TANA07: Data Mining using Matrix Methods
... to the text collection contained in files of directory (or file) FILENAME. Each document must be separeted by a blank line (or another delimiter that is defined by OPTIONS argument) in each file. [A, DICTIONARY] = TMG(FILENAME) returns also the dictionary for the ...
... to the text collection contained in files of directory (or file) FILENAME. Each document must be separeted by a blank line (or another delimiter that is defined by OPTIONS argument) in each file. [A, DICTIONARY] = TMG(FILENAME) returns also the dictionary for the ...
3. Linear function
... given below). Linear functions are the main functions in linear algebra. We study them in ...
... given below). Linear functions are the main functions in linear algebra. We study them in ...
An Arithmetic for Matrix Pencils: Theory and New Algorithms
... • For a matrix pencil by λE − A, a nonzero vector x ∈ Cn is an eigenvector if for some nonzero pair (ε, α) ∈ C \ {(0, 0)} εEx = αAx. If α = 0, then x corresponds to an infinite eigenvalue. If α 6= 0, then x corresponds to the finite eigenvalue λ = ε/α. • The columns of X ∈ Cn×k span a a right deflat ...
... • For a matrix pencil by λE − A, a nonzero vector x ∈ Cn is an eigenvector if for some nonzero pair (ε, α) ∈ C \ {(0, 0)} εEx = αAx. If α = 0, then x corresponds to an infinite eigenvalue. If α 6= 0, then x corresponds to the finite eigenvalue λ = ε/α. • The columns of X ∈ Cn×k span a a right deflat ...
Modeling and learning continuous-valued stochastic processes with
... If A = (R m ; (a)a2E ; w0) is a minimal-dimensional OOM of X , then m is called the dimension of the process X . The dimension of a process is a fundamental characteristic of its stochastic complexity (see [9] for a probability theoretic account of this dimension). If A is a minimal-dimensional OO ...
... If A = (R m ; (a)a2E ; w0) is a minimal-dimensional OOM of X , then m is called the dimension of the process X . The dimension of a process is a fundamental characteristic of its stochastic complexity (see [9] for a probability theoretic account of this dimension). If A is a minimal-dimensional OO ...
MATH10212 Linear Algebra Lecture Notes Textbook
... no free variables, or (ii) infinitely many solutions, when there is at least one free variable. Using row reduction to solve a linear system 1. Write the augmented matrix of the system. 2. Use the row reduction algorithm to obtain an equivalent augmented matrix in echelon form. Decide whether the sy ...
... no free variables, or (ii) infinitely many solutions, when there is at least one free variable. Using row reduction to solve a linear system 1. Write the augmented matrix of the system. 2. Use the row reduction algorithm to obtain an equivalent augmented matrix in echelon form. Decide whether the sy ...
CHAPTER 2: Linear codes
... Are there some other cosets in this case? Theorem Suppose C is a linear [n,k] -code over GF(q). Then (a) every vector of V(n,k) is in some coset of C, (b) every coset contains exactly qk elements, (c) two cosets are either disjoint or identical. Linear codes ...
... Are there some other cosets in this case? Theorem Suppose C is a linear [n,k] -code over GF(q). Then (a) every vector of V(n,k) is in some coset of C, (b) every coset contains exactly qk elements, (c) two cosets are either disjoint or identical. Linear codes ...
CHAPTER 2: Linear codes
... Are there some other cosets in this case? Theorem Suppose C is a linear [n,k] -code over GF(q). Then (a) every vector of V(n,k) is in some coset of C, (b) every coset contains exactly qk elements, (c) two cosets are either disjoint or identical. Linear codes ...
... Are there some other cosets in this case? Theorem Suppose C is a linear [n,k] -code over GF(q). Then (a) every vector of V(n,k) is in some coset of C, (b) every coset contains exactly qk elements, (c) two cosets are either disjoint or identical. Linear codes ...
Incremental Eigenanalysis for Classification
... for examples. Note that the rank of the covariance matrx may be less than the number of observations. An alternative approach to computing the eigenmodel is to use singular value decomposition (SVD). We must also make clear the difference between batch and incremental methods for computing eigenspac ...
... for examples. Note that the rank of the covariance matrx may be less than the number of observations. An alternative approach to computing the eigenmodel is to use singular value decomposition (SVD). We must also make clear the difference between batch and incremental methods for computing eigenspac ...
Chapter 1 - Princeton University Press
... there are only two dates, which we will call today and tomorrow, but which could equally well be called this week and next week, this year and next year, or now and in 10 min. The essential feature of our two-date, one-period model is that no investment decisions are taken between the two dates. One ...
... there are only two dates, which we will call today and tomorrow, but which could equally well be called this week and next week, this year and next year, or now and in 10 min. The essential feature of our two-date, one-period model is that no investment decisions are taken between the two dates. One ...
Random Projection Estimation of Discrete-Choice Models
... The ideas of random projection were popularized in the Machine Learning literature on dimensionality reduction (Achlioptas (2003); Dasgupta and Gupta (2003); Vempala (2000)). As these papers point out, both by mathematical derivations and computational simulations, random projection allows computati ...
... The ideas of random projection were popularized in the Machine Learning literature on dimensionality reduction (Achlioptas (2003); Dasgupta and Gupta (2003); Vempala (2000)). As these papers point out, both by mathematical derivations and computational simulations, random projection allows computati ...
On the Comparison of Fisher Information of the Weibull and GE
... Table 5: The trace of the Fisher information matrices of WE(β,1) and GE(α̃, λ̃) are reported in columns 4 and 5, when the data are right truncated at T ≈ mean and the loss of information from the complete sample are also presented in the same column. The corresponding total asymptotic variances of t ...
... Table 5: The trace of the Fisher information matrices of WE(β,1) and GE(α̃, λ̃) are reported in columns 4 and 5, when the data are right truncated at T ≈ mean and the loss of information from the complete sample are also presented in the same column. The corresponding total asymptotic variances of t ...
On the Kemeny constant and stationary distribution vector
... well–known, the iterates of a Markov chain with transition matrix A converge to the (unique) left Perron vector w of A, normalised so that wT 1 = 1. That eigenvector w, which is known as the stationary distribution vector for the Markov chain, thus carries information about the long–term behaviour o ...
... well–known, the iterates of a Markov chain with transition matrix A converge to the (unique) left Perron vector w of A, normalised so that wT 1 = 1. That eigenvector w, which is known as the stationary distribution vector for the Markov chain, thus carries information about the long–term behaviour o ...
Chapter 4 Linear codes
... If a generator matrix in standard form exists for a linear code C, it is unique, and any generator matrix can be brought to the standard from by the following operations: (R1) Permutation of rows. (R2) Multiplication of a row by a non-zero scalar. (R3) Adding a scalar multiple of one row to another ...
... If a generator matrix in standard form exists for a linear code C, it is unique, and any generator matrix can be brought to the standard from by the following operations: (R1) Permutation of rows. (R2) Multiplication of a row by a non-zero scalar. (R3) Adding a scalar multiple of one row to another ...
Introduction to Linear Algebra using MATLAB Tutorial
... MATLAB can be used in two basic modes. In the Command Window, you can use it interactively; you type a command or expression and get an immediate result. You can also write programs, using scripts and functions (both of which are stored in M-files). This document does not describe the programming co ...
... MATLAB can be used in two basic modes. In the Command Window, you can use it interactively; you type a command or expression and get an immediate result. You can also write programs, using scripts and functions (both of which are stored in M-files). This document does not describe the programming co ...
Linear Combinations and Linearly Independent Sets of Vectors
... • Let p(x) = x2 − 3x + 2 and q(x) = 2x2 − 1. To see if S = {p(x),q(x)} is linearly independent, we set a linear combination of the vectors equal to This gives us a set of equations, one for each power of x: ...
... • Let p(x) = x2 − 3x + 2 and q(x) = 2x2 − 1. To see if S = {p(x),q(x)} is linearly independent, we set a linear combination of the vectors equal to This gives us a set of equations, one for each power of x: ...
Matrix Decomposition and its Application in Statistics
... Thus LU decomposition is not unique. Since we compute LU decomposition by elementary transformation so if we change L then U will be changed such that A=LU To find out the unique LU decomposition, it is necessary to put some restriction on L and U matrices. For example, we can require the lower tria ...
... Thus LU decomposition is not unique. Since we compute LU decomposition by elementary transformation so if we change L then U will be changed such that A=LU To find out the unique LU decomposition, it is necessary to put some restriction on L and U matrices. For example, we can require the lower tria ...
Max algebra and the linear assignment problem
... by a ⊕ b := max(a, b) and a ⊗ b := a + b offers an attractive way for modelling discrete event systems and optimization problems in production and transportation. Moreover, it shows a strong similarity to classical linear algebra: for instance, it allows a consideration of linear equation systems an ...
... by a ⊕ b := max(a, b) and a ⊗ b := a + b offers an attractive way for modelling discrete event systems and optimization problems in production and transportation. Moreover, it shows a strong similarity to classical linear algebra: for instance, it allows a consideration of linear equation systems an ...
Research Article Missing Value Estimation for
... Subsequently, the missing values in every target gene are estimated by matrix B and the coefficient vector X. LLS is put into an iterative framework in the proposed method; that is, the estimated values by LLS are reused to form the temporary matrix in every iteration, and matrices A and B are refin ...
... Subsequently, the missing values in every target gene are estimated by matrix B and the coefficient vector X. LLS is put into an iterative framework in the proposed method; that is, the estimated values by LLS are reused to form the temporary matrix in every iteration, and matrices A and B are refin ...
Chapter 3 Linear Codes
... code C in Fqn is a q n−k × q k array listing all the cosets of Fqn in which the first row consists of the code C with 0 on the extreme left, and the other rows are the cosets ui + C, each arranged in corresponding order, with the coset leader on the left. Remark. The standard array may be constructe ...
... code C in Fqn is a q n−k × q k array listing all the cosets of Fqn in which the first row consists of the code C with 0 on the extreme left, and the other rows are the cosets ui + C, each arranged in corresponding order, with the coset leader on the left. Remark. The standard array may be constructe ...
Compressed sensing and best k-term approximation
... optimality as expressed by (1.11). In this section, we shall see that (1.11) can be reformulated as a property of the null space N of Φ. As it was already remarked in the proof of Lemma 2.1, this null space has codimension not larger than n. We shall also need to consider sections of Φ obtained by k ...
... optimality as expressed by (1.11). In this section, we shall see that (1.11) can be reformulated as a property of the null space N of Φ. As it was already remarked in the proof of Lemma 2.1, this null space has codimension not larger than n. We shall also need to consider sections of Φ obtained by k ...
Eigen Values & Eigen Vectors
... Name of the Topic Matrices and Linear system of equations: Elementary row transformations – Rank – Echelon form, Normal form – Solution of Linear Systems – Direct Methods – LU Decomposition from Gauss Elimination – Solution of Tridiagonal systems – Solution of Linear Systems. Eigen values, Eigen vec ...
... Name of the Topic Matrices and Linear system of equations: Elementary row transformations – Rank – Echelon form, Normal form – Solution of Linear Systems – Direct Methods – LU Decomposition from Gauss Elimination – Solution of Tridiagonal systems – Solution of Linear Systems. Eigen values, Eigen vec ...
A Random Matrix–Theoretic Approach to Handling Singular
... the fact that exact statistics are not available. The first term typically decreases with increasing dimensionality parameter, L, which the second term increases with L. Instead of performing the estimation using one value of the dimensionality-reducing matrix, Φ, one can average the estimator (17) ...
... the fact that exact statistics are not available. The first term typically decreases with increasing dimensionality parameter, L, which the second term increases with L. Instead of performing the estimation using one value of the dimensionality-reducing matrix, Φ, one can average the estimator (17) ...
Bernard Hanzon and Ralf L.M. Peeters, “A Faddeev Sequence
... linear dynamical models the Fisher information matrix is in fact a Riemannian metric tensor and it can also be obtained in symbolic form by solving a number of Lyapunov and Sylvester equations. For further information on these issues the reader is referred to [9, 4, 5]. One straightforward approach ...
... linear dynamical models the Fisher information matrix is in fact a Riemannian metric tensor and it can also be obtained in symbolic form by solving a number of Lyapunov and Sylvester equations. For further information on these issues the reader is referred to [9, 4, 5]. One straightforward approach ...
Ordinary least squares
In statistics, ordinary least squares (OLS) or linear least squares is a method for estimating the unknown parameters in a linear regression model, with the goal of minimizing the differences between the observed responses in some arbitrary dataset and the responses predicted by the linear approximation of the data (visually this is seen as the sum of the vertical distances between each data point in the set and the corresponding point on the regression line - the smaller the differences, the better the model fits the data). The resulting estimator can be expressed by a simple formula, especially in the case of a single regressor on the right-hand side.The OLS estimator is consistent when the regressors are exogenous and there is no perfect multicollinearity, and optimal in the class of linear unbiased estimators when the errors are homoscedastic and serially uncorrelated. Under these conditions, the method of OLS provides minimum-variance mean-unbiased estimation when the errors have finite variances. Under the additional assumption that the errors be normally distributed, OLS is the maximum likelihood estimator. OLS is used in economics (econometrics), political science and electrical engineering (control theory and signal processing), among many areas of application. The Multi-fractional order estimator is an expanded version of OLS.