Weighted random sampling and reconstruction in general multivariate trigonometric polynomial spaces

2021 ◽  
pp. 1-20
Author(s):  
Wei Li ◽  
Jun Xian
Keyword(s):  
Signal Processing ◽  
Probability Distribution ◽  
Trigonometric Polynomial ◽  
Random Sampling ◽  
Reconstruction Formula ◽  
Sampling And Reconstruction ◽  
Sampling Set

The set of sampling and reconstruction in trigonometric polynomial spaces will play an important role in signal processing. However, in many applications, the frequencies in trigonometric polynomial spaces are not all integers. In this paper, we consider the problem of weighted random sampling and reconstruction of functions in general multivariate trigonometric polynomial spaces. The sampling set is randomly selected on a bounded cube with a probability distribution. We obtain that with overwhelming probability, the sampling inequality holds and the explicit reconstruction formula succeeds for all functions in the general multivariate trigonometric polynomial spaces when the sampling size is sufficiently large.

Random sampling and reconstruction in multiply generated shift-invariant spaces

2019 ◽  
Vol 17 (02) ◽  
pp. 323-347 ◽  
Author(s):  
Jianbin Yang
Keyword(s):  
Random Sampling ◽  
Stability Problem ◽  
Approximate Formula ◽  
Reconstruction Algorithm ◽  
Finite Dimensional ◽  
Shift Invariant Spaces ◽  
The Stability ◽  
Sampling And Reconstruction ◽  
Invariant Spaces ◽  
Sampling Set

Shift-invariant spaces play an important role in approximation theory, wavelet analysis, finite elements, etc. In this paper, we consider the stability and reconstruction algorithm of random sampling in multiply generated shift-invariant spaces [Formula: see text]. Under some decay conditions of the generator [Formula: see text], we approximate [Formula: see text] with finite-dimensional subspaces and prove that with overwhelming probability, the stability of sampling set conditions holds uniformly for all functions in certain compact subsets of [Formula: see text] when the sampling size is sufficiently large. Moreover, we show that this stability problem is connected with properties of the random matrix generated by [Formula: see text]. In the end, we give a reconstruction algorithm for the random sampling of functions in [Formula: see text].

scholarly journals Isometric Signal Processing under Information Geometric Framework

Entropy ◽  
2019 ◽  
Vol 21 (4) ◽  
pp. 332 ◽  
Author(s):  
Hao Wu ◽  
Yongqiang Cheng ◽  
Hongqiang Wang
Keyword(s):  
Signal Processing ◽  
Probability Distribution ◽  
Geometric Structure ◽  
Information Geometry ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Statistical Manifold ◽  
Intrinsic Parameter ◽  
Geometric Framework ◽  
Necessary And Sufficient

Information geometry is the study of the intrinsic geometric properties of manifolds consisting of a probability distribution and provides a deeper understanding of statistical inference. Based on this discipline, this letter reports on the influence of the signal processing on the geometric structure of the statistical manifold in terms of estimation issues. This letter defines the intrinsic parameter submanifold, which reflects the essential geometric characteristics of the estimation issues. Moreover, the intrinsic parameter submanifold is proven to be a tighter one after signal processing. In addition, the necessary and sufficient condition of invariant signal processing of the geometric structure, i.e., isometric signal processing, is given. Specifically, considering the processing with the linear form, the construction method of linear isometric signal processing is proposed, and its properties are presented in this letter.

scholarly journals Analysis of an Algorithm for Random Sampling

1994 ◽  
Vol 8 (2) ◽  
pp. 153-168
Author(s):  
Catherine B. Hurley ◽  
Hosam M. Mahmoud
Keyword(s):  
Phase Transitions ◽  
Probability Distribution ◽  
Infinite Series ◽  
Random Sampling ◽  
Geometric Distribution ◽  
Hash Table ◽  
Asymptotically Linear ◽  
Computer File ◽  
Two Phase ◽  
Order Of Magnitude

We analyze a standard algorithm for sampling m items without replacement from a computer file of n records. The algorithm repeatedly selects a record at random from the file, rejecting records that have previously been selected, until m records are obtained. The running time of the algorithm has two components: a rejection component and a search component. We show that the probability distribution of the rejection component undergoes an infinite series of phase transitions, depending on the order of magnitude of m relative to n. We identify an infinite number of ranges of m, each with a different behavior. The rejection component is distributed as a linear combination of Poisson-like random variables. The search component is customarily done using a hash table with separate chaining. The analysis of the hashing scheme in this problem differs from previous hashing analyses, as the number of lookups in the hash table for each insertion has a geometric distribution. We show that the average overall cost of searching is asymptotically linear with only two phase transitions in the coefficient of linearity.

scholarly journals Statistical Signal Processing by Using the Higher-Order Correlation between Sound and Vibration and Its Application to Fault Detection of Rotational Machine

2008 ◽  
Vol 2008 ◽  
pp. 1-7 ◽  
Author(s):  
Hisako Masuike ◽  
Akira Ikuta
Keyword(s):  
Signal Processing ◽  
Probability Distribution ◽  
Time Domain ◽  
Statistical Signal Processing ◽  
Higher Order ◽  
Orthogonal Expansion ◽  
Mutual Relationship ◽  
Diagnosis Method ◽  
Correlation Information ◽  
Multidimensional Probability

In this study, a stochastic diagnosis method based on the changing information of not only a linear correlation but also a higher-order nonlinear correlation is proposed in a form suitable for online signal processing in time domain by using a personal computer, especially in order to find minutely the mutual relationship between sound and vibration emitted from rotational machines. More specifically, a conditional probability hierarchically reflecting various types of correlation information is theoretically derived by introducing an expression on the multidimensional probability distribution in orthogonal expansion series form. The effectiveness of the proposed theory is experimentally confirmed by applying it to the observed data emitted from a rotational machine driven by an electric motor.

Random sampling

2019 ◽  
pp. 71-92
Author(s):  
Max A. Little
Keyword(s):  
Machine Learning ◽  
Signal Processing ◽  
Statistical Model ◽  
Random Sampling ◽  
Joint Distribution ◽  
Optimization Problems ◽  
Viable Alternative ◽  
Statistical Machine Learning ◽  
Digital Signals ◽  
Large Numbers

This chapter provides an overview of generating samples from random variables with a given (joint) distribution, and using these samples to find quantities of interest from digital signals. This task plays a fundamental role in many problems in statistical machine learning and signal processing. For example, effectively simulating the behaviour of the statistical model offers a viable alternative to optimization problems arising from some models for signals with large numbers of variables.

Chow's defense of null-hypothesis testing: Too traditional?

1998 ◽  
Vol 21 (2) ◽  
pp. 199-199
Author(s):  
Robert W. Frick
Keyword(s):  
Probability Distribution ◽  
Hypothesis Testing ◽  
Sample Size ◽  
Random Sampling ◽  
Null Hypothesis ◽  
Interpretation Of Probability ◽  
Null Hypothesis Testing

I disagree with several of Chow's traditional descriptions and justifications of null hypothesis testing: (1) accepting the null hypothesis whenever p > .05; (2) random sampling from a population; (3) the frequentist interpretation of probability; (4) having the null hypothesis generate both a probability distribution and a complement of the desired conclusion; (5) assuming that researchers must fix their sample size before performing their study.

On the Psychology of Ampliative Inference

1992 ◽  
Vol 3 (2) ◽  
pp. 131-135 ◽  
Author(s):  
Tracy S. Myers ◽  
Daniel N. Osherson
Keyword(s):  
Probability Distribution ◽  
Random Sampling ◽  
Point Of View ◽  
Event Space ◽  
Principle Of Maximum Entropy ◽  
Correct Guess ◽  
Psychological Point ◽  
Familiar Form ◽  
Ampliative Inference ◽  
Principle Of Maximum

It is sometimes necessary to guess which probability distribution governs random sampling over a given event space. When the correct guess cannot be deduced from information available about the space, the problem is said to require ampliative inference. The most familiar form of ampliative inference is represented in the principle of maximum entropy. We examine this principle from a descriptive, psychological point of view.

scholarly journals Stochastic Signal Processing for Sound Environment System with Decibel Evaluation and Energy Observation

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Akira Ikuta ◽  
Hisako Orimoto
Keyword(s):  
Signal Processing ◽  
Probability Distribution ◽  
Linear Correlation ◽  
External Noise ◽  
Structural Method ◽  
Observation Data ◽  
Stochastic Signal ◽  
Input And Output ◽  
Sound Environment ◽  
Stochastic Signal Processing

In real sound environment system, a specific signal shows various types of probability distribution, and the observation data are usually contaminated by external noise (e.g., background noise) of non-Gaussian distribution type. Furthermore, there potentially exist various nonlinear correlations in addition to the linear correlation between input and output time series. Consequently, often the system input and output relationship in the real phenomenon cannot be represented by a simple model using only the linear correlation and lower order statistics. In this study, complex sound environment systems difficult to analyze by using usual structural method are considered. By introducing an estimation method of the system parameters reflecting correlation information for conditional probability distribution under existence of the external noise, a prediction method of output response probability for sound environment systems is theoretically proposed in a suitable form for the additive property of energy variable and the evaluation in decibel scale. The effectiveness of the proposed stochastic signal processing method is experimentally confirmed by applying it to the observed data in sound environment systems.

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