Hypothesis Testing in Feedforward Networks with Broadcast Failures

Consider a countably infinite set of nodes, which sequentially make decisions between two given hypotheses. Each node takes a measurement of the underlying truth, observes the decisions from some immediate predecessors, and makes a decision between the given hypotheses. We consider two classes of broadcast failures: 1) each node broadcasts a decision to the other nodes, subject to random erasure in the form of a binary erasure channel; 2) each node broadcasts a randomly flipped decision to the other nodes in the form of a binary symmetric channel. We are interested in whether there exists a decision strategy consisting of a sequence of likelihood ratio tests such that the node decisions converge in probability to the underlying truth. In both cases, we show that if each node only learns from a bounded number of immediate predecessors, then there does not exist a decision strategy such that the decisions converge in probability to the underlying truth. However, in case 1, we show that if each node learns from an unboundedly growing number of predecessors, then the decisions converge in probability to the underlying truth, even when the erasure probabilities converge to 1. We also derive the convergence rate of the error probability. In case 2, we show that if each node learns from all of its previous predecessors, then the decisions converge in probability to the underlying truth when the flipping probabilities of the binary symmetric channels are bounded away from 1/2. In the case where the flipping probabilities converge to 1/2, we derive a necessary condition on the convergence rate of the flipping probabilities such that the decisions still converge to the underlying truth. We also explicitly characterize the relationship between the convergence rate of the error probability and the convergence rate of the flipping probabilities

The mass ratio distribution of short period double degenerate stars

Short period double degenerates (DDs) are close white dwarf - white dwarf binary stars which are the result of the evolution of interacting binary stars. We present the first definitive measurements of the mass ratio for two DDs, WD0136+768 and WD1204+450, and an improved measurement of the mass ratio for WD0957-666. We compare the properties of the 6 known DDs with measured mass ratios to the predictions of various theoretical models. We confirm the result that standard models for the formation of DDs do not predict sufficient DDs with mass ratios near 1. We also show that the observed difference in cooling ages between white dwarfs in DDs is a useful constraint on the initial mass ratio of the binary. A more careful analysis of the properties of the white dwarf pair WD1704+481.2 leads us to conclude that the brighter white dwarf is older than its fainter companion. This is the opposite of the usual case for DDs and is caused by the more massive white dwarf being smaller and cooling faster. The mass ratio in the sense (mass of younger star)/(mass of older star) is then 1.43+-0.06 rather than the value 0.70+-0.03 given previously.Comment: Accepted for publication in MNRA

The triple degenerate star WD1704+481

WD1704+481 is a visual binary in which both components are white dwarfs. We present spectra of the H-alpha line of both stars which show that one component (WD1704+481.2 = Sanduleak B = GR 577) is a close binary with two white dwarf components. Thus, WD1704+481 is the first known triple degenerate star. From radial velocity measurements of the close binary we find an orbital period of 0.1448d, a mass ratio, q=Mbright/Mfaint of q=0.70+-0.03 and a difference in the gravitational redshifts of 11.5+-2.3km/s. The masses of the close pair of white dwarfs predicted by the mass ratio and gravitational redshift difference combined with theoretical cooling curves are 0.39+-0.05 solar mass and 0.56+-0.07 solar masses. WD1704+481 is therefore also likely to be the first example of a double degenerate in which the less massive white dwarf is composed of helium and the other white dwarf is composed of carbon and oxygen.Comment: 5 pages, 4 figure

Submodularity and Optimality of Fusion Rules in Balanced Binary Relay Trees

We study the distributed detection problem in a balanced binary relay tree, where the leaves of the tree are sensors generating binary messages. The root of the tree is a fusion center that makes the overall decision. Every other node in the tree is a fusion node that fuses two binary messages from its child nodes into a new binary message and sends it to the parent node at the next level. We assume that the fusion nodes at the same level use the same fusion rule. We call a string of fusion rules used at different levels a fusion strategy. We consider the problem of finding a fusion strategy that maximizes the reduction in the total error probability between the sensors and the fusion center. We formulate this problem as a deterministic dynamic program and express the solution in terms of Bellman's equations. We introduce the notion of stringsubmodularity and show that the reduction in the total error probability is a stringsubmodular function. Consequentially, we show that the greedy strategy, which only maximizes the level-wise reduction in the total error probability, is within a factor of the optimal strategy in terms of reduction in the total error probability

Detection Performance in Balanced Binary Relay Trees with Node and Link Failures

We study the distributed detection problem in the context of a balanced binary relay tree, where the leaves of the tree correspond to $N$ identical and independent sensors generating binary messages. The root of the tree is a fusion center making an overall decision. Every other node is a relay node that aggregates the messages received from its child nodes into a new message and sends it up toward the fusion center. We derive upper and lower bounds for the total error probability $P_N$ as explicit functions of $N$ in the case where nodes and links fail with certain probabilities. These characterize the asymptotic decay rate of the total error probability as $N$ goes to infinity. Naturally, this decay rate is not larger than that in the non-failure case, which is $\sqrt N$. However, we derive an explicit necessary and sufficient condition on the decay rate of the local failure probabilities $p_k$ (combination of node and link failure probabilities at each level) such that the decay rate of the total error probability in the failure case is the same as that of the non-failure case. More precisely, we show that $\log P_N^{-1}=\Theta(\sqrt N)$ if and only if $\log p_k^{-1}=\Omega(2^{k/2})$

Orbital periods of the binary sdB stars PG0940+068 and PG1247+554

We have used the radial velocity variations of two sdB stars previously reported to be binaries to establish their orbital periods. They are PG0940+068, (P=8.33d) and PG1247+554 (P=0.599d). The minimum masses of the unseen companions, assuming a mass of 0.5 solar masses for the sdB stars, are 0.090 +/- 0.003 solar masses for PG1247+554 and 0.63 +/- 0.02 solar masses for PG0940+068. The nature of the companions is not constrained further by our data.Comment: 5 pages, 2 figure

Performance of the Colorado wind-profiling network, part 1.5A

The Wave Propagation Laboratory (WPL) has operated a network of radar wind Profilers in Colorado for about 1 year. The network consists of four VHF (50-MHz) radars and a UHF (915-MHz) radar. The Platteville VHF radar was developed by the Aeronomy Laboratory (AL) and has been operated jointly by WPL and AL for several years. The other radars were installed between February and May 1983. Experiences with these radars and some general aspects of tropospheric wind measurements with Doppler radar are discussed