결측을 포함한 반복측정자료 모형에서 결측자료 메커니즘의 영향

송주원 한국자료분석학회 2010 Journal of the Korean Data Analysis Society Vol.12 No.3

Missing values often occur in repeatedly measured data due to dropouts and other reasons. To conduct an analysis of repeatedly measured data, models include correlations among different time points for the same subject. It is known that this model does not delete any observed values due to missingness of other time points and provides unbiased estimates of the parameters if missing data mechanism is MAR(Litte and Rubin, 2002). The analysis of repeatedly measured data often follows a restrictive approach that includes only variables of interests and covariates, since this parsimonious model is simple and easy to explain. On the other hand, a simulation study by Collins, Schafer, and Kam(2001) shows that a restrictive model could provide biased parameter estimates if it does not include an explanatory variable which is related to both a response variable and missingness. Here, we conducted a simulation study to compare this restrictive model without the explanatory variable and the model with the explanatory variable for repeatedly measured missing data. Since the restrictive model may provide biased parameter estimates, it is more appropriate to select explanatory variables by considering the missing data mechanism as well. It is also indicated that the size of biases depends on the correlation among repeated measured variables. 반복측정자료에서는 관측개체의 중도탈락 등의 원인으로 인하여 결측이 흔히 발생한다. 반복측정자료를 분석하기 위하여 동일한 개체에 대한 반복측정된 다른 시점의 값들 사이의 연관성을 포함하는 모형이 적합되는데 이 모형은 결측자료 하에서 정보의 손실이 없고 결측자료 메커니즘이 MAR(Little and Rubin, 2002)을 따른다면 모수의 추정에 편향(bias)이 발생하지 않는다고 알려져 있다. 반복측정자료의 분석에는 주요 관심 대상이 되는 변수 및 공변량 만을 모형에 포함시켜 분석하는 제한적 모형(restrictive model)이 흔히 사용되는데 이는 절약모형(parsimonious model)이 해석 및 설명하기 쉽기 때문이다. 반면, Collins, Schafer, and Kam(2001)은 횡단면 분석을 위한 모형에서 반응변수 및 결측 발생과 연관된 설명변수가 포함되지 않은 제한적 모형이 결측자료에 적용된다면 모수에 편향이 발생할 수 있다는 점을 모의실험을 통해 보였다. 본 연구에서는 결측을 포함한 반복측정자료에서 반응변수 및 결측 발생과 모두 연관되어 있지만 주요 관심 대상이 되는 설명 변수와 연관되지 않아 공변량이 아닌 변수가 모형에 포함되지 않는 제한적 모형과 이 변수를 모형에 포함하는 모형에서 모수 추정에 편이가 발생하는 지를 모의실험을 통해 비교하였다. 제한적 모형의 경우 모수 추정에 편향이 발생하는 것으로 나타나 결측자료 분석에서는 결측자료 메커니즘을 고려하여 연관된 변수를 포함하는 모형이 적절하게 나타났다. 하지만 모수 추정의 편이는 동일한 개체에 대하여 반복측정된 값들 사이의 연관성이 높아짐에 따라 작아지는 것으로 나타났다.

Quantized H∞ Control for a Class of 2-D Systems with Missing Measurements

Xuhui Bu,Jiaqi Liang,Zhongsheng Hou,Junqi Yang 제어·로봇·시스템학회 2017 International Journal of Control, Automation, and Vol.15 No.2

In this paper, the problem of quantized H¥ control is investigated for a class of 2-D systems described byRoesser model with missing measurements. The measurement missing of system state is described by a sequenceof random variables obeying the Bernoulli distribution. Meanwhile, the state measurements are quantized by logarithmicquantizer before being communicated. By introducing a new 2-D Lyapunov-like function, a sufficientcondition is derived to guarantee stochastically stable and H¥ performance of the closed-loop 2-D system, wherethe method of sector-bounded uncertainties is utilized to deal with quantization error. Based on the condition, thequantized H¥ control can be designed by using linear matrix inequality technique. A simulation example is alsogiven to illustrate the proposed method.

Unbiased FIR Filtering with Incomplete Measurement Information

Dong Ki Ryu,Chang Joo Lee,박상규,MyotaegLim 제어·로봇·시스템학회 2020 International Journal of Control, Automation, and Vol.18 No.2

This paper proposes an unbiased filter with finite impulse response (FIR) structure for linear discrete time systems in state space form with incomplete measurement information. The measurements are transmitted from the plant to the FIR filter imperfectly due to random packet loss or sensor faults. The Bernoulli random process is used to describe the missing measurement details, and the missing data is replaced with recently transmitted data on the missing horizon. The missing horizon can hold the assumption for finite measurement of the FIR filter. Two examples are provided to demonstrate the proposed unbiased FIR (UFIR) filter robustness against temporary model uncertainty and consecutive missing measurement data compared with existing filters considering missing measurement.

Unscented Kalman Filtering for Nonlinear State Estimation with Correlated Noises and Missing Measurements

Long Xu,Kemao Ma,Hongxia Fan 제어·로봇·시스템학회 2018 International Journal of Control, Automation, and Vol.16 No.3

The unscented Kalman filtering problem is investigated for a class of nonlinear discrete stochastic systems subject to correlated noises and missing measurements. Here, a random variable obeying Bernoulli distribution with known conditional probability is introduced to depict the phenomenon of missing measurements occurring in a stochastic way. Due to taking the correlation of noises into account, a one-step predictor is designed by applying the innovative analysis and unscented transformation approach. And then, based on one-step predictor and the minimum mean square error principle, a new unscented Kalman filtering algorithm is proposed such that, for the correlated noises and missing measurements, the filtering error is minimized. By solving the recursive matrix equation, the filter gain matrices and the error covariance matrices can be obtained and the proposed results can be easily verified by using the standard numerical software. We finally provide a numerical example to show the performance of the proposed approach.

A Robust Finite-Horizon Kalman Filter for Uncertain Discrete Time-Varying Systems with State-Delay and Missing Measurements

Jun-Hui Zheng,Jian-Fen Liu 보안공학연구지원센터 2016 International Journal of Grid and Distributed Comp Vol.9 No.3

In this paper, a robust kalman filter is designed for the uncertainty time-varying discrete systems with state delay in process and output matrices combined with the possibility of missing measurements. The uncertainties are expected in the process, output and white noise covariance matrices. A formula for a candidate upper bound on the actual state estimation error variances for all admissible parameter uncertainties and possible missing measurements is obtained. The filter parameters are optimized to give a minimal upper bound on the state estimation error covariance for all admissible uncertainties and missing measurements.

Optimal and Suboptimal Minimum-Variance Filtering in Networked Systems with Mixed Uncertainties of Random Sensor Delays, Packet Dropouts and Missing Measurements

Maryam Moayedi,Yeng Chai Soh,Yung Kuan Foo 제어·로봇·시스템학회 2010 International Journal of Control, Automation, and Vol.8 No.6

In this paper the Kalman filtering problem for networked stochastic linear discrete-time systems with random measurement delays, packet dropouts and missing measurements is studied. Based on a quasi Markov-chain approach, a unified/combined model is developed to accommodate random delay, packet dropout and missing measurement. Two approaches for constructing a filter via the linear matrix inequality approach are proposed. Simulation studies are then conducted to evaluate the effec-tiveness of the constructed estimators.

Sampled-data H_~ fuzzy filtering for nonlinear systems with missing measurements

Koo, G.B.,Park, J.B.,Joo, Y.H. North-Holland ; Elsevier Science Ltd 2017 FUZZY SETS AND SYSTEMS Vol.316 No.-

<P>In this paper, a sampled-data H-infinity fuzzy filtering problem is considered for nonlinear systems with missing measurements. The nonlinear sampled-data system and missing measurements are assumed to be represented by a Takagi-Sugeno (T-S) fuzzy system and an independent, identically distributed Bernoulli random process, respectively. Based on the fuzzy system, the H-infinity fuzzy filtering problem is formulated to design the sampled-data fuzzy filter. By using the exponential mean-square stability definition, the stability condition with an H-infinity performance is guaranteed for the fuzzy system with the sampled-data fuzzy filter, and its sufficient condition is converted into the linear matrix inequality (LMI) format. Finally, an example is provided to verify the effectiveness of the proposed fuzzy filtering technique. (C) 2016 Elsevier B. V. All rights reserved.</P>

A fuzzy filter with missing measurement for observer-based T-S fuzzy models

Sun Young Noh,Jin Bae Park,Young Hoon Joo 제어로봇시스템학회 2010 제어로봇시스템학회 국제학술대회 논문집 Vol.2010 No.10

This paper is concerned with the problem of a fuzzy filter of nonlinear system with missing measurements. The nonlinear system is represented by a Takagi-Sugeno(TS) fuzzy model. The system measurements may be unavailable at any sample time and the probability of the occurrence of missing data is assumed to be known. The purpose of this problem is to design a linear filter such that, the error state of the filtering process is mean square bounded. A basisdependent Lyapunov function approach is developed to design the fuzzy filter, and it is developed the upper bound of a fuzzy filter gain of the estimation error subject to some LMI constraints. In this situation, the estimation error due to persistent bounded disturbances. Finally, an illustrative numerical example is provided to show the effectiveness of the proposed approach.

l∞ Fuzzy Filter Design for Nonlinear Systems with Missing Measurements: Fuzzy Basis-dependent Lyapunov Function Approach

박진배,노선영,구근범,주영훈 제어·로봇·시스템학회 2016 International Journal of Control, Automation, and Vol.14 No.2

In this paper, l∞ fuzzy filtering problem is dealt for nonlinear systems with both persistent boundeddisturbances and missing probabilistic sensor information. The Takagi–Sugeno (T–S) fuzzy model is adopted torepresent a nonlinear dynamic system. The measurement output is assumed to contain randomly missing data,which is modeled by a Bernoulli distributed with a known conditional probability. To design the l∞ fuzzy filter andguarantee tracking performance, the effect of the perturbation against persistent bounded disturbances is reducedby using the minimum l∞ performance. By using the fuzzy basis-dependent Lyapunov function approach, a sufficientcondition is established that ensure the mean square exponential stability of the filtering error. The proposedsufficient condition is represented as some linear matrix inequalities (LMIs), and the filter gain is obtained by thesolution to a set of LMIs. Finally, the effectiveness of the proposed design method is shown via an example.

Optimized Distributed Fusion Filtering for Uncertain Nonlinear Systems With Missing Measurements: Algorithm Design and Boundedness Analysis

Zhibin Hu,Jun Hu,Junhua Du,Hongjian Liu,Jun Qi 제어·로봇·시스템학회 2023 International Journal of Control, Automation, and Vol.21 No.10

This paper investigates the optimized distributed fusion filtering (DFF) problem for a class of nonlinear discrete time-varying stochastic systems with randomly occurring uncertainty (ROU) and missing measurements (MMs). The stochastic nonlinearity is depicted in terms of statistical means. The phenomena of the ROU and MMs are considered during the modelling of state equation and measurement output respectively, which are characterized by Bernoulli distributed random variables. In order to deal with the effect of the parameter uncertainty, the method that the local estimation error covariances and cross-covariances from all estimators at every sample time are replaced by their upper bounds is adopted. Moreover, the minimum upper bounds for each filtering error covariance (FEC) are obtained by designing the corresponding filter gains. Based on the local filters, a new robust DFF algorithm is developed via the matrix-weighted fusion method. In addition, a sufficient condition concerning on the performance analysis of the developed algorithm is given, which can show that the boundedness of the upper bound for each FEC is guaranteed. Finally, a numerical example is provided to manifest the usefulness of the developed distributed fusion algorithm.