arXiv subjects

Jiashun Jin

Publications and source records attributed to Jiashun Jin.

49 records · Page 3Linked to original sources

arXiv2013

Optimal classification in sparse Gaussian graphic model

Consider a two-class classification problem where the number of features is much larger than the sample size. The features are masked by Gaussian noise with mean zero and covariance matrix $Σ$, where the precision matrix $Ω=Σ^{-1}$ is unknown but is presumably sparse. The useful features, also unknown, are sparse and each contributes weakly (i.e., rare and weak) to the classification decision. By obtaining a reasonably good estimate of $Ω$, we formulate the setting as a linear regression model. We propose a two-stage classification method where we first select features by the method of Innovated Thresholding (IT), and then use the retained features and Fisher's LDA for classification. In this approach, a crucial problem is how to set the threshold of IT. We approach this problem by adapting the recent innovation of Higher Criticism Thresholding (HCT). We find that when useful features are rare and weak, the limiting behavior of HCT is essentially just as good as the limiting behavior of ideal threshold, the threshold one would choose if the underlying distribution of the signals is known (if only). Somewhat surprisingly, when $Ω$ is sufficiently sparse, its off-diagonal coordinates usually do not have a major influence over the classification decision. Compared to recent work in the case where $Ω$ is the identity matrix [Proc. Natl. Acad. Sci. USA 105 (2008) 14790-14795; Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 367 (2009) 4449-4470], the current setting is much more general, which needs a new approach and much more sophisticated analysis. One key component of the analysis is the intimate relationship between HCT and Fisher's separation. Another key component is the tight large-deviation bounds for empirical processes for data with unconventional correlation structures, where graph theory on vertex coloring plays an important role.

stat.ML↗

arXiv2012

UPS delivers optimal phase diagram in high-dimensional variable selection

Consider a linear model $Y=Xβ+z$, $z\sim N(0,I_n)$. Here, $X=X_{n,p}$, where both $p$ and $n$ are large, but $p>n$. We model the rows of $X$ as i.i.d. samples from $N(0,\frac{1}{n}Ω)$, where $Ω$ is a $p\times p$ correlation matrix, which is unknown to us but is presumably sparse. The vector $β$ is also unknown but has relatively few nonzero coordinates, and we are interested in identifying these nonzeros. We propose the Univariate Penalization Screeing (UPS) for variable selection. This is a screen and clean method where we screen with univariate thresholding and clean with penalized MLE. It has two important properties: sure screening and separable after screening. These properties enable us to reduce the original regression problem to many small-size regression problems that can be fitted separately. The UPS is effective both in theory and in computation.

math.ST↗

arXiv2010

Innovated higher criticism for detecting sparse signals in correlated noise

Higher criticism is a method for detecting signals that are both sparse and weak. Although first proposed in cases where the noise variables are independent, higher criticism also has reasonable performance in settings where those variables are correlated. In this paper we show that, by exploiting the nature of the correlation, performance can be improved by using a modified approach which exploits the potential advantages that correlation has to offer. Indeed, it turns out that the case of independent noise is the most difficult of all, from a statistical viewpoint, and that more accurate signal detection (for a given level of signal sparsity and strength) can be obtained when correlation is present. We characterize the advantages of correlation by showing how to incorporate them into the definition of an optimal detection boundary. The boundary has particularly attractive properties when correlation decays at a polynomial rate or the correlation matrix is Toeplitz.

math.ST↗

arXiv2010

Robustness and accuracy of methods for high dimensional data analysis based on Student's t statistic

Student's $t$ statistic is finding applications today that were never envisaged when it was introduced more than a century ago. Many of these applications rely on properties, for example robustness against heavy tailed sampling distributions, that were not explicitly considered until relatively recently. In this paper we explore these features of the $t$ statistic in the context of its application to very high dimensional problems, including feature selection and ranking, highly multiple hypothesis testing, and sparse, high dimensional signal detection. Robustness properties of the $t$-ratio are highlighted, and it is established that those properties are preserved under applications of the bootstrap. In particular, bootstrap methods correct for skewness, and therefore lead to second-order accuracy, even in the extreme tails. Indeed, it is shown that the bootstrap, and also the more popular but less accurate $t$-distribution and normal approximations, are more effective in the tails than towards the middle of the distribution. These properties motivate new methods, for example bootstrap-based techniques for signal detection, that confine attention to the significant tail of a statistic.

stat.ME↗

arXiv2010

Optimal rates of convergence for estimating the null density and proportion of nonnull effects in large-scale multiple testing

An important estimation problem that is closely related to large-scale multiple testing is that of estimating the null density and the proportion of nonnull effects. A few estimators have been introduced in the literature; however, several important problems, including the evaluation of the minimax rate of convergence and the construction of rate-optimal estimators, remain open. In this paper, we consider optimal estimation of the null density and the proportion of nonnull effects. Both minimax lower and upper bounds are derived. The lower bound is established by a two-point testing argument, where at the core is the novel construction of two least favorable marginal densities $f_1$ and $f_2$. The density $f_1$ is heavy tailed both in the spatial and frequency domains and $f_2$ is a perturbation of $f_1$ such that the characteristic functions associated with $f_1$ and $f_2$ match each other in low frequencies. The minimax upper bound is obtained by constructing estimators which rely on the empirical characteristic function and Fourier analysis. The estimator is shown to be minimax rate optimal. Compared to existing methods in the literature, the proposed procedure not only provides more precise estimates of the null density and the proportion of the nonnull effects, but also yields more accurate results when used inside some multiple testing procedures which aim at controlling the False Discovery Rate (FDR). The procedure is easy to implement and numerical results are given.

math.ST↗

arXiv2009

Revisiting Marginal Regression

The lasso has become an important practical tool for high dimensional regression as well as the object of intense theoretical investigation. But despite the availability of efficient algorithms, the lasso remains computationally demanding in regression problems where the number of variables vastly exceeds the number of data points. A much older method, marginal regression, largely displaced by the lasso, offers a promising alternative in this case. Computation for marginal regression is practical even when the dimension is very high. In this paper, we study the relative performance of the lasso and marginal regression for regression problems in three different regimes: (a) exact reconstruction in the noise-free and noisy cases when design and coefficients are fixed, (b) exact reconstruction in the noise-free case when the design is fixed but the coefficients are random, and (c) reconstruction in the noisy case where performance is measured by the number of coefficients whose sign is incorrect. In the first regime, we compare the conditions for exact reconstruction of the two procedures, find examples where each procedure succeeds while the other fails, and characterize the advantages and disadvantages of each. In the second regime, we derive conditions under which marginal regression will provide exact reconstruction with high probability. And in the third regime, we derive rates of convergence for the procedures and offer a new partitioning of the ``phase diagram,'' that shows when exact or Hamming reconstruction is effective.

math.ST↗

arXiv2009

A generalized Fourier approach to estimating the null parameters and proportion of non-null effects in large-scale multiple testing

In a recent paper (Efron (2004)), Efron pointed out that an important issue in large-scale multiple hypothesis testing is that the null distribution may be unknown and need to be estimated. Consider a Gaussian mixture model, where the null distribution is known to be normal but both null parameters--the mean and the variance--are unknown. We address the problem with a method based on Fourier transformation. The Fourier approach was first studied by Jin and Cai (2007), which focuses on the scenario where any non-null effect has either the same or a larger variance than that of the null effects. In this paper, we review the main ideas in Jin and Cai (2007), and propose a generalized Fourier approach to tackle the problem under another scenario: any non-null effect has a larger mean than that of the null effects, but no constraint is imposed on the variance. This approach and that in \cite{JC} complement with each other: each approach is successful in a wide class of situations where the other fails. Also, we extend the Fourier approach to estimate the proportion of non-null effects. The proposed procedures perform well both in theory and in simulated data.

math.ST↗

arXiv2008

Feature selection by Higher Criticism thresholding: optimal phase diagram

We consider two-class linear classification in a high-dimensional, low-sample size setting. Only a small fraction of the features are useful, the useful features are unknown to us, and each useful feature contributes weakly to the classification decision -- this setting was called the rare/weak model (RW Model). We select features by thresholding feature $z$-scores. The threshold is set by {\it higher criticism} (HC). Let $\pee_i$ denote the $P$-value associated to the $i$-th $z$-score and $\pee_{(i)}$ denote the $i$-th order statistic of the collection of $P$-values. The HC threshold (HCT) is the order statistic of the $z$-score corresponding to index $i$ maximizing $(i/n - \pee_{(i)})/\sqrt{\pee_{(i)}(1-\pee_{(i)})}$. The ideal threshold optimizes the classification error. In \cite{PNAS} we showed that HCT was numerically close to the ideal threshold. We formalize an asymptotic framework for studying the RW model, considering a sequence of problems with increasingly many features and relatively fewer observations. We show that along this sequence, the limiting performance of ideal HCT is essentially just as good as the limiting performance of ideal thresholding. Our results describe two-dimensional {\it phase space}, a two-dimensional diagram with coordinates quantifying "rare" and "weak" in the RW model. Phase space can be partitioned into two regions -- one where ideal threshold classification is successful, and one where the features are so weak and so rare that it must fail. Surprisingly, the regions where ideal HCT succeeds and fails make the exact same partition of the phase diagram. Other threshold methods, such as FDR threshold selection, are successful in a substantially smaller region of the phase space than either HCT or Ideal thresholding.

math.ST↗

arXiv2008

Estimation and confidence sets for sparse normal mixtures

For high dimensional statistical models, researchers have begun to focus on situations which can be described as having relatively few moderately large coefficients. Such situations lead to some very subtle statistical problems. In particular, Ingster and Donoho and Jin have considered a sparse normal means testing problem, in which they described the precise demarcation or detection boundary. Meinshausen and Rice have shown that it is even possible to estimate consistently the fraction of nonzero coordinates on a subset of the detectable region, but leave unanswered the question of exactly in which parts of the detectable region consistent estimation is possible. In the present paper we develop a new approach for estimating the fraction of nonzero means for problems where the nonzero means are moderately large. We show that the detection region described by Ingster and Donoho and Jin turns out to be the region where it is possible to consistently estimate the expected fraction of nonzero coordinates. This theory is developed further and minimax rates of convergence are derived. A procedure is constructed which attains the optimal rate of convergence in this setting. Furthermore, the procedure also provides an honest lower bound for confidence intervals while minimizing the expected length of such an interval. Simulations are used to enable comparison with the work of Meinshausen and Rice, where a procedure is given but where rates of convergence have not been discussed. Extensions to more general Gaussian mixture models are also given.

math.ST↗

arXiv2008

Properties of higher criticism under strong dependence

The problem of signal detection using sparse, faint information is closely related to a variety of contemporary statistical problems, including the control of false-discovery rate, and classification using very high-dimensional data. Each problem can be solved by conducting a large number of simultaneous hypothesis tests, the properties of which are readily accessed under the assumption of independence. In this paper we address the case of dependent data, in the context of higher criticism methods for signal detection. Short-range dependence has no first-order impact on performance, but the situation changes dramatically under strong dependence. There, although higher criticism can continue to perform well, it can be bettered using methods based on differences of signal values or on the maximum of the data. The relatively inferior performance of higher criticism in such cases can be explained in terms of the fact that, under strong dependence, the higher criticism statistic behaves as though the data were partitioned into very large blocks, with all but a single representative of each block being eliminated from the dataset.

math.ST↗

arXiv2007

Asymptotic minimaxity of False Discovery Rate thresholding for sparse exponential data

We apply FDR thresholding to a non-Gaussian vector whose coordinates X_i, i=1,..., n, are independent exponential with individual means $μ_i$. The vector $μ=(μ_i)$ is thought to be sparse, with most coordinates 1 but a small fraction significantly larger than 1; roughly, most coordinates are simply `noise,' but a small fraction contain `signal.' We measure risk by per-coordinate mean-squared error in recovering $\log(μ_i)$, and study minimax estimation over parameter spaces defined by constraints on the per-coordinate p-norm of $\log(μ_i)$: $\frac{1}{n}\sum_{i=1}^n\log^p(μ_i)\leq η^p$. We show for large n and small $η$ that FDR thresholding can be nearly Minimax. The FDR control parameter 0 1/2 prevents near minimaxity. These conclusions mirror those found in the Gaussian case in Abramovich et al. [Ann. Statist. 34 (2006) 584--653]. The techniques developed here seem applicable to a wide range of other distributional assumptions, other loss measures and non-i.i.d. dependency structures.

math.ST↗

arXiv2006

Estimating the Null and the Proportion of non-Null effects in Large-scale Multiple Comparisons

An important issue raised by Efron in the context of large-scale multiple comparisons is that in many applications the usual assumption that the null distribution is known is incorrect, and seemingly negligible differences in the null may result in large differences in subsequent studies. This suggests that a careful study of estimation of the null is indispensable. In this paper, we consider the problem of estimating a null normal distribution, and a closely related problem, estimation of the proportion of non-null effects. We develop an approach based on the empirical characteristic function and Fourier analysis. The estimators are shown to be uniformly consistent over a wide class of parameters. Numerical performance of the estimators is investigated using both simulated and real data. In particular, we apply our procedure to the analysis of breast cancer and HIV microarray data sets. The estimators perform favorably in comparison to existing methods.

math.ST↗

arXiv2004

Higher criticism for detecting sparse heterogeneous mixtures

Higher criticism, or second-level significance testing, is a multiple-comparisons concept mentioned in passing by Tukey. It concerns a situation where there are many independent tests of significance and one is interested in rejecting the joint null hypothesis. Tukey suggested comparing the fraction of observed significances at a given α-level to the expected fraction under the joint null. In fact, he suggested standardizing the difference of the two quantities and forming a z-score; the resulting z-score tests the significance of the body of significance tests. We consider a generalization, where we maximize this z-score over a range of significance levels 0<α\leqα_0. We are able to show that the resulting higher criticism statistic is effective at resolving a very subtle testing problem: testing whether n normal means are all zero versus the alternative that a small fraction is nonzero. The subtlety of this ``sparse normal means'' testing problem can be seen from work of Ingster and Jin, who studied such problems in great detail. In their studies, they identified an interesting range of cases where the small fraction of nonzero means is so small that the alternative hypothesis exhibits little noticeable effect on the distribution of the p-values either for the bulk of the tests or for the few most highly significant tests. In this range, when the amplitude of nonzero means is calibrated with the fraction of nonzero means, the likelihood ratio test for a precisely specified alternative would still succeed in separating the two hypotheses.

math.ST↗