SU\G(𝔸)/K·U
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All published works (25)

Signed-Perturbed Sums Estimation of ARX Systems: Exact Coverage and Strong Consistency

2025-05-28
Algo Carè , Erik Weyer , Balázs Csanád Csáji +1 more

Signed-Perturbed Sums Estimation of ARX Systems: Exact Coverage and Strong Consistency (Extended Version)

2024-02-18
Algo Carè , Erik Weyer , Balázs Csanád Csáji +1 more
Sign-Perturbed Sums (SPS) is a system identification method that constructs confidence regions for the unknown system parameters. In this paper, we study SPS for ARX systems, and establish that the … Sign-Perturbed Sums (SPS) is a system identification method that constructs confidence regions for the unknown system parameters. In this paper, we study SPS for ARX systems, and establish that the confidence regions are guaranteed to include the true model parameter with exact, user-chosen, probability under mild statistical assumptions, a property that holds true for any finite number of observed input-output data. Furthermore, we prove the strong consistency of the method, that is, as the number of data points increases, the confidence region gets smaller and smaller and will asymptotically almost surely exclude any parameter value different from the true one. In addition, we also show that, asymptotically, the SPS region is included in an ellipsoid which is marginally larger than the confidence ellipsoid obtained from the asymptotic theory of system identification. The results are theoretically proven and illustrated in a simulation example.
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A Sample Based Algorithm for Constructing Guaranteed Confidence Ellipsoids for Linear Regression Models with Deterministic Regressor

2024-01-01
Xiaopuwen Wang , Erik Weyer

Stochastic Co-design of Storage and Control for Water Distribution Systems

2023-01-01
Ye Wang , Erik Weyer , Chris Manzie +2 more
Water distribution systems (WDSs) are typically designed with a conservative estimate of the ability of a control system to utilize the available infrastructure. The controller is designed and tuned after … Water distribution systems (WDSs) are typically designed with a conservative estimate of the ability of a control system to utilize the available infrastructure. The controller is designed and tuned after a WDS has been laid out, a methodology that may introduce unnecessary conservativeness in both system design and control, adversely impacting operational efficiency and increasing economic costs. To address these limitations, we introduce a method to simultaneously design infrastructure and develop control parameters, the co-design problem, with the aim of improving the overall efficiency of the system. Nevertheless, the co-design of a WDS is a challenging task given the presence of stochastic variables (e.g. water demands and electricity prices). In this paper, we propose a tractable stochastic co-design method to design the best tank size and optimal control parameters for WDS, where the expected operating costs are established based on Markov chain theory. We also give a theoretical result showing that the average long-run operating cost converges to the expected operating cost with probability~1. Furthermore, this method is not only applicable to greenfield projects for the co-design of WDSs but can also be utilized to improve the operations of existing WDSs in brownfield projects. The effectiveness and applicability of the co-design method are validated through three illustrative examples and a real-world case study in South Australia.
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Granger causality from quantized measurements

2022-06-08
Salman Ahmadi , Girish N. Nair , Erik Weyer
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Improved Pump Setpoint Selection Using a Calibrated Hydraulic Model of a High-Pressure Irrigation System

2022-01-01
Yu Wang , Qi Zhao , Wenyan Wu +3 more
This paper presents a case study of the operational management of the Robinvale high-pressure piped irrigation water delivery system (RVHPS) in Australia. Based on datasets available, improved pump setpoint selection … This paper presents a case study of the operational management of the Robinvale high-pressure piped irrigation water delivery system (RVHPS) in Australia. Based on datasets available, improved pump setpoint selection using a calibrated hydraulic model is investigated. The first step was to implement pre-processing of measured flow and pressure data to identify errors in the data and possible faulty sensors. An EPANET hydraulic simulation model was updated with calibrated pipe roughness height values by using the processed pressure and flow data. Then, new pump setpoints were selected using the calibrated model given the actual measured demands such that the pressures in the network were minimized subject to required customer service standards. Based on a two-day simulation, it was estimated that 4.7% savings in pumping energy cost as well as 4.7% reduction in greenhouse gas emissions can be achieved by applying the new pump setpoints.
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Improved Pump Setpoint Selection Using a Calibrated Hydraulic Model of a High-Pressure Irrigation System

2022-01-01
Yu Wang , Qi Zhao , Wenyan Wu +3 more
This paper presents a case study of the operational management of the Robinvale high-pressure piped irrigation water delivery system (RVHPS) in Australia. Based on datasets available, improved pump setpoint selection … This paper presents a case study of the operational management of the Robinvale high-pressure piped irrigation water delivery system (RVHPS) in Australia. Based on datasets available, improved pump setpoint selection using a calibrated hydraulic model is investigated. The first step was to implement pre-processing of measured flow and pressure data to identify errors in the data and possible faulty sensors. An EPANET hydraulic simulation model was updated with calibrated pipe roughness height values by using the processed pressure and flow data. Then, new pump setpoints were selected using the calibrated model given the actual measured demands such that the pressures in the network were minimized subject to required customer service standards. Based on a two-day simulation, it was estimated that 4.7% savings in pumping energy cost as well as 4.7% reduction in greenhouse gas emissions can be achieved by applying the new pump setpoints.
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Minimizing Pumping Energy Cost in Real-Time Operations of Water Distribution Systems Using Economic Model Predictive Control

2021-05-13
Ye Wang , Kevin Too Yok , Wenyan Wu +3 more
Optimizing pump operations is a challenging task for real-time management of water distribution systems (WDS). With suitable pump scheduling, pumping costs can be significantly reduced. In this research, a novel … Optimizing pump operations is a challenging task for real-time management of water distribution systems (WDS). With suitable pump scheduling, pumping costs can be significantly reduced. In this research, a novel economic model predictive control (EMPC) framework for real-time management of WDS is proposed. Optimal pump operations are selected based on predicted system behavior over a receding time horizon with the aim to minimize the total pumping energy cost. Time-varying electricity tariffs are considered while all the required water demands are satisfied. The novelty of this framework is to choose the number of pumps to operate in each pump station as decision variables in order to optimize the total pumping energy costs. By using integer programming, the proposed EMPC is applied to a benchmark case study, the Richmond Pruned network. The simulation with an EPANET hydraulic simulator is implemented. Moreover, a comparison of the results obtained using the proposed EMPC with those obtained using trigger-level control demonstrates significant economic benefits of the proposed EMPC.
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Granger Causality from Quantized Measurements

2021-01-01
Salman Ahmadi , Girish N. Nair , Erik Weyer
An approach is proposed for inferring Granger causality between jointly stationary, Gaussian signals from quantized data. First, a necessary and sufficient rank criterion for the equality of two conditional Gaussian … An approach is proposed for inferring Granger causality between jointly stationary, Gaussian signals from quantized data. First, a necessary and sufficient rank criterion for the equality of two conditional Gaussian distributions is proved. Assuming a partial finite-order Markov property, a characterization of Granger causality in terms of the rank of a matrix involving the covariances is presented. We call this the causality matrix. The smallest singular value of the causality matrix gives a lower bound on the distance between the two conditional Gaussian distributions appearing in the definition of Granger causality and yields a new measure of causality. Then, conditions are derived under which Granger causality between jointly Gaussian processes can be reliably inferred from the second order moments of quantized measurements. A necessary and sufficient condition is proposed for Granger causality inference under binary quantization. Furthermore, sufficient conditions are introduced to infer Granger causality between jointly Gaussian signals through measurements quantized via non-uniform, uniform or high resolution quantizers. Apart from the assumed partial Markov order and joint Gaussianity, this approach does not require the parameters of a system model to be identified. No assumptions are made on the identifiability of the jointly Gaussian random processes through the quantized observations. The effectiveness of the proposed method is illustrated by simulation results.
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Identification of FIR Systems With Binary Input and Output Observations

2020-12-03
Alex S. Leong , Erik Weyer , Girish N. Nair
This article considers the identification of finite-impulse response systems, where information about the inputs and outputs of the system undergoes quantization into binary values before transmission to the estimator. In … This article considers the identification of finite-impulse response systems, where information about the inputs and outputs of the system undergoes quantization into binary values before transmission to the estimator. In the case where the thresholds of the input and output quantizers can be adapted, we propose identification schemes that are strongly consistent for Gaussian distributed inputs and noises. The algorithms are based on the idea that certain joint probabilities of the unquantized signals can be estimated from the binary signals, and the system parameters can then be inferred from these estimates. The algorithms and their properties are illustrated in simulation examples.
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Granger Causality of Gaussian Signals from Noisy or Filtered Measurements

2020-01-01
Salman Ahmadi , Girish N. Nair , Erik Weyer
This paper investigates the assessment of Granger causality (GC) between jointly Gaussian signals based on noisy or filtered measurements. To do so, a recent rank condition for inferring GC between … This paper investigates the assessment of Granger causality (GC) between jointly Gaussian signals based on noisy or filtered measurements. To do so, a recent rank condition for inferring GC between jointly Gaussian stochastic processes is exploited. Sufficient conditions are derived under which GC can be reliably inferred from the second order moments of the noisy or filtered measurements. This approach does not require a model of the underlying Gaussian system to be identified. The noise signals are not required to be Gaussian or independent, and the filters may be noncausal or nonminimum-phase, as long as they are stable.

Finite Sample Confidence Region for EIV Systems Using Regression Model

2019-01-01
Masoud Moravej Khorasani , Erik Weyer

Identification of FIR Systems with Binary Input and Output Observations

2018-09-18
Alex S. Leong , Erik Weyer , Girish N. Nair
This paper considers the identification of FIR systems, where information about the inputs and outputs of the system undergoes quantization into binary values before transmission to the estimator. In the … This paper considers the identification of FIR systems, where information about the inputs and outputs of the system undergoes quantization into binary values before transmission to the estimator. In the case where the thresholds of the input and output quantizers can be adapted, but the quantizers have no computation and storage capabilities, we propose identification schemes which are strongly consistent for Gaussian distributed inputs and noises. This is based on exploiting the correlations between the quantized input and output observations to derive nonlinear equations that the true system parameters must satisfy, and then estimating the parameters by solving these equations using stochastic approximation techniques. If, in addition, the input and output quantizers have computational and storage capabilities, strongly consistent identification schemes are proposed which can handle arbitrary input and noise distributions. In this case, some conditional expectation terms are computed at the quantizers, which can then be estimated based on binary data transmitted by the quantizers, subsequently allowing the parameters to be identified by solving a set of linear equations. The algorithms and their properties are illustrated in simulation examples.
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Identification of FIR Systems with Binary Input and Output Observations

2018-01-01
Alex S. Leong , Erik Weyer , Girish N. Nair
This paper considers the identification of FIR systems, where information about the inputs and outputs of the system undergoes quantization into binary values before transmission to the estimator. In the … This paper considers the identification of FIR systems, where information about the inputs and outputs of the system undergoes quantization into binary values before transmission to the estimator. In the case where the thresholds of the input and output quantizers can be adapted, but the quantizers have no computation and storage capabilities, we propose identification schemes which are strongly consistent for Gaussian distributed inputs and noises. This is based on exploiting the correlations between the quantized input and output observations to derive nonlinear equations that the true system parameters must satisfy, and then estimating the parameters by solving these equations using stochastic approximation techniques. If, in addition, the input and output quantizers have computational and storage capabilities, strongly consistent identification schemes are proposed which can handle arbitrary input and noise distributions. In this case, some conditional expectation terms are computed at the quantizers, which can then be estimated based on binary data transmitted by the quantizers, subsequently allowing the parameters to be identified by solving a set of linear equations. The algorithms and their properties are illustrated in simulation examples.
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Asymptotic properties of SPS confidence regions

2017-05-23
Erik Weyer , Marco C. Campi , Balázs Csanád Csáji
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On the identification of FIR systems with binary input and output observations

2016-12-01
Alex S. Leong , Erik Weyer , Girish N. Nair
This paper considers the identification of FIR systems, where the inputs and outputs of the system undergoes quantization into binary values before transmission to the system identifier. Provided that the … This paper considers the identification of FIR systems, where the inputs and outputs of the system undergoes quantization into binary values before transmission to the system identifier. Provided that the thresholds of the input and output quantizers can be adapted, we propose identification schemes which are strongly consistent for Gaussian distributed inputs and noises. Identification schemes are given both for the case where the mean and variance of the input distribution are known, and when they are unknown.
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Sign-Perturbed Sums (SPS) with Instrumental Variables for the Identification of ARX Systems - Extended Version

2015-09-15
Valerio Volpe , Balázs Csanád Csáji , Algo Carè +2 more
We propose a generalization of the recently developed system identification method called Sign-Perturbed Sums (SPS). The proposed construction is based on the instrumental variables estimate and, unlike the original SPS, … We propose a generalization of the recently developed system identification method called Sign-Perturbed Sums (SPS). The proposed construction is based on the instrumental variables estimate and, unlike the original SPS, it can construct non-asymptotic confidence regions for linear regression models where the regressors contain past values of the output. Hence, it is applicable to ARX systems, as well as systems with feedback. We show that this approach provides regions with exact confidence under weak assumptions, i.e., the true parameter is included in the regions with a (user-chosen) exact probability for any finite sample. The paper also proves the strong consistency of the method and proposes a computationally efficient generalization of the previously proposed ellipsoidal outer-approximation. Finally, the new method is demonstrated through numerical experiments, using both real-world and simulated data.
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Sign-Perturbed Sums (SPS) with Instrumental Variables for the Identification of ARX Systems - Extended Version

2015-01-01
Valerio Volpe , Balázs Csanád Csáji , Algo Carè +2 more
We propose a generalization of the recently developed system identification method called Sign-Perturbed Sums (SPS). The proposed construction is based on the instrumental variables estimate and, unlike the original SPS, … We propose a generalization of the recently developed system identification method called Sign-Perturbed Sums (SPS). The proposed construction is based on the instrumental variables estimate and, unlike the original SPS, it can construct non-asymptotic confidence regions for linear regression models where the regressors contain past values of the output. Hence, it is applicable to ARX systems, as well as systems with feedback. We show that this approach provides regions with exact confidence under weak assumptions, i.e., the true parameter is included in the regions with a (user-chosen) exact probability for any finite sample. The paper also proves the strong consistency of the method and proposes a computationally efficient generalization of the previously proposed ellipsoidal outer-approximation. Finally, the new method is demonstrated through numerical experiments, using both real-world and simulated data.
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Sign-Perturbed Sums: A New System Identification Approach for Constructing Exact Non-Asymptotic Confidence Regions in Linear Regression Models

2014-11-10
Balázs Csanád Csáji , Marco C. Campi , Erik Weyer
We propose a new system identification method, called Sign-Perturbed Sums (SPS), for constructing non-asymptotic confidence regions under mild statistical assumptions. SPS is introduced for linear regression models, including but not … We propose a new system identification method, called Sign-Perturbed Sums (SPS), for constructing non-asymptotic confidence regions under mild statistical assumptions. SPS is introduced for linear regression models, including but not limited to FIR systems, and we show that the SPS confidence regions have exact confidence probabilities, i.e., they contain the true parameter with a user-chosen exact probability for any finite data set. Moreover, we also prove that the SPS regions are star convex with the Least-Squares (LS) estimate as a star center. The main assumptions of SPS are that the noise terms are independent and symmetrically distributed about zero, but they can be nonstationary, and their distributions need not be known. The paper also proposes a computationally efficient ellipsoidal outer approximation algorithm for SPS. Finally, SPS is demonstrated through a number of simulation experiments.
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Finite sample properties of system identification methods

2003-01-22
Erik Weyer , Marco C. Campi
We study the performance of system identification methods on a finite data sample. Our results are of the following form: with a probability not less than 1-/spl delta/, minimising the … We study the performance of system identification methods on a finite data sample. Our results are of the following form: with a probability not less than 1-/spl delta/, minimising the empirical identification cost leads to an estimate which is within an accuracy /spl epsiv/ from the theoretical optimal estimate. Explicit expressions for the accuracy /spl epsiv/ are derived, revealing its dependence on the data generation characteristics and the choices made in the system identification procedure. This paper presents a finite sample identification theory applicable to a general linear time-invariant setting.

Non-asymptotic confidence ellipsoids for the least squares estimate

2002-11-11
Erik Weyer , Marco C. Campi
In this paper we consider the finite sample properties of least squares system identification, and we derive nonasymptotic confidence ellipsoids for the estimate. Unlike asymptotic theory, the obtained confidence ellipsoids … In this paper we consider the finite sample properties of least squares system identification, and we derive nonasymptotic confidence ellipsoids for the estimate. Unlike asymptotic theory, the obtained confidence ellipsoids are valid for a finite number of data points. The probability that the estimate belongs to a certain ellipsoid has a natural dependence on the volume of the ellipsoid, the data generating mechanism, the model order and the number of data points available.

Non-asymptotic confidence ellipsoids for the least-squares estimate

2002-09-01
Erik Weyer , Marco C. Campi

NON-ASYMPTOTIC QUALITY ASSESSMENT OF THE LEAST SQUARES ESTIMATE

2002-01-01
Su Ki Ooi , Marco C. Campi , Erik Weyer

A Recursive Decentralized Parameter Estimator for a General Linear SISO-System

1991-07-01
Erik Weyer

Convergence Aspects of Some Robust Estimators Based Upon Prefiltering of the Input-Output Data

1990-01-01
Rolf Henriksen , Erik Weyer
Abstract By prefiltering the input/output data and employing certain decentralized estimation techniques, it is possible to improve the robustness of some estimators significantly. Earlier papers on these techniques have been … Abstract By prefiltering the input/output data and employing certain decentralized estimation techniques, it is possible to improve the robustness of some estimators significantly. Earlier papers on these techniques have been focused on local convergence properties of certain bootstrap estimators based upon these techniques. This paper is devoted to (1) global convergence properties. and (2) convergence rates when the underlying system is stiff.
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Common Coauthors

Coauthor Papers Together
Marco C. Campi 10
Girish N. Nair 7
Balázs Csanád Csáji 6
Algo Carè 4
Angus R. Simpson 4
Alex S. Leong 4
Salman Ahmadi 3
Wenyan Wu 3
Ye Wang 2
Ailsa Willis 2
Chris Manzie 2
Qi Zhao 2
Yu Wang 2
Masoud Moravej Khorasani 1
Lisa J. Blinco 1
Rolf Henriksen 1
Valerio Volpe 1
Su Ki Ooi 1
Kevin Too Yok 1
Valerio Volpe 1
Xiaopuwen Wang 1

Commonly Cited References

Estimation of quantized linear errors-in-variables models

1995-10-01
Vikram Krishnamurthy

Finite sample properties of system identification methods

2003-01-22
Erik Weyer , Marco C. Campi
We study the performance of system identification methods on a finite data sample. Our results are of the following form: with a probability not less than 1-/spl delta/, minimising the … We study the performance of system identification methods on a finite data sample. Our results are of the following form: with a probability not less than 1-/spl delta/, minimising the empirical identification cost leads to an estimate which is within an accuracy /spl epsiv/ from the theoretical optimal estimate. Explicit expressions for the accuracy /spl epsiv/ are derived, revealing its dependence on the data generation characteristics and the choices made in the system identification procedure. This paper presents a finite sample identification theory applicable to a general linear time-invariant setting.

Sign-Perturbed Sums: A New System Identification Approach for Constructing Exact Non-Asymptotic Confidence Regions in Linear Regression Models

2014-11-10
Balázs Csanád Csáji , Marco C. Campi , Erik Weyer
We propose a new system identification method, called Sign-Perturbed Sums (SPS), for constructing non-asymptotic confidence regions under mild statistical assumptions. SPS is introduced for linear regression models, including but not … We propose a new system identification method, called Sign-Perturbed Sums (SPS), for constructing non-asymptotic confidence regions under mild statistical assumptions. SPS is introduced for linear regression models, including but not limited to FIR systems, and we show that the SPS confidence regions have exact confidence probabilities, i.e., they contain the true parameter with a user-chosen exact probability for any finite data set. Moreover, we also prove that the SPS regions are star convex with the Least-Squares (LS) estimate as a star center. The main assumptions of SPS are that the noise terms are independent and symmetrically distributed about zero, but they can be nonstationary, and their distributions need not be known. The paper also proposes a computationally efficient ellipsoidal outer approximation algorithm for SPS. Finally, SPS is demonstrated through a number of simulation experiments.
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Perturbed datasets methods for hypothesis testing and structure of corresponding confidence sets

2014-11-06
Sándor Kolumbán , István Vajk , J. Schoukens
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Nonparametric Statistics for Stochastic Processes

1998-01-01
Denis Bosq

Nonparametric Statistics for Stochastic Processes: Estimation and Prediction

2000-11-01
John Odencrantz , Denis Bosq
Synopsis.- 1. Inequalities for mixing processes.- 2. Density estimation for discrete time processes.- 3. Regression estimation and prediction for discrete time processes.- 4. Kernel density estimation for continuous time processes.- … Synopsis.- 1. Inequalities for mixing processes.- 2. Density estimation for discrete time processes.- 3. Regression estimation and prediction for discrete time processes.- 4. Kernel density estimation for continuous time processes.- 5. Regression estimation and prediction in continuous time.- 6. The local time density estimator.- 7. Implementation of nonparametric method and numerical applications.- References.
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Mitigating the effects of measurement noise on Granger causality

2007-03-29
Hariharan Nalatore , Mingzhou Ding , Govindan Rangarajan
Computing Granger causal relations among bivariate experimentally observed time series has received increasing attention over the past few years. Such causal relations, if correctly estimated, can yield significant insights into … Computing Granger causal relations among bivariate experimentally observed time series has received increasing attention over the past few years. Such causal relations, if correctly estimated, can yield significant insights into the dynamical organization of the system being investigated. Since experimental measurements are inevitably contaminated by noise, it is thus important to understand the effects of such noise on Granger causality estimation. The first goal of this paper is to provide an analytical and numerical analysis of this problem. Specifically, we show that, due to noise contamination, (1) spurious causality between two measured variables can arise and (2) true causality can be suppressed. The second goal of the paper is to provide a denoising strategy to mitigate this problem. Specifically, we propose a denoising algorithm based on the combined use of the Kalman filter theory and the expectation-maximization algorithm. Numerical examples are used to demonstrate the effectiveness of the denoising approach.
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Asymptotic properties of SPS confidence regions

2017-05-23
Erik Weyer , Marco C. Campi , Balázs Csanád Csáji
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Estimation of the Parameters in Stationary Autoregressive Processes after Hard Limiting

1980-03-01
Benjamin Kedem
Abstract Estimates of the parameters in normal autoregressive (AR(p)) processes may be obtained as functions of certain runs and subsequences in the associated clipped 0 − 1 processes. For example, … Abstract Estimates of the parameters in normal autoregressive (AR(p)) processes may be obtained as functions of certain runs and subsequences in the associated clipped 0 − 1 processes. For example, the parameter in AR (1) is a function of the number of 1 runs only. Equivalently, this parameter can be estimated by counting only the number of axis crossings by the process. The estimates are obtained by a modification of the likelihood function of the clipped data. The loss of information because of hard limiting results in a loss of efficiency relative to the usual maximum likelihood estimates. Nevertheless, for large records our procedure yields quick estimates that do perform well.

Probability and Random Processes

2001-05-31
Geoffrey Grimmett , David Stirzaker
Abstract This third edition of this successful text gives a rigorous and extensive introduction to probability theory and an account in some depth of the most important random processes. It … Abstract This third edition of this successful text gives a rigorous and extensive introduction to probability theory and an account in some depth of the most important random processes. It includes various topics which are suitable for undergraduate courses, but are not routinely taught. It is suitable for students of probability at all levels. There are four main aims: 1) to provide a thorough but straightforward account of basic probability, giving the reader a natural feel for the subject unburdened by oppressive technicalities, 2) to discuss important random processes in depth with many examples. 3) to cover a range of important but less routine topics, 4) to impart to the beginner the flavour of more advanced work. The book begins with basic ideas common to many undergraduate courses in mathematics, statistics and the sciences; it concludes with topics usually found at graduate level. The ordering and numbering of material in this third edition has been mostly preserved from the second. Minor alterations and additions have been added for clearer exposition. Highlights include new sections on sampling and Markov chain Monte Carlo, geometric probability, coupling and Poisson approximation, large deviations, spatial Poisson processes, renewal-reward, queuing networks, stochastic calculus, Itô's formula and option pricing in the Black- Scholes model for financial markets. In addition there are many (nearly 400) new exercises and problems that are entertaining and instructive; their solutions can be found in the companion volume 'One Thousand Exercises in Probability', (OUP).

On the Sensitivity of Granger Causality to Errors‐In‐Variables, Linear Transformations and Subsampling

2018-09-23
Brian D. O. Anderson , Manfred Deistler , Jean–Marie Dufour
This article studies the sensitivity of Granger causality to the addition of noise, the introduction of subsampling, and the application of causal invertible filters to weakly stationary processes. Using canonical … This article studies the sensitivity of Granger causality to the addition of noise, the introduction of subsampling, and the application of causal invertible filters to weakly stationary processes. Using canonical spectral factors and Wold decompositions, we give general conditions under which additive noise or filtering distorts Granger‐causal properties by inducing (spurious) Granger causality, as well as conditions under which it does not. For the errors‐in‐variables case, we give a continuity result, which implies that: a ‘small’ noise‐to‐signal ratio entails ‘small’ distortions in Granger causality. On filtering, we give general necessary and sufficient conditions under which ‘spurious’ causal relations between (vector) time series are not induced by linear transformations of the variables involved. This also yields transformations (or filters) which can eliminate Granger causality from one vector to another one. In a number of cases, we clarify results in the existing literature, with a number of calculations streamlining some existing approaches.
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Estimating the Directed Information and Testing for Causality

2016-08-31
Ioannis Kontoyiannis , Maria Skoularidou
The problem of estimating the directed information rate between two discrete processes (X <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> ) and (Y <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> ) via the plug-in (or maximum-likelihood) estimator is … The problem of estimating the directed information rate between two discrete processes (X <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> ) and (Y <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> ) via the plug-in (or maximum-likelihood) estimator is considered. When the joint process ((X <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> , Y <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> )) is a Markov chain of a given memory length, the plug-in estimator is shown to be asymptotically Gaussian and to converge at the optimal rate O(1/√n) under appropriate conditions; this is the first estimator that has been shown to achieve this rate. An important connection is drawn between the problem of estimating the directed information rate and that of performing a hypothesis test for the presence of causal influence between the two processes. Under fairly general conditions, the null hypothesis, which corresponds to the absence of causal influence, is equivalent to the requirement that the directed information rate be equal to zero. In that case, a finer result is established, showing that the plug-in converges at the faster rate O(1/n) and that it is asymptotically χ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> -distributed. This is proved by showing that this estimator is equal to (a scalar multiple of) the classical likelihood ratio statistic for the above hypothesis test. Finally, it is noted that these results facilitate the design of an actual likelihood ratio test for the presence or absence of causal influence.
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Bootstrap Tests for Regression Models

2009-01-01
L. G. Godfrey
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Directed Information Graphs

2015-09-22
Christopher J. Quinn , Negar Kiyavash , Todd P. Coleman
We propose a graphical model for representing networks of stochastic processes, the minimal generative model graph. It is based on reduced factorizations of the joint distribution over time. We show … We propose a graphical model for representing networks of stochastic processes, the minimal generative model graph. It is based on reduced factorizations of the joint distribution over time. We show that under appropriate conditions, it is unique and consistent with another type of graphical model, the directed information graph, which is based on a generalization of Granger causality. We demonstrate how directed information quantifies Granger causality in a particular sequential prediction setting. We also develop efficient methods to estimate the topological structure from data that obviate estimating the joint statistics. One algorithm assumes upper bounds on the degrees and uses the minimal dimension statistics necessary. In the event that the upper bounds are not valid, the resulting graph is nonetheless an optimal approximation in terms of Kullback-Leibler (KL) divergence. Another algorithm uses near-minimal dimension statistics when no bounds are known, but the distribution satisfies a certain criterion. Analogous to how structure learning algorithms for undirected graphical models use mutual information estimates, these algorithms use directed information estimates. We characterize the sample-complexity of two plug-in directed information estimators and obtain confidence intervals. For the setting when point estimates are unreliable, we propose an algorithm that uses confidence intervals to identify the best approximation that is robust to estimation error. Last, we demonstrate the effectiveness of the proposed algorithms through the analysis of both synthetic data and real data from the Twitter network. In the latter case, we identify which news sources influence users in the network by merely analyzing tweet times.
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SYMMETRIC GAUGE FUNCTIONS AND UNITARILY INVARIANT NORMS

1960-01-01
L. Mirsky
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Convex Optimization

2004-03-01
Stephen Boyd , Lieven Vandenberghe
Convex optimization problems arise frequently in many different fields. A comprehensive introduction to the subject, this book shows in detail how such problems can be solved numerically with great efficiency. … Convex optimization problems arise frequently in many different fields. A comprehensive introduction to the subject, this book shows in detail how such problems can be solved numerically with great efficiency. The focus is on recognizing convex optimization problems and then finding the most appropriate technique for solving them. The text contains many worked examples and homework exercises and will appeal to students, researchers and practitioners in fields such as engineering, computer science, mathematics, statistics, finance, and economics.

Finite Sample Confidence Regions for Parameters in Prediction Error Identification using Output Error Models

2008-01-01
Arnold J. den Dekker , Xavier Bombois , Paul M.J. Van den Hof

Algebraic geometry of Gaussian Bayesian networks

2007-09-07
Seth Sullivant
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Memory-universal prediction of stationary random processes

1998-01-01
Dharmendra S. Modha , E. Masry
We consider the problem of one-step-ahead prediction of a real-valued, stationary, strongly mixing random process (Xi)/sub i=-/spl infin///sup /spl infin//. The best mean-square predictor of X/sub 0/ is its conditional … We consider the problem of one-step-ahead prediction of a real-valued, stationary, strongly mixing random process (Xi)/sub i=-/spl infin///sup /spl infin//. The best mean-square predictor of X/sub 0/ is its conditional mean given the entire infinite past (X/sub i/)/sub i=-/spl infin///sup -1/. Given a sequence of observations X/sub 1/, X/sub 2/, X/sub N/, we propose estimators for the conditional mean based on sequences of parametric models of increasing memory and of increasing dimension, for example, neural networks and Legendre polynomials. The proposed estimators select both the model memory and the model dimension, in a data-driven fashion, by minimizing certain complexity regularized least squares criteria. When the underlying predictor function has a finite memory, we establish that the proposed estimators are memory-universal: the proposed estimators, which do not know the true memory, deliver the same statistical performance (rates of integrated mean-squared error) as that delivered by estimators that know the true memory. Furthermore, when the underlying predictor function does not have a finite memory, we establish that the estimator based on Legendre polynomials is consistent.

Some recent development in a concept of causality

1988-09-01
Clive W. J. Granger

Errors-in-Variables Methods in System Identification

2018-01-01
Torsten Söderström
This book presents an overview of the different errors-in-variables (EIV) methods that can be used for system identification. This book presents an overview of the different errors-in-variables (EIV) methods that can be used for system identification.

Asymptotic theory of mixed time averages and kth-order cyclic-moment and cumulant statistics

1995-01-01
A.V. Dandawaté , Georgios B. Giannakis
We generalize Parzen's (1961) analysis of "asymptotically stationary" processes to mixtures of deterministic, stationary, nonstationary, and generally complex time series. Under certain mixing conditions expressed in terms of joint cumulant … We generalize Parzen's (1961) analysis of "asymptotically stationary" processes to mixtures of deterministic, stationary, nonstationary, and generally complex time series. Under certain mixing conditions expressed in terms of joint cumulant summability, we show that time averages of such mixtures converge in the mean-square sense to their ensemble averages. We additionally show that sample averages of arbitrary orders are jointly complex normal and provide their covariance expressions. These conclusions provide us with statistical tools that treat random and deterministic signals on a common framework and are helpful in defining generalized moments and cumulants of mixed processes. As an important consequence, we develop consistent and asymptotically normal estimators for time-varying, and cyclic-moments and cumulants of kth-order cyclostationary processes and provide computable variance expressions. Some examples are considered to illustrate the salient features of the analysis.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Validity of Time Reversal for Testing Granger Causality

2016-02-18
Irene Winkler , Danny Panknin , Daniel Bartz +2 more
Inferring causal interactions from observed data is a challenging problem, especially in the presence of measurement noise. To alleviate the problem of spurious causality, Haufe <etal xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"/> (2013) proposed … Inferring causal interactions from observed data is a challenging problem, especially in the presence of measurement noise. To alleviate the problem of spurious causality, Haufe <etal xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"/> (2013) proposed to contrast measures of information flow obtained on the original data against the same measures obtained on time-reversed data. They show that this procedure, time-reversed Granger causality (TRGC), robustly rejects causal interpretations on mixtures of independent signals. While promising results have been achieved in simulations, it was so far unknown whether time reversal leads to valid measures of information flow in the presence of true interaction. Here, we prove that, for linear finite-order autoregressive processes with unidirectional information flow between two variables, the application of time reversal for testing Granger causality indeed leads to correct estimates of information flow and its directionality. Using simulations, we further show that TRGC is able to infer correct directionality with similar statistical power as the net Granger causality between two variables, while being much more robust to the presence of measurement noise.
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Statistical Adjustment of Data

1944-04-21
L. A. Walford , W. Edwards Deming

Van Vleck correction generalization for complex correlators with multilevel quantization

2016-01-01
L. Benkevitch , R. J. Cappallo , Kirby A. Baker +4 more
Remote sensing with phased antenna arrays is based on measurement of the cross-correlations between the signals from each antenna pair. Digital correlators have systematic errors due to the quantization losses. … Remote sensing with phased antenna arrays is based on measurement of the cross-correlations between the signals from each antenna pair. Digital correlators have systematic errors due to the quantization losses. The correlation errors allow substantial abatement based on the assumption that the analog signals are stochastic processes sampled from a statistical distribution (usually the Gaussian). The correlation correction technique is named after Van Vleck who was the first to apply it to two-level clipping quantizers. The correction is especially important for high correlation levels, e.g. in studies of solar radio emissions. We offer a generalized method that for every antenna pair inputs the quantized signals' covariance and standard deviations, and outputs high-precision estimates of the analog correlation. Although correlation correction methods have been extensively investigated in the past, there are several problems that, as far as we know, have not been published yet. We consider a very general quantization scheme with arbitrary set of transition thresholds and output levels, and our correction method is designed for correlations obtained from signals with generally unequal standard deviations. We also provide a method for estimation of the analog standard deviation from the quantized one for subsequent use in the correlation correction. We apply the correction to the the complex-valued analytic signals, overwhelmingly used in modern remote sensing systems with arrays of antennas. The approach is valid not only for analytic signals with the imaginary part being the Hilbert transform of the real one, but also for more general, circularly symmetric complex processes whose real and imaginary parts may have arbitrary relationships to each other. This work was motivated by the need for greater precision in analysis of data from the Murchison Widefield Array (MWA).
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Identification of FIR Systems with Binary Input and Output Observations

2018-09-18
Alex S. Leong , Erik Weyer , Girish N. Nair
This paper considers the identification of FIR systems, where information about the inputs and outputs of the system undergoes quantization into binary values before transmission to the estimator. In the … This paper considers the identification of FIR systems, where information about the inputs and outputs of the system undergoes quantization into binary values before transmission to the estimator. In the case where the thresholds of the input and output quantizers can be adapted, but the quantizers have no computation and storage capabilities, we propose identification schemes which are strongly consistent for Gaussian distributed inputs and noises. This is based on exploiting the correlations between the quantized input and output observations to derive nonlinear equations that the true system parameters must satisfy, and then estimating the parameters by solving these equations using stochastic approximation techniques. If, in addition, the input and output quantizers have computational and storage capabilities, strongly consistent identification schemes are proposed which can handle arbitrary input and noise distributions. In this case, some conditional expectation terms are computed at the quantizers, which can then be estimated based on binary data transmitted by the quantizers, subsequently allowing the parameters to be identified by solving a set of linear equations. The algorithms and their properties are illustrated in simulation examples.
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THE GEOMETRY OF RANDOM FIELDS

1982-01-01
Wilfrid S. Kendall
By Robert J. Adler: pp. 280. £17.50. (John Wiley & Sons Ltd., 1981.) By Robert J. Adler: pp. 280. £17.50. (John Wiley & Sons Ltd., 1981.)

Learning Wireless Networks’ Topologies Using Asymmetric Granger Causality

2017-12-25
Mihir Laghate , Danijela Čabrić
Sharing spectrum with a communicating incumbent user (IU) network requires avoiding interference to IU receivers. But since receivers are passive when in the receive mode and cannot be detected, the … Sharing spectrum with a communicating incumbent user (IU) network requires avoiding interference to IU receivers. But since receivers are passive when in the receive mode and cannot be detected, the network topology can be used to predict the potential receivers of a currently active transmitter. For this purpose, this paper proposes a method to detect the directed links between IUs of time multiplexing communication networks from their transmission start and end times. It models the response mechanism of commonly used communication protocols using Granger causality: The probability of an IU starting a transmission after another IU's transmission ends increases if the former is a receiver of the latter. This paper proposes a nonparametric test statistic for detecting such behavior. To help differentiate between a response and the opportunistic access of the available spectrum, the same test statistic is used to estimate the response time of each link. The causal structure of the response is studied through a discrete time Markov chain that abstracts the IUs' medium access protocol and focuses on the response time and response probability of 2 IUs. Through NS-3 simulations, it is shown that the proposed algorithm outperforms existing methods in accurately learning the topologies of infrastructure-based networks and that it can infer the directed data flow in ad hoc networks with finer time resolution than an existing method.
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On the Sample Complexity of the Linear Quadratic Regulator

2019-08-05
Sarah Dean , Horia Mania , Nikolai Matni +2 more
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Generalized reduced rank tests using the singular value decomposition

2005-05-05
Frank Kleibergen , Richard Paap
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Efficient Learning of Distributed Linear-Quadratic Control Policies

2020-01-01
Salar Fattahi , Nikolai Matni , Somayeh Sojoudi
Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 3 October 2019Accepted: 20 July 2020Published online: 01 October 2020Keywordsoptimal distributed control, networked … Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 3 October 2019Accepted: 20 July 2020Published online: 01 October 2020Keywordsoptimal distributed control, networked systems, constrained optimizationAMS Subject Headings49N10, 49N05, 49M37, 93C55, 90C26Publication DataISSN (print): 0363-0129ISSN (online): 1095-7138Publisher: Society for Industrial and Applied MathematicsCODEN: sjcodc

Distribution-free uncertainty quantification for kernel methods by gradient perturbations

2019-06-27
Balázs Csanád Csáji , Krisztián Balázs Kis
We propose a data-driven approach to quantify the uncertainty of models constructed by kernel methods. Our approach minimizes the needed distributional assumptions, hence, instead of working with, for example, Gaussian … We propose a data-driven approach to quantify the uncertainty of models constructed by kernel methods. Our approach minimizes the needed distributional assumptions, hence, instead of working with, for example, Gaussian processes or exponential families, it only requires knowledge about some mild regularity of the measurement noise, such as it is being symmetric or exchangeable. We show, by building on recent results from finite-sample system identification, that by perturbing the residuals in the gradient of the objective function, information can be extracted about the amount of uncertainty our model has. Particularly, we provide an algorithm to build exact, non-asymptotically guaranteed, distribution-free confidence regions for ideal, noise-free representations of the function we try to estimate. For the typical convex quadratic problems and symmetric noises, the regions are star convex centered around a given nominal estimate, and have efficient ellipsoidal outer approximations. Finally, we illustrate the ideas on typical kernel methods, such as LS-SVC, KRR, $$\varepsilon $$ -SVR and kernelized LASSO.
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Nonparametric Statistics for Stochastic Process: Estimation and Prediction

2000-09-01
MTW , D. Boso

Granger Causality of Gaussian Signals from Noisy or Filtered Measurements

2020-01-01
Salman Ahmadi , Girish N. Nair , Erik Weyer
This paper investigates the assessment of Granger causality (GC) between jointly Gaussian signals based on noisy or filtered measurements. To do so, a recent rank condition for inferring GC between … This paper investigates the assessment of Granger causality (GC) between jointly Gaussian signals based on noisy or filtered measurements. To do so, a recent rank condition for inferring GC between jointly Gaussian stochastic processes is exploited. Sufficient conditions are derived under which GC can be reliably inferred from the second order moments of the noisy or filtered measurements. This approach does not require a model of the underlying Gaussian system to be identified. The noise signals are not required to be Gaussian or independent, and the filters may be noncausal or nonminimum-phase, as long as they are stable.

Granger causality from quantized measurements

2022-06-08
Salman Ahmadi , Girish N. Nair , Erik Weyer
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Rademacher Processes and Bounding the Risk of Function Learning

2000-01-01
Vladimir Koltchinskii , Dmitriy Panchenko
We construct data dependent upper bounds on the risk in function learning problems. The bounds are based on local norms of the Rademacher process indexed by the underlying function class, … We construct data dependent upper bounds on the risk in function learning problems. The bounds are based on local norms of the Rademacher process indexed by the underlying function class, and they do not require prior knowledge about the distribution of training examples or any specific properties of the function class. Using Talagrand's type concentration inequalities for empirical and Rademacher processes, we show that the bounds hold with high probability that decreases exponentially fast when the sample size grows. In typical situations that are frequently encountered in the theory of function learning, the bounds give nearly optimal rate of convergence of the risk to zero.

Identification of FIR Systems with Binary Input and Output Observations

2018-01-01
Alex S. Leong , Erik Weyer , Girish N. Nair
This paper considers the identification of FIR systems, where information about the inputs and outputs of the system undergoes quantization into binary values before transmission to the estimator. In the … This paper considers the identification of FIR systems, where information about the inputs and outputs of the system undergoes quantization into binary values before transmission to the estimator. In the case where the thresholds of the input and output quantizers can be adapted, but the quantizers have no computation and storage capabilities, we propose identification schemes which are strongly consistent for Gaussian distributed inputs and noises. This is based on exploiting the correlations between the quantized input and output observations to derive nonlinear equations that the true system parameters must satisfy, and then estimating the parameters by solving these equations using stochastic approximation techniques. If, in addition, the input and output quantizers have computational and storage capabilities, strongly consistent identification schemes are proposed which can handle arbitrary input and noise distributions. In this case, some conditional expectation terms are computed at the quantizers, which can then be estimated based on binary data transmitted by the quantizers, subsequently allowing the parameters to be identified by solving a set of linear equations. The algorithms and their properties are illustrated in simulation examples.
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Foundations of Modern Probability

2002-01-01
Olav Kallenberg
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Non-asymptotic state-space identification of closed-loop stochastic linear systems using instrumental variables

2023-06-10
Szabolcs Szentpéteri , Balázs Csanád Csáji
The paper suggests a generalization of the Sign-Perturbed Sums (SPS) finite sample system identification method for the identification of closed-loop observable stochastic linear systems in state-space form. The solution builds … The paper suggests a generalization of the Sign-Perturbed Sums (SPS) finite sample system identification method for the identification of closed-loop observable stochastic linear systems in state-space form. The solution builds on the theory of matrix-variate regression and instrumental variable methods to construct distribution-free confidence regions for the state-space matrices. Both direct and indirect identification are studied, and the exactness as well as the strong consistency of the construction are proved. Furthermore, a new, computationally efficient ellipsoidal outer-approximation algorithm for the confidence regions is proposed. The new construction results in a semidefinite optimization problem which has an order-of-magnitude smaller number of constraints, as if one applied the ellipsoidal outer-approximation after vectorization. The effectiveness of the approach is also demonstrated empirically via a series of numerical experiments.
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Statistical Learning Theory for Control: A Finite-Sample Perspective

2023-11-14
Anastasios Tsiamis , Ingvar Ziemann , Nikolai Matni +1 more
Learning algorithms have become an integral component to modern engineering solutions. Examples range from self-driving cars and recommender systems to finance and even critical infrastructure, many of which are typically … Learning algorithms have become an integral component to modern engineering solutions. Examples range from self-driving cars and recommender systems to finance and even critical infrastructure, many of which are typically under the purview of control theory. While these algorithms have already shown tremendous promise in certain applications <xref ref-type="bibr" rid="ref1" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[1]</xref> , there are considerable challenges, in particular, with respect to guaranteeing safety and gauging fundamental limits of operation. Thus, as we integrate tools from machine learning into our systems, we also require an integrated theoretical understanding of how they operate in the presence of dynamic and system-theoretic phenomena. Over the past few years, intense efforts toward this goal—an integrated theoretical understanding of learning, dynamics, and control—have been made. While much work remains to be done, a relatively clear and complete picture has begun to emerge for (fully observed) linear dynamical systems. These systems already allow for reasoning about concrete failure modes, thus helping to indicate a path forward. Moreover, while simple at a glance, these systems can be challenging to analyze. Recently, a host of methods from learning theory and high-dimensional statistics, not typically in the control-theoretic toolbox, have been introduced to our community. This tutorial survey serves as an introduction to these results for learning in the context of unknown linear dynamical systems (see "Summary"). We review the current state of the art and emphasize which tools are needed to arrive at these results. Our focus is on characterizing the sample efficiency and fundamental limits of learning algorithms. Along the way, we also delineate a number of open problems. More concretely, this article is structured as follows. We begin by revisiting recent advances in the finite-sample analysis of system identification. Next, we discuss how these finite-sample bounds can be used downstream to give guaranteed performance for learning-based offline control. The final technical section discusses the more challenging online control setting. Finally, in light of the material discussed, we outline a number of future directions.
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Permutation Tests: A Practical Guide to Resampling Methods for Testing Hypotheses

1995-08-01
Minoo Niknian
"Permutation Tests: A Practical Guide to Resampling Methods for Testing Hypotheses." Technometrics, 37(3), pp. 341–342 "Permutation Tests: A Practical Guide to Resampling Methods for Testing Hypotheses." Technometrics, 37(3), pp. 341–342

High Dimensional Probability II

2000-01-01
Evarist Giné , David M. Mason , Jon A. Wellner
I. Measures on General Spaces and Inequalities.- Stochastic inequalities and perfect independence.- Prokhorov-LeCam-Varadarajan's compactness criteria for vector measures on metric spaces.- On measures in locally convex spaces.- II. Gaussian Processes.- … I. Measures on General Spaces and Inequalities.- Stochastic inequalities and perfect independence.- Prokhorov-LeCam-Varadarajan's compactness criteria for vector measures on metric spaces.- On measures in locally convex spaces.- II. Gaussian Processes.- Karhunen-Loeve expansions for weighted Wiener processes and Brownian bridges via Bessel functions.- Extension du theoreme de Cameron-Martin aux translations aleatoires. II. Integrabilite des densites.- III. Limit Theorems.- Rates of convergence for Levy's modulus of continuity and Hinchin's law of the iterated logarithm.- On the limit set in the law of the iterated logarithm for U-statistics of order two.- Perturbation approach applied to the asymptotic study of random operators.- A uniform functional law of the logarithm for a local Gaussian process.- Strong limit theorems for mixing random variables with values in Hilbert space and their applications.- IV. Local Times.- Local time-space calculus and extensions of Ito's formula.- Local times on curves and surfaces.- V. Large, Small Deviations.- Large deviations of empirical processes.- Small deviation estimates for some additive processes.- VI. Density Estimation.- Convergence in distribution of self-normalized sup-norms of kernel density estimators.- Estimates of the rate of approximation in the CLT for L1-norm of density estimators.- VII. Statistics via Empirical Process Theory.- Statistical nearly universal Glivenko-Cantelli classes.- Smoothed empirical processes and the bootstrap.- A note on the asymptotic distribution of Berk-Jones type statistics under the null hypothesis.- A note on the smoothed bootstrap.

Bootstrap Testing of the Rank of a Matrix via Least-Squared Constrained Estimation

2013-11-14
François Portier , Bernard Delyon
AbstractTo test if an unknown matrix M0 has a given rank (null hypothesis noted H0), we consider a statistic that is a squared distance between an estimator and the submanifold … AbstractTo test if an unknown matrix M0 has a given rank (null hypothesis noted H0), we consider a statistic that is a squared distance between an estimator and the submanifold of fixed-rank matrix. Under H0, this statistic converges to a weighted chi-squared distribution. We introduce the constrained bootstrap (CS bootstrap) to estimate the law of this statistic under H0. An important point is that even if H0 fails, the CS bootstrap reproduces the behavior of the statistic under H0. As a consequence, the CS bootstrap is employed to estimate the nonasymptotic quantile for testing the rank. We provide the consistency of the procedure and the simulations shed light on the accuracy of the CS bootstrap with respect to the traditional asymptotic comparison. More generally, the results are extended to test whether an unknown parameter belongs to a submanifold of the Euclidean space. Finally, the CS bootstrap is easy to compute, it handles a large family of tests and it works under mild assumptions.KEY WORDS: Dimension reductionHypothesis testingRank estimation
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An Introduction to the Bootstrap

1994-05-15
Bradley Efron , Robert Tibshirani
This article presents bootstrap methods for estimation, using simple arguments. Minitab macros for implementing these methods are given. This article presents bootstrap methods for estimation, using simple arguments. Minitab macros for implementing these methods are given.

Foundations of Modern Probability

2021-01-01
Olav Kallenberg
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Unified Set Membership theory for identification, prediction and filtering of nonlinear systems

2011-08-29
M. Milanese , Carlo Novara

Rank inequalities for positive semidefinite matrices

1996-11-01
Michael Lundquist , Wayne Barrett

Bootstrap Procedures under some Non-I.I.D. Models

1988-12-01
Regina Y. Liu
It is shown in this article that the classical i.i.d. bootstrap remains a valid procedure for estimating the sampling distributions of certain symmetric estimators of location, as long as the … It is shown in this article that the classical i.i.d. bootstrap remains a valid procedure for estimating the sampling distributions of certain symmetric estimators of location, as long as the random observations are independently drawn from distributions with (essentially) a common location. This may be viewed as a robust property of the classical i.i.d. bootstrap. Also included is a study of the second order properties of a different bootstrap procedure proposed by Wu in the context of heteroscedasticity in regression.
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Digital‐correlation techniques in radio science

1973-08-01
Jon B. Hagen , D. T. Farley
Various digital techniques for estimating the autocorrelation function of a signal are analyzed, with emphasis on methods applicable to radio‐astronomy and incoherent‐scatter studies of the ionosphere. The importance of coarse … Various digital techniques for estimating the autocorrelation function of a signal are analyzed, with emphasis on methods applicable to radio‐astronomy and incoherent‐scatter studies of the ionosphere. The importance of coarse (one‐ or two‐bit) quantization is stressed, as is the effect of oversampling, the use of a sampling interval less than the inverse of twice the signal bandwidth. For applications such as incoherent scatter in which the signal strength varies with range (i.e., time), multibit by one‐bit multiplication combined with oversampling is probably the optimum strategy, although it requires 32% more integration time than the full multibit by multibit technique. For very high speed sampling of a signal of which the mean amplitude is constant, a three‐level by three‐level technique offers many advantages. It is nearly as simple to implement as conventional one‐bit correlation but requires considerably less integration time to achieve equal statistical accuracy (only 26% more than full multibit correlation, when sampled at twice the Nyquist rate, as compared to 146% for a one‐bit correlator with sampling at only the Nyquist rate).

Resampling Methods: A Practical Guide to Data Analysis

2000-08-01
Yogendra P. Chaubey , Phillip I. Good
The goal of this book is to introduce statistical methodology-estimation, hypothesis, testing and classification-to a wide applied audience through resampling from existing data via the bootstrap, and estimation or cross-validation … The goal of this book is to introduce statistical methodology-estimation, hypothesis, testing and classification-to a wide applied audience through resampling from existing data via the bootstrap, and estimation or cross-validation methods. The book provides an accessible introduction and practical guide to the power, simplicity and veritability of the bootstrap, cross-validation and permutation tests. Industrial statistical consultants, professionals and researchers will find the book's methods and software imimediately helpful. (unvollstandig)) This Second edition is a practical guide to data analysis using the bootstrap, cross-validation, and permutation tests. It is an essential resource for industrial statisticians, statistical consultants, and research professionals in science, engineering, and technology. Only requiring minimal mathematics beyond algebra, it provides a table-free introduction to data analysis utilizing numerous exercizes, practical data sets, and freely available statistical shareware. Topics and features: *Thoroughly revised text features more practical examples plus an additional chapter devoted to regression and data mining techniques and their limitations *Uses resampling approach to introduction statistics *A Practical presentation that covers all three sampling methods - bootstrap, density-estimation, and permutations *Includes systematic guide to help one select correct procedure for a particular application *Detailed coverage of all three statistical methodologies - classification, estimation, and hypothesis testing *Suitable for classroom use and individual, self-study purposes *Numerous practical examples using popular computer programs such as SAS, Stata, and StatXact *Useful appendices with computer programs and code to develop own methods *Downloadable freeware from author's website: http://users.oco.net/drphilgood/resamp.htm With its accessable style and intuitive topic development, the book is an excellent basic resource and guide to the power, simplicity and versatility of bootstrap, cross-validation and permutation tests. Students, professionals, and researchers will find it a particularly useful guide to modern resampling methods and their applications.
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