Error Exponent Research Articles

Appointed-time guaranteed adaptive fault-tolerant attitude tracking for quadrotors with aperiodic data updating

In this paper, an appointed-time guaranteed adaptive fault-tolerant attitude tracking for quadrotors with aperiodic data updating is concerned, where inertial moment variation, extraneous disturbance and unknown actuator faults are simultaneously involved. An adaptive parameter estimation law is updated to identify the partial effectiveness loss of actuation and alleviate the estimation burden of suggested observer. Then through filtering manipulations upon aperiodic system states, a novel event-triggered unknown system dynamics estimator (ETUSDE) is developed to learn the remaining uncertainties with exponential error decaying and decreased updating cost, where an event-triggering criterion is incorporated to schedule the transmission instants for control behavior and angular rate, and the explicit formulation between estimation error upper boundary and design coefficients is derived. An appointed-time prescribed performance control (APPC) is introduced to allow for pregiven attitude behaviors. By fusion of adaptive law, ETUSDE and APPC, an adaptive event-triggered attitude control policy is synthesized that delivers appointed-time convergence, accurate steady response and decreased sampling burden. Meanwhile, Lyapunov analysis shows that all the error variables contained in the closed-loop system are ultimately uniformly bounded. Eventually, experiments on a quadrotor attitude test rig are conducted to confirm the superiorities of suggested control strategy.

The effect of corneal power distribution on axial elongation in children using three different orthokeratology lens designs

PurposeTo investigate the correlation between spatial corneal power distribution and one-year axial length (AL) elongation using three ortho-k lens designs by a unified mathematical method. MethodsA total of 137 subjects were included: 42 with Euclid lenses, 28 with DRL lenses, and 67 with CRT lenses. AL elongation, Xmax, Ymax and power exponent were compared among the three groups. One-year relative corneal refractive power change (RCRPC) was calculated by a polynomial function and a monomial function. Factors including age, baseline spherical equivalent refractive error (SER), Xmax, Ymax and power exponent was tested against one-year AL growth in a stepwise multiple linear regression model. ResultsThe power exponent (F = 7.29, P = 0.0012) and Xmax (F = 62.88, P < 0.0001) of the DRL group was significantly smaller than that of the other two lens groups. Ymax was not significantly different among three lens groups (F = 1.18, P = 0.31). The one-year AL elongation of the DRL group (0.09 ± 0.14 mm) was significantly slower than that of the Euclid group (0.26 ± 0.14 mm, P = 0.002) and CRT group (0.32 ± 0.18 mm, P < 0.0001). AL elongation was significantly correlated with Xmax (standardized β = 0.196, P = 0.003), power exponent (standardized β = 0.644, P < 0.001), and age (standardized β = -0.263, P < 0.001), with R2 being 0.608. ConclusionA smaller and more aspheric treatment zone may be beneficial for reducing axial elongation in children undergoing ortho-k treatment, regardless of their baseline myopic refractive error.

Interactive Nearest Lattice Point Search in a Distributed Setting: Two Dimensions

The nearest lattice point problem in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {R}^{n}$ </tex-math></inline-formula> is formulated in a distributed network with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> nodes. The objective is to minimize the probability that an incorrect lattice point is found, subject to a constraint on inter-node communication. Algorithms with a single as well as an unbounded number of rounds of communication are considered for the case <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n=2$ </tex-math></inline-formula> . For the algorithm with a single round, expressions are derived for the error probability as a function of the total number of communicated bits. We observe that the error exponent depends on the lattice structure and that zero error requires an infinite number of communicated bits. In contrast, with an infinite number of allowed communication rounds, the nearest lattice point can be determined without error with a finite average number of communicated bits and a finite average number of rounds of communication. In two dimensions, the hexagonal lattice, which is most efficient for communication and compression, is found to be the most expensive in terms of communication cost.

Pricing efficiency in livestock auction markets: A two‐tier frontier approach

AbstractWe estimate the pricing efficiency of a livestock auction market using a two‐tier stochastic frontier model. This model captures pricing inefficiency of a hedonic pricing model for livestock via exponential error terms above and below the hedonic model specification. The direction of pricing inefficiencies, whether above or below the hedonic pricing model, represent surplus transfer from buyers to sellers or from sellers to buyers, respectively. Using data from a 7‐year period in a livestock auction market, we found that on average livestock buyers extract about 6% of surplus from sellers. That is, on average, livestock auction market underprices by about 6% below its full information price. Buyer concentration also affects the degree of surplus transfer. At low levels of buyer concentration, when the top four‐buyer market concentration is &lt;30%, there is an overpricing of only 1.23% over the full information price, extracting surplus from buyers to sellers. At higher buyer concentrations, buyers can extract up to 7% of surplus from sellers.

Estimating Analytical Errors of Glomerular Filtration Rate Measurement.

Few studies are available on how to optimize time points for sampling and how to estimate effects of analytical uncertainty when glomerular filtration rate (GFR) is calculated. We explored the underlying regression mathematics of how analytical variation of a kidney filtration marker affects 1-compartment, slope-and-intercept GFR calculations, using 2 or 3 time points following a bolus injection, and used this to examine the results from 731 routine 3-point iohexol plasma clearance measurements. GFR calculations inflated analytical uncertainty if the time points were taken too late after the bolus injection and too close after each other. The uncertainty in GFR calculation was, however, the same as the analytical uncertainty if optimal time points were used. The middle of the 3 samples was of little value. The first sample should be taken as early as possible after the distribution phase. Sampling before the patient specific half-life of the kidney filtration marker resulted in an exponential error inflation whereas no error inflation was seen when sampling occurred later than 2 half-lives. Theoretical GFR uncertainty could be lowered 2.6-fold if individually optimized time points for sampling had been used in our 731 clearance measurements. Using Taylor expansions to approximate the moments of transformed random variables, the uncertainty of an individual GFR measurement could be calculated in a simple enough way to be applicable by laboratory software. We provide a theoretical foundation to select patient-optimal time points that may both limit errors and allow calculation of GFR uncertainty.

On the Reliability Function for a BSC with Noiseless Feedback at Zero Rate

We consider the transmission of nonexponentially many messages through a binary symmetric channel with noiseless feedback. We obtain an upper bound for the best decoding error exponent. Combined with the corresponding known lower bound, this allows us to find the reliability function for this channel at zero rate.

Noiseless linear amplification in quantum target detection using Gaussian states

Quantum target detection aims to utilise quantum technologies to achieve performances in target detection not possible through purely classical means. Quantum illumination is an example of this, based on signal–idler entanglement, promising a potential 6 dB advantage in error exponent over its optimal classical counterpart. So far, receiver designs achieving this optimal reception remain elusive with many proposals based on Gaussian processes appearing unable to utilise quantum information contained within Gaussian state sources. This paper considers the employment of a noiseless linear amplifier at the detection stage of a quantum illumination-based quantum target detection protocol. Such a non-Gaussian amplifier offers a means of probabilistically amplifying an incoming signal without the addition of noise. Considering symmetric hypothesis testing, the quantum Chernoff bound is derived and limits on detection error probability is analysed for both the two-mode squeezed vacuum state and the coherent state classical benchmark. Our findings show that in such a scheme the potential quantum advantage is amplified even in regimes where quantum illumination alone offers no advantage, thereby extending its potential use. The same cannot be said for coherent states, whose performances are generally bounded by that without amplification.

Fractional Rectified Linear Unit Activation Function and Its Variants

This paper focuses on deriving and validating the fractional-order form of rectified linear unit activation function and its linear and nonlinear variants. The linear variants include the leaky and parametric, whereas the nonlinear variants include the exponential, sigmoid-weighted, and Gaussian error functions. Besides, a standard formula has been created and used while developing the fractional form of linear variants. Moreover, different expansion series such as Maclaurin and Taylor have been used while designing the fractional version of nonlinear variants. A simulation study has been conducted to validate the performance of all the developed fractional activation functions utilizing a single and multilayer neural network model and to compare them with their conventional counterparts. In this simulation study, a neural network model has been created to predict the system-generated power of a Texas wind turbine. The performance has been evaluated by varying the activation function in the hidden and output layers with the developed functions for single and multilayer networks.

Decoder Ties Do Not Affect the Error Exponent of the Memoryless Binary Symmetric Channel

The generalized Poor-Verdú error lower bound established by Chang <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> (2020) for multihypothesis testing is studied in the classical channel coding context. It is proved that for any sequence of block codes sent over the memoryless binary symmetric channel (BSC), the minimum probability of error (under maximum likelihood decoding) has a relative deviation from the generalized bound that grows at most linearly in blocklength. This result directly implies that for arbitrary codes used over the BSC, decoder ties can only affect the subexponential behavior of the minimum probability of error.

Risk sensitive filtering with randomly delayed measurements

Conventional Bayesian estimation requires an accurate stochastic model of a system. However, this requirement is not always met in many practical cases where the system is not completely known or may differ from the assumed model. For such a system, we consider a scenario where the measurements are transmitted to a remote location using a common communication network and due to which, a delay is introduced while receiving the measurements. The delay that we consider here is random and one step maximum at a given time instant. For such a scenario, this paper develops a robust estimator for a linear Gaussian system by minimizing the risk sensitive error criterion that is defined as an expectation of the accumulated exponential quadratic error. The criteria for the stability of the risk sensitive Kalman filter (RSKF) are derived and the results are used to study the stability of the developed filter. Further, it is assumed that the latency probability related to delay is not known and it is estimated by maximizing the likelihood function. Simulation results suggest that the proposed filter shows acceptable performance under the nominal conditions, and it performs better than the Kalman filter for randomly delayed measurements and the RSKF in presence of both the model uncertainty and random delays.

Combined Dissipative and Hamiltonian Confinement of Cat Qubits

Quantum error correction with biased-noise qubits can drastically reduce the hardware overhead for universal and fault-tolerant quantum computation. Cat qubits are a promising realization of biased-noise qubits as they feature an exponential error bias inherited from their non-local encoding in the phase space of a quantum harmonic oscillator. To confine the state of an oscillator to the cat qubit manifold, two main approaches have been considered so far: a Kerr-based Hamiltonian confinement with high gate performances, and a dissipative confinement with robust protection against a broad range of noise mechanisms. We introduce a new combined dissipative and Hamiltonian confinement scheme based on two-photon dissipation together with a Two-Photon Exchange (TPE) Hamiltonian. The TPE Hamiltonian is similar to Kerr nonlinearity, but unlike the Kerr it only induces a bounded distinction between even- and odd-photon eigenstates, a highly beneficial feature for protecting the cat qubits with dissipative mechanisms. Using this combined confinement scheme, we demonstrate fast and bias-preserving gates with drastically improved performance compared to dissipative or Hamiltonian schemes. In addition, this combined scheme can be implemented experimentally with only minor modifications of existing dissipative cat qubit experiments.

Motion control for a differential vehicle with variable point of interest. Application: Smart cane control

This article addresses an unified solution to the trajectory tracking and path following problems for differential drive mobile robots (DDMR) considering a point of interest (PoI) with variable location relative to the vehicle. The mobile robot is modeled with an extended kinematic model avoiding typical singularities of this kind of vehicles, and allowing a straightforward definition of the corresponding inverse kinematics controller (IKC). This classical IKC fulfills the control objective with exponential error convergence but with the shortcoming of generating backward navigation when the PoI is located behind the DDMR, which is undesirable in some practical applications where the forward navigation must be preserved. This situation is theoretically analyzed, concluding that even though both forward and backward navigations correspond to equilibrium points of the closed loop, the stability of the forward navigation requires a PoI located in front of the DDMR, and the stability of the backward navigation requires a PoI located behind the DDMR. Finally, the article presents novel alternative controllers in order to always fulfill the motion objectives with stable forward navigation. Simulation results are presented to show the performance of the proposed controllers, and a real application of a robotic cane guiding its user is experimentally developed.

Analysis on Optimal Error Exponents of Binary Classification for Source with Multiple Subclasses

We consider a binary classification problem for a test sequence to determine from which source the sequence is generated. The system classifies the test sequence based on empirically observed (training) sequences obtained from unknown sources and . We analyze the asymptotic fundamental limits of statistical classification for sources with multiple subclasses. We investigate the first- and second-order maximum error exponents under the constraint that the type-I error probability for all pairs of distributions decays exponentially fast and the type-II error probability is upper bounded by a small constant. In this paper, we first give a classifier which achieves the asymptotically maximum error exponent in the class of deterministic classifiers for sources with multiple subclasses, and then provide a characterization of the first-order error exponent. We next provide a characterization of the second-order error exponent in the case where only has multiple subclasses but does not. We generalize our results to classification in the case that and are a stationary and memoryless source and a mixed memoryless source with general mixture, respectively.

Optimal Adaptive Strategies for Sequential Quantum Hypothesis Testing

We consider sequential hypothesis testing between two quantum states using adaptive and non-adaptive strategies. In this setting, samples of an unknown state are requested sequentially and a decision to either continue or to accept one of the two hypotheses is made after each test. Under the constraint that the number of samples is bounded, either in expectation or with high probability, we exhibit adaptive strategies that minimize both types of misidentification errors. Namely, we show that these errors decrease exponentially (in the stopping time) with decay rates given by the measured relative entropies between the two states. Moreover, if we allow joint measurements on multiple samples, the rates are increased to the respective quantum relative entropies. We also fully characterize the achievable error exponents for non-adaptive strategies and provide numerical evidence showing that adaptive measurements are necessary to achieve our bounds.

Universal Decoding for the Typical Random Code and for the Expurgated Code

We provide two results concerning the optimality of the stochastic-mutual information (SMI) decoder, which chooses the estimated message according to a posterior probability mass function, which is proportional to the exponentiated empirical mutual information induced by the channel output sequence and the different codewords. First, we prove that the error exponents of the typical random codes under the optimal maximum likelihood (ML) decoder and the SMI decoder are equal. As a corollary to this result, we also show that the error exponents of the expurgated codes under the ML and the SMI decoders are equal. These results strengthen the well-known result due to Csiszár and Körner, according to which, the ML and the maximum-mutual information (MMI) decoders achieve equal random-coding error exponents, since the error exponents of the typical random code and the expurgated code are strictly higher than the random-coding error exponents, at least at low coding rates. The universal optimality of the SMI decoder, in the random-coding error exponent sense, is easily proven by commuting the expectation over the channel noise and the expectation over the ensemble. This commutation can no longer be carried out, when it comes to typical and expurgated exponents. Therefore, the proof of the universal optimality of the SMI decoder must be completely different and it turns out to be highly non-trivial.

Improved unscented Kalman filter algorithm-based rapid identification of thermal errors of machine tool spindle for shortening thermal equilibrium time

Thermal equilibrium of machine tools is needed before machining to ensure the accuracy. To shorten the warm-up time and improve the machining efficiency, a rapid identification of thermal errors is proposed. It can predict the thermal errors in the shortest identification time so that the thermal error compensation can be started in advance to shorten the thermal equilibrium time. A Singular Value Decomposition-Adaptive Unscented Kalman Filter algorithm (SVD-AUKF) is proposed with singular value decomposition and adaptive rule for the fast identification. A general exponential thermal error model is established and applied into the SVD-AUKF. RMSE of the thermal error data within each time is calculated for the evaluation to obtain the shortest identification time. K hysteresis rule, Δt judgment rule and maximum time rule are proposed for the prediction accuracy. Different experiments are proposed with two machine tools to testify the effectiveness of the shortest identification time.

On the antiderivatives of xp/(1 \u2212 x) with an application to optimize loss functions for classification with neural networks

Supervised learning in neural nets means optimizing synaptic weights W such that outputs y(x;W) for inputs x match as closely as possible the corresponding targets t from the training data set. This optimization means minimizing a loss function {mathscr{L}}(mathbf {W}) that usually motivates from maximum-likelihood principles, silently making some prior assumptions on the distribution of output errors y −t. While classical crossentropy loss assumes triangular error distributions, it has recently been shown that generalized power error loss functions can be adapted to more realistic error distributions by fitting the exponent q of a power function used for initializing the backpropagation learning algorithm. This approach can significantly improve performance, but computing the loss function requires the antiderivative of the function f(y) := yq− 1/(1 − y) that has previously been determined only for natural qin mathbb {N}. In this work I extend this approach for rational q = n/2m where the denominator is a power of 2. I give closed-form expressions for the antiderivative {int limits } f(y) dy and the corresponding loss function. The benefits of such an approach are demonstrated by experiments showing that optimal exponents q are often non-natural, and that error exponents q best fitting output error distributions vary continuously during learning, typically decreasing from large q > 1 to small q < 1 during convergence of learning. These results suggest new adaptive learning methods where loss functions could be continuously adapted to output error distributions during learning.

On Distributed Learning With Constant Communication Bits

In this paper, we study a distributed learning problem constrained by constant communication bits. Specifically, we consider the distributed hypothesis testing (DHT) problem where two distributed nodes are constrained to transmit a constant number of bits to a central decoder. In such settings, we show that in order to achieve the optimal error exponents, it suffices to consider the empirical distributions of observed data sequences and encode them to the transmission bits. With such a coding strategy, we develop a geometric approach in the distribution spaces and establish an inner bound of error exponent regions. In particular, we show the optimal achievable error exponents and coding schemes for the following cases: (i) both nodes can transmit <inline-formula> <tex-math notation="LaTeX">$\log _{2}3$ </tex-math></inline-formula> bits; (ii) one of the nodes can transmit 1 bit, and the other node is not constrained; (iii) the joint distribution of the nodes are conditionally independent given one hypothesis. Furthermore, we provide several numerical examples for illustrating the theoretical results. Our results provide theoretical guidance for designing practical distributed learning rules, and the developed approach also reveals new potentials for establishing error exponents for DHT with more general communication constraints.

Joint Communication and Binary State Detection

We study a basic joint communication and sensing setup from an information theoretic perspective. A transmitter sends an encoded message through a pair of noisy channels connected to a receiver and a sensor. The receiver is interested in decoding the message from its noisy observation. The sensor has access to the message as side information, and instead is interested in estimating an unknown yet fixed binary state of its channel. This basic setup models scenarios where a wireless transceiver broadcasts an information bearing waveform, to be decoded by possibly more than one receiver, and then observes a modulated echo of the waveform from which it wishes to detect an element or object in the environment. We consider binary symmetric and Gaussian settings with multiplicative binary states, and establish the fundamental trade-off between reliable message communication and efficient state detection in these settings. This trade-off is captured by the message communication rate against the state detection error exponent. The results give insights into the benefits of carrying out communication and sensing jointly.

Efficient computations of continuous action densities of states for lattice models

The Logarithmic Linear Relaxation (LLR) algorithm is an efficient method for computing densities of states for systems with a continuous spectrum. A key feature of this method is exponential error reduction, which allows us to evaluate the density of states of a system over hundreds of thousands of orders of magnitude with a fixed level of relative accuracy. As a consequence of exponential error reduction, the LLR method provides a robust alternative to traditional Monte Carlo calculations in cases in which states suppressed by the Boltzmann weight play nevertheless a relevant role, e.g., as transition regions between dominant configuration sets. After reviewing the algorithm, we will show an application in U(1) Lattice Gauge Theory that has enabled us to obtain the most accurate estimate of the critical coupling with modest computational resources, defeating exponential tunneling times between metastable vacua. As a further showcase, we will then present an application of the LLR method to the decorrelation of the topological charge in SU(3) Lattice Gauge Theory near the continuum limit. Finally, we will review in general applications of the LLR algorithm to systems affected by a strong sign problem and discuss the case of the Bose gas at finite chemical potential.