Indicator vector

Abstract: This paper extends deterministic models for Boolean regression within a Bayesian framework. For a given binary criterion variable Y and a set of k binary predictor variables X1,…, Xk, a Boolean regression model is a conjunctive (or disjunctive) logical combination consisting of a subset S of the X variables, which predicts Y. Formally, Boolean regression models include a specification of a k-dimensional binary indicator vector (θ1,…,θk) with θj = 1 iff Xj ∈ S. In a probabilistic extension, a parameter π is added which represents the probability of the predicted value ${\hat y_i}$ and the observed value yi differing (for any observation i). Within a Bayesian framework, a posterior distribution of the parameters (θ1,…, θk, π) is looked for. The advantages of such a Bayesian approach include a proper account of the uncertainty in the model estimates and various possibilities for model checking (using posterior predictive checks). We illustrate this method with an example using real data.

Abstract: 1. Given a universeM ofm items, the indicator vector x of a set S is a vector x ∈ {0, 1}, in which x(i) = 1 iff i ∈ S. A set function f : {0, 1} −→ R assigns a value to each set. The Lovasz extension f of f extends its domain to the convex set [0, 1] (allowing fractional values to the coordinates). Its value is defined as the following expectation: f(x) = Eλ[f(xλ)], where λ is chosen uniformly in the range [0, 1], and x(i) = 1 if xλ(i) ≥ λ and x(i) = 0 otherwise. Observe that f is equal to f on integer points. Show that if f is submodular then the corresponding f is a convex function. Namely, the region f(x) ≤ t is convex.

Abstract: The traditional high voltage switchgear (HVS) state evaluation model mostly adopts electrical test, live detection and historical data, neglecting the influence of real-time operation data of HVS composition equipment on the state evaluation results. This paper proposes a HVS operation state evaluation model based on fuzzy set-valued statistics method and kernel vector space model based on electrical test data and on-line monitoring data. First of all, according to the components of high voltage switchgear, the operation state of HVS is described and the evaluation index system is established. Secondly, the fuzzy set-valued statistics method is used to construct the mathematical model of evaluation index weight. Then, the kernel vector space model is introduced, and the Gaussian kernel function is used to map the sample to the features of the high-dimensional feature space. The indicator vector of the sample data and the ideal indicator vector of the high-voltage switchgear operation status level standard are defined in the high-dimensional feature space, and the angle-weighted cosine between the two vectors is calculated as the closeness of the sample to the standard status level, and then the high-voltage switchgear operation status level is obtained. Finally, the real data of a power supply company in western China are simulated. The results show that the greater the closeness degree is, the closer the HVS corresponding to the sample is to the normal state, on the contrary, the smaller the closeness degree is, the closer the HVS is to the fault state.

Abstract: We study the robust mean estimation problem in high dimensions, where $\alpha <0.5$ fraction of the data points can be arbitrarily corrupted. Motivated by compressive sensing, we formulate the robust mean estimation problem as the minimization of the $\ell_0$-`norm' of the outlier indicator vector, under second moment constraints on the inlier data points. We prove that the global minimum of this objective is order optimal for the robust mean estimation problem, and we propose a general framework for minimizing the objective. We further leverage the $\ell_1$ and $\ell_p$ $(0