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Maximum a posteriori state sequence estimation using polyhedral parameterization.

機(jī)譯：使用多面體參數(shù)化的最大后驗(yàn)狀態(tài)序列估計(jì)。

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For a discrete-time Markov process observed in memory-less noise, the recursive equations for maximum a posteriori probability state sequence (MAPSS) estimation can be applied using either continuous or discrete state spaces. For a linear system model on a continuous state space with additive white Gaussian noise, MAPSS estimation results in the Kalman filter (KF) and fixed interval smoother. For a Markov chain on a finite discrete state space, MAPSS estimation results in the Viterbi algorithm (VA). For applications on a continuous state space where linearity and additive white Gaussian noise assumptions may be invalid, the KF cannot be used to implement the MAPSS estimator. In this case, the continuous state space may be quantized onto a discrete state space grid so that a VA estimator can be implemented. Unfortunately, the computational requirements for a VA estimator become excessive when continuous state spaces are quantized onto grids, even for systems of moderate dimensionality.; The problem addressed in this research is the implementation of the algorithms for MAPSS estimation on a continuous state space using polyhedral parameterization. Several methods to represent polyhedra were investigated; it was found that algorithms for MAPSS estimation favor hyperplane representation. In hyperplane representation, polyhedral facets are created by the convex combination of closed linear half spaces defined by support hyperplanes. Hyperplane representation results in smaller data structures, relaxed numeric precision requirements, and simpler algorithms when compared to other methods of representing polyhedra.; The most computationally expensive operation in the MAPSS estimator is maximum convolution. However, maximum convolution of two polyhedral functions is greatly simplified with maximum transforms. Specifically, maximum convolution is replaced by the inverse transform of the sum of two transformed polyhedral functions. The maximum transform of a polyhedral function produces another polyhedral function with hyperplane abscissa and slopes interchanged and hyperplane ordinates and z-intercepts interchanged with negation.; For this research, polyhedral-based algorithms are developed for the MAPSS estimator. Computer simulations quantify the performance of KF, VA, and polyhedral MAPSS estimators for linear systems with state space dimensions of 1, 2, and 3. Hyperplane count is tracked. For similar performance levels, polyhedral estimation significantly reduces memory usage and computational load needed for VA estimation.
機(jī)譯：對(duì)于在無(wú)內(nèi)存噪聲中觀察到的離散時(shí)間馬爾可夫過(guò)程，可以使用連續(xù)或離散狀態(tài)空間來(lái)應(yīng)用最大后驗(yàn)概率狀態(tài)序列（MAPSS）估計(jì)的遞歸方程。對(duì)于具有加性高斯白噪聲的連續(xù)狀態(tài)空間上的線性系統(tǒng)模型，MAPSS估計(jì)可得出卡爾曼濾波器（KF）和固定間隔平滑器。對(duì)于有限離散狀態(tài)空間上的馬爾可夫鏈，MAPSS估計(jì)在維特比算法（VA）中產(chǎn)生。對(duì)于在線性和加性高斯白噪聲假設(shè)可能無(wú)效的連續(xù)狀態(tài)空間上的應(yīng)用，不能使用KF來(lái)實(shí)現(xiàn)MAPSS估計(jì)器。在這種情況下，可以將連續(xù)狀態(tài)空間量化到離散狀態(tài)空間網(wǎng)格上，從而可以實(shí)現(xiàn)VA估計(jì)器。不幸的是，當(dāng)將連續(xù)狀態(tài)空間量化到網(wǎng)格上時(shí)，即使對(duì)于中等維數(shù)的系統(tǒng)，VA估計(jì)器的計(jì)算需求也變得過(guò)大。本研究解決的問(wèn)題是使用多面體參數(shù)化在連續(xù)狀態(tài)空間上進(jìn)行MAPSS估計(jì)的算法的實(shí)現(xiàn)。研究了幾種表示多面體的方法。發(fā)現(xiàn)用于MAPSS估計(jì)的算法有利于超平面表示。在超平面表示中，通過(guò)由支撐超平面定義的封閉線性半空間的凸組合來(lái)創(chuàng)建多面體面。與其他表示多面體的方法相比，超平面表示導(dǎo)致較小的數(shù)據(jù)結(jié)構(gòu)，寬松的數(shù)值精度要求和更簡(jiǎn)單的算法。 MAPSS估計(jì)器中計(jì)算上最昂貴的操作是最大卷積。但是，通過(guò)最大變換極大地簡(jiǎn)化了兩個(gè)多面體函數(shù)的最大卷積。具體地，最大卷積被兩個(gè)變換的多面體函數(shù)之和的逆變換代替。多面體函數(shù)的最大變換產(chǎn)生另一個(gè)多面體函數(shù)，其中超平面橫坐標(biāo)和斜率互換，超平面縱坐標(biāo)和 z 截距互換而取反。對(duì)于這項(xiàng)研究，為MAPSS估計(jì)器開(kāi)發(fā)了基于多面體的算法。對(duì)于狀態(tài)空間尺寸為1、2和3的線性系統(tǒng)，計(jì)算機(jī)仿真可以量化KF，VA和多面MAPSS估計(jì)器的性能。跟蹤超平面計(jì)數(shù)。對(duì)于相似的性能水平，多面估算可顯著減少VA估算所需的內(nèi)存使用量和計(jì)算負(fù)荷。

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