Continuous-time Estimation Research Articles

Optimal Advertising When Envisioning a Product-Harm Crisis

How should forward-looking managers plan advertising if they envision a product-harm crisis in the future? To address this question, we propose a dynamic model of brand advertising in which, at each instant, a nonzero probability exists for the occurrence of a crisis event that damages the brand's baseline sales and may enhance or erode marketing effectiveness when the crisis occurs. Because managers do not know when the crisis will occur, its random time of occurrence induces a stochastic control problem, which we solve analytically in closed form. More importantly, the envisioning of a possible crisis alters managers' rate of time preference: anticipation enhances impatience. That is, forward-looking managers discount the present—even when the crisis has not occurred—more than they would in the absence of crisis. Building on this insight, we then derive the optimal feedback advertising strategies and assess the effects of crisis likelihood and damage rate. We discover the crossover interaction: the optimal precrisis advertising decreases, but the postcrisis advertising increases as the crisis likelihood (or damage rate) increases. In addition, we develop a new continuous-time estimation method to simultaneously estimate sales dynamics and feedback strategies using discrete-time data. Applying the method to market data from the Ford Explorer's rollover recall, we furnish evidence to support the proposed model. We detect compensatory effects in parametric shift: ad effectiveness increases, but carryover effect decreases (or vice versa). We also characterize the crisis occurrence distribution that shows that Ford Explorer should anticipate a crisis in 2.1 years and within 6.3 years at the 95% confidence level. Finally, we find a remarkable correspondence between the observed and optimal advertising decisions.

The macrodynamics of financial fragility within a hard peg arrangement

The paper presents an open-economy macrodynamical growth model with the aim of giving an endogenous characterisation to the process that leads a small country with a currency-board arrangement to accumulate dangerously high levels of external debt and become vulnerable to macroeconomic instability. The macrodynamics of the model results from the combination of the commitment to maintain the peg – that makes liquidity closely dependent on the dynamics of foreign reserves – and the non-linear real and financial interactions that drives the pro-cyclical behaviour of the economy. Within this context, the external financing ease during an economic upswing leads to debt-supported growth and financial fragility; the consequent deterioration of profitability expectations brings about a capital reversal that, in the absence of monetary stabilisation tools, makes the currency arrangement unsustainable. A financial crisis may thus turn into a currency crisis. We run a continuous-time estimation of a non-linear differential equations system for Argentina during the years of the currency-board arrangement. We find that two steady-state solutions exist. The local stability and sensitivity analysis show that both equilibria are unstable and that the qualitative nature of the equilibria depends in particular on lenders' responsiveness to the degree of leverage. We also show that relaxing the assumption on the currency arrangement and allowing for an autonomous monetary policy makes both equilibria stable.

Frequency-domain weighted non-linear least-squares estimation of continuous-time, time-varying systems

A frequency-domain least-squares estimator is presented for identifying linear, continuous-time, time-varying dynamical systems. The model considered is a linear, ordinary differential equation whose coefficients vary as polynomials in time. A frequency-domain approach is used, thus allowing the user to determine easily the frequency band(s) of interest. It is shown that the bias errors because of windowing and sampling the continuous-time signals can be modelled by a polynomial function of the frequency. The regression matrices of the estimators are shown to be very efficiently computed using the fast Fourier transform algorithm and its inverse. The total least-squares, generalised total least-squares and weighted, non-linear least-squares estimators are constructed. The latter two are shown to be consistent. The estimators are illustrated on simulation and measurement data.

Improved estimator of the continuous-time kernel estimator

There have been many papers presenting kernel density estimators for a strictly stationary continuous time process observed over the time interval [0, T]. However the estimators do not satisfy the property of mean-square continuity if the process is mean-square continuous. In this paper we present a modified kernel estimator and substantiate that the modified estimator satisfies the property of mean-square continuity. In a simulation study the results show the modified estimator is better than the original estimator in some cases.

Estimating the Continuous-Time Dynamics of Energy and Fat Metabolism in Mice

The mouse has become the most popular organism for investigating molecular mechanisms of body weight regulation. But understanding the physiological context by which a molecule exerts its effect on body weight requires knowledge of energy intake, energy expenditure, and fuel selection. Furthermore, measurements of these variables made at an isolated time point cannot explain why body weight has its present value since body weight is determined by the past history of energy and macronutrient imbalance. While food intake and body weight changes can be frequently measured over several weeks (the relevant time scale for mice), correspondingly frequent measurements of energy expenditure and fuel selection are not currently feasible. To address this issue, we developed a mathematical method based on the law of energy conservation that uses the measured time course of body weight and food intake to estimate the underlying continuous-time dynamics of energy output and net fat oxidation. We applied our methodology to male C57BL/6 mice consuming various ad libitum diets during weight gain and loss over several weeks and present the first continuous-time estimates of energy output and net fat oxidation rates underlying the observed body composition changes. We show that transient energy and fat imbalances in the first several days following a diet switch can account for a significant fraction of the total body weight change. We also discovered a time-invariant curve relating body fat and fat-free masses in male C57BL/6 mice, and the shape of this curve determines how diet, fuel selection, and body composition are interrelated.

Fault Detection Schemes for Continuous-Time Stochastic Dynamical Systems

In this paper the fault detection (FD) task in stochastic continuous-time dynamical systems is addressed. A new family of FD approaches is presented, which is based on the application of hypothesis testing on continuous-time estimators. The given FD schemes are widely analyzed in the framework of their characteristics, such as fault detectability, false alarms and missed detection. A collection of sufficient detectability conditions are given for a class of faults (referred here as generic), characterizing the faults which can be detected with certain formalized guarantee by the given FD schemes, and providing also an upper bound for the detection time in a probabilistic sense. The application and comparative performance of these FD approaches is illustrated for different faults in a simulation example.

Discrete-Time Estimation of Second-Order Statistics of Generalized Almost-Cyclostationary Processes

In this paper, the problem of estimating second-order statistics of continuous-time generalized almost-cyclostationary (GACS) processes is addressed. GACS processes in the wide sense have autocorrelation function almost-periodic in time whose generalized Fourier series expansion has both frequencies and coefficients that depend on the lag shifts. Almost-cyclostationary (ACS) processes are obtained as a special case when the frequencies do not depend on the lag shifts. ACS processes filtered by Doppler channels and communications signals with time-varying parameters are further examples. It is shown that continuous-time GACS processes do not have a discrete-time counterpart. The discrete-time cyclic cross-correlogram of the discrete-time ACS processes obtained by uniformly sampling GACS processes is considered as estimator of samples of the continuous-time cyclic cross-correlation function. The asymptotic performance analysis is carried out by resorting to the hybrid cyclic cross-correlogram which is partially continuous-time and partially discrete-time. Its mean-square consistency and asymptotic complex Normality as the number of data-samples approaches infinity and the sampling period approaches zero are proved under mild conditions on the regularity of the Fourier series coefficients and the finite or practically finite memory of the processes expressed in terms of summability of cumulants. It is shown that the asymptotic properties of the hybrid cyclic cross-correlogram are coincident with those of the continuous-time cyclic cross-correlogram. Hence, discrete-time estimation does not give rise to any loss in asymptotic performance with respect to continuous-time estimation.

The refined instrumental variable method

This paper describes the unified Refined Instrumental Variable approach to the time domain identification and estimation of both discrete-time (RIV) and continuous-time (RIVC) transfer function models. It demonstrates how this approach yields parameter estimates with optimal statistical properties for the Box-Jenkins and hybrid Box-Jenkins model forms on which the associated RIV and RIVC estimation algorithms are based. The performance of the algorithms, which can be implemented in en-bloc or recursive form, is evaluated by Monte-Carlo simulation analysis and their practical utility is illustrated by a number of practical environmental examples.

Regression Estimation and Prediction in Continuous Time

Regression Estimation and Prediction in Continuous Time

Analyzing Functional Connectivity Using a Network Likelihood Model of Ensemble Neural Spiking Activity

Analyzing the dependencies between spike trains is an important step in understanding how neurons work in concert to represent biological signals. Usually this is done for pairs of neurons at a time using correlation-based techniques. Chornoboy, Schramm, and Karr (1988) proposed maximum likelihood methods for the simultaneous analysis of multiple pair-wise interactions among an ensemble of neurons. One of these methods is an iterative, continuous-time estimation algorithm for a network likelihood model formulated in terms of multiplicative conditional intensity functions. We devised a discrete-time version of this algorithm that includes a new, efficient computational strategy, a principled method to compute starting values, and a principled stopping criterion. In an analysis of simulated neural spike trains from ensembles of interacting neurons, the algorithm recovered the correct connectivity matrices and interaction parameters. In the analysis of spike trains from an ensemble of rat hippocampal place cells, the algorithm identified a connectivity matrix and interaction parameters consistent with the pattern of conjoined firing predicted by the overlap of the neurons' spatial receptive fields. These results suggest that the network likelihood model can be an efficient tool for the analysis of ensemble spiking activity.

Identification and estimation of continuous-time, data-based mechanistic (DBM) models for environmental systems

Initially, the paper provides an introduction to the main aspects of existing time-domain methods for identifying linear continuous-time models from discrete-time data and shows how one of these methods has been applied to the identification and estimation of a model for the transportation and dispersion of a pollutant in a river. It then introduces a widely applicable class of new, nonlinear, State Dependent Parameter (SDP) models. Finally, the paper describes how this SDP approach has been used to identify, estimate and control a nonlinear differential equation model of global carbon cycle dynamics and global warming.

Vitesses optimale et suroptimale des polygones de fréquences pour les processus à temps continu

This Note deals with density estimation in continuous-time. Then under mild regularity and asymptotic independence conditions, the mean integrated square error achieves the same optimal rate T − 4 / 5 of convergence to zero as in the i.i.d. case. Under a local assumption weaker than Castellana–Leadbetter's [Stochastic Process. Appl. 21 (1986) 179–193], we obtain the parametric rate T −1 . To cite this article: F.-X. Lejeune, C. R. Acad. Sci. Paris, Ser. I 341 (2005).

Capture-Recapture Experiments with Time Heterogeneous Capture Probabilities in Continuous Time: A Counting Process Approach

A martingale estimator of the size of a closed population using captures at discrete capture occasions with time heterogeneous capture probabilities is extended to experiments in continuous time. The martingale approach to the analysis of counting processes is then exploited to derive the large sample properties of the estimator. Conditions are given under which the estimator arising in the limit of discrete time capture experiments has the same asymptotic distribution as the continuous time estimator.

Optimal sampling for density estimation in continuous time

Abstract. In the framework of nonparametric density estimation, first, we give optimal sampling schemes of continuous time processes. Next, we study effects of known or small errors‐in‐variables on such samplings. Throughout the paper, various simulations are also presented.

Comparing forecasting ability of parametric and non-parametric methods: an application with Canadian monthly interest rates

The primary objective of this article is to compare the forecasting ability of some recent parametric and non-parametric estimation methods by using monthly Canadian interest rate data between 1964:1–1999:1. The two-factor continuous time term structure model of Brennan and Schwartz was estimated where the first factor represents the short rate and the second factor the long rate using the continuous time estimation procedures developed by Bergstrom. The interest rates using the multivariate GARCH model developed by Engle and Kroner, and two non-parametric estimation methods namely, non-parametric kernel smoothing and the artificial neural networks was modelled. For the short-term rates, it has been found that, the Bergstrom's method and the artificial neural networks model have marginally better forecasting performance than that of the linear benchmark. For the long-term rates, none of the methods produced better forecasting precision than that of the benchmark.

Spatiotemporal reconstruction of list-mode PET data.

We describe a method for computing a continuous time estimate of tracer density using list-mode positron emission tomography data. The rate function in each voxel is modeled as an inhomogeneous Poisson process whose rate function can be represented using a cubic B-spline basis. The rate functions are estimated by maximizing the likelihood of the arrival times of detected photon pairs over the control vertices of the spline, modified by quadratic spatial and temporal smoothness penalties and a penalty term to enforce nonnegativity. Randoms rate functions are estimated by assuming independence between the spatial and temporal randoms distributions. Similarly, scatter rate functions are estimated by assuming spatiotemporal independence and that the temporal distribution of the scatter is proportional to the temporal distribution of the trues. A quantitative evaluation was performed using simulated data and the method is also demonstrated in a human study using 11C-raclopride.

Chaos and the exchange rate

Chaotic exchange rate models are structural models built in discrete time (difference equations), and show that with orthodox assumptions (PPP, interest parity, etc) and introducing plausible nonlinearities in the dynamic equations, it is possible to obtain a model capable of giving rise to chaotic motion. However, none of these models is estimated, and the conclusions are based on simulations: the empirical validity of these models is not tested. In this paper, a continuous time (the exchange rate is obviously a continuous variable) exchange rate model is built as a non-linear set of three differential equations and its theoretical properties (steady state, stability, etc,) analysed. The model is then econometrically estimated in continuous time with Italian data and examined for the possible presence of chaotic motion. This paper also shows that the continuous time estimation of economic models built as systems of nonlinear differential equations is a very powerful tool in the hands of the profession.

ALGORITHM OF PARAMETERS' IDENTIFICATION OF POLYHARMONIC FUNCTION

Abstracts: The problem of on-line continuous-time estimation of parameters of a poly-harmonic function is considered. The analytical conditions of existence of the solution are presented and the design procedure of the estimation algorithm is proposed. The presented results were obtained on the basis of the theory of adaptive systems and identification theory, mathematical theory of stability as well as approaches of the linear optimal control.

Application of time series in medical research

This article reviews the mortality data of chronic liver disease and cirrhosis, as well as tracheal, bronchial and lung cancers, in Hungary by methods of time series analysis. The methods of time series of death rates of chronic liver disease and cirrhosis as well as tracheal, bronchial and lung cancers and their reliability, are analysed from data available from WHO. The author used ARIMA models (autoregressive and integrated moving average models) and auto- and cross-correlation functions to study the substantial role an exogenous environmental factor has on incidences ofdeath. The confidence intervals of autoregressive (AR) coefficient are compared to the standard normal distribution, the estimation of White’s theory and the continuous time estimation model.On the basis of the analysis, it may be concluded that chronic liver disease and cirrhosis can be influenced by an exogenous environmental factor, however, this relation cannot be demonstrated for deaths due to tracheal, bronchial or lung cancers. In each case, the continuous estimation of the AR(1) coefficients give the best results.The paper demonstrates how the presented methods can be applied to agricultural science.

Continuous-time estimation of an endogenous growth model of an open economy

Most recent theoretical research on endogenous economic growth has been conducted with continuous-time models that embody the assumption of an intertemporally optimizing representative agent. Yet virtually none of these models has been confronted with empirical data. In this paper we make use of a model of an open economy recently developed by Turnovsky to demonstrate one approach to the estimation and analysis of such models. We present the results of the estimation, analyze the estimated model's stability properties around its steady-state solution, and explore the model's behavior in response to alternative tax policies.