Estimation Of Statistical Moments Research Articles

Parameters of Spike Trains Observed in a Short Time Window

We study the estimation of statistical moments of interspike intervals based on observation of spike counts in many independent short time windows. This scenario corresponds to the situation in which a target neuron occurs. It receives information from many neurons and has to respond within a short time interval. The precision of the estimation procedures is examined. As the model for neuronal activity, two examples of stationary point processes are considered: renewal process and doubly stochastic Poisson process. Both moment and maximum likelihood estimators are investigated. Not only the mean but also the coefficient of variation is estimated. In accordance with our expectations, numerical studies confirm that the estimation of mean interspike interval is more reliable than the estimation of coefficient of variation. The error of estimation increases with increasing mean interspike interval, which is equivalent to decreasing the size of window (less events are observed in a window) and with decreasing the number of neurons (lower number of windows).

Multilevel design of experiments for statistical moment and probability calculation

A procedure for statistical moment estimation and reliability analysis using design of experiment (DOE) is proposed. A numerical method of finding the optimal levels and weights of DOE for statistical moment estimation is established and applied to three- and five-level cases. The four statistical moments of the system response function are then calculated from the full-factorial DOE, and the probability distribution of the system response function is obtained using the empirical distribution systems such as the Pearson system. The proposed method is tested through several examples and compared with other analysis methods, including the previous developments of a three-level full-factorial design. The results show that it relieves much of the difficulties met in the previous method and provides good accuracy compared to other methods for various input distributions.

A generalized dimension‐reduction method for multidimensional integration in stochastic mechanics

AbstractA new, generalized, multivariate dimension‐reduction method is presented for calculating statistical moments of the response of mechanical systems subject to uncertainties in loads, material properties, and geometry. The method involves an additive decomposition of an N‐dimensional response function into at most S‐dimensional functions, where S≪N; an approximation of response moments by moments of input random variables; and a moment‐based quadrature rule for numerical integration. A new theorem is presented, which provides a convenient means to represent the Taylor series up to a specific dimension without involving any partial derivatives. A complete proof of the theorem is given using two lemmas, also proved in this paper. The proposed method requires neither the calculation of partial derivatives of response, as in commonly used Taylor expansion/perturbation methods, nor the inversion of random matrices, as in the Neumann expansion method. Eight numerical examples involving elementary mathematical functions and solid‐mechanics problems illustrate the proposed method. Results indicate that the multivariate dimension‐reduction method generates convergent solutions and provides more accurate estimates of statistical moments or multidimensional integration than existing methods, such as first‐ and second‐order Taylor expansion methods, statistically equivalent solutions, quasi‐Monte Carlo simulation, and the fully symmetric interpolatory rule. While the accuracy of the dimension‐reduction method is comparable to that of the fourth‐order Neumann expansion method, a comparison of CPU time suggests that the former is computationally far more efficient than the latter. Copyright © 2004 John Wiley & Sons, Ltd.

A univariate dimension-reduction method for multi-dimensional integration in stochastic mechanics

This paper presents a new, univariate dimension-reduction method for calculating statistical moments of response of mechanical systems subject to uncertainties in loads, material properties, and geometry. The method involves an additive decomposition of a multi-dimensional response function into multiple one-dimensional functions, an approximation of response moments by moments of single random variables, and a moment-based quadrature rule for numerical integration. The resultant moment equations entail evaluating N number of one-dimensional integrals, which is substantially simpler and more efficient than performing one N-dimensional integration. The proposed method neither requires the calculation of partial derivatives of response, nor the inversion of random matrices, as compared with commonly used Taylor expansion/perturbation methods and Neumann expansion methods, respectively. Nine numerical examples involving elementary mathematical functions and solid-mechanics problems illustrate the proposed method. Results indicate that the univariate dimension-reduction method provides more accurate estimates of statistical moments or multidimensional integration than first- and second-order Taylor expansion methods, the second-order polynomial chaos expansion method, the second-order Neumann expansion method, statistically equivalent solutions, the quasi-Monte Carlo simulation, and the point estimate method. While the accuracy of the univariate dimension-reduction method is comparable to that of the fourth-order Neumann expansion, a comparison of CPU time suggests that the former is computationally far more efficient than the latter.

Systematic synthesis of parallel architectures for the computation of higher order cumulants

Fine granularity parallel architectures for the efficient estimation of higher order statistics (HOS) are systematically derived in this paper. A unified methodology for constructing locally recursive algorithms and space–time linear mapping operators that lead to highly pipelined architectures consisting of multiple, tightly coupled array stages is discussed first. Then a farm of processors is synthesized that consumes second and fourth order moment estimates to produce the fourth order cumulants. The unified array synthesis methodology allows for the characterization of all valid solutions and the derivation of closed-form expressions for the permissible linear scheduling functions thus facilitating the search for a design instance meeting the architect's specified objectives. Achieving minimum latency and an optimal space–time matching between the farm and the moments generator architecture (derived in [E.S. Manolakos, H.M. Stellakis, Systematic synthesis of parallel architectures for the real-time estimation of higher order statistical moments, Parallel Algorithms and Applications (to appear)]) were the two main specifications driving the synthesis. A linear array solution, that is simpler to interface with the moments generator at the expense of adding some control complexity is also derived. As a result, a two-stage integrated VLSI architecture, that may accept data samples from the host and compute in real-time all non-redundant moment and cumulant terms, up to the fourth order is now possible.

On Estimates of Second Statistical Moments for a Class of Fuzzy Random Vectors

On Estimates of Second Statistical Moments for a Class of Fuzzy Random Vectors

A simplified stochastic model for infiltration into a heterogeneous soil forced by random precipitation

A simplified stochastic infiltration model is presented, aimed at representing the rainfall-runoff transformation in the presence of heterogeneity in the soil and with precipitation as a random variable with complex temporal evolution. Such a model is based on a simple water mass balance of a surface soil layer, considered as a non-linear reservoir. The explicit inclusion of spatial heterogeneity allows the model to be used in sub-grid parametrizations at a variety of scales, from the distributed modelling of the hydrological response of small watersheds to the representation of surface mass fluxes in General Circulation Models. An approximate solution procedure is developed, which allows the estimation of statistical moments of the soil effective saturation and runoff inside discrete time steps where the hydraulic saturated conductivity and the rainfall intensity are taken as random variables with known probability density functions. As a first test of the proposed model, two different simulations, relative to two soils with different hydraulic conductivity distributions, are presented and discussed. A year long record of hourly averaged rainfall intensities, as measured by a tipping bucket gauge in Central Italy, is taken as the main input. The main finding is that the non-linear nature of the soil filter is such that, for random precipitation intensity, the coefficient of variation of the runoff is always higher than that of precipitation. Such a non-linear variability enhancement, due mainly to the threshold character of the soil mass balance equation, tends to be slightly dampened by the variability of the hydraulic saturated conductivity.

Estimation of Statistical Moments for Desferrioxamine and Its Iron and Aluminium Chelates: Contribution to Optimisation of Therapy in Uremic Patients

We investigated the best time of administration of desferrioxamine (DFO) with respect to the dialysis session, using the approach of the stochastic dynamic system, integrated with the classical pharmacokinetic models. In the 6 patients studied, the mean arrival times of DFO, aluminoxamine (AlO) and ferrioxamine (FO) were, respectively, 193, 1,350 and 126 min, the mean residence times were 1,048, infinite, 1,190 min, respectively. AlO serum levels reach steady state in a mean time of 7 h and 22 min and remain stable in the interdialytic period. FO achieves a peak at the end of DFO infusion and declines during the interdialytic period. DFO, AlO and FO persist a very long time in the body of the uremic patient, thus the dialysis session should be administered when AlO and FO reach steady state. With a dose of 5-10 mg/kg b.w. of DFO, we propose to start the dialysis 8-12 h after the infusion if the main purpose is to treat Al overload or 2-3 h after the infusion if the main purpose is the treatment of hemosiderosis.

Improved stability analysis of the response of a duffing oscillator under filtered white noise

The applicability of the equivalent linearisation method for the analysis of non-linear oscillators subjected to narrowband random excitation is discussed. It is shown that when jumps between competing response states occur, the choice of an appropriate value for the kurtosis of the response process is crucial for a reliable estimation of local statistical moments and for a more accurate stochastic stability analysis. Furthermore, a modification is proposed which takes into account the influence of certain non-linear terms in the variational equation for the cases of strong non-linearities. The theoretical predictions correlate well with the results based on extensive digital simulations.

Estimation of Statistical Moments of Spatial Field Averages for Soil Properties and Crop Yields

AbstractCrop yield variability often results from spatial variability of soil properties. The quantities of practical interest are the spatial averages of these properties over the entire field domain. In this study, correlation scales and the conditional and unconditional first two moments of field averages were estimated, by the maximum likelihood (ML) and Conditional Multi‐Variate Normal methods and numerical approach, for four soil properties, two soil variables, and two crop‐yield components of three field crops. The results show the significant contribution of the conditional approach in reducing the variances of the spatial average yield and soil properties by about an order of magnitude. It is also observed that the estimated conditional expectations of the spatial averages are practically identical to the unconditional one, although the maps of the conditional expectation over the field differ from the maps of the unconditional one. The integral scales of crop yields were found to be roughly estimated from the integral scales of the related soil properties. The use of conditional probability as a practical tool for reducing prediction uncertainty and the potential of conditional analysis to help make practical decisions are demonstrated. The principal practical conclusion is that, although unconditional analysis may be much simpler than the conditional approach, the latter may be more advantageous.

Estimation of statistical moments and steady-state volume of distribution for a drug given by intravenous infusion.

Although it is generally recognized that estimates of the area under the drug concentration vs. time curve (AUC) after a dose is rather insensitive to curve-fitting procedures, little is known about estimates of mean residence time (MRT) or volume of distribution at steady-state (Vss), both of which can be derived from area estimates. This question is of particular concern when a drug is given as a short-term constant rate i.v. infusion since the infusion phase is often ignored and blood sampling restricted to the postinfusion period. Two nonexperimental methods for approximating concentration data during infusion termed the linear approximation method and the imaginary bolus method were found to be useful under certain conditions. Both methods provide reasonable estimates of AUC and the area under the first moment of the drug concentration-time curve (AUMC) for drugs with a wide range of pharmacokinetic characteristics. The imaginary bolus method was found to be the better of the two for estimation of MRT and to be widely applicable for this purpose. Vss proved to be highly sensitive to the approximation methods; although the imaginary bolus method is superior to the linear approximation method for estimating Vss, it does not work particularly well for drugs with pronounced multicompartment characteristics. In general, accurate estimation of Vss of drugs given by short-term i.v. infusion requires that at least one drug concentration be determined during infusion.

Statistics of normal mode amplitudes in a random ocean. I. Theory

A statistical theory of acoustic propagation in a model random ocean, valid in the limit of low acoustic frequency, is presented. A random internal-wave model gives sound-speed fluctuations δc (r,z,t) about a deterministic profile ? (z). Using normal modes of ? (z) as a basis, the theory gives quantitative estimates of statistical moments of the mode amplitudes ψn(r,t), which are randomly coupled via δc. Invoking a quasistatic approximation, the theory treats time as a parameter. From any initial (r=0) distribution of modal powers ‖ψn‖2, the evolution of their averages to an equilibrium is predicted by ’’coupled power’’ equations. The theory makes similar predictions for average fluctuations of the modal powers about their means. In the equilibrium limit, the theory gives the full probability distribution of the ψn.