Chromatographic peak alignment using derivative dynamic time warping

Chromatogram overlays are frequently used to monitor inter‐batch performance of bioprocess purification steps. However, the objective analysis of chromatograms is difficult due to peak shifts caused by variable phase durations or unexpected process holds. Furthermore, synchronization of batch process data may also be required prior to performing multivariate analysis techniques. Dynamic time warping was originally developed as a method for spoken word recognition, but shows potential in the objective analysis of time variant signals, such as manufacturing data. In this work we will discuss the application of dynamic time warping with a derivative weighting function to align chromatograms to facilitate process monitoring and fault detection. In addition, we will demonstrate the utility of this method as a preprocessing step for multivariate model development. © 2013 American Institute of Chemical Engineers Biotechnol. Prog., 29: 394–402, 2013     

Robust depth-based estimation in the time warping model

In functional data analysis, the time warping model aims at representing a set of curves exhibiting phase and amplitude variation with respect to a common continuous process. Many biological processes, when observed across the time among different individuals, fit into this concept. The observed curves are modeled as the composition of an "amplitude process," which governs the common behavior, and a "warping process" that induces time distortion among the individuals. We aim at characterizing the first one. Because of the phase variation present among the curves, classical sample statistics computed on the observed sample provide poor representations of the amplitude process. Existing methods to estimate the mean behavior of the amplitude process consist on aligning the curves, that is, eliminating time variation, before estimation. However, since they rely on the use of sample means, they are very sensitive to the presence of outliers. In this article, we propose the use of a functional depth-based median as a robust estimator of the central behavior of the amplitude process. We investigate its properties in the time warping model, and we evaluate its performance in different simulation studies where we compare it to existing estimators, and we show its robustness against atypical observations. Finally, we illustrate its use with real data on a yeast time course microarray data set.     

Symmetric time warping, Boltzmann pair probabilities and functional genomics

Given two time series, possibly of different lengths, time warping is a method to construct an optimal alignment obtained by stretching or contracting time intervals. Unlike pairwise alignment of amino acid sequences, classical time warping, originally introduced for speech recognition, is not symmetric in the sense that the time warping distance between two time series is not necessarily equal to the time warping distance of the reversal of the time series. Here we design a new symmetric version of time warping, and present a formal proof of symmetry for our algorithm as well as for one of the variants of Aach and Church [1]. We additionally design quadratic time dynamic programming algorithms to compute both the forward and backward Boltzmann partition functions for symmetric time warping, and hence compute the Boltzmann probability that any two time series points are aligned. In the future, with the availability of increasingly long and accurate time series gene expression data, our algorithm can provide a sense of biological significance for aligned time points – e.g. our algorithm could be used to provide evidence that expression values of two genes have higher Boltzmann probability (say) in the G1 and S phase than in G2 and M phases. Algorithms, source code and web interface, developed by the first author, are made publicly available via the Boltzmann Time Warping web server at bioinformatics.bc.edu/clotelab/. Research partially supported by National Science Foundation grant DBI-0543506.     

Hierarchical grid transformation for image warping in the analysis of two-dimensional electrophoresis gels

Two-dimensional electrophoresis is a widely used method for separating a large number of proteins from complex protein mixtures and for revealing differential patterns of protein expressions. In the computer-assisted proteome research, the comparison of protein separation profiles involves several heuristic steps, ranging from protein spot detection to matching of unknown spots. An important prerequisite for efficient protein spot matching is the image warping step, where the geometric relationship between the gel profiles is modeled on the basis of a given set of known corresponding spots, so-called landmarks, and the locations of unknown spots are predicted using the optimized model. Traditionally, polynomial functions together with least squares optimization has been used, even though this approach is known to be incapable of modeling all the complex distortions inherent in electrophoretic data. To satisfy the need of more flexible gel distortion correction, a hierarchical grid transformation method with stochastic optimization is presented. The method provides an adaptive multiresolution model between the gels, and good correction performance in the practical cross-validation tests suggests that automatic warping of gel images could be based on this approach. We believe that the proposed model also has significance in the ultimate comparison of corresponding protein spots since the matching process should benefit from the closeness of the true spot pairs.     

The Encoding of Individual Identity in Dolphin Signature Whistles: How Much Information Is Needed?

Bottlenose dolphins (Tursiops truncatus) produce many vocalisations, including whistles that are unique to the individual producing them. Such “signature whistles” play a role in individual recognition and maintaining group integrity. Previous work has shown that humans can successfully group the spectrographic representations of signature whistles according to the individual dolphins that produced them. However, attempts at using mathematical algorithms to perform a similar task have been less successful. A greater understanding of the encoding of identity information in signature whistles is important for assessing similarity of whistles and thus social influences on the development of these learned calls. We re-examined 400 signature whistles from 20 individual dolphins used in a previous study, and tested the performance of new mathematical algorithms. We compared the measure used in the original study (correlation matrix of evenly sampled frequency measurements) to one used in several previous studies (similarity matrix of time-warped whistles), and to a new algorithm based on the Parsons code, used in music retrieval databases. The Parsons code records the direction of frequency change at each time step, and is effective at capturing human perception of music. We analysed similarity matrices from each of these three techniques, as well as a random control, by unsupervised clustering using three separate techniques: k-means clustering, hierarchical clustering, and an adaptive resonance theory neural network. For each of the three clustering techniques, a seven-level Parsons algorithm provided better clustering than the correlation and dynamic time warping algorithms, and was closer to the near-perfect visual categorisations of human judges. Thus, the Parsons code captures much of the individual identity information present in signature whistles, and may prove useful in studies requiring quantification of whistle similarity.     

Where biology meets; or how science advances: Presidential Address to the Linnean Society delivered at the Anniversary Meeting, 24th May 1985

Major advances in 'unrestricted' sciences like biology commonly occur when individual scientists (or techniques) cross conventional discipline boundaries; intra-discipline studies are essential for the consolidation and progress of the science, but are less likely to produce significant insights. 'Restricted' (or exact) sciences ignore variation, and are probably less sensitive to warping from specialization. This generalization is illustrated by recent controversies in evolutionary biology, particularly the neutralism debates of the 1970s, where over-rigid adherence to theoretical models and unjustified assumptions about the effects of gene action were made. The consequence of some of these is shown by considering genetic changes in house mouse (Mus domesticus) populations which were used to demonstrate apparent drift operating on neutral traits, whereas longitudinal studies of closed populations proved that strong natural selection may operate; a proper understanding of genetical forces requires a knowledge both of the history of particular populations and of environmental pressures varying in time and space.     

Building integral projection models with nonindependent vital rates

Population dynamics are functions of several demographic processes including survival, reproduction, somatic growth, and maturation. The rates or probabilities for these processes can vary by time, by location, and by individual. These processes can co‐vary and interact to varying degrees, e.g., an animal can only reproduce when it is in a particular maturation state. Population dynamics models that treat the processes as independent may yield somewhat biased or imprecise parameter estimates, as well as predictions of population abundances or densities. However, commonly used integral projection models (IPMs) typically assume independence across these demographic processes. We examine several approaches for modelling between process dependence in IPMs and include cases where the processes co‐vary as a function of time (temporal variation), co‐vary within each individual (individual heterogeneity), and combinations of these (temporal variation and individual heterogeneity). We compare our methods to conventional IPMs, which treat vital rates independent, using simulations and a case study of Soay sheep (Ovis aries). In particular, our results indicate that correlation between vital rates can moderately affect variability of some population‐level statistics. Therefore, including such dependent structures is generally advisable when fitting IPMs to ascertain whether or not such between vital rate dependencies exist, which in turn can have subsequent impact on population management or life‐history evolution.     

Biological variation of glucose and insulin includes a deterministic chaotic component

Serial data of glucose and insulin values of individual patients vary over short periods of time; this phenomenon has been called biological variation. The classic homeostatic control model assumes that the physiological mechanisms maintaining the concentrations of glucose and insulin are linear. The only deviations over a short period of time one should observe are in relation to a glucose load or major hormonal disturbance. Otherwise, the values of these analytes should be constant and any variations seen are due to random disturbances. We investigated previously published serial data (three for glucose and one for insulin) with nonlinear analytical methods, such as embedding space, correlation dimension, Lyapunov exponents, singular value decomposition and phase portraits, as well as linear methods, such as power spectra and autocorrelation functions. The power spectra failed to show dominant frequencies, but the autocorrelation functions showed significant correlation, consistent with a deterministic process. The correlation dimension was finite, around 4.0, the first Lyapunov exponent was positive, indicative of a deterministic chaotic process. Furthermore, the phase portraits showed directional flow. Therefore, the short-term biological variation observed for analytes arises from nonlinear, deterministic chaotic behavior instead of random variation.     

If at first you don't succeed… Studies of ontogeny shed light on the cognitive demands of habitual tool use

Many species use tools, but the mechanisms underpinning the behaviour differ between species and even among individuals within species, depending on the variants performed. When considering tool use ‘as adaptation’, an important first step is to understand the contribution made by fixed phenotypes as compared to flexible mechanisms, for instance learning. Social learning of tool use is sometimes inferred based on variation between populations of the same species but this approach is questionable. Specifically, alternative explanations cannot be ruled out because population differences are also driven by genetic and/or environmental factors. To better understand the mechanisms underlying routine but non-universal (i.e. habitual) tool use, we suggest focusing on the ontogeny of tool use and individual variation within populations. For example, if tool-using competence emerges late during ontogeny and improves with practice or varies with exposure to social cues, then a role for learning can be inferred. Experimental studies help identify the cognitive and developmental mechanisms used when tools are used to solve problems. The mechanisms underlying the route to tool-use acquisition have important consequences for our understanding of the accumulation in technological skill complexity over the life course of an individual, across generations and over evolutionary time.     

Controlling behavioral experiments with a new programming language (SORCA) for microcomputer systems

A new programming language SORCA has been defined and a compiler has been written for Z80-based microcomputer systems with CP/M operating system. The language was developed to control behavioral experiments by external stimuli and by time schedule in real-time. Eight binary hardware input lines are sampled cyclically by the computer and can be used to sense switches, level detectors and other binary information, while 8 binary hardware output lines, that are cyclically updated, can be used to control relays, lamps, generate tones or for other purposes. The typical reaction time (cycle time) of a SORCA-program is 500 microseconds to 1 ms. All functions can be programmed as often as necessary. Included are the basic logic functions, counters, timers, majority gates and other complex functions. Parameters can be given as constants or as a result of a step function or of a random process (with Gaussian or equal distribution). Several tasks can be performed simultaneously. In addition, results of an experiment (e.g., number of reactions or latencies) can be measured and printed out on request or automatically. The language is easy to learn and can also be used for many other control purposes.     

Evaluation of brain-stem auditory evoked potentials using dynamic time warping

Dynamic time warping is a procedure whereby portions of a temporal sequence of values are stretched or shrunk to make it similar to another sequence. This procedure can be used to align the brain-stem auditory evoked potentials recorded from different subjects prior to averaging. The resultant warp-average more closely resembles the wave form of a typical subject than the conventional average. Dynamic time warping can also be used to compare one brain-stem auditory evoked potential to another. This comparison can show the differences that result from changes in a stimulus parameter such as intensity or repetition rate. When a patient's wave form is compared to a normal template, warping can identify the peaks in the patient's wave form that correspond most closely to the peaks in the normal template. Compared to an experienced human interpreter, warping is very accurate in identifying the waves of normal brain-stem auditory evoked potentials (error rate between 0 and 4%) and reasonably accurate in identifying the peaks in abnormal wave forms (error rate between 3 and 18%).     

Morphological measurement of the SEP using a dynamic time warping algorithm

A dynamic time warping technique was created to align the components of digitally high-pass (300 Hz–2500 Hz) filtered somatosensory evoked potentials evoked by median nerve stimulation recorded with a bipolar cephalic montage. A cost function was assigned related to the amount of warping necessary to match a standard wave derived from 24 normal subjects. Its value ranged from 0.525 to 2.456 (mean 1.305±0.501). This contrasted with a mean of 5.089±4.277 (range 0.701–13.972) derived from 25 patients with definite (n = 24) or possible (n = 1) multiple sclerosis chosen on the basis of having few or no clinical abnormalities at the time of testing. Fourteen (56%) of the patients had cost functions that were 3 or more S.D.s above the normal mean as compared to 3 (12%) having prolonged latency of the N19 peak. When used in combination, the cost function and latency yielded 60% abnormalities; 5 times higher than latency measurement alone.     

Finding the one: optimal choosiness under sequential mate choice

When mates are encountered sequentially, each encounter involves a decision whether to reject the current suitor and risk not finding a better mate, or to accept them despite their flaws. I provide a flexible framework for modelling optimal choosiness when mate encounters occur unpredictably in time. The model allows for temporal variation in the fitness benefits of mating, including seasonal breeding conditions, accrual of mate search costs, survival of the choosing individual or senescence of gametes. The basic optimality framework can be applied iteratively to obtain mate choice equilibria in dynamically evolving populations. My model predicts that individuals should be choosier when the average rate of mate encounters is high, but that choosiness should decline over time as the likelihood of future mate encounters decreases. When mate encounters are uncertain, there is a trade‐off between reproductive timing and mate choice (the ‘when’ and the ‘who’). Mate choice may be selected against when reproductive timing is highly important (e.g. when breeding conditions show a narrow peak in time). This can even lead to step‐shaped mate choice functions, where individuals abruptly switch from rejecting to accepting all suitors as peak breeding conditions approach. The model contributes to our understanding of why individuals may not express mate preferences, even when there is substantial variation in mate quality.     

Individual-based models for stage structured populations: formulation of “no regression” development equations

Individual-based models describe the growth dynamics of a population by performing numerical simulations of the life histories of its individuals. The life of an individual is determined by the basic processes of development, reproduction and mortality. In this paper the model equations for the development process are stochastic difference equations with discrete time and describe the time evolution of the status of an individual, in terms of a physiological age. We address the formulation of development models, when “regression” effects (defined as negative development) on the status of an individual are forbidden; this is a natural assumption when the physiological age is defined in terms of an abstract non-decreasing indicator measuring the maturity or the percentage of development. Different stochastic models of the development process are presented, and their behaviours are analyzed by varying the stochasticity level, which takes into account the degree of intraspecific variability. Moreover, remarks on the choice of the time step are reported.     

Time-synchronized clustering of gene expression trajectories

Current clustering methods are routinely applied to gene expressiontime course data to find genes with similar activation patternsand ultimately to understand the dynamics of biological processes.As the dynamic unfolding of a biological process often involvesthe activation of genes at different rates, successful clusteringin this context requires dealing with varying time and shapepatterns simultaneously. This motivates the combination of anovel pairwise warping with a suitable clustering method todiscover expression shape clusters. We develop a novel clusteringmethod that combines an initial pairwise curve alignment toadjust for time variation within likely clusters. The cluster-specifictime synchronization method shows excellent performance overstandard clustering methods in terms of cluster quality measuresin simulations and for yeast and human fibroblast data sets.In the yeast example, the discovered clusters have high concordancewith the known biological processes.     

An extension of the Cormack-Jolly-Seber model for continuous covariates with application to Microtus pennsylvanicus

Recent developments in the Cormack-Jolly-Seber (CJS) model for analyzing capture-recapture data have focused on allowing the capture and survival rates to vary between individuals. Several methods have been developed in which capture and survival are functions of auxiliary variables that may be discrete, constant over time, or apply to the population as a whole, but the problem has not been solved for continuous covariates that vary with both time and individual. This article proposes a new method to handle such covariates by modeling changes over time via a diffusion process and using logistic functions to link the variable to the CJS capture and survival rates. Bayesian methods are used to estimate the model parameters. The method is applied to study the effect of body mass on the survival of the North American meadow vole, Microtus pennsylvanicus.     

Morphing the feature-based multi-blocks of normative/healthy vertebral geometries to scoliosis vertebral geometries: development of personalized finite element models

Personalized Finite Element (FE) models and hexahedral elements are preferred for biomechanical investigations. Feature-based multi-block methods are used to develop anatomically accurate personalized FE models with hexahedral mesh. It is tedious to manually construct multi-blocks for large number of geometries on an individual basis to develop personalized FE models. Mesh-morphing method mitigates the aforementioned tediousness in meshing personalized geometries every time, but leads to element warping and loss of geometrical data. Such issues increase in magnitude when normative spine FE model is morphed to scoliosis-affected spinal geometry. The only way to bypass the issue of hex-mesh distortion or loss of geometry as a result of morphing is to rely on manually constructing the multi-blocks for scoliosis-affected spine geometry of each individual, which is time intensive. A method to semi-automate the construction of multi-blocks on the geometry of scoliosis vertebrae from the existing multi-blocks of normative vertebrae is demonstrated in this paper. High-quality hexahedral elements were generated on the scoliosis vertebrae from the morphed multi-blocks of normative vertebrae. Time taken was 3 months to construct the multi-blocks for normative spine and less than a day for scoliosis. Efforts taken to construct multi-blocks on personalized scoliosis spinal geometries are significantly reduced by morphing existing multi-blocks.     

Estimating summary statistics in the spike-train space

Estimating sample averages and sample variability is important in analyzing neural spike trains data in computational neuroscience. Current approaches have focused on advancing the use of parametric or semiparametric probability models of the underlying stochastic process, where the probabilistic distribution is characterized at each time point with basic statistics such as mean and variance. To directly capture and analyze the average and variability in the observation space of the spike trains, we focus on a data-driven approach where statistics are defined and computed in a function space in which the spike trains are viewed as individual points. Based on the definition of a “Euclidean” metric, a recent paper introduced the notion of the mean of a set of spike trains and developed an efficient algorithm to compute it under some restrictive conditions. Here we extend this study by: (1) developing a novel algorithm for mean computation that is quite general, and (2) introducing a notion of covariance of a set of spike trains. Specifically, we estimate the covariance matrix using the geometry of the warping functions that map the mean spike train to each of the spike trains in the dataset. Results from simulations as well as a neural recording in primate motor cortex indicate that the proposed mean and covariance successfully capture the observed variability in spike trains. In addition, a “Gaussian-type” probability model (defined using the estimated mean and covariance) reasonably characterizes the distribution of the spike trains and achieves a desirable performance in the classification of the spike trains.     

Breeding status influences timing but not duration of moult in the Northern Fulmar Fulmarus glacialis

Seabirds are key marine top predator species that are often used as indicators of the environmental quality of the oceans. Their breeding phenology has been studied extensively, but their pelagic habits mean less is known about the phenology of other events during the non-breeding period. Here, we used miniaturized saltwater immersion light-based geolocators (GLS) to investigate moult phenology in individuals with known breeding histories in a population of Northern Fulmar Fulmarus glacialis in Orkney, Scotland. As seabirds spend more time on the water during moult, moulting periods can be identified from patterns of variation in the amount of time that birds are in contact with saltwater. Estimates of daily variation in this behaviour during the non-breeding period were based upon wet/dry sensors and then modelled to characterize the timing of the moult. Light-based geolocation provided information on the areas used by each individual during its moult period. Inter-individual variability in moult timing was investigated in relation to sex and breeding success in the previous summer. We found a sex difference in the location of the moult, but not in its timing. However, the timing of moult did differ between individuals that had succeeded or failed in their previous breeding attempt, with successful breeders moulting the latest. In contrast, the duration of moult did not depend on prior reproductive success, but there was evidence of inter-annual variation in moult duration. GLS studies have provided a step change in our understanding of the at-sea distribution of pelagic seabirds. Our work highlights how activity data from these devices can add value to such studies by identifying key phases of the annual cycle, and locations at these times, when seabirds may be at particular risk. Furthermore, our findings indicate that individual and inter-annual variation in breeding success may influence phenological patterns in other phases of the Northern Fulmar annual cycle.