is slow walking more stable several efforts have been made to study gait stability using measures derived from nonlinear timeseries analysis the maximum finite time lyapunov exponent lmax quantifies how a system responds to an infinitesimally small perturbation recent studies suggested that slow walking leads to lower lmax values and thus is more stable than fast walking but these studies suffer from methodological limitations we studied the effects of walking speed on the amount of kinematic variability and stability in human walking trunk motions of 15 healthy volunteers were recorded in 3d during 2 min of treadmill walking at different speeds from those time series maximum lyapunov exponents indicating short term and long term divergence ls stride and ll stride and mean standard deviation meansd were calculated ls stride showed a linear decrease with increasing speed for forward backward ap movements and quadratic effects inverted u shaped for medio lateral ml and up down vt movements ll stride showed a quadratic effect inverted u shaped of walking speed for ap movements a linear decrease for ml movements and a linear increase for vt movements moreover positive correlations between ls and meansd were found for all directions while ll stride and meansd were correlated negatively in the ap direction the different effects of walking speed on ls stride and ll stride for the different planes suggest that slow walking is not necessarily more stable than fast walking the absence of a consistent pattern of correlations between ll stride and meansd over the three directions suggests that variability and stability reflect at least to a degree different properties of the dynamics of walking

comparison of different state space definitions for local dynamic stability analyses abstract measures of local dynamic stability such as the local divergence exponent s quantify how quickly small perturbations deviate from an attractor that defines the motion when the governing equations of motion are unknown an attractor can be reconstructed by defining an appropriate state space however state space definitions are not unique and accepted methods for defining state spaces have not been established for biomechanical studies this study first determined how different state space definitions affected s for the lorenz attractor since exact theoretical values were known a priori values of s exhibited errors s decreased post fatigue for all 6 state spaces differences were statistically significant for 3 of these state spaces for state spaces defined using delay embedding increasing the embedding dimension decreased s in both the lorenz and experimental data overall our findings suggest that direct numerical comparisons between studies that use different state space definitions should be made with caution however trends across experimental comparisons appear to persist biomechanical state spaces constructed using positions and velocities or delay reconstruction of individual states are likely to provide consistent results

a semiparametric proportional odds regression model for the analysis of current status data abstract an approach to covariate analysis of current status data is described the method is applicable when the logit of the conditional probability of survival given the covariates is some increasing function of time plus a linear combination of the covariates the approach is based on approximating the infinite dimensional nuisance parameter the baseline log odds of failure with a step function and carrying out a maximum likelihood procedure the resulting finite dimensional parameter estimates for the regression parameters are shown to be asymptotically normal and semiparametric efficient numerical studies with small and moderate samples are discussed

methods for comparing cumulative hazard functions in a semi proportional hazard model graphical methods based on the analysis of differences between log cumulative hazard functions are considered for a two group semi proportional hazard model which allows for interaction between treatments and covariates confidence procedures and test statistics that can be used to test for interaction and for main effects are developed their use is illustrated by applying them to the analysis of kidney transplant data from the university of california san francisco

maximum likelihood estimation in the proportional odds model abstract we consider maximum likelihood estimation of the parameters in the proportional odds model with right censored data the estimator of the regression coefficient is shown to be asymptotically normal with efficient variance the maximum likelihood estimator of the unknown monotonic transformation of the survival time converges uniformly at a parametric rate to the true transformation estimates for the standard errors of the estimated regression coefficients are obtained by differentiation of the profile likelihood and are shown to be consistent a likelihood ratio test for the regression coefficient is also considered

on kubota s dirichlet series kubota 19 showed how the theory of eisenstein series on the higher metaplectic covers of sl2 which he discovered can be used to study the analytic properties of dirichlet series formed with n th order gauss sums in this paper we will prove a functional equation for such dirichlet series in the precise form required by the companion paper 2 closely related results are in eckhardt and patterson 10 the kubota dirichlet series are the entry point to a fascinating universe their residues for example are mysterious if n 3 though there is tantalizing evidence that these residues exhibit a rich structure that can only be partially glimpsed at this time when n 4 the residues are the fourier coefficients of the biquadratic theta function that were studied by suzuki 23 suzuki found that he could only determine some of the coefficients this failure to determine all the coefficients was explained in terms of the failure of uniqueness of whittaker models for the generalized theta series by deligne 9 and by kazhdan and patterson 15 on the other hand patterson 22 conjectured that the mysterious coefficients are essentially square roots of gauss sums evidence for patterson s conjecture is discussed in bump and hoffstein 6 and in eckhardt and patterson 10 where the conjecture is refined in light of numerical data partial proofs were given by suzuki in 24 and 25 another set of conjectures relevant to the mysterious coefficients of n th order theta functions were given by bump and hoffstein who considered theta functions on the n fold covers of glr for arbitrary r they are expressed as identities between rankin selberg convolutions of generalized theta series and whittaker coefficients of eisenstein series on the metaplectic group but they boil down to properties of the residues of kubota dirichlet series and their higher rank generalizations see bump and hoffstein 6 bump 4 and hoffstein 12 these conjectures are different from the patterson conjecture and there are other considerations which suggest that there may be further unproved relations beyond those described in the conjectures of patterson and bump and hoffstein

estimation and testing in a two sample generalized odds rate model abstract the generalized odds ratio for a survival variable t is defined as t t c c 1 1 sc t sc t for c 0 and log s t for c 0 where s t pr t t this ratio coincides with the integrated hazard for c 0 and the odds ratio for c 1 when the distribution of t depends on a covariate vector we assume that conditionally on the covariates log t t c is linear in the covariates this model is a generalization of the proportional hazard model phm which has an interpretation both as a phm with random nuisance effects clayton and cuzick 1986 and as a proportional odds rate model with the odds rate defined from the response times of series systems harrington and fleming 1982 and bickel 1986a considered rank tests for this semiparametric model clayton and cuzick 1986 considered estimation we use the odds rate representation to define a class of estimates of the proportionality parameter in the two sample case we show that the estimates are consistent and asymptotica

effects of external trunk loads on lumbar spine stability abstract stability of the lumbar spine is an important factor in determining spinal response to sudden loading using two different methods this study evaluated how various trunk load magnitudes and directions affect lumbar spine stability the first method was a quick release procedure in which effective trunk stiffness and stability were calculated from trunk kinematic response to a resisted force release the second method combined trunk muscle emg data with a biomechanical model to calculate lumbar spine stability twelve subjects were tested in trunk flexion extension and lateral bending under nine permutations of vertical and horizontal trunk loading the vertical load values were set at 0 20 and 40 of the subject s body weight bw the horizontal loads were 0 10 and 20 of bw effective spine stability as obtained from quick release experimentation increased significantly p

concurrent validity of a wearable imu for objective assessments of functional movement quality and control of the lumbar spine abstract inertial measurement units imus are being recognized in clinical and rehabilitation settings for their ability to assess movement related disorders of the spine for better guidance of treatment planning and tracking of recovery this study evaluated the mbientlab metamotionr imus relative to vicon motion capture equipment in measuring local dynamic stability of the spine quantified using maximum finite time lyapunov exponent max lumbopelvic coordination quantified using mean absolute relative phase marp and intersegmental motor variability quantified using deviation phase dp of lumbopelvic segments in 10 participants during 35 cycles of repetitive spine flexion extension fe intraclass correlations were strong between systems when using both the fe angle time series and the sum of squares ss time series to measure local dynamic stability 0 807 icc 2 1 m a x f e 0 919 0 738 icc 2 1 m a x s s 0 868 sagittal plane lumbopelvic coordination 0 961 icc 2 1 marp 0 963 and sagittal plane lumbopelvic variability 0 961 icc 2 1 dp 0 963 it was concluded that the metamotionr imus can be reliably used for measuring features associated with spine movement quality and motor control during a repetitive fe task future work will assess the reliability of sensor placement performance during multi directional movements and ability to discern clinical and healthy populations based on assessment of movement quality and control

sieve estimation for the proportional odds failure time regression model with interval censoring abstract estimation of the proportional odds failure time regression model with interval censoring is considered conditions that allow for positive information for the regression parameter are discussed the efficient score is characterized by a fredholm equation of the second kind the sieve maximum likelihood estimator for the finite dimensional regression parameter is shown to be asymptotically normal with n convergence rate and to achieve the information bound data analysis and simulations assist in clarifying our thoughts regarding the choice of sieve for finite sample problems

weyl group multiple dirichlet series ii the stable case to each reduced root system of rank r and each sufficiently large integer n we define a family of multiple dirichlet series in r complex variables whose group of functional equations is isomorphic to the weyl group of the coefficients in these dirichlet series exhibit a multiplicativity that reduces the specification of the coefficients to those that are powers of a single prime p for each p the number of nonzero such coefficients is equal to the order of the weyl group and each nonzero coefficient is a product of n th order gauss sums the root system plays a basic role in the combinatorics underlying the proof of the functional equations

on the implications of interpreting the stability index a spine example quantifying the stability of the spinal column offers a perspective on the effectiveness of the motor control strategy to ensure a stable spine and minimize the risk of injury from experiencing an unstable event there are essentially three energy based methods of calculating a stability index for the lumbar spine all three methods involve mathematical manipulation of an 18 x 18 hessian matrix the purpose of this paper was to consider the mathematical implications for the three methods of determining a single stability index and examine the effects of biological factors such as muscle activation in each of these methods the first approach computes the hessian s determinant and is thought to represent a more global or average perspective on stability a second approach computes the smallest eigenvalue of the hessian matrix to determine the weakest link of the spine the final method determines an average critical stiffness difference for the spine and is intended to effectively determines how far a human spine is from instability and allows comparison between tasks this study shows that the same interpretation of stability is achieved via all three computational approaches they agree as to whether the spine is stable or not however they appear to differ in their sensitivity to the effect of muscle activation patterns

a practical method for calculating largest lyapunov exponents from small data sets detecting the presence of chaos in a dynamical system is an important problem that is solved by measuring the largest lyapunov exponent lyapunov exponents quantify the exponential divergence of initially close state space trajectories and estimate the amount of chaos in a system we present a new method for calculating the largest lyapunov exponent from an experimental time series the method follows directly from the definition of the largest lyapunov exponent and is accurate because it takes advantage of all the available data we show that the algorithm is fast easy to implement and robust to changes in the following quantities embedding dimension size of data set reconstruction delay and noise level furthermore one may use the algorithm to calculate simultaneously the correlation dimension thus one sequence of computations will yield an estimate of both the level of chaos and the system complexity

a direct comparison of spine rotational stiffness and dynamic spine stability during repetitive lifting tasks stability of the spinal column is critical to bear loads allow movement and at the same time avoid injury and pain however there has been a debate in recent years as to how best to define and quantify spine stability with the outcome being that different methods are used without a clear understanding of how they relate to one another therefore the goal of the present study was to directly compare lumbar spine rotational stiffness calculated with an emg driven biomechanical model to local dynamic spine stability calculated using lyapunov analyses of kinematic data during a series of continuous dynamic lifting challenges twelve healthy male subjects performed 30 repetitive lifts under three varying load and three varying rate conditions with an increase in the load lifted constant rate there was a significant increase in mean maximum and minimum spine rotational stiffness po 0 001 and a significant increase in local dynamic stability po 0 05 both stability measures were moderately to strongly related to one another r 0 55 to 0 71 with an increase in lifting rate constant load there was also a significant increase in mean and maximum spine rotational stiffness po 0 01 however there was a non significant decrease in the minimum rotational stiffness and a non significant decrease in local dynamic stability p40 05 weak linear relationships were found for the varying rate conditions r 0 02 to 0 27 the results suggest that spine rotational stiffness and local dynamic stability are closely related to one another as they provided similar information when movement rate was controlled however based on the results from the changing lifting rate conditions it is evident that both models provide unique information and that future research is required to completely understand the relationship between the two models using both techniques concurrently may provide the best information regarding the true effects of in stability under different loading and movement scenarios and in comparing healthy and clinical populations

kinematic variability and local dynamic stability of upper body motions when walking at different speeds a ubiquitous characteristic of elderly and patients with gait disabilities is that they walk slower than healthy controls many clinicians assume these patients walk slower to improve their stability just as healthy people slow down when walking across ice however walking slower also leads to greater variability which is often assumed to imply deteriorated stability if this were true then slowing down would be completely antithetical to the goal of maintaining stability this study sought to resolve this paradox by directly quantifying the sensitivity of the locomotor system to local perturbations that are manifested as natural kinematic variability eleven young healthy subjects walked on a motorized treadmill at five different speeds three dimensional movements of a single marker placed over the first thoracic vertebra were recorded during continuous walking mean stride to stride standard deviations and maximum finite time lyapunov exponents were computed for each time series to quantify the variability and local dynamic stability respectively of these movements quadratic regression analyses of the dependent measures vs walking speed revealed highly significant u shaped trends for all three mean standard deviations but highly significant linear trends with significant or nearly significant quadratic terms for five of the six finite time lyapunov exponents subjects exhibited consistently better local dynamic stability at slower speeds for these five measures these results support the clinically based intuition that people who are at increased risk of falling walk slower to improve their stability even at the cost of increased variability

testing the proportional odds model under random censoring summary in practical applications it is not uncommon for the hazard functions of two groups to converge with time one approach that allows for converging hazard functions is the proportional odds model we develop a procedure for testing the proportional odds assumption when the available data consist of two independent random samples of ran domly right censored lifetimes asymptotic normality of the test statistic is proved and the procedure is applied to two well known datasets the effective significance level and power of the proposed test are assessed through a simulation study the objective of this paper is to develop a test for comparison of survival times under random censorship when the hazard functions for the two groups of interest are not proportional through time but their ratio appears to be tending to 1 such a situation is encountered for example when the initial effects of treatment or the differences between stages of the disease at diagnosis tend to diminish with time and the hazard rates of different groups of patients become more similar a decaying treatment effect was found in the analysis of university of california san francisco s kidney transplant data by dabrowska et al 1992 and in the analysis of malignant melanoma data by andersen et al 1993 ch 7 frequently converging hazard functions are modelled by extending cox s proportional hazards model to include time dependent explanatory variables or by introducing a frailty parameter we refer to andersen et al 1993 ch 7 9 hougaard 2000 ch 2 7 and klein moeschberger 2003 ch 9 13 for detailed discussion an alternative approach to the problem of converging hazard functions is through the pro portional odds model if fg t is the true survival function underlying the data for groupi i and fi t then qi t for i are the true odds of a group iindividual surviving beyond time t we say that the two groups satisfy a proportional odds model if b2 t for all t and some constant if aksk is the hazard

analysis of survival data by the proportional odds model a model is presented for the analysis of lifetime data in which the rates of mortality for separate groups of patients converge with time a non parametric estimate is given for the survivor function the theoretical basis for the model assumes that prognostic factors have a multiplicative effect on the odds against survival beyond any given time the model is fitted to data using maximum likelihood estimation and an example of its use in the analysis of a lung cancer trial is given

the effect of unstable loading versus unstable support conditions on spine rotational stiffness and spine stability during repetitive lifting lumbar spine stability has been extensively researched due to its necessity to facilitate load bearing human movements and prevent structural injury the nature of certain human movement tasks are such that they are not equivalent in levels of task stability i e the stability of the external environment the goal of the current study was to compare the effects of dynamic lift instability administered through both the load and base of support on the dynamic stability maximal lyapunov exponents and stiffness emg driven model of the lumbar spine during repeated sagittal lifts fifteen healthy males performed 23 repetitive lifts with varying conditions of instability at the loading and support interfaces an increase in spine rotational stiffness occurred during unstable support scenarios resulting in an observed increase in mean and maximum euclidean norm spine rotational stiffness p 0 0011 significant stiffening effects were observed in unstable support conditions about all lumbar spine axes with the exception of lateral bend relative to a stable control lifting trial the addition of both an unstable load as well as an unstable support did not result in a significant change in the local dynamic stability of the lumbar spine p 0 5592 the results suggest that local dynamic stability of the lumbar spine represents a conserved measure actively controlled at least in part by trunk muscle stiffening effects it is evident therefore that local dynamic stability of the lumbar spine can be modulated effectively within a young healthy population however this may not be the case in a patient population

does the personal lift assist device affect the local dynamic stability of the spine during lifting abstract the personal lift assist device plad is an on body ergonomic aid that reduces low back physical demands through the restorative moment of an external spring element which possesses a mechanical advantage over the erector spinae although the plad has proven effective at reducing low back muscular demand spinal moments and localized muscular fatigue during laboratory and industrial tasks the effects of the device on the neuromuscular control of spinal stability during lifting have yet to be assessed thirty healthy subjects 15m 15f performed repetitive lifting for three minutes at a rate of 10 lifts per minute with and without the plad maximum finite time lyapunov exponents representing short term max s and long term max l divergence were calculated from the measured trunk kinematics to estimate the local dynamic stability of the lumbar spine using a mixed design repeated measures anova it was determined that wearing the plad did not significantly change max s np 0 335 p 0 321 p 0 225 but did significantly reduce max l np 0 0024 p 0 0011 p 0 014 2 0 197 there were no between subject effects of sex or significant interactions p 0 720 the present results indicated that max s was not statistically different between the device conditions but that the plad significantly reduced max l to a negative stable value this shows that subjects neuromuscular systems were able to respond to local perturbations more effectively when wearing the device reflecting a more stable control of spinal movements these findings are important when recommending the plad for long term industrial or clinical use