Random Multiplier Research Articles

On linear codes with random multiplier vectors and the maximum trace dimension property

Abstract Let C C be a linear code of length n n and dimension k k over the finite field F q m {{\mathbb{F}}}_{{q}^{m}} . The trace code Tr ( C ) {\rm{Tr}}\left(C) is a linear code of the same length n n over the subfield F q {{\mathbb{F}}}_{q} . The obvious upper bound for the dimension of the trace code over F q {{\mathbb{F}}}_{q} is m k mk . If equality holds, then we say that C C has maximum trace dimension. The problem of finding the true dimension of trace codes and their duals is relevant for the size of the public key of various code-based cryptographic protocols. Let C a {C}_{{\boldsymbol{a}}} denote the code obtained from C C and a multiplier vector a ∈ ( F q m ) n {\boldsymbol{a}}\in {\left({{\mathbb{F}}}_{{q}^{m}})}^{n} . In this study, we give a lower bound for the probability that a random multiplier vector produces a code C a {C}_{{\boldsymbol{a}}} of maximum trace dimension. We give an interpretation of the bound for the class of algebraic geometry codes in terms of the degree of the defining divisor. The bound explains the experimental fact that random alternant codes have minimal dimension. Our bound holds whenever n ≥ m ( k + h ) n\ge m\left(k+h) , where h ≥ 0 h\ge 0 is the Singleton defect of C C . For the extremal case n = m ( h + k ) n=m\left(h+k) , numerical experiments reveal a closed connection between the probability of having maximum trace dimension and the probability that a random matrix has full rank.

A Novel and Fast Memory Perturbation Method to Increase Exploration in Particle Swarm Optimization Algorithm

Particle Swarm Optimization (PSO), one of the versatile nature-inspired optimization algorithm, continue to suffer from premature convergence despite the numerous amount of research trying to improve this algorithm. Many research had tried to address this issue but often use a complex algorithm which tax on computational time and complexity. This research introduced a novel perturbation method to mitigate premature convergence / to increase exploration while keeping the computational cost at a minimum. The particles’ memories (i.e the position of personal and global best) are modified by a random multiplier which in turn will ‘perturb’ the particles’ velocity. The implementation of this novel perturbation method in early iterations had resulted in 100% success rate in finding global optima in multimodal benchmark tests including the Rastrigin problem - whereas the original PSO failed in all benchmark tests - without adding a significant amount of computational complexity and time.

Fatigue crack growth assessment method subject to model uncertainty

Fatigue crack growth (FCG) is an important degradation process of many critical mechanical equipment. Probabilistic FCG models are often used to account for the variability among FCG process conditions. In the well-known model of Yang and Manning, a deterministic FCG model is randomized by multiplying the crack growth rate with a random multiplier assumed to obey a lognormal distribution and unknown parameters are jointly estimated through Maximum likelihood estimation. By so doing, the modeling error due to inappropriate choice of the deterministic FCG model and that due to unsuitable assignment of the probability distribution of the random multiplier cannot be distinguished. Besides, the model uncertainty of the random multiplier is not explicitly considered. In this paper, a two-step least-square estimation method is proposed, which estimates the unknown parameters in the deterministic FCG model at first, and generates a sample set for the estimation of the random multiplier considering model uncertainty by way of Bayesian model selection. In Bayesian model selection, three types of Bayes factor are considered to select the appropriate candidate model and a simulation experiment is carried out to guide their selection. The effectiveness and feasibility of the proposed method are illustrated through two case studies using the real FCG datasets.

An improved multi-branch residual network based on random multiplier and adaptive cosine learning rate method

Deep residual networks have emerged as a leading technique showing the accuracy and excellent convergence performance. However, because of overfitting and the vanishing gradient problems, we cannot achieve better results by increasing the model parameters. In this study, we propose a novel architecture that has a multi-branch residual network. The shortcut branch of residual networks can ease off the vanishing gradient. The random function and the structure of a multi-branch network improve the fitting ability. The dropout function can weaken overfitting. The proposed method also adopts the adaptive cosine learning rate method and variate batch size to improve the test accuracy. Some experimental investigations are set up to explore the impact of the following factors on the performance: the sequence of the layers, random numbers and different batch sizes, etc. We report that the results achieve 2.63% and 14.2% Top-1 error on CIFAR-10 and CIFAR-100.

Pushing Through or Slacking Off? Heterogeneity in the Reaction to Rank Feedback

This paper studies heterogeneity in the reaction to rank feedback. In a laboratory experiment, individuals take part in a series of dynamic real-effort contests with intermediate feedback. To solve the identification problem in estimating the causal effect of rank feedback on subsequent effort provision we implement a random multiplier in the first round of each contest. The realization of this multiplier then serves as a valid instrument for rank feedback. While rank feedback has a robust effect on subsequent effort provision on average, an explicit analysis of between-subject heterogeneity reveals that a substantial fraction of participants in fact react entirely opposite than the aggregated results indicate. We further show that this heterogeneity has consequences for overall outcomes, thereby arguing that heterogeneous sensitivities to rank feedback could have implications for the design of various policies in education and organizations.

Interactions and constraints in model species response to environmental heteroscedasticity

Increasing environmental variability could exacerbate the effects of climate change on ecological processes such as population dynamics, or positive and negative effects (favorable or unfavorable weather) could balance. Such a balance could depend on constraints of the processes. Biological and spatial constraints are represented in a spatially explicit individual based simulation of an ecotone reduced to two species on a single environmental gradient. The effects of climate amelioration are simulated from a plant's-eye-view by increasing the establishment and decreasing the mortality rates. Variability is introduced as a random multiplier of these rates, and the strength of the variation is increased through the period of climate change. The biological constraints limit change in the rates, and the extent of the simulation grid represents a spatial constraint. A small increase in environmental variation, multiplied through time with climate change, increases extinction rates. The biological and spatial constraints have little effect on the response of populations. Instead, competition, based on the form of the species response functions to the environmental gradient at the point where they intersect, determines differences in population responses. Positive and negative variations in the environment do not balance because the responses are hierarchical and asymmetric. Differences persist because extinction during a negative anomaly cannot be reversed by a later positive one.

Random multipliers numerically stabilize Gaussian and block Gaussian elimination: Proofs and an extension to low-rank approximation

We study two applications of standard Gaussian random multipliers. At first we prove that with a probability close to 1 such a multiplier is expected to numerically stabilize Gaussian elimination with no pivoting as well as block Gaussian elimination. Then, by extending our analysis, we prove that such a multiplier is also expected to support low-rank approximation of a matrix without customary oversampling. Our test results are in good accordance with this formal study. The results remain similar when we replace Gaussian multipliers with random circulant or Toeplitz multipliers, which involve fewer random parameters and enable faster multiplication. We formally support the observed efficiency of random structured multipliers applied to approximation, but we still continue our research in the case of elimination. We specify a narrow class of unitary inputs for which Gaussian elimination with no pivoting is numerically unstable and then prove that, with a probability close to 1, a Gaussian random circulant multiplier does not fix numerical stability problems for such inputs. We also prove that the power of the random circulant preprocessing increases if we also include random permutations.

The comparison of sensitivity analysis of hydrological uncertainty estimates by GLUE and Bayesian method under the impact of precipitation errors

The input uncertainty is as significant as model error, which affects the parameter estimation, yields bias and misleading results. This study performed a comprehensive comparison and evaluation of uncertainty estimates according to the impact of precipitation errors by GLUE and Bayesian methods using the Metropolis Hasting algorithm in a validated conceptual hydrological model (WASMOD). It aims to explain the sensitivity and differences between the GLUE and Bayesian method applied to hydrological model under precipitation errors with constant multiplier parameter and random multiplier parameter. The 95 % confidence interval of monthly discharge in low flow, medium flow and high flow were selected for comparison. Four indices, i.e. the average relative interval length, the percentage of observations bracketed by the confidence interval, the percentage of observations bracketed by the unit confidence interval and the continuous rank probability score (CRPS) were used in this study for sensitivity analysis under model input error via GLUE and Bayesian methods. It was found that (1) the posterior distributions derived by the Bayesian method are narrower and sharper than those obtained by the GLUE under precipitation errors, but the differences are quite small; (2) Bayesian method performs more sensitive in uncertainty estimates of discharge than GLUE according to the impact of precipitation errors; (3) GLUE and Bayesian methods are more sensitive in uncertainty estimate of high flow than the other flows by the impact of precipitation errors; and (4) under the impact of precipitation, the results of CRPS for low and medium flows are quite stable from both GLUE and Bayesian method while it is sensitive for high flow by Bayesian method.

Synthetic turbulence constructed by spatially randomized fractal interpolation

A spatially randomized fractal interpolation algorithm to construct synthetic fields with statistical properties close to real turbulence is proposed. It improves the previous works on fractal interpolation so that the position mapping between large and small scales is chosen randomly and the stretching factors are drawn from any chosen random multiplier model. In particular, using different parameters of Log-Poisson model, synthetic fields with absolute scaling properties close to SL94 model (for fully developed turbulence), K41 model and MB2000 model (for magnetohydrodynamic turbulence) are obtained, respectively. To model real turbulence fields which do not obey absolute scaling laws but ESS scaling laws, a refined technique is added. It is shown that both the velocity structure functions and the moments of subgrid-scale stress can be precisely predicted when the scale invariance is broken.

Impact of Species Uncertainty Perturbation on the Solution Stability of Positive Matrix Factorization of Atmospheric Particulate Matter Data

Statistical measures for evaluating the similarity of different source apportionment solutions are proposed. The sensitivity of positive matrix factorization to small perturbations in species measurement uncertainty estimates is examined using fine particulate matter measurements on organic carbon, elemental carbon, ions, and metals at the St. Louis-Midwest Supersite. A perturbed uncertainty matrix is created by multiplying each original uncertainty value by a random multiplier generated from a log-normal distribution with a mean of 1 and a standard deviation (and CV) equal to either 0.25, 0.50, or 0.75. The relative errors in reproducing the average contribution estimates from the perturbed data are generally highest for the gasoline exhaust, with the relative error (expressed as a percentage of the "true" value) exceeding 30% for all three perturbation scenarios. The most stable estimates of average source contribution were associated with secondary sulfate and secondary nitrate, with relative errors always less than 4%. Averaged over all 10 sources, the average values for our measure of relative error for the three scenarios are 8%, 14%, and 17%, respectively. Relative errors associated with day-to-day estimates of source contributions can be more than double the size of the relative errors associated with estimates of average source contributions, with errors for four of 10 source contributions exceeding 30% for the largest-perturbation scenario. The stability of source profile estimates in our simulation varies greatly between sources, with a mean correlation between perturbed gasoline exhaust profiles and the true profile equal to only 59% for the largest-perturbation scenario. The process used for evaluation is a tool that may be used to assess the stability of solutions in source apportionment studies.

On invariant tori of full dimension for 1D periodic NLS

Consider the NLS with periodic boundary conditions in 1D (0.1)iut+Δu+Mu±ɛu|u|4=0, where M is a random Fourier multiplier defined by (0.2)Mu^(n)=Vnu^(n) and (Vn)n∈Z are independently chosen in [-1,1]. The quintic nonlinearity in (0.1) is unimportant and may be replaced by u|u|p-2,p∈2Z,p⩾4. We give a proof of the following fact. Theorem. For appropriate M, (0.1) has an invariant tori T (of full dimension) satisfying 12e-r|n|<|qn|<2e-r|n|(n∈Z,q∈T) (r>0 is arbitrary). Remark. The statement holds in fact for most (Vn)n∈Z∈[-1,1]Z, although not explicitly proven here. Written in Fourier modes (qn)n∈Z, the Hamiltonian corresponding to (0.1) is given by (0.3)H(q,q¯)=∑(n2+Vn)|qn|2+ɛ∑n1-n2+n3-n4+n5-n6=0qn1q¯n2qn3q¯n4qn5q¯n6. The proof of Theorem 1 will proceed along the ‘usual’ KAM scheme where the perturbation is eventually removed by consecutive canonical transformations of phase space. The most relevant literature in the present context of an infinite dimensional phase space are the papers of Frohlich et al. [Frohlich, Spencer, Wayne, Localization in disordered, nonlinear dynamical systems, J. Statist. Phys. 42 (1986) 247–274] and especially Poschel [Poschel, Small divisors with spatial structure in infinite dimensional Hamiltonian systems, CMP 127 (1990) 351–393] on disordered systems. Both [Frohlich, Spencer, Wayne, Localization in disordered, nonlinear dynamical systems, J. Statist. Phys. 42 (1986) 247–274, Poschel, Small divisors with spatial structure in infinite dimensional Hamiltonian systems, CMP 127 (1990) 351–393] consider Hamiltonians with short-range interactions and hence these results do not apply to our problem. It turns out, however that the scheme, as elaborated on in great detail in [Poschel, Small divisors with spatial structure in infinite dimensional Hamiltonian systems, CMP 127 (1990) 351–393], is still applicable to (0.3), due to special arithmetical features as will be explained in the next section. Roughly speaking, the key point is the following observation. Let (ni) be a finite set of modes, |n1|⩾|n2|⩾⋯ and (0.4)n1-n2+n3-⋯=0. In the case of a ‘near’ resonance, there is also a relation (0.5)n12-n22+n32-⋯=o(1). Unless n1=n2, one may then control |n1|+|n2| from (0.4), (0.5) by ∑j⩾3|nj|. This feature is specifically 1-dimensional and we do not know at this time how to prove a 2D-analogue of Theorem 1, considering for instance the cubic NLS iut+Δu±u|u|2=0 on T2. It should also be pointed out that almost periodic solutions on a full set of frequencies for NLS and NLW in 1D were constructed in earlier works (see [Bourgain, Construction of approximative and almost periodic solutions of perturbed linear Schrodinger and wave equations, GAFA 6 (2) (1996) 201–230] and [Poschel, On the construction of almost periodic solutions for nonlinear Schrodinger equations, Ergodic Theory Dynamical Systems 22 (5) (2002) 1537–1559]). These invariant tori (of full dimension) were obtained by successive small perturbations of finite-dimensional tori, resulting in very strong compactness properties and in fact a nonexplicit decay rate of the action variables In for n→∞. On the other hand, the construction in this paper (similarly to [Poschel, Small divisors with spatial structure in infinite dimensional Hamiltonian systems, CMP 127 (1990) 351–393]) treats all Fourier modes at once and requires explicit and realistic decay conditions. The multiplier M=(Vn) in (0.3) is to be considered as a parameter and (0.1) a parameter-dependent equation. The role of this parameter is essential to ensure appropriate nonresonance properties of the (modulated) frequencies along the iteration. In the absence of exterior parameters, these conditions need to be realized from amplitude–frequency modulation and suitable restriction of the action-variables. This problem is harder. Indeed, a fast decay of the action-variables (enhancing convergence of the process) allows less frequency modulation and worse small divisors (cf. [Bourgain, On diffusion in high-dimensional Hamiltonian systems and PDE, J. Anal. Math. 80 (2000) 1–35]).

Galton--Watson Branching Processes in a Random Environment I: Limit Theorems

Let $Z_n$~be the number of individuals at time n in a branching process in a random environment generated by independent identically distributed random probability generating functions~$f_0(s), f_1(s),\ldots, f_n(s),\ldots\;$. Let \begin{eqnarray*} X_i&=&\log f_{i-1}^{\prime}(1),\qquad i=0,1,2,\ldots\,; \\ S_0 &=&0,\quad S_n=X_1+\cdots+X_n,\qquad n\ge 1. \end{eqnarray*} It is shown that if $Z_n$ is, in a sense, ``critical," then there exists a limit in distribution $$ \lim_{n\to\infty}\exp\Big\{-\min_{0\le j\le n}S_j\Big\}\, {\bf P}\{Z_n > 0\,\big|\, f_0,\ldots,f_{n-1}\}=\zeta, $$ where $\zeta$ is a proper random variable positive with probability~1. In addition, it is shown that for a ``typical" realization of the environment the number of individuals~$Z_n$ given $\{Z_n > 0\}$ grows as $\exp\{S_n-\min_{0\le j\le n}S_j\}$ (up to a positive finite random multiplier).

Random small Hamming weight products with applications to cryptography

There are many cryptographic constructions in which one uses a random power or multiple of an element in a group or a ring. We describe a fast method to compute random powers and multiples in certain important situations including powers in the Galois field F 2 n , multiples on Koblitz elliptic curves, and multiples in NTRU convolution polynomial rings. The underlying idea is to form a random exponent or multiplier as a product of factors, each of which has low Hamming weight when expanded as a sum of powers of some fast operation.

A variance reducing multiplier for Monte Carlo integrations

A random multiplier for variance reduction is investigated. Its efficiency depends on the correlation of the integrand and the reference function. However the multiplier can be used in quasi-Monte Carlo integrations also.

Gradients of potential fields on rough surfaces: Perturbative calculation of the singularity distribution function f( alpha ) for small surface dimension.

An exact solution to the Laplace equation with Dirichlet boundary conditions on a simple boundary is used in an iterative fashion to study the case of a stochastic rough surface with fractal dimension D=2+\ensuremath{\epsilon}. For small \ensuremath{\epsilon}, an analytic expression is derived for the spectrum of singularities f(\ensuremath{\alpha}) of the gradient of the potential normal to the boundary, using the random multiplier approach. A binomial approximation to the multiplicative process is shown to severely underestimate f(\ensuremath{\alpha}) for low values of \ensuremath{\alpha}.

Results of Recent U.S. Outer Continental Shelf Lease Bidding

Summary The larger number of leases offered in area-wide sales has lowered the average number of bids per lease. Yet, the number of competing bids for leases of the same bidder interest has been stable and almost equal to that for non-area-wide sales. Data The SPE Senior Technical Editor recently asked pertinent questions in JPT about competition in Outer pertinent questions in JPT about competition in Outer Continental Shelf (OCS) Sale 72, the first Gulf of Mexico area-wide offering. The use of a mixed Poisson distribution model (3MP) that groups leases into three classes (no, low, and high bidder interest) triggered the questions. Table 1 shows 3MP results for Sale 72 plus subsequent area-wide sales and other recent sales. The Poisson means of competing bids were stable for leases with the same level of bidder interest, but a much larger fraction of leases were in the no bidder interest class in the area-wide sales. An opposite and wrong conclusion can be drawn comparing only the average number of bids per lease. A current OCS lease policy issue is whether more leases should be offered with a royalty or profit share stipulation. That stipulation which engenders more competition is more desirable. Two reports from the U.S. General Accounting Office (GAO) use an average number of bids per lease to measure competition. Table 2 shows 3MP results for three sales with both fixed royalty and profit share offerings. Profit share offerings were almost devoid of no-bidder-interest leases. Poisson means were about the same or higher for fixed royalty as compared to profit share leases with the same bidder interest. Opposite conclusions would be drawn using the GAO measure of competition. The Senior Technical Editor also noted that brevity is a merit in these writings. Without resorting to the sophisticated syntax of the burgeoning discipline of model validation, the validation of 3MP, previously expunged for brevity, will be addressed using a crude analogy. Suppose you have some homing pigeons and want to know how well individual pigeons can get home. Presumably some pigeons may be more sensitive to Presumably some pigeons may be more sensitive to dishevelment than others. The better pigeon would find its way home even when subjected to severe dishevelment. This is, in essence, exactly how 3MP was tested. Known underlying distributions were disheveled by a random multiplier. The Sept. 12, 1972, OCS sale, which was the worst-fitting of the 31 sales in our original data, provided an outlying reference. When distributions provided an outlying reference. When distributions disheveled severely as compared to the outlying reference reproduced the underlying 3MP parameters, 3MP was accepted. Validation by reproduction of underlying parameters from disheveled data has no intrinsic connection with statistical goodness of fit. This is obvious from the homing pigeon analogy. One also could measure the directness of a pigeon's path from the start to its home. Some pigeons might flit back and forth but still arrive home. Others might take a direct path ending far from home. Any of a multitude of statistical goodness-of-fit tests could measure a pigeon's directness of path; however, all are meaningless or, at best, peripheral to assessing a pigeon's ability to get home. Thus, goodness-of-fit tests pigeon's ability to get home. Thus, goodness-of-fit tests can be ignored. Questions about the correct use of the right data for 3MP studies cannot be ignored, however. I was the originator of the Lease Production and Revenue data base and obtained the source data for the 31 sales studied originally from that data base. Many investigators received the data and obtained results agreeing with ours. The exception was the work referred to but not presented, no doubt for brevity, by Bruckner and presented, no doubt for brevity, by Bruckner and Nachstheim in the July 1983 JPT. When I saw their detailed work, I concluded there was some calculational error. Neither that work nor their data appear available now. The reason for the difference can only be surmised with no resolution possible. Fortunately, that indeterminate status need not prevail for 3MP results given here. All raw data are available. Detailed 3MP results along with the computer program (in BASIC) that produced them are available. Brevity precludes their inclusion here. precludes their inclusion here. JPT P. 1509

State estimation for cox processes on general spaces

Let N be an observable Cox process on a locally compact space E directed by an unobservable random measure M. Techniques are presented for estimation of M, using the observations of N to calculate conditional expectations of the form E [ M]| F A], where F A is the σ–algebra generated by the restriction of N to A. We introduce a random measure whose distribution depends on N A , from which we obtain both exact estimates and a recursive method for updating them as further observations become available. Application is made to the specific cases of estimation of an unknown, random scalar multiplier of a known measure, of a symmetrically distributed directing measure M and of a Markov–directed Cox process on R . By means of a Poisson cluster representation, the results are extended to treat the situation where N is conditionally additive and infinitely divisible given M.

Calculation of error probability in an optical communication channel in the presence of intersymbol interference, using a characteristic function method

In this paper the error rate of an optical channel is calculated. The channel considered consists of a laser or light emitting diode as the optical source and a glass fibre as transmission medium. The transmitted optical pulses are spread in the fibre causing intersymbol interference. The receiver is assumed to be an avalanche photodiode which is considered as a p-n junction and a random multiplier whose electron probability distribution function is known, and a filter. In calculating the probability of error, their characteristic functions are multiplied and the probability density function of all the noises put together is truncated to a finite interval and described in a trigonometric series form. The total characteristic function is related to the Fourier coefficients of the periodic probability density function, resulting in an expression that is then used, for a given bit rate, to determine the worst case and average probability of error as functions of transmitted power or signal-to-noise ratio, for different amounts of intersymbol interference.

A Random Signal Multiplier for Turbulence Measurements

A Random Signal Multiplier for Turbulence Measurements

Maximum-Likelihood Estimation of Parameters Subject to Restraints

The estimation of a parameter lying in a subset of a set of possible parameters is considered. This subset is the null space of a well-behaved function and the estimator considered lies in the subset and is a solution of likelihood equations containing a Lagrangian multiplier. It is proved that, under certain conditions analogous to those of Cramer, these equations have a solution which gives a local maximum of the likelihood function. The asymptotic distribution of this `restricted maximum likelihood estimator' and an iterative method of solving the equations are discussed. Finally a test is introduced of the hypothesis that the true parameter does lie in the subset; this test, which is of wide applicability, makes use of the distribution of the random Lagrangian multiplier appearing in the likelihood equations.