Functional Data Analysis Techniques Research Articles

Day of week effects on diurnal ozone/NOx cycles and transportation emissions in Southern California

Diurnal cycles of ground-level ozone and its precursor NOx concentrations stem from and reflect complex temporal patterning of many underlying factors, including transportation emissions. Investigating the complexity of diurnal ozone/NOx cycles at a finer temporal resolution allows a better understanding of ozone dynamics and helps in designing ozone control strategies. This study applied functional data analysis techniques to hourly resolved ozone and NOx measurement data from the 1997 Southern California Ozone Study. Functional analysis of variance on diurnal ozone/NOx cycles for urban and rural monitoring sites confirmed, in a new continuous functional form, the ozone weekend effect. Functional data analysis also allows for a direct examination of day-of-week effects on ozone formation/destruction rates. Comparisons of Sunday ozone rates to those on weekdays demonstrate earlier, faster, and longer duration of ozone accumulation on Sunday. The results are further interpreted from the transportation emissions perspective using hourly resolved weigh-in-motion traffic data.

Functional data analysis of joint coordination in the development of vertical jump performance

Mastery of complex motor skills requires effective development of inter-segment coordination patterns. These coordination patterns can be described and quantified using various methods, including descriptive angle–angle diagrams, conjugate cross-correlations, vector coding, normalized root mean squared error techniques and, as in this study, functional data analysis procedures. Lower limb kinematic data were obtained for 49 children performing the vertical jump. Participants were assigned to developmental stages using the criteria of Gallahue and Ozmun (2002). Inter-segment joint coordination data consisting of pairs of joint angle–time data were smoothed using B-splines and the resulting bivariate functions were analysed using functional principal component analysis and stepwise discriminant analysis. The results of the analysis showed that the knee–hip joint coordination pattern was most effective at discriminating between developmental stages. The results provide support for the application of functional data analysis techniques in the analysis of joint coordination or time series type data.

Anticipatory motion in piano performance

Recent motor control studies of piano performance address the anticipatory movements of pianists fingers. The reported study investigated anticipatory motion under manipulations of performance rate or tempo. It was tested whether faster tempi require larger anticipatory preparation, as suggested in piano pedagogy literature. Sixteen skilled pianists repeatedly performed short isochronous melodies from memory at four different tempi (from 500 to 143 ms inter-onset intervals). A passive 3-D motion capture system tracked the movements of 40 markers on the hand, fingers, and the piano keys. The melodies manipulated both the repositioning of certain fingers (thumb and pinkie finger) on the keyboard and the distance between repeating finger movements (three or six tones). Functional data analysis techniques were applied to the analysis of motion trajectories. Data analyses are presented that indicate that the repositioning movement of a finger toward its next goal (keypress) begins sooner at faster tempi than at slower tempi in the x plane (sideways motion on the keyboard). Applications to piano practice will be discussed. [Work supported by the Austrian Science Fund, NSERC, and the Canada Research Chairs program.]

Application of functional data analysis techniques to study knee biomechanics in patients with degenerative arthritis before and after total knee replacement

Application of functional data analysis techniques to study knee biomechanics in patients with degenerative arthritis before and after total knee replacement

Cross-modal interactions in the perception of musical performance

We investigate the dynamics of sensory integration for perceiving musical performance, a complex natural behavior. Thirty musically trained participants saw, heard, or both saw and heard, performances by two clarinetists. All participants used a sliding potentiometer to make continuous judgments of tension (a measure correlated with emotional response) and continuous judgments of phrasing (a measure correlated with perceived musical structure) as performances were presented. The data analysis sought to reveal relations between the sensory modalities (vision and audition) and to quantify the effect of seeing the performances on participants' overall subjective experience of the music. In addition to traditional statistics, functional data analysis techniques were employed to analyze time-varying aspects of the data. The auditory and visual channels were found to convey similar experiences of phrasing but different experiences of tension through much of the performances. We found that visual information served both to augment and to reduce the experience of tension at different points in the musical piece (as revealed by functional linear modeling and functional significance testing). In addition, the musicians' movements served to extend the sense of phrasing, to cue the beginning of new phrases, to indicate musical interpretation, and to anticipate changes in emotional content. Evidence for an interaction effect suggests that there may exist an emergent quality when musical performances are both seen and heard. The investigation augments knowledge of human communicative processes spanning language and music, and involving multiple modalities of emotion and information transfer.

Forecasting age‐specific breast cancer mortality using functional data models

Accurate estimates of future age-specific incidence and mortality are critical for allocation of resources to breast cancer control programmes and evaluation of screening programmes. The purpose of this study is to apply functional data analysis techniques to model age-specific breast cancer mortality time trends, and forecast entire age-specific mortality functions using a state-space approach. We use annual unadjusted breast cancer mortality rates in Australia, from 1921 to 2001 in 5 year age groups (45 to 85+). We use functional data analysis techniques where mortality and incidence are modelled as curves with age as a functional covariate varying by time. Data are smoothed using non-parametric smoothing methods then decomposed (using principal components analysis) to estimate basis functions that represent the functional curve. Period effects from the fitted coefficients are forecast then multiplied by the basis functions, resulting in a forecast mortality curve with prediction intervals. To forecast, we adopt a state-space approach and an automatic modelling framework for selecting among exponential smoothing methods.Overall, breast cancer mortality rates in Australia remained relatively stable from 1960 to the late 1990s, but have declined over the last few years. A set of four basis functions minimized the mean integrated squared forecasting error and account for 99.3 per cent of variation around the mean mortality curve. Twenty year forecasts suggest a continuing decline, but at a slower rate, and stabilizing beyond 2010. Forecasts show a decline in all age groups with the greatest decline in older women. The proposed methods have the potential to incorporate important covariates such as hormone replacement therapy and interventions to represent mammographic screening. This would be particularly useful for evaluating the impact of screening on mortality and incidence from breast cancer.

Normalizing temporal patterns to analyze sit-to-stand movements by using registration of functional data

Functional data analysis techniques provide an alternative way of representing movement and movement variability as a function of time. In particular, the registration of functional data provides a local normalization of time functions. This normalization transforms a set of curves, records of repeated trials, yielding a new set of curves that only vary in terms of amplitude. Therefore, main events occur at the “same time” for all transformed curves and interesting features of individual recordings remain after averaging processes. This paper presents an application of the registration process to the analysis of the vertical forces exerted on the ground by both feet during the sit-to-stand movement. This movement is particularly interesting in functional evaluations related to balance control, lower extremity dysfunction or low-back pain.

Functional principal component analysis of fMRI data.

We describe a principal component analysis (PCA) method for functional magnetic resonance imaging (fMRI) data based on functional data analysis, an advanced nonparametric approach. The data delivered by the fMRI scans are viewed as continuous functions of time sampled at the interscan interval and subject to observational noise, and are used accordingly to estimate an image in which smooth functions replace the voxels. The techniques of functional data analysis are used to carry out PCA directly on these functions. We show that functional PCA is more effective than is its ordinary counterpart in recovering the signal of interest, even if limited or no prior knowledge of the form of hemodynamic function or the structure of the experimental design is specified. We discuss the rationale and advantages of the proposed approach relative to other exploratory methods, such as clustering or independent component analysis, as well as the differences from methods based on expanded design matrices.

Functional data analysis of the dynamics of the monthly index of nondurable goods production

Functional data analysis techniques pioneered by Ramsay and Bernard Silverman are used to analyze the dynamics of a monthly nonseasonally adjusted index of production. After determining the order (three) of the ordinary differential equation that describes the data and estimating its coefficients; the time variation of each vector of coefficients is examined in detail to determine the evolutionary dynamics of each series over a 70-year span. The dynamical properties of the solution paths are also analyzed for their qualitative dynamics. The resulting models are used to examine the interaction between seasonal dynamics, dynamics at business cycle frequencies, and long-term growth.

Modelling Human Gestures and Control Behaviour from Measured Data

Approaches to modelling human control behaviour and gestures from measurements are reviewed. Multiple-model and nonparametric statistical models can be used to model and recognise human gestures, and control behaviour. Functional Data Analysis (FDA) techniques with nonparametric models, and regularisation, can be used to explore the data. The applicability of the approach is demonstrated in a gesture recognition test based on complex hand movements.

Functional Components of Variation in Handwriting

Abstract Functional data analysis techniques are used to analyze a sample of handwriting in Chinese. The goals are (a) to identify a differential equation that satisfactorily models the data's dynamics, and (b) to use the model to classify handwriting samples taken from differential individuals. After preliminary smoothing and registration steps, a second-order linear differential equation, for which the forcing function is small, is found to provide a good reconstruction of the original script records. The equation is also able to capture a substantial amount of the variation in the scripts across replication. The cross-validated classification process is 100% effective for the samples analyzed.

Functional Components of Variation in Handwriting

Functional data analysis techniques are used to analyze a sample of handwriting in Chinese. The goals are (a) to identify a differential equation that satisfactorily models the data's dynamics, and (b) to use the model to classify handwriting samples taken from differential individuals. After preliminary smoothing and registration steps, a second-order linear differential equation, for which the forcing function is small, is found to provide a good reconstruction of the original script records. The equation is also able to capture a substantial amount of the variation in the scripts across replication. The cross-validated classification process is 100% effective for the samples analyzed.

Functional data analyses of lip motion.

The vocal tract's motion during speech is a complex patterning of the movement of many different articulators according to many different time functions. Understanding this myriad of gestures is important to a number of different disciplines including automatic speech recognition, speech and language pathologies, speech motor control, and experimental phonetics. Central issues are the accurate description of the shape of the vocal tract and determining how each articulator contributes to this shape. A problem facing all of these research areas is how to cope with the multivariate data from speech production experiments. In this paper techniques are described that provide useful tools for describing multivariate functional data such as the measurement of speech movements. The choice of data analysis procedures has been motivated by the need to partition the articulator movement in various ways: end effects separated from shape effects, partitioning of syllable effects, and the splitting of variation within an articulator site from variation from between sites. The techniques of functional data analysis seem admirably suited to the analyses of phenomena such as these. Familiar multivariate procedures such as analysis of variance and principal components analysis have their functional counterparts, and these reveal in a way more suited to the data the important sources of variation in lip motion. Finally, it is found that the analyses of acceleration were especially helpful in suggesting possible control mechanisms. The focus is on using these speech production data to understand the basic principles of coordination. However, it is believed that the tools will have a more general use.

Functional data analyses of lip motion

The vocal tract’s motion during speech is a complex patterning of the movement of many different articulators according to many different time functions. Central issues are the accurate description of the shape of the vocal tract and determining how each articulator contributes to this shape. Techniques are described that provide useful tools for describing multivariate functional data such as the measurement of speech movements. The choice of data analysis procedures has been motivated by the need to partition the articulator movement in various ways: movement start- and end-effects separated from shape effects, effects due to different syllables, and the splitting of within-sensor variation from between-sensor overall variation. The techniques of functional data analysis seem admirably suited to the analyses of phenomena such as these. Familiar multivariate procedures such as analysis of variance and principal components analysis have their functional counterparts, and these reveal in a way more suited to the data the important sources of variation in lip motion. Finally, it is found that the analyses of acceleration were especially revealing in considering the character of possible control mechanisms. [Work supported by NIH Grant DC-00594.]

A Functional Data Analysis of the Pinch Force of Human Fingers

SUMMARY The ability of the human thumb and forefinger to adapt the pinch force to the static and dynamic characteristics of the object being grasped is one of the marvels of human physiology. We analyse a sample of records of the force applied during a brief squeeze by functional data analysis techniques in which familiar statistical concepts are adapted to observations that are functional in character. Except for scale, a graph of these force impulses closely resembles a log-normal density function, and this has a plausible physiological rationale. Specially adapted smoothing spline approximations along with a functional version of principal components analysis reveal that the residual variation is essentially one dimensional in structure, and that the force functions can be described by a simple linear differential equation incorporating the effects of drag or viscosity in the joints and muscles involved.