fluctuation theory of connectivities for subcritical random cluster models we develop a fluctuation theory of connectivities for subcritical random cluster models the theory is based on a comprehensive nonperturbative probabilistic description of long connected clusters in terms of essentially one dimensional chains of irreducible objects statistics of local observables for example displacement over such chains obey classical limit laws and our construction leads to an effective random walk representation of percolation clusters the results include a derivation of a sharp ornstein zernike type asymptotic formula for two point functions a proof of analyticity and strict convexity of inverse correlation length and a proof of an invariance principle for connected clusters under diffusive scaling in two dimensions duality considerations enable a reformulation of these results for supercritical nearest neighbor random cluster measures in particular for nearest neighbor potts models in the phase transition regime accordingly we prove that in two dimensions potts equilibrium crystal shapes are always analytic and strictly convex and that the interfaces between different phases are always diffusive thus no roughening transition is possible in the whole regime where our results apply our results hold under an assumption of exponential decay of finite volume wired connectivities assumption 1 2 below in rectangular domains that is conjectured to hold in the whole subcritical regime the latter is known to be true in any dimensions when q 1 q 2 and when q is sufficiently large in two dimensions assumption 1 2 holds whenever there is an exponential decay of connectivities in the infinite volume measure by duality this includes all supercritical nearest neighbor potts models with positive surface tension between ordered phases

multitask quantile regression under the transnormal model abstractwe consider estimating multitask quantile regression under the transnormal model with focus on high dimensional setting we derive a surprisingly simple closed form solution through rank based covariance regularization in particular we propose the rank based l1 penalization with positive definite constraints for estimating sparse covariance matrices and the rank based banded cholesky decomposition regularization for estimating banded precision matrices by taking advantage of the alternating direction method of multipliers nearest correlation matrix projection is introduced that inherits sampling properties of the unprojected one our work combines strengths of quantile regression and rank based covariance regularization to simultaneously deal with nonlinearity and nonnormality for high dimensional regression furthermore the proposed method strikes a good balance between robustness and efficiency achieves the oracle like convergence rate and provides the provable prediction interval und

on the gibbs states of the noncritical potts model on z 2 we prove that all gibbs states of the q state nearest neighbor potts model on mathbb z 2 below the critical temperature are convex combinations of the q pure phases in particular they are all translation invariant to achieve this goal we consider such models in large finite boxes with arbitrary boundary condition and prove that the center of the box lies deeply inside a pure phase with high probability our estimate of the finite volume error term is of essentially optimal order which stems from the brownian scaling of fluctuating interfaces the results hold at any supercritical value of the inverse temperature beta beta c q log left 1 sqrt q right

determining the number of factors in the general dynamic factor model this article develops an information criterion for determining the number q of common shocks in the general dynamic factor model developed by forni et al as opposed to the restricted dynamic model considered by bai and ng and by amengual and watson our criterion is based on the fact that this number q is also the number of diverging eigenvalues of the spectral density matrix of the observations as the number n of series goes to infinity we provide sufficient conditions for consistency of the criterion for large n and t where t is the series length we show how the method can be implemented and provide simulations and empirics illustrating its very good finite sample performance application to real data adds a new empirical facet to an ongoing debate on the number of factors driving the u s economy

forecasting using principal components from a large number of predictors this article considers forecasting a single time series when there are many predictors n and time series observations t when the data follow an approximate factor model the predictors can be summarized by a small number of indexes which we estimate using principal components feasible forecasts are shown to be asymptotically efficient in the sense that the difference between the feasible forecasts and the infeasible forecasts constructed using the actual values of the factors converges in probability to 0 as both n and t grow large the estimated factors are shown to be consistent even in the presence of time variation in the factor model

conformal invariance in random cluster models i holmorphic fermions in the ising model we construct discrete holomorphic observables in the ising model at criticality and show that they have conformally covariant scaling limits as mesh of the lattice tends to zero in the sequel those observables are used to construct conformally invariant scaling limits of interfaces though ising model is often cited as a classical example of conformal invariance it seems that ours is the first paper where it is actually established

high dimensional covariance matrix estimation using a factor model abstract high dimensionality comparable to sample size is common in many statistical problems we examine covariance matrix estimation in the asymptotic framework that the dimensionality p tends to as the sample size n increases motivated by the arbitrage pricing theory in finance a multi factor model is employed to reduce dimensionality and to estimate the covariance matrix the factors are observable and the number of factors k is allowed to grow with p we investigate the impact of p and k on the performance of the model based covariance matrix estimator under mild assumptions we have established convergence rates and asymptotic normality of the model based estimator its performance is compared with that of the sample covariance matrix we identify situations under which the factor approach increases performance substantially or marginally the impacts of covariance matrix estimation on optimal portfolio allocation and portfolio risk assessment are studied the asymptotic results are supported by a thorough simulation study

quenched large deviations for simple random walks on percolation clusters including long range correlations we prove a quenched large deviation principle ldp for a simple random walk on a supercritical percolation cluster srwpc on mathbb z d d geq 2 the models under interest include classical bernoulli bond and site percolation as well as models that exhibit long range correlations like the random cluster model the random interlacement and the vacant set of random interlacements for d geq 3 and the level sets of the gaussian free field d geq 3 inspired by the methods developed by kosygina et al commun pure appl math 59 1489 1521 2006 for proving quenched ldp for elliptic diffusions with a random drift and by yilmaz commun pure appl math 62 8 1033 1075 2009 and rosenbluth quenched large deviations for multidimensional random walks in a random environment a variational formula ph d thesis nyu arxiv 0804 1444v1 for similar results regarding elliptic random walks in random environment we take the point of view of the moving particle and prove a large deviation principle for the quenched distribution of the pair empirical measures of the environment markov chain in the non elliptic case of srwpc via a contraction principle this reduces easily to a quenched ldp for the distribution of the mean velocity of the random walk and both rate functions admit explicit variational formulas the main difficulty in our set up lies in the inherent non ellipticity as well as the lack of translation invariance stemming from conditioning on the fact that the origin belongs to the infinite cluster we develop a unifying approach for proving quenched large deviations for srwpc based on exploiting coercivity properties of the relative entropies in the context of convex variational analysis combined with input from ergodic theory and invoking geometric properties of the supercritical percolation cluster

on the critical percolation probabilities we prove that the critical probabilities of site percolation on the square lattice satisfy the relation p c p 1 furthermore we prove the continuity of the function percolation probability

projected principal component analysis in factor models this paper introduces a projected principal component analysis projected pca which employees principal component analysis to the projected smoothed data matrix onto a given linear space spanned by covariates when it applies to high dimensional factor analysis the projection removes noise components we show that the unobserved latent factors can be more accurately estimated than the conventional pca if the projection is genuine or more precisely when the factor loading matrices are related to the projected linear space when the dimensionality is large the factors can be estimated accurately even when the sample size is finite we propose a flexible semi parametric factor model which decomposes the factor loading matrix into the component that can be explained by subject specific covariates and the orthogonal residual component the covariates effects on the factor loadings are further modeled by the additive model via sieve approximations by using the newly proposed projected pca the rates of convergence of the smooth factor loading matrices are obtained which are much faster than those of the conventional factor analysis the convergence is achieved even when the sample size is finite and is particularly appealing in the high dimension low sample size situation this leads us to developing nonparametric tests on whether observed covariates have explaining powers on the loadings and whether they fully explain the loadings the proposed method is illustrated by both simulated data and the returns of the components of the s p 500 index

counting coloured planar maps differential equations we address the enumeration of q coloured planar maps counted by the number of edges and the number of monochromatic edges we prove that the associated generating function is differentially algebraic that is satisfies a non trivial polynomial differential equation with respect to the edge variable we give explicitly a differential system that characterizes this series we then prove a similar result for planar triangulations thus generalizing a result of tutte dealing with their proper q colourings in statistical physics terms we solve the q state potts model on random planar lattices this work follows a first paper by the same authors where the generating function was proved to be algebraic for certain values of q including q 1 2 and 3 it is known to be transcendental in general in contrast our differential system holds for an indeterminate q for certain special cases of combinatorial interest four colours proper q colourings maps equipped with a spanning forest we derive from this system in the case of triangulations an explicit differential equation of order 2 defining the generating function for general planar maps we also obtain a differential equation of order 3 for the four colour case and for the self dual potts model

sliced inverse regression for dimension reduction abstract modern advances in computing power have greatly widened scientists scope in gathering and investigating information from many variables information which might have been ignored in the past yet to effectively scan a large pool of variables is not an easy task although our ability to interact with data has been much enhanced by recent innovations in dynamic graphics in this article we propose a novel data analytic tool sliced inverse regression sir for reducing the dimension of the input variable x without going through any parametric or nonparametric model fitting process this method explores the simplicity of the inverse view of regression that is instead of regressing the univariate output variable y against the multivariate x we regress x against y forward regression and inverse regression are connected by a theorem that motivates this method the theoretical properties of sir are investigated under a model of the form y f 1 x k x e where the k s are the unknown

maximum likelihood estimation and inference for high dimensional generalized factor models with application to factor augmented regressions abstract this paper reestablishes the main results in bai 2003 and bai and ng 2006 for generalized factor models with slightly stronger conditions on the relative magnitude of n number of subjects and t number of time periods convergence rates of the estimated factor space and loading space and asymptotic normality of the estimated factors and loadings are established under mild conditions that allow for linear logit probit tobit poisson and some other single index nonlinear models the probability density mass function is allowed to vary across subjects and time thus mixed models are also allowed for for factor augmented regressions this paper establishes the limit distributions of the parameter estimates the conditional mean and the forecast when factors estimated from nonlinear mixed data are used as proxies for the true factors

eigenvalue ratio test for the number of factors this paper proposes two new estimators for determining the number of factors r in static approximate factor models we exploit the well known fact that the r largest eigenvalues of the variance matrix of n response variables grow unboundedly as n increases while the other eigenvalues remain bounded the new estimators are obtained simply by maximizing the ratio of two adjacent eigenvalues our simulation results provide promising evidence for the two estimators

determining the dimensionality in sliced inverse regression abstract a general regression problem is one in which a response variable can be expressed as some function of one or more different linear combinations of a set of explanatory variables as well as a random error term sliced inverse regression is a method for determining these linear combinations in this article we address the problem of determining how many linear combinations are involved procedures based on conditional means and conditional covariance matrices as well as a procedure combining the two approaches are considered in each case we develop a test that has an asymptotic chi squared distribution when the vector of explanatory variables is sampled from an elliptically symmetric distribution

mixing times of critical two dimensional potts models we study dynamical aspects of the q state potts model on an n n box at its critical c q heat bath glauber dynamics and cluster dynamics such as swendsen wang that circumvent low temperature bottlenecks are all expected to undergo critical slowdowns in the presence of periodic boundary conditions the inverse spectral gap which in the subcritical regime is o 1 should at criticality be polynomial in n for 1 4 in accordance with the predicted discontinuous phase transition this was confirmed for q 2 the ising model by the second author and sly and for sufficiently large q by borgs et al here we show that the following holds for the critical potts model on the torus for q 3 the inverse gap of glauber dynamics is no 1 for q 4 it is at most no log n and for every q 4 in the phase coexistence regime the inverse gaps of both glauber dynamics and swendsen wang dynamics are exponential in n for free or monochromatic boundary conditions and large q we show that the dynamics at criticality is faster than on the torus unlike the ising model where free periodic boundary conditions induce similar dynamical behavior at all temperatures the inverse gap of swendsen wang dynamics is exp no 1 2017 wiley periodicals inc

determining the number of factors in approximate factor models in this paper we develop some econometric theory for factor models of large dimensions the focus is the determination of the number of factors r which is an unresolved issue in the rapidly growing literature on multifactor models we first establish the convergence rate for the factor estimates that will allow for consistent estimation of r we then propose some panel criteria and show that the number of factors can be consistently estimated using the criteria the theory is developed under the framework of large cross sections n and large time dimensions t no restriction is imposed on the relation between n and t simulations show that the proposed criteria have good finite sample properties in many configurations of the panel data encountered in practice

on sliced inverse regression with high dimensional covariates sliced inverse regression is a promising method for the estimation of the central dimension reduction subspace cdr space in semiparametric regression models it is particularly useful in tackling cases with high dimensional covariates in this article we study the asymptotic behavior of the estimate of the cdr space with high dimensional covariates that is when the dimension of the covariates goes to infinity as the sample size goes to infinity strong and weak convergence are obtained we also suggest an estimation procedure of the bayes information criterion type to ascertain the dimension of the cdr space and derive the consistency a simulation study is conducted

the critical probability for random voronoi percolation in the plane is 1 2 we study percolation in the following random environment let z be a poisson process of constant intensity on 2 and form the voronoi tessellation of 2 with respect to z colour each voronoi cell black with probability p independently of the other cells we show that the critical probability is 1 2 more precisely if p 1 2 then the union of the black cells contains an infinite component with probability 1 while if p 1 2 then the distribution of the size of the component of black cells containing a given point decays exponentially these results are analogous to kesten s results for bond percolation in 2 the result corresponding to harris theorem for bond percolation in 2 is known zvavitch noted that one of the many proofs of this result can easily be adapted to the random voronoi setting for kesten s results none of the existing proofs seems to adapt the methods used here also give a new and very simple proof of kesten s theorem for 2 we hope they will be applicable in other contexts as well

strict inequality for critical values of potts models and random cluster processes we prove that the critical value c of a ferromagnetic potts model is astrictly decreasing function of the strengths of interaction of the process this is achieved in the more general context of the random cluster representation of fortuin and kasteleyn by deriving and utilizing a formula which generalizes the technique known in percolation theory as russo s formula as a byproduct of the method we present a general argument for showing that at any given point on the critical surface of a multiparameter process the values of a certain critical exponent do not depend on the direction of approach of that point our results apply to all random cluster processes satisfying the fkg inequality

sufficient forecasting using factor models we consider forecasting a single time series when there is a large number of predictors and a possible nonlinear effect the dimensionality was first reduced via a high dimensional approximate factor model implemented by the principal component analysis using the extracted factors we develop a novel forecasting method called the sufficient forecasting which provides a set of sufficient predictive indices inferred from high dimensional predictors to deliver additional predictive power the projected principal component analysis will be employed to enhance the accuracy of inferred factors when a semi parametric approximate factor model is assumed our method is also applicable to cross sectional sufficient regression using extracted factors the connection between the sufficient forecasting and the deep learning architecture is explicitly stated the sufficient forecasting correctly estimates projection indices of the underlying factors even in the presence of a nonparametric forecasting function the proposed method extends the sufficient dimension reduction to high dimensional regimes by condensing the cross sectional information through factor models we derive asymptotic properties for the estimate of the central subspace spanned by these projection directions as well as the estimates of the sufficient predictive indices we further show that the natural method of running multiple regression of target on estimated factors yields a linear estimate that actually falls into this central subspace our method and theory allow the number of predictors to be larger than the number of observations we finally demonstrate that the sufficient forecasting improves upon the linear forecasting in both simulation studies and an empirical study of forecasting macroeconomic variables

high dimensional semiparametric gaussian copula graphical models ing the spearman s rho and kendall s tau we prove that the nonparanormal skeptic achieves the optimal parametric rates of convergence for both graph recovery and parameter estimation this result suggests that the nonparanormal graphical models can be used as a safe replacement of the popular gaussian graphical models even when the data are truly gaussian besides theoretical analysis we also conduct thorough numerical simulations to compare the graph recovery performance of dierent estimators under both ideal and noisy settings the proposed methods are then applied on a largescale genomic dataset to illustrate their empirical usefulness the r package huge implementing the proposed methods is available on the comprehensive r archive network http cran r project org

noisy matrix completion understanding statistical guarantees for convex relaxation via nonconvex optimization this paper studies noisy low rank matrix completion given partial and noisy entries of a large low rank matrix the goal is to estimate the underlying matrix faithfully and efficiently arguably o

on almost linearity of low dimensional projections from high dimensional data this paper studies the shapes of low dimensional projections from high dimensional data after standardization let mathbf x be a p dimensional random variable with mean zero and identity covariance for a projection beta mathbf x beta 1 find another direction b so that the regression curve of b mathbf x against beta mathbf x is as nonlinear as possible we show that when the dimension of mathbf x is large for most directions beta even the most nonlinear regression is still nearly linear our method depends on the construction of a pair of p dimensional random variables mathbf w 1 mathbf w 2 called the rotational twin and its density function with respect to the standard normal density with this we are able to obtain closed form expressions for measuring deviation from normality and deviation from linearity in a suitable sense of average as an interesting by product from a given set of data we can find simple unbiased estimates of e f beta mathbf x t phi 1 t 1 2 and e lbrack e mathbf x mid beta beta mathbf x t 2 t 2 f 2 beta mathbf x t phi 2 1 t rbrack where phi 1 is the standard normal density f beta mathbf x is the density for beta mathbf x and the e is taken with respect to the uniformly distributed beta this is achieved without any smoothing and without resorting to any laborious projection procedures such as grand tours our result is related to the work of diaconis and freedman the impact of our result on several fronts of data analysis is discussed for example it helps establish the validity of regression analysis when the link function of the regression model may be grossly wrong a further generalization which replaces beta mathbf x by b mathbf x with b beta 1 ldots beta k for k randomly selected orthonormal vectors beta i i 1 ldots k helps broaden the scope of application of sliced inverse regression sir

the three pass regression filter a new approach to forecasting using many predictors we forecast a single time series using many predictor variables with a new estimator called the three pass regression lter 3prf it is calculated in closed form and conveniently represented as a set of ordinary least squares regressions 3prf forecasts converge to the infeasible best forecast when both the time dimension and cross section dimension become large this requires only specifying the number of relevant factors driving the forecast target regardless of the total number of common including potentially irrelevant factors driving the cross section of predictors we derive inferential theory including limiting distributions for estimated relevant factors predictive coefcients and forecasts and provide consistent standard error estimators we explore two empirical applications forecasting macroeconomic aggregates with a large panel of economic indices and forecasting stock market aggregates with many individual assets price dividend ratios these combined with a range of monte carlo experiments demonstrate the 3prf s forecasting power

sufficient direction factor model and its application to gene expression quantitative trait loci discovery rapid improvement in technology has made it relatively cheap to collect genetic data however statistical analysis of existing data is still much cheaper thus secondary analysis of single nucleotide polymorphism snp data i e reanalysing existing data in an effort to extract more information is an attractive and cost effective alternative to collecting new data we study the relationship between gene expression and snps through a combination of factor analysis and dimension reduction estimation to take advantage of the flexibility in traditional factor models where the latent factors are not required to be normal we recommend using semiparametric sufficient dimension reduction methods in the joint estimation of the combined model the resulting estimator is flexible and has superior performance relative to the existing estimator which relies on additional assumptions on the latent factors we quantify the asymptotic performance of the proposed parameter estimator and perform inference by assessing the estimation variability and by constructing confidence intervals the new results enable us to identify for the first time statistically significant snps concerning gene snp relations in lung tissue from genotype tissue expression data

on the coupling time of the heat bath process for the fortuin kasteleyn random cluster model we consider the coupling from the past implementation of the random cluster heat bath process and study its random running time or coupling time we focus on hypercubic lattices embedded on tori in dimensions one to three with cluster fugacity at least one we make a number of conjectures regarding the asymptotic behaviour of the coupling time motivated by rigorous results in one dimension and monte carlo simulations in dimensions two and three amongst our findings we observe that for generic parameter values the distribution of the appropriately standardized coupling time converges to a gumbel distribution and that the standard deviation of the coupling time is asymptotic to an explicit universal constant multiple of the relaxation time perhaps surprisingly we observe these results to hold both off criticality where the coupling time closely mimics the coupon collector s problem and also at the critical point provided the cluster fugacity is below the value at which the transition becomes discontinuous finally we consider analogous questions for the single spin ising heat bath process

influence and sharp threshold theorems for monotonic measures the influence theorem for product measures on the discrete space 0 1 n may be extended to probability measures with the property of monotonicity which is equivalent to strong positive association corresponding results are valid for probability measures on the cube 0 1 n that are absolutely continuous with respect to lebesgue measure these results lead to a sharp threshold theorem for measures of random cluster type and this may be applied to box crossings in the two dimensional random cluster model

forecasting economic time series using targeted predictors abstract this paper studies two refinements to the method of factor forecasting first we consider the method of quadratic principal components that allows the link function between the predictors and the factors to be non linear second the factors used in the forecasting equation are estimated in a way to take into account that the goal is to forecast a specific series this is accomplished by applying the method of principal components to targeted predictors selected using hard and soft thresholding rules our three main findings can be summarized as follows first we find improvements at all forecast horizons over the current diffusion index forecasts by estimating the factors using fewer but informative predictors allowing for non linearity often leads to additional gains second forecasting the volatile one month ahead inflation warrants a high degree of targeting to screen out the noisy predictors a handful of variables notably relating to housing starts and interest rates are found to have systematic predictive power for inflation at all horizons third the targeted predictors selected by both soft and hard thresholding changes with the forecast horizon and the sample period holding the set of predictors fixed as is the current practice of factor forecasting is unnecessarily restrictive

phase separation in random cluster models ii the droplet at equilibrium and local deviation lower bounds we study the droplet that results from conditioning the planar subcritical fortuin kasteleyn random cluster model on the presence of an open circuit 0 encircling the origin and enclosing an area of at least or exactly n2 we consider local deviation of the droplet boundary measured in a radial sense by the maximum local roughness mlr 0 this being the maximum distance from a point in the circuit 0 to the boundary conv 0 of the circuit s convex hull and in a longitudinal sense by what we term maximum facet length namely the length of the longest line segment of which the polygon conv 0 is formed we prove that there exists a constant c 0 such that the conditional probability that the normalised quantity n 1 3 log n 2 3 mlr 0 exceeds c tends to 1 in the high n limit and that the same statement holds for n 2 3 log n 1 3 mfl 0 to obtain these bounds we exhibit the random cluster measure conditional on the presence of an open circuit trapping high area as the invariant measure of a markov chain that resamples sections of the circuit boundary we analyse the chain at equilibrium to prove the local roughness lower bounds alongside complementary upper bounds provided in 14 the fluctuations mlr 0 and mfl 0 are determined up to a constant factor

density and uniqueness in percolation two results on site percolation on thed dimensional lattice d 1 arbitrary are presented in the first theorem we show that for stationary underlying probability measures each infinite cluster has a well defined density with probability one the second theorem states that if in addition the probability measure satisfies the finite energy condition of newman and schulman then there can be at most one infinite cluster with probability one the simple arguments extend to a broad class of finite dimensional models including bond percolation and regular lattices

phase separation in random cluster models iii circuit regularity we study the droplet that results from conditioning the subcritical fortuin kasteleyn random cluster model on the presence of an open circuit encircling the origin and enclosing an area of at least or exactly n 2 in this paper we prove that the resulting circuit is highly regular we define a notion of a regeneration site in such a way that for any such site v the circuit cuts through the radial line segment through v only at v we show that provided that the conditioned circuit is centred at the origin in a natural sense the set of regeneration sites reaches into all parts of the circuit with maximal distance from one such site to the next being at most logarithmic in n with high probability the result provides a flexible control on the conditioned circuit that permits the use of surgical techniques to bound its fluctuations and as such it plays a crucial role in the derivation of bounds on the local fluctuation of the circuit carried out in hammond arxiv 1001 1527 2009 arxiv 1001 1528 2009

decay of correlations in random cluster models we prove exponential decay for the tail of the radius r of the cluster at the origin for subcritical random cluster models under an assumption slightly weaker than that here d is the number of dimensions specifically if throughout the subcritical phase then for some 0 this implies the exponential decay of the two point correlation function of subcritical potts models subject to a hypothesis of at least polynomial decay of this function similar results are known already for percolation and ising models and for potts models when the number q of available states is sufficiently large indeed the hypothesis of polynomial decay has been proved rigorously for these cases in two dimensions the hypothesis that is weaker than requiring that the susceptibility be finite i e that the two point function be summable the principal new technique is a form of russo s formula for random cluster models reported by bezuidenhout grimmett and kesten for the current application this leads to an analysis of a first passage problem for random cluster models and a proof that the associated time constant is strictly positive if and only if the tail of r decays exponentially

factor modeling for high dimensional time series inference for the number of factors this paper deals with the factor modeling for high dimensional time series based on a dimension reduction viewpoint under stationary settings the inference is simple in the sense that both the number of factors and the factor loadings are estimated in terms of an eigenanalysis for a nonnegative definite matrix and is therefore applicable when the dimension of time series is on the order of a few thousands asymptotic properties of the proposed method are investigated under two settings i the sample size goes to infinity while the dimension of time series is fixed and ii both the sample size and the dimension of time series go to infinity together in particular our estimators for zero eigenvalues enjoy faster convergence or slower divergence rates hence making the estimation for the number of factors easier in particular when the sample size and the dimension of time series go to infinity together the estimators for the eigenvalues are no longer consistent however our estimator for the number of the factors which is based on the ratios of the estimated eigenvalues still works fine furthermore this estimation shows the so called blessing of dimensionality property in the sense that the performance of the estimation may improve when the dimension of time series increases a two step procedure is investigated when the factors are of different degrees of strength numerical illustration with both simulated and real data is also reported

phase separation in random cluster models i uniform upper bounds on local deviation this is the first in a series of three papers that addresses the behaviour of the droplet that results in the percolating phase from conditioning the planar fortuin kasteleyn random cluster model on the presence of an open dual circuit 0 encircling the origin and enclosing an area of at least or exactly n2 by the fortuin kasteleyn representation the model is a close relative of the droplet formed by conditioning the potts model on an excess of spins of a given type we consider local deviation of the droplet boundary measured in a radial sense by the maximum local roughness mlr 0 this being the maximum distance from a point in the circuit 0 to the boundary of the circuit s convex hull and in a longitudinal sense by what we term maximum facet length mlf 0 namely the length of the longest line segment of which the polygon is formed the principal conclusion of the series of papers is the following uniform control on local deviation that there are constants 0 c c such that the conditional probability that the normalized quantity n 1 3 log n 2 3mlr lies in the interval c c tends to 1 in the high n limit and that the same statement holds for n 2 3 log n 1 3 mlf in this way we confirm the anticipated n1 3 scaling of maximum local roughness and provide a sharp logarithmic power law correction this local deviation behaviour occurs by means of locally gaussian effects constrained globally by curvature and we believe that it arises in many radially defined stochastic interface models including growth models belonging to the kardar parisi zhang universality class

principal components estimation and identification of static factors a b s t r a c t it is known that the principal component estimates of the factors and the loadings are rotations of the un derlying latent factors and loadings we study conditions under which the latent factors can be estimated asymptotically without rotation we derive the limiting distributions for the estimated factors and factor loadings when n and t are large and make precise how identification of the factors affects inference based on factor augmented regressions we also consider factor models with additive individual and time effects the asymptotic analysis can be modified to analyze identification schemes not considered in this analysis

variational bayesian inversion of synthetic 3d controlled source electromagnetic geophysical data abstractinversion of controlled source electromagnetic data is dealt with for a geophysical application the goal is to retrieve a map of conductivity of an unknown body embedded in a layered underground from measurements of the scattered electric field that results from its interaction with a known interrogating wave this constitutes an inverse scattering problem whose associated forward problem is described by means of electric field domain integral equations the inverse problem is solved in a bayesian framework in which prior information is introduced via a gauss markov potts model this model describes the body as being composed of a finite number of different materials distributed into compact homogeneous regions the posterior distribution of the unknowns is approached by means of the variational bayesian approximation as a separable distribution that minimizes the kullback leibler divergence with respect to the posterior law thus we get a parametric model for the distributions of the induced cu

nonparametric regression with nonparametrically generated covariates we analyze the statistical properties of nonparametric regression estimators using covariates which are not directly observable but have be estimated from data in a preliminary step these so called generated covariates appear in numerous applications including two stage nonparametric regression estimation of simultaneous equation models or censored regression models yet so far there seems to be no general theory for their impact on the final estimator s statistical properties our paper provides such results we derive a stochastic expansion that characterizes the influence of the generation step on the final estimator and use it to derive rates of consistency and asymptotic distributions accounting for the presence of generated covariates

sharpness of the phase transition for continuum percolation in mathbb r 2 r 2 we study the phase transition of random radii poisson boolean percolation around each point of a planar poisson point process we draw a disc of random radius independently for each point the behavior of this process is well understood when the radii are uniformly bounded from above in this article we investigate this process for unbounded and possibly heavy tailed radii distributions under mild assumptions on the radius distribution we show that both the vacant and occupied sets undergo a phase transition at the same critical parameter lambda c moreover the techniques we develop in this article can be applied to other models such as the poisson voronoi and confetti percolation

interfaces in the potts model i pirogov sinai theory of the fortuin kasteleyn representation we develop a new analysis of the order disorder transition in ferromagnetic potts models for large numberq of spin states we use the pirogov sinai theory which we adapt to the fortuin kasteleyn representation of the models this theory applies in a rather direct way in our approach and leads to a system of non interacting contours with small activities as a consequence simpler and more natural techniques are found allowing us to recover previous results on the bulk properties of the model which then extend to non integer values ofq and to deal with non translation invariant boundary conditions this will be applied in a second part of this work to study the behaviour of the interfaces at the transition point

the self dual point of the two dimensional random cluster model is critical for q 1 we prove a long standing conjecture on random cluster models namely that the critical point for such models with parameter q 1 on the square lattice is equal to the self dual point p sd q sqrt q 1 sqrt q this gives a proof that the critical temperature of the q state potts model is equal to log 1 sqrt q for all q 2 we further prove that the transition is sharp meaning that there is exponential decay of correlations in the sub critical phase the techniques of this paper are rigorous and valid for all q 1 in contrast to earlier methods valid only for certain given q the proof extends to the triangular and the hexagonal lattices as well

exponential decay of connection probabilities for subcritical voronoi percolation in mathbb r d r d we prove that for voronoi percolation on mathbb r d d ge 2 there exists p c p c d in 0 1 such that for dimension 2 this result offers a new way of showing that p c 2 1 2 this paper belongs to a series of papers using the theory of algorithms to prove sharpness of the phase transition see 10 11

prediction by supervised principal components in regression problems where the number of predictors greatly exceeds the number of observations conventional regression techniques may produce unsatisfactory results we describe a technique called supervised principal components that can be applied to this type of problem supervised principal components is similar to conventional principal components analysis except that it uses a subset of the predictors selected based on their association with the outcome supervised principal components can be applied to regression and generalized regression problems such as survival analysis it compares favorably to other techniques for this type of problem and can also account for the effects of other covariates and help identify which predictor variables are most important we also provide asymptotic consistency results to help support our empirical findings these methods could become important tools for dna microarray data where they may be used to more accurately diagnose and treat cancer

minimum principle for indefinite mean field free energies the extension of bogoliubov jr s minimax principle to biaxial nematic liquid crystals has recently made it possible to derive the universal mean field phase diagram for all quadrupolar interactions between nematogenic biaxial molecules including those rendering the mean field free energy fancyscript f 0 indefinite to justify this extension so far largely based on heuristic arguments we prove here a minimum principle that also applies when fancyscript f 0 is indefinite bogoliubov jr s minimax principle then emerges as a characterization of this minimum principle the theory presented here is more general than its application to biaxial nematic liquid crystals could reveal it essentially builds upon a concavity property of fancyscript f 0 in the order tensor associated with the repulsive component of the molecular interaction

inferential theory for factor models of large dimensions this paper develops an inferential theory for factor models of large dimensions the principal components estimator is considered because it is easy to compute and is asymptotically equivalent to the maximum likelihood estimator if normality is assumed we derive the rate of convergence and the limiting distributions of the estimated factors factor loadings and common components the theory is developed within the framework of large cross sections n and a large time dimension t to which classical factor analysis does not apply we show that the estimated common components are asymptotically normal with a convergence rate equal to the minimum of the square roots of n and t the estimated factors and their loadings are generally normal although not always so the convergence rate of the estimated factors and factor loadings can be faster than that of the estimated common components these results are obtained under general conditions that allow for correlations and heteroskedasticities in both dimensions stronger results are obtained when the idiosyncratic errors are serially uncorrelated and homoskedastic a necessary and sufficient condition for consistency is derived for large n but fixed t

dynamic factor models with infinite dimensional factor spaces one sided representations abstract factor model methods recently have become extremely popular in the theory and practice of large panels of time series data those methods rely on various factor models which all are particular cases of the generalized dynamic factor model gdfm introduced in forniet al 2000 that paper however rests on brillinger s dynamic principal components the corresponding estimators are two sided filters whose performance at the end of the observation period or for forecasting purposes is rather poor no such problem arises with estimators based on standard principal components which have been dominant in this literature on the other hand those estimators require the assumption that the space spanned by the factors has finite dimension in the present paper we argue that such an assumption is extremely restrictive and potentially quite harmful elaborating upon recent results by anderson and deistler 2008a b on singular stationary processes with rational spectrum we obtain one sided representations for the gdfm without assuming finite dimension of the factor space construction of the corresponding estimators is also briefly outlined in a companion paper we establish consistency and rates for such estimators and provide monte carlo results further motivating our approach

nearest comoment estimation with unobserved factors we propose a minimum distance estimator for the higher order comoments of a multivariate distribution exhibiting a lower dimensional latent factor structure we derive the influence function of the proposed estimator and prove its consistency and asymptotic normality the simulation study confirms the large gains in accuracy compared to the traditional sample comoments the empirical usefulness of the novel framework is shown in applications to portfolio allocation under non gaussian objective functions and to the extraction of factor loadings in a dataset with mental ability scores

phases coexistence and surface tensions for the potts model theq states potts model exhibits a first order phase transition at some inverse temperature t between ordered and disordered phases forq large as proved in 1 in space dimension 2 we use theduality transformation as aninternal symmetry of the partition function at t to derive an estimate on the probability of a contour this enables us to prove the preceding result and the following new results i the discontinuity of the mass gap at t ii the existence of astrictly positive surface tension between two ordered phases up to t iii the existence of a non zero surface tension between an ordered and the disordered phase at t

improved mixing condition on the grid for counting and sampling independent sets the hard core model has received much attention in the past couple of decades as a lattice gas model with hard constraints in statistical physics a multicast model of calls in communication networks and as a weighted independent set problem in combinatorics probability and theoretical computer science in this model each independent set i in a graph g is weighted proportionally to i for a positive real parameter for large computing the partition function namely the normalizing constant which makes the weighting a probability distribution on a finite graph on graphs of maximum degree 3 is a well known computationally challenging problem more concretely let lambda c mathbb t delta denote the critical value for the so called uniqueness threshold of the hard core model on the infinite regular tree recent breakthrough results of weitz proceedings of the 38th annual acm symposium on theory of computing stoc pp 140 149 2006 and sly proceedings of the 51st annual ieee symposium on foundations of computer science focs pp 287 296 2010 have identified lambda c mathbb t delta as a threshold where the hardness of estimating the above partition function undergoes a computational transition we focus on the well studied particular case of the square lattice mathbb z 2 and provide a new lower bound for the uniqueness threshold in particular taking it well above lambda c mathbb t 4 our technique refines and builds on the tree of self avoiding walks approach of weitz resulting in a new technical sufficient criterion of wider applicability for establishing strong spatial mixing and hence uniqueness for the hard core model our new criterion achieves better bounds on strong spatial mixing when the graph has extra structure improving upon what can be achieved by just using the maximum degree applying our technique to mathbb z 2 we prove that strong spatial mixing holds for all 2 3882 improving upon the work of weitz that held for 27 16 1 6875 our results imply a fully polynomial deterministic approximation algorithm for estimating the partition function as well as rapid mixing of the associated glauber dynamics to sample from the hard core distribution