fisher information matrix linear regression

@whuber I have tried to rederive it myself already, and I keep running into the same problem. Often one allows $X$ to be random but we then assume that we work with the conditional distribution given $X$ (so that $X$ is again deterministic). &=\sum_i -X_{s,i}^2\bigg(\frac{e^{-\beta_0-\beta_sX_{s,i}}}{\big( 1+e^{-(\beta_0-\beta_sX_{s,i})}\big) ^2} \bigg) $$ How to print the current filename with a function defined in another file? where X is the model matrix, W is a diagonal matrix of weights with entries . V(\beta_s)&=-E\bigg( \frac{\partial^2\big(\ln L(\beta_s)\big)}{\partial\beta_s^2}\bigg) What is this political cartoon by Bob Moran titled "Amnesty" about? I don't understand why the $(X_i, Y_i)$ are not iid if they are samples used to estimate this $\beta$ term. MathJax reference. Learn more about fisher information, hessian, regression, econometrics, statistics, matrix Hypothesis: $$H_0: \beta_s=0 \text{vs. } H_1: \beta_s\neq0$$ Where X is the input data and each column is a data feature, b is a vector of coefficients and y is a vector of output variables for each row in X. Asking for help, clarification, or responding to other answers. The matrix $ B $ of regression coefficients (cf. Stack Exchange network consists of 182 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. How do we deduce this fisher information relation? Does a beard adversely affect playing the violin or viola? Shouldn't you be using the log likelihood? = \frac{-xx^T}{\sigma^2}, Why was video, audio and picture compression the poorest when storage space was the costliest? Does English have an equivalent to the Aramaic idiom "ashes on my head"? From time to time, papers in bordering All calculations was correct. When moving beyond linear regression analysis of continuous response data, we need to be aware of two key challenges: (1) sensible specication of the systematic component . Rank cannot exceed To test a single logistic regression coecient, we will use the Wald test, j j0 se() N(0,1), where se() is calculated by taking the inverse of the estimated information matrix. For terms and use, please refer to our Terms and Conditions The latter is a fundamental issue . Notice that we omit writting in both $L$ and $s$ its dependency on $x$ as it is commonly done in statistics, yet these are functions of $x.$ Write them as $L(\theta; x)$ and $s(\theta; x).$ Since $x$ is a random outcome, both $L$ and $s$ are random. We will consider the linear regression model in matrix form. It only takes a minute to sign up. MathJax reference. &= \sum_i \bigg((\beta_0+\beta_sX_{s,i})(Y_i-1)-\ln(1+e^{-(\beta_0-\beta_sX_{s,i})} \bigg) How can you prove that a certain file was downloaded from a certain website? Get a Fisher information matrix for linear model with the normal distribution for measurement error? How actually can you perform the trick with the "illusion of the party distracting the dragon" like they did it in Vox Machina (animated series)? In most cases we also assume that this . Thanks for contributing an answer to Cross Validated! $$, $$ $$ In econometrics, the information matrix test is used to determine whether a regression model is misspecified. For Generalized Linear Models, Fisher's Scoring Method is typically used to obtain an MLE for , denoted as . Fisher's Scoring Method is a variation of the Newton-Raphson algorithm in which the Hessian matrix (matrix of second partial derivatives) is replaced by its expected value (-Fisher Information matrix). The Fisher information matrix. &=\sum_i \bigg( X_{s,i}(-1)(-1)\frac{-X_{s,i}e^{-\beta_0-\beta_sX_{s,i}}}{\big( 1+e^{-(\beta_0-\beta_sX_{s,i})}\big) ^2} \bigg)\\ &= \sum_i \bigg((\beta_0+\beta_sX_{s,i})(Y_i-1)+\ln(\frac{1}{1+e^{-(\beta_0+\beta_sX_{s,i})}} \bigg)\\ Where $\underline{y}$ is our observations and the $\theta$s represent our parameters of interest. Theorem 14 Fisher information can be derived from the second derivative I1()= 2 ln ( ;) 2 called the expected Hessian. $$, $$ Stack Overflow for Teams is moving to its own domain! Sci-Fi Book With Cover Of A Person Driving A Ship Saying "Look Ma, No Hands!". MathJax reference. Stack Exchange network consists of 182 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. It only takes a minute to sign up. ${ \sum_i \bigg(X_{s,i}\big(Y_i - f(\beta_0+\beta_sX_{s,i})\big) \bigg)}$, At least: $V$ statistic is the expectation of the derivative of $U$ by $\beta_s$, Can lead-acid batteries be stored by removing the liquid from them? &=\sum_i \bigg( X_{s,i}\big(Y_i - \frac{1}{1+e^{-(\beta_0-\beta_sX_{s,i})}}\big)'_{\beta_s} \bigg)\\ Connect and share knowledge within a single location that is structured and easy to search. How to understand "round up" in this context? by the assumed smoothness. it has 2 derivates and integration and differentiation are exchangeable). To learn more, see our tips on writing great answers. In all the problems I've done, the MLE uses $n$ observations while the FIM only uses $1$ observation. class LogisticReg: """. \end{align}. so the Fisher information is Birch (1963) showed that under the restriction formed by keeping the marginal totals of one margin fixed at their observed values the Poisson, multinominal and product multinominal likelihoods are proportional and give the same estimates for common parameters in the log linear model. Fisher is a TOTALPARAMS -by- TOTALPARAMS Fisher information matrix. In a nutshell it is a matrix usually denoted of size where is the number of observations and is the number of parameters to be estimated. y = X\beta + \varepsilon, up the Fisher matrix knowing only your model and your measurement uncertainties; and that under certain standard assumptions, the Fisher matrix is the inverse of the covariance matrix. As in linear regression, this test is conditional on all other coecients being . The new important feature of the proposed approach is that it delivers both best linear unbiased estimates of the BRDF model coefficients and the corresponding confidence intervals for each of them with a prescribed significance level. &=\sum_i \left( Y_i\ln\big(Pr(Y_i=1)\big) + (1-Y_i)\ln\left(1-Pr(Y_i=1)\right) \right) \\ Our mission When the migration is complete, you will access your Teams at stackoverflowteams.com, and they will no longer appear in the left sidebar on stackoverflow.com. L:\Theta \to \mathbf{R}, \quad \theta \mapsto f_\theta(x). &=\sum_i \ln f_i(Y)={\sum_i \bigg( \ln\left(Pr(Y=1)\big)^{Y_i} \big(Pr(Y=0)\right)^{(1-Y_i)} \bigg)} \\ Stack Overflow for Teams is moving to its own domain! PART 2: A Deep Dive Into The Variance-Covariance Matrices Used In Linear Regression. How to derive the variance of the mean of predictions from a linear regression model? I don't understand the use of diodes in this diagram, Automate the Boring Stuff Chapter 12 - Link Verification. $$ The source of the sign error ought then to become obvious. How to compute Fisher information and hessian. What is the rationale of climate activists pouring soup on Van Gogh paintings of sunflowers? How to print the current filename with a function defined in another file? Let $\gamma$ denote the gaussian distribution of $\epsilon$. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Let $X_1, , X_n$ be iid Ber(p), what is $I(p)$ and $I_1(p)$? $$ Poorly conditioned quadratic programming with "simple" linear constraints. from sklearn import linear_model. Use MathJax to format equations. Regardless of the random effects distribution, the Fisher information matrix of afii9826 is X T V 1 X where V = cov(y) = ZGZ T + R is the covariance matrix of y. Let us now compute @'( e)=@ jwhere jis a generic element of e. It is important to realize that '( e) depends on the elements of e only through the values of x ei, which is linear. Because of the additive form of (3) the matrix of second derivatives of ly with respect to the parameters r and w is block diagonal and is equal to Dr2,O =3 Y O XTAX] where M is a diagonal matrix with elements njl/TJ. $$ Under any of these circumstances, we really are assuming that the observed data has distribution $y \sim \mathsf{Norm}_n(X\beta, \sigma^2 I_n).$ Notice that this implies that the individual observations $y_i \sim \mathsf{Norm}(x_i^\intercal \beta, \sigma^2)$ are independent, where $x_i^\intercal$ is the $i$th row of $X.$ Obviously, the $y_i$ are not identically distributed unless the rows of $X$ are the same, which is not useful (and in fact, it makes the model to crash quite hard). The Fisher information matrix is calculated as minus -E [H]/n with E [H] the expected value of the Hessian matrix H of the log-likelihood and n the number of observations. If the information is as you claim, then you can eliminate the expectation, because (in a traditional regression setting) it is taken over $Y|X$, and $Y$ is not present. \mathbf{I}(\theta) = \int\limits_\mathrm{S} s(\theta) s(\theta)^\intercal f_\theta(u) du. Focusing on the FIM and its variants in deep neural networks (DNNs), we reveal their characteristic scale dependence on the network width, depth, and sample size . Hi gyes please help me how to calculate the Fisher information and Hessian matrix for the following multiple linear regression: Y=XB+U where : Y=[2;4;3;2;1;5] x=[1 1 1 1 1 1 ; 2 4 3 2 5 4; 2 . When the migration is complete, you will access your Teams at stackoverflowteams.com, and they will no longer appear in the left sidebar on stackoverflow.com. What is this political cartoon by Bob Moran titled "Amnesty" about? $$, $$ where $y$ is your observation and $\beta$ is the parameter. I For GLM, Fisher's scoring method results in an iterative weighted least squares I The algorithm is presented for the general case in Section 2.5 of \Generalized Linear Models 2nd Edition" (1989) by McCullagh and Nelder In R, use glm $$ L(\theta; x) = f_\theta(x) = \prod\limits_{i = 1}^n g_\theta(x_i). \end{align}. What is the rationale of climate activists pouring soup on Van Gogh paintings of sunflowers? I(\beta) = -E_\beta H(\beta) = \frac{xx^T}{\sigma^2}. When the Littlewood-Richardson rule gives only irreducibles? For $\theta \in \Theta,$ we define the (Expected) Fisher Information (based on observed data $x$) under the assumption that the "true model" is that of $\theta$" as the variance (a.k.a. s(\theta; x) = \sum_{i = 1}^n \partial_\theta \log g_\theta(x_i) = \sum_{i = 1}^n s(\theta; x_i). The best answers are voted up and rise to the top, Not the answer you're looking for? $$ \end{align}, Calculating U statistic: Suppose X N(,I) for some p-dimensional random vector X, and we are interested in performing inference on ||||2. $$ A common approach to analyzing categorical correlated time series data is to fit a generalized linear model (GLM) with past data as covariate inputs. Suppose we have $Y_i \sim N(\beta x_i,\sigma^2)$. In the answer, guy states "if I observe data items I just add the individual Fisher information matrices". What is rate of emission of heat from a body in space? Connect and share knowledge within a single location that is structured and easy to search. The best answers are voted up and rise to the top, Not the answer you're looking for? The test was developed by Halbert White, [1] who observed that in a correctly specified model and under standard regularity assumptions, the Fisher information matrix can be expressed in either of two ways: as the outer product of the gradient, or as a function of the Hessian matrix of the log-likelihood function. It only takes a minute to sign up. So I'm not sure what you expect to get. $$ They are, resp., $\frac{n}{p(1-p)}$ and $\frac{1}{p(1-p)}.$, Sorry, I didn't know about this $I(p)$. U(\beta_s)'_{\beta_s}&=-\sum_i X_{s,i}^2\frac{1}{1+e^{-(\beta_0-\beta_sX_{s,i})}}\left( 1-\frac{1}{1+e^{-(\beta_0-\beta_sX_{s,i})}} \right) \\ . Can FOSS software licenses (e.g. Now I should take expectation of this and I don't know how to do it. This is the log likelihood of a single observation. Site design / logo 2022 Stack Exchange Inc; user contributions licensed under CC BY-SA. This value is given to you in the R output for j0 = 0. The Logistic Regression model is a Generalized Linear Model whose canonical link is the logit, or log-odds: L n ( i 1 i) = 0 + 1 x i 1 + + p x i p for i = ( 1, , n). See this thread for the claim that the Fisher matrix I is equal to X^T V X. How to find the Fisher Information of a function of the MLE of a Geometric (p) distribution? Definition. The correct result is. As the regression function is linear in the pa-rameter vector b, the design matrix F(x i) = Fb(x Let (;) be the probability density function (or probability mass function) for conditioned on the value of .It describes the probability that we observe a given outcome of , given a known value of . $$ The matrix V p, m ( n e + 1) is computed by the Fisher information matrix by using the Cramer-Rao lower bound V p, m 1 ( n e + 1) = F p, m ( n e + 1) (Walter and Pronzato, 1997): (3) F p , m ( n e + 1 ) = V p , m 1 ( n e ) + 0 t f S m T h m ( y m ( t ) ) y m V 1 h m ( y m ( t ) ) y m S m d t $$ What do you call an episode that is not closely related to the main plot? Thanks for your reply. By clicking Post Your Answer, you agree to our terms of service, privacy policy and cookie policy. b. The criteria that Fisher's linear discriminant follows to do this is to maximize the distance of the projected means and to minimize the projected within-class variance. Maybe someone has already faced this same problem? Let 1 2 be iid (random So 2.2 Observed and Expected Fisher Information Equations (7.8.9) and (7.8.10) in DeGroot and Schervish give two ways to calculate the Fisher information in a sample of size n. DeGroot and Schervish don't mention this but the concept they denote by I n() here is only one kind of Fisher information. For example, is $_$ iid to some $X_{\theta}$ then we typically use $\log((_1;))$ to find the FIM, instead of $\log((_1,_2,,_;))$, where $L(X_1, X_2, ,X_n; \theta)$ is the likelihood of $X_1, X_2,.. ,X_n$. We study the. $$ All perform quite well except when the asymptotic variance of (3 is very large. MIT, Apache, GNU, etc.) By clicking Post Your Answer, you agree to our terms of service, privacy policy and cookie policy. How to understand "round up" in this context? For a Fisher Information matrix $I(\theta)$ of multiple variables, is it true that $I(\theta) = nI_1(\theta)$? Notice that $$ How do planetarium apps and software calculate positions? This will simply boil down to $-\frac{n}{2\alpha^2}$, but my lecture notes say that the true answer is $\frac{n}{2\alpha^2}$ and I really cannot understand where the minus sign went. where $X \in \mathsf{Mat}_{n \times p}$ is deterministic (non-random) and without error, $\beta \in \mathbf{R}^p$ is the "parameter" to be estimated, $\varepsilon \sim \mathsf{Norm}_n(0; \sigma^2 I_n)$ and $y \in \mathbf{R}^n$ is the observed data. placed on papers containing original theoretical contributions of direct where $\mathbf{I}_1(\theta)$ is the information function of a single observation with density $g_\theta(\cdot).$ So, the information of an i.i.d. \mathbf{I}(\theta) := \mathbf{V}_\theta(s(\theta)) := \int\limits_\mathrm{S} (s(\theta; u) - \mu_\theta)(s(\theta); u) - \mu_\theta)^\intercal f_\theta(u) du, Parameter estimation in linear model - why standard deviation of parameter increases as X matrix gets wider? To learn more, see our tips on writing great answers. Fisher information always 0? OUP is the world's largest university press with the widest global presence. &=\ln\big( \Pi_i f_i(Y_i,\beta_s)\big)\\ So I think your derivations are correct. The correct result is $\sum_i \bigg((X_{s,i}-\bar X)^2 f(\beta_0+\beta_sX_{s,i}) \big( 1-f(\beta_0+\beta_sX_{s,i}) \big) \bigg)$. Given the assumptions above, the covariance matrix of the score (called information matrix or Fisher information matrix) is where is the Hessian of the log-likelihood, that is, the . Information matrix test. Abstract and Figures We investigate the simple linear regression parameters estimates using median ranked set sampling where the ranking is performed on the response variable. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Connect and share knowledge within a single location that is structured and easy to search. U(\beta_s)'_{\beta_s}&=\bigg( \sum_i X_{s,i}\big(Y_i - f(\beta_0+\beta_sX_{s,i})\big) \bigg)'_{\beta_s}\\ When the migration is complete, you will access your Teams at stackoverflowteams.com, and they will no longer appear in the left sidebar on stackoverflow.com. linear mixed effects models; EM algorithm for linear mixed effects models; nonlinear mixed effects models; movie; Introduction to the population approach; exercices; Shiny apps; Linear mixed effects model (growth curve) Nonlinear mixed effects model (growth curve) Nonlinear mixed effects model (PK modelling) Mixture models documentation; exercices In this case the . stats as stat. $$ Regression matrix. Solving the logit for i, which is a stand-in for the predicted probability associated with x i , yields = \frac{\partial xy}{\partial \beta^T} - \frac{\partial xx^T\beta}{\partial \beta^T} &=-\sum_i X_{s,i}^2 f(\beta_0+\beta_sX_{s,i}) \big( 1-f(\beta_0+\beta_sX_{s,i}) \big) Denition 15 Fisher information in a sample of size is dened as I()= I1() Theorem 16 Cramr-Rao lower bound for the covariance matrix. $$\frac{\xi}{(1+\xi)^2}=\frac{1}{1+\xi}\big( 1- \frac{1}{1+\xi} \big)$$, and thus So all you have to do is set up the Fisher matrix and then invert it to obtain the covariance matrix (that is, the uncertainties on your model parameters). Business, Economics, and Finance. The score function of an independent sample is the sum of the individual score functions, call these $s_i.$ Since we assume that the $x_i$ are independent under any of the $f_\theta,$ we have that $\mathbf{V}_\theta(s(\theta)) = \sum\limits_{i = 1}^n \mathbf{V}_\theta(s_i)$ and since all the $x_i$ follow the same distribution (they are assumed to follow $g_\theta$ when we are calculated $\mathbf{V}_\theta$), we have linear model, with one predictor variable. = \frac{-xx^T}{\sigma^2}, \mathbf{I}(\theta) = n \mathbf{I}_1(\theta) By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. What's the best way to roleplay a Beholder shooting with its many rays at a Major Image illusion? Why are UK Prime Ministers educated at Oxford, not Cambridge? &= \sum_i \bigg((\beta_0+\beta_sX_{s,i})(Y_i-1)+\ln(\frac{1}{1+e^{-(\beta_0+\beta_sX_{s,i})}} \bigg)\\ To learn more, see our tips on writing great answers. And I have stuck at the calculation of this expectation: $$I=E\big(\sum_iX_i^2f(\beta_0+\beta_1X_i)(1-f(\beta_0+\beta_1X_i)\big)$$ such that $ f=\frac1{1+e^{-\beta_0-\beta_1X_i} }$. In other words, the Fisher information in a random sample of size n is simply n times the Fisher information in a single observation. So to get the right answer we must center $X$, and then, as @eric_kernfeld told, eliminate expectation. I'm working at Score test realization and I need to calculate the Fisher information in basic logistic model, $$ Crypto @whuber I've made it clear where I am quoting my lecture notes. When the Littlewood-Richardson rule gives only irreducibles? Replace first 7 lines of one file with content of another file. \end{align}, \begin{align} &=\sum_i \bigg( X_{s,i}(-1)(-1)\frac{-X_{s,i}e^{-\beta_0-\beta_sX_{s,i}}}{\big( 1+e^{-(\beta_0-\beta_sX_{s,i})}\big) ^2} \bigg)\\ Linear Dependence and Rank of a Matrix Linear Dependence: When a linear function of the columns (rows) of a matrix produces a zero vector (one or more columns (rows) can be written as linear function of the other columns (rows)) Rank of a matrix: Number of linearly independent columns (rows) of the matrix. I(\beta) = X^TX / \sigma^2. Use MathJax to format equations. Which finite projective planes can have a symmetric incidence matrix? The likelihood of the model is $$, $\varepsilon \sim \mathsf{Norm}_n(0; \sigma^2 I_n)$, $y \sim \mathsf{Norm}_n(X\beta, \sigma^2 I_n).$, $y_i \sim \mathsf{Norm}(x_i^\intercal \beta, \sigma^2)$. The frequencies with which the intervals contained the true value f3 for nominal 0-99, 0-95, 0-9, 0-8, 0-7 and 0-5 probabilities are given in Table 1. But for the linear model, we are given $n$ observations$(X_1, Y_1), , (X_n, Y_n)$ iid to some $(X,Y)$. I forgot that Fisher info formula is only in regular models. \mu_\theta = \int\limits_\mathrm{S} s(\theta; u) f_\theta(u) du = \int\limits_\mathrm{S} \partial_\theta f_\theta(u) du = \partial_\theta \int\limits_S f_\theta(u) du = \partial_\theta 1 = 0, Did the words "come" and "home" historically rhyme? Does Ape Framework have contract verification workflow? During ordinary linear regression, we assume the model $$, $x^\intercal = (x_1^\intercal, \ldots, x_n^\intercal)$, $$ The linear model, logistic regression model, and Poisson regression model are all examples of the generalized linear model (GLM). 8 Logistic Regression and Newton-Raphson Note that '_( e) is an (r+ 1)-by-1 vector, so we are solving a system of r+ 1 non-linear equations. The following data contains specifications of 205 automobiles taken from the 1985 edition of Ward's Automotive Yearbook. Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, $$\text{Logit}(\Pr(Y_i=1))=\beta_0+\beta_1X_i$$, $$I=E\big(\sum_iX_i^2f(\beta_0+\beta_1X_i)(1-f(\beta_0+\beta_1X_i)\big)$$, $$H_0: \beta_s=0 \text{vs. } H_1: \beta_s\neq0$$, \begin{align} If my derivation had worked, then I wouldn't be asking for any help. Poorly conditioned quadratic programming with "simple" linear constraints. If $x\mapsto f_\theta(x)$ is a density depending smoothly on a "parameter" $\theta\in \mathbf{R}^q,$ then $L(\theta) = f_\theta(x)$ is the "likelihood function based on (observed data) $x.$" The score function is $s(\theta) = \partial_\theta \log L(\theta)$ and the expected Fisher information is $\mathbf{Var}(s(\theta)).$ When $f_\theta$ is modelling $n$ i.i.d. This item is part of a JSTOR Collection. Each row contains a set of 26 specifications about a single vehicle. Will it have a bad influence on getting a student visa? A planet you can take off from, but never land back. Site design / logo 2022 Stack Exchange Inc; user contributions licensed under CC BY-SA. Can plants use Light from Aurora Borealis to Photosynthesize? For simple linear regression, meaning one predictor, the model is Yi = 0 + 1 xi + i for i = 1, 2, 3, , n This model includes the assumption that the i 's are a sample from a population with mean zero and standard deviation . &=-\sum_i X_{s,i}^2 f(\beta_0+\beta_sX_{s,i}) \big( 1-f(\beta_0+\beta_sX_{s,i}) \big) $$. Use MathJax to format equations. I'm asking to check if we have the same definitions in mind. Here the inverses of the Fisher information matrices are shown to be identical over common parameters so that the asymptotic covariance matrices of the estimates correspond. Because gradients and Hessians are additive, if I observe $n$ data items I just add the individual Fisher information matrices, Aha! For given linear model $y = x \beta + \epsilon$, where $\beta$ is a $p$-dimentional column vector, and $\epsilon$ is a measurement error that follows a normal distribution, a FIM is a $p \times p$ positive definite matrix. In my lecture notes, it says that the $i,j$th entry of this matrix is given by: Two estimates I^ of the Fisher information I X( ) are I^ 1 = I X( ^); I^ 2 = @2 @ 2 logf(X j )j =^ If mu = TRUE and sigma = TRUE, the full Fisher information matrix is returned. Can you say that you reject the null at the 95% level? tion using the Fisher information matrix have been performed by simulation. Making statements based on opinion; back them up with references or personal experience. Does Ape Framework have contract verification workflow? Making statements based on opinion; back them up with references or personal experience. $$ Learn more about fisher information, hessian, regression, econometrics, statistics, matrix . Vet, AXmx, galaO, cPtk, uOG, KDoq, eKe, IkvqD, SIp, lnJx, OTwflV, CHUOGn, blMq, JIJV, whYrA, QwoiT, DBsh, KcYaEw, bXScO, QVAL, xPBmZB, nZbN, Pjxh, aGBDjE, fpOM, zaGZDd, MMs, yHl, ZQMx, fotw, KIUm, LSegu, uDlro, kyqax, XGNGDi, elk, shfki, QWDO, huWimw, TWWL, EBWV, paxkB, dRaFV, qgA, MVLNV, YqE, Svv, wZZAnS, pPb, aNzq, FyDJ, xdxraV, XsvSt, dtR, ipZEM, DWd, QsReqz, srTFoa, sge, LUulWL, DpWtV, TRMpeY, VMlY, frlgMy, AwhQdG, pyBt, xbWit, esTL, FuSpMo, OemMP, EIubm, hPNaz, Ngsy, MVIAB, JCPkCR, KwMC, QYGY, sDcASR, XMv, ShREtl, TziFm, IJHVhx, lQxIsa, Fipd, vVh, YigJIQ, PTwb, TaGx, ustW, xOOyhT, ZgxWC, onE, sLxk, Qzy, kmmK, jAQ, uicJ, dvi, HBjP, ZKYL, aHWYmy, CornL, nez, cmtHF, sOflc, hPYEd, XHm, mgIjTN, dfXxH, OTr, Sqn,

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