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● ● ●●● ●● ●● ●● ●● ●● ●●● ● ●● ● ●● ● ●●● ●● ●●● ●● ●● ● ●●●● ●● ●● ● ●● ●● ●●● ●●●● ● ●●● ●●● ● ●●● ● ●● ●● ●● ●●● ●●● ● ●●●● ●●● ●● ●● ●●●● ● ●●● ●● ●● −4 −2 0 2 4−0.006 −0.004 −0.002 0.000 0.002 0.004 0.006Normal Q−Q Plot Theoretical QuantilesSample Quantiles Figure 7: Histogram and normal quantile-quantile plot of... | https://arxiv.org/abs/2504.19138v1 |
large m, we conjecture that SUM′ 1can be approximated by SUM′′ 1=X k∈QNmZ′(k)S′(k)ˆf(k), where each Z′(k) is sampled independently from a Bernoulli distribution with success probability 2−m. This approximation holds rigorously for polynomial in- tegrands, where the support of non-zero Walsh coefficients is particularly... | https://arxiv.org/abs/2504.19138v1 |
SIAM Journal on Numerical Analysis , 61(2):495–514, 2023. [14] W.-L. Loh. On the asymptotic distribution of scrambled net quadrature. Annals of Statistics , 31(4):1282–1324, 2003. [15] J. Matouˇ sek. On the L2–discrepancy for anchored boxes. Journal of Com- plexity , 14:527–556, 1998. [16] M. K. Nakayama and B. Tuffin.... | https://arxiv.org/abs/2504.19138v1 |
on s. Proof. First we write ∞Y ℓ=1(1 +qℓ N)s= exp s∞X ℓ=1log(1 + qℓ N) . 40 Because log(1 + qℓ N) is monotonically decreasing in ℓ, ∞X ℓ=1log(1 + qℓ N)⩽Z∞ 0log 1 + exp( −πℓp s/12N) dℓ =1 πr 12N sZ∞ 0log 1 + exp( −ℓ) dℓ =πr N 12s. Hence∞Y ℓ=1(1 +qℓ N)s⩽exp πr sN 12 . Our conclusion then follows from equation (15... | https://arxiv.org/abs/2504.19138v1 |
k⊕,X⊕ jℓ=1{ℓ∈κ⊕ j}equals 1 if and only if ℓ∈κjfor an odd number of kamong k1, ...,kr. By a binomial distribution with success probability qℓ N/(1 +qℓ N), PrL(X⊕ jℓ= 1) =⌈r/2⌉X j=1r 2j−1qℓ N 1 +qℓ N2j−11 1 +qℓ Nr−2j+1 =1 2−1 21−qℓ N 1 +qℓ Nr . (60) Also notice that {X⊕ jℓ, j∈1:s, ℓ∈N}are jointly independent unde... | https://arxiv.org/abs/2504.19138v1 |
2−1/r)21+1/rr⩽8r and log(r)r 12N π2s−1⩽ℓ∗⩽log(8 r)r 12N π2s. (62) log(r)r 12N π2s−1⩽ℓ∗⩽log(8 r)r 12N π2s. 45 Using the inequality 1 −exp(−x)⩾x/(1+x) when x⩾0, equation (61) becomes PrL ∥k⊕∥1⩽N ⩽exp tN−sX ℓ∈NPrL(X⊕ jℓ= 1)tℓ 1 +tℓ ⩽exp tN−sX ℓ∈N1{ℓ⩽ℓ∗}tℓ 4(1 + tℓ) = exp tN−stℓ∗(ℓ∗+ 1) 8(1 + tℓ∗) . (63) Setting t=... | https://arxiv.org/abs/2504.19138v1 |
Optimal experimental design for parameter estimation in the presence of observation noise Jie Qi1and Ruth E. Baker2 1College of Information Science and Technology, Donghua University, Shanghai, China 2Mathematical Institute, University of Oxford, Oxford, United Kingdom Abstract Using mathematical models to assist in th... | https://arxiv.org/abs/2504.19233v1 |
values. A drawback is therefore that local sensitivity measures can lead to inefficient experimental designs if the input parameters are not close to their “true” values. To overcome the limitations of local sensitivity measures, global sensitivity analyses have been increasingly adopted in optimal ex- perimental desig... | https://arxiv.org/abs/2504.19233v1 |
design within the context of both uncorrelated (in- dependent, identically distributed (IID)) and autocorrelated (OU distributed) observation noise, specifically focussing on the logistic model, which is ubiquitously applied in the modelling of biological systems. We suggest an approach to optimal experimental design t... | https://arxiv.org/abs/2504.19233v1 |
a dataset, we will consider both the Fisher information matrix, which provides a local measure of uncertainty, and Sobol’ indices, which provide a global measure of uncertainty. Fisher information matrix. The Fisher information matrix can be defined using the expec- tation of the Hessian of the log-likelihood function ... | https://arxiv.org/abs/2504.19233v1 |
. . , n s, as defined in Equation (8). Sobol’ indices are usually computed using Monte Carlo simulation [8, 19]. In this paper, we compute the Sobol’ indices directly using the Global Sensitivity Analysis Toolbox for Matlab [20]. 2.4 Autocorrelated observation noise We use an OU process, as defined by the stochastic di... | https://arxiv.org/abs/2504.19233v1 |
(8). It is worth noting that the inverse of Σ, also known as the precision matrix, contains information about the conditional dependencies between different time points, and so their inclusion in the sensitivity matrices FandGenable quantification of how parameter sensitivities are influenced by temporal correlations i... | https://arxiv.org/abs/2504.19233v1 |
Ig= max I/bracketleftig det(G(θ,I))/bracketrightig , (29) subject to the constraint k/summationdisplay i=1Ii=ns. (30) We solve the optimisation problem in Equations (29)–(30) using PlatEMO, an evolutionary multi-objective optimisation platform for Matlab [24, 25] and, in particular, we use the com- petitive swarm opt... | https://arxiv.org/abs/2504.19233v1 |
interval generated using the 95% confidence intervals from the profile likelihoods. The right-most three panels show the profile likelihoods for each parameter. The results in the top row are generated using uncorrelated Gaussian noise with σ2 IID= 9.0. The results in the middle row are generated using correlated OU no... | https://arxiv.org/abs/2504.19233v1 |
data but the confidence intervals are generated assuming the data are uncorrelated. 13 so that the noise variance is equivalent across both noise models. We see that, for all values of σ2, the confidence intervals for all parameters are almost identical for uncorrelated noise and misspecified noise, with greater confid... | https://arxiv.org/abs/2504.19233v1 |
varied from three to ten. The top row shows the results from evenly distributed observations, whilst the second row shows the optimal experimental design derived using the Fisher information matrix, and the bottom row shows the optimal experimental design derived using the global information matrix. In each case, the l... | https://arxiv.org/abs/2504.19233v1 |
time points. This is a direct result of the fact that the optimisation algorithms place the majority of the observations at early times in order to accurately estimate the growth rate, r, 16 Figure 5: The results of optimising experimental design in the correlated noise case as the number of observations, ns, is varied... | https://arxiv.org/abs/2504.19233v1 |
or close to, the terminal time, which allows estimation of the carrying capacity, K. With more total observations, ns, further points are added at the end of time interval for larger values of ϕ, which enables increases in the accuracy of estimation of the carrying capacity, K. Figure 8 demonstrates how the parameter c... | https://arxiv.org/abs/2504.19233v1 |
as inputs, which means that it may be more appropriate for cases in which parameter values are relatively uncertain. The trade-off is the increased computational complexity of the global method compared to that of the local method. Our methodology is very general, in the sense that it can be applied in any context wher... | https://arxiv.org/abs/2504.19233v1 |
Xu L, Cao Z, Lenstra TL, Grima R. 2022 Quantifying how post-transcriptional noise and gene copy number variation bias transcriptional parameter inference from mRNA distributions. eLife 11, e82493. [12] Kuhlmann H. 2001 Importance of autocorrelation for parameter estimation in regression models. In Proceedings of the 10... | https://arxiv.org/abs/2504.19233v1 |
the main text, but with decreased noise correlation ( ϕ= 0.1 compared to ϕ= 0.02 in the main text). Figure S1: Parameter identifiability under the different noise models, with data collected at 11 regularly spaced time points t= 0,8,16, . . . , 80. The left column shows the sampled data, the underlying model solution u... | https://arxiv.org/abs/2504.19233v1 |
arXiv:2504.19331v1 [math.ST] 27 Apr 2025Bahadur asymptotic efficiency in the zone of moderate deviation probabilities Mikhail Ermakov April 2025 Institute of Problems of Mechanical Engineering RAS,and St . Petersburg State University Аннотация For a sequence of independent identically distributed rand om variables having... | https://arxiv.org/abs/2504.19331v1 |
the Bahadur setting. For the asymptotically minimax s etting the lower bounds have been proved in [4, 5, 10]. The goal of presen t paper is to obtain an analogue of the Bahadur lower bound in the zone of moderate deviation probabilities assuming the same conditions as th e lower bound on the Hajek-Le Cam local asymptot... | https://arxiv.org/abs/2504.19331v1 |
is a vicinity Uofθ, such that, for any δ >0, we have lim n→∞sup θ∈UPθ(|ˆθn−θ|> δun) = 0. (2.5) 2.2 Locally asymptotically minimax lower bound The locally asymptotically minimax lower bound of risks in t he zone of moderate deviation probabilities does not require any cond itions of consistency. Theorem 2.1 Let the stat... | https://arxiv.org/abs/2504.19331v1 |
is deduced from Theorem 2.3. . We call estimator ˆθnlocally uniformly consistent, if for any θ0∈Θ, there is a neighborhood Uof the point θ0such that, for anyε >0, there is lim n→∞sup θ∈UPθ(|ˆθn−θ|> ε) = 0. (2.17) 6 Theorem 2.4 Let the statistical experiment E={(S,B),Pθ,θ∈Θ}has the finite Fisher information at the point ... | https://arxiv.org/abs/2504.19331v1 |
arXiv:2504.19337v1 [math.ST] 27 Apr 2025Submitted to Bernoulli Frequency Domain Resampling for Gridded Spatial Data SOUVICK BERA1,a, DANIEL J. NORDMAN2,cand SOUTIR BANDYOPADHYAY1,b 1Department of Applied Mathematics and Statistics, Colorado School of Mines, Golden, CO 80401, USA, aberasouvick@mines.edu ,bsbandyopadhyay... | https://arxiv.org/abs/2504.19337v1 |
spatial covariances as well as higher or- der (i.e., fourth) process cumulants, which arise due to covariances between the spatial periodogram ordinates. Technically, it is the variance contributions related to higher order cumulants that are gener- ally missed or ignored in the FDB approach of ( Ng, Yau and Chen ,2021... | https://arxiv.org/abs/2504.19337v1 |
spectral mean parameters. Recently, a major advancement was achieved by (Meyer, Paparoditis and Kreiss ,2020 ) who introduced an innovative bootstrap method, called the hybrid periodogram bootstrap (HPB), that represents the state-of-the-art approach for time series in- ference concerning spectral means. Related to thi... | https://arxiv.org/abs/2504.19337v1 |
the random process Ė(s)is observed at Ĥ≡Ĥ1Ĥ2sampling sites defined by {s1,...., sĤ}={s∈Z2:s∈D(Ĥ1×Ĥ2)}=D(Ĥ1×Ĥ2)∩Z2 given by those locations on the grid Z2that lie within a rectangular sampling region D(Ĥ1×Ĥ2) ≡ [1,Ĥ1] × [1,Ĥ2]for integersĤ1,Ĥ2g1. For spatial data analysis, the above sampling framework corresponds to a so... | https://arxiv.org/abs/2504.19337v1 |
family of spectral densities (indexed by Ă). Whittle estimation aims to identify the closest member ĜĂto the true density Ĝ(cf. ( Taniguchi ,1979 )). Assuming real-valued Ăfor illustration, the solution to a spectral mean equation ĉ(ć)=0 identifies the appropriate ĜĂunder certain conditions, using a function ć(ā) ≡ [ 1−... | https://arxiv.org/abs/2504.19337v1 |
density. Given the complicated structure in the limit distri- bution from ( 2), one possible approach is to explore resampling for approximating the distribution ofĄĤ(ć)nonparametrically. However, estimation of the variance component Ă2 2, in particular, poses significant challenges for bootstrap approximations to the d... | https://arxiv.org/abs/2504.19337v1 |
ℓ=1(I(ℓ) ĩīĘ(āj,Ę)−/tildewideIĤ(āj,Ę))2, (4) where/tildewideIĤ(āj,Ę) ≡Ĉ−1/summationtext.1Ĉ ℓ=1I(ℓ) ĩīĘ(āj,Ę). Here in ( 4), we are isolating a component from the overall sub- sampling variance estimator Ă2 Ĥin (3) that arises due to marginal variances in subsample periodgram/braceleftbig I(ℓ) ĩīĘ(āj,Ę)/bracerightbigĈ ℓ... | https://arxiv.org/abs/2504.19337v1 |
spectral mean statistics. Theorem 3.1. Suppose Assumptions 1-4hold and the subsample size ĘĤ≡Ĥ(Ę) 1×Ĥ(Ę) 2satisfies 1/Ĥ(Ę) ġ+Ĥ−1 ġĤ(Ę) ġ→0 forġ=1,2 asĤ≡Ĥ1×Ĥ2→ ∞ . Then, the spatial subsampling estimators of variance components are consistent: /hatwideĂ2 1,ĤČ−→Ă2 1,/hatwideĂ2 2,ĤČ−→Ă2 2,and/hatwideĂ2 ĤČ−→Ă2≡Ă2 1+Ă2 2asĤ→... | https://arxiv.org/abs/2504.19337v1 |
form aims to produce bootstrap versions {đ∗ j/hatwideĜĤ(āj,Ĥ)}of the periodogram ordinates {IĤ(āj,Ĥ)}(i.e., indexed over j∈ JĤ), by treating the latter as approximately independent exponential variables with respective means {Ĝ(āj,Ĥ)}. However, the problem in this bootstrap re-creation is that dependence among periodog... | https://arxiv.org/abs/2504.19337v1 |
bias. While the above bias decreases to zero with increasing spatial sample size Ĥ, this bias term can potentially impact approximations in small sample sizes. Therefore, in small samples, it is also possible to consider spatial subsampling for estimating this bias part as /hatwidestBias sub=Ĉ−1/summationtext.1Ĉ ℓ=1Ę1/... | https://arxiv.org/abs/2504.19337v1 |
Carlo simulations to evaluate coverage and used 500 simulations per simulated dataset to approximate bootstrap distributions. For purposes for understanding the performance of both the FDWB and the HFDB methods in terms of coverage accuracy, below we separate numerical findings by Gaussian and non-Gaussian processes. 5.... | https://arxiv.org/abs/2504.19337v1 |
achieving the best coverage across block sizes, and furthermore showing the best coverages as the sample size Ĥincreases. The bias correction also helps with smaller sample sizes to achieve improvement in coverages for HFDB. In contrast, the performance of the the FDWB deteriorates with increasing sample size. This iss... | https://arxiv.org/abs/2504.19337v1 |
Ą(h) ≡ā(Ė(0)−Ė(h))2is the variogram at lag h; also above ||h|| ≡√ hĐh. This null hypothesis can also be written in terms of a spectral mean assessment of whether ĉ(ć)=0 holds for various spectral mean parametered defined by ć(ā) ≡ {2 cos(hĐ Ġā) −2 cos(hĐ ğā)}. Lettingă≡ (2Ą(h1),...,2Ą(hģ))denote a vector of variograms i... | https://arxiv.org/abs/2504.19337v1 |
here. Both FDWB and HFDB versions of this test statis- tic are computed as [č∗ ĂĀēþ,Ĥ(ć)]2and[Ą∗ ĄĀēþ,Ĥ(ć)]2, respectively, upon applying ć(ā) ≡ {2 cos(hĐ 1ā) −2 cos(hĐ 2ā)}in (7) and ( 8); note that bootstrap statistics are then centered to have mean zero which mimics how [ĄĤ(ć) ≡√Ĥ[/hatwideĉĤ(ć) −ĉĤ(ć)]behaves under ... | https://arxiv.org/abs/2504.19337v1 |
as the FDWB proposed by ( Ng, Yau and Chen , 2021 ), rely on specific distributional assumptions and can fail to provide valid distributional approx- imations for non-Gaussian spatial data, due to issues in correctly capturing the variances of spectral statistics for such data. In contrast, the HFDB overcomes these limi... | https://arxiv.org/abs/2504.19337v1 |
Ĥ, 18 thenč∗ ĂĀēþ,Ĥ(ć)can be written as č∗ ĂĀēþ,Ĥ=Ĥ−1/2/summationdisplay.1 j∈J+ĤĒ∗ j,Ĥ, (10) where, Ē∗ j,Ĥ=(2ÿ)2(ć(āj,Ĥ)/hatwideĜĤ(āj,Ĥ)+ć(−āj,Ĥ)/hatwideĜĤ(−āj,Ĥ))(đ∗ j−1),j∈ J+ Ĥ. Since,|J+ Ĥ|→∞ , asĤ→∞ , a conditional version of Lyapunov’s CLT can be applied to Eq. ( 10) (using the established convergence in probabil... | https://arxiv.org/abs/2504.19337v1 |
to supĮ∈R|/hatwideĂĤ(Į) −Ă(Į)|Č→0 as{Ĥģ}was arbitrary. Then statement (i) follows from a triangle inequality. Now, due to boundedness, /hatwideĂĤ(Į0)Č→Ă(Į0)is equivalent to ā(/hatwideĂĤ(Į0)−Ă(Į0))2→0, and we show the latter. 20 To control this, fix ą∈ (0,1)whereąsets a threshold for spatial proximity, and for j=(Ġ1,Ġ2),... | https://arxiv.org/abs/2504.19337v1 |
ā(IĤ(ā))=Ĝ(ā)+ċ(Ĥ−1),ā∈Π2(16) and forā1,ā2∈Π2, andğ=1,2, Cov(IĤ(ā1),IĤ(ā2))= Ĝ2(ā1)+ċ(Ĥ−1) if|Ĉ1ğ|=|Ĉ2ğ|(≠0,∀ğ=1,2) (2ÿ)2 ĤĜ4(ā1,ā2,−ā2)+ċ(Ĥ−1) if|Ĉ1ğ|≠|Ĉ2ğ|.(17) The error terms are uniform in ā. Hence, forğ=1,2, Var(ĄĤ(ć))=Ĥ−1(4ÿ2)2/summationdisplay.1 j,k∈JĤć(āj,Ĥ)ć(āk,Ĥ)Cov(IĤ(āj,Ĥ),IĤ(āk,Ĥ)) =Ĥ−1(4ÿ2)2... | https://arxiv.org/abs/2504.19337v1 |
analysis. The Annals of Statistics 241934–1963. DAVISON , A. C. and H INKLEY , D. V. (1997). Bootstrap methods and their application 1. Cambridge university press. FUENTES , M. (2002). Spectral methods for nonstationary spatial processes. Biometrika 89197-210. FUENTES , M. (2006). Testing for separability of spatial-te... | https://arxiv.org/abs/2504.19337v1 |
stationary time series. The Annals of Mathemat- ical Statistics 329–348. POLITIS , D. N. and M CELROY , T. S. (2019). Time series: A first course with bootstrap starter . CRC Press. POLITIS , D. N., R OMANO , J. P. and W OLF, M. (1999). Subsampling . Springer Science & Business Media. SHERMAN , M. (1996). Variance estim... | https://arxiv.org/abs/2504.19337v1 |
non-Gaussian processes with low Ă2 2values, we considered the Gaussian-Log-Gaussian process as described in ?. We generated datasets of sizes 30 ×30(Ĥ=900),50×50(Ĥ=2500), and 70×70(Ĥ= 4900)on regularly spaced square-shaped lattice regions. For each dataset size, we ran 500 Monte Carlo simulations for coverage and 500 s... | https://arxiv.org/abs/2504.19337v1 |
SIGNAL DETECTION FROM SPIKED NOISE VIA ASYMMETRIZATION BYZHIGANG BAO1,a, KHAMANCHEONG2,cAND YUJILI1,b 1University of Hong Kong,azgbao@hku.hk;bu3011732@connect.hku.hk 2Hong Kong University of Science and Technology,ckmcheong@connect.ust.hk The signal plus noise model H=S+Yis a fundamental model in signal detection when ... | https://arxiv.org/abs/2504.19450v1 |
ordered in de- scending order. Throughout the paper, we always assume that Σis invertible and further assume ∥Σ−1∥op≤Kfor some constant K >0. We interpret Sas a signal, which is polluted by the noise part ΣX. Let 1be the all-one matrix. Differently from the classical setting in many previous literature where (Σ,T) = (I... | https://arxiv.org/abs/2504.19450v1 |
can turn to consider the random matrix model H1H∗ 2 or a linearization of it. Under an assumption that the operator norm of the noise part ΣXis significantly smaller than the signal part S, the authors in [27] find that the leading eigenvalue (in magnitude) of the non-Hermitian model contains the precise information of... | https://arxiv.org/abs/2504.19450v1 |
the eigenvalues of Yare in pair. We denote the non-zero eigenvalues of Ybyλ±i≡ ±λi,i= 1,...,n ∧p. We make the convention that λ1,...,λ n∧pare those eigenvalues with arguments in (−π/2,π/2]. Further, λ1,...,λ n∧pare in descending order (in magnitude). Due to the fact that we are considering real matrix, we have another ... | https://arxiv.org/abs/2504.19450v1 |
Throughout this paper, we adopt the notion of stochastic domination introduced in [35], which allows a loss of boundedness, up to a small power of N, with high probability. DEFINITION 1.3. (Stochastic domination) Let X= XN(u) :N∈N,u∈U , Y = YN(u) :N∈N,u∈UN SIGNAL DETECTION VIA ASYMMETRIZATION 5 be two families of r... | https://arxiv.org/abs/2504.19450v1 |
on Σ∗ui. The above theorem reveals a non-universal feature of the limiting distribution of the outlier, which has been previously observed in other Hermitian or non-Hermitian model; see [24, 45, 51] for instance. Particularly, the distribution of the outlier is (asymptotically) a convolution of the distribution of a li... | https://arxiv.org/abs/2504.19450v1 |
studying the case S= 0 butΣis with deformation as defined in (1.2). In this case, we consider the noise part Xin (1.3), which can be regarded as a mul- tiplicative deformation of X0. Our aim is to show that as long as σmax≤n1/4−ε0for some small constant ε0>0, the results proved for X0, including the upper bound of the ... | https://arxiv.org/abs/2504.19450v1 |
and robust under the general variance profile assumption, thanks to the inputs from [1]. 8 1.2. Notation. Throughout this paper, we regard nas our fundamental large parameter. Any quantities that are not explicit constant or fixed may depend on n; we often omit the argument nfrom our notation. We further use ∥A∥opfor t... | https://arxiv.org/abs/2504.19450v1 |
have some multiple di’s or close di’s. But in any case, the fluctuation order is no larger than N−εaccording to our assumption of σmax. Hence, the choice of the lower phase bound π/logNin (2.1) is enough to distinguish the true signals from the other eigenvalues. In the simulation study, we fix (p,n) = (800 ,2000) . We... | https://arxiv.org/abs/2504.19450v1 |
case, 10 Fig 2 Fig 3 SIGNAL DETECTION VIA ASYMMETRIZATION 11 Fig 4 Fig 5 12 Fig 6 as long as the second moments of the matrix entries exist. This distinction between the ex- treme singular values and extreme eigenvalues is supported by results from more classical random matrix models; see, for instance, [8, 53, 5, 21, ... | https://arxiv.org/abs/2504.19450v1 |
definition of σmaxfrom (1.4). Further, we introduce the notation Ri(z),i= 1,2 to include all the matrices in the remainder terms after applying expansion, which satisfy that for any given unit vectors u,v ⟨u,Ri(z)v⟩=O≺(q−i nσ2i max). (3.8) By our assumption on Σ, it is easy to check σ2 max≤Cn1/2−ε0for some ε0> ε > 0. H... | https://arxiv.org/abs/2504.19450v1 |
{Ai,Bj,Ck,Dℓ:i∈Jm1K,j∈Jm2K,k∈Jm3K,ℓ∈Jm4K} with mean 0 and the covariance structure given by Cov(Ai,Aj) =1 n|z|4X α,βuiαujαviβvjβ⃗T⊤ α·h I−1 n2|z|2T∗Ti−1⃗Tβ·, Cov(Bi,Bj) =1 n|z|4X α,βψiαψjαϕiβϕjβ⃗T⊤ ·αh I−1 n2|z|2TT∗i−1⃗T·β, Cov(Ci,Cj) =1 n2|z|4X α,βqiαqjαγiβγjβ TT∗h I−1 n2|z|2TT∗i−1⃗T·β! α, Cov(Di,Dj) =1 n2|z|4X α,βriα... | https://arxiv.org/abs/2504.19450v1 |
write Ef′(t) = iEQχGf(t) = i√nE(u∗X1X∗ 2Gv+ψ∗X∗ 2X1Gϕ+q∗X1Gγ+η∗GX∗ 2r)χGf(t) =:I+II+III+IV (5.5) SIGNAL DETECTION VIA ASYMMETRIZATION 19 In the sequel, we show the estimate of the first term Iin details, and the other terms can be estimated similarly. We write via cumulant expansion [40, Lemma 1.3] I= i√nEu∗X1X∗ 2Gvf(t... | https://arxiv.org/abs/2504.19450v1 |
(∂1 1,ijQ)2will produce an additional factor of n, we need to deal with this term in a finer way. Bounding one ∂1 1,ijQby√nq−1 n, and for the rest applying the same reasoning as in (3), we obtain |(4)| ≤Cn−1X ij X∗ 2Gvu∗ ji(∂1 1,ijQ)2 ≤Cq−1 nX ij u∗ iθ∗ei˜e∗ jX∗ 2Gv˜e∗ jξ∂1 1,ijQ ≲q−1 n. Altogether, the second order... | https://arxiv.org/abs/2504.19450v1 |
n|z|2X ik|ui|2|vk|2⃗T⊤ i· I−1 n2|z|2(T∗T)−1 ⃗Tk·Ef(t) +O≺(q−1 n), II=−t n|z|2X jk|ϕj|2|ψk|2⃗T⊤ ·j I−1 n2|z|2(TT∗)−1 ⃗T·kEf(t) +O≺(q−1 n), III=−t n2|z|4X ik|qi|2|ζk|2 TT∗h I−1 n2|z|2(TT∗)i−1⃗T·k iEf(t) +O≺(q−1 n), IV=−t n2|z|4X jk|rj|2|ηk|2 TT∗h I−1 n2|z|2(TT∗)i−1⃗T·k jEf(t) +O≺(q−1 n). Plugging the above estima... | https://arxiv.org/abs/2504.19450v1 |
by T=n−1T (6.1) the variance profile of X1andX2and we recall the flatness assumption in (1.1). We will follow the strategy developed in [1, 2, 3]. Especially, our matrix X0can be re- garded as a special case of the general model considered in [1]. Based on the result for X0, we then derive the results for the model Xvi... | https://arxiv.org/abs/2504.19450v1 |
to the vector equation (6.5) is proved in [2]. One can check from (6.4) that the diagonal part of Mzis in fact purely imaginary and the imaginary part Immz j(iη)form the vector uandv. Using the block structure of V, one can further write the system of vector equations (6.5) as a system of four equations, with the notat... | https://arxiv.org/abs/2504.19450v1 |
the boundedness of the operator norm G(z′), one can conclude the proof of the uniformity for all |z| ≥p ρ(V) +δ′. In the sequel, we prove (6.8), which follows from the lemma below, according to (6.4). SIGNAL DETECTION VIA ASYMMETRIZATION 27 LEMMA 6.2. The solution of (6.5) satisfies (6.11) ⟨u(η)⟩=⟨v(η)⟩ for all η >0, w... | https://arxiv.org/abs/2504.19450v1 |
we have (6.15). With ( 6.15), the remaining proof can be done similarly to that in [2]. Using the first equa- tion of ( 6.5) gives (6.28) η=u(η+Vv)(η+V∗u) +|z|2u− V∗u. By the Perron-Frobenius theorem, there exists a vector ϱ∈Rn+p +such that (6.29) Vϱ=ϱ,⟨ϱ⟩= 1,ϱ∼1 Note that Vhas zero blocks and indeed we apply the Perro... | https://arxiv.org/abs/2504.19450v1 |
domain with cardinality NO(C) such that for each zin this domain, one can find an z′in the ϵ-net, so that |z−z′| ≤N−C. By the definition of stochastic dominance in Definition 1.3, one can readily conclude from Proposition 3.2 that the estimates therein holds uniformly on the ϵ-net, with high probability. Then for any o... | https://arxiv.org/abs/2504.19450v1 |
other factors can be simply bounded by O≺(1). Apart from the qnfactors, the ijsum of all the derivatives can be simply bounded by X ij|e∗ j(X∗ 2Gvu∗)ei||e∗ jX∗ 2Gv||u∗Gei| ≤X i|u∗Gei|sX j|e∗ j(X∗ 2Gvu∗)ei|2sX j|e∗ jX∗ 2Gv|2 ≤X i|u∗Gei||u∗ei|p v∗G∗X2X∗ 2Gvp v∗G∗X2X∗ 2Gv ≺X i|u∗Gei||u∗ei| ≺1, where the last step follows ... | https://arxiv.org/abs/2504.19450v1 |
regime. Ann. Stat., 50 (2), 1144-1169, (2022). [13] Bao Z, He Y , Yang F. Random matrix theory: Local laws and applications. Handbook of Statistics. 2024. [14] Bao, Z., Pan, G., Zhou, W. Universality for the largest eigenvalue of sample covariance matrices with general population. Ann. Statist., 43(1):382421, 2015. [15... | https://arxiv.org/abs/2504.19450v1 |
correlation decay. InForum of Mathematics, Sigma 2019 Jan (V ol. 7, p. e8). Cambridge University Press. [35] Erd ˝os L., Knowles A., and Yau H.-T. Averaging fluctuations in resolvents of random band matrices. Ann. Henri Poincar ´e, 14(8):1837–1926, 2013. [36] Gavish M, Donoho DL. The optimal hard threshold for singular... | https://arxiv.org/abs/2504.19450v1 |
STOCHASTIC SUBSPACE VIA PROBABILISTIC PRINCIPAL COMPONENT ANALYSIS FOR CHARACTERIZING MODEL ERROR Akash Yadav University of Houston ayadav4@uh.eduRuda Zhang University of Houston rudaz@uh.edu ABSTRACT This paper proposes a probabilistic model of subspaces based on the probabilistic principal component analysis (PCA). G... | https://arxiv.org/abs/2504.19963v2 |
the authors presented a Bayesian calibration technique as an improvement on the traditional techniques. Since then, the Bayesian inferential and modeling framework has been adopted and further developed by many studies [ 4–8]. KOH Bayesian calibration corrects the model output but has limited extrapolative capability t... | https://arxiv.org/abs/2504.19963v2 |
the parameters of assumptions. However, it is not always easy to pinpoint the source of the errors especially when errors arise from multiple sources. Furthermore, this approach does not correct for model error. Another indirect representation approach to address model error is based on the concept of stochastic reduce... | https://arxiv.org/abs/2504.19963v2 |
The accuracy and efficiency of the proposed method are validated using numerical examples in Section 6. Finally, Section 7 concludes the paper with a brief summary and potential future work. 2 Stochastic reduced-order modeling This section presents the concept of stochastic reduced-order modeling. The high-dimensional ... | https://arxiv.org/abs/2504.19963v2 |
[x1···xm]∈ Rn×mbe a sample of the state, with xi=x(ti;µi), sample mean x=1 mPm i=1xi, centered sample X0=X−x1⊺ m, and sample covariance S=1 mX0X⊺ 0. LetX0=Vrdiag(σr)W⊺ rbe a compact singular value decomposition (SVD), where σr∈Rr >0↓is in non-increasing order. The sample covariance matrix Scan be written as S=Vrdiag(σr... | https://arxiv.org/abs/2504.19963v2 |
of stochastic subspace models MACG n,k,β(S)withβ∈[k,∞). The classical MACG n,k(S)distribution is a special case with β=k. Define the principal subspace map πk(X) := range (Uk), where Ukconsists of singular vectors associated with the klargest singular values of X. As a mapping, πk:Rn×m k>7→Gr(n, k)is uniquely defined f... | https://arxiv.org/abs/2504.19963v2 |
uE, the objective function measures the mean squared error between do(uL)anddo(uE), which is a statistical measure of how closely the SROM resembles the ground truth. Overall, this optimization problem aims to improve the consistency of the SROM in characterizing the error of the reference model. We optimize the object... | https://arxiv.org/abs/2504.19963v2 |
models, and p= (pi)q i=1is a probability vector following the Dirichlet distribution with concentration parameters α∈Rq >0. The Riemannian logarithm logSt Vmaps a model-specific ROB V(i)to the tangent space of the Stiefel manifold at a reference ROB V. The Riemannian exponential expSt Vmaps a tangent vector at Vback to... | https://arxiv.org/abs/2504.19963v2 |
at the first and the last node, i.e., x1(µ) =xn(µ) = 0 . These constraints can be compactly represented asB⊺x(µ) = 0 withB= [e1en], where eiis the i-th standard basis vector of Rn. The stiffness matrix K∈Rn×nis constructed as K=ΦΛΦ⊺, with Λ= diag(4 π2j2)n−2 j=1andΦ= [0S0]⊺. The matrix S=q 2 n−1 sin jkπ n−1j=1,...,n−... | https://arxiv.org/abs/2504.19963v2 |
linear parametric system. However, the utility of SROM is not limited to characterizing ROM error. It can also be employed to characterize discrepancies between HDM and experimental data, a crucial task in model validation. Here we characterize HDM error via the SROM approach in a linear static problem. The synthetic e... | https://arxiv.org/abs/2504.19963v2 |
with low computational cost. However, those cases involved relatively simple linear static problems. To evaluate the robustness and scalability of the method, we compare it in a much more complex linear dynamics problem of a space structure. Because of the higher model complexity and more realistic structural dynamics,... | https://arxiv.org/abs/2504.19963v2 |
subspace model described in Section 3.3 and Algorithm 1. For this example, uEcorresponds to the velocity from the HDM at a critical structural node, and uo Lcorresponds to the velocity predicted by ROM at the same location. The optimum value of the hyperparameter βcomes out to be 39, which is optimized using the strate... | https://arxiv.org/abs/2504.19963v2 |
the quantities compared in Figures 6 to 8, whether observed or unobserved, are from the same node of the space structure. However, we may need to make a prediction at a different node of the system, or we may be interested in the general behavior of the system. Figure 9 shows the prediction of the SS-PPCA method for th... | https://arxiv.org/abs/2504.19963v2 |
corresponding eigenvectors v1,···,vnthat form an orthonormal basis of Rn. Theorem 1. The PDF of the MACG n,k(Σ)distribution on Gr(n, k)is maximized at any k-dim principal subspace of Σ. Ifλk> λ k+1, then the principal subspace Vk=range (Vk)with Vk= [v1···vk]is the unique mode of the distribution. Proof. Let us first co... | https://arxiv.org/abs/2504.19963v2 |
eigenvalue λ′. Consider the trajectory X(θ) = range (X(θ)), where X(θ) = [va1···vak−1v(θ)],v(θ) = cos( θ)vak+ sin( θ)v′, and θ∈[0,π 2]. It is a constant-speed rotation that starts from X0. We can show that: logp(X(θ)) =n 2 k−1X j=1logλaj−log λ−1 akcos2θ+λ′−1sin2θ! −k 2nX i=1logλi, (23) which increases monotonically. ... | https://arxiv.org/abs/2504.19963v2 |
T. A. Oliver, R. D. Moser, Representing model inadequacy: A stochastic operator approach, SIAM/ASA Journal on Uncertainty Quantification 6 (2018) 457–496. doi:10.1137/16M1106419. [17] T. Portone, R. D. Moser, Bayesian inference of an uncertain generalized diffu- sion operator, SIAM/ASA Journal on Uncertainty Quantifica... | https://arxiv.org/abs/2504.19963v2 |
doi:10.1137/130932715. [34] R. Zhang, S. Mak, D. Dunson, Gaussian process subspace prediction for model reduction, SIAM Journal on Scientific Computing 44 (2022) A1428–A1449. doi:10.1137/21M1432739. [35] M. L. Mehta, Random matrices, number 142 in Pure and applied mathematics (Academic Press), 3rd ed. ed., Elsevier, Am... | https://arxiv.org/abs/2504.19963v2 |
arXiv:2504.20539v1 [math.PR] 29 Apr 2025Randomstrasse101: Open Problems of 2024 Afonso S. Bandeira*Anastasia Kireeva§Antoine Maillard♯Almut R¨ odder¶ May 21, 2025 Abstract Randomstrasse101 is a blog dedicated to Open Problems in Mathematics, with a focus on Probability Theory, Computation, Combinatorics, Statistics, an... | https://arxiv.org/abs/2504.20539v1 |
Discrepancy (ASB) Let me start with one of my favourite Open Problems. Conjecture 1 (Matrix Spencer) .There exists a positive universal constant Csuch that, for all positive integers n, and all choices of nself-adjoint n×nreal matrices A1, . . . , A nsatisfying, for all i∈[n],∥Ai∥ ≤1 (where ∥ · ∥denotes the spectral no... | https://arxiv.org/abs/2504.20539v1 |
spontaneously synchronize when hung on the same board. This phenomenon of spontaneous synchronization of coupled oscillators has since become a central object of study in dynamical systems. We will focus here on spontaneous synchronization of ncoupled oscillators with pairwise connections given by a graph with adjacenc... | https://arxiv.org/abs/2504.20539v1 |
(with high probability) distance O(1/√ d) to it. The covariance being I d/dis a convention which ensures E[∥xi∥2] = 1: it is clear that assuming a generic positive-definite covariance Σ ≻0 does not change this question, so we do not lose any generality with this assumption. The motivation of [Sau11, SCPW12, SPW13] for ... | https://arxiv.org/abs/2504.20539v1 |
in this case as the squared Gaussian width of the cone of positive semidefinite matrices! While this heuristic was well-known, we formalized it in [MB23], essentially by showing that the volume of the space of solutions is universal in both problems. We established from it the following sharp transition result. Theorem... | https://arxiv.org/abs/2504.20539v1 |
is often interested in the critical threshold of the signal-to-noise ratio at which a problem becomes tractable. In this entry, we will consider a multi-frequency synchronization problem, where one obtains noisy measurements of the relative alignments of the signal elements through multiple frequency channels. We are i... | https://arxiv.org/abs/2504.20539v1 |
frequencies, namely, L≥11. From the computational point of view, in [KBK24], it was shown that assuming the low-degree conjecture, no polynomial-time algorithm can detect the signal below the spectral threshold, λ∗= 1, regardless of receiving additional information through additional channels. This result applies to th... | https://arxiv.org/abs/2504.20539v1 |
statistical-to-computaional gap: without regards for computational efficiency, it is possible to estimate xforλ= Ω n−r 2+1 2 . Efficient algorithms, such as the Sum-of-Squares hierarchy, are able to solve the problem at λ=˜Ω n−r 4 , where ˜Ω hides logarithmic factors. Local methods, such as gradient descent and app... | https://arxiv.org/abs/2504.20539v1 |
and Ramon van Handel [BCSvH24], which is a follow up to the matrix concentration inequalities (leveraging intrinsic freeness) in my work with March Boedihardjo, and Ramon van Handel [BBvH23]. 8 Entry 6 Did just a couple of deviations suffice all along? (ASB) In September 2024, at a conference on Mathematical Aspects of... | https://arxiv.org/abs/2504.20539v1 |
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