One for All and All for One:Regression Checks With Many Regressors

We develop a novel approach to build checks of parametric regression models when many regressors are present, based on a class of rich enough semiparametric alternatives, namely single-index models. We propose an omnibus test based on the kernel method that performs against a sequence of directional nonparametric alternatives as if there was one regressor only, whatever the number of regressors. This test can be viewed as a smooth version of the integrated conditional moment (ICM) test of Bierens. Qualitative information can be easily incorporated in the procedure to enhance power. Our test is little sensitive to the smoothing parameter and performs better than several known lack-of-fit tests in multidimensional settings, as illustrated by extensive simulations and an application to a cross-country growth regression.Dimensionality, Hypothesis testing, Nonparametric methods

Bandpass filters for unconstrained target recognition and their implementation in coherent optical correlators

An up-dateable correlator is simulated which is based on the non-degenerate four wave mixing (NDFWM) interaction in the photorefractive material bismuth silicon oxide (Bi12SiO20). Specifically, it is shown that variable bandpass filters can be implemented directly in the correlator by adjusting the relative strengths of the signal and reference beams used to write the Fourier transform hologram into the photorefractive. The synthetic discriminant function (SDF) method of grey-level multiplexing is reviewed. A bandpass modification of this technique is used in the design of a multiplexed filter for the recognition of an industrial test component from a limited number of known stable state orientations when viewed from an overhead camera position. Its performance in this task when implemented in the up-dateable correlator is assessed through simulation. The conclusion of this work is that filter multiplexing must be used judiciously for orientation invariant recognition. Only a limited number of images, typically under ten, may be multiplexed into each filter since correlation peak heights and peak-to-sidelobe ratios inevitably progressively deteriorate as images are added to the filter. The effect of severe amplitude disruptions in the frequency plane on correlation peak localisation is examined. In two or higher dimensions simulations show the localisation is very robust to this disruption; an analysis is developed to indicate the reason for this. The effect is exploited by the implementation of an algorithm that locally removes the spatial frequencies that exhibit close phase matching between intra- and inter-class images. The inter-class response can be forced to zero while simultaneously improving the intra-class tolerance to orientation changes. The technique is assessed through simulation with images of two types of motor vehicle, in a variety of orientations, and shown to be effective in improving discrimination and intra-class tolerance for examples in which these were initially very poor. Bandpass filters are experimentally implemented in a joint transform correlator (JTC) based on a NDFWM interaction in Bi12SiO20. The JTC is described and its full bandwidth performance initially assessed. As anticipated from the previous considerations, inter-class discrimination was high but the intra-class tolerance very poor due to the high sensitivity of the filter. The difference of Gaussian approximation to a Laplacian of a Gaussian filter is described and its experimental implementation in the JTC detailed. Experimental results are presented for the orientation independent recognition of a car while maintaining discrimination against another car. An intra-class to inter-class correlation ratio of 7.5 dB was obtained as a best case and 3.6 dB as a worst case, the intra-class variation being at 11 ° increments in orientation at zero elevation angle. The results are extrapolated to estimate that approximately 80 filters would be required for a full 2 steradian orientation coverage. The implementation of the frequency removal technique and the Wiener filter in the JTC is briefly considered in conclusion to this work

Matched wavelet construction and its application to target detection

This dissertation develops a new wavelet design technique that produces a wavelet that matches a desired signal in the least squares sense. The Wavelet Transform has become very popular in signal and image processing over the last 6 years because it is a linear transform with an infinite number of possible basis functions that provides localization in both time (space) and frequency (spatial frequency). The Wavelet Transform is very similar to the matched filter problem, where the wavelet acts as a zero mean matched filter. In pattern recognition applications where the output of the Wavelet Transform is to be maximized, it is necessary to use wavelets that are specifically matched to the signal of interest. Most current wavelet design techniques, however, do not design the wavelet directly, but rather, build a composite wavelet from a library of previously designed wavelets, modify the bases in an existing multiresolution analysis or design a multiresolution analysis that is generated by a scaling function which has a specific corresponding wavelet. In this dissertation, an algorithm for finding both symmetric and asymmetric matched wavelets is developed. It will be shown that under certain conditions, the matched wavelets generate an orthonormal basis of the Hilbert space containing all finite energy signals. The matched orthonormal wavelets give rise to a pair of Quadrature Mirror Filters (QMF) that can be used in the fast Discrete Wavelet Transform. It will also be shown that as the conditions are relaxed, the algorithm produces dyadic wavelets which when used in the Wavelet Transform provides significant redundancy in the transform domain. Finally, this dissertation develops a shift, scale and rotation invariant technique for detecting an object in an image using the Wavelet Radon Transform (WRT) and matched wavelets. The detection algorithm consists of two levels. The first level detects the location, rotation and scale of the object, while the second level detects the fine details in the object. Each step of the wavelet matching algorithm and the object detection algorithm is demonstrated with specific examples

Frequency domain parameter identification and the statistical properties of frequency response estimates

Frequency domain techniques in systems theory have their origins in Heavyside's operational calculus (Heavyside, 1889). Such work was later developed by Foster and Campbell (1931), Brune (1931), Nyquist (1932), Black (1934), Darlington (1939) and subsequently Bode (1948). This interest in the frequency domain was due to its appeal to the intuition of the engineer. The dominance of frequency domain techniques was subsequently eroded from the late 1950s through the 1960s by the influence of the space programmes. The space systems being analysed were based on strong theoretical foundations with well-defined sets of differential equations. The analysis led to the development of the state-space methods which were able to cope with the multivariable problems and were amenable to numerical solution. As a result of these developments, control engineering was largely dominated by the state-space approach and the associated areas of LQG optimal control, Kaiman-Bucy filters, observability and controllability. Two factors led to a resurgence of interest amongst academics in the development of frequency domain techniques in the 1970s and 1980s. The first was the development of the Fast Fourier Transform (FFT) (Cooley & Tookey, 1965). This provided an efficient method of analysing the Fourier transforms of signals and allowed the development of spectral methods of obtaining frequency response estimates. The collection of data was greatly speeded up and this enabled frequency domain methods to be increasingly applied to on-line control problems. The second factor was that the developments in the time domain were never fully embraced by practicing engineers in traditional control environments

Signal-to-noise ratio estimation in digital computer simulation of lowpass and bandpass systems with applications to analog and digital communications, volume 3

Techniques are developed to estimate power gain, delay, signal-to-noise ratio, and mean square error in digital computer simulations of lowpass and bandpass systems. The techniques are applied to analog and digital communications. The signal-to-noise ratio estimates are shown to be maximum likelihood estimates in additive white Gaussian noise. The methods are seen to be especially useful for digital communication systems where the mapping from the signal-to-noise ratio to the error probability can be obtained. Simulation results show the techniques developed to be accurate and quite versatile in evaluating the performance of many systems through digital computer simulation

A Generalized Portmanteau Goodness-of-fit Test for Time Series Models

We present a goodness of fit test for time series models based on the discrete spectral average estimator. Unlike current tests of goodness of fit, the asymptotic distribution of our test statistic allows the null hypothesis to be either a short or long range dependence model. Our test is in the frequency domain, is easy to compute and does not require the calculation of residuals from the fitted model. This is especially advantageous when the fitted model is not a finite order autoregressive model. The test statistic is a frequency domain analogue of the test by Hong (1996) which is a generalization of the Box-Pierce (1970) test statistic. A simulation study shows that our test has power comparable to that of Hong's test and superior to that of another frequency domain test by Milhoj (1981).Statistics Working Papers Serie

Function estimation of irregularly spaced data with long memory dependence

We examine the problem of estimating an underlying function from collected data. The methods considered include parametric regression, density estimation, kernel estimation, wavelet regression, and specific results from when our underlying function f\left(x\right) is a member of the Besov or the Triebel spaces. Then we consider the problem of long memory error in several settings, including data which is equally spaced, data which is unequally spaced, and data which is a member of the Holder class and several other spaces. Ultimately we focus on three different problems. The first involves using linear interpolation or local averaging to account for the problem of irregularly spaced data. The second involves using a function H to reorder the data in a more general space. The third involves solving the problem in the matrix setting and considers the use of penalty functions. This method leads to general equations which describe the Mean Square Error in terms of Oracle risk. All three of these problems attempt to bound the Mean Integrated Square Error when the data is subject to long memory error