id int64 0 25.6k | text stringlengths 0 4.59k |
|---|---|
15,100 | can be efficiently implemented using dedicated computing hardware such as graphical processor units (gpusand other parallel computing architecture note also that many matrix computations that run in quadratic time can be replaced with linear-time componentwise multiplication specificallymultiplication of vector with di... |
15,101 | finallynote that to obtain the gradient ` /th of the training losswe simply need to loop algorithm over all the training examplesas follows algorithm computing the gradient of the training loss inputtraining set {(xi yi )}ni= weight matrices and bias vectors {wl bl } = =thactivation functions {sl } = outputthe gradient... |
15,102 | methods for training neural networks have been studied for long timeyet it is only recently that there have been sufficient computational resources to train them effectively the training of neural networks requires minimization of training loss` (th) ni= ci (th)which is typically difficult high-dimensional optimization... |
15,103 | activation functionwe will use small positive values to ensure that its derivative is not zero zero derivative of the activation function prevents the propagation of information useful for computing good search direction recall that computation of the gradient of the training loss via algorithm requires averaging over ... |
15,104 | the levenberg-marquardt algorithm is not suitable for networks with large number of parametersbecause the cost of the matrix computations becomes prohibitive for instanceobtaining the levenberg-marquardt search direction in ( usually incurs an ( cost in additionthe levenberg-marquardt algorithm is applicable only when ... |
15,105 | in addition to { }we will make use of the vectors { defined via the recursionr : - > - ( at the final iteration tthe bfgs updating formula ( can be rewritten in the formct > - ct- at- ut dt > by iterating the recursion ( backwards to we can writect > > > = that iswe can express ct in terms of the initial and the entire... |
15,106 | algorithm limited-memory bfgs update inputbfgs history list { }hj= initial and input outputd -ch uwhere ct ut dt > ct- ut > ut dt > for hh do /backward recursion to compute - ui di ti > ui ti for do ui (ti > qdi /compute ( /compute recursion ( return -qthe value of -ch in summarya quasi-newton algorithm with limited-m... |
15,107 | contained in the hessian matrix of ` (thin addition to the bfgs methodthere are other ways in which we can exploit the history of past gradient computations one approach is to use the normal approximation methodin which the hessian of ` at tht is approximated via ui > ( ht = - + where ut- + ut are the most recently com... |
15,108 | momentum method yet another computationally cheap approach is the momentum methodin which the steepest descent iteration ( is modified to tht+ tht at ut dt where dt tht tht- and is tuning parameter this strategy frequently performs better than the "vanillasteepest descent methodbecause the search direction is less like... |
15,109 | nextthe initialize method generates random initial weight matrices and bias vecl tors {wl bl } = specificallyall parameters are initialized with values distributed according to the standard normal distribution def initialize (pw_sig )wb [[]]len( )[[]]len(pfor in range ( len( )) [ ]w_sig np random randn ( [ ] [ - ] [ ]w... |
15,110 | delta [lgr_s[lgr_c for in range ( , - ) , dci_dbl delta [ldci_dwl delta [la[ - ---sum up over samples ---dc_db [ldc_db [ldci_dbl / dc_dw [ldc_dw [ldci_dwl / delta [ - gr_s[ - [lt delta [lreturn dc_dw dc_db loss as explained in section it is sometimes more convenient to collect all the weight matrices and bias vectors {... |
15,111 | len(xnum_epochs print (epoch batch loss"print ("for epoch in range ( num_epochs + )batch_idx np random choice (nbatch_size batch_x xbatch_idx reshape - , batch_y =ybatch_idx reshape - , dc_dw dc_db loss backward ( , ,batch_x batch_y d_beta list vec (dc_dw dc_db loss_arr append (loss flatten ([ ]ifepoch == or np mod(epo... |
15,112 | fit batch loss output input iteration figure left panelthe fitted neural network with training loss of ` (gt right panelthe evolution of the estimated lossb ` (gt (th))over the steepest-descent iterations image classification in this sectionwe will use the package pytorchwhich is an open-source machine learning library... |
15,113 | import matplotlib pyplot as plt from torch utils data import dataset dataloader from pil import image import torch nn functional as ###############################################################data loader class ###############################################################class loaddata dataset )def __init__ (self f... |
15,114 | of the first flat hidden layer should besecond convolution layer } - ( { first convolution layer where the multiplication by follows from the fact that the second convolution layer has output channels having said thatthe flat_fts variable determines the number of output layers of the convolution block this number is us... |
15,115 | learning parameters num_epochs learning_rate device torch device ('cpu 'use this to run on cpu device torch device ('cuda 'use this to run on gpu instance of the conv net cnn cnn (cnn todevice device #loss function and optimizer criterion nn crossentropyloss (optimizer torch optim adam(cnn parameters (lrlearning_rate t... |
15,116 | epoch training loss epoch training loss epoch training loss epoch training loss epoch training loss epoch training loss test accuracy of the model on the , training test images finallywe evaluate the network performance using the test data set typical minibatch loss as function of iteration is shown in figure and the p... |
15,117 | projection pursuit is network with one hidden layer that can be written asprojection pursuit (xs (ox)where is univariate smoothing cubic spline if we use squared-error loss with tn {yi xi }ni= we need to minimize the training loss yi (oxi = with respect to and all cubic smoothing splines this training of the network ... |
15,118 | algorithm stochastic gradient descent with reshuffling inputtraining set tn {(xi yi )}ni= initial weight matrices and bias vectors {wl bl } = th activation functions {sl } = learning rates { outputthe parameters of the trained learner and epoch while stopping condition is not met do iid draw un ( let be the permutation... |
15,119 | suppose that [ uh- ]where all rn are column vectors and we have computed (in uu)- via the qr factorization method in exercise if the columns of matrix are updated to [ uh- uh ]show that the inverse (in uu)- can be updated in ( ntime (rather than computed from scratch in ( ntimededuce that the computing cost of updating... |
15,120 | show that the bfgs formula ( is the solution to the constrained optimization problemcbfgs argmin ( ) subject to da awhere is the kullback-leibler discrepancy defined in ( on the other handshow that the dfp formula ( is the solution to the constrained optimization problemcdfp argmin subject to da ad( consider again the ... |
15,121 | consider again the python implementation of the polynomial regression in section where the stochastic gradient descent was used for training using the polynomial regression data setimplement and run the following four alternative training methods(athe steepest-descent algorithm (bthe levenberg-marquardt algorithm in co... |
15,122 | inear lgebra and unctional nalysis the purpose of this appendix is to review some important topics in linear algebra and functional analysis we assume that the reader has some familiarity with matrix and vector operationsincluding matrix multiplication and the computation of determinants vector spacesbasesand matrices ... |
15,123 | using basis of vwe can thus represent each vector as row or column of numbers [ an or ( an standard basis transpose typicallyvectors in rn are represented via the standard basisconsisting of unit vectors (pointse ( )en ( as consequenceany point ( xn rn can be representedusing the standard basisas row or column vector o... |
15,124 | suppose [ an ]where the {ai form basis of rn take any vector [ xn ]> with respect to the standard basis (we write subscript to stress thisthen the representation of this vector with respect to is simply - xwhere - is the inverse of athat isthe matrix such that aa- - in where in is the -dimensional identity matrix to se... |
15,125 | geometricallythe determinant of square matrix [ an is the (signedvolume of the parallelepiped ( -dimensional parallelogramdefined by the columns an that isthe set of points ni= ai ai where ai the easiest way to compute determinant of general matrix is to apply simple operations to the matrix that potentially reduce its... |
15,126 | corresponding to the blue parallelogram the linear operation that transforms the red to the blue parallelogram can be thought of as succession of two linear transformations the first is to transform the coordinates of points on the red parallelogram (in standard basisto the basis formed by the columns of secondrelative... |
15,127 | definition moore-penrose pseudo-inverse moore-penrose pseudo-inverse the moore-penrose pseudo-inverse of real matrix rnxp is defined as the unique matrix ar pxn that satisfies the conditions aaa aaaa (aa)aa (aa)aa we can write aexplicitly in terms of when has full column or row rank for examplewe always have aaaa(aa)((... |
15,128 | if and are perpendicularthen pythagorastheorem holds|| || hx yx yi hxxi hxyi hyyi || || || || pythagorastheorem ( basis { vn of rn in which all the vectors are pairwise perpendicular and have norm is called an orthonormal (short for orthogonal and normalizedbasis for examplethe standard basis is orthonormal orthonormal... |
15,129 | the (euclideaninner product of and (viewed as column vectorsis now defined as hxyi hermitian unitary xi yi = which is no longer symmetrichxyi hyxi note that this generalizes the real-valued inner product the determinant of complex matrix is defined exactly as in ( as consequencedet(adet(aa complex matrix is said to be ... |
15,130 | for any point rn the point in that is closest to is its orthogonal projection pxas the following theorem shows theorem orthogonal projection and minimal distance let { uk be an orthonormal basis of subspace and let be the orthogonal projection matrix onto the solution to the minimization program min kx yk yv is px that... |
15,131 | span { { rand the eigenspace corresponding to is span { the algebraic and geometric multiplicities are in this case any pair of vectors taken from and forms basis for figure shows how and are transformed to av and av respectively - - - - figure the dashed arrows are the unit eigenvectors (blueand (redof matrix their tr... |
15,132 | therefore det( and hence that is an eigenvalue of alet be an eigenvector corresponding to thenaw lw orequivalentlywa lwfor this reasonwe call wthe left-eigenvector of for eigenvalue if is (right-eigenvector of athen its adjoint vis usually not left-eigenvectorunless aa aa(such matrices are called normala real symmetric... |
15,133 | eigenvalues and eigenvectors theorem eigenvalues of hermitian matrix any hermitian matrix has real eigenvalues the corresponding matrix of normalized eigenvectors is unitary matrix prooflet be hermitian matrix by theorem there exists unitary matrix such that - au twhere is upper triangular it follows that the adjoint (... |
15,134 | - - - figure the eigenvectors and eigenvalues of aadetermine the principal axes of the ellipse the following definition generalizes the notion of positivity of real variable to that of (hermitianmatrixproviding crucial concept for multivariate differentiation and optimizationsee appendix definition positive (semi)defin... |
15,135 | proofthe matrix is both hermitian (by definitionand real (by assumptionand hence it is symmetric by theorem we can write qdqwhere is the diagonal matrix of (realeigenvalues of by theorem all eigenvalues are non-negativeand thus their square root is real-valued nowdefine dwhere is defined as the diagonal matrix whose el... |
15,136 | swap the first row with the row having the maximal absolute value element in the first column make every other element in the first column equal to by adding appropriate multiples of the first row to the other rows suppose that at has plu decomposition pt lt ut then it is easy to check that ## at > pt- ( pt > ct /at lt... |
15,137 | forward substitution backward substitution the first equation can be solved efficiently via forward substitutionand the second via backward substitutionas illustrated in the following example example (solving linear equations with an lu decompositionlet plu be the same as in example we wish to solve ax [ ]firstsolving ... |
15,138 | whichafter similar calculation as the one aboveyields ##in onxk ( bd- )- onxk in -bd- - ik - - ik okxn - okxn ( the upper-left block of - from ( must be the same as the upper-left block of - from ( )leading to the woodbury identity( bd- )- - - ( ca- )- ca- ( det( bd- cdet(ddet(adet( ca- ( woodbury identity from ( and t... |
15,139 | thereforeby evolving the recursive relationships up until pwe obtain- - | | - - - - = > - - > - - = these expressions will allow us to easily compute - - and | | provided the following quantities are availableckj : - - sinceby theorem we can write- - - ak- - - - ak- ak- ak- > - - - ak- jthe quantities {ckj can be compu... |
15,140 | as consequence of theorem the solution to the linear system ax can be computed in ( ntime via- - cd [ bif pthe sherman-morrison recursion can frequently be much faster than the ( direct solution via the lu decomposition method in section in summarythe following algorithm computes the matrices and in theorem via the rec... |
15,141 | proofthe proof is by inductive construction for nlet ak be the left-upper submatrix of an with :[ ]we have > ae by the positive-definiteness of it follows that suppose that ak- has cholesky factorization lk- > - with lk- having strictly positive diagonal elementswe can construct cholesky factorization of ak as follows ... |
15,142 | qr decomposition and the gram-schmidt procedure let be an matrixwhere thenthere exists matrix rnxp satisfying qq and an upper triangular matrix pxp such that qr this is the qr decomposition for real-valued matrices when has full column ranksuch decomposition can be obtained via the gram-schmidt procedurewhich construct... |
15,143 | for some numbers ri ii denoting the corresponding upper triangular matrix [ri by rwe have in matrix notationr qr [ [ ar pp which yields qr decomposition the qr decomposition can be used to efficiently solve least-squares problemsthis will be shown shortly it can also be used to calculate the determinant of the matrix a... |
15,144 | ln by theorem the matrix [ vn of right-eigenvectors is unitary matrix define the -th singular value as si li and suppose lr are all greater than and lr+ ln in particularavi for let ui avi /si thenfor ij rhui * ui * aavi si singular value li { { jsi we can extend ur to an orthonormal basis { um of cm ( using the gramsch... |
15,145 | example (ellipseswe continue example using the svd method of the module numpy linalgwe obtain the following svd matrices for matrix - - usand - - figure shows the columns of the matrix us as the two principal axes of the ellipse that is obtained by applying matrix to the points of the unit circle practical method to co... |
15,146 | and (blxb ( from ( we get xl/ by substituting it in ( )we arrive at (xx)- yand hence is given by bxl (xx)- (xx)- xy an example python code is given below svdexample py from numpy import diag zeros vstack from numpy random import rand seed from numpy linalg import svd pinv seed ( rand( ,py rand( , , ,vt svd(xsi diag ( /... |
15,147 | - -( - -( - - -( - an- - an- an- general square toeplitz matrix is completely determined by the elements along its first row and column if is also hermitian ( aa)then clearly it is determined by only elements if we define the matrices ** ** and - ** ** then ( is satisfied with -( - - - - - -( - : -( - an- - an- - an- -... |
15,148 | wheredenoting the -dimensional column vector of zeros by we have that - - - - clearlythe matrix is (sparsetoeplitz matrix circulant matrix is special toeplitz matrix which is obtained from vector by circularly permuting its indices as followsc cn- cn- ( cn- cn- cn- cn- note that is completely determined by the elements... |
15,149 | clearlypk > and pk pk ik holdso that in fact pk is an orthogonal matrix we can solve the linear system an xn an in ( time recursivelyas follows assume that we have somehow solved for the upper block ak xk ak and now we wish to solve for the ( ( block#ak pk ak ak ak+ xk+ ak+ = > pk ak+ thereforea ak+ > pk ak ak pk ak - ... |
15,150 | thenwe have the following factorization of an theorem diagonalization of toeplitz correlation matrix an for real-valued symmetric positive-definite toeplitz matrix an of the form ( )we have ln an > dn where ln is the lower diagonal matrix ( and dn :diag( bn- is diagonal matrix proofwe give proof by induction obviouslyl... |
15,151 | note that it is possible to avoid the explicit construction of the lower triangular matrix in ( via the following modification of algorithm which only stores an extra vector at each recursive step of the levinson-durbin algorithm algorithm solving an xn bn with (nmemory cost inputfirst row [ > - of matrix an and right-... |
15,152 | converges to some hthat isk fn as sequence that satisfies ( is called cauchy sequence complete inner product space is called hilbert space the most fundamental hilbert space of functions is the space an in-depth introduction to requires some measure theory [ for our purposesit suffices to assume that rd and that on mea... |
15,153 | the norm of every fi is that ish fi fi for all the fi are orthogonalthat ish fi for orthonormal basis it follows then that the fi are linearly independentthat isthe only linear combination (xthat is equal to fi (xfor all is the one where ai and for an orthonormal system fi is called an orthonormal basis if there is no ... |
15,154 | figure fourier approximations of the unit step function on the interval ( )truncating the infinite sum in ( to and termsgiving the dotted bluedashed redand solid green curvesrespectively starting from any countable basiswe can use the gram-schmidt procedure to obtain an orthonormal basisas illustrated in the following ... |
15,155 | - - - figure the first normalized legendre polynomials as the legendre polynomials form an orthonormal basis of ( {- dx)they can be used to approximate arbitrary functions in this space for examplefigure shows an approximation using the first legendre polynomials ( of the fourier expansion of the indicator function on ... |
15,156 | by the recurrence ( )gn+ ( ( )gn (xngn- ( ) with ( and ( xfor the hermite polynomials are obtained when using instead the density of the standard normal distributionw(xe- / px these polynomials satisfy the recursion gn+ (xxgn (xdgn (xdx hermite polynomials with ( note that the hermite polynomials as defined above have ... |
15,157 | converselysuppose is continuous in particularit is continuous at (the zeroelement of vthustake and let and be as in ( for any let /( kgkv as khkv / dit follows from ( that kahkw kagkw kgkv rearranging the last inequality gives kagkw /dkgkv showing that is bounded theorem riesz representation theorem any bounded linear ... |
15,158 | definition (multivariatefourier transform the fourier transform of (realor complex-valuedfunction (rd is the function defined as ( : - (xdx rd fourier transform rd the fourier transform is continuousuniformly bounded (since (rd imr plies that ( ) rd ( )dx )and satisfies limktke ( ( result known as the riemann-lebesgue ... |
15,159 | convolutionlet fg (rd be real or complex valued functions their convolutionf gis defined as )(xf (yg( ydyrd and is also in (rd moreoverthe fourier transform satisfies gf [ dualitylet and both be in (rd then [ ]](tf (- product formulalet fg (rd and denote by fe their respective fourier transforms then gf (rd )and (zg(zd... |
15,160 | where - ( - on- ( - ( - ) the matrix is so-called vandermonde matrixand is clearly symmetric ( fmoreoverfn is in fact unitary matrix and hence its inverse is simply its complex conjugate fn thusf- / and we have that the inverse discrete fourier transform (idftis given by - -st ( xt = or in terms of matrices and vectors... |
15,161 | compute [ zn ln- ] compute steps and are (up to constantsin the form of an idftand step is in the form of dft these are computable via the fft (section in ( ln ntime step is dot product computable in (ntime thusthe circular convolution can be computed with the aid of the fft in ( ln ntime one can also efficiently compu... |
15,162 | the essence of the algorithm lies in the following observation suppose then one can express any index appearing in ( via pair ( )with where { and { similarlyone can express any index appearing in ( via pair ( )with where { and { identifying xt xt , and , we may re-express ( as xt , - - = , = ( observe that (because or ... |
15,163 | ultivariate ifferentiation and ptimization the purpose of this appendix is to review various aspects of multivariate differentiation and optimization we assume the reader is familiar with differentiating realvalued function multivariate differentiation for multivariate function that maps vector [ xn ]to real number ( )... |
15,164 | matrix of jacobi the transpose of this matrix is known as the matrix of jacobi of at (sometimes called the frechet derivative of at )that isf # ( :( fm fm fm xn if we define ( : (xand take the "vector/vectorderivative of with respect to xwe obtain the matrix of second-order partial derivatives of xm xm ( :( xm xm xm he... |
15,165 | example (scalar/matrix derivative via the woodbury identitylet tr - where xa rnxn we now prove that - -ax- to show thisapply the woodbury matrix identity to an infinitesimal perturbationx euof xand take to obtain the following( )- - - - ( - )- - -- - - thereforeas tr ( )- tr - --tr - - -tr - ax- nowsuppose that is an a... |
15,166 | it follows immediately that if is symmetricthat isa athen (xax ax and (xax to prove ( )first observe that (xxax ni= nj= ai xi which is quadratic form in xis real-valuedwith xx aik xi ak ai xi xk xk = = = = the first term on the right-hand side is equal to the -th element of axwhereas the second term equals the -th elem... |
15,167 | howin neighborhood of xthe function can be approximated by linear functionf ( hf (xj ( )hand similarly for (ythe well-known chain rule of calculus simply states that the derivative of the composition is the matrix product of the derivatives of and that ischain rule gf (xj (yj (xgf rn figure function composition the blu... |
15,168 | is an orthogonal matrix and diag( is the diagonal matrix of eigenvalues of the eigenvalues are strictly positivesince is positive definite denoting the columns of by (qi )we have li > aqi tr qi aq> ( from the properties of determinantswe have :ln |aln | qln(| | | |pp ln |di= ln li we can thus write ln |ax ln li li ln l... |
15,169 | heref is the objective functionand {gi and {hi are given functions so that hi ( and gi ( represent the equality and inequality constraintsrespectively the region where the objective function is defined and where all the constraints are satisfied is called the feasible region an optimization problem without constraints ... |
15,170 | subgradient the vector in ( may not be unique and is referred to as subgradient of one of the crucial properties of convex function is that jensen' inequality holds (see exercise in ) (xf (ex)for any random vector directional derivative example (convexity and directional derivativethe directional derivative of multivar... |
15,171 | example (convexity and differentiabilityif is continuously differentiable multivariate functionthen is convex if and only if the univariate function ( : ( ) [ is convex function for any and in the interior of the domain of this property provides an alternative definition for convexity of multivariate and differentiable... |
15,172 | table some common classes of optimization problems name (xconstraints linear program (lpcx ax and inequality form lp cx ax quadratic program (qp ax bx dx dex convex qp ax bx dx dex convex program (xconvex {gi ( )convex{hi ( )of the form > bi ( recognizing convex optimization problems or those that can be transformed to... |
15,173 | theorem karush-kuhn-tucker (kktconditions necessary condition for point xto be local minimizer of (xin the optimization problem ( is the existence of an aand bsuch that (xab (xab gi ( * * gi ( ki ki for convex programs we have the following important results [ ] every local solution xto convex programming problem is gl... |
15,174 | strong duality linearly constrained problemsif the primal is infeasible (does not have solution satisfying the constraints)then the dual is either infeasible or unbounded converselyif the dual is infeasible then the primal has no solution of crucial importance is the strong duality theoremwhich states that for convex p... |
15,175 | starting with an initial guess most minimization and root-finding algorithms create sequence using the iterative updating rulext+ xt at dt ( where at is (typically smallstep sizecalled the learning rateand the vector dt is the search direction at step the iteration ( continues until the sequence {xt is deemed to have c... |
15,176 | algorithm newton-raphson for finding roots of ( inputan initial guess and stopping error outputthe approximate root of ( while ( ) and budget is not exhausted do solve the linear system (xd ( - while ( ) ( - ak ( ) do / xx+ad return we can adapt root-finding newton-like method in order to minimize differentiable functi... |
15,177 | downside with all newton-like methods is that at each step they require the calculation and inversion of an hessian matrixwhich has computing time of ( )and is thus infeasible for large one way to avoid this cost is to use quasi-newton methodsdescribed next quasi-newton methods the idea behind quasi-newton methods is t... |
15,178 | in practical implementationwe keep single copy of in memory and apply the bfgs update to it at every iteration note that if the current is symmetricthen so is the updated matrix moreoverthe bfgs update is matrix of rank two since the kullback-leibler divergence is not symmetricit is possible to flip the roles of and in... |
15,179 | |{zcauchy-schwartz kuk uthe steepest descent iterationxt+ xt at (xt )still requires suitable choice of the learning rate at an alternative way to think about the iteration is to assume that the learning rate is always unityand that at each iteration we use an inverse hessian matrix of the form at for some positive cons... |
15,180 | hessian approximation is invertiblewe add suitable diagonal matrix to obtain the regularized version of the approximationt ui > ht = - + with this full-rank approximation of the hessianthe newton search direction in ( becomes- dt ui > ut ( = - + thusdt can be computed in ( ntime via the sherman-morrison algorithm furth... |
15,181 | unlike the search direction ( for newton-like algorithmsthe search direction of gauss-newton algorithm does not require the computation of hessian matrix observe that in the gauss-newton approach we determine dt by viewing the search direction as coefficients in linear regression with feature matrix gt and response -et... |
15,182 | table some transformations to eliminate constraints penalty functions constrained unconstrained > > exp(yy ( asin (yunfortunatelyan unconstrained minimization method used in combination with these transformations is rarely effective insteadit is more common to use penalty functions the overarching idea of penalty funct... |
15,183 | suppose that ( has inequality constraints only barrier functions are an important example of penalty functions that can handle inequality constraints the prototypical example is logarithmic barrier function which gives the unconstrained optimizatione (xf (xn ln(- ( )) = such that the minimizer of tends to the minimizer... |
15,184 | where is the hessian of at xd :diag ( /( :[ -iis an ( nmatrixand )is an diagonal matrix#- : - furtherwe define hn :( be- )- ( )- using this notation and applying the matrix blockwise inversion formula ( )we obtain the inverse of the matrix of jacobihn #- - - hn -hn be - - bhn - - bhn be- be -dhn dhn dhn thereforethe se... |
15,185 | algorithm approximating xargmin > (xwith logarithmic barrier inputan initial guess and stopping error outputthe approximate nonnegative minimizer xn of xb /sdx while kdxk and budget is not exhausted do compute the gradient and the hessian of at /ss /sw if ( diag( ) then /if cholesky successful compute the cholesky fact... |
15,186 | robability and tatistics the purpose of this is to establish the baseline probability and statistics background for this book we review basic concepts such as the sum and product rules of probabilityrandom variables and their probability distributionsexpectationsindependenceconditional probabilitytransformation rulesli... |
15,187 | sum rule union bound elementary event when ( holds as an equalityit is often referred to as the sum rule of probability it simply states that if an event can happen in number of different but not simultaneous waysthe probability of that event is the sum of the probabilities of the comprising events if the events are al... |
15,188 | all probabilities involving random variable can be computedin principlefrom its cumulative distribution function (cdf)defined by (xp[ ] cumulative distribution function for example [ bp[ bp[ af(bf(afigure shows generic cdf note that any cdf is right-continuousincreasingand lies between and figure cumulative distributio... |
15,189 | continuous cdf cdf fc is called continuous if there exists positive function such that for all fc (xf (udu ( pdf note that such an fc is differentiable (and hence continuouswith derivative the function is called the probability density function (continuous pdfby the fundamental theorem of integrationwe have [ bf(bf(af ... |
15,190 | table commonly used continuous distributions name notation uniform [abnormal (us gamma gamma(alf ( - - la xa- -lx ( - la - - -lx (ainverse gamma invgamma(alx parameters [aba ral ral exponential exp(ll -lx rl> beta beta(abg( ba- ( ) - ( ) ( [ ab weibull weib(alal (lx) - -(lxral pareto pareto(alral student tn al ( lx)-( ... |
15,191 | expectation expectation it is often useful to consider different kinds of numerical characteristics of random variable one such quantity is the expectationwhich measures the "averagevalue of the distribution the expectation (or expected value or meanof random variable with pdf denoted by ex or [ (and sometimes )is defi... |
15,192 | it is sometimes useful to consider the moment generating function of random variable this is the function defined by (se sx moment generating function ( the moment generating functions of two random variables coincide if and only if the random variables have the same distributionsee also theorem example (moment generat... |
15,193 | conditioning and independence conditional probabilities and conditional distributions are used to model additional information on random experiment independence is used to model lack of such information conditional probability conditional probability suppose some event ohm occurs given this factevent will occur if and ... |
15,194 | the concept of independence can also be formulated for random variables random variables are said to be independent if the events {xi xi }{xin xin are independent for all finite choices of distinct indices in and values xi xin an important characterization of independent random variables is the following (for proofsee ... |
15,195 | -sx sy cov(xy sx sy cov(ax byza cov(xzb cov(yz cov(xxs var[ ys cov(xy if and are independentthen cov(xy as consequence of properties and we have that for any sequence of independent random variables xn with variances var[ an xn expectation vector ( for any choice of constants an for random column vectorssuch as [ xn ]i... |
15,196 | conditional density and conditional expectation suppose and are both discrete or both continuouswith joint pdf and suppose fx ( thenthe conditional pdf of given is given by fy| ( xf (xyfx (xfor all conditional pdf ( in the discrete casethe formula is direct translation of ( )with fy| ( xp[ xin the continuous casea simi... |
15,197 | theorem linear transformation if has an expectation vector and covariance matrix then the expectation vector of is uz ux ( and the covariance matrix of is ( ifin additiona is an invertible matrix and is continuous random vector with pdf then the pdf of the continuous random vector ax is given by (zf ( - zz rn det( )( w... |
15,198 | theorem transformation rule let be an -dimensional vector of continuous random variables with pdf let ( )where is an invertible mapping with inverse - and matrix of jacobi that isthe matrix of partial derivatives of thenat (xthe random vector has pdf (zf (xf - ( )det( - ( ))| rn det( ( ))( prooffor fixed xlet ( )and th... |
15,199 | which has determinant since (cos th sin thr it follows by the transformation rule ( that the joint pdf of and th is given by fr,th (rthfx, (xyr re th ( ) by integrating out th and rrespectivelywe find fr (rr - / and fth (th /( psince fr,th is the product of fr and fth the random variables and th are independent normal ... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.