This thesis explores forecasting for intermittent demand requirements. Intermittent demand occurs at random, with some time periods showing no demand. In addition, demand, when it occurs, may not be for a single unit or a constant size. Consequently, intermittent demand creates significant problems in the supply and manufacturing environment as far as forecasting and inventory control are concerned. A certain confusion is shared amongst academics and practitioners about how intermittent demand (or indeed any other demand pattern that cannot be reasonably represented by the normal distribution) is defined. As such, we first construct a framework that aims at facilitating the conceptual categorisation of what is termed, for the purposes of this research, “non-normal” demand patterns. Croston (1972) proposed a method according to which intermittent demand estimates can be built from constituent elements, namely the demand size and inter-demand interval. The method has been claimed to provide unbiased estimates and it is regarded as the “standard” approach to dealing with intermittence. In this thesis we show that Croston’s method is biased. The bias is quantified and two new estimation procedures are developed based on Croston’s concept of considering both demand sizes and inter-demand intervals. Consequently the issue of variability of the intermittent demand estimates is explored and finally Mean Square Error (MSE) expressions are derived for all the methods discussed in the thesis. The issue of categorisation of the demand patterns has not received sufficient academic attention thus far, even though, from the practitioner’s standpoint it is appealing to switch from one estimator to the other according to the characteristics of the demand series under concern. Algebraic comparisons of MSE expressions result in universally applicable (and theoretically coherent) categorisation rules, based on which, “non-normal” demand patterns can be defined and estimators be selected. All theoretical findings are checked via simulation on theoretically generated demand data. The data is generated upon the same assumptions considered in the theoretical part of the thesis. Finally, results are generated using a large sample of empirical data. Appropriate accuracy measures are selected to assess the forecasting accuracy performance of the estimation procedures discussed in the thesis. Moreover, it is recognised that improvements in forecasting accuracy are of little practical value unless they are translated to an increased customer service level and/or reduced inventory cost. In consequence, an inventory control system is specified and the inventory control performance of the estimators is also assessed on the real data. The system is of the periodic order-up-to-level nature. The empirical results confirm the practical validity and utility of all our theoretical claims and demonstrate the benefits gained when Croston’s method is replaced by an estimator developed during this research, the Approximation method.

In wireless communications, channel estimation is necessary for coherent symbol detection. This thesis considers a network which consists of two transceivers communicating with the help of a relay applying the amplify-and-forward (AF) relaying scheme. The training based channel estimation technique is applied to the proposed network where the numbers of the training sequence transmitted by the two transceivers, are different. All three terminals are equipped with a single antenna for signal transmission and reception. Communication between the transceivers is carried out in two phases. In the first phase, each transceiver sends a transmission block of data embedded with known training symbols to the relay. In the second phase, the relay retransmits an amplified version of the received signal to both transceivers. Estimates of the channel coefficients are obtained using the Maximum Likelihood (ML) estimator. The performance analysis of the derived estimates are carried out in terms of the mean squared error (MSE) and we determine conditions required to increase the estimation accuracy. / UOIT

Effort to estimate unknown time-varying channels as a part of high-speed mobile communication systems is of interest especially for next-generation wireless systems. The high computational complexity of the optimal Wiener estimator usually makes its use impractical in fast time-varying channels. As a powerful candidate, the adaptive least mean squares (LMS) algorithm offers a computationally efficient solution with its simple first-order weight-vector update equation. However, the performance of the LMS algorithm deteriorates in time-varying channels as a result of the eigenvalue disparity, i.e., spread, of the input correlation matrix in such chan nels. In this work, we incorporate the L MS algorithm into the well-known bidirectional processing idea to produce an extension called the bidirectional LMS. This algorithm is shown to be robust to the adverse effects of time-varying channels such as large eigenvalue spread. The associated tracking performance is observed to be very close to that of the optimal Wiener filter in many cases and the bidirectional LMS algorithm is therefore referred to as near-optimal. The computational complexity is observed to increase by the bidirectional employment of the LMS algorithm, but nevertheless is significantly lower than that of the optimal Wiener filter. The tracking behavior of the bidirectional LMS algorithm is also analyzed and eventually a steady-state step-size dependent mean square error (MSE) expression is derived for single antenna flat-fading channels with various correlation properties. The aforementioned analysis is then generalized to include single-antenna frequency-selective channels where the so-called ind ependence assumption is no more applicable due to the channel memory at hand, and then to multi-antenna flat-fading channels. The optimal selection of the step-size values is also presented using the results of the MSE analysis. The numerical evaluations show a very good match between the theoretical and the experimental results under various scenarios. The tracking analysis of the bidirectional LMS algorithm is believed to be novel in the sense that although there are several works in the literature on the bidirectional estimation, none of them provides a theoretical analysis on the underlying estimators. An iterative channel estimation scheme is also presented as a more realistic application for each of the estimation algorithms and the channel models under consideration. As a result, the bidirectional LMS algorithm is observed to be very successful for this real-life application with its increased but still practical level of complexity, the near-optimal tracking performa nce and robustness to the imperfect initialization.

We address the problem of image denoising for additive white Gaussian noise (AWGN), Poisson noise, and Chi-squared noise scenarios. Thermal noise in electronic circuitry in camera hardware can be modeled as AWGN. Poisson noise is used to model the randomness associated with photon counting during image acquisition. Chi-squared noise statistics are appropriate in imaging modalities such as Magnetic Resonance Imaging (MRI). AWGN is additive, while Poisson noise is neither additive nor multiplicative. Although Chi-squared noise is derived from AWGN statistics, it is non-additive. Mean-square error (MSE) is the most widely used metric to quantify denoising performance. In parametric denoising approaches, the optimal parameters of the denoising function are chosen by employing a minimum mean-square-error (MMSE) criterion. However, the dependence of MSE on the noise-free signal makes MSE computation infeasible in practical scenarios. We circumvent the problem by adopting an MSE estimation approach. The ground-truth-independent estimates of MSE are Stein’s unbiased risk estimate (SURE), Poisson unbiased risk estimate (PURE) and Chi-square unbiased risk estimate (CURE) for AWGN, Poison and Chi-square noise models, respectively. The denoising function is optimized to achieve maximum noise suppression by minimizing the MSE estimates. We have chosen the bilateral filter as the denoising function. Bilateral filter is a nonlinear edge-preserving smoother. The performance of the bilateral filter is governed by the choice of its parameters, which can be optimized to minimize the MSE or its estimate. However, in practical scenarios, MSE cannot be computed due to inaccessibility of the noise-free image. We derive SURE, PURE, and CURE in the context of bilateral filtering and compute the parameters of the bilateral filter that yield the minimum cost (SURE/PURE/CURE). On processing the noisy input with bilateral filter whose optimal parameters are chosen by minimizing MSE estimates (SURE/PURE/CURE), we obtain the estimate closest to the ground truth. We denote the bilateral filter with optimal parameters as SURE-optimal bilateral filter (SOBF), PURE-optimal bilateral filter (POBF) and CURE-optimal bilateral filter (COBF) for AWGN, Poisson and Chi-Squared noise scenarios, respectively. In addition to the globally optimal bilateral filters (SOBF and POBF), we propose spatially adaptive bilateral filter variants, namely, SURE-optimal patch-based bilateral filter (SPBF) and PURE-optimal patch-based bilateral filter (PPBF). SPBF and PPBF yield significant improvements in performance and preserve edges better when compared with their globally-optimal counterparts, SOBF and POBF, respectively. We also propose the SURE-optimal multiresolution bilateral filter (SMBF) where we couple SOBF with wavelet thresholding. For Poisson noise suppression, we propose PURE-optimal multiresolution bilateral filter (PMBF), which is the Poisson counterpart of SMBF. We com-pare the performance of SMBF and PMBF with the state-of-the-art denoising algorithms for AWGN and Poisson noise, respectively. The proposed multiresolution-based bilateral filtering techniques yield denoising performance that is competent with that of the state-of-the-art techniques.

Η παρούσα μεταπτυχιακή διατριβή εντάσσεται ερευνητικά στην περιοχή της Στατιστικής Θεωρίας Αποφάσεων και ειδικότερα στην εκτίμηση των παραμέτρων στο μοντέλο της διπαραμετρικής εκθετικής κατανομής με παράμετρο θέσης μ και παράμετρο κλίμακος σ. Θεωρούμε το πρόβλημα εκτίμησης των παραμέτρων κλίμακας μ και θέσης σ, όταν μ≤c, όπου c είναι μία γνωστή σταθερά. Αποδεικνύουμε ότι σε σχέση με το κριτήριο του Μέσου Τετραγωνικού Σφάλματος (ΜΤΣ), οι βέλτιστοι αναλλοίωτοι εκτιμητές των μ και σ, είναι μη αποδεκτοί όταν μ≤c, και προτείνουμε βελτιωμένους. Επίσης συγκρίνουμε του εκτιμητές αυτούς σε σχέση με το κριτήριο του Pitman. Επιπλέον, προτείνουμε εκτιμητές που είναι καλύτεροι από τους βέλτιστους αναλλοίωτους εκτιμητές, όταν μ≤c, ως προς την συνάρτηση ζημίας LINEX. Τέλος, η θεωρία που αναπτύσσεται εφαρμόζεται σε δύο ανεξάρτητα δείγματα προερχόμενα από εκθετική κατανομή. / The present master thesis deals with the estimation of the location parameter μ and the scale parameter σ of the two-parameter exponential distribution. We consider the problem of estimation of locasion parameter μ and the scale parameter σ, when it is known apriori that μ≤c, where c is a known constant. We establish that with respect to the mean square error (mse) criterion the best affine estimators of μ and σ in the absence of information μ≤c are inadmissible and we propose estimators which are better than these estimators. Also, we compare these estimators with respect to the Pitman Nearness criterion. We propose estimators which are better than the standard estimators in the unrestricted case with respect to the suitable choise of LINEX loss. Finally, the theory developed is applied to the problem of estimating the location and scale parameters of two exponential distributions when the location parameters are ordered.

Tidigare studier visar på att blodtrycket hos gravida sjunker under andra trimestern och sedanökar i ett senare skede av graviditeten. Högt blodtryck hos gravida kan medföra hälsorisker, vilket gör mätningar av blodtryck relevanta. Dock uppstår det osäkerhet då olika personer inom vården hanterar blodtrycksmätningarna på olika sätt. Delar av vårdpersonalen avrundarmätvärden och andra gör det inte, vilket kan leda till svårigheter att tolkablodtrycksutvecklingen. I uppsatsen behandlas ett dataset innehållandes blodtrycksvärden hos gravida genom att skatta nio olika linjära regressionsmodeller med blandade effekter. Därefter genomförs en simuleringsstudie med syfte att undersöka hur mätproblem orsakat av avrundningar påverkar parameterskattningar och modellval i en linjär blandad modell. Slutsatsen är att blodtrycksavrundningarna inte påverkar typ 1-felet men påverkar styrkan. Dock innebär inte detta något problem vid fortsatt analys av blodtrycksvärdena i det verkliga datasetet.