Package `nnet`
... a formula expression as for regression models, of the form response ~ predictors. The response should be a factor or a matrix with K columns, which will be interpreted as counts for each of K classes. A log-linear model is fitted, with coefficients zero for the first class. An offset can be included ...
... a formula expression as for regression models, of the form response ~ predictors. The response should be a factor or a matrix with K columns, which will be interpreted as counts for each of K classes. A log-linear model is fitted, with coefficients zero for the first class. An offset can be included ...
nnet
... a formula expression as for regression models, of the form response ~ predictors. The response should be a factor or a matrix with K columns, which will be interpreted as counts for each of K classes. A log-linear model is fitted, with coefficients zero for the first class. An offset can be included ...
... a formula expression as for regression models, of the form response ~ predictors. The response should be a factor or a matrix with K columns, which will be interpreted as counts for each of K classes. A log-linear model is fitted, with coefficients zero for the first class. An offset can be included ...
THE HURWITZ THEOREM ON SUMS OF SQUARES BY LINEAR
... (More generally, Lj and Lk are anti-commuting for any different j, k > 2.) Viewing L3 and L4 as linear operators not on Cn but on the vector space U 1, their anticommutativity on U forces dim U = n/2 to be even by Lemma 2.1, so n must be a multiple of 4. This eliminates the choice n = 6 and conclude ...
... (More generally, Lj and Lk are anti-commuting for any different j, k > 2.) Viewing L3 and L4 as linear operators not on Cn but on the vector space U 1, their anticommutativity on U forces dim U = n/2 to be even by Lemma 2.1, so n must be a multiple of 4. This eliminates the choice n = 6 and conclude ...
LINEAR TRANSFORMATIONS
... Exercise 1 (Rotations in the plane). Consider the function which, given a vector v in the plane, produces as output the same vector rotated (anti-clockwise) through an angle θ: we write Rθ v for this new vector. What is Rθ (v + w), in terms of Rθ v and Rθ w? What is Rθ (λv) in terms of Rθ v? Exercis ...
... Exercise 1 (Rotations in the plane). Consider the function which, given a vector v in the plane, produces as output the same vector rotated (anti-clockwise) through an angle θ: we write Rθ v for this new vector. What is Rθ (v + w), in terms of Rθ v and Rθ w? What is Rθ (λv) in terms of Rθ v? Exercis ...
PRINCIPAL COMPONENT ANALYSIS
... The definition and derivation of the principal component analysis is described. In between Lagrange multipliers for finding a maximum of a function with constraints and eigenvalues and eigenvectors are explained, because these ideas are needed in the derivation. ...
... The definition and derivation of the principal component analysis is described. In between Lagrange multipliers for finding a maximum of a function with constraints and eigenvalues and eigenvectors are explained, because these ideas are needed in the derivation. ...
Chapter 13 - Mathematical Marketing
... Unweighted least squares would not solve the third problem, namely heteroskedasticity. There are a variety of other estimation schemes for probit regression that would deal with this problem, but now we turn our attention to a very widely used model for choice data, the logit model. Note that in Equ ...
... Unweighted least squares would not solve the third problem, namely heteroskedasticity. There are a variety of other estimation schemes for probit regression that would deal with this problem, but now we turn our attention to a very widely used model for choice data, the logit model. Note that in Equ ...
Lecture 25 March 24 Wigner
... of Ak as paths of length k from a vertex back to itself, where the value of a path is the product of the labels along the path. We can classify paths by their structure: in some paths, every edge is traversed at least twice, and in others, there is at least one edge which is traversed only once. Let ...
... of Ak as paths of length k from a vertex back to itself, where the value of a path is the product of the labels along the path. We can classify paths by their structure: in some paths, every edge is traversed at least twice, and in others, there is at least one edge which is traversed only once. Let ...
An aggregator point of view on NL-Means
... Peyr3 for a review). While intuitive and enlightening, those points of view have not yet permited to justify mathematically the performance of the NL-Means methods. We propose to look at those methods with a different eye so as to propose a different path to their mathematical justification. We cons ...
... Peyr3 for a review). While intuitive and enlightening, those points of view have not yet permited to justify mathematically the performance of the NL-Means methods. We propose to look at those methods with a different eye so as to propose a different path to their mathematical justification. We cons ...
The Multivariate Gaussian Distribution
... In the case of the multivariate Gaussian density, the argument of the exponential function, − 21 (x − µ)T Σ−1 (x − µ), is a quadratic form in the vector variable x. Since Σ is positive definite, and since the inverse of any positive definite matrix is also positive definite, then for any non-zero ve ...
... In the case of the multivariate Gaussian density, the argument of the exponential function, − 21 (x − µ)T Σ−1 (x − µ), is a quadratic form in the vector variable x. Since Σ is positive definite, and since the inverse of any positive definite matrix is also positive definite, then for any non-zero ve ...
Linear Algebra, II
... Explanation of the conversion factor in the change of variables formula • Setup: Assume we are given a change of variables x = T(u), where x = hx1 , . . . , xn i is the standard rectangular coordinate system in Rn , u = hu1 , . . . , un i denotes another coordinate system in Rn , and T is the conver ...
... Explanation of the conversion factor in the change of variables formula • Setup: Assume we are given a change of variables x = T(u), where x = hx1 , . . . , xn i is the standard rectangular coordinate system in Rn , u = hu1 , . . . , un i denotes another coordinate system in Rn , and T is the conver ...
Multivariate Analysis (Slides 2)
... we need to cover before we look at many multivariate analysis methods. • This material will include topics that you are likely to have seen in courses in probability and linear algebra. ...
... we need to cover before we look at many multivariate analysis methods. • This material will include topics that you are likely to have seen in courses in probability and linear algebra. ...
Week Two True or False
... If A is an m × n matrix and if the equation Ax = b is inconsistent for some b in Rm , then A cannot have a pivot position in every row. TRUE Linear Algebra, David Lay ...
... If A is an m × n matrix and if the equation Ax = b is inconsistent for some b in Rm , then A cannot have a pivot position in every row. TRUE Linear Algebra, David Lay ...
Slide 1
... diagonal elements during the vector-vector product. So one Jacobi step becomes one matrix-vector product, one vector-vector product and one vector subtract. ...
... diagonal elements during the vector-vector product. So one Jacobi step becomes one matrix-vector product, one vector-vector product and one vector subtract. ...
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 ...
Notes
... Internally, Matlab uses column major layout — all the entries of the first column of a matrix are listed first in memory, then all the entries of the second column, and so on. This is actually visible at the user level in some contexts. For example, when I enter A as above, the Matlab expression A(6 ...
... Internally, Matlab uses column major layout — all the entries of the first column of a matrix are listed first in memory, then all the entries of the second column, and so on. This is actually visible at the user level in some contexts. For example, when I enter A as above, the Matlab expression A(6 ...
Basics for Math 18D, borrowed from earlier class
... (9) Nul (T ) = {v ∈ V : T (v) = 0} – all vectors in the domain which are sent to zero. (a) T is one to one iff Nul (T ) = {0} . (b) be able to find a basis for Nul (A) ⊂ Rn when A is a m × n matrix. (10) Ran (T ) = {T (v) ∈ W : v ∈ V } – the range of T. Equivalently, w ∈ Ran (T ) iff there exists a ...
... (9) Nul (T ) = {v ∈ V : T (v) = 0} – all vectors in the domain which are sent to zero. (a) T is one to one iff Nul (T ) = {0} . (b) be able to find a basis for Nul (A) ⊂ Rn when A is a m × n matrix. (10) Ran (T ) = {T (v) ∈ W : v ∈ V } – the range of T. Equivalently, w ∈ Ran (T ) iff there exists a ...
A Brief report of: Integrative clustering of multiple
... where X is the mean-centered expression matrix of dimension p×n (no intercept), Z = ( z1,..., zK−1)’ is the cluster indicator matrix of dimension (K − 1)×n as defined in 1.1, W is the coefficient matrix of dimension p×(K − 1), and ε=(ε1,...,εp)’ is a set of independent error terms with zero mean and ...
... where X is the mean-centered expression matrix of dimension p×n (no intercept), Z = ( z1,..., zK−1)’ is the cluster indicator matrix of dimension (K − 1)×n as defined in 1.1, W is the coefficient matrix of dimension p×(K − 1), and ε=(ε1,...,εp)’ is a set of independent error terms with zero mean and ...
1 The Covariance Matrix
... there always exists an orhtonormal set of eigenvectors of Σ. It is often convenient to work in an orthonormal coordinate system where the coordinate axes are eigenvectors of Σ. In this coordinte system we have that Σ is a diagonal matrix with Σi,i = λi , the eigenvalue of coordinate i. In the coorin ...
... there always exists an orhtonormal set of eigenvectors of Σ. It is often convenient to work in an orthonormal coordinate system where the coordinate axes are eigenvectors of Σ. In this coordinte system we have that Σ is a diagonal matrix with Σi,i = λi , the eigenvalue of coordinate i. In the coorin ...
restrictive (usually linear) structure typically involving aggregation
... the first situation in which there exist many consistent solutions. One can think of the entire set of consistent solutions as y = yR + yN where yR is the linear combination of the rows of A consistent with AyR = x and yN = NTk where k is a (n-r)-length vector of free variables or weights on the bas ...
... the first situation in which there exist many consistent solutions. One can think of the entire set of consistent solutions as y = yR + yN where yR is the linear combination of the rows of A consistent with AyR = x and yN = NTk where k is a (n-r)-length vector of free variables or weights on the bas ...
Linear Algebra Libraries: BLAS, LAPACK - svmoore
... This level contains matrix-matrix operations of the form as well as solving for triangular matrices , among other things. This level contains the widely used General Matrix Multiply (GEMM) operation. ...
... This level contains matrix-matrix operations of the form as well as solving for triangular matrices , among other things. This level contains the widely used General Matrix Multiply (GEMM) operation. ...
Sketching as a Tool for Numerical Linear Algebra (slides)
... Open Questions Recent monograph in NOW Publishers D. Woodruff, “Sketching as a Tool for Numerical Linear Algebra” Other types of low rank approximation: (Spectral) How quickly can we find a rank k matrix A’, so that |A-A’|2 · (1+ε) |A-Ak|2, w.h.p., where Ak = argminrank k matrices B |A-B|2 (R ...
... Open Questions Recent monograph in NOW Publishers D. Woodruff, “Sketching as a Tool for Numerical Linear Algebra” Other types of low rank approximation: (Spectral) How quickly can we find a rank k matrix A’, so that |A-A’|2 · (1+ε) |A-Ak|2, w.h.p., where Ak = argminrank k matrices B |A-B|2 (R ...
LINEAR TRANSFORMATIONS Math 21b, O. Knill
... that the column vector ~v1 , ~vi , ~vn are the images of the standard vectors ~e1 = ...
... that the column vector ~v1 , ~vi , ~vn are the images of the standard vectors ~e1 = ...
Similarity - U.I.U.C. Math
... vector v ∈ V can be written in exactly one way as a linear combination of the basis vectors of X: v = c1 x1 + · · · + cn xn , where all ci ∈ R; we call the n × 1 column vector (c1 , . . . , cn )> the coordinate list of v wrt X (with respect to X). Of course, every vector w ∈ W has a coordinate list ...
... vector v ∈ V can be written in exactly one way as a linear combination of the basis vectors of X: v = c1 x1 + · · · + cn xn , where all ci ∈ R; we call the n × 1 column vector (c1 , . . . , cn )> the coordinate list of v wrt X (with respect to X). Of course, every vector w ∈ W has a coordinate list ...
COMPRESSIVE NONSTATIONARY SPECTRAL ESTIMATION
... TF spectrum is effectively supported in a few relatively small regions of the TF plane (corresponding to TF localized signal components) and thus almost zero in the rest of the TF plane. In [8], we proposed a “compressive” estimator of the Wigner-Ville spectrum that exploits TF sparsity to reduce th ...
... TF spectrum is effectively supported in a few relatively small regions of the TF plane (corresponding to TF localized signal components) and thus almost zero in the rest of the TF plane. In [8], we proposed a “compressive” estimator of the Wigner-Ville spectrum that exploits TF sparsity to reduce th ...
3x − 5y = 3 −4x + 7y = 2 2 1 −2 5 3 5 −2 14 2 −4 3 15
... (a) Describe row operations that would transform the first column of A so that it has a leading 1 at the top, with 0’s below. ...
... (a) Describe row operations that would transform the first column of A so that it has a leading 1 at the top, with 0’s below. ...
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.