ppt - IBM Research
... – Alice has random d x d sign matrix A-1 – b is a standard basis vector ei – Alice computes A = (A-1)-1 and puts it into the stream. Solution x to minx ||Ax=b|| is i-th column of A-1 – Bob can isolate entries of A-1, solving indexing ...
... – Alice has random d x d sign matrix A-1 – b is a standard basis vector ei – Alice computes A = (A-1)-1 and puts it into the stream. Solution x to minx ||Ax=b|| is i-th column of A-1 – Bob can isolate entries of A-1, solving indexing ...
Section 1.9 23
... The standard matrix of a linear transformation from R2 to R2 that reflects points through thehorizontal axis, the vertical a 0 , where a and d are axis, or the orign has the form 0 d ±1 TRUE We can check this by checking the images of the basis vectors. ...
... The standard matrix of a linear transformation from R2 to R2 that reflects points through thehorizontal axis, the vertical a 0 , where a and d are axis, or the orign has the form 0 d ±1 TRUE We can check this by checking the images of the basis vectors. ...
Procrustes distance
... This routine computes the true Procrustes distance between each pair of specimens in the dataset. Procrustes distance is reported as an angle in radians, representing the geodesic distance between two specimens in Procrustes space. As written here, the routine will print pairwise distances in the SA ...
... This routine computes the true Procrustes distance between each pair of specimens in the dataset. Procrustes distance is reported as an angle in radians, representing the geodesic distance between two specimens in Procrustes space. As written here, the routine will print pairwise distances in the SA ...
Generalized Linear Models For The Covariance Matrix of
... Pan, J.X. and Mackenzie, G. (2003). Model selection for joint mean-covariance structures in longitudinal studies. Biometrika, 90, 239-249. Pourahmadi, M. (2001). Foundations of Time Series Analysis and Prediction Theory, John Wiley, New York. Pourahmadi, M. and Daniels, M. (2002). Dyanamic condition ...
... Pan, J.X. and Mackenzie, G. (2003). Model selection for joint mean-covariance structures in longitudinal studies. Biometrika, 90, 239-249. Pourahmadi, M. (2001). Foundations of Time Series Analysis and Prediction Theory, John Wiley, New York. Pourahmadi, M. and Daniels, M. (2002). Dyanamic condition ...
lect9
... • Johnson-Lindenstrauss lemma: Given ε>0, and an integer n, let k be a positive integer such that k≥k0=O(ε-2 logn). For every set X of n points in Rd there exists F: RdRk such that for all xi, xj єX (1-ε)||xi - xj||2≤ ||F(xi )- F(xj)||2≤ (1+ε)||xi - xj||2 ...
... • Johnson-Lindenstrauss lemma: Given ε>0, and an integer n, let k be a positive integer such that k≥k0=O(ε-2 logn). For every set X of n points in Rd there exists F: RdRk such that for all xi, xj єX (1-ε)||xi - xj||2≤ ||F(xi )- F(xj)||2≤ (1+ε)||xi - xj||2 ...
Lecture 16 - Math TAMU
... to an n×n matrix A if B = S −1 AS for some nonsingular n×n matrix S. Remark. Two n×n matrices are similar if and only if they represent the same linear operator on Rn with respect to different bases. Theorem If A and B are similar matrices then they have the same (i) determinant, (ii) trace = the su ...
... to an n×n matrix A if B = S −1 AS for some nonsingular n×n matrix S. Remark. Two n×n matrices are similar if and only if they represent the same linear operator on Rn with respect to different bases. Theorem If A and B are similar matrices then they have the same (i) determinant, (ii) trace = the su ...
Sample examinations Linear Algebra (201-NYC-05) Winter 2012
... 18. a. If T is injective then the dimension of the kernel of T is zero. Since dim ker T + rank T = n by the rank formula, it follows that the dimension of the range of T (i.e., the rank of T ) is n. b. If T is surjective then its rank is m, so (as in Part a) the rank formula implies that the dimensi ...
... 18. a. If T is injective then the dimension of the kernel of T is zero. Since dim ker T + rank T = n by the rank formula, it follows that the dimension of the range of T (i.e., the rank of T ) is n. b. If T is surjective then its rank is m, so (as in Part a) the rank formula implies that the dimensi ...
notes
... where r is the rank of A. This decomposition of A is called the singular value decomposition, or SVD. The values σi , for i = 1, 2, . . . , n, are the singular values of A. The columns of U are the left singular vectors, and the columns of V are the right singular vectors. An alternative decompositi ...
... where r is the rank of A. This decomposition of A is called the singular value decomposition, or SVD. The values σi , for i = 1, 2, . . . , n, are the singular values of A. The columns of U are the left singular vectors, and the columns of V are the right singular vectors. An alternative decompositi ...
The Linear Algebra Version of the Chain Rule 1
... 1) Remember that n × k and k × m yields n × m. Thus one can think of plumbing pipes: you can plumb them together only if they fit. After fitting them together the ends in the middle are eliminated, leaving only the outer ends. 2) The matrix product is associative. 3) In general, if AB makes sense, t ...
... 1) Remember that n × k and k × m yields n × m. Thus one can think of plumbing pipes: you can plumb them together only if they fit. After fitting them together the ends in the middle are eliminated, leaving only the outer ends. 2) The matrix product is associative. 3) In general, if AB makes sense, t ...
ex.matrix - clic
... This distance matrix can be used to produce a spatial layout of the distances between customers on the basis of the distances just calculated. This is done via Multi-Dimensional Scaling – a technique that produces a visualization in two dimensions of the distances between points in a multi-dimension ...
... This distance matrix can be used to produce a spatial layout of the distances between customers on the basis of the distances just calculated. This is done via Multi-Dimensional Scaling – a technique that produces a visualization in two dimensions of the distances between points in a multi-dimension ...
sup-3-Learning Linear Algebra
... % load Y with a 4x6 matrix full of random numbers between 0 and 1 % The random numbers are uniformly distributed on [0,1] Y=rand(4,6) % And to load a single random number just use r=rand % randn: % load Y with a 4x6 matrix full of random numbers with a Gaussian % distribution with zero mean and a va ...
... % load Y with a 4x6 matrix full of random numbers between 0 and 1 % The random numbers are uniformly distributed on [0,1] Y=rand(4,6) % And to load a single random number just use r=rand % randn: % load Y with a 4x6 matrix full of random numbers with a Gaussian % distribution with zero mean and a va ...
Notes
... Any symmetric positive definite matrix A can be written as A QQ where Q is some nonsingular matrix. For example, consider the covariance matrix 2V . This matrix is positive def. So its inverse also is. So we can decompose it V 1 QQ . We premultiply both sides of ...
... Any symmetric positive definite matrix A can be written as A QQ where Q is some nonsingular matrix. For example, consider the covariance matrix 2V . This matrix is positive def. So its inverse also is. So we can decompose it V 1 QQ . We premultiply both sides of ...
5 Least Squares Problems
... (0) Set up the problem by computing A∗ A and A∗ b. (1) Compute the Cholesky factorization A∗ A = R∗ R. (2) Solve the lower triangular system R∗ w = A∗ b for w. (3) Solve the upper triangular system Rx = w for x. The operations count for this algorithm turns out to be O(mn2 + 13 n3 ). Remark The solu ...
... (0) Set up the problem by computing A∗ A and A∗ b. (1) Compute the Cholesky factorization A∗ A = R∗ R. (2) Solve the lower triangular system R∗ w = A∗ b for w. (3) Solve the upper triangular system Rx = w for x. The operations count for this algorithm turns out to be O(mn2 + 13 n3 ). Remark The solu ...
GINI-Coefficient and GOZINTO
... GINI-Coefficient and GOZINTO-Graph (Workshop) (Two economic applications of secondary school mathematics) Josef Böhm, ACDCA & DERIVE User Group, [email protected] Abstract: GINI-Coefficient together with LORENZ-curve and GOZINTO-Graphs are economic applications of secondary school mathematics. They ...
... GINI-Coefficient and GOZINTO-Graph (Workshop) (Two economic applications of secondary school mathematics) Josef Böhm, ACDCA & DERIVE User Group, [email protected] Abstract: GINI-Coefficient together with LORENZ-curve and GOZINTO-Graphs are economic applications of secondary school mathematics. They ...
Talk - IBM Research
... • SyT only has support on |T| coordinates • Let G µ [n]\T be such that each i 2 G hashes to a bucket containing a j 2 T • || = || · |SyT|2¢|SyG|2
...
... • SyT only has support on |T| coordinates • Let G µ [n]\T be such that each i 2 G hashes to a bucket containing a j 2 T • |
1.9 matrix of a linear transformation
... 35. If T: Rn →Rm maps Rn onto Rm , then its standard matrix A has a pivot in each row, by Theorem 12 and by Theorem 4 in Section 1.4. So A must have at least as many columns as rows, so m n. When T is one-to-one, A must have a pivot in each column, by Theorem 12, so m n. 37. [M] There is no pivot in ...
... 35. If T: Rn →Rm maps Rn onto Rm , then its standard matrix A has a pivot in each row, by Theorem 12 and by Theorem 4 in Section 1.4. So A must have at least as many columns as rows, so m n. When T is one-to-one, A must have a pivot in each column, by Theorem 12, so m n. 37. [M] There is no pivot in ...
PDF
... With these assumptions, there exists a unique solution, which can be obtained from the above matrix by back substitution. For the general case, the termination procedure is somewhat more complicated. First recall that a matrix is in echelon form if each row has more leading zeros than the rows above ...
... With these assumptions, there exists a unique solution, which can be obtained from the above matrix by back substitution. For the general case, the termination procedure is somewhat more complicated. First recall that a matrix is in echelon form if each row has more leading zeros than the rows above ...
Kernel Methods
... • Only the observations (training set) close to the query point are considered for regression computation • While regressing, an observation point gets a weight that decreases as its distance from the query point increases ...
... • Only the observations (training set) close to the query point are considered for regression computation • While regressing, an observation point gets a weight that decreases as its distance from the query point increases ...
(Slide 1) Question 10
... embrittlement mechanisms of chemical elements influence is proposed. The model includes three basic mechanisms influencing to radiation embrittlement for RPV steels: matrix damage, irradiation induced precipitation and element segregation. (Slide 2) 1. As neutrons interact with the crystalline struc ...
... embrittlement mechanisms of chemical elements influence is proposed. The model includes three basic mechanisms influencing to radiation embrittlement for RPV steels: matrix damage, irradiation induced precipitation and element segregation. (Slide 2) 1. As neutrons interact with the crystalline struc ...
Sum of Squares seminar- Homework 0.
... Here is some reading and exercises that I would like you to do before the course. Feel free to collaborate with others while solving those. You don’t need to submit them, or even write the solutions down properly or anything — just make sure you know the material. Also, please don’t hesitate to emai ...
... Here is some reading and exercises that I would like you to do before the course. Feel free to collaborate with others while solving those. You don’t need to submit them, or even write the solutions down properly or anything — just make sure you know the material. Also, please don’t hesitate to emai ...
Bump Hunting with Non-Gaussian Kernels
... It is well known that the number of modes of a kernel density estimator is monotone nonincreasing in the bandwidth if the kernel is a Gaussian density. There is numerical evidence of nonmonotonicity in the case of some non-Gaussian kernels, but little additional information is available. The present ...
... It is well known that the number of modes of a kernel density estimator is monotone nonincreasing in the bandwidth if the kernel is a Gaussian density. There is numerical evidence of nonmonotonicity in the case of some non-Gaussian kernels, but little additional information is available. The present ...
AEMAA Course Outline - Hedland Senior High School
... 2.1.4 with the aid of an appropriate graphical display (chosen from dot plot, stem plot, bar chart or histogram), describe the distribution of a numerical data set in terms of modality (uni or multimodal), shape (symmetric versus positively or negatively skewed), location and spread and outliers, an ...
... 2.1.4 with the aid of an appropriate graphical display (chosen from dot plot, stem plot, bar chart or histogram), describe the distribution of a numerical data set in terms of modality (uni or multimodal), shape (symmetric versus positively or negatively skewed), location and spread and outliers, an ...
Notes 11: Dimension, Rank Nullity theorem
... in unknows x, y can be solved. This says that our set spans R2 . • The only relations of the form au + bv = 0 are the ones with a = b = 0. This says that the set is independent. Example 3. Let ei denote the element of Rn with all zero entries except in the i-th position where a 1 occurs. Then S = {e ...
... in unknows x, y can be solved. This says that our set spans R2 . • The only relations of the form au + bv = 0 are the ones with a = b = 0. This says that the set is independent. Example 3. Let ei denote the element of Rn with all zero entries except in the i-th position where a 1 occurs. Then S = {e ...
Introduction; matrix multiplication
... The point of the next few lectures will be to introduce these themes (4A) and (some of) our techniques in the context of two very simple operations: multiplication of a matrix by a vector, and multiplication of two matrices I will assume that you have already had a course in linear algebra and that ...
... The point of the next few lectures will be to introduce these themes (4A) and (some of) our techniques in the context of two very simple operations: multiplication of a matrix by a vector, and multiplication of two matrices I will assume that you have already had a course in linear algebra and that ...
Sketching as a Tool for Numerical Linear Algebra Lecture 1
... 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 ...
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.