Optimal Data-based Kernel Estimation of Evolutionary Spectra

Complex demodulation of evolutionary spectra is formulated as a two-dimensional kernel smoother in the time-frequency domain. In the first stage, a tapered Fourier transform, $y_{nu}(f,t)$, is calculated. Second, the log-spectral estimate, $\hat{\theta}_{\nu}(f,t) \equiv \ln(|y_{nu}(f,t)|^2$, is smoothed. As the characteristic widths of the kernel smoother increase, the bias from temporal and frequency averaging increases while the variance decreases. The demodulation parameters, such as the order, length, and bandwidth of spectral taper and the kernel smoother, are determined by minimizing the expected error. For well-resolved evolutionary spectra, the optimal taper length is a small fraction of the optimal kernel half-width. The optimal frequency bandwidth, $w$, for the spectral window scales as $w^2 \approx \lambda_F/ \tau$, where $\tau$ is the characteristic time, and $\lambda_F$ is the characteristic frequency scale-length. In contrast, the optimal half-widths for the second stage kernel smoother scales as $h \approx 1/(\tau \lambda_F)^{1 \over ( p+2) }$, where $p$ is the order of the kernel smoother. The ratio of the optimal frequency half-width to the optimal time half-width satisfies $h_F / h_T ~ (|\partial_t ^p \theta | / |\partial_f^p \theta|)$. Since the expected loss depends on the unknown evolutionary spectra, we initially estimate $|\partial_t^p \theta|^2$ and $|\partial_f^p \theta|^2$ using a higher order kernel smoothers, and then substitute the estimated derivatives into the expected loss criteria

Improved Asymptotics for Zeros of Kernel Estimates via a Reformulation of the Leadbetter-Cryer Integral

The expected number of false inflection points of kernel smoothers is evaluated. To obtain the small noise limit, we use a reformulation of the Leadbetter-Cryer integral for the expected number of zero crossings of a differentiable Gaussian process

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

Optimal Estimation of Dynamically Evolving Diffusivities

The augmented, iterated Kalman smoother is applied to system identification for inverse problems in evolutionary differential equations. In the augmented smoother, the unknown, time-dependent coefficients are included in the state vector, and have a stochastic component. At each step in the iteration, the estimate of the time evolution of the coefficients is linear. We update the slowly varying mean temperature and conductivity by averaging the estimates of the Kalman smoother. Applications include the estimation of anomalous diffusion coefficients in turbulent fluids and the plasma rotation velocity in plasma tomography

Piecewise Convex Function Estimation: Representations, Duality and Model Selection

We consider spline estimates which preserve prescribed piecewise convex properties of the unknown function. A robust version of the penalized likelihood is given and shown to correspond to a variable halfwidth kernel smoother where the halfwidth adaptively decreases in regions of rapid change of the unknown function. When the convexity change points are prescribed, we derive representation results and smoothness properties of the estimates. A dual formulation is given which reduces the estimate is reduced to a finite dimensional convex optimization in the dual space

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

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, $\tau_E = 0.07192 M^{1/2}$ $(R/a)^{-0.221} R^{1.568} \kappa^{.3} I_p^{.904} B_t^{.201} \bar{n}^{0.106} P^{-0.493}$, is similar to ITER89P with a slightly stronger size dependence

Piecewise Convex Function Estimation and Model Selection

Given noisy data, function estimation is considered when the unknown function is known apriori to consist of a small number of regions where the function is either convex or concave. When the regions are known apriori, the estimate is reduced to a finite dimensional convex optimization in the dual space. When the number of regions is unknown, the model selection problem is to determine the number of convexity change points. We use a pilot estimator based on the expected number of false inflection points.Comment: arXiv admin note: text overlap with arXiv:1803.0390

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.Comment: Better in Mathematics, Spectral Theory, General, or Numerical Analysi

Adaptive Smoothing of the Log-Spectrum with Multiple Tapering

A hybrid estimator of the log-spectral density of a stationary time series is proposed. First, a multiple taper estimate is performed, followed by kernel smoothing the log-multiple taper estimate. This procedure reduces the expected mean square error by $(\pi^2/ 4)^{4/5}$ over simply smoothing the log tapered periodogram. A data adaptive implementation of a variable bandwidth kernel smoother is given