This document discusses using diffusion kernels on single nucleotide polymorphism (SNP) data embedded in non-Euclidean metric spaces. It defines kernels as weighting functions that provide a similarity metric based on a chosen distance metric, such as Euclidean or Manhattan distance. The document explores using graphs to represent SNP data, with nodes for each genotype and edges connecting similar genotypes. It describes constructing diffusion kernels on these graphs by taking Cartesian graph products of one-dimensional genotype graphs. This allows modeling SNP data as grids where similarity decreases as the number of differing loci increases.
This document discusses vector algebra concepts including scalar and vector quantities, vector operations like addition and subtraction, and different coordinate systems used to represent vectors such as Cartesian, cylindrical, and spherical coordinates. Key topics covered include defining a vector using basis vectors in a coordinate system, calculating the magnitude and direction of a vector, adding and subtracting vectors using graphical representations, computing the dot and cross products of vectors, and converting between Cartesian, cylindrical, and spherical coordinate representations of vectors.
This document contains a chapter on functions with 30 math exercises. The exercises involve evaluating functions, determining domains and ranges, analyzing graphs of functions, and solving word problems involving functions.
1. The document discusses concepts related to expectation and variance of random variables including expected value, variance, moments, and examples of calculating these for different probability distributions like uniform, normal, exponential, and Rayleigh.
2. Problems at the end provide examples of computing expected value, variance, and cumulative distribution function for random variables following different distributions. Solutions show the calculations and formulas used.
3. Key formulas introduced include definitions of expected value and variance, the relationship between them, and formulas for calculating moments, expected value, and variance for specific distributions. Examples demonstrate applying the concepts and formulas to problems.
Green’s Function Solution of Non-homogenous Singular Sturm-Liouville ProblemIJSRED
This document discusses solving non-homogeneous singular Sturm-Liouville problems using Green's function methods. It begins with an introduction to Green's functions and their use in solving differential equations with singularities. It then provides examples of applying Green's functions to solve two specific singular Sturm-Liouville problems - Bessel's equation and a second-order differential equation with a singular point at 0. The document derives the Green's function for each example problem and uses it to find the solution that satisfies the given boundary conditions.
This document provides examples of calculating area under curves using the integral. It contains 19 practice problems that involve slicing areas vertically or horizontally and integrating to find the area. The key steps are to identify intersection points, set up integrals using appropriate bounds, and evaluate the integrals to obtain area estimates.
The document describes the Jacobi iterative method for solving systems of linear equations. It begins with an initial estimate for the solution variables, inserts them into the equations to get updated estimates, and repeats this process iteratively until the estimates converge to the desired solution. As an example, it applies the method to a set of 3 equations in 3 unknowns, showing the estimates after each iteration getting progressively closer to the exact solution obtained using Gaussian elimination. A Fortran program implementing the Jacobi method is also presented.
This document contains solutions to problems from calculus and multivariable calculus courses. It begins with single variable calculus problems involving tangent lines, integrals, derivatives, and infinite series. The second part involves problems related to parametric equations, vectors, planes, cylinders, and graphing surfaces. The last part contains problems involving level curves, least squares regression, and using computer algebra systems to plot functions.
This document discusses vector algebra concepts including scalar and vector quantities, vector operations like addition and subtraction, and different coordinate systems used to represent vectors such as Cartesian, cylindrical, and spherical coordinates. Key topics covered include defining a vector using basis vectors in a coordinate system, calculating the magnitude and direction of a vector, adding and subtracting vectors using graphical representations, computing the dot and cross products of vectors, and converting between Cartesian, cylindrical, and spherical coordinate representations of vectors.
This document contains a chapter on functions with 30 math exercises. The exercises involve evaluating functions, determining domains and ranges, analyzing graphs of functions, and solving word problems involving functions.
1. The document discusses concepts related to expectation and variance of random variables including expected value, variance, moments, and examples of calculating these for different probability distributions like uniform, normal, exponential, and Rayleigh.
2. Problems at the end provide examples of computing expected value, variance, and cumulative distribution function for random variables following different distributions. Solutions show the calculations and formulas used.
3. Key formulas introduced include definitions of expected value and variance, the relationship between them, and formulas for calculating moments, expected value, and variance for specific distributions. Examples demonstrate applying the concepts and formulas to problems.
Green’s Function Solution of Non-homogenous Singular Sturm-Liouville ProblemIJSRED
This document discusses solving non-homogeneous singular Sturm-Liouville problems using Green's function methods. It begins with an introduction to Green's functions and their use in solving differential equations with singularities. It then provides examples of applying Green's functions to solve two specific singular Sturm-Liouville problems - Bessel's equation and a second-order differential equation with a singular point at 0. The document derives the Green's function for each example problem and uses it to find the solution that satisfies the given boundary conditions.
This document provides examples of calculating area under curves using the integral. It contains 19 practice problems that involve slicing areas vertically or horizontally and integrating to find the area. The key steps are to identify intersection points, set up integrals using appropriate bounds, and evaluate the integrals to obtain area estimates.
The document describes the Jacobi iterative method for solving systems of linear equations. It begins with an initial estimate for the solution variables, inserts them into the equations to get updated estimates, and repeats this process iteratively until the estimates converge to the desired solution. As an example, it applies the method to a set of 3 equations in 3 unknowns, showing the estimates after each iteration getting progressively closer to the exact solution obtained using Gaussian elimination. A Fortran program implementing the Jacobi method is also presented.
This document contains solutions to problems from calculus and multivariable calculus courses. It begins with single variable calculus problems involving tangent lines, integrals, derivatives, and infinite series. The second part involves problems related to parametric equations, vectors, planes, cylinders, and graphing surfaces. The last part contains problems involving level curves, least squares regression, and using computer algebra systems to plot functions.
The document provides solutions to problems from a discrete-time signal processing textbook. It includes:
1) Solutions to convolution problems graphically representing signals and their convolution.
2) Derivations of impulse responses from system functions using the z-transform.
3) Analyses of signals as eigenfunctions and determining if systems are linear and time-invariant.
4) Solutions involving filtering, modulation, and determining system properties from inputs and outputs.
This document contains a problem set in quantitative methods with 17 questions covering topics in linear algebra including: solving systems of linear equations using Gauss-Jordan elimination; determining the inverse of matrices; finding the null space and row/column spaces of matrices; determining if sets of vectors are linearly independent/dependent or span vector spaces; and identifying if sets of vectors form bases. The problem set is assigned by Manimay Sengupta for the Monsoon Semester 2012 at South Asian University.
The document provides solutions to three problems involving finite element analysis. Problem 1 shows the derivation of the weak form of a partial differential equation. Problem 2 sets up finite element models for a bar with two elements, assembles the stiffness matrix and load vector, and solves for displacements and stresses. Problem 3 models a problem with two linear elements, derives the finite element equations, and solves for displacements and stresses, then compares to an exact solution and a single quadratic element solution.
NUMERICAL METHODS -Iterative methods(indirect method)krishnapriya R
The document discusses two iterative methods for solving systems of linear equations: Gauss-Jacobi and Gauss-Seidel. Gauss-Jacobi solves each equation separately using the most recent approximations for the other variables. Gauss-Seidel updates each variable with the most recent values available. The document provides an example applying both methods to solve a system of three equations. Gauss-Seidel converges faster, requiring fewer iterations than Gauss-Jacobi to achieve the same accuracy. Both methods are useful alternatives to direct methods like Gaussian elimination when round-off errors are a concern.
system of algebraic equation by Iteration methodAkhtar Kamal
The document discusses iterative methods for solving systems of linear equations, specifically Jacobi's method and Gauss-Seidel method. It provides examples of using these methods to solve several systems of 3 equations with 3 unknowns. For each system, it shows rewriting the equations in a form suitable for the methods, choosing initial approximations, iterating to obtain better approximations, and concluding when subsequent iterations yield identical results to 3 significant digits.
This document discusses operations on interval-valued fuzzy graphs. It begins with an introduction to fuzzy graphs and definitions related to fuzzy graph theory. It then presents the main results which prove properties of the union and join operations on interval-valued fuzzy graphs. Specifically, it proves that the union of two interval-valued fuzzy graphs G1 and G2 is isomorphic to their join, and vice versa. It also discusses subgraphs, complements, and neighbors in fuzzy graph theory.
1. The document discusses a universal Bayesian measure for arbitrary data that is either discrete or continuous.
2. It presents Ryabko's measure for continuous variables and generalizes it using the Radon-Nikodym theorem to define density functions for both discrete and continuous random variables.
3. It then shows that given a universal histogram sequence, the normalized log ratio of the true density function to this generalized measure converges to zero, providing a universal Bayesian solution to the problem.
FITTED OPERATOR FINITE DIFFERENCE METHOD FOR SINGULARLY PERTURBED PARABOLIC C...ieijjournal
In this paper, we study the numerical solution of singularly perturbed parabolic convection-diffusion type
with boundary layers at the right side. To solve this problem, the backward-Euler with Richardson
extrapolation method is applied on the time direction and the fitted operator finite difference method on the
spatial direction is used, on the uniform grids. The stability and consistency of the method were established
very well to guarantee the convergence of the method. Numerical experimentation is carried out on model
examples, and the results are presented both in tables and graphs. Further, the present method gives a more
accurate solution than some existing methods reported in the literature.
The document discusses inner product spaces. Some key points:
- An inner product is a function that associates a number (<u,v>) with each pair of vectors (u,v) in a vector space, satisfying certain properties like symmetry and homogeneity.
- An inner product space is a vector space with an additional inner product structure.
- Properties of inner products include positivity (<v,v>≥0), linearity, and defining the norm (||v||) of a vector.
- Examples show the weighted Euclidean inner product satisfies the inner product properties and define the unit sphere in an inner product space.
This document contains an exercise set with 46 problems involving real numbers, intervals, and inequalities. The problems cover topics such as determining whether numbers are rational or irrational, solving equations, graphing inequalities on number lines, factoring polynomials, and solving compound inequalities.
The document provides instructions for a mathematics test that is 1 1/2 hours long and consists of 75 questions worth a total of 225 marks. For each correct answer, 3 marks are awarded, and for each wrong answer, 1 mark is deducted. It then lists 34 math problems as sample questions on topics including relations, functions, complex numbers, matrices, series, and calculus.
This document is the student solutions manual for the 4th edition of the textbook "Essential Mathematics for Economic Analysis" by Knut Sydsæter, Peter Hammond, and Arne Strøm. The manual provides detailed solutions to problems marked in the textbook. The preface states that the solutions should be used along with any abbreviated answers in the textbook. The authors welcome feedback to improve accuracy. The manual is organized by chapter and covers topics like algebra, equations, functions, differentiation, integration, optimization, and matrices.
Vector Calculus and Linear Algebra (2110015) covers topics in inner products including the Cauchy-Schwarz inequality, orthogonal vectors, the Pythagorean theorem, orthogonal projections, the Gram-Schmidt process, and least squares solutions. The document provides examples and explanations of these concepts, such as finding the orthogonal projection of a vector onto a subspace and using the normal equations to solve the least squares problem for an inconsistent system of linear equations.
2012 mdsp pr11 ica part 2 face recognitionnozomuhamada
The document describes using independent component analysis (ICA) for face recognition. ICA is applied to a data matrix of face images to extract statistically independent basis images that represent local facial features. These basis images can then be used as a feature vector to identify faces. Specifically, ICA is applied to a training set of 425 face images to extract 25 statistically independent component basis images. These basis images provide local facial features that can be used to represent faces for recognition.
This document contains solutions to exercises involving double integration using Cartesian and polar coordinates. It includes 8 exercises with solutions involving double integrals over various regions in 2D planes. The solutions calculate the double integrals using different orders and techniques of integration, including changing to polar coordinates.
This document contains an exercise set from a chapter on functions. It includes 35 multi-part math problems testing concepts like domains and ranges of functions, rates of change, and word problems involving temperature, speed, and geometric shapes. The problems cover skills like determining maximum/minimum values, solving equations, and sketching graphs of functions.
Here are the key steps:
1) Choose u and dv based on LIPET:
u = ex
dv = cos x dx
2) Find du and v:
du = ex dx
v = sin x
3) Apply integration by parts formula:
∫uex dx = uv - ∫vdu
= exsinx - ∫sinxexdx
4) Repeat integration by parts on the second term:
∫sinxexdx = excosx - ∫-cosxexdx
5) Combine like terms:
exsinx + excosx - ∫excosxdx
6) The integral on the right is
Whole-genome prediction of complex traits using kernel methodsGota Morota
This document outlines a kernel-based whole-genome prediction method and applies it to dairy cattle, wheat, and dairy cow health trait data. It first describes using kernel methods to predict complex traits from genotypes, including whole-genome marker regression and kernel-based regression. It then applies these methods to different livestock and crop datasets, comparing predictive accuracy across parametric and non-parametric kernels like Gaussian, diffusion, and multiple kernel learning. Results show the kernel-based approach can capture additive and non-additive genetic effects and improve prediction of complex traits.
NGS由来ゲノムワイド多型マーカ構築とそのRDF注釈情報統合化
Eli Kaminuma1, Takatomo Fujisawa1, Takako Mochizuki1, Yasuhiro Tanizawa1, Atsushi Toyoda1, Asao Fujiyama1, Nori Kurata1, Tokurou Shimizu2, Yasukazu Nakamura1
1. National Institute of Genetics, SOKENDAI ; 1111 Yata, Mishima, Shizuoka, 411-8540 Japan.
2. National Institute of Fruit Tree Science; Okitsu Nakacho, Shizuoka, 424-0292 Japan
BMB2013(第36回日本分子生物学会年会)ポスター 3P-0030
2013年12月5日
The document provides solutions to problems from a discrete-time signal processing textbook. It includes:
1) Solutions to convolution problems graphically representing signals and their convolution.
2) Derivations of impulse responses from system functions using the z-transform.
3) Analyses of signals as eigenfunctions and determining if systems are linear and time-invariant.
4) Solutions involving filtering, modulation, and determining system properties from inputs and outputs.
This document contains a problem set in quantitative methods with 17 questions covering topics in linear algebra including: solving systems of linear equations using Gauss-Jordan elimination; determining the inverse of matrices; finding the null space and row/column spaces of matrices; determining if sets of vectors are linearly independent/dependent or span vector spaces; and identifying if sets of vectors form bases. The problem set is assigned by Manimay Sengupta for the Monsoon Semester 2012 at South Asian University.
The document provides solutions to three problems involving finite element analysis. Problem 1 shows the derivation of the weak form of a partial differential equation. Problem 2 sets up finite element models for a bar with two elements, assembles the stiffness matrix and load vector, and solves for displacements and stresses. Problem 3 models a problem with two linear elements, derives the finite element equations, and solves for displacements and stresses, then compares to an exact solution and a single quadratic element solution.
NUMERICAL METHODS -Iterative methods(indirect method)krishnapriya R
The document discusses two iterative methods for solving systems of linear equations: Gauss-Jacobi and Gauss-Seidel. Gauss-Jacobi solves each equation separately using the most recent approximations for the other variables. Gauss-Seidel updates each variable with the most recent values available. The document provides an example applying both methods to solve a system of three equations. Gauss-Seidel converges faster, requiring fewer iterations than Gauss-Jacobi to achieve the same accuracy. Both methods are useful alternatives to direct methods like Gaussian elimination when round-off errors are a concern.
system of algebraic equation by Iteration methodAkhtar Kamal
The document discusses iterative methods for solving systems of linear equations, specifically Jacobi's method and Gauss-Seidel method. It provides examples of using these methods to solve several systems of 3 equations with 3 unknowns. For each system, it shows rewriting the equations in a form suitable for the methods, choosing initial approximations, iterating to obtain better approximations, and concluding when subsequent iterations yield identical results to 3 significant digits.
This document discusses operations on interval-valued fuzzy graphs. It begins with an introduction to fuzzy graphs and definitions related to fuzzy graph theory. It then presents the main results which prove properties of the union and join operations on interval-valued fuzzy graphs. Specifically, it proves that the union of two interval-valued fuzzy graphs G1 and G2 is isomorphic to their join, and vice versa. It also discusses subgraphs, complements, and neighbors in fuzzy graph theory.
1. The document discusses a universal Bayesian measure for arbitrary data that is either discrete or continuous.
2. It presents Ryabko's measure for continuous variables and generalizes it using the Radon-Nikodym theorem to define density functions for both discrete and continuous random variables.
3. It then shows that given a universal histogram sequence, the normalized log ratio of the true density function to this generalized measure converges to zero, providing a universal Bayesian solution to the problem.
FITTED OPERATOR FINITE DIFFERENCE METHOD FOR SINGULARLY PERTURBED PARABOLIC C...ieijjournal
In this paper, we study the numerical solution of singularly perturbed parabolic convection-diffusion type
with boundary layers at the right side. To solve this problem, the backward-Euler with Richardson
extrapolation method is applied on the time direction and the fitted operator finite difference method on the
spatial direction is used, on the uniform grids. The stability and consistency of the method were established
very well to guarantee the convergence of the method. Numerical experimentation is carried out on model
examples, and the results are presented both in tables and graphs. Further, the present method gives a more
accurate solution than some existing methods reported in the literature.
The document discusses inner product spaces. Some key points:
- An inner product is a function that associates a number (<u,v>) with each pair of vectors (u,v) in a vector space, satisfying certain properties like symmetry and homogeneity.
- An inner product space is a vector space with an additional inner product structure.
- Properties of inner products include positivity (<v,v>≥0), linearity, and defining the norm (||v||) of a vector.
- Examples show the weighted Euclidean inner product satisfies the inner product properties and define the unit sphere in an inner product space.
This document contains an exercise set with 46 problems involving real numbers, intervals, and inequalities. The problems cover topics such as determining whether numbers are rational or irrational, solving equations, graphing inequalities on number lines, factoring polynomials, and solving compound inequalities.
The document provides instructions for a mathematics test that is 1 1/2 hours long and consists of 75 questions worth a total of 225 marks. For each correct answer, 3 marks are awarded, and for each wrong answer, 1 mark is deducted. It then lists 34 math problems as sample questions on topics including relations, functions, complex numbers, matrices, series, and calculus.
This document is the student solutions manual for the 4th edition of the textbook "Essential Mathematics for Economic Analysis" by Knut Sydsæter, Peter Hammond, and Arne Strøm. The manual provides detailed solutions to problems marked in the textbook. The preface states that the solutions should be used along with any abbreviated answers in the textbook. The authors welcome feedback to improve accuracy. The manual is organized by chapter and covers topics like algebra, equations, functions, differentiation, integration, optimization, and matrices.
Vector Calculus and Linear Algebra (2110015) covers topics in inner products including the Cauchy-Schwarz inequality, orthogonal vectors, the Pythagorean theorem, orthogonal projections, the Gram-Schmidt process, and least squares solutions. The document provides examples and explanations of these concepts, such as finding the orthogonal projection of a vector onto a subspace and using the normal equations to solve the least squares problem for an inconsistent system of linear equations.
2012 mdsp pr11 ica part 2 face recognitionnozomuhamada
The document describes using independent component analysis (ICA) for face recognition. ICA is applied to a data matrix of face images to extract statistically independent basis images that represent local facial features. These basis images can then be used as a feature vector to identify faces. Specifically, ICA is applied to a training set of 425 face images to extract 25 statistically independent component basis images. These basis images provide local facial features that can be used to represent faces for recognition.
This document contains solutions to exercises involving double integration using Cartesian and polar coordinates. It includes 8 exercises with solutions involving double integrals over various regions in 2D planes. The solutions calculate the double integrals using different orders and techniques of integration, including changing to polar coordinates.
This document contains an exercise set from a chapter on functions. It includes 35 multi-part math problems testing concepts like domains and ranges of functions, rates of change, and word problems involving temperature, speed, and geometric shapes. The problems cover skills like determining maximum/minimum values, solving equations, and sketching graphs of functions.
Here are the key steps:
1) Choose u and dv based on LIPET:
u = ex
dv = cos x dx
2) Find du and v:
du = ex dx
v = sin x
3) Apply integration by parts formula:
∫uex dx = uv - ∫vdu
= exsinx - ∫sinxexdx
4) Repeat integration by parts on the second term:
∫sinxexdx = excosx - ∫-cosxexdx
5) Combine like terms:
exsinx + excosx - ∫excosxdx
6) The integral on the right is
Whole-genome prediction of complex traits using kernel methodsGota Morota
This document outlines a kernel-based whole-genome prediction method and applies it to dairy cattle, wheat, and dairy cow health trait data. It first describes using kernel methods to predict complex traits from genotypes, including whole-genome marker regression and kernel-based regression. It then applies these methods to different livestock and crop datasets, comparing predictive accuracy across parametric and non-parametric kernels like Gaussian, diffusion, and multiple kernel learning. Results show the kernel-based approach can capture additive and non-additive genetic effects and improve prediction of complex traits.
NGS由来ゲノムワイド多型マーカ構築とそのRDF注釈情報統合化
Eli Kaminuma1, Takatomo Fujisawa1, Takako Mochizuki1, Yasuhiro Tanizawa1, Atsushi Toyoda1, Asao Fujiyama1, Nori Kurata1, Tokurou Shimizu2, Yasukazu Nakamura1
1. National Institute of Genetics, SOKENDAI ; 1111 Yata, Mishima, Shizuoka, 411-8540 Japan.
2. National Institute of Fruit Tree Science; Okitsu Nakacho, Shizuoka, 424-0292 Japan
BMB2013(第36回日本分子生物学会年会)ポスター 3P-0030
2013年12月5日
TEDx Manchester: AI & The Future of WorkVolker Hirsch
TEDx Manchester talk on artificial intelligence (AI) and how the ascent of AI and robotics impacts our future work environments.
The video of the talk is now also available here: https://youtu.be/dRw4d2Si8LA
The document provides information about linear equations in two variables of the form ax + by + c = 0. It gives two example equations, 3x - y = 7 and 2x + 3y = 1, and asks questions about their coefficients, constant terms, and solutions for given x- and y-values. It then shows the graphs of the two equations and finds their intersection point. It also solves systems of two linear equations and finds their intersection points. Finally, it provides two homework problems about finding the points where a line intersects the axes and finding the vertices and area of a rectangle.
IIT Jam math 2016 solutions BY TrajectoryeducationDev Singh
The document contains a mathematics exam question paper with 10 single mark questions (Q1-Q10) and 20 two mark questions (Q11-Q30). The questions cover topics like sequences, linear transformations, integrals, permutations, differential equations etc. Some key questions asked about the nature of a sequence involving sines, order of a permutation, evaluating a limit, checking if a differential equation is exact etc. and provided solutions to them.
Calculus Early Transcendentals 10th Edition Anton Solutions Manualnodyligomi
This document contains a table of contents for a calculus textbook, outlining 15 chapters that cover topics from limits and continuity to vector calculus and partial derivatives. It also includes 3 appendix sections on graphing functions, trigonometry review, and solving polynomial equations.
Different kind of distance and Statistical DistanceKhulna University
A short brief of distance and statistical distance which is core of multivariate analysis.................you will get here some more simple conception about distances and statistical distance.
This document discusses concepts related to 2D geometry including:
- Formulas for calculating the distance between two points and the slope of a line joining two points.
- Formulas for finding the midpoint, internal and external divisions of a line segment, and the area of a triangle.
- Relationships between the slopes of parallel and perpendicular lines and determining if three points are collinear based on slopes.
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.
Howard, anton calculo i- um novo horizonte - exercicio resolvidos v1cideni
This document contains exercises related to functions and graphs. Exercise set 1.1 contains word problems involving various functional relationships and graphs. Exercise set 1.2 involves evaluating and sketching functions, determining domains and ranges, and identifying piecewise functions. Exercise set 1.3 involves selecting appropriate axis ranges and scales to graph functions over specified domains.
The document presents a Green's function-based method for transient analysis of multiconductor transmission lines. It begins with an introduction to existing time-domain modeling techniques and their issues. It then describes modeling transmission lines as a vector Sturm-Liouville problem and using the spectral representation of the Green's function to solve it. Numerical results are presented for lines with both frequency-independent and dependent parameters. The method provides a rational model representation of transmission line behavior.
This is the entrance exam paper for ISI MSQE Entrance Exam for the year 2008. Much more information on the ISI MSQE Entrance Exam and ISI MSQE Entrance preparation help available on http://crackdse.com
Amth250 octave matlab some solutions (2)asghar123456
This document contains the solutions to 5 questions regarding numerical analysis techniques. Question 1 finds the zeros of a function graphically and numerically. Question 2 finds the millionth zero of tan(x)-x. Question 3 examines the convergence rates of Newton's method for various functions. Question 4 applies Newton's method to find the inverse of a function. Question 5 finds the maximum of a function using golden section search and parabolic interpolation.
FITTED OPERATOR FINITE DIFFERENCE METHOD FOR SINGULARLY PERTURBED PARABOLIC C...ieijjournal
In this paper, we study the numerical solution of singularly perturbed parabolic convection-diffusion type
with boundary layers at the right side. To solve this problem, the backward-Euler with Richardson
extrapolation method is applied on the time direction and the fitted operator finite difference method on the
spatial direction is used, on the uniform grids. The stability and consistency of the method were established
very well to guarantee the convergence of the method. Numerical experimentation is carried out on model
examples, and the results are presented both in tables and graphs. Further, the present method gives a more
accurate solution than some existing methods reported in the literature.
This document discusses operations on multiple random variables, including:
- The expected value of a function of two random variables X and Y is the sum of the expected values of the functions.
- Joint moments describe the relationship between multiple random variables and can be used to find properties like covariance and correlation.
- Two random variables are jointly Gaussian if their joint density function follows a specific form, and properties of Gaussian random variables include being fully defined by their first and second moments.
- Transformations of multiple random variables, such as applying a linear transformation, preserve properties like expected value and covariance if the original variables were Gaussian.
The Gaussian or normal distribution is one of the most widely used in statistics. Estimating its parameters using
Bayesian inference and conjugate priors is also widely used. The use of conjugate priors allows all the results to be
derived in closed form. Unfortunately, different books use different conventions on how to parameterize the various
distributions (e.g., put the prior on the precision or the variance, use an inverse gamma or inverse chi-squared, etc),
which can be very confusing for the student. In this report, we summarize all of the most commonly used forms. We
provide detailed derivations for some of these results; the rest can be obtained by simple reparameterization. See the
appendix for the definition the distributions that are used.
1. The document discusses various operations that can be performed on signals including time reversal, time shifting, time scaling, amplitude scaling, signal addition, and signal multiplication.
2. Examples are provided to demonstrate how to graphically represent signals and how the different operations change the signals.
3. Key steps are outlined for performing each operation including reversing the time axis, delaying or advancing signals, compressing or expanding the time axis, amplifying or attenuating signal amplitude, adding or multiplying signal values.
Second-order Cosmological Perturbations Engendered by Point-like MassesMaxim Eingorn
R. Brilenkov and M. Eingorn, Second-order cosmological perturbations engendered by point-like masses, ApJ 845 (2017) 153: http://iopscience.iop.org/article/10.3847/1538-4357/aa81cd
In the ΛCDM framework, presenting nonrelativistic matter inhomogeneities as discrete massive particles, we develop the second‐order cosmological perturbation theory. Our approach relies on the weak gravitational field limit. The derived equations for the second‐order scalar, vector, and tensor metric corrections are suitable at arbitrary distances, including regions with nonlinear contrasts of the matter density. We thoroughly verify fulfillment of all Einstein equations, as well as self‐consistency of order assignments. In addition, we achieve logical positive results in the Minkowski background limit. Feasible investigations of the cosmological backreaction manifestations by means of relativistic simulations are also outlined.
The document describes the decoding of BCH codes through syndrome calculation and Berlekamp's iterative algorithm. It discusses how the syndrome is calculated from the received vector, and how the syndrome components relate to the error pattern. Berlekamp's algorithm determines the error-location polynomial σ(x) iteratively by ensuring its coefficients satisfy the Newton identities relating it to the syndrome at each step, until σ(x) is obtained after 2t steps.
Additional Mathematics form 4 (formula)Fatini Adnan
This document provides a summary of various math formulae for Form 4 students in Malaysia, including:
1. Functions, quadratic equations, and quadratic functions
2. Simultaneous equations, indices and logarithms, and coordinate geometry
3. Statistics, circular measures, and differentiation
It lists common formulae for topics like the quadratic formula, completing the square, differentiation rules, and measures of central tendency and dispersion. The document is intended as a study guide for students to review essential formulae.
This document provides a summary of various math formulae for Form 4 students in Malaysia, including:
1. Functions, quadratic equations, and quadratic functions
2. Simultaneous equations, indices and logarithms, and coordinate geometry
3. Statistics, circular measures, and differentiation
It lists common formulae for topics like the quadratic formula, completing the square, differentiation rules, and measures of central tendency and dispersion. The document is intended as a study guide for students to review essential formulae.
This document provides a summary of various math formulae for Form 4 students in Malaysia, including:
1. Functions, quadratic equations, and quadratic functions
2. Simultaneous equations, indices and logarithms, and coordinate geometry
3. Statistics, circular measures, and differentiation
It lists common formulae for topics like the quadratic formula, completing the square, differentiation rules, and measures of central tendency and dispersion. The document is intended as a study guide for students to review essential formulae.
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Diffusion kernels on SNP data embedded in a non-Euclidean metric
1. Diffusion kernels on SNP data embedded in a
non-Euclidean metric
Animal Breeding & Genomics Seminar
Gota Morota
April 10, 2012
1 / 37
2. Kernel functions
Definition
A kernel is a weighting function which provides a similarity metric
1. define a function that measures distance (metric) for
genotypes
2. compute a similarity based on this metric space
⇓
function of a distance under certain metric space f(||x − x ||)
• Euclidean distance
• Manhattan distance
• Mahalanobis distance
• Minkowski distance
2 / 37
3. Metric (Distance function)
Definition
A function which defines a distance between two points
If one picks Euclidean metric, the Mat´ern covariance function
offers flexible kernels
K(x, x ) = σ2
K
21−ν
Γ(ν)
√
2ν(||x − x ||/h)ν
K(||x − x ||/h)
• Gaussian Kernel: ν = ∞, exp(−θ(||x − x ||2
))
• Exponentail Kernel: ν = 1
2 , exp(−θ(||x − x ||))
⇓
A choice of a metric determines characteristics of a kernel
3 / 37
4. Euclidean Metric
Definition
The distance function given by the Pythagorean theorem
(a2
+ b2
= c2
)
Euclidean distance on R2
xi = (xi1, xi2), xj = (xj1, xj2)
||xi − xj|| = (xi1 − xj1)2 + (xi2 − xj2)2
Figure 1: Euclidean distance
between two points A and B
Euclidean distance on Rp
||xi − xj|| = (xi1 − xj1)2 + · · · + (xik − xjk )2 + · · · + (xip − xjp)2
4 / 37
5. Euclidean space
Euclidean distance is a metric on a metric space callled Euclidean
space
Figure 2: 3-dimensional Euclidean
space. −∞ ≤ (X, Y, Z) ≤ ∞
Suppose, we observed two
individuals with 3 SNP
genotypes.
• ID1 = x1 = (0,2,2)
• ID2 = x2 = (2,1,0)
Euclidean distance on R3
||x1 − x2|| = (0 − 2)2 + (2 − 1)2 + (2 − 0)2 = 3
5 / 37
6. Metric on graphs
A graph is consisted of vertices and edges
0 1 2
012
0
1
2
1st Genotype
2ndGenotype
3rdGenotype
6 / 37
8. Metric on graphs (continue)
Two individuals with 3 SNP genotypes previously shown.
• ID1 = x1 = (0,2,2), ID2 = x2 = (2,1,0)
0 1 2
012
0
1
2
1st Genotype
2ndGenotype
3rdGenotype
(2,1,0)
(0,2,2)
8 / 37
9. The purpose of this study
1. Is the Euclidean distance adequate for genotypes?
2. The metric on graphs seems to be given by the Manhattan
distance, but how to express the degree of similarity?
• Embed SNP data in a non-Euclidean metric space
• Define a metric for discrete genotypes on graphs and
construct a kernel on this metric
⇓
Develope a kernel that is suited for all kinds of kernel-based
genomic analyses
9 / 37
10. Diffusion on one-dimensional graphs (Z1
3)
We have three possible genotypes, 0 (aa), 1 (Aa) and 2 (’AA’).
0 − 1 − 2 (1)
0 − 1
/
2
(2)
10 / 37
11. Diffusion on one-dimensional graphs (Z1
3)
We have three possible genotypes, 0 (aa), 1 (Aa) and 2 (’AA’).
0 − 1 − 2 (1)
0 − 1
/
2
(2)
1. Graph (1) path graph
• genotype 1’s (’Aa’) influence diffuses to genotype 0 (’aa’) and
2 (’AA’)
• genotype 0’s (’aa’) influence diffuses to only genotype 1 (’Aa’)
• genotype 2’s (’AA’) influence diffuses to only genotype 1 (’Aa’)
2. Graph (2) complete graph
• the distance from genotype 0 (’aa’) to genotype 2 (’AA’) is the
same as that from 0 (’aa’) to 1 (’Aa’).
10 / 37
12. Diffusion on one-dimensional graphs (Z1
3)
We have three possible genotypes, 0 (aa), 1 (Aa) and 2 (’AA’).
0 − 1 − 2 (1)
0 − 1
/
2
(2)
1. Graph (1) path graph
• genotype 1’s (’Aa’) influence diffuses to genotype 0 (’aa’) and
2 (’AA’)
• genotype 0’s (’aa’) influence diffuses to only genotype 1 (’Aa’)
• genotype 2’s (’AA’) influence diffuses to only genotype 1 (’Aa’)
2. Graph (2) complete graph
• the distance from genotype 0 (’aa’) to genotype 2 (’AA’) is the
same as that from 0 (’aa’) to 1 (’Aa’).
• more reasonable to assume that genotype ’Aa’ is closer than
’aa’ to ’AA’ which has two copies of the ’A’ allele.
• genotype 0 (’aa’) requires two mutations to become genotype
2 (’AA’), while genotype 1 (’Aa’) requires only one mutation 10 / 37
13. Diffusion on two-dimensional graphs (Z2
3)
Two-dimensional graphs are given by the Cartesian graph product
( ) of the 2 one-dimensional graphs 0 - 1 - 2.
0 − 1 − 2 0 − 1 − 2 (3)
Let Γ1 and Γ2 be two graphs. Consider a graph with vertex set
V(Γ1) × V(Γ2), with vertices (x, x ) ∈ V(Γ1) and (y, y ) ∈ V(Γ2).
Cartesian graph product
The Cartesian graph product connects two vertices (x, y) and
(x , y ) if only if x = x , y ∼ y or y = y , x ∼ x , where “∼” means
connected.
11 / 37
14. Example of the Cartesian graph product ( )
Cartesian graph product of the 2 one-dimensional graphs
0 − 1 − 2 0 − 1 − 2
Fisrt, list all possible configuration of vertices
02 12 22
01 11 21
00 10 20
12 / 37
15. Example of the Cartesian graph product ( ) (continue)
Cartesian graph product of the 2 one-dimensional graph
0 − 1 − 2 0 − 1 − 2
The Cartesian graph product connects two vertices (x, y) and
(x , y ) if only if x = x , y ∼ y or y = y , x ∼ x , where “∼” means
connected.
• 0 = 0, 0 ∼ 1 → connected
• 0 = 0, 1 ∼ 2 → connected
02 12 22
01 11 21
00 10 20
⇒
13 / 37
16. Example of the Cartesian graph product ( ) (continue)
Cartesian graph product of the 2 one-dimensional graph
0 − 1 − 2 0 − 1 − 2
The Cartesian graph product connects two vertices (x, y) and
(x , y ) if only if x = x , y ∼ y or y = y , x ∼ x , where “∼” means
connected.
• 0 = 0, 0 ∼ 1 → connected
• 0 = 0, 1 ∼ 2 → connected
02 12 22
01 11 21
00 10 20
⇒
13 / 37
17. Example of the Cartesian graph product ( ) (continue)
Cartesian graph product of the 2 one-dimensional graph
0 − 1 − 2 0 − 1 − 2
The Cartesian graph product connects two vertices (x, y) and
(x , y ) if only if x = x , y ∼ y or y = y , x ∼ x , where “∼” means
connected.
• 0 = 0, 0 ∼ 1 → connected
• 0 = 0, 1 ∼ 2 → connected
02 12 22
01 11 21
00 10 20
⇒
02 12 22
|
01 11 21
|
00 10 20
13 / 37
18. Example of the Cartesian graph product ( ) (continue)
Cartesian graph product of the 2 one-dimensional graphs
0 − 1 − 2 0 − 1 − 2
The Cartesian graph product connects two vertices (x, y) and
(x , y ) if only if x = x , y ∼ y or y = y , x ∼ x , where “∼” means
connected.
• 0 = 0, 0 ∼ 1 → connected
• 0 1, 0 1 → not connected
• 0 1, 0 2 → not connected
02 12 22
|
01 11 21
|
00 10 20
⇒
14 / 37
19. Example of the Cartesian graph product ( ) (continue)
Cartesian graph product of the 2 one-dimensional graphs
0 − 1 − 2 0 − 1 − 2
The Cartesian graph product connects two vertices (x, y) and
(x , y ) if only if x = x , y ∼ y or y = y , x ∼ x , where “∼” means
connected.
• 0 = 0, 0 ∼ 1 → connected
• 0 1, 0 1 → not connected
• 0 1, 0 2 → not connected
02 12 22
|
01 11 21
|
00 10 20
⇒
14 / 37
20. Example of the Cartesian graph product ( ) (continue)
Cartesian graph product of the 2 one-dimensional graphs
0 − 1 − 2 0 − 1 − 2
The Cartesian graph product connects two vertices (x, y) and
(x , y ) if only if x = x , y ∼ y or y = y , x ∼ x , where “∼” means
connected.
• 0 = 0, 0 ∼ 1 → connected
• 0 1, 0 1 → not connected
• 0 1, 0 2 → not connected
02 12 22
|
01 11 21
|
00 10 20
⇒
02 12 22
|
01 11 21
|
00 − 10 20
14 / 37
21. Diffusion on two-dimensional graphs (Z2
3) (continue)
A graph from the Cartesian graph product between path graphs of
any size takes the form of a grid.
02 − 12 − 22
| | |
01 − 11 − 21
| | |
00 − 10 − 20
A SNP grid of p loci is a p dimensional grid with vertices in Zp
3
, with
two vertices x and x adjacent if and only if
p
i=1
|xi − xi | = 1.
i.e., two vertices are adjacent if and only if just one SNP locus
differs by 1.
15 / 37
22. Diffusion on three-dimensional graphs (Z3
3
)
Cartesian graph product of the 3 one-dimensional graphs.
0 − 1 − 2 0 − 1 − 2 0 − 1 − 2
In general, the p-dimensional SNP grid graph is
p
i=1
Γ, where
Γ = 0 − 1 − 2.
0 1 2
012
0
1
2
1st Genotype
2ndGenotype
3rdGenotype
(2,1,2)
(2,0,1)
(0,1,2)
(0,2,0)
(0,1,0)
(0,1,1)
(1,0,0) (2,0,0)
(1,1,0) (2,1,0)
(1,2,0) (2,2,0)
(1,0,1)
(1,1,1) (2,1,1)
(0,2,1) (1,2,1) (2,2,1)
(0,2,2) (1,2,2) (2,2,2)
(1,0,2)
(1,1,2)
16 / 37
23. Graph Laplacians
The Laplacian of a graph 0 − 1 − 2 is
L(Γ) = −A(Γ) + Λ
= −
0 1 0
1 0 1
0 1 0
+
1 0 0
0 2 0
0 0 1
=
1 −1 0
−1 2 −1
0 −1 1
where A is an adjacency matrix and Λ is a diagonal matrix with
Λii = n
j=1 Aij.
17 / 37
25. Diffusion on graphs at time t
• kx is a function which measures the spread of ’influence’ of
the genotype x over other genotypes.
• k˜x(0, x) = 1x=˜x(x), at time 0.
• define the time t diffusion of the ’influence’ of genotype ˜x on
genotype x to be
k˜x(t, x) = k˜x(t − 1, x) +
|x−x |=1
α(k˜x(t − 1, x ) − k˜x(t − 1, x))
19 / 37
26. Diffusion on graphs at time t (continue)
k˜x(t, x) = k˜x(t − 1, x) +
|x−x |=1
α(k˜x(t − 1, x ) − k˜x(t − 1, x))
• x = (0, 1, 2) is the genotype code, α = (0.1, 0.2) is the
diffusion rate.
• k˜x(t, x) is the time t diffusion of the influence of genotype ˜x on
genotype x.
α= 0.1 α = 0.2
x = 0 1 2 x= 0 1 2
k1(0, x) 0 1 0 k1(0, x) 0 1 0
k1(1, x) 0.1 0.8 0.1 k1(1, x) 0.2 0.6 0.2
k1(2, x) 0.17 0.66 0.17 k1(2, x) 0.28 0.44 0.28
k1(3, x) 0.219 0.562 0.219 k1(3, x) 0.312 0.376 0.312
k1(15, x) 0.331 0.336 0.331 k1(15, x) 0.333 0.333 0.333
20 / 37
27. Diffusion on graphs at time t (continue)
Writing in vector form, with k˜x(t, x) = [k˜x(t)]x, we get
k˜x(t) = k˜x(t − 1) + αHk˜x(t − 1)
= (I + αH)k˜x(t − 1)
= (I + αH)t
k˜x(0)
• H is the negative of the graph Laplacian
• in order to make ’time’ continuous, let α = θh (θ > 0) and
t = 1/h.
• by using a small h, we can achieve a discretization of the
’diffusion time’
lim
h→0
(I + θhH(Γ))1/h
= exp(θH)
=
∞
k=0
θk
k!
Hk
= I + θH +
θ2
2
H2
+
θ3
3!
H3
+ · · · +
θn
n!
Hn
+ · · ·
21 / 37
28. Diffusion kernels
Definition
Suppose a graph Γ with a graph Laplacian L(Γ). Then exp(θH(Γ))
or exp(−θL(Γ)) is called the diffusion kernel or heat kernel for
graph Γ, where θ is a rate of diffusion.
Here putting K = exp(θH) and taking the derivative with respect to
θ gives,
d
dθ
K = HK (4)
which is a diffusion equation (heat equation) on a graph with
H = −L(Γ).
22 / 37
29. Gaussian kernels
Definition
A Gaussian kernel is a space continuous diffusion kernel
• in order to make ’space’ continuous, we create an infinite
number of ’fake’ genotypes between and outside of 0 and 2
• i.e., consider genotypes such as 1.23 or −10.5.
• each genotype x is connected to only two genotypes, x + dx
and x − dx for some infinitesimal dx.
• H becomes an infinite matrix, and H(x, x ) is −2 for x = x
and 1 for x + dx, x − dx.
H(Γ) =
−1 1 0
1 −2 1
0 1 −1
⇒
Infinite matrix with diagonal
elements equal to -2 and 1 for its
neighbors and 0 otherwise
23 / 37
30. Gaussian kernels (continue)
• a vector of genotypes: x = (−∞, · · · , x − dx, x, x + dx, · · · , ∞)
• an influence function:
f = (f(−∞), · · · , f(x − dx), f(x), f(x + dx), · · · , f(∞))
• Approximating dx by h, and dividing H by h2
, HfT
/h2
indexed
by the genotype x will be
1
h2
[H(x, ·)fT
] =
f(x + h) − 2f(x) + f(x − h)
h2
=
f(x+h)−f(x)
h −
f(x)−f(x−h)
h
h
f (x)
• Thus, with space continuity, H acts like d
dx2 . Using this analogy
back in (4), we get the heat equation.
d
dθ
Kθ(x) =
d
dx2
Kθ(x)
24 / 37
31. Gaussian kernels (continue)
• The solution to this partial differential equation (PDE) with
Dirac delta initial condition of concentration on x = 0,
k0(x) = 1x=0, is given by
Gθ(x) =
1
√
4πθ
exp −
x2
4θ
• This is a Gaussian density in one dimensional space with
θ = σ2
e/2.
• With the initial condition K0(x) = f(x), the solution to this PDE
is
Kθ(x) =
R
f(x )Gθ(x − x )dx
This kernel gθ(x, x ) = G(x − x ) is the Gaussian kernel with
bandwidth θ.
25 / 37
32. Gaussian kernels (continue)
• For example, allowing additional genotypes (0.25, 0.50, 0.75,
1.25, 1.50, 1.75).
• now, x ∈ R9
instead of x ∈ Z3
0 1 2
012
0
1
2
1st Genotype
2ndGenotype
3rdGenotype
(0,1.75,2)
26 / 37
33. Computation of diffusion kernels
Kernel notation:
• K as the kernel matrix indexed by the observed covariates
• K for the infinite dimensional kernel for the Gaussian, and the
3p
× 3p
dimensional kernel for the diffusion kernel
Gaussian kernels
• K: infinite dimensional
kernel
• K = exp(−θ(||x − x ||2
))
• we have a closed form for K,
so no need to deal with K
Diffusion kernels
• K: 3p
× 3p
dimensional
kernel
• K: is there any way to
directly compute K so that
we don’t need to deal with
K?
• closed form for K?
27 / 37
34. Computation of diffusion kernels (continue)
Let K1(θ) and K2(θ) be the kernels for the two graphs Γ1 and Γ2.
The diffusion kernel for Γ = Γ1 Γ2 is
K1(θ) ⊗ K2(θ).
were ⊗ is the tensor product.
Suppose, Γ1 = 0 − 1 − 2, K(Γ1) is a diffusion kernel on Γ1.
SNP grid graph on p dimensions
• p
i=1
Γ1
SNP grid kernel on p dimensions
•
p
i=1 K(Γ1)
⇓
We just need to compute K(Γ1) = exp(−θL(Γ1)) and take the
tensor product p times!
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35. Matrix exponentiation
Γ1 = 0 − 1 − 2
H =
1 −1 0
−1 2 −1
0 −1 1
We make use of matrix diagonalization H = TDT−1
to obtain
Kθ = exp(θH)
= T exp(θD)T−1
=
1
6
e−3θ + 3e−θ + 2 −2e−3θ + 2 e−3θ − 3e−θ + 2
−2e−3θ + 2 4e−3θ + 2 −2e−3θ + 2
e−3θ − 3e−θ + 2 −2e−3θ + 2 e−3θ + 3e−θ + 2
Here, exp(θD) becomes simple componentwise exponentiation
because D is a diagonal matrix of eigenvalues.
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36. Diffusion kernels indexed by the observed covariates
Symmetric property
Kθ(x, x ) =
−2e−3θ + 2 if |xi − xi
| = 1
e−3θ − 3e−θ + 2 if |xi − xi
| = 2
e−3θ + 3e−θ + 2 if xi = xi
, x 1
4e−3θ + 2 if xi = xi
= 1
Thus,
K
⊗p
θ (x, x ) ∝
p
i=1
(e−3θ
− 3e−θ
+ 2)δ|xi−xi
|=2 + (−2e−3θ
+ 2)δ|xi−xi
|=1
+ (e−3θ
+ 3e−θ
+ 2)δxi=xi
1 + (4e−3θ
+ 2)δxi=xi
=1
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37. Diffusion kernels indexed by the observed covariates
(continue)
• let x and x be an SNP data for p loci; ns be the number of loci
for which |xi − xi
| = s
• let m11 be the number of loci for which xi = xi
= 1, i.e., m11 is
the number of loci that two individuals share heterozygous
states.
Using the fact that
n1 + n0 + n2 = p,
K
⊗p
θ (x, x ) =(−2e−3θ
+ 2)n1
(e−3θ
− 3e−θ
+ 2)n2
(e−3θ
+ 3e−θ
+ 2)n0−m11
(4e−3θ
+ 2)m11
∝
(−2e−3θ + 2)n1 (e−3θ − 3e−θ + 2)n2 (4e−3θ + 2)m11
(e−3θ + 3e−θ + 2)n1+n2+m11
We obtain a SNP grid kernel.
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38. Example of computing a diffusion kernel
Two individuals with 3 SNP genotypes previously shown.
• ID1 = x1 = (0,2,2)
• ID2 = x2 = (2,1,0)
0 1 2
012
0
1
2
1st Genotype
2ndGenotype
3rdGenotype
(2,1,0)
(0,2,2)
Since
Kθ(x, x ) =
−2e−3θ + 2 if |xi − xi
| = 1
e−3θ − 3e−θ + 2 if |xi − xi
| = 2
e−3θ + 3e−θ + 2 if xi = xi
, x 1
4e−3θ + 2 if xi = xi
= 1
Similarity between ID1 and ID2 is
K⊗3
θ (x, x ) =
(−2e−3θ + 2)1
(e−3θ − 3e−θ + 2)2
(e−3θ + 3e−θ + 2)1+2
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39. Diffusion kernels for binary genotypes
Here, x ∈ Zp
2
Γ = 0 − 2
L(Γ) = −H(Γ)
=
1 −1
−1 1
K
⊗p
θ (x, x ) ∝
1 − exp(−2θ)
1 + 2 exp(−2θ)
d(x,x )
where d(x, x ) is the Hamming distance, that is, number of
coordinates at which x and x differ.
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40. Applications
A SNP kernel can be used in DNA-based genomic analyses
including
• regressions
• classifications
• kernel association studies
• kernel principal component analyses
Application of using the diffusion kernel on real data
• 7902 Holstein bulls (USDA-ARS AIPL)
• 43382 SNPs
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41. Diffusion kernels based on for different θ
K(i,i')
Frequency
0.10 0.15 0.20 0.25 0.30 0.35
0.0e+001.0e+07
θ = 10
K(i,i')
Frequency
0.45 0.50 0.55 0.60 0.65 0.70
0.0e+001.0e+07
θ = 11
K(i,i')
Frequency
0.74 0.78 0.82 0.86
0.0e+006.0e+061.2e+07
θ = 12
K(i,i')
Frequency
0.90 0.91 0.92 0.93 0.94 0.95
0.0e+006.0e+061.2e+07
θ = 13
Figure 8: Elements of four diffusion kernels based on four different
bandwidth parameters (θ).
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42. Conclusion
Diffusion kernels
• various graph structures can be used to represent sets of
discrete random variables, such as genotypes
• defines the distance between two vertices, and projects this
information into a more interpretable Rn
• matrix exponentiation of the graph Laplacian
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43. Conclusion
Diffusion kernels
• various graph structures can be used to represent sets of
discrete random variables, such as genotypes
• defines the distance between two vertices, and projects this
information into a more interpretable Rn
• matrix exponentiation of the graph Laplacian
• which senario, the Gaussian can approximate the diffusion
kernel well?
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44. Conclusion
Diffusion kernels
• various graph structures can be used to represent sets of
discrete random variables, such as genotypes
• defines the distance between two vertices, and projects this
information into a more interpretable Rn
• matrix exponentiation of the graph Laplacian
• which senario, the Gaussian can approximate the diffusion
kernel well?
R package ’dkDNA’ will be available on CRAN soon
• SNP grid kernel
• binary grid kernel
• other DNA structures/polymorphisms in future
• written in Fortran
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