Trace Moment Research Articles

Magic squares, the symmetric group and Möbius randomness

Diaconis and Gamburd computed moments of secular coefficients in the CUE ensemble. We use the characteristic map to give a new combinatorial proof of their result. We also extend their computation to moments of traces of symmetric powers, where the same result holds but in a wider range. Our combinatorial proof is inspired by gcd matrices, as used by Vaughan and Wooley and by Granville and Soundararajan. We use these CUE computations to suggest a conjecture about moments of characters sums twisted by the Liouville (or by the Möbius) function, and establish a version of it in function fields. The moral of our conjecture (and its verification in function fields) is that the Steinhaus random multiplicative function is a good model for the Liouville (or for the Möbius) function twisted by a random Dirichlet character. We also evaluate moments of secular coefficients and traces of symmetric powers, without any condition on the size of the matrix. As an application we give a new formula for a matrix integral that was considered by Keating, Rodgers, Roditty-Gershon and Rudnick in their study of the k-fold divisor function.

Scaling and intermittency in time of urban precipitation in Warsaw, Poland

Study regionA network of 25 rain gauges roughly covering an area of 517.2 km2 of Warsaw, in Poland. Study focusScaling and intermittency of data series of rainfall intensity are investigated for each of the network rain gauges using five multifractal framework methods: spectral density analysis, functional box-counting, trace moment, probability distribution/multiple scaling, and double trace moment. Synthetic precipitation series are then generated by means of continuous universal random cascade models. Cascade generators are defined by universal parameters calculated for selected Warsaw rain gauges, or rain gauge clusters displaying similarity of multifractal parameters. The generated synthetic precipitation time series are subject to statistical evaluation by comparing the complementary cumulative distribution function and the intermittency calculated for synthetic versus observed data. Finally, a novel filtering algorithm is proposed to correct the intermittency characteristics of synthetic precipitation time series. New hydrological insightsBased on our analyses, the temporal structure of the recorded Warsaw precipitation time series is found to be a multifractal set characterized by a scaling behaviour over a wide range of scales. Furthermore, it has been observed that most Warsaw rain gauges, have a distinct similarity of multifractal properties. It has also been demonstrated that, for the first time in Poland, the universal continuous cascades could be used in practice for the generation of synthetic rainfall series at fine (1-min) temporal resolution

Moments of traces of Frobenius of higher order Dirichlet $L$-functions over $\mathbb F_q[T

We study the moments of ${\rm Tr}(\Theta _\chi)$ as $\chi $ runs over Dirichlet characters defined over $\mathbb F_q[T]$ of fixed order $r$. In particular, we show that after an appropriate normalization, the $q$-limit of the power sum moments behaves lik

Analytic and numerical bootstrap for one-matrix model and “unsolvable” two-matrix model

We propose the relaxation bootstrap method for the numerical solution of multi-matrix models in the large N limit, developing and improving the recent proposal of H. Lin. It gives rigorous inequalities on the single trace moments of the matrices up to a given “cutoff” order (length) of the moments. The method combines usual loop equations on the moments and the positivity constraint on the correlation matrix of the moments. We have a rigorous proof of applicability of this method in the case of the one-matrix model where the condition of positivity of the saddle point solution appears to be equivalent to the presence of supports of the eigenvalue distribution only on the real axis and only with positive weight. We demonstrate the numerical efficiency of our method by solving the analytically “unsolvable” two-matrix model with tr[A, B]2 interaction and quartic potentials, even for solutions with spontaneously broken discrete symmetry. The region of values for computed moments allowed by inequalities quickly shrinks with the increase of the cutoff, allowing the precision of about 6 digits for generic values of couplings in the case of ℤ2 symmetric solutions. Our numerical data are checked against the known analytic results for particular values of parameters.

A Novel Polarimetric PSI Method Using Trace Moment-Based Statistical Properties and Total Power Interferogram Construction

With the launch of various multipolarimetric satellites, many scholars have introduced the persistent scatterer (PS)-oriented polarimetric optimization methods and extended the persistent scatterer interferometry (PSI) method to multipolarimetric data configuration, called polarimetric PSI (PolPSI) technology. Most PolPSI methods mainly take the amplitude dispersion index (ADI) as the optimization criterion and evaluate the temporal amplitude stationarity of each polarimetric channel for finding an optimal one. However, due to the unstable statistical characteristics of the quality indicator, many non-PS pixels are easily mistaken for the PS candidates (PSCs), and the performance of interferometric phase optimization is also limited. To overcome these restrictions, in this article, a novel PolPSI method is proposed based on the following two improved innovations. First, in terms of PSC selection, the trace moment (TM)-based statistical properties of time-series polarimetric coherency matrices are utilized for selecting the scatterers with the temporal polarimetric stationarity. Second, in terms of interferometric phase optimization, all interferometric coherency matrices of multipolarization channels are added up together to construct the total power (TP) interferogram for suppressing the effect of speckle noise and decorrelation. In the experiment, 13 scenes of quad-polarization ALOS PALSAR-1 image are selected to verify the algorithm’s effectiveness. The experimental results demonstrate that the proposed PolPSI method can better improve the deformation monitoring performance in three aspects than both the single-polarimetric HH and traditional exhaustive search polarimetric optimization (ESPO) methods, including phase quality improvement, density of PSs, and computational efficiency.

Symmetric function theory and unitary invariant ensembles

Representation theory and the theory of symmetric functions have played a central role in random matrix theory in the computation of quantities such as joint moments of traces and joint moments of characteristic polynomials of matrices drawn from the circular unitary ensemble and other circular ensembles related to the classical compact groups. The reason is that they enable the derivation of exact formulas, which then provide a route to calculating the large-matrix asymptotics of these quantities. We develop a parallel theory for the Gaussian Unitary Ensemble (GUE) of random matrices and other related unitary invariant matrix ensembles. This allows us to write down exact formulas in these cases for the joint moments of the traces and the joint moments of the characteristic polynomials in terms of appropriately defined symmetric functions. As an example of an application, for the joint moments of the traces, we derive explicit asymptotic formulas for the rate of convergence of the moments of polynomial functions of GUE matrices to those of a standard normal distribution when the matrix size tends to infinity.

Estimation of Equivalent Number of Looks in Time-Series Pol(In)SAR Data

As an essential parameter in synthetic aperture radar (SAR) images, the equivalent number of looks (ENL) not only indicates the speckle noise level in multi-look SAR data but also can be used for evaluating the region homogeneity level. Currently, time-series polarimetric (interferometric) SAR (TSPol(In)SAR) data are increasingly abundant, but traditional equivalent number of looks (ENL) estimators only use polarimetric information from a mono-temporal observation and do not consider the temporal characteristics or interferometric coherence of ground targets. Therefore, this paper puts forward four novel ENL estimators to overcome the restrictions of inadequate observation information. Firstly, based on the traditional trace moment estimator for polarimetric SAR data (TM-PolSAR), we extend it to both PolInSAR and TSPolInSAR data and then propose both TM-PolInSAR and TM-TSPolInSAR estimators, respectively. Secondly, for both TSPolSAR and single-reference TSPolInSAR data, we estimate the ENL by stacking the trace moments (STM) of multitemporal coherency matrices, called STM-TSPolSAR and STM-TSPolInSAR estimators, respectively. Therefore, these proposed ENL estimators can effectively deal with most of the requirements of TSPol(In)SAR data types in practical applications, mainly including statistical distribution modeling and region homogeneity evaluation. The simulation and real experiments detailedly compare the proposed four ENL estimators to the classical TM-PolSAR estimator and quantitatively analyze the estimation performance. The proposed estimators have obtained the ENL with less bias and standard deviation than the traditional estimator, especially in case of small spatial samples coherency matrices. Additionally, these STM-TSPolSAR, STM-TSPolInSAR, and TM-TSPolInSAR estimators have provided more effective statistical characteristics with the increase of the time-series size. It has been demonstrated that the proposed STM-TSPolSAR estimator considers the time-varying polarimetric characteristics of the crop and detects many edges that the traditional estimator cannot discover, which means a superior capability of region homogeneity evaluation.

Applications of directional wavelets, Universal Multifractals and anisotropic scaling in ensemble nowcasting; a review of methods with case‐studies

AbstractThis article begins with a review of the scale decomposition and stochastic noise generation aspects of the ensemble nowcasting process, as used in the Short Term Ensemble Prediction System (STEPS). Alternative methods are then suggested, with case‐study examples, which could lead to enhanced performance and may also be applicable in other fields. The standard FFT‐band‐pass‐filtering scale decomposition, essential to this process, is shown to be practically the same as a redundant Continuous Wavelet Transform (CWT). This immediately makes accessible a number of more advanced methods from the wavelet literature. Particular attention is given to directional wavelet transforms and their use for analysing anisotropic scaling behaviour, common in weather images. With a view to numerically efficient operational usage, the focus is then shifted to Discrete Wavelet Transforms (DWTs). While the standard DWT has a number of disadvantages, including limited orientational selectivity in two dimensions, the relatively recent Dual Tree Complex WT (DTCWT) overcomes most of these. Examples are given that show how its six‐fold directionality can be exploited, to create synthetic features that replace the unpredictable small‐scale parts of a (weather‐radar‐based) extrapolation nowcast with more realistic spatially varying anisotropy. Further, the Bounded Log‐normal Cascade noise model, as used in STEPS, is reconsidered and a proposal to use Universal Multifractal realizations is presented. For the noise parameters, some operationally viable estimation methods are considered, such as the Double Trace Moment technique and a structure‐function‐based approach, which could work on readily available DTCWT coefficients.

Everyday Conversions: Islam, Domestic Work, and South Asian Migrant Women in Kuwait

Everyday Conversions: Islam, Domestic Work, and South Asian Migrant Women in Kuwait

Everyday Conversions: Islam, Domestic Work, and South Asian Migrant Women in Kuwait

Everyday Conversions: Islam, Domestic Work, and South Asian Migrant Women in Kuwait

Parameters effective on estimating a nonstationary mixed-phase wavelet using cumulant matching approach

Seismic inversion is a process performed to remove the effects of propagated wavelets in order to recover the acoustic impedance. To obtain valid velocity and density values related to subsurface layers through the inversion process, it is highly essential to perform reliable wavelet estimation such as cumulant matching approach. For this purpose, the seismic data were windowed in this work in such a way that two consecutive windows were only one sample apart. Also, we did not consider any fixed wavelet for any window and let the phase of each wavelet rotate in each sample in the window. Comparing the fourth order cumulant of the whitened trace and fourth-order moment of the all-pass operator in each window generated a cost function that should be minimized with a non-linear optimization method. In this regard, parameters effective on the estimation of the nonstationary mixed-phase wavelets were tested over the created nonstationary seismic trace at 0.82s and 1.6s. Besides, we compared the consequences of each parameter on estimated wavelets at two mentioned times. The parameters studied in this work are window length, taper type, the number of iteration, signal-to-noise ratio, bandwidth to central frequency ratio, and Q factor. The results show that applying the optimum values of the effective parameters, the average correlation of the estimated mixed-phase wavelets with the original ones is about 87%. Moreover, the effectiveness of the proposed approach was examined on a synthetic nonstationary seismic section with variable Q factor values alongside the time and offset axis. Eventually, the cumulant matching method was applied on a cross line of the migrated data from a 3D data set of an oilfield in the Persian Gulf. Also, the effect of the wrong Q estimation on the estimated mixed-phase wavelet was considered on the real data set. It is concluded that the accuracy of the estimated wavelet relied on the estimated Q and more than 10% error in the estimated value of Q is acceptable. Eventually, an 88% correlation was found between the estimated mixed-phase wavelets and the original ones for three horizons. The estimated wavelets applied to the data and the result of deconvolution processes was presented.

Yearning to Give Back: Searching for Social Purpose in Computer Science and Engineering.

Computing is highly segregated and stratified by gender. While there is abundant scholarship investigating this problem, emerging evidence suggests that a hierarchy of value exists between the social and technical dimensions of Computer Science and Engineering (CSE) and this plays a role in the underrepresentation of women in the field. This ethnographic study of women's experiences in computing offers evidence of a systemic preference for the technical dimensions of computing over the social and a correlation between gender and social aspirations. Additionally, it suggests there is a gap between the exaltation of computing's social contributions and the realities of them. My participants expressed a yearning to contribute to the collective well-being of society using their computing skills. I trace moments of rupture in my participants' stories, moments when they felt these aspirations were in conflict with the cultural values in their organizations. I interpret these ruptures within a consideration of yearning, a need my participants had to contribute meaningfully to society that remained unfulfilled. The yearning to align one's altruistic values with one's careers aspirations in CSE illuminates an area for greater exploration on the path to realizing gender equity in computing. I argue that before a case can be made that careers in computing do indeed contribute to social and civil engagements, we must first address the meaning of the social within the values, ideologies and practices of CSE institutions and next, develop ways to measure and evaluate the field's contributions to society.

Gaussian distribution of short sums of trace functions over finite fields

AbstractWe show that under certain general conditions, short sums of ℓ-adic trace functions over finite fields follow a normal distribution asymptotically when the origin varies, generalising results of Erdős–Davenport, Mak–Zaharescu and Lamzouri. In particular, this applies to exponential sums arising from Fourier transforms such as Kloosterman sums or Birch sums, as we can deduce from the works of Katz. By approximating the moments of traces of random matrices in monodromy groups, a quantitative version can be given as in Lamzouri's article, exhibiting a different phenomenon than the averaging from the central limit theorem.

Bimatrix variate gamma-beta distributions

ABSTRACTTwo bimatrix distributions with beta and gamma marginals are introduced. Various properties (including product moments of determinants and traces, entropies, marginal distributions) are derived. Parameter estimation by the method of maximum likelihood is discussed. The performance and efficiencies of the maximum likelihood estimators and the associated confidence intervals are assessed by simulation. The efficiencies are compared versus those for the maximum likelihood estimators and the associated confidence intervals based on matrix variate gamma distributions. A discussion of possible applications of the bimatrix distributions is given.

Texture-Invariant Estimation of Equivalent Number of Looks Based on Trace Moments in Polarimetric Radar Imagery

This letter introduces a novel estimator of equivalent number of looks (ENL) that can be applied to any distribution of texture model, i.e., an estimator that is texture invariant. The novel estimator is the Development of Trace Moments (DTM), which cancels the textural variation using trace moments. Five forms of the DTM estimator using submatrices are presented and compared with each other. The results show that the full-dimensional matrix form seems to be the best in performance and computational complexity. The experiments were performed using simulated and real data. The comparisons among all the existing methods of ENL estimation in the product model of the clutter, such as K-distribution and G0 distribution, show the performance of the DTM estimator to be the best if there is a sufficient number of samples. The global and local ENL estimations of the real data of San Francisco are analyzed, and the results agree with the simulated case. This shows that the DTM always gives a good result, particularly in the global estimation of ENL. Therefore, it can be concluded that the DTM estimator is robust to any distribution model, with low computational complexity and high accuracy, particularly in wide areas with similar scattering mechanism.

Moments of traces of circular beta-ensembles

Let $\theta_1,\ldots,\theta_n$ be random variables from Dyson's circular $\beta$-ensemble with probability density function $\operatorname {Const}\cdot\prod_{1\leq j<k\leq n}|e^{i\theta_j}-e^{i\theta _k}|^{\beta}$. For each $n\geq2$ and $\beta>0$, we obtain some inequalities on $\mathbb{E}[p_{\mu}(Z_n)\bar{p_{\nu}(Z_n)}]$, where $Z_n=(e^{i\theta_1},\ldots,e^{i\theta_n})$ and $p_{\mu}$ is the power-sum symmetric function for partition $\mu$. When $\beta=2$, our inequalities recover an identity by Diaconis and Evans for Haar-invariant unitary matrices. Further, we have the following: $ \lim_{n\to\infty}\mathbb{E}[p_{\mu}(Z_n)\bar{p_{\nu}(Z_n)}]= \delta_{\mu\nu}(\frac{2}{\beta})^{l(\mu)}z_{\mu}$ for any $\beta>0$ and partitions $\mu,\nu$; $\lim_{m\to\infty}\mathbb{E}[|p_m(Z_n)|^2]=n$ for any $\beta>0$ and $n\geq2$, where $l(\mu)$ is the length of $\mu$ and $z_{\mu}$ is explicit on $\mu$. These results apply to the three important ensembles: COE ($\beta=1$), CUE ($\beta=2$) and CSE ($\beta=4$). We further examine the nonasymptotic behavior of $\mathbb{E}[|p_m(Z_n)|^2]$ for $\beta=1,4$. The central limit theorems of $\sum_{j=1}^ng(e^{i\theta_j})$ are obtained when (i) $g(z)$ is a polynomial and $\beta>0$ is arbitrary, or (ii) $g(z)$ has a Fourier expansion and $\beta=1,4$. The main tool is the Jack function.

General Moments of the Inverse Real Wishart Distribution and Orthogonal Weingarten Functions

We study a random positive definite symmetric matrix distributed according to a real Wishart distribution. We compute general moments of the random matrix and of its inverse explicitly. To do so, we employ the orthogonal Weingarten function, which was recently introduced in the study of Haar-distributed orthogonal matrices. As applications, we give formulas for moments of traces of a Wishart matrix and its inverse.

The collaborative invention of meaning: A short history of evolving ideas

In this paper I trace moments of spontaneous insight occurring among a group of young children in an environment where they are encouraged and supported in their efforts to collaboratively interrogate their own and one another’s ideas. I argue that noticing, confronting, and responding to enigmas, surprises, and evolving meanings as they are happening, enhances learning while also revealing musical structures that are often hidden in the assumptions that inform our notations and analytic categories, thus influencing our teaching. The examples focus particularly on young children at work learning to transform the elusiveness of organized actions — clapping a rhythm, walking a line, drumming — into explanations and descriptions that hold still to be looked at and upon which to reflect.

Compound Real Wishart and q-Wishart Matrices

We introduce a family of matrices with noncommutative entries that generalize the classical real Wishart matrices. With the help of the Brauer product, we derive a nonasymptotic expression for the moments of traces of monomials in such matrices; the expression is quite similar to the formula derived in [9, Theorem 2.1; Asymptotic normality for traces of polynomials in independent complex Wishart matrices; Probability Theory and Related fields, 140, 2008] for independent complex Wishart matrices.We then analyze the fluctuations about the Marchenko-Pastur law. We show that after centering by the mean, traces of real symmetric polynomials in q-Wishart matrices converge in distribution, and we identify the asymptotic law as the normal law when q = 1, and as the semicircle law when q = 0.

Estimation of the Equivalent Number of Looks in Polarimetric Synthetic Aperture Radar Imagery

This paper addresses estimation of the equivalent number of looks (ENL), an important parameter in statistical modeling of multilook synthetic aperture radar (SAR) images. Two new ENL estimators are discovered by looking at certain moments of the multilook polarimetric covariance matrix, which is commonly used to represent multilook polarimetric SAR (PolSAR) data, and assuming that the covariance matrix is complex Wishart distributed. First, a second-order trace moment provides a polarimetric extension of the ENL definition and also a matrix-variate version of the conventional ENL estimator. The second estimator is obtained from the log-determinant matrix moment and is also shown to be the maximum likelihood estimator under the Wishart model. It proves to have much lower variance than any other known ENL estimator, whether applied to single-polarization or PolSAR data. Moreover, this estimator is less affected by texture and thus provides more accurate results than other estimators should the assumption of Gaussian statistics for the complex scattering coefficients be violated. These are the first known estimators to use the full covariance matrix as input, rather than individual intensity channels, and therefore to utilize all the statistical information available. We finally demonstrate how an ENL estimate can be computed automatically from the empirical density of small sample estimates calculated over a whole scene. We show that this method is more robust than procedures where the estimate is calculated in a manually selected region of interest.