Hunting for Foxes with Sheaves

Hunting for Foxes with Sheaves Michael Robinson Introduction Michael Robinson is an Associate Professor in the Department of Mathemat- To a radio amateur (or “ham”), fox hunting has nothing ics and Statistics at American University. His email address is michaelr to do with animals. It is a sport in which individuals race @american.edu. All figures can be reproduced using The Jupyter Notebook at: https://github For permission to reprint this article, please contact: .com/kb1dds/foxsheaf. [email protected]. Communicated by Notices Associate Editor Emilie Purvine. DOI: https://doi.org/10.1090/noti1867 MAY 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 661 each other to locate a hidden radio transmitter on a known We are left with the need for a general deterministic frequency. Since hams are encouraged to design and build method for finding the fox from a small number ofmea- their own equipment, the typical fox hunt involves a vari- surements. This article explains how to meet this need us- ety of different receivers and antennas with different ca- ing sheaves, mathematical objects that describe local consis- pabilities. Some of these can display the received signal tency within data. We can perform data fusion for any sheaf, strength from the hidden transmitter (loosely measuring though the fox hunting problem will guide our selection distance to the transmitter), while others estimate the com- of the specific sheaf we need and will be the context for pass bearing. Both of these estimates vary in accuracy and its interpretation. The resulting fox hunting sheaf is mod- in precision depending on terrain, environmental condi- ular; different sensors or models of their performance can tions, equipment quality, and the skill of the operator. be substituted easily without changing how their data are Fox hunting also serves the purpose of preparing radio analyzed. amateurs for emergency or disaster operations. Because This article explains how to model a collection of sen- disaster operations require the concerted efforts of mul- sors in the section called “Formalizing the Sensors in the tiple radio operators, it seems fitting to explore how the Fox Hunt” so that they can be combined into a sheaf model sport changes if fox hunting becomes cooperative. When in “Formalizing the Interactions between Sensors.” Once participants combine their estimates of distance and bear- the sheaf is constructed, we show how to locate the fox ing, how much faster can they find the transmitter? Part transmitter in “Consistency Radius: Where is the Fox?” and of the challenge of fox hunting is that measurements are determine if there are actually multiple fox transmitters in taken infrequently, only once every few minutes. To win “Local Consistency Radius: Finding Multiple Foxes.” Fi- the hunt, every minute must count! In the most demand- nally, since applied sheaf theory is still in its infancy, “Fron- ing scenario, each sensor only gets to take one measure- tiers”points the reader to some interesting directions for ment of the fox transmitter. future study. Locating the fox transmitter from a collection of different sensors is a model-based data fusion problem: combining Formalizing the Sensors in the Fox Hunt disparate local observations into a global inference. With- Each receiver (or sensor) 퐴 used in the fox hunt produces a out a model that describes how signals from the transmit- signal report concerning its observation of the fox transmitter arrive at each receiver, the signal reports are not helpful ter. Signal reports may be of different types, depending on for locating the transmitter. Even with such a model, the the sensor. For instance, the strength of the received signal effects of terrain, the transmitter’s antenna system, and the is typically reported as a single real number. In contrast, a environment can cause substantial differences between the compass bearing is reported as an angle, properly an ele- modeled signal and an actual received signal. Therefore, it ment of the metric space 푆1—the unit circle. To handle is important that we remain even-handed about assump- both of these situations (and more), let us suppose that a tions of the quality of the estimates and the quality of the signal report is an element of a pseudometric space 퐷퐴, de- model. pending on the sensor 퐴. The sensor 퐴 produces reports Though there are many techniques for solving data fu- through a continuous measurement function sion problems, they broadly fall into two categories: (1) 2 2 푀퐴 ∶ ℝ × 퐶퐹 × ℝ × 퐶퐴 → 퐷퐴, problem-specific deterministic methods, and (2) general statistical methods. Well-crafted problem-specific determin- depending on fox transmitter location (in the plane), the istic methods are very effective. Because of the physics of fox transmitter equipment settings 퐶퐹 (such as transmitter radio propagation, it is not too difficult to construct a de- power and antenna orientation), the receiver location in terministic method specifically for locating a fox transmit- the plane, and the receiver equipment settings 퐶퐴 (such as ter. However, problem-specific methods often carry hid- antenna orientation). den assumptions that make it hard to transfer useful tech- Our sensor data will be drawn from a parameterized dis- niques to another problem. Worse, the bookkeeping as- tribution 푆퐴, in which the noise level 휎 is taken as a pa- sociated with all combinations of sensors grows exponen- rameter. To ensure consistency between the deterministic tially as more sensors are deployed. Regardless of their at- model and the stochastic one, our stochastic models sat- traction, problem-specific methods for solving data fusion isfy ′ ′ ′ ′ problems are costly and difficult to manage. Statistical 푆퐴(푥, 푦, 푐푓, 푥 , 푦 , 푐퐴; 휎) → 훿푀퐴(푥,푦,푐푓,푥 ,푦 ,푐퐴) as 휎 → 0, methods automate the bookkeeping and tailoring needed in which 훿 is the unit impulse at 훼. for a problem-specific method, but they usually require 훼 There are typically two kinds of sensors that are used in many observations to produce accurate results. Since we radio fox hunting: calibrated signal strength meters and are only using one measurement from each sensor, statis- directional antennas. Given knowledge of the fox trans- tical methods are not the best option. mitter’s power output, a calibrated signal strength meter 662 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 5 Figure 1. Simulated received power for a transmitter located at the top of a mountain using (a) the Longley-Rice model [8] and (b) the simpliﬁed model in (1). Colors are on a Figure 2. The author’s son holding a direction-sensitive logarithmic scale, in decibels referenced to the power of 1 antenna (a Yagi-Uda array) built by the author for 144 MHz. milliwatt (dBm). can help estimate the distance from the receiver to the fox. where 퐼0 is the modified Bessel function of the first kind. A directional antenna tells the operator the direction from The Rice distribution models the distance between the ori- which the fox’s signal appears to be the strongest. gin and a point in the plane drawn from a bivariate Gauss- Calibrated signal strength (RSSI) sensors. A received sig- ian with mean 푣 and standard deviation 휎. Assuming the nal strength indication (RSSI) sensor measures the amount received signal is a combination of the inverse square law of power absorbed by its antenna from the fox transmit- and Rician fading, the RSSI signal reports 푃 will follow the ter. To model this accurately requires careful specification distribution of the terrain and any obstacles between the transmitter ′ ′ and receiver. The Longley-Rice model is popular among 푆푅푆푆퐼(푃; 푥, 푦, 푝, 푥 , 푦 ) (2) engineers because it incorporates the effect of terrain and 1 2 = (푅푖푐푒 (푃; 2푀 (푥, 푦, 푝, 푥′, 푦′)푅, 휎)) , atmospheric losses on the received power, and its predic- 2푅 √ 푅푆푆퐼 tions are realistic [8]. Figure 1(a) shows the received signal power predicted by the Longley-Rice model for a fox trans- in which antenna characteristic impedance 푅 (typically 50Ω mitter placed on a mountain. for amateur equipment) and root mean square noise volt- While the realism of the Longley-Rice model is a bene- age 휎 are assumed to be known in advance and constant. fit, its precise specification is quite complicated. Noneof Bearing sensors. A bearing sensor measures the angle be- this complexity is necessary to demonstrate our approach, tween true north (or some other convenient, global direc- because all our analyses are modular. A different measure- tion) and the apparent direction of arrival of signals from ment function can be substituted later if desired without the fox transmitter. Most amateur bearing sensors con- changing the analysis techniques. Therefore, we will sim- sist of an antenna (like the one shown in Figure 2) that ply model the power transferred from transmitter to re- is preferentially sensitive to signals arriving from a specific ceiver by an inverse-square law (Figure 1(b)). This corre- direction. The operator rotates the antenna until the sig- sponds to a measurement function in which there is only nal strength is greatest, and then records its direction. For fox transmitter power 푝 ∈ 퐶퐹 = [0, ∞) and no receiver strong signals, the operator can also block incoming sig- configuration, nals by holding a hand held radio against the chest. When 푝 (1) the operator slowly turns around until the signal strength 푀 (푥, 푦, 푝, 푥′, 푦′) = , 푅푆푆퐼 4휋 ((푥 − 푥′)2 + (푦 − 푦′)2) is minimized, the transmitter is then directly behind the ′ ′ operator! where (푥, 푦) is the transmitter location and (푥 , 푦 ) is the A useful antenna for fox hunting produces a response receiver location. like one of the two shown in Figure 3 (p.664).

Load more