Parameters estimation for asymmetric bifurcating autoregressive processes with missing data

A phase transition for large values of bifurcating autoregressive models

We describe the asymptotic behavior of the number $Z_n[a_n,\infty)$ of individuals with a large value in a stable bifurcating autoregressive process. The study of the associated first moment $\mathbb{E}(Z_n[a_n,\infty))$ is equivalent to the annealed large deviation problem $\mathbb{P}(Y_n\geq a_n)$, where $Y$ is an autoregressive process in a random environment and $a_n\rightarrow \infty$. The population with large values and the trajectorial behavior of $Z_n[a_n,\infty)$ is obtained from the ancestral paths associated to the large deviations of $Y$ together with its environment. The study of…

缺失数据下的顺序无关自回归建模

Order-Agnostic Autoregressive Modelling with Missing Data

本文通过缺失数据视角重新审视顺序无关自回归模型，提出缺失感知训练框架，并利用其条件密度估计进行主动信息获取，在多个基准上优于传统插补方法。

Deviation inequalities for bifurcating Markov chains on Galton-Watson tree

We provide deviation inequalities for properly normalized sums of bifurcating Markov chains on Galton-Watson tree. These processes are extension of bifurcating Markov chains (which was introduced by Guyon to detect cellular aging from cell lineage) in case the index set is a binary Galton-Watson process. As application, we derive deviation inequalities for the least-squares estimator of autoregressive parameters of bifurcating autoregressive processes with missing data. These processes allow, in case of cell division, to take into account the cell's death. The results are obtained under an uni…

Adaptive estimation for bifurcating Markov chains

In a first part, we prove Bernstein-type deviation inequalities for bifurcating Markov chains (BMC) under a geometric ergodicity assumption, completing former results of Guyon and Bitseki Penda, Djellout and Guillin. These preliminary results are the key ingredient to implement nonparametric wavelet thresholding estimation procedures: in a second part, we construct nonparametric estimators of the transition density of a BMC, of its mean transition density and of the corresponding invariant density, and show smoothness adaptation over various multivariate Besov classes under $L^p$-loss error, f…

Autoregressive Wild Bootstrap Inference for Nonparametric Trends

In this paper we propose an autoregressive wild bootstrap method to construct confidence bands around a smooth deterministic trend. The bootstrap method is easy to implement and does not require any adjustments in the presence of missing data, which makes it particularly suitable for climatological applications. We establish the asymptotic validity of the bootstrap method for both pointwise and simultaneous confidence bands under general conditions, allowing for general patterns of missing data, serial dependence and heteroskedasticity. The finite sample properties of the method are studied in…

Iterative Neural Autoregressive Distribution Estimator (NADE-k)

Training of the neural autoregressive density estimator (NADE) can be viewed as doing one step of probabilistic inference on missing values in data. We propose a new model that extends this inference scheme to multiple steps, arguing that it is easier to learn to improve a reconstruction in $k$ steps rather than to learn to reconstruct in a single inference step. The proposed model is an unsupervised building block for deep learning that combines the desirable properties of NADE and multi-predictive training: (1) Its test likelihood can be computed analytically, (2) it is easy to generate inde…

A Robbins Monro algorithm for nonparametric estimation of NAR process with Markov-Switching: consistency

We consider nonparametric estimation for functional autoregressive processes with Markov switching. First, we study the case where complete data is available; i.e. when we observe the Markov switching regime. Then we estimate the regression function in each regime using a Nadaraya-Watson type estimator. Second, we introduce a nonparametric recursive algorithm in the case of hidden Markov switching regime. Our algorithm restores the missing data by means of a Monte-Carlo step and estimate the regression function via a Robbins-Monro step. Consistency of the estimators are proved in both cases. F…

Imputation for High-Dimensional Linear Regression

We study high-dimensional regression with missing entries in the covariates. A common strategy in practice is to \emph{impute} the missing entries with an appropriate substitute and then implement a standard statistical procedure acting as if the covariates were fully observed. Recent literature on this subject proposes instead to design a specific, often complicated or non-convex, algorithm tailored to the case of missing covariates. We investigate a simpler approach where we fill-in the missing entries with their conditional mean given the observed covariates. We show that this imputation sc…

Limit theorems for bifurcating Markov chains. Application to the detection of cellular aging

We propose a general method to study dependent data in a binary tree, where an individual in one generation gives rise to two different offspring, one of type 0 and one of type 1, in the next generation. For any specific characteristic of these individuals, we assume that the characteristic is stochastic and depends on its ancestors' only through the mother's characteristic. The dependency structure may be described by a transition probability $P(x,dy dz)$ which gives the probability that the pair of daughters' characteristics is around $(y,z)$, given that the mother's characteristic is $x$. N…

基于卷积神经网络的非高斯空间自回归模型建模空间极值

Modeling Spatial Extremes using Non-Gaussian Spatial Autoregressive Models via Convolutional Neural Networks

本文提出一种基于卷积神经网络的非高斯空间自回归建模框架，用于高效估计SAR模型参数并捕捉空间极值行为，成功应用于北美年最大降水量数据分析。

Bayesian modeling of temporal dependence in large sparse contingency tables

In many applications, it is of interest to study trends over time in relationships among categorical variables, such as age group, ethnicity, religious affiliation, political party and preference for particular policies. At each time point, a sample of individuals provide responses to a set of questions, with different individuals sampled at each time. In such settings, there tends to be abundant missing data and the variables being measured may change over time. At each time point, one obtains a large sparse contingency table, with the number of cells often much larger than the number of indi…

Fast and stable modification of the Gauss-Newton method for low-rank signal estimation

The weighted nonlinear least-squares problem for low-rank signal estimation is considered. The problem of constructing a numerical solution that is stable and fast for long time series is addressed. A modified weighted Gauss-Newton method, which can be implemented through the direct variable projection onto a space of low-rank signals, is proposed. For a weight matrix which provides the maximum likelihood estimator of the signal in the presence of autoregressive noise of order $p$ the computational cost of iterations is $O(N r^2 + N p^2 + r N \log N)$ as $N$ tends to infinity, where $N$ is the…