scholarly journals Relationship among Continuous Probability Distributions and Interpolation

2021 ◽  
pp. 196-210
Author(s):  
Kelachi P. Enwere ◽  
Uchenna P. Ogoke
Keyword(s):  
Probability Distribution ◽  
Probability Distributions ◽  
Linear Interpolation ◽  
Data Set ◽  
Statistical Probability ◽  
Practical Applications ◽  
Gamma Distributions ◽  
Chi Squared ◽  
Interpolation Techniques ◽  
The Difference

Aims: The Study seeks to determine the relationship that exists among Continuous Probability Distributions and the use of Interpolation Techniques to estimate unavailable but desired value of a given probability distribution. Study Design: Statistical Probability tables for Normal, Student t, Chi-squared, F and Gamma distributions were used to compare interpolated values with statistical tabulated values. Charts and Tables were used to represent the relationships among the five probability distributions. Methodology: Linear Interpolation Technique was employed to interpolate unavailable but desired values so as to obtain approximate values from the statistical tables. The data were analyzed for interpolation of unavailable but desired values at 95% a-level from the five continuous probability distribution. Results: Interpolated values are as close as possible to the exact values and the difference between the exact value and the interpolated value is not pronounced. The table and chart established showed that relationships do exist among the Normal, Student-t, Chi-squared, F and Gamma distributions. Conclusion: Interpolation techniques can be applied to obtain unavailable but desired information in a data set. Thus, uncertainty found in a data set can be discovered, then analyzed and interpreted to produce desired results. However, understanding of how these probability distributions are related to each other can inform how best these distributions can be used interchangeably by Statisticians and other Researchers who apply statistical methods employed in practical applications.

BOUNDS ON EXTROPY WITH VARIATIONAL DISTANCE CONSTRAINT

2018 ◽  
Vol 33 (2) ◽  
pp. 186-204 ◽  
Author(s):  
Jianping Yang ◽  
Wanwan Xia ◽  
Taizhong Hu
Keyword(s):  
Probability Distribution ◽  
Confidence Interval ◽  
Upper Bound ◽  
Probability Distributions ◽  
Analytic Formula ◽  
Distance Constraint ◽  
Variation Distance ◽  
The Difference ◽  
Variational Distance

The relation between extropy and variational distance is studied in this paper. We determine the distribution which attains the minimum or maximum extropy among these distributions within a given variation distance from any given probability distribution, obtain the tightest upper bound on the difference of extropies of any two probability distributions subject to the variational distance constraint, and establish an analytic formula for the confidence interval of an extropy. Such a study parallels to that of Ho and Yeung [3] concerning entropy. However, the proofs of the main results in this paper are different from those in Ho and Yeung [3]. In fact, our arguments can simplify several proofs in Ho and Yeung [3].

Shannon information as a measure of distortion in coordination polyhedra

2005 ◽  
Vol 38 (1) ◽  
pp. 152-157 ◽  
Author(s):  
Erwin Lalik
Keyword(s):  
Probability Distribution ◽  
Uniform Distribution ◽  
Information Content ◽  
Distortion Function ◽  
Shannon Information ◽  
Bond Orders ◽  
Coordination Polyhedra ◽  
Chi Squared ◽  
Real Value ◽  
The Difference

Variations in the distribution of bond orders within coordination polyhedra in crystals have been used as a scale of departure from perfect polyhedral symmetry and a measure of polyhedral distortion. The distribution of bond orders mathematically resembles the probability distribution; the bond orders can be normalized to their sum within a polyhedron and used to calculate the Shannon information content that depends on the degree of their departure from uniform distribution. The difference in the Shannon information between a perfect and a distorted polyhedron can be defined as a function taking bond orders as arguments to yield a non-negative real value unambiguously ascribed to a polyhedron. The so-defined distortion function is continuous and vanishes for all the bond orders being equal in the case of a perfect polyhedron. For small distortions it can be reduced to a form analogous to the well known chi-squared function.

Angle-domain common-image gathers from anisotropic migration

Geophysics ◽  
2007 ◽  
Vol 72 (2) ◽  
pp. S81-S91 ◽  
Author(s):  
Biondo Biondi
Keyword(s):  
Complex I ◽  
Theoretical Development ◽  
Isotropic Case ◽  
Data Set ◽  
Practical Applications ◽  
Wavefield Continuation ◽  
The Difference ◽  
Contrast Group ◽  
Vti Media ◽  
The Relationship

I present a general methodology for computing angle-domain common-image gathers (ADCIGs) in conjunction with anisotropic wavefield-continuation migration. The method is based on transforming the prestack image from the subsurface-offset domain to the angle domain using slant stacks. The processing sequence is the same as that for computing ADCIGs for the isotropic case, though the interpretation of the relationship between the slopes measured in the prestack image and the aperture angles is more complex. I demonstrate that the slopes measured by performing slant stacks along the subsurface-offset axis of the prestack image provide a good approximation of the phase aperture angles, and they are exactly equal to the phase aperture angles for flat reflectors in vertical transversly isotropic (VTI) media. In the general case of dipping reflectors, the angles computed using slant stacks can be easily corrected by applying the relationships that I present in this paper, and the accurate aperture angles can be determined as a function of the reflector dip and anisotropic slowness at the reflector. I derive these relationships from both plane-wave and ray viewpoints. This theoretical development links the kinematics in ADCIGs with migration-velocity errors. I apply the proposed method to compute ADCIGs from the prestack image obtained by anisotropic migration of a 2D line recorded in the Gulf of Mexico. I analyze the error introduced by neglecting the difference between the true phase aperture angle and the angle computed through slant stacks, showing that, at least for this data set, these errors are negligible and can be safely ignored. In contrast, group aperture angles can be quite different from phase aperture angles; thus, ignoring the distinction between these two angles can be detrimental to practical applications.

Study of Cubic Splines and Fourier Series as Interpolation Techniques for Filling in Short Periods of Missing Building Energy Use and Weather Data

Solar Energy ◽  
2002 ◽  
Author(s):  
Juan-Carlos Baltazar ◽  
David E. Claridge
Keyword(s):  
Time Series ◽  
Fourier Series ◽  
Energy Use ◽  
Linear Interpolation ◽  
Cubic Splines ◽  
Dew Point ◽  
Weather Data ◽  
Data Set ◽  
Data Points ◽  
Interpolation Techniques

A study of cubic splines and Fourier series as interpolation techniques for filling in missing data in energy and meteorological time series is presented. The followed procedure created artificially missing points (pseudo-gaps) in measured data sets and was based on the local behavior of the data set around those pseudo-gaps. Five variants of the cubic spline technique and 12 variants of Fourier series were tested and compared with linear interpolation, for filling in gaps of 1 to 6 hours of data in 20 samples of energy use and weather data. Each of the samples is at least one year in length. The analysis showed that linear interpolation is superior to the spline and Fourier series techniques for filling in 1–6 hour gaps in time series dry bulb and dew point temperature data. For filling 1–6 hour gaps in building cooling and heating use, the Fourier series approach with 24 data points before and after each gap and six constants was found to be the most suitable. In cases where there are insufficient data points for the application of this approach, simple linear interpolation is recommended.

scholarly journals Probability Distribution of Philippine Daily Rainfall Data

2017 ◽  
Author(s):  
Vanessa Althea B. Bermudez ◽  
Ariel Bettina B. Abilgos ◽  
Diane Carmeliza N. Cuaresma ◽  
Jomar F. Rabajante
Keyword(s):  
Probability Distribution ◽  
Probability Distributions ◽  
Daily Rainfall ◽  
Rainfall Data ◽  
Economic Systems ◽  
Tropical Country ◽  
Data Set ◽  
Cumulative Rainfall ◽  
The Pacific ◽  
The Pacific Ocean

Philippines as an archipelago and tropical country, which is situated near the Pacific ocean, faces uncertain rainfall intensities. This makes environmental, agricultural and economic systems affected by precipitation difficult to manage. Time series analysis of Philippine rainfall pattern has been previously done, but there is no study investigating its probability distribution. Modeling the Philippine rainfall using probability distributions is essential, especially in managing risks and designing insurance products. Here, daily and cumulative rainfall data (January 1961 - August 2016) from 28 PAGASA weather stations are fitted to probability distributions. Moreover, the fitted distributions are examined for invariance under subsets of the rainfall data set. We observe that the Gamma distribution is a suitable fit for the daily up to the ten-day cumulative rainfall data. Our results can be used in agriculture, especially in forecasting claims in weather index-based insurance.

scholarly journals Maximum rainfall probability distributions pattern in Haryana –A case study

2016 ◽  
Vol 8 (4) ◽  
pp. 2029-2036
Author(s):  
Manoj Kumar ◽  
Chander Shekhar ◽  
Veena Manocha
Keyword(s):  
Probability Distribution ◽  
Coefficient Of Variation ◽  
Probability Distributions ◽  
Monsoon Season ◽  
Daily Rainfall ◽  
Extreme Value Distribution ◽  
Generalized Extreme Value Distribution ◽  
Data Set ◽  
Maximum Rainfall ◽  
Maximum Daily Rainfall

The present study has been undertaken to fit best probability distribution of rainfall in Ambala District of Haryana State. The analysis showed that the maximum daily rainfall among the years ranged between 41mm (1980) to 307.9mm (2009) indicating a very large variation during the period of study. The mean of maximum daily rainfall of all years annually is 112.13mm. The means of monthly and weekly values ranged from 33.10-88.92mm and 8.77- 46.28 mm, respectively. The maximum daily rainfall in a year/monsoon season was307.9 mm and monthly maximum daily rainfall in monsoon season ranged from 105 -307.9mm. The weekly maximum daily rainfall ranged from48 mm-307.9 mm. It was also observed that the minimum among the maximum daily rainfall was 41mm for annual, 34mm for season and 0 in all the months and weeks. The maximum value of coefficient of variation was observed in the first week which indicated a large fluctuation in the rainfall data set and minimum value of coefficient of variation 0.464 was observed for the whole year which shows that fluctuation was minimum for the whole year. Generalized extreme value distribution was found to be best fit probability distribution for most of the periods.

scholarly journals A Discussion on Simple Support Functions

2014 ◽  
Vol 10 ◽  
pp. 13-21
Author(s):  
D.N. Kandekar
Keyword(s):  
Decision Making ◽  
Probability Density Function ◽  
Probability Distribution ◽  
Support Function ◽  
Daily Life ◽  
Probability Distributions ◽  
Support Functions ◽  
Statistical Probability ◽  
Helpful Tool ◽  
Simple Support

Prediction about each and every incident happening in our daily life is impossible. But we can predict about some incidents. Probability is most helpful tool in predicting about outcomes or conclusions of such incidents. Such incidents happened in our life, always follow some known or unknown statistical probability distribution which may consist of simple or complicated probability density function. Therefore with help of probability distributions, we get some blurred idea about the functioning of incidents happening in our life. Using some commonly used probability distributions, we obtain conclusions which are helpful in decision making. Support functions viz. simple support functions are very useful in decision making. In this paper, we quote some results and applications regarding simple support function based on probability transformations.

scholarly journals LONG TERMS AND READABILITY OF PHYSICS SCHOOL TEXT

2017 ◽  
Vol 5 ◽  
pp. 815-819
Author(s):  
Ivana Ĺ korecová ◽  
Aba Teleki ◽  
Ľubomír Zelenický
Keyword(s):  
Probability Distribution ◽  
Strong Correlation ◽  
Test Scores ◽  
Probability Distributions ◽  
Survival Function ◽  
Survival Functions ◽  
School Text ◽  
The Difference ◽  
Random Fluctuations

This article presents a comparison of two physics school texts from the perspective of readability and use of specific terms. The study uses the survival function to associate the readability of physics school text to the length of terms used in the text. First, the study compares the survival functions of two full texts and that of the terms in these texts, and then analyzes the associated relative readability. Next, the results of two cloze tests involving 150 students are compared. The last step investigates the randomness of the differences between the results. The results show a strong correlation between the test scores and the probability distributions of terms used in the school texts. The difference between the probability distribution of the compared texts corresponds with the differences between the appropriate survival functions, where random fluctuations in the frequency of terms are suppressed.

scholarly journals The Riccati-Bernoulli Sub-ODE Technique for Solving the Deterministic (Stochastic) Generalized-Zakharov System

2018 ◽  
Vol 1 (3) ◽  
Author(s):  
Mahmoud A.E. Abdelrahman ◽  
M. A. Sohaly
Keyword(s):  
Traveling Wave ◽  
Probability Distributions ◽  
Random Solution ◽  
Statistical Probability ◽  
Zakharov System ◽  
The Difference ◽  
First Moment ◽  
Difference Effect

This article concerns  with  the  construction of the  analytical traveling  wave so- lutions  for the Generalized-Zakharov System  by the Riccati-Bernoulli Sub- ODE technique. Also, we will discuss this  technique  in random  case by using random  traveling  wave trans- formation  in order  to  find what  is the  effect of the  randomness  input  for this  technique. We presented the Generalized-Zakharov System as an example to show the difference effect between the deterministic and stochastic Riccati-Bernoulli Sub-ODE technique.  The first moment of random solution is computed for different statistical probability distributions.

scholarly journals Modified Likelihood Ratio Test for Sub-mean Vectors with Two-step Monotone Missing Data in Two-sample Problem

2021 ◽  
Vol 50 (1) ◽  
pp. 88-104
Author(s):  
Tamae Kawasaki ◽  
Takashi Seo
Keyword(s):  
Missing Data ◽  
Likelihood Ratio ◽  
Likelihood Ratio Test ◽  
Null Distribution ◽  
Linear Interpolation ◽  
Ratio Test ◽  
Data Set ◽  
Chi Squared ◽  
Mean Vectors ◽  
Modified Likelihood

This article deals with the problem of testing for two normal sub-mean vectors when the data set have two-step monotone missing observations. Under the assumptions that the population covariance matrices are equal, we obtain the likelihood ratio test (LRT) statistic. Furthermore, an asymptotic expansion for the null distribution of the LRT statistic is derived under the two-step monotone missing data by the perturbation method. Using the result, we propose two improved statistics with good chi-squared approximation. One is the modified LRT statistic by Bartlett correction,and the other is the modified LRT statistic using the modification coefficient by linear interpolation. The accuracy of the approximations are investigated by using a Monte Carlo simulation. The proposed methods are illustrated using an example.

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