classification rule is illustrated on three data sets: Leukemia, Colon and Prostate.     

Nonlinear dimensionality reduction: Alternative ordination approaches for extracting and visualizing biodiversity patterns in tropical montane forest vegetation data

Ecological patterns are difficult to extract directly from vegetation data. The respective surveys provide a high number of interrelated species occurrence variables. Since often only a limited number of ecological gradients determine species distributions, the data might be represented by much fewer but effectively independent variables. This can be achieved by reducing the dimensionality of the data. Conventional methods are either limited to linear feature extraction (e.g., principal component analysis, and Classical Multidimensional Scaling, CMDS) or require a priori assumptions on the intrinsic data dimensionality (e.g., Nonmetric Multidimensional Scaling, NMDS, and self organized maps, SOM).In this study we explored the potential of Isometric Feature Mapping (Isomap). This new method of dimensionality reduction is a nonlinear generalization of CMDS. Isomap is based on a nonlinear geodesic inter-point distance matrix. Estimating geodesic distances requires one free threshold parameter, which defines linear geometry to be preserved in the global nonlinear distance structure. We compared Isomap to its linear (CMDS) and nonmetric (NMDS) equivalents. Furthermore, the use of geodesic distances allowed also extending NMDS to a version that we called NMDS-G. In addition we investigated a supervised Isomap variant (S-Isomap) and showed that all these techniques are interpretable within a single methodical framework.As an example we investigated seven plots (subdivided in 456 subplots) in different secondary tropical montane forests with 773 species of vascular plants. A key problem for the study of tropical vegetation data is the heterogeneous small scale variability implying large ranges of β-diversity. The CMDS and NMDS methods did not reduce the data dimensionality reasonably. On the contrary, Isomap explained 95% of the data variance in the first five dimensions and provided ecologically interpretable visualizations; NMDS-G yielded similar results. The main shortcoming of the latter was the high computational cost and the requirement to predefine the dimension of the embedding space. The S-Isomap learning scheme did not improve the Isomap variant for an optimal threshold parameter but substantially improved the nonoptimal solutions.We conclude that Isomap as a new ordination method allows effective representations of high dimensional vegetation data sets. The method is promising since it does not require a priori assumptions, and is computationally highly effective.     

Nonlinear dynamics of heart rate variability during experimental hemorrhage in ketamine-anesthetized rats

Indexes of heart rate variability (HRV) based on linear stochastic models are independent risk factors for arrhythmic death (AD). An index based on a nonlinear deterministic model, a reduction in the point correlation dimension (PD2i), has been shown in both animal and human studies to have a higher sensitivity and specificity for predicting AD. Dimensional reduction subsequent to transient ischemia was examined previously in a simple model system, the intrinsic nervous system of the isolated rabbit heart. The present study presents a new model system in which the higher cerebral centers are blocked chemically (ketamine inhibition of N-methyl-D-aspartate receptors) and the system is perturbed over a longer 15-min interval by continuous hemorrhage. The hypothesis tested was that dimensional reduction would again be evoked, but in association with a more complex relationship between the system variables. The hypothesis was supported, and we interpret the greater response complexity to result from the larger autonomic superstructure attached to the heart. The complexities observed in the nonlinear heartbeat dynamics constitute a new genre of autonomic response, one clearly distinct from a hardwired reflex or a cerebrally determined defensive reaction.     

Spatio-temporal patterns in a mechanical model for mesenchymal morphogenesis

We present an in-depth study of spatio-temporal patterns in a simplified version of a mechanical model for pattern formation in mesenchymal morphogenesis. We briefly motivate the derivation of the model and show how to choose realistic boundary conditions to make the system well-posed. We firstly consider one-dimensional patterns and carry out a nonlinear perturbation analysis for the case where the uniform steady state is linearly unstable to a single mode. In two-dimensions, we show that if the displacement field in the model is represented as a sum of orthogonal parts, then the model can be decomposed into two sub-models, only one of which is capable of generating pattern. We thus focus on this particular sub-model. We present a nonlinear analysis of spatio-temporal patterns exhibited by the sub-model on a square domain and discuss mode interaction. Our analysis shows that when a two-dimensional mode number admits two or more degenerate mode pairs, the solution of the full nonlinear system of partial differential equations is a mixed mode solution in which all the degenerate mode pairs are represented in a frequency locked oscillation.     

Enhanced analysis of landscape structure: Inclusion of transition zones and small-scale landscape elements

Landscape diversity is an essential component of biodiversity because it determines the diversity of habitats and species. It is necessary to adopt an adaptive and realizable landscape structure monitoring model for ecosystem management. However, the existing methods of landscape structure analysis often do not consider the transitions between different types of patch, which are dynamic and play several functional roles in landscape structure. Another component of information that has often been ignored is “small-scale landscape elements” inside patches, which leads to within-patch heterogeneity being underestimated. This paper presents a method for solving these problems by regular landscape monitoring using RapidEye remote sensing data. We incorporate the third altitude dimension in landscape structure analysis, especially for detecting certain small-scale landscape structures such as groves, tree rows, and transitions at forest boundaries. For these detailed structures a number of landscape metrics are computed under a three-dimensional perspective. The results indicate that a more realistic and more precise representation of landscape structure is obtained on the basis of the detailed spatial scale and the true surface geometries of patches. Taking transitions into account helps smooth “edge effects” between patches and leads to more realistic quantification of fragmentation.     

Application of constraint programming techniques for structure prediction of lattice proteins with extended alphabets

MOTIVATION: Predicting the ground state of biopolymers is a notoriously hard problem in biocomputing. Model systems, such as lattice proteins, are simple tools and valuable to test and improve new methods. Best known are models with sequences composed from a binary (hydrophobic and polar) alphabet. The major drawback is the degeneracy, i.e. the number of different ground state conformations. RESULTS: We show how recently developed constraint programming techniques can be used to solve the structure prediction problem efficiently for a higher order alphabet. To our knowledge it is the first report of an exact and computationally feasible solution to model proteins of length up to 36 and without resorting to maximally compact states. We further show that degeneracy is reduced by more than one order of magnitude and that ground state conformations are not necessarily compact. Therefore, more realistic protein simulations become feasible with our model.     

Multidimensional scaling for large genomic data sets

Background  

Multi-dimensional scaling (MDS) is aimed to represent high dimensional data in a low dimensional space with preservation of the similarities between data points. This reduction in dimensionality is crucial for analyzing and revealing the genuine structure hidden in the data. For noisy data, dimension reduction can effectively reduce the effect of noise on the embedded structure. For large data set, dimension reduction can effectively reduce information retrieval complexity. Thus, MDS techniques are used in many applications of data mining and gene network research. However, although there have been a number of studies that applied MDS techniques to genomics research, the number of analyzed data points was restricted by the high computational complexity of MDS. In general, a non-metric MDS method is faster than a metric MDS, but it does not preserve the true relationships. The computational complexity of most metric MDS methods is over O(N ² ), so that it is difficult to process a data set of a large number of genes N, such as in the case of whole genome microarray data.     

Dimension reduction strategies for analyzing global gene expression data with a response

The analysis of global gene expression data from microarrays is breaking new ground in genetics research, while confronting modelers and statisticians with many critical issues. In this paper, we consider data sets in which a categorical or continuous response is recorded, along with gene expression, on a given number of experimental samples. Data of this type are usually employed to create a prediction mechanism for the response based on gene expression, and to identify a subset of relevant genes. This defines a regression setting characterized by a dramatic under-resolution with respect to the predictors (genes), whose number exceeds by orders of magnitude the number of available observations (samples). We present a dimension reduction strategy that, under appropriate assumptions, allows us to restrict attention to a few linear combinations of the original expression profiles, and thus to overcome under-resolution. These linear combinations can then be used to build and validate a regression model with standard techniques. Moreover, they can be used to rank original predictors, and ultimately to select a subset of them through comparison with a background 'chance scenario' based on a number of independent randomizations. We apply this strategy to publicly available data on leukemia classification.