This document describes the Kumaraswamy generalized (Kw-G) distribution, a new family of continuous probability distributions defined on the interval (0,1). The Kw-G distribution is constructed by applying the Kumaraswamy distribution to an existing parent distribution with cumulative distribution function G(x). Properties of the Kw-G distribution such as its probability density function, moments, order statistics, and L-moments are expressed in terms of the parent distribution G(x). Several special cases of the Kw-G distribution are also discussed, including the Kw-normal, Kw-Weibull, and Kw-gamma distributions.
The document discusses a new family of generalized distributions called the Kumaraswamy distributions (Kw-G distributions). These distributions extend common distributions like the normal, Weibull, and gamma distributions by introducing additional shape parameters. The key properties of the Kw-G distributions are:
1) They are defined on the interval (0,1) and are derived by applying a quantile function to a parent continuous distribution G(x).
2) Important special cases are the Kw-normal, Kw-Weibull, and Kw-gamma distributions.
3) Moments and probability weighted moments of the Kw-G distributions can be written as infinite weighted sums of the moments of the parent G distribution.
DISTANCE TWO LABELING FOR MULTI-STOREY GRAPHSgraphhoc
An L (2, 1)-labeling of a graph G (also called distance two labeling) is a function f from the vertex set V (G) to the non negative integers {0,1,…, k }such that |f(x)-f(y)| ≥2 if d(x, y) =1 and | f(x)- f(y)| ≥1 if d(x, y) =2. The L (2, 1)-labeling number λ (G) or span of G is the smallest k such that there is a f with
max {f (v) : vє V(G)}= k. In this paper we introduce a new type of graph called multi-storey graph. The distance two labeling of multi-storey of path, cycle, Star graph, Grid, Planar graph with maximal edges and its span value is determined. Further maximum upper bound span value for Multi-storey of simple
graph are discussed.
The document discusses error analysis for quasi-Monte Carlo methods used for numerical integration. It introduces the concepts of reproducing kernel Hilbert spaces and mean square discrepancy to analyze integration error. Specifically, it shows that the mean square discrepancy of randomized low-discrepancy point sets can be computed in O(n) operations, whereas the standard discrepancy requires O(n^2) operations, making randomized quasi-Monte Carlo methods more efficient for high-dimensional integration problems.
The document discusses characteristics of (γ, 3)-critical graphs. It begins by providing examples of (γ, 3)-critical graphs, such as the circulant graph C12 1, 4 and the Cartesian product Kt Kt . It then shows that a (γ, k)-critical graph is not necessarily (γ, k′)-critical for k ≠ k′ between 1 and 3. The document also verifies properties of (γ, 3)-critical graphs, such as not having vertices of degree 3. It concludes by proving characteristics about (γ, 3)-critical graphs that are paths, including that they have no vertices in V+ and satisfy other properties.
This document proposes a theoretical framework for analyzing the probability of successful decoding in single-relay networks using network coding. It defines key terms like random linear network coding and presents two theorems:
1) The probability that two randomly generated coding matrices at a source and relay are simultaneously full rank is given by a formula involving the dimensions and number of common rows of the matrices.
2) The probability of successful decoding at two destinations in a network defined by certain parameters is calculated as the sum of probabilities involving the coding matrices and dimensions at each stage of transmission through the source, relay, and destinations.
Numerical results are presented to validate the theoretical analysis.
This document presents a space-time diagram in the unit disk that represents events, worldlines, and inertial motion in a 4-dimensional space-time. It defines variables to represent spatial and temporal intervals between events using cross ratios and angles in the disk. Calculations show that locally, this representation satisfies the Minkowski metric and derives the velocity addition law and time dilation formula. The diagram separates space-time into regions for matter, antimatter, and tachyons. Appendices introduce related non-Euclidean models and an application to conformal optics.
1. The displacement field for a body is given. The displaced position of a point originally at (1, 2, 3) is calculated.
2. The strain matrix and strain in a given direction are calculated for a given displacement field.
3. The strains are calculated at a point for a given displacement field in several directions.
Tailored Bregman Ball Trees for Effective Nearest NeighborsFrank Nielsen
This document presents an improved Bregman ball tree (BB-tree++) for efficient nearest neighbor search using Bregman divergences. The BB-tree++ speeds up construction using Bregman 2-means++ initialization and adapts the branching factor. It also handles symmetrized Bregman divergences and prioritizes closer nodes. Experiments on image retrieval with SIFT descriptors show the BB-tree++ outperforms the original BB-tree and random sampling, providing faster approximate nearest neighbor search.
The document discusses a new family of generalized distributions called the Kumaraswamy distributions (Kw-G distributions). These distributions extend common distributions like the normal, Weibull, and gamma distributions by introducing additional shape parameters. The key properties of the Kw-G distributions are:
1) They are defined on the interval (0,1) and are derived by applying a quantile function to a parent continuous distribution G(x).
2) Important special cases are the Kw-normal, Kw-Weibull, and Kw-gamma distributions.
3) Moments and probability weighted moments of the Kw-G distributions can be written as infinite weighted sums of the moments of the parent G distribution.
DISTANCE TWO LABELING FOR MULTI-STOREY GRAPHSgraphhoc
An L (2, 1)-labeling of a graph G (also called distance two labeling) is a function f from the vertex set V (G) to the non negative integers {0,1,…, k }such that |f(x)-f(y)| ≥2 if d(x, y) =1 and | f(x)- f(y)| ≥1 if d(x, y) =2. The L (2, 1)-labeling number λ (G) or span of G is the smallest k such that there is a f with
max {f (v) : vє V(G)}= k. In this paper we introduce a new type of graph called multi-storey graph. The distance two labeling of multi-storey of path, cycle, Star graph, Grid, Planar graph with maximal edges and its span value is determined. Further maximum upper bound span value for Multi-storey of simple
graph are discussed.
The document discusses error analysis for quasi-Monte Carlo methods used for numerical integration. It introduces the concepts of reproducing kernel Hilbert spaces and mean square discrepancy to analyze integration error. Specifically, it shows that the mean square discrepancy of randomized low-discrepancy point sets can be computed in O(n) operations, whereas the standard discrepancy requires O(n^2) operations, making randomized quasi-Monte Carlo methods more efficient for high-dimensional integration problems.
The document discusses characteristics of (γ, 3)-critical graphs. It begins by providing examples of (γ, 3)-critical graphs, such as the circulant graph C12 1, 4 and the Cartesian product Kt Kt . It then shows that a (γ, k)-critical graph is not necessarily (γ, k′)-critical for k ≠ k′ between 1 and 3. The document also verifies properties of (γ, 3)-critical graphs, such as not having vertices of degree 3. It concludes by proving characteristics about (γ, 3)-critical graphs that are paths, including that they have no vertices in V+ and satisfy other properties.
This document proposes a theoretical framework for analyzing the probability of successful decoding in single-relay networks using network coding. It defines key terms like random linear network coding and presents two theorems:
1) The probability that two randomly generated coding matrices at a source and relay are simultaneously full rank is given by a formula involving the dimensions and number of common rows of the matrices.
2) The probability of successful decoding at two destinations in a network defined by certain parameters is calculated as the sum of probabilities involving the coding matrices and dimensions at each stage of transmission through the source, relay, and destinations.
Numerical results are presented to validate the theoretical analysis.
This document presents a space-time diagram in the unit disk that represents events, worldlines, and inertial motion in a 4-dimensional space-time. It defines variables to represent spatial and temporal intervals between events using cross ratios and angles in the disk. Calculations show that locally, this representation satisfies the Minkowski metric and derives the velocity addition law and time dilation formula. The diagram separates space-time into regions for matter, antimatter, and tachyons. Appendices introduce related non-Euclidean models and an application to conformal optics.
1. The displacement field for a body is given. The displaced position of a point originally at (1, 2, 3) is calculated.
2. The strain matrix and strain in a given direction are calculated for a given displacement field.
3. The strains are calculated at a point for a given displacement field in several directions.
Tailored Bregman Ball Trees for Effective Nearest NeighborsFrank Nielsen
This document presents an improved Bregman ball tree (BB-tree++) for efficient nearest neighbor search using Bregman divergences. The BB-tree++ speeds up construction using Bregman 2-means++ initialization and adapts the branching factor. It also handles symmetrized Bregman divergences and prioritizes closer nodes. Experiments on image retrieval with SIFT descriptors show the BB-tree++ outperforms the original BB-tree and random sampling, providing faster approximate nearest neighbor search.
From planar maps to spatial topology change in 2d gravityTimothy Budd
The document summarizes a talk on generalized causal dynamical triangulations (CDT) in two dimensions. It introduces the generalized CDT model, which allows spatial topology to change in time. It describes how generalized CDT can be solved by viewing causal quadrangulations as labeled trees. It also discusses bijections between labeled quadrangulations and labeled planar maps via Schaeffer's algorithm, which allow counting the numbers of objects in the generalized CDT model.
The document discusses shortest path algorithms for weighted graphs. It introduces Dijkstra's algorithm and the Bellman-Ford algorithm for finding shortest paths. Dijkstra's algorithm works for graphs with non-negative edge weights, while Bellman-Ford can handle graphs with negative edge weights. The document also describes how to find shortest paths in directed acyclic graphs and compute all-pairs shortest paths.
This document discusses Gaussian quadrature formulas, which approximate definite integrals of functions by using weighted sums of function values at specified points. It presents the one-point, two-point, and three-point Gaussian quadrature formulas. The one-point formula is exact for polynomials up to degree 1, the two-point formula is exact for polynomials up to degree 3, and the three-point formula is exact for polynomials up to degree 5. Examples are provided to demonstrate applying the formulas.
The document discusses minimum spanning trees (MST) and two algorithms for finding them: Prim's algorithm and Kruskal's algorithm. It begins by defining an MST as a spanning tree (connected acyclic graph containing all vertices) with minimum total edge weight. Prim's algorithm grows a single tree by repeatedly adding the minimum weight edge connecting the growing tree to another vertex. Kruskal's algorithm grows a forest by repeatedly merging two components via the minimum weight edge connecting them. Both algorithms produce optimal MSTs by adding only "safe" edges that cannot be part of a cycle.
The document summarizes the Frame-Stewart algorithm for solving the generalized Tower of Hanoi puzzle with n disks on k pegs. It begins by introducing the standard 3-peg Tower of Hanoi puzzle and recursive solution. It then describes Henry Dudeney's 4-peg variation and the Frame-Stewart algorithm from 1939 for solving the problem with n disks on any number of pegs k. The algorithm uses recursion and finding the optimal partition of disks to minimize the number of moves. The document proves properties of the number of additional moves between problems sizes.
This document summarizes the derivation of an evidence lower bound (ELBO) for latent LSTM allocation, a model that uses an LSTM to determine topic assignments in a topic modeling framework. It expresses the ELBO as terms related to the variational posterior distributions over topics and topics proportions, the generative process of words given topics, and the LSTM's prediction of topic assignments. It also describes how to optimize the ELBO with respect to the variational and LSTM parameters through gradient ascent.
A total dominating set D of graph G = (V, E) is a total strong split dominating set if the induced subgraph < V-D > is totally disconnected with atleast two vertices. The total strong split domination number γtss(G) is the minimum cardinality of a total strong split dominating set. In this paper, we characterize total strong split dominating sets and obtain the exact values of γtss(G) for some graphs. Also some inequalities of γtss(G) are established.
TopicRNN is a generative model for documents that:
1. Draws a topic vector from a standard normal distribution and uses it to generate words in a document.
2. Computes a lower bound on the log marginal likelihood of words and stop word indicators.
3. Approximates the expected values in the lower bound using samples from an inference network that models the approximate posterior distribution over topics.
Patch Matching with Polynomial Exponential Families and Projective DivergencesFrank Nielsen
This document presents a method called Polynomial Exponential Family-Patch Matching (PEF-PM) to solve the patch matching problem. PEF-PM models patch colors using polynomial exponential families (PEFs), which are universal smooth positive densities. It estimates PEFs using a Score Matching Estimator and accelerates batch estimation using Summed Area Tables. Patch similarity is measured using a statistical projective divergence called the symmetrized γ-divergence. Experiments show PEF-PM handles noise robustly, symmetries, and outperforms baseline methods.
The document presents an algorithm to find an optimal L(2,1)-labeling for triangular windmill graphs. It begins with definitions of triangular windmill graphs, L(2,1)-labelings, and the Chang-Kuo algorithm. The Chang-Kuo algorithm is then applied to obtain an L(2,1)-labeling of a triangular windmill graph W(3,n) by iteratively finding and labeling maximal 2-stable sets. The maximum label used is the labeling number λ(G).
This document discusses subspace clustering with missing data. It summarizes two algorithms for solving this problem: 1) an EM-type algorithm that formulates the problem probabilistically and iteratively estimates the subspace parameters using an EM approach. 2) A k-means form algorithm called k-GROUSE that alternates between assigning vectors to subspaces based on projection residuals and updating each subspace using incremental gradient descent on the Grassmannian manifold. It also discusses the sampling complexity results from a recent paper, showing subspace clustering is possible without an impractically large sample size.
The document introduces new classes of odd graceful graphs called m-shadow graphs and m-splitting graphs. It proves that m-shadow graphs of paths, complete bipartite graphs, and symmetric products of paths and null graphs are odd graceful. It also proves that m-splitting graphs of paths, stars, and symmetric products of paths and null graphs are odd graceful. Examples are provided to illustrate the theories.
The student reflects on completing a math project for their calculus course as a way to study for an upcoming exam. They acknowledge that they procrastinated significantly but were able to cover a broad range of calculus concepts through multi-step word problems selected from different units. While the assignment did not dramatically increase their knowledge, it helped reinforce some details and connections between topics. The student resolves to select deadlines more wisely and stop procrastinating for future projects.
Generalized CDT as a scaling limit of planar mapsTimothy Budd
Generalized causal dynamical triangulations (generalized CDT) is a model of two-dimensional quantum gravity in which a limited number of spatial topology changes is allowed to occur. After identifying the model as a scaling limit of random quadrangulations, I will show how it can be solved using a bijection between quadrangulations and trees. Another bijection relating quadrangulations to planar maps allows us to interpret generalized CDT as a scaling limit or random planar maps with a restriction on the number of faces. Finally I will show how this interpretation clarifies certain mysterious identities in generalized CDT amplitudes. (This talk is largely based on arXiv:1302.1763.)
The document describes a discrete-time Kalman filter implemented in MATLAB to estimate the position of an underwater vehicle using sensor measurements. It presents the state space modeling equations used in the filter, including modifying the state vector to address non-linearities in the direction measurement. Simulation results using a carefully designed trajectory show the filter provides estimates with errors generally within a few meters for position, a few centimeters for velocity bias, and a few meters for range over 1000 iterations.
This document discusses 2D geometric transformations, including translation, rotation, scaling, and their matrix representations using homogeneous coordinates. It provides the transformation equations and routines for translating, rotating, and scaling polygons. Key points covered include:
- The basic equations for 2D translation, rotation, and scaling
- Using matrix multiplication to represent sequences of transformations
- Expanding 2D coordinates to 3x3 homogeneous coordinates matrices
- The transformation matrices for translation, rotation, and scaling
- Calculating the inverse of transformation matrices
LADDER AND SUBDIVISION OF LADDER GRAPHS WITH PENDANT EDGES ARE ODD GRACEFULFransiskeran
The ladder graph plays an important role in many applications as Electronics, Electrical and Wireless
communication areas. The aim of this work is to present a new class of odd graceful labeling for the ladder
graph. In particular, we show that the ladder graph Ln with m-pendant Ln mk1 is odd graceful. We also
show that the subdivision of ladder graph Ln with m-pendant S(Ln) mk1 is odd graceful. Finally, we
prove that all the subdivision of triangular snakes ( k snake ) with pendant edges
1
( ) k S snake mk are odd graceful.
This document discusses graph algorithms and directed acyclic graphs (DAGs). It explains that the edges in a graph can be identified as tree, back, forward, or cross edges based on the color of vertices during depth-first search (DFS). It also defines DAGs as directed graphs without cycles and describes how to perform a topological sort of a DAG by inserting vertices into a linked list based on their finishing times from DFS. Finally, it discusses how to find strongly connected components (SCCs) in a graph using DFS on the original graph and its transpose.
Kriging is an optimal interpolation technique that estimates values at unmeasured locations based on measured data from nearby locations. It assigns weights to surrounding data points based on their distances and spatial covariance, accounting for clustering of data points. Simple kriging assumes a known mean and estimates values as a weighted average of residuals from the mean at nearby locations, with weights chosen to minimize the estimation variance. The document provides an example of applying simple kriging to estimate porosity using six nearby data points.
This document presents a new generalized Lindley distribution (NGLD). The NGLD contains the gamma, exponential, and Lindley distributions as special cases. Statistical properties of the NGLD like the hazard function, moments, and moment generating function are derived. Maximum likelihood estimation is discussed to estimate the parameters of the NGLD. Two real data sets are analyzed to illustrate the usefulness of the new distribution.
From planar maps to spatial topology change in 2d gravityTimothy Budd
The document summarizes a talk on generalized causal dynamical triangulations (CDT) in two dimensions. It introduces the generalized CDT model, which allows spatial topology to change in time. It describes how generalized CDT can be solved by viewing causal quadrangulations as labeled trees. It also discusses bijections between labeled quadrangulations and labeled planar maps via Schaeffer's algorithm, which allow counting the numbers of objects in the generalized CDT model.
The document discusses shortest path algorithms for weighted graphs. It introduces Dijkstra's algorithm and the Bellman-Ford algorithm for finding shortest paths. Dijkstra's algorithm works for graphs with non-negative edge weights, while Bellman-Ford can handle graphs with negative edge weights. The document also describes how to find shortest paths in directed acyclic graphs and compute all-pairs shortest paths.
This document discusses Gaussian quadrature formulas, which approximate definite integrals of functions by using weighted sums of function values at specified points. It presents the one-point, two-point, and three-point Gaussian quadrature formulas. The one-point formula is exact for polynomials up to degree 1, the two-point formula is exact for polynomials up to degree 3, and the three-point formula is exact for polynomials up to degree 5. Examples are provided to demonstrate applying the formulas.
The document discusses minimum spanning trees (MST) and two algorithms for finding them: Prim's algorithm and Kruskal's algorithm. It begins by defining an MST as a spanning tree (connected acyclic graph containing all vertices) with minimum total edge weight. Prim's algorithm grows a single tree by repeatedly adding the minimum weight edge connecting the growing tree to another vertex. Kruskal's algorithm grows a forest by repeatedly merging two components via the minimum weight edge connecting them. Both algorithms produce optimal MSTs by adding only "safe" edges that cannot be part of a cycle.
The document summarizes the Frame-Stewart algorithm for solving the generalized Tower of Hanoi puzzle with n disks on k pegs. It begins by introducing the standard 3-peg Tower of Hanoi puzzle and recursive solution. It then describes Henry Dudeney's 4-peg variation and the Frame-Stewart algorithm from 1939 for solving the problem with n disks on any number of pegs k. The algorithm uses recursion and finding the optimal partition of disks to minimize the number of moves. The document proves properties of the number of additional moves between problems sizes.
This document summarizes the derivation of an evidence lower bound (ELBO) for latent LSTM allocation, a model that uses an LSTM to determine topic assignments in a topic modeling framework. It expresses the ELBO as terms related to the variational posterior distributions over topics and topics proportions, the generative process of words given topics, and the LSTM's prediction of topic assignments. It also describes how to optimize the ELBO with respect to the variational and LSTM parameters through gradient ascent.
A total dominating set D of graph G = (V, E) is a total strong split dominating set if the induced subgraph < V-D > is totally disconnected with atleast two vertices. The total strong split domination number γtss(G) is the minimum cardinality of a total strong split dominating set. In this paper, we characterize total strong split dominating sets and obtain the exact values of γtss(G) for some graphs. Also some inequalities of γtss(G) are established.
TopicRNN is a generative model for documents that:
1. Draws a topic vector from a standard normal distribution and uses it to generate words in a document.
2. Computes a lower bound on the log marginal likelihood of words and stop word indicators.
3. Approximates the expected values in the lower bound using samples from an inference network that models the approximate posterior distribution over topics.
Patch Matching with Polynomial Exponential Families and Projective DivergencesFrank Nielsen
This document presents a method called Polynomial Exponential Family-Patch Matching (PEF-PM) to solve the patch matching problem. PEF-PM models patch colors using polynomial exponential families (PEFs), which are universal smooth positive densities. It estimates PEFs using a Score Matching Estimator and accelerates batch estimation using Summed Area Tables. Patch similarity is measured using a statistical projective divergence called the symmetrized γ-divergence. Experiments show PEF-PM handles noise robustly, symmetries, and outperforms baseline methods.
The document presents an algorithm to find an optimal L(2,1)-labeling for triangular windmill graphs. It begins with definitions of triangular windmill graphs, L(2,1)-labelings, and the Chang-Kuo algorithm. The Chang-Kuo algorithm is then applied to obtain an L(2,1)-labeling of a triangular windmill graph W(3,n) by iteratively finding and labeling maximal 2-stable sets. The maximum label used is the labeling number λ(G).
This document discusses subspace clustering with missing data. It summarizes two algorithms for solving this problem: 1) an EM-type algorithm that formulates the problem probabilistically and iteratively estimates the subspace parameters using an EM approach. 2) A k-means form algorithm called k-GROUSE that alternates between assigning vectors to subspaces based on projection residuals and updating each subspace using incremental gradient descent on the Grassmannian manifold. It also discusses the sampling complexity results from a recent paper, showing subspace clustering is possible without an impractically large sample size.
The document introduces new classes of odd graceful graphs called m-shadow graphs and m-splitting graphs. It proves that m-shadow graphs of paths, complete bipartite graphs, and symmetric products of paths and null graphs are odd graceful. It also proves that m-splitting graphs of paths, stars, and symmetric products of paths and null graphs are odd graceful. Examples are provided to illustrate the theories.
The student reflects on completing a math project for their calculus course as a way to study for an upcoming exam. They acknowledge that they procrastinated significantly but were able to cover a broad range of calculus concepts through multi-step word problems selected from different units. While the assignment did not dramatically increase their knowledge, it helped reinforce some details and connections between topics. The student resolves to select deadlines more wisely and stop procrastinating for future projects.
Generalized CDT as a scaling limit of planar mapsTimothy Budd
Generalized causal dynamical triangulations (generalized CDT) is a model of two-dimensional quantum gravity in which a limited number of spatial topology changes is allowed to occur. After identifying the model as a scaling limit of random quadrangulations, I will show how it can be solved using a bijection between quadrangulations and trees. Another bijection relating quadrangulations to planar maps allows us to interpret generalized CDT as a scaling limit or random planar maps with a restriction on the number of faces. Finally I will show how this interpretation clarifies certain mysterious identities in generalized CDT amplitudes. (This talk is largely based on arXiv:1302.1763.)
The document describes a discrete-time Kalman filter implemented in MATLAB to estimate the position of an underwater vehicle using sensor measurements. It presents the state space modeling equations used in the filter, including modifying the state vector to address non-linearities in the direction measurement. Simulation results using a carefully designed trajectory show the filter provides estimates with errors generally within a few meters for position, a few centimeters for velocity bias, and a few meters for range over 1000 iterations.
This document discusses 2D geometric transformations, including translation, rotation, scaling, and their matrix representations using homogeneous coordinates. It provides the transformation equations and routines for translating, rotating, and scaling polygons. Key points covered include:
- The basic equations for 2D translation, rotation, and scaling
- Using matrix multiplication to represent sequences of transformations
- Expanding 2D coordinates to 3x3 homogeneous coordinates matrices
- The transformation matrices for translation, rotation, and scaling
- Calculating the inverse of transformation matrices
LADDER AND SUBDIVISION OF LADDER GRAPHS WITH PENDANT EDGES ARE ODD GRACEFULFransiskeran
The ladder graph plays an important role in many applications as Electronics, Electrical and Wireless
communication areas. The aim of this work is to present a new class of odd graceful labeling for the ladder
graph. In particular, we show that the ladder graph Ln with m-pendant Ln mk1 is odd graceful. We also
show that the subdivision of ladder graph Ln with m-pendant S(Ln) mk1 is odd graceful. Finally, we
prove that all the subdivision of triangular snakes ( k snake ) with pendant edges
1
( ) k S snake mk are odd graceful.
This document discusses graph algorithms and directed acyclic graphs (DAGs). It explains that the edges in a graph can be identified as tree, back, forward, or cross edges based on the color of vertices during depth-first search (DFS). It also defines DAGs as directed graphs without cycles and describes how to perform a topological sort of a DAG by inserting vertices into a linked list based on their finishing times from DFS. Finally, it discusses how to find strongly connected components (SCCs) in a graph using DFS on the original graph and its transpose.
Kriging is an optimal interpolation technique that estimates values at unmeasured locations based on measured data from nearby locations. It assigns weights to surrounding data points based on their distances and spatial covariance, accounting for clustering of data points. Simple kriging assumes a known mean and estimates values as a weighted average of residuals from the mean at nearby locations, with weights chosen to minimize the estimation variance. The document provides an example of applying simple kriging to estimate porosity using six nearby data points.
This document presents a new generalized Lindley distribution (NGLD). The NGLD contains the gamma, exponential, and Lindley distributions as special cases. Statistical properties of the NGLD like the hazard function, moments, and moment generating function are derived. Maximum likelihood estimation is discussed to estimate the parameters of the NGLD. Two real data sets are analyzed to illustrate the usefulness of the new distribution.
This document discusses nonparametric density estimation techniques. It begins by noting limitations of histograms for estimating densities and then introduces the kernel density estimator. The kernel density estimator smooths the empirical density by placing a kernel (such as the normal density) over each data point. The smoothing parameter determines the width of the kernels and impacts the tradeoff between bias and variance. Optimal choices minimize the integrated mean squared error. Kernel density estimators can be used to estimate multivariate densities and hazard rates without parametric assumptions about the underlying distributions. Nonparametric regression similarly estimates relationships without specifying a functional form.
The Odd Generalized Exponential Log Logistic Distributioninventionjournals
We propose a new lifetime model, called the odd generalized exponential log logistic distribution (OGELLD).We obtain some of its mathematical properties. Some structural properties of the new distribution are studied. The maximum likelihood method is used for estimating the model parameters and the Fisher’s information matrix is derived. We illustrate the usefulness of the proposed model by applications to real lifetime data.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
This document outlines an approach to studying time correlations of conserved fields in anharmonic chains using nonlinear fluctuating hydrodynamics. It introduces the BS model, which has two conserved fields - displacement and potential energy. The dynamics of these fields can be approximated by a two-component stochastic Burgers equation. Classifying the universality classes of this equation's correlation functions allows insights into the original anharmonic chain model. Numerical results for specific potentials are also discussed.
This document discusses state-space realizations of linear time-invariant (LTI) systems. It begins by introducing state-space representations using matrices A, B, C, and D. It then discusses the concept of equivalent state-space representations that have the same transfer function through transformations. The document also introduces the concepts of zero-state equivalence and companion forms. It concludes by discussing conditions for a transfer function to have a state-space realization and provides a method to obtain a realization using a block companion form.
I am Driss Fumio. I am a Multivariate Methods Assignment Expert at statisticsassignmentexperts.com. I hold a Master’s Degree in Statistics, from New Brunswick University, Canada. I have been helping students with their assignments for the past 14 years. I solve assignments related to Multivariate Methods. Visit statisticsassignmentexperts.com or email info@statisticsassignmentexperts.com. You can also call on +1 678 648 4277 for any assistance with Multivariate Methods Assignments.
This is the entrance exam paper for ISI MSQE Entrance Exam for the year 2010. Much more information on the ISI MSQE Entrance Exam and ISI MSQE Entrance preparation help available on http://crackdse.com
This document discusses optimal control problems for stochastic sequential machines (SSMs). It begins by introducing SSMs and defining their components. It then formulates the optimal control problem for processes represented by SSMs, proving the principle of optimality. Using dynamic programming, it derives the Bellman equation to find the optimal control solution. In conclusions, it shows that the Bellman equation and principle of optimality apply to obtaining the optimal control for processes modeled as SSMs.
1. The document provides instructions to solve problems related to digital waveguide oscillators, digital lattice filters, and other discrete-time linear systems. Students are asked to write state space equations, find eigenvalues, compute responses, and represent systems using different forms such as state space and block diagrams. MATLAB code is provided to help with computations.
2. Students must analyze cascaded and parallel systems, check controllability and observability, and represent pulse transfer functions using state space, direct form, cascade form, and other block diagram representations. They are also asked to transform state space representations between different coordinate systems.
This document provides solutions to 8 practice problems involving concepts of probability and statistics. The problems cover key distributions like binomial, Poisson, negative binomial, geometric, uniform, hypergeometric, beta, and gamma. For each problem, the document identifies the appropriate distribution, states the required probability, and shows the step-by-step workings. Key results include formulas for moment generating functions, means, variances, and probabilities for events related to these common distributions.
Fixed points theorem on a pair of random generalized non linear contractionsAlexander Decker
1) The document presents a fixed point theorem for a pair of random generalized non-linear contraction mappings involving four points of a separable Banach space.
2) It proves that if two random operators A1(w) and A2(w) satisfy a certain inequality involving upper semi-continuous functions, then there exists a unique random variable η(w) that is the common fixed point of A1(w) and A2(w).
3) As an example, the theorem is applied to prove the existence of a solution in a Banach space to a random non-linear integral equation of the form x(t;w) = h(t;w) + integral of k
This document summarizes some statistical models used for calibrating imperfect mathematical models. It discusses three main approaches:
1. Gaussian stochastic process (GaSP) calibration, which models bias as a Gaussian process. This is commonly used but can produce inconsistent parameter estimates.
2. L2 calibration, which estimates reality separately from the model before estimating parameters. However, it does not use model information.
3. Scaled Gaussian stochastic process (S-GaSP) calibration, which constrains the GaSP to have a fixed L2 norm. This satisfies predicting reality and calibrated parameters. The S-GaSP is equivalent to penalized kernel ridge regression.
The document analyzes the nonparametric regression setting
1) Probability is defined as a set function that satisfies three axioms: non-negativity, the probability of the sample space is 1, and countable additivity.
2) Conditional probability is the probability of an event B given that event A has occurred, defined as P(B|A)=P(A∩B)/P(A). Events A and B are independent if P(B|A)=P(B) and P(A|B)=P(A).
3) Bayes' theorem gives the probability of an event A given that event B has occurred as P(A|B)=P(A)P(B|A)/P(B).
This first lecture describes what EMT is. Its history of evolution. Main personalities how discovered theories relating to this theory. Applications of EMT . Scalars and vectors and there algebra. Coordinate systems. Field, Coulombs law and electric field intensity.volume charge distribution, electric flux density, gauss's law and divergence
This document discusses intensity transformation and spatial filtering in digital image processing. It covers spatial domain vs transform domain processing, and various spatial domain intensity transformation functions including image negatives, log transformations, power-law (gamma) transformations, and piecewise-linear transformations. Histogram processing techniques like histogram equalization, histogram matching, and local histogram processing are also introduced. Examples are provided to illustrate different intensity transformations and histogram matching.
Main Java[All of the Base Concepts}.docxadhitya5119
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
Walmart Business+ and Spark Good for Nonprofits.pdfTechSoup
"Learn about all the ways Walmart supports nonprofit organizations.
You will hear from Liz Willett, the Head of Nonprofits, and hear about what Walmart is doing to help nonprofits, including Walmart Business and Spark Good. Walmart Business+ is a new offer for nonprofits that offers discounts and also streamlines nonprofits order and expense tracking, saving time and money.
The webinar may also give some examples on how nonprofits can best leverage Walmart Business+.
The event will cover the following::
Walmart Business + (https://business.walmart.com/plus) is a new shopping experience for nonprofits, schools, and local business customers that connects an exclusive online shopping experience to stores. Benefits include free delivery and shipping, a 'Spend Analytics” feature, special discounts, deals and tax-exempt shopping.
Special TechSoup offer for a free 180 days membership, and up to $150 in discounts on eligible orders.
Spark Good (walmart.com/sparkgood) is a charitable platform that enables nonprofits to receive donations directly from customers and associates.
Answers about how you can do more with Walmart!"
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
-------------------------------------------------------------------------------
Find out more about ISO training and certification services
Training: ISO/IEC 27001 Information Security Management System - EN | PECB
ISO/IEC 42001 Artificial Intelligence Management System - EN | PECB
General Data Protection Regulation (GDPR) - Training Courses - EN | PECB
Webinars: https://pecb.com/webinars
Article: https://pecb.com/article
-------------------------------------------------------------------------------
For more information about PECB:
Website: https://pecb.com/
LinkedIn: https://www.linkedin.com/company/pecb/
Facebook: https://www.facebook.com/PECBInternational/
Slideshare: http://www.slideshare.net/PECBCERTIFICATION
Temple of Asclepius in Thrace. Excavation resultsKrassimira Luka
The temple and the sanctuary around were dedicated to Asklepios Zmidrenus. This name has been known since 1875 when an inscription dedicated to him was discovered in Rome. The inscription is dated in 227 AD and was left by soldiers originating from the city of Philippopolis (modern Plovdiv).
Chapter wise All Notes of First year Basic Civil Engineering.pptxDenish Jangid
Chapter wise All Notes of First year Basic Civil Engineering
Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
Buildings: Selection of site for Buildings, Layout of Building Plan, Types of buildings, Plinth area, carpet area, floor space index, Introduction to building byelaws, concept of sun light & ventilation. Components of Buildings & their functions, Basic concept of R.C.C., Introduction to types of foundation
Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
Text Books:
1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1
What is Digital Literacy? A guest blog from Andy McLaughlin, University of Ab...
Kumaraswamy disribution
1. KUMARASWAMY DISTRIBUTIONS: A NEW
FAMILY OF GENERALIZED DISTRIBUTIONS
Pankaj Das
Roll No: 20394
M.Sc.(Agricultural Statistics)
Chairman: Dr. Amrit Kumar Paul
2. Contents
Introduction
Conversion of a distribution into Kw-G distribution
Some Special Kw generalized distributions
Properties of Kw generalized distributions
Parameter estimation
Relation to the Beta distribution
Applications
References
2
3. Introduction
Beta distributions are very versatile and a variety of uncertainties can be usefully
modeled by them. In practical situation, many of the finite range distributions
encountered can be easily transformed into the standard beta distribution.
In econometrics, many times the data are modeled by finite range distributions.
Generalized beta distributions have been widely studied in statistics and numerous
authors have developed various classes of these distributions
Eugene et al. (2002) proposed a general class of distributions for a random variable
defined from the beta random variable by employing two parameters whose role is
to introduce skewness and to vary tail weight.
3
4. Introduction
Nadarajaha and Kotz (2004) introduced the beta Gumbel distribution, Nadarajaha
and Gupta (2004) proposed the beta Frechet distribution and Nadarajaha and Kotz
(2004) worked with the beta exponential distribution.
However, all these works lead to some mathematical difficulties because the beta
distribution is not fairly tractable and, in particular, its cumulative distribution
function (cdf) involves the incomplete beta function ratio.
Poondi Kumaraswamy (1980) proposed a new probability distribution for
variables that are lower and upper bounded.
4
5. Introduction
In probability and statistics, the Kumaraswamy's double bounded distribution is a
family of continuous probability distributions defined on the interval (0, 1)
differing in the values of their two non-negative shape parameters, a and b.
Eugene et al (2004) and Jones (2004) constructed a new class of Kumaraswamy
generalized distribution (Kw-G distribution) on the interval (0,1). The probability
density function (pdf) and the cdf with two shape parameters a >0 and b > 0
defined by
5
-1 -1
( ) (1- ) ( ) 1-(1- )a a b a b
f x abx x and F x x (1)
where x
6. Conversion of a distribution into Kw-G
distribution
Let a parent continuous distribution having cdf G(x) and pdf g(x). Then by applying
the quantile function on the interval (0, 1) we can construct Kw-G distribution
(Cordeiro and de Castro, 2009). The cdf F(x) of the Kw-G is defined as
Where a > 0 and b > 0 are two additional parameters whose role is to introduce
skewness and to vary tail weights.
Similarly the density function of this family of distributions has a very simple form
6
( ) 1 {1 ( ) }a b
F x G x (2)
1 1
( ) ( ) ( ) {1 ( ) }a a b
f x abg x G x G x
(3)
7. Some Special Kw generalized distributions
Kw- normal:
The Kw-N density is obtained from (3) by taking G (.) and g (.) to be the cdf and pdf of
the normal distribution, so that
where is a location parameter, σ > 0 is a scale parameter, a, b > 0 are shape
parameters, and and Ф (.) are the pdf and cdf of the standard normal distribution,
respectively.
A random variable with density f (x) above is denoted by X ~ Kw-N
7
1 1
( ) ( ){ ( )} {1 ( ) }a a bab x x x
f x
(4)
(.)
,x
8. Some Special Kw generalized distributions
Kw-Weibull:
The cdf of the Weibull distribution with parameters β > 0 and c > 0 is for x > 0.
Correspondingly, the density of the Kw-Weibull distribution, say Kw-W (a, b, c,
β), reduces to
Here x, a, b, c, β > 0
8
1 1 1
( ) exp{ ( ) }[1 exp{ ( ) }] {1 [1 exp{ ( ) }] }c c c c a c a b
f x abc x x x x
(5)
9. Some Special Kw generalized distributions
Kw-gamma:
Let Y be a gamma random variable with cdf G(y) for y, α, β > 0, where Г(-) is
the gamma function and is the incomplete gamma function. The
density of a random variable X following a Kw-Ga distribution, say X ~ Kw-Ga
(a, b, β, α), can be expressed as
Where x, α, β, a, b >0
9
1
0
( )
z
t
z t e dt
1
1 1
( ) ( ) { ( ) ( )}
( )
x
a a b
x xab
ab x e
f x
(6)
10. Figure 1. Some possible shapes of density function of Kw-G distribution. (a)
Kw-normal (a, b, 0, 1) and (b) Kw- gamma (a, b, 1, α) density functions
(dashed lines represent the parent distributions)
10
Graphical representation of Kw- G
11. A general expansion for the density function
Cordeiro and de Castro (2009) elaborate a general expansion of the distribution.
For b > 0 real non-integer, the form of the distribution
where the binomial coefficient is defined for any real. From the above expansion and
formula (3), we can write the Kw-G density as
Where the coefficients are
and
11
1 1
0
{1 ( ) } ( 1) ( ) ( )a b i b ai
i
i
G x G x
(7)
( 1) 1
0
( ) ( ) ( )a i
i
i
f x g x wG x
(8)
1
( , ) ( 1) ( )i b
i i iw w a b ab
0
0i
i
w
12. General formulae for the moments
The s-th moment of the Kw-G distribution can be expressed as an infinite
weighted sum of PWMs of order (s, r) of the parent distribution G..
We assume Y and X following the baseline G and Kw-G distribution, respectively.
The s-th moment of X, say µ's, can be expressed in terms of the (s, r)-th PWMs
of Y for r = 0, 1 ..., as defined by Greenwood et al. (1979).
For a= integer
12
{ }
rs
sr E Y G Y
'
, ( 1) 1
0
s r s a r
r
w
(9)
13. General formulae for the moments
Whereas for a real non integer the formula
The moments of the Kw-G distribution are calculated in terms of infinite weighted
sums of PWMs of the G distribution
13
'
, , ,
, 0 0
s i j r s r
i j r
w
(10)
14. Probability weighted moments
The (s,r)-th PWM of X following the Kw-G distribution, say, is formally defined
by
This formula also can be written in the following form
the (s,m+l)-th PMW of G distribution and the coefficients
14
,
Kw
s r
, { ( ) } ( ) ( )Kw s r s r
s r E X F X x F x f x dx
(11)
, , , , ,
, , 0 0
( , )Kw
s r r m u v l s m l
m u v l
v
p a b w
,
0 0
( , ) ( )( 1) ( 1) ( )( )( )
u
u k mr l kb ma l
r m k m l r
k m l r
p a b
15. Order statistics
The density of the i-th order statistic, for i = 1,..., n, from i.i.d. random variables
X1,... ,Xn following any Kw-G distribution, is simply given by
Where B(.,.) denote the beta function and then
15
:i nf x
1 1
:
( )
( ) {1 ( )}
( , 1)
i
i
n
n
f x
F x Fx
B n i
f x
i
(13)1 ( 1) 1
( ) ( ) [1 {1 ( ) } ]{1 ( ) }
( , 1)
i a b a b n iab
g x G x G x G x
B i n i
1
0
:
( )
( 1) ( ) ( )
( , 1)
n i
j n i i j
j
i n j
f x
F x
B i n i
f x
(14)
16. Order Statistics
After expanding all the terms of equation (14) we get the following two forms
When a = non integer
When a = integer
Hence, the ordinary moments of order statistics of the Kw-G distribution can be
written as infinite weighted sums of PWMs of the G distribution
16
, , , 1
0 , , 0 0
:
( )
( 1) ( ) ( , ) ( )
( , 1)
n i v
j n i r t
j u v t r i j
j r u v t
i n
g x
w p a bx G x
B i
f
i n
(15)
( 1) 1
,: 1
0 , 0
( )
( 1) ( ) ( )
( , 1)
n i
j n i a u r
j u r i j
r
n
j
i
u
g x
w p abG x
B i n
f x
i
(16)
17. L moments
In statistics, L-moments are a sequence of statistics used to summarize the shape of
probability distribution. They can be estimated by linear combinations of order
statistics.
The L-moments have several theoretical advantages over the ordinary moments.
They exist whenever the mean of the distribution exists, even though some higher
moments may not exist.
They are able to characterize a wider range of distributions and, when estimated
from a sample, are more robust to the effects of outliers in the data.
L-moments can be used to calculate quanties that analogous to SD, skewness and
kurtosis , termed as L-scale, L-skewness and L-kurtosis respectively.
17
18. L-moments
The L-moments are linear functions of expected order statistics defined as
the first four L-moments are
, ,
and
18
1
1 1 : 1
0
( 1) ( 1) ( ) ( )
r
k r
r k r k r
k
r E X
1 1:1( )E X 2 2:2 1:2
1
( )
2
E X X 3 3:3 2:3 1:3
1
( 2 )
3
E X X X
4 4:4 3:4 2:4 1:4
1
( 3 3 )
4
E X X X X
(17)
19. L-moments
The L-moments can also be calculated in terms of PWMs given in (12) as
In particular
19
1 1,
0
( 1) ( )( )r k r r k Kw
r k k k
k
(18)
1 1,0 2 1:1 1:0 3 1:2 1:1 1:0, 2 , 6 6Kw Kw Kw Kw Kw Kw
4 1:3 1:2 1:1 1:020 30 12Kw Kw Kw Kw
20. Mean deviations
Mean deviation denotes the amount of scatter in a population. This is evidently
measured to some extent by the totality of deviations from the mean and median.
Let X ∼ Kw-G (a, b). The mean deviations about the mean (δ1(X)) and about the
median (δ2(X)) can be expressed as
and
Where ,M = median, is come from pdf and
20
1 ' ' '
1 1 1 1 1( ) ( ) 2 ( ) 2 ( )X E X F T '
2 1( ) ( ) 2 ( )X E X M T M
'
1 ( )E X '
1( )F
( ) ( )
z
T z xf x dx
21. Parameter Estimation
Let γ be the p-dimensional parameter vector of the baseline distribution in
equations (2) and (3). We consider independent random variables X1,..., Xn,
each Xi following a Kw-G distribution with parameter vector θ = (a,b, γ). The
log-likelihood function for the model parameters obtained from (3) is
The elements of the score vector are given by
21
( )
1 1 1
( ) {log( ) log( )} log{ ( ; )} ( 1) log{ ( ; )} ( 1) log{1 ( ; ) }
n n n
a
i i i
i i i
n a b g x a G x b G x
1
( 1) ( ; )( )
log{ ( ; )}{1 }
1 ( ; )
an
i
i a
i i
b G xd n
G x
da a G x
22. Parameter Estimation
and
These partial derivatives depend on the specified baseline distribution. Numerical
maximization of the log-likelihood above is accomplished by using the RS method
(Rigby and Stasinopoulos, 2005) available in the gamlss package in R.
22
1
( )
log{1 ( ; ) }
n
a
i
i
d n
G x
db b
1
( ; ) ( ; )( ) 1 1 ( 1)
[ {1 }
( ; ) ( ; ) ( ; ) 1
n
i i
a
ij i i i
dg x dG xd a b
d g x d G x d G x
23. Relation to the Beta distribution
The density function of beta distribution is defined as
The density function of Kw-G distribution is defined as
When b=1, both of them are identical.
23
1 11
( ) ( ) ( ) {1 ( )}
( , )
a b
f x g x G x G x
B a b
1 1
( ) ( ) ( ) {1 ( ) }a a b
f x abg x G x G x
24. Relation to the Beta distribution
Let is a Kumaraswamy distributed random variable with parameters a and b.
Then is the a-th root of a suitably defined Beta distributed random variable.
Let denote a Beta distributed random variable with parameters and .
One has the following relation between and .
With equality in distribution,
24
,a bX
,a bX
1,bY 1 b
,a bX 1,bY
1/
, 1,
a
a b bX Y
1 1 1 1/
, 1, 1,
0 0
{ } (1 ) (1 ) { } { }
a
x x
a a b b a a
a b b bP X x abt t dt b t dt P Y x P Y x
25. Advantages of Kw-G distribution
Jones (2008) explored the background and genesis of the Kw distribution and, more
importantly, made clear some similarities and differences between the beta and Kw
distributions.
He highlighted several advantages of the Kw distribution over the beta distribution:
The normalizing constant is very simple;
Simple explicit formulae for the distribution and quantile functions which do
not involve any special functions;
A simple formula for random variate generation;
Explicit formulae for L-moments and simpler formulae for moments of order
statistics
25
26. Application
The superiority of some new Kw-G distributions proposed here as compared
with some of their sub-models.
We give two applications (uncensored and censored data) using well- known
data sets to demonstrate the applicability of the proposed regression model.
26
27. Application 1(Censored data)
This is an example with data from adult numbers of Flour beetle (T. confusum)
cultured at 29°C presented by Cordeiro and de Castro (2009).
Analysis is done in R console.
The required package is gamlss package.
Table 1 gives AIC values in increasing order for some fitted distributions and the
MLEs of the parameters together with its standard errors. According to AIC, the
beta normal and Kw-normal distributions yield slightly different fittings,
outperforming the remaining selected distributions.
27
28. Application 1
The fitted distributions superimposed to the histogram of the data in Figure 3
reinforce the result in Table 1 for the gamma distribution.
Further for the comparison between observed and expected frequencies we
construct Table 2. The mean absolute deviation between expected and observed
frequencies reaches the minimum value for the Kw-normal distribution.
Based on the values of the LR statistic , the Kw-gamma and the Kw-exponential
distributions are not significantly different yielding LR = 1.542 (1 d.f., p-value =
0.214). Comparing the Kw-gamma and the gamma distributions, we find a
significant difference (LR = 6.681, 2 d.f., p-value = 0.035)
28
29. Application 2 (uncensored data)
In this section,we compare the results of Nadarajaha et al (2011).
They fits some distributions to a voltage data set which gives the times of failure
and running times for a sample of devices from a field-tracking study of a larger
system.
At a certain point in time, 30 electric units were installed in normal service
conditions. Two causes of failure were observed for each unit that failed: the
failure caused by an accumulation of randomly occurring damage from power-
line voltage spikes during electric storms and failure caused by normal product
wear.
The required numerical evaluations were implemented using the SAS procedure
NLMIXED.
29
30. Application 2
Table 3 lists the MLEs (and the corresponding standard errors in parentheses) of
the parameters and the values of the following statistics for some fitted models:
AIC (Akaike information criterion), BIC (Bayesian information criterion) and
CAIC (Consistent Akaike information criterion).
These results indicate that the Kw-Weibull model has the lowest AIC, CAIC and
BIC values among all fitted models, and so it could be chosen as the best model.
In order to assess whether the model is appropriate, plots of the histogram of the
data Figure 4.
We conclude that the Kw-XGT distribution fits well to these data.
30
31. Conclusion
Following the idea of the class of beta generalized distributions and the
distribution by Kumaraswamy, we define a new family of Kw generalized (Kw-G)
distributions to extend several widely-known distributions such as the normal,
Weibull, gamma and Gumbel distributions.
We show how some mathematical properties of the Kw-G distributions are readily
obtained from those of the parent distributions.
The moments of the Kw-G distribution can be expressed explicitly in terms of
infinite weighted sums of probability weighted moments (PWMs) of the G
distribution
31
32. Conclusion
We discuss maximum likelihood estimation and inference on the parameters. The
maximum likelihood estimation in Kw-G distributions is much simpler than the
estimation in beta generalized distributions
We also show the feasibility of the Kw-G distribution in case of Environmental
data (both censored data and Uncensored data) with applications.
So we can conclude that the Kumaraswamy distribution: new family of
generalized distribution can be used in environmental data.
32
33. References
Azzalini, A. (1985). A class of distributions which includes the normal ones.
Scandinavian Journal of Statistics. 12:171-178.
Barakat, H. M. and Abdelkader, Y. H. (2004). Computing the moments of order
statistics from nonidentical random variables. Statistical Methods and
Applications. 13:15-26.
Barlow, R. E. and Proschan, F. (1975). Statistical theory of reliability and life
testing: probability models. Holt, Rinehart and Winston, New York, London.
Cordeiroa, Gauss M. and Castrob, Mario de (2009). A new family of generalized
distributions. Journal of Statistical Computation & Simulation. 79: 1-17.
33
34. References
Eugene, N., Lee, C., and Famoye, F. (2002). Beta-normal distribution and its
applications. Communications in Statistics. Theory and Methods. 31:497-
512.
Fletcher, S. C. and Ponnambalam, K. (1996). Estimation of reservoir yield and
storage distribution using moments analysis. Journal of Hydrology. 182:
259-275.
Greenwood, J. A., Landwehr, J. M., Matalas, N. C. and Wallis, J. R. (1979).
Probability weighted moments - definition and relation to parameters of
several distributions expressable in inverse form. Water Resources
Research. 15:1049-1054.
Hosking, J. R. M. (1990). L-moments: analysis and estimation of distributions
using linear combinations of order statistics. Journal of the Royal Statistical
Society. Series B.52:105-124.
34
35. References
Jones, M. C. (2004). Families of distributions arising from distributions of order
statistics (with discussion). Test. 13:1-43.
Jones, M. C. (2008). Kumaraswamy's distribution: A beta-type distribution with
some tractability advantages. Statistical Methodology. 6:70-81.
Kumaraswamy, P. (1980). Generalized probability density-function for double-
bounded random- processes. Journal of Hydrology. 462:79-88.
Leadbetter, M.R., Lindgren, G. and Rootzén, H. (1987). Extremes and Related
Properties of Random Sequences and Processes. Springer, New York,
London.
35
36. References
Nadarajaha, S. and Gupta, A. K. (2004). The beta Frechet distribution. Far East
Journal of Theoretical Statistics. 14:15-24.
Nadarajaha, S. and Kotz, S. (2006). The beta exponential distribution. Reliability
Engineering & System Safety. 91: 689-697.
Nadarajaha, S., Cordeirob, Gauss M. and Ortegac, Edwin M. M. (2011). General
results for the Kumaraswamy-G distribution. Journal of Statistical
Computation and Simulation. 81: 1-29.
Rigby, R. A. and Stasinopoulos, D. M.(2005). Generalized additive models for
location, scale and shape (with discussion). Applied Statistics. 54:507-554.
36
37. References
R Development Core Team. (2009). R: A Language and Environment for Statistical
Computing. R Foundation for Statistical Computing. Vienna, Austria.
Sundar, V. and Subbiah, K. (1989). Application of double bounded probability
density-function for analysis of ocean waves. Ocean Engineering. 16:193-
200.
Seifi, A., Ponnambalam, K. and Vlach, J. (2000). Maximization of manufacturing
yield of systems with arbitrary distributions of component values. Annals
of Operations Research. 99:373- 383.
Stasinopoulos, D. M. and. Rigby, R. A. (2007). Generalized additive models for
location scale and shape (GAMLSS) in R. Journal of Statistical Software.
23:1-46.
37
39. Probability weighted moments
A distribution function F = F(x) = P(X ≤ x) may be characterized by probability
weighted moments, which are defined as
where i, j, and k are real numbers. If j = k = 0 and i is a nonnegative integer, then
represents the conventional moment about the origin of order i.
If exists and X is a continuous function of F, then exists for all
nonnegative real numbers j and k.
39
1
, ,
0
[ (1 ) ] [ ( )] (1 )i j k i j k
i j k E X F F x F F F dF
,0,0i,0,0i
,0,0i
41. Probability weighted moments
Application:(Barakat and Abdelkader, 2004)
The summarization and description of theoretical probability distributions
Estimation of parameters and quantiles of probability distributions and
hypothesis testing for probability distributions
Nonparametric estimation of the underlying distribution of an observed sample
41
42. Probability weighted moments
Conditions for application of PWM: (Greenwood et al,1979)
1. Distributions that can be expressed in inverse form, particularly those that can only
be expressed may present problems in deriving explicit expressions for their
parameters as functions of conventional moments.
2. When the estimated characteristic parameters of a distribution fitted by central
moments are often marked less accurate.
42
43. AIC (Akaike's Information Criterion)
An index used in a number of areas as an aid to choosing between competing models.
It is defined as
Where L is the likelihood function for an estimated model with p parameters.
The index takes into account both the statistical goodness of fit and the number of
parameters that have to be estimated to achieve this particular degree of fit, by
imposing a penalty for increasing the number of parameters.
Lower values of the index indicate the preferred model, that is, the one with the
fewest parameters that still provides an adequate fit to the data.
L + p- ln=AIC
43
44. Bayesian Information Criterion (BIC)
The Bayesian information criterion (BIC) or Schwarz criterion (also SBC, SBIC) is
a criterion for model selection among a finite set of models. It is based, in part, on
the likelihood function and it is closely related to the Akaike information criterion
(AIC).
The formula is
where n is the sample size, Lp is the maximized log-likelihood of the model and p is
the number of parameters in the model.
The index takes into account both the statistical goodness of fit and the number of
parameters that have to be estimated to achieve this particular degree of fit, by
imposing a penalty for increasing the number of parameters.
n+ pL- p ln2
44
45. Consistent Akaike information criterion (CAIC)
• Bozdogan (1987) reviews a number of criteria that he terms ‘dimension consistent’
or CAIC, i.e. consistent AIC.
• The formula of CAIC is
• The dimension-consistent criteria were derived with the objective that the order of
the true model was estimated in an asymptotically unbiased (i.e. consistent)
manner
• there is an interest in parameter estimation where bias is low and where precision
is high (i.e. parsimony).
45
^
CAIC 2log [ ( )] [log ( ) 1]e eL p n
46. 46 Table 1 : AIC values in increasing order for some fitted distributions and the
MLEs of the parameters together with its standard errors
48. 48
Table 2: Observed and expected frequencies of adult numbers for T. confusum cultured
at 29°C and mean absolute deviation (MAD) between the frequencies
49. Table 3: lists the MLEs of the parameters and the values of the following statistics
for some fitted models:49