arXiv subjects
Publications and source records attributed to Kurt S. Riedel.
The restricted maximum likelihood (REML) estimator of the dispersion matrix for random coefficient models is rewritten in terms of the sufficient statistics of the individual regressions.
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.
Given noisy data, function estimation is considered when the unknown function is known a priori to consist of a small number of regions where the function is either convex or concave. When the number of regions is unknown, the model selection problem is to determine the number of convexity change points. For kernel estimates in Gaussian noise, the number of false change points is evaluated as a function of the smoothing parameter. To ensure that the number of false convexity change points tends to zero, the smoothing level must be larger than is generically optimal for minimizing the mean integrated square error (MISE). A two-stage estimator is proposed and shown to achieve the optimal rate of convergence of the MISE. In the first-stage, the number and location of the change points is estimated using strong smoothing. In the second-stage, a constrained smoothing spline fit is performed with the smoothing level chosen to minimize the MISE. The imposed constraint is that a single change point occur in a region about each empirical change point from the first-stage estimate. This constraint is equivalent to the requirement that the third derivative of the second-stage estimate have a single sign in a small neighborhood about each first-stage change point. The change points from the second-stage are in a neighborhood of the first-stage change points, but need not be at the identical locations.
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.
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.
An input pair $(A,B)$ is triangular input normal if and only if $A$ is triangular and $AA^* + BB^* = I_n$, where $I_n$ is theidentity matrix. Input normal pairs generate an orthonormal basis for the impulse response. Every input pair may be transformed to a triangular input normal pair. A new system representation is given: $(A,B)$ is triangular normal and $A$ is a matrix fraction, $A=M^{-1}N$, where $M$ and $N$ are triangular matrices of low bandwidth. For single input pairs, $M$ and $N$ are bidiagonal and an explicit parameterization is given in terms of the eigenvalues of $A$. This band fraction structure allows for fast updates of state space systems and fast system identification. When A has only real eigenvalues, one state advance requires $3n$ multiplications for the single input case.
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-multitaper estimate. This procedure reduces the expected mean square error by $({π^2 \over 4})^{.8}$ over simply smoothing the log tapered periodogram. The optimal number of tapers is $O(N^{8/15})$. A data adaptive implementation of a variable bandwidth kernel smoother is given. When the spectral density is discontinuous, one sided smoothing estimates are used.
The adaptive identification of the impulse response of an innovation filter is considered. The impulse response is a finite sum of known basis functions with unknown coefficients. These unknown coefficients are estimated using a pseudolinear regression. This estimate is implemented using a square root algorithm based on a displacement rank structure. When the initial conditions have low displacement rank, the filter update is $O(n)$. If the filter architecture is chosen to be triangular input balanced, the estimation problem is well-conditioned and a simple, low rank initialization is available.
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.
We determine the expected error by smoothing the data locally. Then we optimize the shape of the kernel smoother to minimize the error. Because the optimal estimator depends on the unknown function, our scheme automatically adjusts to the unknown function. By self-consistently adjusting the kernel smoother, the total estimator adapts to the data. Goodness of fit estimators select a kernel halfwidth by minimizing a function of the halfwidth which is based on the average square residual fit error: $ASR(h)$. A penalty term is included to adjust for using the same data to estimate the function and to evaluate the mean square error. Goodness of fit estimators are relatively simple to implement, but the minimum (of the goodness of fit functional) tends to be sensitive to small perturbations. To remedy this sensitivity problem, we fit the mean square error %goodness of fit functional to a two parameter model prior to determining the optimal halfwidth. Plug-in derivative estimators estimate the second derivative of the unknown function in an initial step, and then substitute this estimate into the asymptotic formula.
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, $\hatθ_ν(f,t) \equiv \ln(|y_{nu}(f,t)|^2$, 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 $w^2 \approx λ_F/ τ$, where $τ$ is the characteristic time, and $λ_F$ is the characteristic frequency scale-length. In contrast, the optimal half-widths for the second stage kernel smoother scales as $h \approx 1/(τλ_F)^{1 \over ( p+2) }$, where $p$ is the order of the kernel smoother. The ratio of the optimal frequency half-width to the optimal time half-width satisfies $h_F / h_T ~ (|\partial_t ^p θ| / |\partial_f^p θ|)$. Since the expected loss depends on the unknown evolutionary spectra, we initially estimate $|\partial_t^p θ|^2$ and $|\partial_f^p θ|^2$ using a higher order kernel smoothers, and then substitute the estimated derivatives into the expected loss criteria.
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 $(π^2/ 4)^{4/5} $ over simply smoothing the log tapered periodogram. A data adaptive implementation of a variable bandwidth kernel smoother is given.