The EM algorithm

Our dataset has 52 bacteria of three genera (Corynebacterium: 10, Mycobacterium: 31, Rhodococcus: 11), each of which has both 16S and gyrB sequences. For simplicity, let us call these genera as class 1-3, respectively. For 16S and gyrB, we computed the second order count kernel, which is the dot product of bimer counts (Tsuda et al., 2002). Each kernel matrix is normalized such that the norm of each sample in the feature space becomes one. The kernel matrices of gyrB and 16S can be seen in Figure 2 (b) and (c), respectively. For reference, we show an ideal matrix as Figure 2 (a), which indicates the true classes. In our scenario, for a considerable number of bacteria, gyrB sequences are not available as in Figure 2 (d). We will complete the missing entries by the em algorithm with the spectral variants of the 16S matrix. When the em algorithm converges, we end up with two matrices: the completed matrix on data manifold D and the estimated matrix on model manifold M . The completed and estimated matrices are shown in Figure 2 (e) and (f), respectively. These two matrices are in general not the same, because the two manifolds may not have intersection.

In our work, we introduce a spacial interval arithmetic that always produces results that are smaller (in the sense that it is contained) than the traditional interval arithmetic [, ]. This arithmetic enables the extension of the Kalman filter as well as the EM algorithm to interval setting in a true sense. In our restricted interval arithmetic, the interval Kalman filter we introduce here is optimal. However, with respect to the more general interval arithmetic, our interval Kalman filter is suboptimal.

The EM algorithm is a general purpose algorithm for maximum likeli- hood estimation in a wide variety of situations where the likelihood of the ob- served data is intractable but the joint likelihood of the observed and missing data has a simple form (see [2]). The algorithm works for any augmentation scheme however appropriately defined missing data can lead to easier and more efficient implementation. Without loss of generality we assume two data points and denote the observed data by v obs = {V 0 = u, V t = w}.

In this paper, we study the use of so-called word trigger pairs for short: word triggers Bahl et al., 1984, Lau and Rosenfeld, 1993, Tillmann and Ney, 1996 to improve an existing languag[r]

DOI: 10.4236/jamp.2018.61002 12 Journal of Applied Mathematics and Physics algorithm has an obvious shortcoming: it is very sensitive to the initial value. Therefore, in order to get the parameter estimation of the closest to the true val- ue, we have to find a method to initialize the EM algorithm. We can list several usual methods with initialization: random center, hierarchical clustering, k-means algorithm and so on [1]. As a result of the k-means clustering algo- rithm is also a kind of dynamic iterative algorithm and decides the classification number by subjective factors. Further, it is accordant with EM algorithm for pa- rameter estimation of finite mixture model. Hence we can use outlier detection based on proximity to remove the outliers in order to reduce the influence of noise for the parameter estimation. Then, a rough grouping of the rest of the mixed data is given by k-means clustering. Finally, a rough estimate of parame- ters is given based on packet data.

Unfortunately, there’s no explicit solution to (2.2), even for the simplest mixture models such as normal mixtures. Consequently, the solution has to be obtained by algo- rithms. There are two general approaches to solve (2.2): the Newton-Raphson algorithm (McHugh (1956)) and the Expectation-Maximization (EM) algorithm (Dempster et al. (1977)). It is well known that for mixture models, the Newton-Raphson method con- verges faster than the EM algorithm but cannot guarantee convergence. Thus, the EM algorithm, known as the EM algorithm, is a more popular approach for analyzing mix- ture models. Readers may refer to McLachlan and Krishnan (2007) for a comprehensive introduction of the EM algorithm and its properties.

The EM algorithm is widely used to develop iterative parameter estimation procedures for statisti- cal models. In cases where these procedures strictly follow the EM formulation, the convergence properties of the estimation procedures are well understood. In some instances there are practical reasons to develop procedures that do not strictly fall within the EM framework. We study EM vari- ants in which the E-step is not performed exactly, either to obtain improved rates of convergence, or due to approximations needed to compute statistics under a model family over which E-steps cannot be realized. Since these variants are not EM procedures, the standard (G)EM convergence results do not apply to them. We present an information geometric framework for describing such algorithms and analyzing their convergence properties. We apply this framework to analyze the convergence properties of incremental EM and variational EM. For incremental EM, we discuss conditions under these algorithms converge in likelihood. For variational EM, we show how the E-step approximation prevents convergence to local maxima in likelihood.

In our case, we deal with the fixed parameter problem and adopt the EM algorithm to determine the distribution of the latent variables in the next expected step. Then, we adopt the clustering approach. This paper proceeds as follows, we first propose in section 1 a mixture of two stochastic volatility models; the ARSV-t and the SVOL model. Secondly, we use in section 2 the Expectation maximization algorithm (EM) in order to estimate the parameter. Finally, we apply in section 3 the clustering in order to classify the database according to the first or the second model.

The stochastic EM (SEM) algorithm proposed by Celeux and Diebolt 12 is also a stochastic version of the EM implementations as a way for executing the E-step by simulation. A very attractive merit of this algorithm is that it replaces the E-Step with an S-Step, which is very easy to implement whatever the underlying distribution and the missing data are. Compared with the Monte Carlo EM algorithm, the SEM algorithm completes the observed sample by replacing each missing datum by a value randomly drawn from the distribution conditional on results from the previous step. The M-step is thus a complete-data maximum likelihood estimation, which is often very easy to solve. The SEM algorithm has been shown to be computationally less burdensome and more appropriate than the EM algorithm in a lot of problems (Celeux and Diebolt 12 , Tregouet et al. 13 , Delignon et al. 14 , Cariou and Chehdi 15 ). It is shown by Nielsen 16 that the SEM algorithm always converges to some local optimum. Some applications of the SEM algorithm suggest that this algorithm tends to converge to the global optimum or a non-significant local optimum (Diebolt and Celeux 17 , Cariou and Chehdi 15 , Svensson and Sjöstedt-de Luna 18 ).

Abstract—This paper describes a new approach to matching geometric structure in 2D point-sets. The novel feature is to unify the tasks of estimating transformation geometry and identifying point-correspondence matches. Unification is realized by constructing a mixture model over the bipartite graph representing the correspondence match and by affecting optimization using the EM algorithm. According to our EM framework, the probabilities of structural correspondence gate contributions to the expected likelihood function used to estimate maximum likelihood transformation parameters. These gating probabilities measure the consistency of the matched neighborhoods in the graphs. The recovery of transformational geometry and hard correspondence matches are interleaved and are realized by applying coupled update operations to the expected log-likelihood function. In this way, the two processes bootstrap one another. This provides a means of rejecting structural outliers. We evaluate the technique on two real-world problems. The first involves the matching of different perspective views of 3.5-inch floppy discs. The second example is furnished by the matching of a digital map against aerial images that are subject to severe barrel distortion due to a line-scan sampling process. We complement these experiments with a sensitivity study based on synthetic data.

This study experiment with two partitioning-based algorithms and a probabilistic model-based algorithm, the clusters formed are shown in Figures 5, 6, and 7. The algorithms operate under similar parameter settings as represented in Table 1. For the purpose of validation of the formed clusters, the results were exported to excel file as shown in the experimental setup in Figure 1 for further computations. The scatter plots of k- means and k-medoids looks much alike and comparing their results to the class-label of the original data being analysed, each has 61% and 62% accuracy respectively as displayed in Table 2. The EM-algorithm shows a different result entirely and the comparisons show that it is 51.8% accurate. In general, the three algorithms are very fast, while k-means remain the fastest among them.

The EM (expectation-maximization)algorithm is a well- known tool for iterative maximum likelihood estimation. The earlier EM methods for the state space model were developed by Shumway and Stoffer (1982) and Wat- son and Engle (1983). In addition to Newton-Raphson, Shumway and Stoffer (1982)presented conceptually sim- pler estimation procedure based on the EM algorithm (Dempster et al.1977). The basic idea is that if we can observe the states, X n = x 0 , x 1 , ...., x n and Y n =

Although the EM algorithm [3] has been widely used to solve optimization problems in many machine learning algorithms, such as the K-Means for clustering, the EM has a severe limitation, known as the initial value problem, in which solutions can vary depending on the initial parameter setting. To overcome such a problem, we have applied the Deterministic Annealing (DA) algorithm to GTM to find more robust answers against random initial values.

populations, the EM-based method obtains either the best or the second best results. This is because, in these situations, the EM algorithm starts with good enough starting points. The conclusion is that, when we are faced with accurate enough experts’ opinions, the EM-based method obtains reasonable results. It may be asked, what if we do not know anything about the accuracy level of the opinions at hand? The EM-based method may get stuck in a local optimum far from true experts’ accuracies. This was the main stimulus for proposing an alternative approach, the marginalization- based scoring function, in Section 5. It is clear from the tables that the marginalization-based score obtains more robust results.

Simulation results are shown in Figs. 5-6 and Table 2. Figure 5 shows the system trajectories after 20 training cycles (solid line: desired trajectory; dashed line: system actual output). Comparison results of RMSE between IEMGA and other algorithms are shown in Fig. 6 (dashed line: GA, dash-dotted line: EM, and solid line: IEMGA). Obviously, the IEMGA algorithm has better performance in RMSE than the EM algorithm and GA algorithm. The computation time of EM and IEMGA are 614.182 seconds and 25.138, respectively. IEMGA enhances the performance of EM in computation effort. Table 2 shows the comparison results of RMSE and computation time. From Table 2, IEMGA algorithm has best approximation result (Mean RMSE: 0.5814). We see that the best, worst, and mean RMSEs of

It is clear from the experimental results that the performance of the proposed approach of EMFPCM is better in terms of clustering accuracy, mean squared error, execution time and conver[r]

The transmission study produced narrow beam attenuation coefficients for 9 'Tc which were used together with the ML-EM algorithm to perform non-uniform attenuation compensation of an em[r]

Mario G.C.A. Cimino, Beatrice Lazzerini, Francesco Marcelloni was proposed the TS system to build a dissimilarity matrix which is fed as input to an unsupervised fuzzy relational clustering algorithm, denoted any relation clustering algorithm (ARCA), which partitions the data set based on the proximity of the vectors containing the dissimilarity values between each pattern and all the other patterns in the data set. [2]. Christy Maria Joy1, S. Leela was proposed the Hierarchical Fuzzy Relational Clustering Algorithm and it is a hybrid method which is a combination of general hierarchical clustering concepts with fuzzy relation models that is existing fuzzy clustering algorithms.This method used Cosine similarity.[4]. This paper Vasileios Hatzivassiloglou, Judith L. Klavans was present a statistical similarity measuring and clustering tool, SIMFINDER , that organizes small pieces of [5].

Diabetes mellitus or simply diabetes is a disease caused due to the increase level of blood glucose. Various available traditional methods for diagnosing diabetes are based on physical and chemical tests. These methods can have errors due to different uncertainties. A number of Data mining algorithms were designed to overcome these uncertainties. Among these algorithms, amalgam KNN and ANFIS provides higher classification accuracy than the existing approaches. The main data mining algorithms discussed in this paper are EM algorithm, KNN algorithm, K-means algorithm, amalgam KNN algorithm and ANFIS algorithm. EM algorithm is the expectation-maximization algorithm used for sampling, to determine and maximize the expectation in successive iteration cycles. KNN algorithm is used for classifying the objects and used to predict the labels based on some closest training examples in the feature space. K means algorithm follows partitioning methods based on some input parameters on the datasets of n objects. Amalgam combines both the features of KNN and K means with some additional processing. ANFIS is the Adaptive Neuro Fuzzy Inference System which combines the features of adaptive neural network and Fuzzy Inference System. The data set chosen for classification and experimental simulation is based on Pima Indian Diabetic Set from University of California, Irvine (UCI) Repository of Machine Learning databases.

Abstract—This paper presents a fast and effective electromagnetic reconstruction algorithm with phaseless data under weak scattering conditions. The proposed algorithm is based on the phaseless data multiplicative regularized contrast sources inversion method (PD- MRCSI). We recast the weak scattering problem as an optimization problem in terms of the undetermined contrast and contrast sources. Using the conjugate gradient iterative method, the problem is solved by alternately updating the contrast sources and the contrast. Additionally, this method can combine with the PD-MRCSI method. Taking advantage of the properties of fast convergence of this algorithm and stable convergence of PD-MRCSI method, the combined technique makes image reconstructions more fast and effective. Although the method is derived from weak scattering situation, it is also useful for the case which weak scattering approximation is not satisfied. The synthetic numerical reconstruction results, as well as experimental reconstruction results, presented that the proposed method is a very fast and effective reconstruction algorithm.