Nonparametric Identification Research Articles

Kernel-based identification with frequency domain side-information

This paper discusses the problem of system identification when frequency domain side-information is available. We mainly consider the case where the side-information is provided as the H∞-norm of the system being bounded by a given scalar. This framework allows considering different forms of frequency domain side-information, such as the dissipativity of the system. We propose a nonparametric identification approach for estimating the impulse response of the system under the given side-information. The estimation problem is formulated as a constrained optimization in a stable reproducing kernel Hilbert space, where suitable constraints are considered for incorporating the desired frequency domain features. The resulting optimization has an infinite-dimensional feasible set with an infinite number of constraints. We show that this problem is a well-defined convex program with a unique solution. We propose a heuristic that tightly approximates this unique solution. The proposed approach is equivalent to solving a finite-dimensional convex quadratically constrained quadratic program. The efficiency of the discussed method is verified by several numerical examples.

Nonparametric identification of random coefficients in aggregate demand models for differentiated products

Summary This paper studies nonparametric identification in market-level demand models for differentiated products with heterogeneous consumers. We consider a general class of models that allows for the individual-specific coefficients to vary continuously across the population and give conditions under which the density of these coefficients, and hence also functionals such as the fractions of individuals who benefit from a counterfactual intervention, is identified.

Beyond Nyquist in Frequency Response Function Identification: Applied to Slow-Sampled Systems

Fast-sampled models are essential for control design, e.g., to address intersample behavior. The aim of this paper is to develop a non-parametric identification technique for fast-sampled models of systems that have relevant dynamics and actuation above the Nyquist frequency of the sensor, such as vision-in-the-loop systems. The developed method assumes smoothness of the frequency response function, which allows to disentangle aliased components through local models over multiple frequency bands. The method identifies fast-sampled models of slowly-sampled systems accurately in a single identification experiment. Finally, an experimental example demonstrates the effectiveness of the technique.

Non-parametric Identification of Posterior and Reasoning Errors in Isolated and Social Environments

Non-parametric Identification of Posterior and Reasoning Errors in Isolated and Social Environments

О некотором классе уравнений Вольтерра I рода в задаче идентификации линейной нестационарной динамической системы

This paper proposes an approach to the identification of a nonstationary linear dynamic system. Its input-output mathematical model is presented as a Volterra equation of the first kind. The problem of nonparametric identification of Volterra kernels is solved on the basis of an active experiment using test piecewise linear signals (that have a rising front). The problem statement is based on the conditions for modeling the dynamics of technical devices in the energy and power industry. The choice of an admissible family of input signals is driven by the complexity of generating piecewise-constant type signals for real energy objects. The original problem is reduced to solving Volterra integral equations of the first kind with two variable integration limits. A formula for the inversion of the integral equations under study is constructed. Sufficient conditions are obtained for the solvability of the corresponding equations with respect to Volterra kernels in the class of continuous functions.

Physics-informed sparse identification of bistable structures

The design of bistable structures is a hot topic in the last decade due to its wide application in smart actuators, energy harvesters, flexible robotics, etc. The characterization of the nonlinear restoring force of bistable structures plays a significant role in modeling and enhancing dynamic performance. However, the traditional nonparametric identification methods may have insufficient accuracy or even be invalid because of numerical differentiation procedures and static fitting. Besides, the modern data-driven sparse regression identification methods rely highly on the assumed nonlinear basis functions and lack interpretability. In this paper, a physics-informed sparse identification method is proposed for the nonlinear restoring force identification of bistable structures. The function of the nonlinear restoring force is physically informed by the derived equation of the Hilbert transform and parameter fitting. Furthermore, sparse identification is conducted based on the free vibration responses of the bistable vibrator. The numerical studies verify the effectiveness of the proposed algorithm under the noise level of 30 dB. Experimental measurement is conducted on a magnetic coupled bistable beam to perform the model identification. It has been demonstrated that the reconstructed dynamic response and nonlinear restoring force both keep in good agreement with the measured ones.

Time-varying unobserved heterogeneity in earnings shocks

This paper considers the transitory-permanent model for the earnings process, and allows for time-varying individual-specific unobserved heterogeneity in each shock. The cross-sectional heterogeneity in each shock is drawn from an unknown distribution at each time period. Sufficient conditions for the nonparametric identification of the cross-sectional density functions of the heterogeneity are provided, under different assumptions on the time series behavior of the transitory shock. The method proposed is then applied to earnings data to document a high degree of cross-sectional heterogeneity in each shock.

Nonparametric identification and estimation with discrete instruments and regressors

In a nonparametric instrumental regression model in which the regressor and instrument are discretely distributed, we strengthen the conventional moment independence assumption between instrument and residual to include higher order moments. We give conditions under which a function of interest is partially identified when the regressor has more mass points than the instrument. We also show that the function is point identified as long as certain latent parameters corresponding to conditional moments of the residual lie outside of a set of measure zero. We give an asymptotically normal estimator for the structural function when it is point identified. We also provide a straightforward method for inference under partial identification. These perform well in Monte-Carlo simulations and in two empirical applications.

Nonparametric identification and estimation of the extended Roy model

We propose a new identification method for the extended Roy model, in which the agents maximize their utility rather than just their outcome. We nonparametrically identify the joint distribution of potential outcomes, which is of great importance in causal inference. We exploit the extended Roy model structure and the monotonicity assumption but do not require any functional form assumption nor any support assumption. The identification is achieved by matching the indifferent agents across choices, who are identified by the local instrumental variable method. Based on the identification result, we propose an easy-to-implement nonparametric simulation-based estimator and derive its convergence rate. An empirical illustration on Malawian farmers’ hybrid maize adoption is provided.

Complete effect decomposition for an arbitrary number of multiple ordered mediators with time-varying confounders: A method for generalized causal multi-mediation analysis.

Causal mediation analysis is advantageous for mechanism investigation. In settings with multiple causally ordered mediators, path-specific effects have been introduced to specify the effects of certain combinations of mediators. However, most path-specific effects are unidentifiable. An interventional analog of path-specific effects is adapted to address the non-identifiability problem. Moreover, previous studies only focused on cases with two or three mediators due to the complexity of the mediation formula in a large number of mediators. In this study, we provide a generalized definition of traditional path-specific effects and interventional path-specific effects with a recursive formula, along with the required assumptions for nonparametric identification. Subsequently, a general approach is developed with an arbitrary number of multiple ordered mediators and with time-varying confounders. All methods and software proposed in this study contribute to comprehensively decomposing a causal effect confirmed by data science and help disentangling causal mechanisms in the presence of complicated causal structures among multiple mediators.

Problem of Modeling and Control of Multi-Dimensional Timeless Processes with Dependent Output Variables

In this paper, we consider the problems of modeling and controlling multidimensional inertialess systems with delay under nonparametric uncertainty. We are talking about the multidimensional processes which are in conditions where the form of parametric equations for various channels of the object is absent due to lack of a priori information. The main emphasis is placed on the case when the components of the vector of output variables are randomly connected in an unforeseen way. In the case of a stochastic dependence of the output variables, the mathematical description of the object is reduced to a system of implicit equations, the form of which is unknown. Therefore, the main task of modeling is to find the predicted values of output variables from known input variables. When managing a multidimensional object, an important feature is the definition of setting actions. The main thing here is that the setting influences of the control system should not be chosen arbitrarily from the corresponding areas, but depending on the definition of the previous ones. Nonparametric identification and control algorithms for multidimensional systems are proposed. Computational experiments that show the efficiency of using the proposed nonparametric identification and control algorithms are presented.

Nonparametric identification and estimation of all-pay auction and contest models

AbstractIn this paper, I study the nonparametric identification and estimation of multi-unit all-pay auctions of incomplete information. First, I consider the setting where multiple goods are allocated among several risk-neutral participants with independent private values (IPV). I prove the nonparametric identification of the model and derive two different consistent estimators of the distribution of bidder valuations. The first estimator is based on the classical structural approach similar to that of Guerre et al. (Econometrica 68(3):525–574, 2000). The second estimator, instead, allows estimation of the quantile function of the bidders’ valuations directly using the quantile density of the bids. Monte Carlo simulations show good small sample property under various assumptions of the number of players and goods. Next, I consider a variety of model extensions: the case of affiliated private values (APV), asymmetric players, the addition of random noise, as well as the case of risk-averse bidders. In contrast to all other scenarios, I prove that the general model with risk-averse bidders is not identified even in the semi-parametric case in which utility function is restricted to belong to the class of functions with constant absolute risk aversion (CARA). On the other hand, I show that the model with risk aversion can be identified if the distribution of valuations is restricted to having fixed support.

Identification of English auctions when losing entrants are not observed

A typical feature of English auctions and negotiated-price markets modeled as English auctions is that only the transaction price and identity of the winner are observed; auction entrants other than the winner are commonly not recorded in the data. Meanwhile, existing identification results for independent private values, including Athey and Haile (2002), require that the set of entrants be observed. This paper fills the gap by establishing nonparametric identification of asymmetric bidders’ value distributions when losing entrants are not observed, under a general class of entry models in which each potential entrant enters the auction with some bidder-specific probability.

Evaluation of instantaneous vibration parameters of a snowboard with a prototype granular dissipator

A container partially filled with loose plastic granules was attached to the shovel of the snowboard to suppress large-amplitude lateral vibrations by dissipating energy through non-conservative multi-granule interactions. A custom laboratory stand allowed to evaluate the performance using a full-scale snowboard deck. The response of the system with a prototype granular dissipator was measured for free lateral vibrations of the initially deflected board and under prescribed sinusoidal base motion. The damping characteristics for different fill ratios of the container were obtained using a direct method of nonparametric identification. The applied Hilbert–Huang transform-based vibration analysis method gave more insight into the board’s damping performance than the logarithmic decrement analysis. The results show that using the granular dissipator with a predestined number of granules increases the damping capacity at large amplitudes but is less effective at small amplitudes. At best, the damping factor was 70% higher when the granular dissipator was used than when the board was damped only intrinsically.

A note on identification of causal effects in cluster randomized trials with post-randomization selection bias

Cluster randomized trials are commonly used for evaluating treatment effects of a cluster-level intervention. Because subjects are often recruited after cluster randomization, a main complication is the potential for selection bias due to the intervention affecting which subjects are identified as being eligible for receiving the actual treatment. In the presence of such post-randomization selection bias, direct comparison of the observed outcomes among eligible subjects between arms does not yield a valid causal effect estimate. In this note, we define average and quantile causal estimands using principal stratification in the context of post-randomization selection bias, and provide nonparametric identification assumptions and formulas for these estimands.

Prices, profits, proxies, and production

This paper studies nonparametric identification and counterfactual bounds for heterogeneous firms that can be ranked in terms of productivity. Our approach works when quantities and prices are latent, rendering standard approaches inapplicable. Instead, we require observation of profits or other optimizing-values such as costs or revenues, and either prices or price proxies of flexibly chosen variables. We extend classical duality results for price-taking firms to a setup with discrete heterogeneity, endogeneity, and limited variation in possibly latent prices. Finally, we show that convergence results for nonparametric estimators may be directly converted to convergence results for production sets.

Preferences and Performance in Simultaneous First-Price Auctions: A Structural Analysis

AbstractMotivated by the prevalence of simultaneous bidding across a wide range of auction markets, we develop and estimate a model of strategic interaction in simultaneous first-price auctions when objects are heterogeneous and bidders have non-additive preferences over combinations. We establish non-parametric identification of primitives in this model under standard exclusion restrictions, providing a basis for both estimation and testing of preferences over combinations. We then apply our model to data on Michigan Department of Transportation (MDOT) highway procurement auctions, quantifying the magnitude of cost synergies and evaluating the performance of the simultaneous first-price mechanism in the MDOT marketplace.

Identification of Quadratic Volterra Polynomials in the “Input–Output” Models of Nonlinear Systems

In this paper, we propose a new algorithm for constructing an integral model of a nonlinear dynamic system of the “input–output” type in the form of a quadratic segment of the Volterra integro-power series (polynomial). We consider nonparametric identification of models using physically realizable piecewise linear test signals in the time domain. The advantage of the presented approach is to obtain explicit formulas for calculating the transient responses (Volterra kernels), which determine the unique solution of the Volterra integral equations of the first kind with two variable integration limits. The numerical method proposed in the paper for solving the corresponding equations includes the use of smoothing splines. An important result is that the constructed identification algorithm has a low methodological error.

Handling parametric assumptions in principal causal effect estimation using Gaussian mixtures.

Given the latent stratum membership, principal stratification models with continuous outcomes naturally fit in the parametric estimation framework of Gaussian mixtures. However, with models that are not nonparametrically identified, relying on parametric mixture modeling has been mostly discouraged as a way of identifying principal effects. This study revisits this rather deserted use of parametric mixture modeling, which may open up various possibilities in principal stratification modeling. The main problem with using the parametric mixture modeling approach is that it is hard to assess the quality of principal effect estimates given its reliance on parametric conditions. As a way of assessing the estimation quality in this situation, this study proposes that we use parametric mixture modeling in two different ways, with and without the assurance of nonparametric identification. The key identifying assumption employed in this study is the moving exclusion restriction, a flexible version of the standard exclusion restriction assumption. This assumption is used as a temporary vehicle to help assess the quality of principal effect estimates obtained relying on parametric mixture modeling. The study presents promising results, showing the possibility of using parametric mixture modeling as an accessible tool for causal inference.

Multiply Robust Estimation of Causal Effects under Principal Ignorability

AbstractCausal inference concerns not only the average effect of the treatment on the outcome but also the underlying mechanism through an intermediate variable of interest. Principal stratification characterizes such a mechanism by targeting subgroup causal effects within principal strata, which are defined by the joint potential values of an intermediate variable. Due to the fundamental problem of causal inference, principal strata are inherently latent, rendering it challenging to identify and estimate subgroup effects within them. A line of research leverages the principal ignorability assumption that the latent principal strata are mean independent of the potential outcomes conditioning on the observed covariates. Under principal ignorability, we derive various nonparametric identification formulas for causal effects within principal strata in observational studies, which motivate estimators relying on the correct specifications of different parts of the observed-data distribution. Appropriately combining these estimators yields triply robust estimators for the causal effects within principal strata. These triply robust estimators are consistent if two of the treatment, intermediate variable and outcome models are correctly specified, and moreover, they are locally efficient if all three models are correctly specified. We show that these estimators arise naturally from either the efficient influence functions in the semiparametric theory or the model-assisted estimators in the survey sampling theory. We evaluate different estimators based on their finite-sample performance through simulation and apply them to two observational studies.