scholarly journals Nonparametric Estimation of Continuously Parametrized Families of Probability Density Functions—Computational Aspects

Algorithms ◽  
2020 ◽  
Vol 13 (7) ◽  
pp. 164 ◽  
Author(s):  
Wojciech Rafajłowicz
Keyword(s):  
Probability Density ◽  
Nonparametric Estimation ◽  
Learning Algorithm ◽  
Estimation Procedure ◽  
Estimation Algorithm ◽  
Real Variable ◽  
Probability Density Functions ◽  
Density Functions ◽  
Computational Aspects ◽  
2D Fft

We consider a rather general problem of nonparametric estimation of an uncountable set of probability density functions (p.d.f.’s) of the form: f ( x ; r ) , where r is a non-random real variable and ranges from R 1 to R 2 . We put emphasis on the algorithmic aspects of this problem, since they are crucial for exploratory analysis of big data that are needed for the estimation. A specialized learning algorithm, based on the 2D FFT, is proposed and tested on observations that allow for estimate p.d.f.’s of a jet engine temperatures as a function of its rotation speed. We also derive theoretical results concerning the convergence of the estimation procedure that contains hints on selecting parameters of the estimation algorithm.

scholarly journals An empirical method for estimating probability density functions of gridded daily minimum and maximum temperature

2013 ◽  
Vol 10 (1) ◽  
pp. 59-64
Author(s):  
C. Lussana
Keyword(s):  
Probability Density ◽  
Extreme Values ◽  
Estimation Procedure ◽  
Empirical Method ◽  
Probability Density Functions ◽  
Optimal Interpolation ◽  
Maximum Temperature ◽  
Density Functions ◽  
Daily Minimum ◽  
Time Period

Abstract. The presented work focuses on the investigation of gridded daily minimum (TN) and maximum (TX) temperature probability density functions (PDFs) with the intent of both characterising a region and detecting extreme values. The empirical PDFs estimation procedure has been realised using the most recent years of gridded temperature analysis fields available at ARPA Lombardia, in Northern Italy. The spatial interpolation is based on an implementation of Optimal Interpolation using observations from a dense surface network of automated weather stations. An effort has been made to identify both the time period and the spatial areas with a stable data density otherwise the elaboration could be influenced by the unsettled station distribution. The PDF used in this study is based on the Gaussian distribution, nevertheless it is designed to have an asymmetrical (skewed) shape in order to enable distinction between warming and cooling events. Once properly defined the occurrence of extreme events, it is possible to straightforwardly deliver to the users the information on a local-scale in a concise way, such as: TX extremely cold/hot or TN extremely cold/hot.

scholarly journals Modeling Diameter Distributions with Six Probability Density Functions in Pinus halepensis Mill. Plantations Using Low-Density Airborne Laser Scanning Data in Aragón (Northeast Spain)

2021 ◽  
Vol 13 (12) ◽  
pp. 2307
Author(s):  
J. Javier Gorgoso-Varela ◽  
Rafael Alonso Ponce ◽  
Francisco Rodríguez-Puerta
Keyword(s):  
Probability Density ◽  
Linear Models ◽  
Pinus Halepensis ◽  
Beta Function ◽  
Probability Density Functions ◽  
Lidar Data ◽  
Density Functions ◽  
Minimum Diameter ◽  
Location Parameters ◽  
Diameter Distributions

The diameter distributions of trees in 50 temporary sample plots (TSPs) established in Pinus halepensis Mill. stands were recovered from LiDAR metrics by using six probability density functions (PDFs): the Weibull (2P and 3P), Johnson’s SB, beta, generalized beta and gamma-2P functions. The parameters were recovered from the first and the second moments of the distributions (mean and variance, respectively) by using parameter recovery models (PRM). Linear models were used to predict both moments from LiDAR data. In recovering the functions, the location parameters of the distributions were predetermined as the minimum diameter inventoried, and scale parameters were established as the maximum diameters predicted from LiDAR metrics. The Kolmogorov–Smirnov (KS) statistic (Dn), number of acceptances by the KS test, the Cramér von Misses (W2) statistic, bias and mean square error (MSE) were used to evaluate the goodness of fits. The fits for the six recovered functions were compared with the fits to all measured data from 58 TSPs (LiDAR metrics could only be extracted from 50 of the plots). In the fitting phase, the location parameters were fixed at a suitable value determined according to the forestry literature (0.75·dmin). The linear models used to recover the two moments of the distributions and the maximum diameters determined from LiDAR data were accurate, with R2 values of 0.750, 0.724 and 0.873 for dg, dmed and dmax. Reasonable results were obtained with all six recovered functions. The goodness-of-fit statistics indicated that the beta function was the most accurate, followed by the generalized beta function. The Weibull-3P function provided the poorest fits and the Weibull-2P and Johnson’s SB also yielded poor fits to the data.

scholarly journals A synergy of the velocity gradients technique and the probability density functions for identifying gravitational collapse in self-absorbing media

2021 ◽  
Vol 502 (2) ◽  
pp. 1768-1784
Author(s):  
Yue Hu ◽  
A Lazarian
Keyword(s):  
Magnetic Fields ◽  
Probability Density ◽  
Column Density ◽  
Mass Density ◽  
Probability Density Functions ◽  
Emission Lines ◽  
Density Functions ◽  
Star Forming ◽  
Velocity Gradients ◽  
Self Gravity

ABSTRACT The velocity gradients technique (VGT) and the probability density functions (PDFs) of mass density are tools to study turbulence, magnetic fields, and self-gravity in molecular clouds. However, self-absorption can significantly make the observed intensity different from the column density structures. In this work, we study the effects of self-absorption on the VGT and the intensity PDFs utilizing three synthetic emission lines of CO isotopologues 12CO (1–0), 13CO (1–0), and C18O (1–0). We confirm that the performance of VGT is insensitive to the radiative transfer effect. We numerically show the possibility of constructing 3D magnetic fields tomography through VGT. We find that the intensity PDFs change their shape from the pure lognormal to a distribution that exhibits a power-law tail depending on the optical depth for supersonic turbulence. We conclude the change of CO isotopologues’ intensity PDFs can be independent of self-gravity, which makes the intensity PDFs less reliable in identifying gravitational collapsing regions. We compute the intensity PDFs for a star-forming region NGC 1333 and find the change of intensity PDFs in observation agrees with our numerical results. The synergy of VGT and the column density PDFs confirms that the self-gravitating gas occupies a large volume in NGC 1333.

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