User mathtick - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T22:53:05Z http://mathoverflow.net/feeds/user/8916 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/119895/multivariate-expansion-in-terms-of-single-variate-products-what-is-the-name-for Multivariate expansion in terms of single variate products: what is the name for this? mathtick 2013-01-25T21:59:32Z 2013-01-27T09:57:24Z <p>In some situations we have access to a representation like this:</p> <p>$f(x,y) = \sum_i u_i(x) v_i(y)$</p> <p>What is this called? (I know when you jam this into PDE get to call it 'separation of variables' but I'm sure it's got a different name in pure math).</p> http://mathoverflow.net/questions/118880/pairwise-gaussian-vs-jointly-gaussian-k-wise-gaussian-vs-n-wise-gaussian Pairwise Gaussian vs Jointly Gaussian (k-wise Gaussian vs n-wise Gaussian) mathtick 2013-01-14T14:44:02Z 2013-01-15T11:42:16Z <p>Suppose $X_i, \; i=1,2,3...,n$ are each Gaussian, then it is not in general true that the set is jointly Gaussian (a multivariate Gaussian).</p> <p>Does a similar statement hold if the variates are <em>pairwise</em> Gaussian? i.e. if we have that $X_i, X_j$ are a bivariate Gaussian for all $i, j$, then it is not in general true that the set is jointly Gaussian. (And I suppose any dimensional variant of this? k-wise Gaussian => n-wise Gaussian)</p> <p>If the statement holds, what is a simple low-dimensional example where you have a pairwise Gaussian system where this does not hold? Maybe this is in Counterexamples in Probability ... but I don't have this available right now. </p> http://mathoverflow.net/questions/107724/how-to-approximate-a-distribution-using-a-random-perturbation-of-the-distribution How to approximate a distribution using a random perturbation of the distribution mathtick 2012-09-20T23:10:07Z 2012-12-12T15:55:00Z <p>Suppose $f(0)=0$ and you want to simulate $f(Z)$ for some random variate $Z$ that you can generate. However, you can only obtain values of $f(Y+Z)$ and $f(Y)$ for some other variate $Y$. This feels like a standard problem that may even have a name. If so what is it called? If not, what can one do to approximate $f(Z)$?</p> <p>I should also note that I'm particularly interested in tail values of $f(Z)$ but any approximation ideas would be useful.</p> <p>I should also, also note that you may not use $Y$ or $Z$ or any metric on them. You can only use the values $f(Y+Z)$ and $f(Y)$ that you simulate. I'm thinking this almost kills any approximation possibilities but maybe there is some way of using the $f(0)=0$ property.</p> http://mathoverflow.net/questions/115676/log-likelihood-of-ito-diffusion log-likelihood of ito diffusion mathtick 2012-12-07T03:55:10Z 2012-12-09T19:13:42Z <p>Consider a diffusion process:</p> <p>$\text{d}X_t = f(X_t)\text{d}t + \text{d}W_t$</p> <p>I've seen it given that the log-likelihood of the path is proportional to the Onsager-Machlup functional</p> <p>$\int_0^T \left(\frac{1}{2} \left|\dot{x}-f(x)\right|^2+\frac{1}{2}\nabla_x\cdot f(x)\right)\text{d}t$.</p> <p>Where does the second term with the divergence come from? I think I once knew this ... is it something to do with the interpretation of $\dot{x}$ or the commutator $[x,\dot{x}]$?</p> <p>The expression can be found, for example, in p. 5 section 3.1 of Hairer et al, "A Bayesian approach to Data Assimilation" <a href="http://www.hairer.org/papers/bayesian.pdf" rel="nofollow">http://www.hairer.org/papers/bayesian.pdf</a>.</p> http://mathoverflow.net/questions/114657/using-symmetries-of-a-r-v-s-distribution-to-boost-samples-and-possibly-do-varian Using symmetries of a r.v.'s distribution to boost samples and possibly do variance reduction mathtick 2012-11-27T12:59:42Z 2012-11-27T12:59:42Z <p>Suppose, for example, you are simulating samples from a (multivariate) Gaussian with mean zero and covariance $\Gamma=BB^T$. If you had generated a sample $x$, you could generate more (dependent) samples by some unitary transformation $U$, I think via $\tilde{x}=BUB^{-1}x$ in this case. If you knew something about the statistic you were trying to estimate (for example, maybe we estimate $\xi=\mathbb{E}(f(x))$ where $f$ is smooth and is //not// invariant to these transformations) you could try to use the correlated samples to do some sort of variance reduction.</p> <p>Does this kind of technique where you use a symmetry of the distribution have a name? Is it even a technique ... maybe it is fatally flawed somehow? Maybe it would just fall under the Control Variate or CRN techniques? Uses of "Gauge invariance" feels like the right area to look in but I don't have much experience in physics.</p> http://mathoverflow.net/questions/105596/marginal-parameter-estimation-in-copula-with-copula-dependence-parameter-known marginal parameter estimation in copula with copula (dependence) parameter known mathtick 2012-08-27T02:39:59Z 2012-08-27T08:43:10Z <p>I've posted this already in stats.stackexchange. I'm not sure what the rules are for cross-posting but mathoverflow seems to be more active.</p> <p>Suppose we have data $x_i, i=1,2,3,...n$ that are <em>dependent</em> and identically distributed with marginal $f(\cdot|\alpha)$. If we model this with the likelihood</p> <p>$L = c(F(x_1|\alpha),F(x_2|\alpha),...F(x_n|\alpha)|\theta)\prod_{i=1}^n f(x_i|\alpha)$</p> <p>and the dependence parameter $\theta$ is known, can we apply some variant of the Expectation Maximization algorithm to estimate $\alpha$ using an iterative procedure with relatively simple steps?</p> <p>For instance, I considered a simple problem with exponential marginals and Gaussian copula (with known correlation), and did something procedural (and hokey). I introduced the unknown independent samples $\tilde{x}_i, i=1,2,...,n$ which you would compute by knowing the correct value of $\alpha$, mapping the $x_i$ to correlated Gaussians $y_i = \Phi^{-1}(F(x_i|\alpha))$ and then "undoing" the correlations $z = B^{-1}y$ and mapping the $z$ forward again to produce the independent $\tilde{x_i}=F^{-1}(\Phi(z_i)|\alpha)$. Here $C=BB^T$ is the correlation matrix. Here $\Phi$ is the (0,1)-normal cdf. If you turn this into an iterative procedure, using $x$ as the initial guess for the independent data, it seems to produce a series that (at least in my trials) converged. However, the whole thing is doubtful since it depends entirely on what you choose for $B$ (only defined up to a unitary matrix). I think it's the unitary invariance of the Gaussian hitting you when you try to basically do an inversion.</p> <p>Is there an obvious way to turn this kind of problem into a sane iterative procedure using simple steps like the EM? I feel like I'm missing a simple trick.</p> http://mathoverflow.net/questions/96007/gaussian-copula-and-the-addition-of-an-identity-matrix/105548#105548 Answer by mathtick for Gaussian Copula and the addition of an Identity matrix mathtick 2012-08-26T16:06:43Z 2012-08-26T16:06:43Z <p>What's a good reference for this derivation (online). Wikipedia only has an advertisement for someone's book as a reference. </p> <p>I find this presentation very confusing, for example I had missing that this was the density (small c) as opposed to the CDF and was obvious not making any sense of it. I can imagine other readers doing the same thing.</p> http://mathoverflow.net/questions/105390/results-for-minimizing-the-norm-w-r-t-a-unitary-matrix Results for minimizing the norm w.r.t a unitary matrix mathtick 2012-08-24T14:04:37Z 2012-08-24T14:53:56Z <p>Suppose $x \in \mathbb{R}^n$, $B,U \in \mathbb{R}^n\times\mathbb{R}^n$ and $U$ a unitary matrix. Define $g_{U}(x) = || BUx||$ where $||.||$ is some norm or norm-ish function on $\mathbb{R}^n$ (not unitarily invariant obviously). How can we choose $U$ in the unitary matrices to minimize $g$? Or what kind of results are there regarding the minimum value?</p> <p>I'm particularly interested in $||.||_{\infty}$, the tail expectation, and maybe also the quantile function (i.e. $||x||$ is the $k$th largest element ... which will not be a norm.)</p> <p>I'm mostly looking to "bind" this problem in the right way so that I can read the most appropriate material. I would imagine this kind of problem has been studied to death in various contexts. Or maybe I'm not seeing that the problem is in fact trivial or I've mis-specified it.</p> http://mathoverflow.net/questions/102777/derivative-of-the-cdf-of-a-family-of-random-variables Derivative of the CDF of a family of random variables mathtick 2012-07-20T22:22:06Z 2012-07-20T23:54:25Z <p>Suppose I have a r.v. $Z = X + \alpha Y$ and that $F_Z$ is the probability distribution function of $Z$. If we think of the probability $p = F_Z(q) = \mathbb{P}(X+\alpha Y &lt; q)$ as a function $p = p(q, \alpha)$, how can we write the derivative $\partial_{\alpha} p(q, \alpha)$ supposing as much regularilty on the distribution of $X$ and $Y$ as we want?</p> <p>I'm loosely thinking of a situation where you know the marginals distributions $F_X$ and $F_Y$ and maybe locally (around $(q, \alpha)$) know some information about the dependence of $X$ and $Y$ (maybe correlation is enough in some cases to build an approximation?). </p> http://mathoverflow.net/questions/100195/multivariate-linear-regression-with-dependent-noise-terms multivariate linear regression with dependent noise terms? mathtick 2012-06-21T01:06:26Z 2012-07-05T13:22:00Z <p>What is it //called// when you are doing linear regression on the problem: $Y = AX+BZ$ where you are given observations Y and X and are assuming Z is independent Gausssians? If you do max-Likelihood I think you end up minimizing something like $\| Y-AX\|_B = (Y-AX)^TC^{-1}(Y-AX)$ where $C = B^T B$.</p> <p>But what is this called and are there any tricks like linear regression or is the standard method just to optimize over the variables with some conditions on the rank of $B$ I believe. </p> http://mathoverflow.net/questions/37277/stochastic-matrix-for-which-a-stochastic-matrix-logarithm-exists/37297#37297 Answer by mathtick for (Stochastic) matrix for which a stochastic matrix logarithm exists? mathtick 2010-08-31T19:08:30Z 2010-08-31T19:08:30Z <p>Steve Hunstman's link above is good:</p> <p>See the part leading up to Theorem 9 for something relevant to applications:</p> <blockquote> <p>The main application of the following theorem may be to establish that certain Markov matrices arising in applications are not embeddable, and hence either that the entries are not numerically accurate or that the underlying process is not autonomous. The theorem is a quantitative strengthening of Lemma 8. It is of limited value except when n is fairly small, but this is often the case in applications.</p> </blockquote> <p>Also the part on regularization for best compromises when matrices are not embeddable.</p> http://mathoverflow.net/questions/37277/stochastic-matrix-for-which-a-stochastic-matrix-logarithm-exists (Stochastic) matrix for which a stochastic matrix logarithm exists? mathtick 2010-08-31T16:28:42Z 2010-08-31T19:08:30Z <p>I think this is basically the inverse question of <a href="http://mathoverflow.net/questions/33230/matrices-whose-exponential-is-stochastic" rel="nofollow">http://mathoverflow.net/questions/33230/matrices-whose-exponential-is-stochastic</a>. </p> <p>i.e. what are sufficient conditions on the matrix representation of an evolution operator of a (finite) discrete Markov chain for it to be embeddable in a continuous Markov chain?</p> <p>I've found some old paper that may answer this (something about embeddability criteria) but I can't access it as it published in a closed-access journal.</p> <p>I hope this is a sane question.</p> http://mathoverflow.net/questions/114657/using-symmetries-of-a-r-v-s-distribution-to-boost-samples-and-possibly-do-varian Comment by mathtick mathtick 2012-11-28T17:29:03Z 2012-11-28T17:29:03Z Good point. In my case (at least as I meant to describe it), the problem lies in the fact the probability distribution has a symmetry but the function that we are estimating breaks this symmetry. Basically, only part of the integrand has the symmetry. http://mathoverflow.net/questions/105390/results-for-minimizing-the-norm-w-r-t-a-unitary-matrix Comment by mathtick mathtick 2012-08-26T02:55:55Z 2012-08-26T02:55:55Z @will: $x$ is fixed. I wasn't very clear about that and wrote the problem in a funny way ... partly due to where it was coming from. http://mathoverflow.net/questions/105390/results-for-minimizing-the-norm-w-r-t-a-unitary-matrix/105393#105393 Comment by mathtick mathtick 2012-08-24T15:43:39Z 2012-08-24T15:43:39Z I think you are maximizing but I the same reasoning applies: to minimize the infinity norm: rotate or reflect the x onto the &quot;diagonal&quot; i.e. something like the &quot;1&quot; vector. For //minimizing// the $k$th worst, I suppose it's trivial. You can always make $BUX=e_1$ which will always have $k$th worst (in absolute value) of zero. For the tail expectation, I mean the average of the $k$ largest components. http://mathoverflow.net/questions/105390/results-for-minimizing-the-norm-w-r-t-a-unitary-matrix Comment by mathtick mathtick 2012-08-24T15:16:13Z 2012-08-24T15:16:13Z Yes, x is fixed. Sorry I should have mentioned that. I've written the problem in a bit of a bizarre way. http://mathoverflow.net/questions/102777/derivative-of-the-cdf-of-a-family-of-random-variables/102788#102788 Comment by mathtick mathtick 2012-08-07T19:43:48Z 2012-08-07T19:43:48Z ... and in case the internet changes by the time you read the previous comment the reference is: Sensitivity analysis of Values at Risk, C. Gourieroux, J.P. Laurent, O. Scaille http://mathoverflow.net/questions/102777/derivative-of-the-cdf-of-a-family-of-random-variables/102788#102788 Comment by mathtick mathtick 2012-08-07T18:56:08Z 2012-08-07T18:56:08Z The appendix here is useful: <a href="http://efinance.org.cn/cn/FEshuo/FE-sensitivity%20analysis%20of%20VAR.pdf" rel="nofollow">efinance.org.cn/cn/FEshuo/&hellip;</a> http://mathoverflow.net/questions/102777/derivative-of-the-cdf-of-a-family-of-random-variables Comment by mathtick mathtick 2012-07-22T01:24:19Z 2012-07-22T01:24:19Z I guess I should say I'm looking for something where you don't use the full joint distribution function. http://mathoverflow.net/questions/37277/stochastic-matrix-for-which-a-stochastic-matrix-logarithm-exists Comment by mathtick mathtick 2010-08-31T16:31:48Z 2010-08-31T16:31:48Z Furthermore, as a modeling bonus question. Is there, in general, a &quot;right&quot; thing to do when you are given one Markov chain and need to pick a transition matrix for a new Markov chain with shorter time intervals?