On the false discovery rates of a frequentist: Asymptotic expansions

Consider a testing problem for the null hypothesis $H_0:\theta\in\Theta_0$. The standard frequentist practice is to reject the null hypothesis when the p-value is smaller than a threshold value $\alpha$, usually 0.05. We ask the question how many of the null hypotheses a frequentist rejects are actually true. Precisely, we look at the Bayesian false discovery rate $\delta_n=P_g(\theta\in\Theta_0|p-value<\alpha)$ under a proper prior density $g(\theta)$. This depends on the prior $g$, the sample size $n$, the threshold value $\alpha$ as well as the choice of the test statistic. We show that the Benjamini--Hochberg FDR in fact converges to $\delta_n$ almost surely under $g$ for any fixed $n$. For one-sided null hypotheses, we derive a third order asymptotic expansion for $\delta_n$ in the continuous exponential family when the test statistic is the MLE and in the location family when the test statistic is the sample median. We also briefly mention the expansion in the uniform family when the test statistic is the MLE. The expansions are derived by putting together Edgeworth expansions for the CDF, Cornish--Fisher expansions for the quantile function and various Taylor expansions. Numerical results show that the expansions are very accurate even for a small value of $n$ (e.g., $n=10$). We make many useful conclusions from these expansions, and specifically that the frequentist is not prone to false discoveries except when the prior $g$ is too spiky. The results are illustrated by many examples.Comment: Published at http://dx.doi.org/10.1214/074921706000000699 in the IMS Lecture Notes--Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

A Sparse Johnson--Lindenstrauss Transform

Dimension reduction is a key algorithmic tool with many applications including nearest-neighbor search, compressed sensing and linear algebra in the streaming model. In this work we obtain a {\em sparse} version of the fundamental tool in dimension reduction --- the Johnson--Lindenstrauss transform. Using hashing and local densification, we construct a sparse projection matrix with just $\tilde{O}(\frac{1}{\epsilon})$ non-zero entries per column. We also show a matching lower bound on the sparsity for a large class of projection matrices. Our bounds are somewhat surprising, given the known lower bounds of $\Omega(\frac{1}{\epsilon^2})$ both on the number of rows of any projection matrix and on the sparsity of projection matrices generated by natural constructions. Using this, we achieve an $\tilde{O}(\frac{1}{\epsilon})$ update time per non-zero element for a $(1\pm\epsilon)$-approximate projection, thereby substantially outperforming the $\tilde{O}(\frac{1}{\epsilon^2})$ update time required by prior approaches. A variant of our method offers the same guarantees for sparse vectors, yet its $\tilde{O}(d)$ worst case running time matches the best approach of Ailon and Liberty.Comment: 10 pages, conference version

An Improved Algorithm for Eye Corner Detection

In this paper, a modified algorithm for the detection of nasal and temporal eye corners is presented. The algorithm is a modification of the Santos and Proenka Method. In the first step, we detect the face and the eyes using classifiers based on Haar-like features. We then segment out the sclera, from the detected eye region. From the segmented sclera, we segment out an approximate eyelid contour. Eye corner candidates are obtained using Harris and Stephens corner detector. We introduce a post-pruning of the Eye corner candidates to locate the eye corners, finally. The algorithm has been tested on Yale, JAFFE databases as well as our created database

State feedback control of power system oscillations

Damping of electromechanical oscillations in power systems is one of the major concerns in the operation of power system since many years. The oscillations may be local to a single generator or generator plant (local oscillations), or they may involve a number of generators widely separated geographically (inter-area oscillations). These oscillations causes improper of the power system incorporating losses. Local oscillations often occur when a fast exciter is used on the generator, and to stabilize these oscillations, Power System Stabilizers (PSS) were developed. Inter-area oscillations may appear as the systems loading is increased across the weak transmission links in the system which characterize these oscillations. If not controlled, these oscillations may lead to total or partial power interruption. Electricité de France developed two state feedback controllers aiming to effectively damp electromechanical oscillations present in power systems. These are Desensitized Four Loop Regulator (DFLR) and Extended Desensitized Four Loop Regulator (EDFLR). The DFLR is designed to damp local lectromechanical oscillations while the EDFLR aims at damping both local and inter-area oscillations. The dynamics of the DFLR and EDLFR are needed to be studies in order to model them. These models are to be incorporated with the generator models to get a power system model with state feedback control. On simulating the system in Simulink with the controllers we will get the power system model with state feedback control and we can observe how these controllers are helpful in damping the oscillations