from time series to linear system part i finite dimensional linear time invariant systems abstract in the first part of this paper the definition of a dynamical system as simply consisting of a family of time series will be developed in this context the notions of linearity time invariance and finite dimensionality will be introduced it will be shown that a given family of time series may be represented by a system of ar equations riw t l rl 1w t l 1 r0w t 0 or equivalently by a finite dimensional linear time invariant system x t 1 ax t bu t y t cx t du t w u y if and only if this family is linear shift invariant and complete or as is equivalent closed in the topology of pointwise convergence this yields a very high level and elegant set of axioms which characterize these familiar objects it is emphasized however that no a priori choice is made as to which components of w are inputs and which are outputs such a separation always exists in any specific linear time invariant model starting from these definitions the structural indices of such systems are introduced and it is shown how an ar representation of a system having a given behaviour can be constructed these results will be used in a modelling context in part ii of the paper

a new l curve for ill posed problems the truncated singular value decomposition is a popular method for the solution of linear ill posed problems the method requires the choice of a truncation index which affects the quality of the computed approximate solution this paper proposes that an l curve which is determined by how well the given data right hand side can be approximated by a linear combination of the first few left singular vectors or functions be used as an aid for determining the truncation index

getting a stochastic process from a conservative lagrangian a first approach abstract the transition probability p v for a stochastic process generated by a conservative lagrangian l l 0 e v is obtained at first order from a perturbation series found using a path integral this p v corresponds to the transition probability for a random walk with a probability density given by the sum of a normal distribution and a perturbation which may be understood as the contribution of the interaction of the random walk with the external field it is also found that the moment generating function for p v can be expressed as the generating function of a normal distribution modified by a perturbation applications of these results to a linear potential a harmonic oscillator potential and an exponentially decaying potential are shown

linear systems and taylor complexes a taylor complex is a graded collection of polynomials satisfying a few simple axioms we show that there is a natural one to one correspondence between the solution sets of linear constant coefficient differential equations and taylor complexes using this correspondence we then give a characterization of the solution sets of linear constant coefficient differential equations which follows closely willems characterization of the solution sets of linear constant coefficient difference equations we also characterize linear differential operators the presentation is given for the case of several variables

statistics of camera based single particle tracking camera based single particle tracking enables quantitative determination of transport properties and provides nanoscale information about material characteristics such as viscosity and elasticity however static localization noise and the blurring of a particle s position over camera integration times introduce artifacts into measurement results even for a particle executing simple diffusion common data analysis methods based on the mean square displacement do not properly account for these effects in this paper we analyze the statistics of tracking data for freely diffusing particles in realistic experimental scenarios we derive a convenient and asymptotically optimal maximum likelihood estimator for the diffusion coefficient and for the magnitude of localization noise together with the corresponding fisher information which bounds the performance of all unbiased estimators we find that the effect of varying the illumination profile during the camera integration time is quantified by a motion blur coefficient r we also find that a double pulse illumination sequence maximizes the information content in some common experimental scenarios our results provide a rigorous theoretical framework and practical experimental recipe for achieving optimal performance in camera based single particle tracking

relative completeness and specifiedness properties of continuous linear dynamical systems the properties of relative completeness and specifiedness are introduced for continuous linear dynamical systems it is shown then that having these two properties and the differentiation invariance property is necessary and sufficient for a continuous linear dynamical system to be represented by means of a linear constant coefficient differential equation

optimal parameters for anomalous diffusion exponent estimation from noisy data the most common way of estimating the anomalous diffusion exponent from single particle trajectories consists in a linear fitting of the dependence of the time averaged mean square displacement on the lag time at the log log scale however various measurement noises that are unavoidably present in experimental data can strongly deteriorate the quality of this estimation procedure and bias the estimated exponent to investigate the impact of noises and to improve the estimation quality we compare three approaches for estimating the anomalous diffusion exponent and check their efficiency on fractional brownian motion corrupted by gaussian noise we discuss how the parameters of both the anomalous diffusion model and the estimation techniques influence the estimated exponent we show that the conventional linear fitting is the least optimal method for the analysis of noisy data

mean square displacement analysis of single particle trajectories with localization error brownian motion in an isotropic medium we examine the capability of mean square displacement msd analysis to extract reliable values of the diffusion coefficient d of a single particle undergoing brownian motion in an isotropic medium in the presence of localization uncertainty the theoretical results supported by simulations show that a simple unweighted least squares fit of the msd curve can provide the best estimate of d provided an optimal number of msd points are used for the fit we discuss the practical implications of these results for data analysis in single particle tracking experiments

optimizing experimental parameters for tracking of diffusing particles we describe how a single particle tracking experiment should be designed in order for its recorded trajectories to contain the most information about a tracked particle s diffusion coefficient the precision of estimators for the diffusion coefficient is affected by motion blur limited photon statistics and the length of recorded time series we demonstrate for a particle undergoing free diffusion that precision is negligibly affected by motion blur in typical experiments while optimizing photon counts and the number of recorded frames is the key to precision building on these results we describe for a wide range of experimental scenarios how to choose experimental parameters in order to optimize the precision generally one should choose quantity over quality experiments should be designed to maximize the number of frames recorded in a time series even if this means lower information content in individual frames

improved estimation of anomalous diffusion exponents in single particle tracking experiments the mean square displacement is a central tool in the analysis of single particle tracking experiments shedding light on various biophysical phenomena frequently parameters are extracted by performing time averages on single particle trajectories followed by ensemble averaging this procedure however suffers from two systematic errors when applied to particles that perform anomalous diffusion the first is significant at short time lags and is induced by measurement errors the second arises from the natural heterogeneity in biophysical systems we show how to estimate and correct these two errors and improve the estimation of the anomalous parameters for the whole particle distribution as a consequence we manage to characterize ensembles of heterogeneous particles even for rather short and noisy measurements where regular time averaged mean square displacement analysis fails we apply this method to both simulations and in vivo measurements of telomere diffusion in 3t3 mouse embryonic fibroblast cells the motion of telomeres is found to be subdiffusive with an average exponent constant in time individual telomere exponents are normally distributed around the average exponent the proposed methodology has the potential to improve experimental accuracy while maintaining lower experimental costs and complexity

statistical analysis of particle trajectories in living cells recent advances in molecular biology and fluorescence microscopy imaging have made possible the inference of the dynamics of molecules in living cells such inference allows us to understand and determine the organization and function of the cell the trajectories of particles e g biomolecules in living cells computed with the help of object tracking methods can be modeled with diffusion processes three types of diffusion are considered i free diffusion ii subdiffusion and iii superdiffusion the mean square displacement msd is generally used to discriminate the three types of particle dynamics we propose here a nonparametric three decision test as an alternative to the msd method the rejection of the null hypothesis i e free diffusion is accompanied by claims of the direction of the alternative subdiffusion or superdiffusion we study the asymptotic behavior of the test statistic under the null hypothesis and under parametric alternatives which are currently considered in the biophysics literature in addition we adapt the multiple testing procedure of benjamini and hochberg to fit with the three decision test setting in order to apply the test procedure to a collection of independent trajectories the performance of our procedure is much better than the msd method as confirmed by monte carlo experiments the method is demonstrated on real data sets corresponding to protein dynamics observed in fluorescence microscopy

statistical and sampling issues when using multiple particle tracking video microscopy can be used to simultaneously track several microparticles embedded in a complex material the trajectories are used to extract a sample of displacements at random locations in the material from this sample averaged quantities characterizing the dynamics of the probes are calculated to evaluate structural and or mechanical properties of the assessed material however the sampling of measured displacements in heterogeneous systems is singular because the volume of observation with video microscopy is finite by carefully characterizing the sampling design in the experimental output of the multiple particle tracking technique we derive estimators for the mean and variance of the probes dynamics that are independent of the peculiar statistical characteristics we expose stringent tests of these estimators using simulated and experimental complex systems with a known heterogeneous structure up to a certain fundamental limitation which we characterize through a material degree of sampling by the embedded probe tracking these estimators can be applied to quantify the heterogeneity of a material providing an original and intelligible kind of information on complex fluid properties more generally we show that the precise assessment of the statistics in the multiple particle tracking output sample of observations is essential in order to provide accurate unbiased measurements

optimal estimation of drift and diffusion coefficients in the presence of static localization error we consider the inference of the drift velocity and the diffusion coefficient of a particle undergoing a directed random walk in the presence of static localization error a weighted least squares fit to mean square displacement msd data is used to infer the parameters of the assumed drift diffusion model for experiments which cannot be repeated we show that the quality of the inferred parameters depends on the number of msd points used in the fitting an optimal number of fitting points popt is shown to exist which depends on the time interval between frames t and the unknown parameters we therefore also present a simple iterative algorithm which converges rapidly toward popt for repeatable experiments the quality depends crucially on the measurement time interval over which measurements are made reflecting the different timescales associated with drift and diffusion an optimal measurement time interval topt exists which depends on the number of measurement points and the unknown parameters and so again we present an iterative algorithm which converges quickly toward topt and is shown to be robust to initial parameter guesses

time averaged quadratic functionals of a gaussian process the characterization of a stochastic process from its single random realization is a challenging problem for most single particle tracking techniques which survey an individual trajectory of a tracer in a complex or viscoelastic medium we consider two quadratic functionals of the trajectory the time averaged mean square displacement msd and the time averaged squared root mean square displacement srms for a large class of stochastic processes governed by the generalized langevin equation with arbitrary frictional memory kernel and harmonic potential the exact formulas for the mean and covariance of these functionals are derived the formula for the mean value can be directly used for fitting experimental data e g in optical tweezers microrheology the formula for the variance and covariance allows one to estimate the intrinsic fluctuations of measured or simulated time averaged msd or srms for choosing the experimental setup appropriately we show that the time averaged srms has smaller fluctuations than the time averaged msd in spite of much broader applications of the latter one the theoretical results are successfully confirmed by monte carlo simulations of the langevin dynamics we conclude that the use of the time averaged srms would result in a more accurate statistical analysis of individual trajectories and more reliable interpretation of experimental data

characterization of linear differential systems in several variables abstract we show that a set of smooth trajectories is the solution set of a linear constant coefficient partial differential equation if and only if it is linear shift invariant and complete by completeness we mean exactly what willems called jet completeness in his automatica paper in 1986

optimal and suboptimal quadratic forms for noncentered gaussian processes individual random trajectories of stochastic processes are often analyzed by using quadratic forms such as time averaged ta mean square displacement msd or velocity auto correlation function vacf the appropriate quadratic form is expected to have a narrow probability distribution in order to reduce statistical uncertainty of a single measurement we consider the problem of finding the optimal quadratic form that minimizes a chosen cumulant moment e g the variance of the probability distribution under the constraint of fixed mean value for discrete noncentered gaussian processes we construct the optimal quadratic form by using the spectral representation of cumulant moments moreover we obtain a simple explicit formula for the smallest achievable cumulant moment that may serve as a quality benchmark for other quadratic forms we illustrate the optimality issues by comparing the optimal variance with the variances of the ta msd and ta vacf of fractional brownian motion superimposed with a constant drift and independent gaussian noise

brief paper axiomatic characterization of linear differential systems and operators in his ieee trans aut contr paper in 1991 willems posed the following question given a set of smooth trajectories when does there exist a linear constant coefficient differential operator whose kernel is precisely the given set we show that the properties that are necessary and sufficient for a set of smooth trajectories to admit a kernel representation are linearity time invariance jet closedness and jet determinedness it is interesting to note that the properties of jet closedness and jet determinedness were introduced by willems himself in his automatica paper in 1986

characterising solution sets of lti differential equations we prove that a set of smooth trajectories is ltid i e the solution set of a constant coefficient linear differential equation if and only if it is the direct sum of a linear time invariant closed controllable part and a linear time invariant finite dimensional part this characterisation does not directly involve derivatives in its formulation it solves a problem opened by willems 1991 we also characterise morphisms between ltid sets as linear time invariant maps which do not increase support

probability distribution of the time averaged mean square displacement of a gaussian process we study the probability distribution of the time averaged mean square displacement of a discrete gaussian process an empirical approximation for the probability density is suggested and numerically validated for fractional brownian motion the optimality of quadratic forms for inferring dynamical and microrheological quantities from individual random trajectories is discussed with emphasis on a reliable interpretation of single particle tracking experiments

when are linear differentiation invariant spaces differential it is shown that a linear differentiation invariant subspace of a c trajectory space is differential i e can be represented as the kernel of a linear constant coefficient differential operator if and only if its mcmillan degree is finite

estimation of the diffusion constant from intermittent trajectories with variable position uncertainties the movement of a particle described by brownian motion is quantified by a single parameter d the diffusion constant the estimation of d from a discrete sequence of noisy observations is a fundamental problem in biological single particle tracking experiments since it can provide information on the environment and or the state of the particle itself via the hydrodynamic radius here we present a method to estimate d that takes into account several effects that occur in practice important for the correct estimation of d and that have hitherto not been combined together for an estimation of d these effects are motion blur from the finite integration time of the camera intermittent trajectories and time dependent localization uncertainty our estimation procedure a maximum likelihood estimation with an information based confidence interval follows directly from the likelihood expression for a discretely observed brownian trajectory that explicitly includes these effects we begin with the formulation of the likelihood expression and then present three methods to find the exact solution each method has its own advantages in either computational robustness theoretical insight or the estimation of hidden variables the fisher information for this likelihood distribution is calculated and analyzed to show that localization uncertainties impose a lower bound on the estimation of d confidence intervals are established and then used to evaluate our estimator on simulated data with experimentally relevant camera effects to demonstrate the benefit of incorporating variable localization errors