Random Time Sequences Research Articles

A random benchmark suite and a new reaction strategy in dynamic multiobjective optimization

In the domain of evolutionary computation, more and more attention has been paid to dynamic multiobjective optimization. Generally, artificial benchmarks are effective tools for the performance evaluation of dynamic multiobjective evolutionary algorithms (DMOEAs). After reviewing existing benchmarks and highlighting their weaknesses, this paper proposes a new benchmark suite to promote the comprehensive testing of algorithms. This proposed benchmark suite has eight random instances in which the randomness is produced by designed random time sequences. Also, this suite introduces challenging but rarely considered characteristics, including diverse features in fitness landscape (e.g. deception, multimodality, and bias) and complex trade-off geometries (e.g. convexity-concavity mixed geometry and disconnected geometry). Empirical studies have shown that the proposed benchmark poses reasonable challenges to DMOEAs in terms of convergence and diversity. Besides, a center matching strategy (CMS) is suggested to track random changes in these problems, which applies the history individual information in a global scope for population prediction. Compared with other reaction strategies, CMS has been demonstrated to be very competitive in dealing with random problems.

Detection of Low‐Frequency Earthquakes in Broadband Random Time Sequences: Are They Independent Events?

AbstractLow‐frequency earthquakes (LFEs) are detected primarily from continuous seismograms using a matched‐filter technique with an impulsive template waveform in a relatively narrow frequency band. However, this method can also detect events from some kinds of random time sequences without clearly isolated events. Here this fact is demonstrated via simple numerical simulations using the synthetic moment accelerations from a model of broadband slow earthquake, the Brownian slow earthquake (BSE) model, and a totally random noise sequence. The matched‐filter technique identifies time sections including relatively isolated pulse‐like fluctuations, as signals in both time sequences, depending on the threshold. These waveforms, stacked relative to the signal timing, show a clear impulse similar to the assumed template in both time sequences, which highlights that we might potentially misinterpret an original time sequence as containing many isolated pulse‐like events. An important difference exists between the BSE model and random noise at frequencies much lower than the analyzed frequency band, with the stacked BSE sequence containing coherent signals at very low frequencies, which are not visible in the noise. Real observations in the Cascadia subduction zone also contain similar coherent signals at low frequencies, suggesting that these LFE signals are coincident with some slow slip. Therefore, so‐called LFEs might potentially be a misinterpretation due to signal processing, or at least they are the tip of the iceberg, with these signals forming a component of a very broadband slow‐earthquake‐like slip process that possibly occurs over sub‐second to multi‐year timescales.

Electrocortical activity during stretch reflex in athletes

To evaluate timing and characteristics of electrocortical activity during stretch reflex evocation of the quadriceps femoris at different body postures. Twenty athletes were subjected to reflex hammer percussions on patellar tendon, in a random time sequence, in seated and supine decubitus postures. 745 ms of acquisition were evaluated for data analysis: 200 ms of pre-stimulation, 45 ms relative to the elapsed time from stimulation to the end of the muscular contraction, and 500 ms of post-stimulation. First, a scalp-map analysis, considering channels data, was performed; second, independent component analysis decomposition was applied to all 16 channels. To highlight the differences among two conditions (pre- and post-stimulation) and two sessions (seated and supine decubitus postures), a two-way analysis of variance was used. Time–frequency analysis has been applied to identify the density of the frequencies as a function of the time, instant by instant. Fast alpha, beta, and gamma frequency bands showed a greater electroencephalographic activity after the stimulation. Seated posture showed a greater activation than the decubitus supine posture. Time frequency analysis highlighted an higher electrocortical activation for 160 ms as well as from about 250–300 ms at all frequency bands after the stimulation. Our findings improve the understanding of the neurophysiological dynamics following the stretch reflex after concussion of the patellar tendon, executed in different postures, considering scalp-map, power spectra, and timing of activation. The use of scalp-map and power spectra analysis techniques represents a sophisticated use of advanced signal processing methodologies to analyze brain activity during movement and in specificity in sport science.

Uniform approximation of the Cox–Ingersoll–Ross process via exact simulation at random times

AbstractIn this paper we uniformly approximate the trajectories of the Cox–Ingersoll–Ross (CIR) process. At a sequence of random times the approximate trajectories will be even exact. In between, the approximation will be uniformly close to the exact trajectory. From a conceptual point of view, the proposed method gives a better quality of approximation in a path-wise sense than standard, or even exact, simulation of the CIR dynamics at some deterministic time grid.

Directional Modulation Based on 4-D Antenna Arrays

Four-dimensional (4-D) antenna arrays are formed by introducing a fourth dimension, time, into traditional antenna arrays. In this paper, a time-modulated 4-D array with constant instantaneous directivity is proposed for directional modulation. The main idea is that the 4-D array transmits correct signal without time modulation in the desired direction, while transmitting time-modulated signals in other directions. As longs as the time modulation frequency is less than the bandwidth of the transmitted signal, the time-modulated signals cannot be demodulated correctly due to the aliasing effect, implying that time-modulated signals go distorted. Thus, the 4-D array can be used to transmit direction-dependent signals in secure wireless communications. The proposed idea is verified by experiments based on AM signal transmission through the 4-D array. Moreover, BPSK signal transmission through the 4-D array is studied and the bit error rate (BER) performance is investigated. Simulation results show that the BERs of time-modulated BPSK (TM-BPSK) signals transmitted through the sidelobes of the 4-D array are much higher than those of BPSK signals and almost keep unchanged even under higher SNR. Finally, two enhanced methods are presented to improve the feasibility of directional modulation by using random time sequences and random time modulation frequency.

Weak Convergence Methods for Approximation of the Evaluation of Path-Dependent Functionals

In many applications, one needs to evaluate a path-dependent objective functional $V$ associated with a continuous-time stochastic process $X$. Due to the nonlinearity and possible lack of Markovian property, more often than not, $V$ cannot be evaluated analytically, and only Monte Carlo simulation or numerical approximation is possible. In addition, such calculations often require the handling of stopping times, the usual dynamic programming approach may fall apart, and the continuity of the functional becomes the main issue. Denoting by $h$ the stepsize of the approximation sequence, this work develops a numerical scheme so that an approximating sequence of path-dependent functionals $V^h$ converges to $V$. By a natural division of labors, the main task is divided into two parts. Given a sequence $X^h$ that converges weakly to $X$, the first part provides sufficient conditions for the convergence of the sequence of path-dependent functionals $V^h$ to $V$. The second part constructs a sequence of approximations $X^h$ using Markov chain approximation methods and demonstrates the weak convergence of $X^h$ to $X$, when $X$ is the solution of a stochastic differential equation. As a demonstration, combining the results of the two parts above, approximation of option pricing for the discrete-monitoring-barrier option underlying stochastic volatility model is provided. Different from the existing literature, the weak convergence analysis is carried out by using the Skorohod topology together with the continuous mapping theorem. The advantage of this approach is that the functional under study may be a function of stopping times, projection of the underlying diffusion on a sequence of random times, and/or maximum/minimum of the underlying diffusion.

Application Study of Load Forecasting of Central Air-Conditioning System Based on Time Sequence Analysis Method

With the characteristics of non-stationarity, non-linearity, time-lag of refrigeration/ heating supplying in minds, load forecasting of central air-conditioning system is carried using time sequence analysis method. Firstly, acquisition sample data of central air-conditioning system is pretreated, and random time sequence AR model of system is formulated. Then, forecasting of AR refrigeration/heating load based on Yule-walker method is conducted. In order to enhance forecasting accuracy, crossover forecasting is introduced into the load forecasting, that is, to use vertical forecasting to follow household demands for load and horizontal forecasting to track changes of weather. Then, weight cross is made to vertical and horizontal forecasting results. Finally, refrigeration/heating load forecasting software of central air-conditioning system is developed, which is used in energy-saving monitoring and control of central air-conditioning system.

Bottom profiling by correlating beam-steered noise sequences

It has already been established that by cross-correlating ambient noise time series received on the upward and downward steered beams of a drifting vertical array one can obtain a subbottom layer profile. Strictly, the time differential of the cross correlation is the impulse response of the seabed. Here it is shown theoretically and by simulation that completely uncorrelated surface noise results in a layer profile with predictable amplitudes proportional to those of an equivalent echo sounder at the same depth as the array. The phenomenon is simulated by representing the sound sources as multiple random time sequences emitted from random locations in a horizontal plane above a vertical array and then accounting for the travel times of the direct and bottom reflected paths. A well-defined correlation spike is seen at the depth corresponding to the bottom reflection despite the fact that the sound sources contain no structure whatsoever. The effects of using simultaneously steered upward and downward conical beams with a tilted or faceted seabed and multiple layers are also investigated by simulation. Experimental profiles are obtained using two different vertical arrays in smooth and rough bottom sites in the Mediterranean. Correlation peak amplitudes follow the theory and simulations closely.

Millimeter-wave Low Sidelobe Time Modulated Linear Arrays with Uniform Amplitude Excitations

The study of millimeter-wave low sidelobe linear arrays with time modulation technique is presented in this paper. Special emphasis is on the realization of low sidelobes with uniform amplitude excitations, which are easier for the practical implementation. Two types of time modulated linear arrays with uniform amplitude excitations, namely, the variable aperture size (VAS) linear arrays with optimized time sequences and the linear arrays with optimized random time sequences were studied. The optimized time sequences are obtained via the differential evolution (DE) algorithm and the genetic algorithm (GA). Many factors that affect the array performance are discussed.

Propagation model for signal fluctuations on transionospheric radio links: Recent developments

The propagation model for signal fluctuations on transionospheric radio links previously developed by the authors has been further extended to be capable of describing some new applications e.g. spatial coherency of the field, or the time-correlation properties of level (log-amplitude) and phase fluctuations and their frequency spectra. In particular, the technique has been developed to also generate random time sequences of the level and phase of the field on transionospheric paths of propagation. Spatial coherency is of importance for differential GPS in the assessment of the additional fluctuational errors in rangefinding with respect to a reference GPS station whereas random time series are necessary for modelling the fluctuational Earth-satellite channel of propagation.

Theory for a wavelet analysis of electrochemical noise in the laplace space with use of two operational frequencies

The image of a random process in the Laplace space may be viewed as resulting from use of a oneway continuous wavelet transform with an exponential as the basic function, i.e. as resulting from the application of the Laplace wavelet. If the Laplace-wavelet variance of an electrochemical noise allows one to determine the Laplace transform of a time-correlation function, i.e. a factual operational spectral density of the noise, then the covariance of two Laplace waveletes of an electrochemical noise, each of which corresponds to its own operational frequency, allows one to verify a local consistency of the initial experimental noise data. The Laplace waveletes are applied rather widely. In fact, any stationary random process and stationary random time sequence can be described with operational noise spectra.

On time- and cycle-stationarity

Consider processes split into cycles by a sequence of random times (called points). The standard Palm relationship between stationary processes with cycles and processes with stationary cycles is produced in two transparent steps: length-biasing and re-centring. It has the following standard intuitive interpretation: the process with stationary cycles behaves like the stationary one conditioned on a point at time zero. A less known modification of this relationship is produced by conditioning on the invariant σ-algebra before length-biasing. It has the following intuitive interpretation: the stationary process behaves like the cycle-stationary one centred at a time chosen at random on the line. The present approach leads to strong conditioning, limit and coupling results motivating these interpretations.

Uniform Cesaro limit theorems for synchronous processes with applications to queues

Let X={ X( t): t⩾0} be a positive recurrent synchronous process (PRS), that is, a process for which there exists an increasing sequence of random times τ={ τ( k)} such that for each k the distribution of θ τ(k)X = def {X(t+τ(k)):t⩾0} is the same and the cycle lengths T n = def τ(n+1)−τ(n) have finite first moment. Such processes (in general) do not converge to steady-state weakly (or in total variation) even when regularity conditions are placed on the cycles (such as non-lattice, spread-out, or mixing). Nonetheless, in the present paper we first show that the distributions of { θ s X: s > 0} are tight in the function space D(0,∞) . Then we investigate conditions under which the Cesaro averaged functionals μ ̄ i = def ( 1 t )∫ t 0E(ƒ(θ sX)) ds converge uniformly (over a class of functions) to π(ƒ), where π is the stationary distribution of X. We show that μ t(ƒ)→π(ƒ) uniformly over ƒ satisfying ‖ƒ‖ ∞⩽1 (total variation convergence). We also show that to obtain uniform convergence over all ƒ satisfying |ƒ|⩽g (gϵL + 1(π) fixed) requires placing further conditions on the PRS. This is in sharp contrast to both classical regenerative processes and discrete time Harris recurrent Markov chains (where renewal theory can be applied) where such uniform convergence holds without any further conditions. For continuous time positive Harris recurrent Markov processes (where renewal theory cannot be applied) we show that these further conditions are in fact automatically satisfied. In this context, applications to queueing models are given.

Digital simulation of group estimation of an energy spectrum according to the entropy-minimax method

Estimation of the energy spectrum of a random time sequence {x(l)} according to the entropy-minimax method (EMM) is implemented in a system of M linear decorrelators having different orders m = 1, 2, ..., M (see [1]). Its computation formula for a sample of size L is (1) and uses the set of squares of the amplitude-frequency responses (AFR) of the decorrelators. The conclusions which have been drawn may be extended to varying degree to other well-known methods of group estimation which do not differ fundamentally in efficiency from the considered Berg method. Thus, the results of the investigation which has been performed of the proposed method for the group estimation of spectra convincingly prove the expediency of its application in cases of restricted samplings and low signal/noise ratios.

Digital generator of corona noise on power line carrier channels

A procedure is presented for the design of a digital simulator which generates a random time sequence having the statistical characteristics of the corona noise that occurs on transmission line carrier channels. The method involves the synthesis of an autoregressive filter excited by a white-noise generator is a part of the overall digital transmission system, which includes the generating and receiving sections, and the PLC (power line carrier) channel. The proposed procedure is applied to simulate corona noise on practical HVAC transmission lines.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Periodic regeneration

We consider stochastic processes Z = ( Z t ) [0,∞), on a general state space, having a certain periodic regeneration property: there is an increasing sequence of random times ( S n ) ∞ 0 such that the post- S n process is conditionally independent of S 0,…, S n given ( S n mod 1) and the conditional distribution does not depend on n. Our basic condition is that the distributions P s(A) = P(S n+1 − S n ∈ A|S n = s), s ∈[0, 1) , have a common component that is absolutely continuous w.r.t. Lebesgue measure. Then Z has the following time-homogeneous regeneration property: there exists a discrete aperiodic renewal process T = ( T n ) ∞ 0 such that the post- T n process is independent of T 0,…, T n and its distribution does not depend on n; this yields weak ergodicity. Further, the Markov chain ( S n mod 1) ∞ 0 has an invariant distribution π [0,1) and it holds that T n+1 − T n has finite first moment if and only if m = ∫ m( P s ) π [0,1)( ds) < ∞ where m( P s ) is the first moment of P s ; this yields periodic ergodicity. Also, some distributional properties of T 0 and T n+1 − T n are established leading to improved ergodic regults. Finally, a uniform key periodic renewal theorem is derived.

Time changes of Markov chains

An increasing sequence of random times { T n , n ⩾ 0} is called a Markov time change if { X( T n )} is a new Markov chain. If the { T n } satisfy certain ‘operational’ requirements such as conditional independence of the T n -past and T n -future given X( T n ), then there is an equivalent, algebraic description of the { T n } in terms of a triple ( T 0, S, Γ), where T 0 and S are splitting times with respect to a set Γ. A further assumption on Γ makes it easy to check that a triple will generate a Markov time change, and it is shown that processes such as last exit processes and excision processes satisfy this assumption.

Efficient FFT Simulation of Digital Time Sequences

A stacked inverse finite Fourier transform (FFT) algorithm is presented that will efficiently synthesize a discrete random time sequence of N values from only N/2 complex values having a desired known spectral representation. This stacked inverse FFT algorithm is compatible with the synthesis of discrete random time sequences that are used with the more desirable periodic-random type of dynamic testing systems used to compute complex-valued transfer functions by the frequency-sweep method. An application to the generation of large random surface gravity waves by a hinged wavemaker in a large-scale wave flume demonstrates excellent agreement between the desired theoretical spectral representation and the smoothed, measured spectral representation for two types of two-parameter theoretical spectra as a result of the lengthier realization made possible by the stacked FFT algorithm.

Limit Theorems for Semi-Markov Processes and Renewal Theory for Markov Chains

With any Harris-recurrent Markov chain one can associate a sequence of random times at which the chain has the same distribution, and the chain can thereby be shown to be equivalent to one having a recurrence point. This idea makes available a regeneration scheme for such chains, which is exploited in this paper to prove the ergodic theorem for semi-Markov processes, and a renewal theorem for Markov chains on a general state space.

A new approach to the limit theory of recurrent Markov chains

Let { X n ; n ⩾ 0 } \{ {X_n};\,n \geqslant 0\} be a Harris-recurrent Markov chain on a general state space. It is shown that there is a sequence of random times { N i ; i ⩾ 1 } \{ {N_i};\,i \geqslant 1\} such that { X N i ; i ⩾ 1 } \{ {X_{{N_i}}};{\text { }}i \geqslant 1\} are independent and identically distributed. This idea is used to show that { X n } \{ {X_n}\} is equivalent to a process having a recurrence point, and to develop a regenerative scheme which leads to simple proofs of the ergodic theorem, existence and uniqueness of stationary measures.