Quantization is a process that maps a continous or discrete set of values into approximations that belong to a smaller set. Quantization is a lossy: some information about the original data is lost in the process. The key to a successful quantization is therefore the selection of an error criterion – such as entropy and signal-to-noise ratio – and the development of optimal quantizers for this criterion.

Quantization

This paper synthesizes the results, methodology, and research conducted concerning the K-means clustering method over the last fifty years. The K-means method is first introduced, various formulations of the minimum variance loss function and alternative loss functions within the same class are outlined, and different methods of choosing the number of clusters and initialization, variable preprocessing, and data reduction schemes are discussed. Theoretic statistical results are provided and various extensions of K-means using different metrics or modifications of the original algorithm are given, leading to a unifying treatment of K-means and some of its extensions. Finally, several future studies are outlined that could enhance the understanding of numerous subtleties affecting the performance of the K-means method.

K‐means clustering: A half‐century synthesis

Principal curves have been defined as "self-consistent" smooth curves which pass through the "middle" of a d-dimensional probability distribution or data cloud. They give a summary of the data and also serve as an efficient feature extraction tool. We take a new approach by defining principal curves as continuous curves of a given length which minimize the expected squared distance between the curve and points of the space randomly chosen according to a given distribution. The new definition makes it possible to theoretically analyze principal curve learning from training data and it also leads to a new practical construction. Our theoretical learning scheme chooses a curve from a class of polygonal lines with k segments and with a given total length to minimize the average squared distance over n training points drawn independently. Convergence properties of this learning scheme are analyzed and a practical version of this theoretical algorithm is implemented. In each iteration of the algorithm, a new vertex is added to the polygonal line and the positions of the vertices are updated so that they minimize a penalized squared distance criterion. Simulation results demonstrate that the new algorithm compares favorably with previous methods, both in terms of performance and computational complexity, and is more robust to varying data models.

/pdf/learning-and-design-of-principal-curves-4gjmqelpvy.pdf

Learning and design of principal curves

A class of procedures based on "impartial trimming" (self-determined by the data) is introduced with the aim of robustifying k-means, hence the associated clustering analysis We include a detailed study of optimal regions, showing that only nonpathological regions can arise from impartial trimming procedures The asymptotic results provided in the paper focus on strong consistency of the suggested methods under widely general conditions A section is devoted to exploring the performance of the procedure to detect anomalous data in simulated data sets

/pdf/trimmed-k-means-an-attempt-to-robustify-quantizers-1to4nqct7b.pdf

Trimmed $k$-means: an attempt to robustify quantizers

The generalized k means method is based on the minimization of the discrepancy between a random variable (or a sample of this random variable) and a set with k points measured through a penalty function Φ. As in the M estimators setting (k = 1), a penalty function, Φ, with unbounded derivative, Ψ, naturally leads to nonrobust generalized k means. However, surprisingly the lack of robustness extends also to the case of bounded Ψ; that is, generalized k means do not inherit the robustness properties of the M estimator from which they came. Attempting to robustify the generalized k means method, the generalized trimmed k means method arises from combining k means idea with a so-called impartial trimming procedure. In this article study generalized k means and generalized trimmed k means performance from the viewpoint of Hampel's robustness criteria; that is, we investigate the influence function, breakdown point, and qualitative robustness, confirming the superiority provided by the trimming. We inc...

Robustness Properties of k Means and Trimmed k Means

The $k$ principal points of a $p$-variate random vector $\mathbf{X}$ are those points $\xi_1, \ldots, \xi_k \in \mathbb{R}^p$ which approximate the distribution of $\mathbf{X}$ by minimizing the expected squared distance of $\mathbf{X}$ from the nearest of the $\xi_j$. Any set of $k$ points $\mathbf{y}_1, \ldots, \mathbf{y}_k$ partitions $\mathbb{R}^p$ into "domains of attraction" $D_1, \ldots, D_k$ according to minimal distance; following Hastie and Stuetzle we call $\mathbf{y}_1, \ldots, \mathbf{y}_k$ self-consistent if $E\lbrack\mathbf{X}\mid\mathbf{X} \in D_j\rbrack = \mathbf{y}_j$ for $j = 1, \ldots, k$. Principal points are a special case of self-consistent points. In this paper we study principal points and self-consistent points of $p$-variate elliptical distributions. The main results are the following: (1) If $k$ self-consistent points of $\mathbf{X}$ span a subspace of dimension $q < p$, then this subspace is also spanned by $q$ principal components, that is, self-consistent points of elliptical distributions exist only in principal component subspaces. (2) The subspace spanned by $k$ principal points of $\mathbf{X}$ is identical with the subspace spanned by the principal components associated with the largest roots. This proves a conjecture of Flury. We also discuss implications of our results for the computation and estimation of principal points.

/pdf/principal-points-and-self-consistent-points-of-elliptical-3dtg0zitdn.pdf

Principal Points and Self-Consistent Points of Elliptical Distributions

Uniqueness of principal points for univariate distributions

Molecular and metabolic mechanisms of bufalin against lung adenocarcinoma: New and comprehensive evidences from network pharmacology, metabolomics and molecular biology experiment

Tumor Laboratory, Jining No. 1 People’s Hospital, Jining Medical University, Jining, Shandong 272000, China Department of Spine Surgery, Jining No. 1 People’s Hospital, Jining Medical University, Jining, Shandong 272000, China Cheeloo College of Medicine, Shandong University, Ji’nan, Shandong 250100, China Department of Pharmacy, Jining No. 1 People’s Hospital, Jining Medical University, Jining, Shandong 272000, China

Advances in Antitumor Mechanism of Evodiamine

Integrated Network Pharmacology Analysis And Experimental Validation to Reveal the Mechanism Of Euphorbia Fischeriana Steud.Extract in the Treatment of HCC

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Papers

Principal Points and Self-Consistent Points of Elliptical Distributions

Uniqueness of principal points for univariate distributions

Molecular and metabolic mechanisms of bufalin against lung adenocarcinoma: New and comprehensive evidences from network pharmacology, metabolomics and molecular biology experiment

Advances in Antitumor Mechanism of Evodiamine

Integrated Network Pharmacology Analysis And Experimental Validation to Reveal the Mechanism Of Euphorbia Fischeriana Steud.Extract in the Treatment of HCC