Showing papers on "Stochastic process published in 2018"

High-Dimensional Probability: An Introduction with Applications in Data Science

Abstract: High-dimensional probability offers insight into the behavior of random vectors, random matrices, random subspaces, and objects used to quantify uncertainty in high dimensions Drawing on ideas from probability, analysis, and geometry, it lends itself to applications in mathematics, statistics, theoretical computer science, signal processing, optimization, and more It is the first to integrate theory, key tools, and modern applications of high-dimensional probability Concentration inequalities form the core, and it covers both classical results such as Hoeffding's and Chernoff's inequalities and modern developments such as the matrix Bernstein's inequality It then introduces the powerful methods based on stochastic processes, including such tools as Slepian's, Sudakov's, and Dudley's inequalities, as well as generic chaining and bounds based on VC dimension A broad range of illustrations is embedded throughout, including classical and modern results for covariance estimation, clustering, networks, semidefinite programming, coding, dimension reduction, matrix completion, machine learning, compressed sensing, and sparse regression

Probability and Random Processes for Electrical and Computer Engineers

Abstract: With updates and enhancements to the incredibly successful first edition, Probability and Random Processes for Electrical and Computer Engineers, Second Edition retains the best aspects of the original but offers an even more potent introduction to probability and random variables and processes Written in a clear, concise style that illustrates the subject’s relevance to a wide range of areas in engineering and physical and computer sciences, this text is organized into two parts The first focuses on the probability model, random variables and transformations, and inequalities and limit theorems The second deals with several types of random processes and queuing theory New or Updated for the Second Edition: A short new chapter on random vectors that adds some advanced new material and supports topics associated with discrete random processes Reorganized chapters that further clarify topics such as random processes (including Markov and Poisson) and analysis in the time and frequency domain A large collection of new MATLAB®-based problems and computer projects/assignments Each Chapter Contains at Least Two Computer Assignments Maintaining the simplified, intuitive style that proved effective the first time, this edition integrates corrections and improvements based on feedback from students and teachers Focused on strengthening the reader’s grasp of underlying mathematical concepts, the book combines an abundance of practical applications, examples, and other tools to simplify unnecessarily difficult solutions to varying engineering problems in communications, signal processing, networks, and associated fields

Speed Limit for Classical Stochastic Processes.

Abstract: We consider the speed limit for classical stochastic Markov processes with and without the local detailed balance condition. We find that, for both cases, a trade-off inequality exists between the speed of the state transformation and the entropy production. The dynamical activity is related to a time scale and plays a crucial role in the inequality. For the dynamics without the local detailed balance condition, we use the Hatano-Sasa entropy production instead of the standard entropy production. Our inequalities consist of the quantities that are commonly used in stochastic thermodynamics and explicitly show underlying physical mechanisms.

Locally stationary random processes

Abstract: A new kind of random process, the locally stationary random process, is defined, which includes the stationary random process as a special case. Numerous examples of locally stationary random processes are exhibited. By the generalized spectral density \Psi(\omega, \omega \prime) of a random process is meant the two-dimensional Fourier transform of the covariance of the process; as is well known, in the case of stationary processes, \Psi(\omega, \omega \prime) reduces to a positive mass distribution on the line \omega = \omega \prime in the \omega, \omega \prime plane, a fact which is the gist of the familiar Wiener-Khintchine relations. In the case of locally stationary random processes, a relation is found between the covariance and the spectral density which constitutes a natural generalization of the Wiener-Khintchine relations.

Quantized Stabilization for T–S Fuzzy Systems With Hybrid-Triggered Mechanism and Stochastic Cyber-Attacks

Abstract: This paper examines quantized stabilization for Takagi–Sugeno (T–S) fuzzy systems with a hybrid-triggered mechanism and stochastic cyber-attacks. A hybrid-triggered scheme, which is described by a Bernoulli variable, is adopted to mitigate the burden of the network. By taking the effect of the hybrid-triggered scheme and stochastic cyber-attacks into consideration, a mathematical model for a closed-loop control system with quantization is constructed. Theorems for main results are developed to guarantee the asymptotical stability of networked control systems by using Lyapunov stability theory and linear matrix inequality techniques. Based on the derived sufficient conditions in theorems, the controller gains are presented in an explicit form. Finally, two practical examples demonstrate the feasibility of designed algorithm.

Improved Stability and Stabilization Results for Stochastic Synchronization of Continuous-Time Semi-Markovian Jump Neural Networks With Time-Varying Delay

Abstract: Continuous-time semi-Markovian jump neural networks (semi-MJNNs) are those MJNNs whose transition rates are not constant but depend on the random sojourn time. Addressing stochastic synchronization of semi-MJNNs with time-varying delay, an improved stochastic stability criterion is derived in this paper to guarantee stochastic synchronization of the response systems with the drive systems. This is achieved through constructing a semi-Markovian Lyapunov–Krasovskii functional together as well as making use of a novel integral inequality and the characteristics of cumulative distribution functions. Then, with a linearization procedure, controller synthesis is carried out for stochastic synchronization of the drive-response systems. The desired state-feedback controller gains can be determined by solving a linear matrix inequality-based optimization problem. Simulation studies are carried out to demonstrate the effectiveness and less conservatism of the presented approach.

Flexible Spacing Adaptive Cruise Control Using Stochastic Model Predictive Control

Abstract: This paper proposes a stochastic model predictive control (MPC) approach to optimize the fuel consumption in a vehicle following context. The practical solution of that problem requires solving a constrained moving horizon optimal control problem using a short-term prediction of the preceding vehicle’s velocity. In a deterministic framework, the prediction errors lead to constraint violations and to harsh control reactions. Instead, the suggested method considers errors, and limits the probability of a constraint violation. A conditional linear Gauss model is developed and trained with real measurements to estimate the probability distribution of the future velocity of the preceding vehicle. The prediction model is used to evaluate two different stochastic MPC approaches. On the one hand, an MPC with individual chance constraints is applied. On the other hand, samples are drawn from the conditional Gaussian model and used for a scenario-based optimization approach. Finally, both developed control strategies are evaluated and compared against a standard deterministic MPC. The evaluation of the controllers shows a significant reduction of the fuel consumption compared with standard adaptive cruise control algorithms.

Finite-Time State Estimation for Recurrent Delayed Neural Networks With Component-Based Event-Triggering Protocol

Abstract: This paper deals with the event-based finite-time state estimation problem for a class of discrete-time stochastic neural networks with mixed discrete and distributed time delays. In order to mitigate the burden of data communication, a general component-based event-triggered transmission mechanism is proposed to determine whether the measurement output should be released to the estimator at certain time-point according to a specific triggering condition. A new concept of finite-time boundedness in the mean square is put forward to quantify the estimation performance by introducing a settling-like time function. The objective of the addressed problem is to construct an event-based state estimator to estimate the neuron states such that, in the presence of both mixed time delays and external noise disturbances, the dynamics of the estimation error is finite-time bounded in the mean square with a prescribed error upper bound. Sufficient conditions are established, via stochastic analysis techniques, to guarantee the desired estimation performance. By solving an optimization problem with some inequality constraints, the explicit expression of the estimator gain matrix is characterized to minimize the settling-like time. Finally, a numerical simulation example is exploited to demonstrate the effectiveness of the proposed estimator design scheme.

Event-Triggered $H_\infty$ State Estimation for Delayed Stochastic Memristive Neural Networks With Missing Measurements: The Discrete Time Case

Abstract: In this paper, the event-triggered $H_\infty $ state estimation problem is investigated for a class of discrete-time stochastic memristive neural networks (DSMNNs) with time-varying delays and missing measurements. The DSMNN is subject to both the additive deterministic disturbances and the multiplicative stochastic noises. The missing measurements are governed by a sequence of random variables obeying the Bernoulli distribution. For the purpose of energy saving, an event-triggered communication scheme is used for DSMNNs to determine whether the measurement output is transmitted to the estimator or not. The problem addressed is to design an event-triggered $H_\infty $ estimator such that the dynamics of the estimation error is exponentially mean-square stable and the prespecified $H_\infty $ disturbance rejection attenuation level is also guaranteed. By utilizing a Lyapunov–Krasovskii functional and stochastic analysis techniques, sufficient conditions are derived to guarantee the existence of the desired estimator, and then, the estimator gains are characterized in terms of the solution to certain matrix inequalities. Finally, a numerical example is used to demonstrate the usefulness of the proposed event-triggered state estimation scheme.

Mean Field Analysis of Neural Networks: A Central Limit Theorem

Abstract: We rigorously prove a central limit theorem for neural network models with a single hidden layer. The central limit theorem is proven in the asymptotic regime of simultaneously (A) large numbers of hidden units and (B) large numbers of stochastic gradient descent training iterations. Our result describes the neural network's fluctuations around its mean-field limit. The fluctuations have a Gaussian distribution and satisfy a stochastic partial differential equation. The proof relies upon weak convergence methods from stochastic analysis. In particular, we prove relative compactness for the sequence of processes and uniqueness of the limiting process in a suitable Sobolev space.

Recursive parameter estimation algorithm for multivariate output-error systems

Abstract: In this paper, we consider the parameter estimation issues of a class of multivariate output-error systems. A decomposition based recursive least squares identification method is proposed using the hierarchical identification principle and the auxiliary model idea, and its convergence is analyzed through the stochastic process theory. Compared with the existing results on parameter estimation of multivariate output-error systems, a distinct feature for the proposed algorithm is that such a system is decomposed into several sub-systems with smaller dimensions so that parameters to be identified can be estimated interactively. The analysis shows that the estimation errors converge to zero in mean square under certain conditions. Finally, in order to show the effectiveness of the proposed approach, some numerical simulations are provided.

Forecasting Functional Time Series with a New Hilbertian ARMAX Model: Application to Electricity Price Forecasting

Abstract: A functional time series is the realization of a stochastic process where each observation is a continuous function defined on a finite interval. These processes are commonly found in electricity markets and are gaining more importance as more market data become available and markets head toward continuous-time marginal pricing approaches. Forecasting these time series requires models that operate with continuous functions. This paper proposes a new functional forecasting method that attempts to generalize the standard seasonal ARMAX time series model to the $L^2$ Hilbert space. The structure of the proposed model is a linear regression where functional parameters operate on functional variables. The variables can be lagged values of the series (autoregressive terms), past observed innovations (moving average terms), or exogenous variables. In this approach, the functional parameters used are integral operators whose kernels are modeled as linear combinations of sigmoid functions. The parameters of each sigmoid are optimized using a Quasi-Newton algorithm that minimizes the sum of squared errors. This novel approach allows us to estimate the moving average terms in functional time series models. The new model is tested by forecasting the daily price profile of the Spanish and German electricity markets and it is compared to other functional reference models.

Event-Based Variance-Constrained ${\mathcal {H}}_{\infty }$ Filtering for Stochastic Parameter Systems Over Sensor Networks With Successive Missing Measurements

Abstract: This paper is concerned with the distributed ${\mathcal {H}}_{\infty }$ filtering problem for a class of discrete time-varying stochastic parameter systems with error variance constraints over a sensor network where the sensor outputs are subject to successive missing measurements The phenomenon of the successive missing measurements for each sensor is modeled via a sequence of mutually independent random variables obeying the Bernoulli binary distribution law To reduce the frequency of unnecessary data transmission and alleviate the communication burden, an event-triggered mechanism is introduced for the sensor node such that only some vitally important data is transmitted to its neighboring sensors when specific events occur The objective of the problem addressed is to design a time-varying filter such that both the ${\mathcal {H}}_{\infty }$ requirements and the variance constraints are guaranteed over a given finite-horizon against the random parameter matrices, successive missing measurements, and stochastic noises By recurring to stochastic analysis techniques, sufficient conditions are established to ensure the existence of the time-varying filters whose gain matrices are then explicitly characterized in term of the solutions to a series of recursive matrix inequalities A numerical simulation example is provided to illustrate the effectiveness of the developed event-triggered distributed filter design strategy

Finite-Time $H_{\infty}$ State Estimation for Discrete Time-Delayed Genetic Regulatory Networks Under Stochastic Communication Protocols

Abstract: This paper investigates the problem of finite-time $H_{\infty }$ state estimation for discrete time-delayed genetic regulatory networks under stochastic communication protocols (SCPs). The network measurements are transmitted from two groups of sensors to a remote state estimator via two independent communication channels of limited bandwidths, and two SCPs are utilized to orchestrate the transmission orders of sensor nodes with aim to avoid data collisions. The estimation error dynamics is modeled by a Markovian switching system with two switching signals. By constructing a transmission-order-dependent Lyapunov–Krasovskii functional and utilizing an up-to-date discrete Wirtinger-based inequality together with the reciprocally convex approach, sufficient conditions are established to guarantee the stochastic finite-time boundedness for the estimation error dynamics with a prescribed $H_{\infty }$ disturbance attenuation level. The parameters of the state estimator are designed by solving a convex optimization problem which minimizes the disturbance attenuation level subject to several inequality constraints. The repressilator model is utilized to illustrate the effectiveness of the design procedure of the proposed state estimator.

Event-Based Fault Detection Filtering for Complex Networked Jump Systems

Abstract: This paper is concerned with the fault detection filtering for complex systems over communication networks subject to nonhomogeneous Markovian parameters. A residual signal is generated that gives a satisfactory estimation of the fault, and an event-triggered scheme is proposed to determine whether the networks should be updated at the trigger instants decided by the event-threshold. Moreover, a random process is employed to model the phenomenon of malicious packet losses. Consequently, a novel method is presented to address the stochastically stability analysis and satisfies a given $H_{2}$ performance index simultaneously. The condition of the existence of the filter design algorithm is derived by a convex optimization approach to estimate the faults and to generate a residual. Finally, the proposed fault detection filtering method is then applied to an industrial nonisothermal continuous stirred tank reactor under realistic network conditions. Simulation results are given to show the effectiveness of the proposed design method and the designed filter.

Stochastic Thermodynamic Interpretation of Information Geometry.

Abstract: In recent years, the unified theory of information and thermodynamics has been intensively discussed in the context of stochastic thermodynamics. The unified theory reveals that information theory would be useful to understand nonstationary dynamics of systems far from equilibrium. In this Letter, we have found a new link between stochastic thermodynamics and information theory well-known as information geometry. By applying this link, an information geometric inequality can be interpreted as a thermodynamic uncertainty relationship between speed and thermodynamic cost. We have numerically applied an information geometric inequality to a thermodynamic model of a biochemical enzyme reaction.

Fault-Tolerant Control of Time-Delay Markov Jump Systems With $It\hat{o}$ Stochastic Process and Output Disturbance Based on Sliding Mode Observer

Abstract: This paper focuses on the fault-tolerant control problem of Markov jump systems (MJS) with $It\hat{o}$ stochastic process and output disturbances. Such a problem widely exists in practical systems such as mobile manipulator systems. Since MJS can suitably describe mobile manipulator systems, in this paper, a new approach based on the MJS model is proposed. First, a proportional-derivative sliding mode observer (SMO) and an observer-based controller are designed and synthesized. Two new theorems are derived to ensure the close-loop stochastic stability and the reachability of the sliding mode surface. Compared with the existing works, the system model is more general, which could describe a larger variety of plants or processes. The controller design procedure is simplified by solving the sliding mode parameters and the controller gain simultaneously with only one linear matrix inequality problem. In addition, the augmented fault vector can be reconstructed by employing a descriptor SMO. Simulations are provided to demonstrate the validity of the derived theorems and the effectiveness of the proposed algorithm.

Stochastic prediction of train delays in real-time using Bayesian networks

Abstract: In this paper we present a stochastic model for predicting the propagation of train delays based on Bayesian networks. This method can efficiently represent and compute the complex stochastic inference between random variables. Moreover, it allows updating the probability distributions and reducing the uncertainty of future train delays in real time under the assumption that more information continuously becomes available from the monitoring system. The dynamics of a train delay over time and space is presented as a stochastic process that describes the evolution of the time-dependent random variable. This approach is further extended by modelling the interdependence between trains that share the same infrastructure or have a scheduled passenger transfer. The model is applied on a set of historical traffic realisation data from the part of a busy corridor in Sweden. We present the results and analyse the accuracy of predictions as well as the evolution of probability distributions of event delays over time. The presented method is important for making better predictions for train traffic, that are not only based on static, offline collected data, but are able to positively include the dynamic characteristics of the continuously changing delays.

Quantized Output Feedback Control for Stochastic Semi-Markov Jump Systems With Unreliable Links

Abstract: This brief is concerned with the problem of quantized output feedback (OF) control for semi-Markov jump systems subject to unreliable links. The jump among the operation modes is subject to the semi-Markov process, where the transition probabilities depend not only on the next system mode but the sojourn time at current mode as well. The communication link is considered to be unreliable, in which the packet dropouts described by a stochastic Bernoulli parameter and signal quantization are taken into account simultaneously. The purpose is to synthesize an OF controller, which ensures that the closed-loop system is mean-square $\sigma $ -error stable. Some sojourn time-dependent criteria are established for the realizability of admissible OF controller. With the help of those criteria, the controller gains could be obtained by finding a solution to a convex optimization problem. Finally, both a numerical example and a PWM-driven boost converter are presented to confirm the availability and feasibility of the proposed approach.

Stochastic Geometry Methods for Modeling Automotive Radar Interference

Abstract: As the use of automotive radar increases, performance limitations associated with radar-to-radar interference will become more significant. In this paper, we employ tools from stochastic geometry to characterize the statistics of radar interference. Specifically, using two different models for the spatial distributions of vehicles, namely, a Poisson point process and a Bernoulli lattice process, we calculate for each case the interference statistics and obtain analytical expressions for the probability of successful range estimation. This paper shows that the regularity of the geometrical model appears to have limited effect on the interference statistics, and so it is possible to obtain tractable tight bounds for the worst case performance. A technique is proposed for designing the duty cycle for the random spectrum access, which optimizes the total performance. This analytical framework is verified using Monte Carlo simulations.

State Estimation for Discrete Time-Delayed Genetic Regulatory Networks With Stochastic Noises Under the Round-Robin Protocols

Abstract: This paper investigates the problem of state estimation for discrete time-delayed genetic regulatory networks with stochastic process noises and bounded exogenous disturbances under the Round-Robin (RR) protocols. The network measurement outputs obtained by two groups of sensors are transmitted to two remote sub-estimators via two independent communication channels, respectively. To lighten the communication loads of the networks and reduce the occurrence rate of data collisions, two RR protocols are utilized to orchestrate the transmission orders of sensor nodes in two groups, respectively. The error dynamics of the state estimation is governed by a switched system with periodic switching parameters. By constructing a transmission-order-dependent Lyapunov-like functional and utilizing the up-to-date discrete Wirtinger-based inequality together with the reciprocally convex approach, sufficient conditions are established to guarantee the exponentially ultimate boundedness of the estimation error dynamics in mean square with a prescribed upper bound on the decay rate. An asymptotic upper bound of the outputs of the estimation errors in mean square is derived and the estimator parameters are then obtained by minimizing such an upper bound subject to linear matrix inequality constraints. The repressilator model is utilized to illustrate the effectiveness of the designed estimator.

A Data-Driven Stochastic Reactive Power Optimization Considering Uncertainties in Active Distribution Networks and Decomposition Method

Abstract: To address the uncertain output of distributed generators for reactive power optimization in active distribution networks, the stochastic programming model is widely used. The model is employed to find an optimal control strategy with minimum expected network loss while satisfying all the physical constraints. Therein, the probability distribution of uncertainties in the stochastic model is always pre-defined by the historical data. However, the empirical distribution can be biased due to a limited amount of historical data and thus result in a suboptimal control decision. Therefore, in this paper, a data-driven modeling approach is introduced to assume that the probability distribution from the historical data is uncertain within a confidence set. Furthermore, a data-driven stochastic programming model is formulated as a two-stage problem, where the first-stage variables find the optimal control for discrete reactive power compensation equipment under the worst probability distribution of the second stage recourse. The second-stage variables are adjusted to uncertain probability distribution. In particular, this two-stage problem has a special structure so that the second-stage problem can be directly decomposed into several small-scale sub-problems, which can be handled in parallel without the information of dual problems. Numerical study on two distribution systems has been performed. Comparisons with the two-stage stochastic and robust approaches demonstrate the effectiveness of the proposal.

Nonfragile State Estimation of Quantized Complex Networks With Switching Topologies

Abstract: This paper considers the nonfragile $H_\infty $ estimation problem for a class of complex networks with switching topologies and quantization effects. The network architecture is assumed to be dynamic and evolves with time according to a random process subject to a sojourn probability. The coupled signal is to be quantized before transmission due to power and bandwidth constraints, and the quantization errors are transformed into sector-bounded uncertainties. The concept of nonfragility is introduced by inserting randomly occurred uncertainties into the estimator parameters to cope with the unavoidable small gain variations emerging from the implementations of estimators. Both the quantizers and the estimators have several operation modes depending on the switching signal of the underlying network structure. A sufficient condition is provided via a linear matrix inequality approach to ensure the estimation error dynamic to be stochastically stable in the absence of external disturbances, and the $H_\infty $ performance with a prescribed index is also satisfied. Finally, a numerical example is presented to clarify the validity of the proposed method.

A Stochastic Model of Cascading Failure Dynamics in Communication Networks

Abstract: In this brief, we propose a stochastic model based on state transition theory to investigate the dynamics of cascading failures in communication networks. We describe the failure events of the nodes in the network as node state transitions. Taking a probabilistic perspective, we focus on two uncertain conditions in the failure propagation process: which node in the network will fail next and how long it will last before the next node state transition takes place. The stochastic model gives each overloaded element a probability of failing, and the failure rate is relevant to the degree of overloading. Moreover, the time dimension is considered in the stochastic process, by removing a node after a time delay when its traffic load exceeds its capacity. We employ this proposed model to study the dynamics of cascading failure evolution in a Barabasi–Albert scale-free network and an Internet AS-level network. Simulation results reveal the effects of the initial failure pattern, community structure and network design parameters on the dynamic propagation of cascading failures.

Variance-Constrained State Estimation for Complex Networks With Randomly Varying Topologies

Abstract: This paper investigates the variance-constrained $H_{\infty}$ state estimation problem for a class of nonlinear time-varying complex networks with randomly varying topologies, stochastic inner coupling, and measurement quantization A Kronecker delta function and Markovian jumping parameters are utilized to describe the random changes of network topologies A Gaussian random variable is introduced to model the stochastic disturbances in the inner coupling of complex networks As a kind of incomplete measurements, measurement quantization is taken into consideration so as to account for the signal distortion phenomenon in the transmission process Stochastic nonlinearities with known statistical characteristics are utilized to describe the stochastic evolution of the complex networks We aim to design a finite-horizon estimator, such that in the simultaneous presence of quantized measurements and stochastic inner coupling, the prescribed variance constraints on the estimation error and the desired $H_{\infty}$ performance requirements are guaranteed over a finite horizon Sufficient conditions are established by means of a series of recursive linear matrix inequalities, and subsequently, the estimator gain parameters are derived A simulation example is presented to illustrate the effectiveness and applicability of the proposed estimator design algorithm

Signature moments to characterize laws of stochastic processes

Abstract: The normalized sequence of moments characterizes the law of any finite-dimensional random variable. We prove an analogous result for path-valued random variables, that is stochastic processes, by using the normalized sequence of signature moments. We use this to define a metric for laws of stochastic processes. This metric can be efficiently estimated from finite samples, even if the stochastic processes themselves evolve in high-dimensional state spaces. As an application, we provide a non-parametric two-sample hypothesis test for laws of stochastic processes.

A Stochastic Computational Multi-Layer Perceptron with Backward Propagation

Abstract: Stochastic computation has recently been proposed for implementing artificial neural networks with reduced hardware and power consumption, but at a decreased accuracy and processing speed. Most existing implementations are based on pre-training such that the weights are predetermined for neurons at different layers, thus these implementations lack the ability to update the values of the network parameters. In this paper, a stochastic computational multi-layer perceptron (SC-MLP) is proposed by implementing the backward propagation algorithm for updating the layer weights. Using extended stochastic logic (ESL), a reconfigurable stochastic computational activation unit (SCAU) is designed to implement different types of activation functions such as the $tanh$ and the rectifier function. A triple modular redundancy (TMR) technique is employed for reducing the random fluctuations in stochastic computation. A probability estimator (PE) and a divider based on the TMR and a binary search algorithm are further proposed with progressive precision for reducing the required stochastic sequence length. Therefore, the latency and energy consumption of the SC-MLP are significantly reduced. The simulation results show that the proposed design is capable of implementing both the training and inference processes. For the classification of nonlinearly separable patterns, at a slight loss of accuracy by 1.32-1.34 percent, the proposed design requires only 28.5-30.1 percent of the area and 18.9-23.9 percent of the energy consumption incurred by a design using floating point arithmetic. Compared to a fixed-point implementation, the SC-MLP consumes a smaller area (40.7-45.5 percent) and a lower energy consumption (38.0-51.0 percent) with a similar processing speed and a slight drop of accuracy by 0.15-0.33 percent. The area and the energy consumption of the proposed design is from 80.7-87.1 percent and from 71.9-93.1 percent, respectively, of a binarized neural network (BNN), with a similar accuracy.