Approximate Maximum Likelihood Research Articles

Tailoring Surface Codes for Highly Biased Noise

The surface code, with a simple modification, exhibits ultra-high error correction thresholds when the noise is biased towards dephasing. Here, we identify features of the surface code responsible for these ultra-high thresholds. We provide strong evidence that the threshold error rate of the surface code tracks the hashing bound exactly for all biases, and show how to exploit these features to achieve significant improvement in logical failure rate. First, we consider the infinite bias limit, meaning pure dephasing. We prove that the error threshold of the modified surface code for pure dephasing noise is $50\%$, i.e., that all qubits are fully dephased, and this threshold can be achieved by a polynomial time decoding algorithm. We demonstrate that the sub-threshold behavior of the code depends critically on the precise shape and boundary conditions of the code. That is, for rectangular surface codes with standard rough/smooth open boundaries, it is controlled by the parameter $g=\gcd(j,k)$, where $j$ and $k$ are dimensions of the surface code lattice. We demonstrate a significant improvement in logical failure rate with pure dephasing for co-prime codes that have $g=1$, and closely-related rotated codes, which have a modified boundary. The effect is dramatic: the same logical failure rate achievable with a square surface code and $n$ physical qubits can be obtained with a co-prime or rotated surface code using only $O(\sqrt{n})$ physical qubits. Finally, we use approximate maximum likelihood decoding to demonstrate that this improvement persists for a general Pauli noise biased towards dephasing. In particular, comparing with a square surface code, we observe a significant improvement in logical failure rate against biased noise using a rotated surface code with approximately half the number of physical qubits.

Inference on Weibull parameters under a balanced two‐sample type II progressive censoring scheme

AbstractThe progressive censoring scheme has received a considerable amount of attention in the last 15 years. During the last few years, joint progressive censoring scheme has gained some popularity. Recently, the authors Mondal and Kundu (“A New Two Sample Type‐II Progressive Censoring Scheme,” Communications in Statistics‐Theory and Methods) introduced a balanced two‐sample type II progressive censoring scheme and provided the exact inference when the two populations are exponentially distributed. In this article, we consider the case when the two populations follow Weibull distributions with the common shape parameter and different scale parameters. We obtain the maximum likelihood estimators of the unknown parameters. It is observed that the maximum likelihood estimators cannot be obtained in explicit forms; hence, we propose approximate maximum likelihood estimators, which can be obtained in explicit forms. We construct the asymptotic and bootstrap confidence intervals of the population parameters. Further, we derive an exact joint confidence region of the unknown parameters. We propose an objective function based on the expected volume of this confidence region, and using that, we obtain the optimum progressive censoring scheme. Extensive simulations have been performed to see the performances of the proposed method, and one real data set has been analyzed for illustrative purposes.

Unsynchronized OFDM network positioning in multipath

Position estimation of a target node based on time-of-flight measurements in multipath environments is challenging, especially when the signal bandwidth is limited, such as is the case in Wi-Fi 802.11. Furthermore, the availability of location based services in today’s wireless networks is still limited, since it requires either the overhead of synchronizing the base stations (BSs) or applying modification in the BSs firmware and hardware to enable round trip delay measurements by two-way transmission. In this paper, an algorithm for location estimation of a target node in a multipath environment based on one-way transmission of OFDM signals from unsynchronized base stations is developed, which enables to estimate the position of a device in a wireless network without requiring any modifications in the BSs. The solution approach is based on using a reference node with known position, which also receives the BSs transmissions, and deriving an approximate maximum likelihood direct position estimation of the target node based on the collected samples at both the target and the reference nodes. We demonstrate the algorithm’s performance with off-the-shelf Wi-Fi 802.11g routers, which are unsynchronized and do not include any dedicated protocol for range or location estimation. The novel algorithm attains a relatively low position root mean square error of 1.6m.

Model-based clustering of time-evolving networks through temporal exponential-family random graph models

Dynamic networks are a general language for describing time-evolving complex systems, and discrete time network models provide an emerging statistical technique for various applications. It is a fundamental research question to detect a set of nodes sharing similar connectivity patterns in time-evolving networks. Our work is primarily motivated by detecting groups based on interesting features of the time-evolving networks (e.g., stability). In this work, we propose a model-based clustering framework for time-evolving networks based on discrete time exponential-family random graph models, which simultaneously allows both modeling and detecting group structure. To choose the number of groups, we use the conditional likelihood to construct an effective model selection criterion. Furthermore, we propose an efficient variational expectation–maximization (EM) algorithm to find approximate maximum likelihood estimates of network parameters and mixing proportions. The power of our method is demonstrated in simulation studies and empirical applications to international trade networks and the collaboration networks of a large research university.

Fusion Rules for Distributed Detection in Clustered Wireless Sensor Networks With Imperfect Channels

In this paper, we investigate fusion rules for distributed detection in large random clustered wireless sensor networks with a three-tier hierarchy; the sensor nodes (SNs), the cluster heads (CHs), and the fusion center (FC). The CHs collect the SNs’ local decisions and relay them to the FC, which then fuses them to reach the ultimate decision. The SN–CH and the CH–FC channels suffer from additive white Gaussian noise. In this context, we derive the optimal log-likelihood ratio (LLR) fusion rule, which turns out to be intractable. So, we develop a sub-optimal linear fusion rule (LFR) that weighs the cluster's data according to both its local detection performance and the quality of the communication channels. In order to implement it, we propose an approximate maximum likelihood based LFR (LFR-aML), which estimates the required parameters for the LFR. We also derive Gaussian-tail upper bounds for the detection and false alarms probabilities for the LFR. Furthermore, an optimal CH transmission power allocation strategy is developed by solving the Karush–Kuhn–Tucker conditions for the related optimization problem. Extensive simulations show that the LFR attains a detection performance near to that of the optimal LLR and confirms the validity of the proposed upper bounds. Moreover, when compared to equal power allocation, simulations show that our proposed power allocation strategy achieves a significant power saving at the expense of a small reduction in the detection performance.

A graph subspace approach to system identification based on errors-in-variables system models

System identification based on the errors-in-variables (EIV) system model has been investigated by a number of people, led by Söderström and others. The total least-squares (TLS) algorithm is now well known, and has been effective for estimating the system parameters. In this paper, we first show that the TLS algorithm computes approximate maximum likelihood estimate (MLE) of the system parameters. Then we propose a graph subspace approach to tackle the same EIV identification problem, and derive a new estimation algorithm that is more general than the TLS algorithm. Two numerical examples are worked out to illustrate the proposed estimation algorithm for the EIV-based system identification.

A Generalized Phase Gradient Autofocus Algorithm

The phase gradient autofocus (PGA) algorithm has seen widespread use and success within the synthetic aperture radar (SAR) imaging community. However, its use and success has largely been limited to collection geometries where either the polar format algorithm (PFA) or range migration algorithm is suitable for SAR image formation. In this paper, a generalized phase gradient autofocus (GPGA) algorithm is developed that is applicable with both the PFA and backprojection algorithm (BPA), thereby directly supporting a wide range of collection geometries and SAR imaging modalities. The GPGA algorithm is shown to preserve the four crucial signal processing steps comprising the PGA algorithm while alleviating the constraint of using a single scatterer per range cut for phase error estimation, which exists with the PGA algorithm. Moreover, the GPGA algorithm, whether using the PFA or BPA, is shown to yield an approximate maximum marginal likelihood estimate of phase errors having marginalized over unknown complex-valued reflectivities of selected scatterers. Simulation and experimental results produced by applying the GPGA algorithm to PFA and BPA SAR images are presented.

Approximate maximum likelihood estimation of a threshold diffusion process

In order to estimate the parameters of a two-regime threshold diffusion process with discretely sampled data, an approximate maximum likelihood method (AMLE) based on approximating the log-likelihood function of the observations is proposed. Both the drift and the diffusion terms are allowed to be either linear or non-linear. In order to choose the most appropriate among these four possibilities, three information criteria are employed. Further, a likelihood ratio test can help to determine whether threshold effects are present. Via simulations, the finite sample performance of the proposed AMLE is compared to an alternative quasi-likelihood estimator and the finite sample performance of the information criteria as well as the likelihood ratio test are studied. Finally, the efficacy of our approach is demonstrated with two financial time series.

Algebraic distributed source localisation algorithm using TDOA and AOA measurements

This study proposes an algebraic distributed source localisation algorithm that combines time difference of arrival (TDOA) and angle of arrival (AOA) measurements. The proposed algorithm uses AOAs to remove the unknown parameters in TDOA equations caused by the specially distributed structure. Then, the observation equations are transformed into a set of pseudo-linear equations and apply linear weighted least square to obtain the source position. The application of weighting matrix can lead to an approximate maximum likelihood estimator and produce a substantial improvement in source localisation accuracy. Both theoretical analysis and simulation results indicate the efficiency of the proposed algorithm and its performance can achieve the Cramer–Rao lower bound at a moderate noise level.

Capacity-Achieving Guessing Random Additive Noise Decoding

We introduce a new algorithm for realizing Maximum Likelihood (ML) decoding in discrete channels with or without memory. In it, the receiver rank orders noise sequences from most likely to least likely. Subtracting noise from the received signal in that order, the first instance that results in a member of the code-book is the ML decoding. We name this algorithm GRAND for Guessing Random Additive Noise Decoding. We establish that GRAND is capacity-achieving when used with random code-books. For rates below capacity we identify error exponents, and for rates beyond capacity we identify success exponents. We determine the scheme's complexity in terms of the number of computations the receiver performs. For rates beyond capacity, this reveals thresholds for the number of guesses by which if a member of the code-book is identified it is likely to be the transmitted code-word. We introduce an approximate ML decoding scheme where the receiver abandons the search after a fixed number of queries, an approach we dub GRANDAB, for GRAND with ABandonment. While not an ML decoder, we establish that the algorithm GRANDAB is also capacity-achieving for an appropriate choice of abandonment threshold, and characterize its complexity, error and success exponents. Worked examples are presented for Markovian noise that indicate these decoding schemes substantially out-perform the brute force decoding approach.

Reliability estimation in a multicomponent stress–strength model under generalized half-normal distribution based on progressive type-II censoring

ABSTRACTIn this paper, based on progressively Type-II censored samples, the problem of estimation of multicomponent stress–strength reliability under generalized half-normal (GHN) distribution is considered. The reliability of a k-component stress-strength system is estimated when both stress and strength variates are assumed to have a GHN distribution with various cases of same and different shape and scale parameters. Different methods such as the maximum likelihood estimates (MLEs) and Bayes estimation are discussed. The expectation maximization algorithm and approximate maximum likelihood methods are proposed to compute the MLE of reliability. The Lindley's approximation method, as well as Metropolis–Hastings algorithm, are applied to compute Bayes estimates. The performance of the proposed procedures is also demonstrated via a Monte Carlo simulation study and an illustrative example.

Exponential Dispersion Process for Degradation Analysis

The methods for analyzing degradation data are usually based on a specific degradation model. However, this may result in a large bias to estimate the reliability of the product when the assumed model is wrong. In this paper, a new stochastic process, called exponential dispersion process, is developed to describe the degradation path of the product's physical or chemical characteristics. Exponential dispersion process includes the most used degradation models (Wiener process, gamma process, inverse Gaussian process, and compound Poisson process) as special cases. Thus, it is useful for suggesting an appropriate degradation model for a specific dataset. We investigate the exponential dispersion model with nonlinear degradation path, and use approximated maximum likelihood method and bootstrap approach to obtain the point and confidence interval estimates of the unknown model parameters. Furthermore, exponential dispersion process models with acceleration factors and random effects are also explored. The developed methodologies are then applied to real data analysis.

Statistical inference for the exponentiated half logistic distribution based on censored data

In this paper, we consider the estimates for the scale parameter of an exponentiated half logistic distribution under progressively Type-II censoring and general progressively Type-II censoring, respectively. Because of the complex form of the traditional maximum likelihood estimator, it is hard to get an explicit solution. Therefore, we propose three new estimation methods based on censored data to obtain approximate estimates of the scale parameter. The first two are approximate maximum likelihood estimation methods based on Taylor expansion and the third is the approximate estimation method based on pivotal quantities we establish. Furthermore, we consider asymptotic confidence intervals based on pivotal quantities. Finally, simulation studies are conducted to compare the performance of the proposed estimators. A real dataset analysis is carried out to illustrate the proposed methods.

Injection of cocaine is associated with a recent HIV outbreak in people who inject drugs in Luxembourg.

BackgroundAn outbreak of HIV infections among people who inject drugs (PWID) started in 2014 in Luxembourg.ObjectivesWe conducted phylogenetic and epidemiological analyses among the PWID infected with HIV in Luxembourg or attending the supervised drug consumption facility (SDCF) to understand the main causes of the outbreak.MethodsBetween January 2013 and December 2017, analysis of medical files were performed from all PWID infected with HIV at the National Service of Infectious Diseases (NSID) providing clinical care nationwide. PWID were interviewed at NSID and SDCF using a standardized questionnaire focused on drug consumption and risk behaviours. The national drug monitoring system RELIS was consulted to determine the frequency of cocaine/heroin use. Transmission clusters were analysed by phylogenetic analyses using approximate maximum-likelihood. Univariate and multivariate logistic regression analyses were performed on epidemiological data collected at NSID and SDCF to determine risk factors associated with cocaine use.ResultsFrom January 2013 to December 2017, 68 new diagnosis of HIV infection reported injecting drug use as the main risk of transmission at NSID. The proportion of female cases enrolled between 2013–2017 was higher than the proportion among cases enrolled prior to 2013. (33% vs 21%, p < 0.05). Fifty six viral sequences were obtained from the 68 PWID newly diagnosed for HIV. Two main transmission clusters were revealed: one HIV-1 subtype B cluster and one CRF14_BG cluster including 37 and 9 patients diagnosed since 2013, respectively. Interviews from 32/68 (47%) newly diagnosed PWID revealed that 12/32 (37.5%) were homeless and 27/32 (84.4%) injected cocaine. Increased cocaine injection was indeed reported by the RELIS participants from 53 to 63% in drug users with services contacts between 2012 and 2015, and from 5 to 22% in SDCF users between 2012 and 2016. Compared with PWID who injected only heroin (n = 63), PWID injecting cocaine and heroin (n = 107) were younger (mean of 38 vs 44 years, p≤0.001), reported more frequent piercing (≤0.001), shared and injected drugs more often (p≤0.01), and were more frequently HIV positive (p<0.05) at SDCF using univariate logistic regression analysis. Finally, in the multivariate analysis, use of heroin and cocaine was independently associated with younger age, piercing, sharing of drugs, and regular consumption (p<0.05).ConclusionsInjecting cocaine is a new trend of drug use in Luxembourg associated with HIV infection in this recent outbreak among PWID.

A Modified Hybrid Gamma and Generalized Pareto Distribution for Precipitation Data

This study introduces a modified hybrid gamma and generalized Pareto distribution. Prior to this, we define a general spliced distribution and its corresponding gamma distribution, which is part of the head, and a generalized Pareto (GP) distribution, which is part of the tail. We then examine the threshold conditions for the modified hybrid gamma and GP distribution and defined probability density function. Also, we derive the negative log-likelihood function of the modified hybrid gamma and GP distribution and estimate approximate maximum likelihood estimates using the differential evolution algorithm for each simulation to minimize it. Moreover, by presenting the mean square error for each sample size, the model is evaluated according to the size of the sample. Finally, we use daily observed summer precipitation for Seoul, Korea, from 1961 to 2011, which includes 4692 data sets. We use 2051 data sets corresponding to wet conditions. As a result, the estimated threshold of the modified hybrid gamma and GP distribution is 0.1455. After deriving Fisher information through the Hessian matrix, we also present the standard error of the maximum likelihood estimator.

Censoring Outliers in Radar Data: An Approximate ML Approach and its Analysis

This paper deals with the problem of censoring outliers in a class of complex multivariate elliptically contoured distributed radar data, which is a vital issue in radar signal processing applications, such as adaptive radar detection and space-time adaptive processing. The maximum likelihood (ML) estimate of the outlier subset is derived resorting to the generalized likelihood function (GLF) criterion. Since the ML estimate involves the solution of a combinatorial problem, a reduced complexity but approximate ML (AML) procedure is also considered. At the analysis stage, the performance of the AML method is evaluated in the presence of both simulated and real radar data, also in comparison with the conventional generalized inner product (GIP) and the reiterative censored GIP (RCGIP) algorithms. The results highlight that the AML technique achieves a satisfactory performance level and can outperform both GIP and RCGIP in some situations.

Approximate MLSE Equalization of SOQPSK-TG in Aeronautical Telemetry

An approximate maximum likelihood sequence estimator (MLSE) equalizer for shaped offset QPSK, version TG (SOQPSK-TG) is derived based on the pulse amplitude modulation (PAM) approximation of SOQPSK-TG using the two principal PAM pulses. The resulting equalizer is a generalization of the equalizer based on the Ungerboeck observation model, where the generalization is to two parallel, cross-correlated symbol streams over parallel intersymbol interference channels. The bit error rate (BER) performance and computational complexity of the approximate MLSE equalizer operating over three example channels, motivated by multipath propagation in aeronautical telemetry and with increasing selectivity, are presented. The BER performance was evaluated using computer simulations and shown to be approximately 5–8 dB better than the best frequency-domain minimum mean-squared error (MMSE) equalizers published in the open literature. The computational complexity, measured by the number of real-valued multiplications, is comparable to that of the frequency-domain MMSE equalizers.

Quantification of uncertainties in thermoelectric properties of materials from a first-principles prediction method: An approach based on Gaussian process regression

We present the electron-phonon averaged via Gaussian process regression (EPA-GPR) method, in which the electron-phonon coupling matrix is represented as a function of two energies and is in turn modeled as a Gaussian process. The EPA-GPR method can be used as an efficient method to estimate thermoelectric properties of materials for fast-screening applications, comparable to the original electron-phonon averaged (EPA) method and the electron-phonon averaged via moving-least-squares (EPA-MLS) method. The EPA-GPR method does not require specification of any open parameter, unlike the other EPA-related methods, since all the hyperparameters in the model can be unambiguously estimated within the type II maximum likelihood (ML-II) approximation. Thus, the EPA-GPR method is a parameter-free estimation method. Additionally, the concept of Gaussian processes in the EPA-GPR method allows us to quantify the uncertainty in estimated properties of thermoelectric materials. One can randomly realize the electron-phonon coupling coefficients from the identified Gaussian process, and those realized samples can be further analyzed in the solution process of the semiclassical Boltzmann transport equation for charge carriers. The results of the semiclassical Boltzmann transport equation provide the statistical properties of the thermoelectric properties of interest. The means, standard deviations, histograms, and confidence intervals of the Seebeck coefficient, the electrical conductivity, and the power factor can be constructed and analyzed. The proposed EPA-GPR method is applied to a $p$-type half-Heusler compound, i.e., HfCoSb, as a case example, the results of which clearly present the advantages of the method.

Harmonic Retrieval Based Baseband Channel Estimation for Millimeter Wave OFDM Systems

For massive multi-input multi-output (MIMO) enhanced millimeter wave (mmWave) frequency-division multiplexing (OFDM) systems, channel estimation (CE) is challenging. In this paper, the baseband CE of mmWave based MIMO-OFDM systems after beam searching is formulated as a harmonic retrieval (HR) problem, where each path of the channel represents a harmonic with its frequency and strength to be estimated. We propose two methods, a windowed orthogonal matching pursuit (window-OMP) method and a windowed discrete Fourier transform (window-DFT) method, to approximately acquire the maximum likelihood (ML) estimate of the baseband CE. The window-OMP method is capable of approximating the ML estimator with high accuracy, while the window-DFT method has a lower complexity and is shown to acquire approximate ML estimate under the assumption that the frequencies of harmonics are sufficiently separated. Theoretical analysis is performed to derive a closed-form Cramer–Rao lower bound as well as to investigate the effect of wrong paths to the estimation accuracy. A simulation study is conducted to investigate the performance of the proposed methods, and the results obtained verify that our methods outperform the existing estimation of signal parameters by rotational invariant techniques based HR method and the conventional interpolation method.

A Variational Inference-Based Detection Method for Repetition Coded Generalized Spatial Modulation

In this paper, we consider a simple coding scheme for spatial modulation, where the same set of active transmit antennas is repeatedly used over consecutive multiple transmissions. Based on a Gaussian approximation, an approximate maximum likelihood (ML) detection problem is formulated to detect the indices of active transmit antennas. We show that the solution to the approximate ML detection problem can achieve a full coding gain. Furthermore, we develop a low-complexity iterative algorithm to solve the problem with low complexity based on a well-known machine learning approach, i.e., variational inference. Simulation results show that the proposed algorithm can have a near ML performance. A salient feature of the proposed algorithm is that its complexity is independent of the number of active transmit antennas, whereas an exhaustive search for the ML problem requires a complexity that grows exponentially with the number of active transmit antennas.