Rate of Convergence in Density Estimation Using Neural Networks

1996 ◽  
Vol 8 (5) ◽  
pp. 1107-1122 ◽  
Author(s):  
Dharmendra S. Modha ◽  
Elias Masry
Keyword(s):  
Neural Networks ◽  
Probability Density Function ◽  
Probability Density ◽  
Rate Of Convergence ◽  
Density Function ◽  
Density Estimation ◽  
Exponential Family ◽  
Density Estimator ◽  
Estimation Scheme ◽  
Hidden Layer

Given N i.i.d. observations {Xi}Ni=1 taking values in a compact subset of Rd, such that p* denotes their common probability density function, we estimate p* from an exponential family of densities based on single hidden layer sigmoidal networks using a certain minimum complexity density estimation scheme. Assuming that p* possesses a certain exponential representation, we establish a rate of convergence, independent of the dimension d, for the expected Hellinger distance between the proposed minimum complexity density estimator and the true underlying density p*.

On the linear exponential family

1968 ◽  
Vol 64 (2) ◽  
pp. 481-483 ◽  
Author(s):  
J. K. Wani
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Scalar Product ◽  
Exponential Family ◽  
Characterization Theorem ◽  
Random Variable ◽  
Moment Generating Function ◽  
The Moment ◽  
Image Position

In this paper we give a characterization theorem for a subclass of the exponential family whose probability density function is given bywhere a(x) ≥ 0, f(ω) = ∫a(x) exp (ωx) dx and ωx is to be interpreted as a scalar product. The random variable X may be an s-vector. In that case ω will also be an s-vector. For obvious reasons we will call (1) as the linear exponential family. It is easy to verify that the moment generating function (m.g.f.) of (1) is given by

scholarly journals A numerical study of entropy and residual entropy estimators based on smooth density estimators for non-negative random variables

2021 ◽  
Vol 54 (2) ◽  
pp. 99-121
Author(s):  
Yogendra P. Chaubey ◽  
Nhat Linh Vu
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Numerical Study ◽  
Random Variables ◽  
Random Variable ◽  
Residual Entropy ◽  
Density Estimator ◽  
Wide Range ◽  
Smooth Density

In this paper, we are interested in estimating the entropy of a non-negative random variable. Since the underlying probability density function is unknown, we propose the use of the Poisson smoothed histogram density estimator to estimate the entropy. To study the per- formance of our estimator, we run simulations on a wide range of densities and compare our entropy estimators with the existing estimators based on different approaches such as spacing estimators. Furthermore, we extend our study to residual entropy estimators which is the entropy of a random variable given that it has been survived up to time $t$.

ROBUST LOCATION AND SPREAD MEASURES FOR NONPARAMETRIC PROBABILITY DENSITY FUNCTION ESTIMATION

2009 ◽  
Vol 19 (05) ◽  
pp. 345-357 ◽  
Author(s):  
EZEQUIEL LÓPEZ-RUBIO
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Covariance Matrix ◽  
Density Estimation ◽  
Function Estimation ◽  
Sample Mean ◽  
Sample Covariance ◽  
Density Function Estimation ◽  
Probability Density Function Pdf

Robustness against outliers is a desirable property of any unsupervised learning scheme. In particular, probability density estimators benefit from incorporating this feature. A possible strategy to achieve this goal is to substitute the sample mean and the sample covariance matrix by more robust location and spread estimators. Here we use the L 1-median to develop a nonparametric probability density function (PDF) estimator. We prove its most relevant properties, and we show its performance in density estimation and classification applications.

scholarly journals Shannon Entropy Estimation for Linear Processes

2020 ◽  
Vol 13 (9) ◽  
pp. 205
Author(s):  
Timothy Fortune ◽  
Hailin Sang
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Shannon Entropy ◽  
Kernel Density ◽  
Linear Process ◽  
Kernel Density Estimator ◽  
Density Estimator ◽  
Linear Processes ◽  
Entropy Estimation

In this paper, we estimate the Shannon entropy S(f)=−E[log(f(x))] of a one-sided linear process with probability density function f(x). We employ the integral estimator Sn(f), which utilizes the standard kernel density estimator fn(x) of f(x). We show that Sn(f) converges to S(f) almost surely and in Ł2 under reasonable conditions.

Structural health monitoring by probability density function of autoregressive-based damage features and fast distance correlation method

2021 ◽  
pp. 107754632110201
Author(s):  
Mohammad Ali Heravi ◽  
Seyed Mehdi Tavakkoli ◽  
Alireza Entezami
Keyword(s):  
Time Series ◽  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Large Scale ◽  
Hypothesis Test ◽  
Correlation Method ◽  
Model Parameters ◽  
Density Estimator ◽  
Distance Correlation

In this article, the autoregressive time series analysis is used to extract reliable features from vibration measurements of civil structures for damage diagnosis. To guarantee the adequacy and applicability of the time series model, Leybourne–McCabe hypothesis test is used. Subsequently, the probability density functions of the autoregressive model parameters and residuals are obtained with the aid of a kernel density estimator. The probability density function sets are considered as damage-sensitive features of the structure and fast distance correlation method is used to make decision for detecting damages in the structure. Experimental data of a well-known three-story laboratory frame and a large-scale bridge benchmark structure are used to verify the efficiency and accuracy of the proposed method. Results indicate the capability of the method to identify the location and severity of damages, even under the simulated operational and environmental variability.

Fully Nonparametric Probability Density Function Estimation Based on Empirical Mode Decomposition

2012 ◽  
Vol 460 ◽  
pp. 189-192
Author(s):  
Hong Ying Hu ◽  
Chun Ming Kan
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Density Estimation ◽  
Empirical Mode Decomposition ◽  
Estimation Method ◽  
Function Estimation ◽  
Mode Decomposition ◽  
Non Stationary Signal ◽  
Density Function Estimation

Empirical Mode Decomposition (EMD) is a non-stationary signal processing method developed recently. It has been applied in many engineering fields. EMD has many similarities with wavelet decomposition. But EMD Decomposition has its own characteristics, especially in accurate rend extracting. Therefore the paper firstly proposes an algorithm of extracting slow-varying trend based on EMD. Then, according to wavelet probability density function estimation method, a new density estimation method based on EMD is presented. The simulations of Gaussian single and mixture model density estimation prove the advantages of the approach with easy computation and more accurate result

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