arXiv subjects

Kurt S. Riedel

Publications and source records attributed to Kurt S. Riedel.

At least 19 recordsLinked to original sources

arXiv2021

Mean and Variance of Brownian Motion with Given Final Value, Maximum and ArgMax: Extended Version

The conditional expectation and conditional variance of Brownian motion is considered given the argmax, B(t|argmax), as well as those with additional information: B(t|close, argmax), B(t|max, argmax), B(t|close, max, argmax) where the close is the final value: B(t=1)=c and t in [0,1]. We compute the expectation and variance of a Brownian meander in time. By splicing together two Brownian meanders, the mean and variance of the constrained process are calculated. Computational results displaying both the expectation and variance in time are presented. Comparison of the simulation with theoretical values are shown when the close and argmax are given.

math.PR↗

arXiv2019

Combinatorial Models of Cross-Country Dual Meets: What is a Big Victory?

Combinatorial/probabilistic models for cross-country dual-meets are proposed. The first model assumes that all runners are equally likely to finish in any possible order. The second model assumes that each team is selected from a large identically distributed population of potential runners and with each potential runner's ranking determined by the initial draw from the combined population.

stat.AP↗

arXiv2018

Quasi-linear Transport in a Sheared Flow Field

The evolution of a passive scalar field is considered for a slowly varying stratified medium, which is convected in an incompressible sheared flow with many overlapping static flux islands. Within the quasilinear/random phase approximation, a multiple scale expansion is made. Due to the rapid spatial variation of the temperature, the "ensemble" averaged/ slowly varying part of the solution is not described by the arithmetic average of the oscillatory evolution equation. The standard Markovian and continuum approximations are shown to be invalid. For times of order $N$, where there are $O(N^2)$ excited modes, most of the time dependent perturbation phase mixes away and the fluid reaches a new saturated state with small time oscillations about the temperature. This saturated state has smaller resonance layers, (corresponding to magnetic islands) than those that occur in the isolated resonant perturbation case. Thus the quasilinear response to the resonant interactions reduces the effective size of the perturbations. The temperature gradient of the saturated state vanishes at all the excited resonance surfaces but has a nonzero average. Thus either the quasi-linear approximation ceases to be valid on long time scales, or the fluid remains essentially in this modified equilibrium and does not evolve diffusively. Thus collisionless, driftless fast particles will not be lost rapidly in equilibria with many small islands.

physics.plasm-ph↗

arXiv2018

Dimensionally Correct Power Law Scaling Expressions for L-mode Confinement

Confinement scalings of divertor and radiofrequency heated discharges are shown to differ significantly from the standard neutral beam heated limiter scaling. The random coefficient two stage regression algorithm is applied to a neutral beam heated limiter subset of the ITER L mode database as well as a combined dataset. We find a scaling similar to Goldston scaling for the NB limiter dataset and a scaling similar to ITER89P for the combined dataset. Various missing value algorithms are examined for the missing $B_t$ scalings. We assume that global confinement can be approximately described a power law scaling. After the second stage, the constraint of collisional Maxwell Vlasov similarity is tested and imposed. When the constraint of collisional Maxwell Vlasov similarity is imposed, the C.I.T. uncertainty is significantly reduced while the I.T.E.R. uncertainty is slightly reduced.

physics.plasm-ph↗

arXiv2018

Spectral Estimation of Plasma Fluctuations I: Comparison of Methods

The relative root mean squared errors (RMSE) of nonparametric methods for spectral estimation is compared for microwave scattering data of plasma fluctuations. These methods reduce the variance of the periodogram estimate by averaging the spectrum over a frequency bandwidth. As the bandwidth increases, the variance decreases, but the bias error increases. The plasma spectra vary by over four orders of magnitude, and therefore, using a spectral window is necessary. We compare the smoothed tapered periodogram with the adaptive multiple taper methods and hybrid methods. We find that a hybrid method, which uses four orthogonal tapers and then applies a kernel smoother, performs best. For 300 point data segments, even an optimized smoothed tapered periodogram has a 24 \% larger relative RMSE than the hybrid method. We present two new adaptive multi-taper weightings which outperform Thomson's original adaptive weighting.

stat.AP↗

arXiv2018

Spectral Estimation of Plasma Fluctuations II: Nonstationary Analysis of ELM Spectra

Several analysis methods for nonstationary fluctuations are described and applied to the edge localized mode (ELM) instabilities of limiter H-mode plasmas. The microwave scattering diagnostic observes poloidal $k_θ$ values of 3.3 cm$^{-1}$, averaged over a 20 cm region at the plasma edge.A short autoregressive filter enhances the nonstationary component of the plasma fluctuations by removing much of the background level of stationary fluctuations. Between ELMs, the spectrum predominantly consists of broad-banded 300-700 kHz fluctuations propagating in the electron diamagnetic drift direction, indicating the presence of a negative electric field near the plasma edge. The time-frequency spectrogram is computed with the multiple taper technique. By using the singular value decomposition of the spectrogram, it is shown that the spectrum during the ELM is broader and more symmetric than that of the stationary spectrum. The ELM period and the evolution of the spectrum between ELMs varies from discharge to discharge. For the discharge under consideration which has distinct ELMs with a 1 msec period, the spectrum has a maximum in the electron drift direction which relaxes to a near constant value %its characteristic shape in the first half millisecond after the end of the ELM and then grows slowly. In contrast, the level of the fluctuations in the ion drift direction increases exponentially by a factor of eight in the five milliseconds~after the ELM. High frequency precursors are found which occur one millisecond before the ELMs and propagate in the ion drift direction. These precursors are very short ($\sim 10 μ$secs), coherent bursts, and they predict the occurrence of an ELM with a high success rate.

physics.plasm-ph↗

arXiv2018

Random Coefficient H-mode Confinement Scalings

The random coefficient two-stage regression algorithm with the collisional Maxwell-Vlasov constraint is applied to the ITER H-mode confinement database. The data violate the collisional Maxwell-Vlasov constraint at the 10-30% significance level, probably owing to radiation losses. The dimensionally constrained scaling, $τ_E = 0.07192 M^{1/2}$ $(R/a)^{-0.221} R^{1.568} κ^{.3} I_p^{.904} B_t^{.201} \bar{n}^{0.106} P^{-0.493}$, is similar to ITER89P with a slightly stronger size dependence.

physics.plasm-ph↗

arXiv2018

Minimum bias multiple taper spectral estimation

Two families of orthonormal tapers are proposed for multi-taper spectral analysis: minimum bias tapers, and sinusoidal tapers $\{ \bf{v}^{(k)}\}$, where $v_n^{(k)}=\sqrt{\frac{2}{N+1}}\sin\frac{πkn}{N+1}$, and $N$ is the number of points. The resulting sinusoidal multitaper spectral estimate is $\hat{S}(f)=\frac{1}{2K(N+1)} \sum_{j=1}^K |y(f+\frac{j}{2N+2}) -y(f-\frac{j}{2N+2})|^2$, where $y(f)$ is the Fourier transform of the stationary time series, $S(f)$ is the spectral density, and $K$ is the number of tapers. For fixed $j$, the sinusoidal tapers converge to the minimum bias tapers like $1/N$. Since the sinusoidal tapers have analytic expressions, no numerical eigenvalue decomposition is necessary. Both the minimum bias and sinusoidal tapers have no additional parameter for the spectral bandwidth. The bandwidth of the $j$th taper is simply $\frac{1}{N}$ centered about the frequencies $\frac{\pm j}{2N+2}$. Thus the bandwidth of the multitaper spectral estimate can be adjusted locally by simply adding or deleting tapers. The band limited spectral concentration, $\int_{-w}^w |V(f)|^2 df$, of both the minimum bias and sinusoidal tapers is very close to the optimal concentration achieved by the Slepian tapers. In contrast, the Slepian tapers can have the local bias, $\int_{-1/2}^{1/2} f^2 |V(f)|^2 df$, much larger than of the minimum bias tapers and the sinusoidal tapers.

stat.ME↗

arXiv2018

Profile Shape Parameterization of JET Electron Temperature and Density Profiles

The temperature and density profiles of the Joint European Torus are parameterized using log-additive models in the control variables. Predictive error criteria are used to determine which terms in the log-linear model to include. The density and temperature profiles are normalised to their line averages ($\bar{n}$ and $\bar{T}$). The normalised Ohmic density shape depends primarily on the parameter $\bar{n}/B_t$, where $B_t$ is the toroidal magnetic field. Both the Low-mode (L-mode) and edge localized mode-free (ELM-free) high mode (H-mode) temperature profile shapes depend strongly on the type of heating power, with ion cyclotron resonant heating producing a more peaked profile than neutral beam injection. Given the heating type dependence, the L-mode temperature shape is nearly independent of the other control variables. The H-mode temperature shape broadens as the effective charge, $Z_{eff}$, increases. The line average L-mode temperature scales as $B_t^{.96}$(Power per particle)$^{.385}$. The L-mode normalized density shape depends primarily on the ratio of line average density, $\bar{n}$, to the edge safety factor, $q_{95}$. As $\bar{n}/q_{95}$ increases, the profile shape broadens. The current, $I_p$, is the most important control variable for the normalized H-mode density. As the current increases, the profile broadens and the gradient at the edge sharpens. Increasing the heating power, especially the ion cyclotron resonant heating, or decreasing the average density, peaks the H-mode density profile slightly.

physics.plasm-ph↗

arXiv2018

A Sherman-Morrison-Woodbury Identity for Rank Augmenting Matrices with Application to Centering

Matrices of the form $\bf{A} + (\bf{V}_1 + \bf{W}_1)\bf{G}(\bf{V}_2 + \bf{W}_2)^*$ are considered where $\bf{A}$ is a $singular$ $\ell \times \ell$ matrix and $\bf{G}$ is a nonsingular $k \times k$ matrix, $k \le \ell$. Let the columns of $\bf{V}_1$ be in the column space of $\bf{A}$ and the columns of $\bf{W}_1$ be orthogonal to $\bf{A}$. Similarly, let the columns of $\bf{V}_2$ be in the column space of $\bf{A}^*$ and the columns of $\bf{W}_2$ be orthogonal to $\bf{A}^*$. An explicit expression for the inverse is given, provided that $\bf{W}_i^* \bf{W}_i$ has rank $k$. %and $\bf{W}_1$ and $\bf{W}_2$ have the same column space. An application to centering covariance matrices about the mean is given.

stat.ME↗

arXiv2018

Generalized Epsilon-Pseudospectra

We generalize $ε$-pseudospectra and the associated computational algorithms to the generalized eigenvalue problem. Rank one perturbations are used to determine the $ε$-pseudospectra.

math.NA↗

arXiv2018

A Hierarchy of Empirical Models of Plasma Profiles and Transport

Two families of statistical models are presented which generalize global confinement expressions to plasma profiles and local transport coefficients. The temperature or diffusivity is parameterized as a function of the normalized flux radius, $\barψ$, and the engineering variables, ${\bf u} = (I_p,B_t,\bar{n},q_{95})^\dagger$. The log-additive temperature model assumes that $\ln [T(\barψ, {\bf u})] =$ $f_0 (\barψ) + f_I (\barψ)\ln[I_p]$ $+ f_B (\barψ) \ln [B_t]$ $+ f_n (\barψ) \ln [ \bar{n}] + f_{q}\ln[q_{95}]$. The unknown $f_i (\barψ)$ are estimated using smoothing splines. A 43 profile Ohmic data set from the Joint European Torus is analyzed and its shape dependencies are described. The best fit has an average error of 152 eV which is 10.5 \% percent of the typical line average temperature. The average error is less than the estimated measurement error bars. The second class of models is log-additive diffusivity models where $\ln [ χ(\barψ, {\bf u})] $ $=\ g_0 (\barψ) + g_I (\barψ) \ln[I_p]$ $+ g_B (\barψ) \ln [B_t ]$ $+ g_n (\barψ) \ln [ \bar{n} ]$. These log-additive diffusivity models are useful when the diffusivity is varied smoothly with the plasma parameters. A penalized nonlinear regression technique is recommended to estimate the $g_i (\barψ)$. The physics implications of the two classes of models, additive log-temperature models and additive log-diffusivity models, are different. The additive log-diffusivity models adjust the temperature profile shape as the radial distribution of sinks and sources. In contrast, the additive log-temperature model predicts that the temperature profile depends only on the global parameters and not on the radial heat deposition.

stat.ME↗

arXiv2018

Optimal Boundary Kernels and Weightings for Local Polynomial Regression

Kernel smoothers are considered near the boundary of the interval. Kernels which minimize the expected mean square error are derived. These kernels are equivalent to using a linear weighting function in the local polynomial regression. It is shown that any kernel estimator that satisfies the moment conditions up to order $m$ is equivalent to a local polynomial regression of order $m$ with some non-negative weight function if and only if the kernel has at most $m$ sign changes. A fast algorithm is proposed for computing the kernel estimate in the boundary region for an arbitrary placement of data points.

stat.ME↗

arXiv2018

Sufficient Conditions for a Linear Estimator to be a Local Polynomial Regression

It is shown that any linear estimator that satisfies the moment conditions up to order $p$ is equivalent to a local polynomial regression of order $p$ with some non-negative weight function if and only if the kernel has at most $p$ sign changes. If the data points are placed symmetrically about the estimation point, a linear weighting function is equivalent to the standard quadratic weighting function.

stat.ME↗

arXiv2018

Block Diagonally Dominant Positive Definite Sub-optimal Filters and Smoothers

We examine stochastic dynamical systems where the transition matrix, $Φ$, and the system noise, $\bfΓ\bf{Q}\bfΓ^T$, covariance are nearly block diagonal. When $\bf{H}^T \bf{R}^{-1} \bf{H}$ is also nearly block diagonal, where $\bf{R}$ is the observation noise covariance and $\bf{H}$ is the observation matrix, our suboptimal filter/smoothers are always positive semi-definite, and have improved numerical properties. Applications for distributed dynamical systems with time dependent pixel imaging are discussed.

stat.ME↗

arXiv2018

Smoothing Spline Growth Curves With Covariates

We adapt the interactive spline model of Wahba to growth curves with covariates. The smoothing spline formulation permits a non-parametric representation of the growth curves. In the limit when the discretization error is small relative to the estimation error, the resulting growth curve estimates often depend only weakly on the number and locations of the knots. The smoothness parameter is determined from the data by minimizing an empirical estimate of the expected error. We show that the risk estimate of Craven and Wahba is a weighted goodness of fit estimate. A modified loss estimate is given, where $σ^2$ is replaced by its unbiased estimate.

stat.ME↗

arXiv2018

Statistical tests for evaluating an earthquake prediction method

The impact of including postcursors in the null hypothesis test is discussed. Unequal prediction probabilities can be included in the null hypothesis test using a generalization of the central limit theorem. A test for determining the enhancement factor over random chance is given. The seismic earthquake signal may preferentially precede earthquakes even if the VAN methodology fails to forecast the earthquakes. We formulate a statistical test for this possibility.

stat.ME↗

arXiv2018

Kernel estimation of the instantaneous frequency

We consider kernel estimators of the instantaneous frequency of a slowly evolving sinusoid in white noise. The expected estimation error consists of two terms. The systematic bias error grows as the kernel halfwidth increases while the random error decreases. For a non-modulated signal, $g(t)$, the kernel halfwidth which minimizes the expected error scales as$h \sim \left[{ σ^2 \over N| \partial_t^2 g^{}|^2 } \right]^{1/ 5}$, where %$A^{(\ell)}$ is the coherent signal at frequency, $f_{\ell}$, $σ^2$ is the noise variance and $N$ is the number of measurements per unit time. We show that estimating the instantaneous frequency corresponds to estimating the first derivative of a modulated signal, $A(t)\exp(iϕ(t))$. For instantaneous frequency estimation, the halfwidth which minimizes the expected error is larger: $h_{1,3} \sim \left[{ σ^2 \over A^2N| \partial_t^3 (e^{i \tildeϕ(t)} )|^2 } \right]^{1/ 7}$. Since the optimal halfwidths depend on derivatives of the unknown function, we initially estimate these derivatives prior to estimating the actual signal.

stat.ME↗