Introduction to Estimation Theory
... variables in question. However, when going from minimum variance and MAP to ML we relaxed the statistical assumptions by considering we knew nothing about the statistics of the variable(s) of interest (w, in that case). Relaxing even further the statistical assumptions for the estimation problem tak ...
... variables in question. However, when going from minimum variance and MAP to ML we relaxed the statistical assumptions by considering we knew nothing about the statistics of the variable(s) of interest (w, in that case). Relaxing even further the statistical assumptions for the estimation problem tak ...
Sketching as a Tool for Numerical Linear Algebra
... Ohm's law V = R ∙ I Find linear function that best fits the data ...
... Ohm's law V = R ∙ I Find linear function that best fits the data ...
A recursive algorithm for computing Cramer-Rao
... [e,;.., 19,]' a real, nonrandom parameter vector residing in columns of the n X n identity matrix, i.e., 8 = [_el;-., _ep],and E, x 8,.Let {P,}, E be a family of probability mea- is the jth unit column vector in R". Using a standard identity for 0 = 0,x sures for a certain random-variable Y taking v ...
... [e,;.., 19,]' a real, nonrandom parameter vector residing in columns of the n X n identity matrix, i.e., 8 = [_el;-., _ep],and E, x 8,.Let {P,}, E be a family of probability mea- is the jth unit column vector in R". Using a standard identity for 0 = 0,x sures for a certain random-variable Y taking v ...
xi. linear algebra
... This leads to a problem, that eigenvalues are not necessarily real numbers, since the term under the radical is not necessarily positive. What we are interested in is the modulus of any complex eigenvalues. If we have a complex number z = x + yi , where x and y are real scalars, the modulus or magn ...
... This leads to a problem, that eigenvalues are not necessarily real numbers, since the term under the radical is not necessarily positive. What we are interested in is the modulus of any complex eigenvalues. If we have a complex number z = x + yi , where x and y are real scalars, the modulus or magn ...
4.3 Least Squares Approximations
... Other least squares problems have more than two unknowns. Fitting by the best parabola has n D 3 coefficients C; D; E (see below). In general we are fitting m data points by n parameters x1 ; : : : ; xn . The matrix A has n columns and n < m. The derivatives of kAx bk2 give the n equations A T Ab ...
... Other least squares problems have more than two unknowns. Fitting by the best parabola has n D 3 coefficients C; D; E (see below). In general we are fitting m data points by n parameters x1 ; : : : ; xn . The matrix A has n columns and n < m. The derivatives of kAx bk2 give the n equations A T Ab ...
SECOND-ORDER VERSUS FOURTH
... where C is the contraction of the SSQ tensor on the matrix A H M A, i.e C = K (A H M A). The last relation is strikingly similar to the one obtained at 2nd-order with Q (M) in place of the array output covariance matrix and C in place of the signal covariance. Since only 4th-order cumulants are used ...
... where C is the contraction of the SSQ tensor on the matrix A H M A, i.e C = K (A H M A). The last relation is strikingly similar to the one obtained at 2nd-order with Q (M) in place of the array output covariance matrix and C in place of the signal covariance. Since only 4th-order cumulants are used ...
Error in dot products; forward and backward error
... bound does not necessarily hold for Strassen’s algorithm). Algorithms whose computed results in floating point correspond to a small relative backward error, such as the standard dot-product and matrix-vector multiplication algorithm, are said to be backward stable. ...
... bound does not necessarily hold for Strassen’s algorithm). Algorithms whose computed results in floating point correspond to a small relative backward error, such as the standard dot-product and matrix-vector multiplication algorithm, are said to be backward stable. ...
A. Melino Large Sample Theory ECO 327 - Lecture Notes
... Application to random variables In statistical applications, we usually have a sample of observations, say ((y1,X1),(y2,X2),...(yn,Xn)). Suppose we think of this as just the first n observations from an infinite sample which we call T. [The set all possible samples is called the sample space and is ...
... Application to random variables In statistical applications, we usually have a sample of observations, say ((y1,X1),(y2,X2),...(yn,Xn)). Suppose we think of this as just the first n observations from an infinite sample which we call T. [The set all possible samples is called the sample space and is ...
The Fundamental Theorem of Linear Algebra
... is not perpendicular to C(A). Then, by the Pythagoras theorem, ke0 k2 = kek2 + kp − p0 k2 > kek2 , so ke0 k > kek for any e0 6= e (see fig. 5). Thus, e is smallest when it is perpendicular to C(A). We need to find such x̂ that e is smallest. e is smallest when it’s orthogonal to C(A), i.e. to all ve ...
... is not perpendicular to C(A). Then, by the Pythagoras theorem, ke0 k2 = kek2 + kp − p0 k2 > kek2 , so ke0 k > kek for any e0 6= e (see fig. 5). Thus, e is smallest when it is perpendicular to C(A). We need to find such x̂ that e is smallest. e is smallest when it’s orthogonal to C(A), i.e. to all ve ...
Matrix
... • There is an m x n matrix 0, such that 0 + A = A for each A • There is an m x n matrix, -A, such that A + (-A) = 0 for each A • k(A + B) = kA + kB • (k+p)A = kA + pA • (kp)A = k(pA) ...
... • There is an m x n matrix 0, such that 0 + A = A for each A • There is an m x n matrix, -A, such that A + (-A) = 0 for each A • k(A + B) = kA + kB • (k+p)A = kA + pA • (kp)A = k(pA) ...
Linear Algebra - 1.4 The Matrix Equation Ax=b
... Matrix-Vector Multiplication Key Concepts to Master Linear combinations can be viewed as a matrix-vector multiplication. Matrix-Vector Multiplication If A is an m × n matrix, with columns a1 , a2 , . . . , an , and if x is in Rn , then the product of A and x, denoted by Ax, is the linear combination ...
... Matrix-Vector Multiplication Key Concepts to Master Linear combinations can be viewed as a matrix-vector multiplication. Matrix-Vector Multiplication If A is an m × n matrix, with columns a1 , a2 , . . . , an , and if x is in Rn , then the product of A and x, denoted by Ax, is the linear combination ...
Slides 2
... The form of Equation 1 looks similar to the linear modelling approach. In particular, we are still assuming a linear relationship between the latent variable Yi∗ and the regressors of the model Xi . The only difference is that we do not observe whether or not Yi∗ is positive. This means that we can ...
... The form of Equation 1 looks similar to the linear modelling approach. In particular, we are still assuming a linear relationship between the latent variable Yi∗ and the regressors of the model Xi . The only difference is that we do not observe whether or not Yi∗ is positive. This means that we can ...
Orthogonal Projections and Least Squares
... (2) If S is a subspace of the inner product space V , then S ⊥ is also a subspace of V . Proof: (1.) Note that 0+0 = 0 is in U ⊕V . Now suppose w1 , w2 ∈ U ⊕V , then w1 = u1 +v1 and w2 = u2 +v2 with ui ∈ U and vi ∈ V and w1 + w2 = (u1 + v1 ) + (u2 + v2 ) = (u1 + u2 ) + (v1 + v2 ). Since U and V are ...
... (2) If S is a subspace of the inner product space V , then S ⊥ is also a subspace of V . Proof: (1.) Note that 0+0 = 0 is in U ⊕V . Now suppose w1 , w2 ∈ U ⊕V , then w1 = u1 +v1 and w2 = u2 +v2 with ui ∈ U and vi ∈ V and w1 + w2 = (u1 + v1 ) + (u2 + v2 ) = (u1 + u2 ) + (v1 + v2 ). Since U and V are ...
Leslie and Lefkovitch matrix methods
... matrix projection reflect the density dependence we know exists in the real world? Obviously, we need to find a way to have the mx and px respond to density, rather than remaining fixed. To add density dependence to the dynamics we construct another matrix to represent density effects. While it is p ...
... matrix projection reflect the density dependence we know exists in the real world? Obviously, we need to find a way to have the mx and px respond to density, rather than remaining fixed. To add density dependence to the dynamics we construct another matrix to represent density effects. While it is p ...
3 5 2 2 3 1 3x+5y=2 2x+3y=1 replace with
... At equilibrium, the amount paid by each sector for the resources it needs will equal the income it earns from the sale of its output. We assume all output is consumed by the sectors (no surplus, no shortage). To determine what prices should be charged by each sector in this equilibrium, we solve a s ...
... At equilibrium, the amount paid by each sector for the resources it needs will equal the income it earns from the sale of its output. We assume all output is consumed by the sectors (no surplus, no shortage). To determine what prices should be charged by each sector in this equilibrium, we solve a s ...
Improved bounds on sample size for implicit matrix trace estimators
... The matrix-dependent bound (10), proved to be sufficient in Theorem 3, provides additional information over (5) about the type of matrices for which the Gaussian estimator is (probabilistically) guaranteed to require only a small sample size: if the eigenvalues of an SPSD matrix are distributed such ...
... The matrix-dependent bound (10), proved to be sufficient in Theorem 3, provides additional information over (5) about the type of matrices for which the Gaussian estimator is (probabilistically) guaranteed to require only a small sample size: if the eigenvalues of an SPSD matrix are distributed such ...
Population structure identification
... • Every n by n symmetric matrix Σ has an eigenvector decomposition Σ=QDQT where D is a diagonal matrix containing eigenvalues of Σ and the columns of Q are the eigenvectors of Σ. • Every m by n matrix A has a singular value decomposition A=USVT where S is m by n matrix containing singular values of ...
... • Every n by n symmetric matrix Σ has an eigenvector decomposition Σ=QDQT where D is a diagonal matrix containing eigenvalues of Σ and the columns of Q are the eigenvectors of Σ. • Every m by n matrix A has a singular value decomposition A=USVT where S is m by n matrix containing singular values of ...
Faster Dimension Reduction By Nir Ailon and Bernard Chazelle
... space via a linear function will produce an approximate representation of the original data. Think of the directions contained in the random space as samples from a population, each offering a slightly different view of a set of vectors, given by their projection therein. The collection of these na ...
... space via a linear function will produce an approximate representation of the original data. Think of the directions contained in the random space as samples from a population, each offering a slightly different view of a set of vectors, given by their projection therein. The collection of these na ...
Fast Monte-Carlo Algorithms for Matrix Multiplication
... Given a row of A – say A(i) – the algorithm computes a good fit for the row A(i) using the rows in R as the basis, by approximately solving ...
... Given a row of A – say A(i) – the algorithm computes a good fit for the row A(i) using the rows in R as the basis, by approximately solving ...
Lecture 2 Matrix Operations
... if A is square, and (square) matrix F satisfies F A = I, then • F is called the inverse of A, and is denoted A−1 • the matrix A is called invertible or nonsingular if A doesn’t have an inverse, it’s called singular or noninvertible by definition, A−1A = I; a basic result of linear algebra is that AA ...
... if A is square, and (square) matrix F satisfies F A = I, then • F is called the inverse of A, and is denoted A−1 • the matrix A is called invertible or nonsingular if A doesn’t have an inverse, it’s called singular or noninvertible by definition, A−1A = I; a basic result of linear algebra is that AA ...
Linear Algebra, Section 1.9 First, some vocabulary: A function is a
... The range or image of f is: {y|y = f (x)} We don’t talk about the codomain in calculus anymore for some reason... Think of the range (or image) as a subset of the codomain. In calculus, we have the following definitions. These are necessary before we can talk about the inverse of a function. • A fun ...
... The range or image of f is: {y|y = f (x)} We don’t talk about the codomain in calculus anymore for some reason... Think of the range (or image) as a subset of the codomain. In calculus, we have the following definitions. These are necessary before we can talk about the inverse of a function. • A fun ...
8. Linear mappings and matrices A mapping f from IR to IR is called
... • quadratic matrix: If m = n, i.e., if the matrix A has as many rows as it has columns, A is called quadratic. • m = 1: A matrix of type (1; n) is a row vector. • n = 1: A matrix of type (m; 1) is a column vector. • m = n = 1: A matrix of type (1; 1) can be identified with a single real number (i.e. ...
... • quadratic matrix: If m = n, i.e., if the matrix A has as many rows as it has columns, A is called quadratic. • m = 1: A matrix of type (1; n) is a row vector. • n = 1: A matrix of type (m; 1) is a column vector. • m = n = 1: A matrix of type (1; 1) can be identified with a single real number (i.e. ...
Covariance - KSU Faculty Member websites
... covariance equal to 4.00. This value is equal to the value of the covariance coefficient computed from the formula expressed in deviation scores computed previously. Lack of Upper and LowerLimits The coefficient of covariance has no upper or lower limits. As will be seen later, this indeterminacy i ...
... covariance equal to 4.00. This value is equal to the value of the covariance coefficient computed from the formula expressed in deviation scores computed previously. Lack of Upper and LowerLimits The coefficient of covariance has no upper or lower limits. As will be seen later, this indeterminacy i ...
Vectors and Matrices in Data Mining and Pattern Recognition
... linear algebra, with the emphasis on data mining and pattern recognition. It depends heavily on the availability of an easy-to-use programming environment that implements the algorithms that we will present. Thus, instead of describing in detail the algorithms, we will give enough mathematical theor ...
... linear algebra, with the emphasis on data mining and pattern recognition. It depends heavily on the availability of an easy-to-use programming environment that implements the algorithms that we will present. Thus, instead of describing in detail the algorithms, we will give enough mathematical theor ...
Lecture7 linear File - Dr. Manal Helal Moodle Site
... Conditional likelihood estimation: choose the weights which make the probability of the observed values y be the highest, given the observations xi ...
... Conditional likelihood estimation: choose the weights which make the probability of the observed values y be the highest, given the observations xi ...
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.