# A Sheaf Theoretical Approach to Uncertainty Quantification of Heterogeneous Geolocation Information

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## Abstract

**:**

## 1. Introduction

- Section 2 provides context around the history of target tracking methodologies and multi-sensor fusion, including the particular role we are advocating specifically for sheaf methods.
- Then in Section 3 we describe the experimental setup for our data collection effort regarding tracking black bears in Asheville, North Carolina. This involved four sensors deployed among the bears, several “dummy” bear collars deployed at fixed locations in the field, and team of scientists charged with tracking their location. This experiment was designed specifically to identify a real tracking task where multi-sensor integration is required, but where the system complexity is not so large as to overwhelm the methodological development and demonstration, while still being sufficiently complexity to demonstrate the value of sheaf-based methods.
- Section 4 continues with a detailed mathematical development of the sheaf-based tracking model. This includes several features novel to information fusion models, including the explicit initial attention to the complexity of sensor interaction in the core sheaf models, but then the ability to equip these models with uncertainty tolerance in the form of approximate sections, and then to measure that both globally via a consistency radius and also at a more fine-grained level in terms of specific sensor dependencies in terms of a consistency filtration.
- Section 5 shows several aspects of our models interpreted through the measured data. First, overall results in terms of bear and dummy collars is provided; then results for a particular bear collar are examined in detail; and finally results for a particular time point in the measured data set is shown to illustrate the consistency filtration in particular.
- While our sheaf models are novel, we sought to compare them to a more traditional modeling approach, introduced in Section 6. Specifically, a statistical approach was developed to process the data once registered into a common coordinate system, a dynamic linear model which was estimated using a Kalman filter. This approach also recovered bear and human locations and sensor accuracies. Comparisons in both form and results are obtained, which demonstrate the role that sheaves as generic integration models can play in conjunction with specific modeling approaches such as these: as noted, all integration models will provably recapitulate some portion of sheaf theory [2], even if they are not first registered into a common coordinate system.
- We conclude in Section 7 with some general observations and discussion.

## 2. History and Context

#### 2.1. Target Tracking Methods

#### 2.2. Multi-Sensor Fusion Methods

#### 2.3. Sheaf Geometry for Fusion

## 3. Tracking Experiment

#### 3.1. Black Bear Study Capture and Monitoring Methodology

#### 3.2. Tracking Exercise

## 4. Sheaf Modeling Methodology

#### 4.1. Simplicial Sheaf Models

**Definition**

**1.**

**abstract simplicial complex**(ASC) over a finite base set U is a collection Δ of subsets of U, for which $\delta \in \Delta $ implies that every subset of δ is also in Δ. We call each $\delta \in \Delta $ with $d+1$ elements a

**d-face**of Δ, referring to the number d as its

**dimension**. Zero dimensional faces (singleton subsets of U) are called

**vertices**, and one dimensional faces are called

**edges**. For $\gamma ,\delta \in \Delta $, we say that γ is a

**face**of δ (written $\gamma \u21dd\delta $) whenever γ is a proper subset of δ.

**Remark**

**1.**

- The GPS reading on the Bear Collar, denoted G;
- The Radio VHF Device receiver, denoted R;
- The $\mathit{Text}$ report, denoted T; and
- The Vehicle GPS, denoted V.

**attachment diagram**corresponding to the ASC. This is a directed acyclic graph, where nodes are faces of the ASC, connected by a directed edge pointing up from a face to its attached face (co-face) of higher dimension.

**Definition**

**2.**

**sheaf$\mathcal{S}$of sets**on an abstract simplicial complex Δ consists of the assignment of

- a set $\mathcal{S}\left(\delta \right)$ to each face δ of Δ (called the
**stalk**at δ), and - a function $\mathcal{S}(\gamma \u21dd\delta ):\mathcal{S}\left(\gamma \right)\to \mathcal{S}\left(\delta \right)$ (called the
**restriction map**from γ to δ) to each inclusion of faces $\gamma \u21dd\delta $, which obeys$$\mathcal{S}(\delta \u21dd\lambda )\circ \mathcal{S}(\gamma \u21dd\delta )=\mathcal{S}(\gamma \u21dd\lambda )\phantom{\rule{4.pt}{0ex}}\mathit{whenever}\phantom{\rule{4.pt}{0ex}}\gamma \u21dd\delta \u21dd\lambda .$$

**sheaf of vector spaces**assigns a vector space to each face and a linear map to each attachment. In addition, a

**sheaf of groups**assigns a group to each face and a group homomorphism to each attachment.

**global section**. Non-agreement is definitely possible, giving rise to the notion of an

**assignment**.

**Definition**

**3.**

**assignment**$\alpha :\Delta \to {\displaystyle \prod _{\delta \in \Delta}}\mathcal{S}\left(\delta \right)$ provides a value $\alpha \left(\delta \right)\in \mathcal{S}\left(\delta \right)$ to each face $\delta \in \Delta $. A

**partial assignment**, β, provides a value for a subset ${\Delta}^{\prime}\subset \Delta $ of faces, $\beta {\Delta}^{\prime}{\displaystyle \prod _{\delta \in {\Delta}^{\prime}}}\mathcal{S}\left(\delta \right)$. An assignment s is called a

**global section**if for each inclusion $\delta \u21dd\lambda $ of faces, $\mathcal{S}(\delta \u21dd\lambda )\left(s\right(\delta \left)\right)=s\left(\lambda \right)$.

#### 4.2. Consistency Structures, Pseudosections, and Approximate Sections

**consistency structure**.

**Definition**

**4.**

**consistency structure**is a triple $(\Delta ,\mathcal{S},\mathit{C})$ where Δ is an abstract simplicial complex, $\mathcal{S}$ is a sheaf over Δ, and $\mathit{C}$ is the assignment to each non-vertex d-face $\lambda \in \Delta ,d>0$, of a function

**standard consistency structure**for a sheaf $\mathcal{S}$, which assigns an equality test to each non-vertex face $\lambda =\{{v}_{1},{v}_{2},\cdots ,{v}_{k}\}$:

**$\u03f5$-approximate consistency structure**for a sheaf $\mathcal{S}$ as follows. For each non-vertex size k face $\lambda =\{{v}_{1},{v}_{2},\cdots ,{v}_{k}\}\in \Delta ,k>1$ define

**Definition**

**5.**

**pseudosection**if for each non-vertex face $\lambda =\{{v}_{1},\cdots ,{v}_{k}\}$

- ${\mathit{C}}_{\lambda}\left([\mathcal{S}(\left\{{v}_{i}\right\}\u21dd\lambda )s\left(\left\{{v}_{i}\right\}\right)\phantom{\rule{2.84526pt}{0ex}}:\phantom{\rule{2.84526pt}{0ex}}i=1,\cdots ,k]\right)=1$, and
- ${\mathit{C}}_{\lambda}([\mathcal{S}(\left\{{v}_{i}\right\}\u21dd\lambda )s\left(\left\{{v}_{i}\right\}\right)\phantom{\rule{2.84526pt}{0ex}}:\phantom{\rule{2.84526pt}{0ex}}i=1,\cdots ,j-1,j+1,\cdots k]\cup \left[s\left(\lambda \right)\right])=1$ for all $j=1,2,\cdots ,k$.

- $\widehat{\sigma}\left([\mathcal{S}(\left\{{v}_{i}\right\}\u21dd\lambda )s\left(\left\{{v}_{i}\right\}\right)\phantom{\rule{2.84526pt}{0ex}}:\phantom{\rule{2.84526pt}{0ex}}i=1,\cdots ,k]\right)\le \u03f5$, and
- $\widehat{\sigma}([\mathcal{S}(\left\{{v}_{i}\right\}\u21dd\lambda )s\left(\left\{{v}_{i}\right\}\right)\phantom{\rule{2.84526pt}{0ex}}:\phantom{\rule{2.84526pt}{0ex}}i=1,\cdots ,j-1,j+1,\cdots k]\cup \left[s\left(\lambda \right)\right])\le \u03f5$ for all $j=1,2,\cdots ,k$.

- $\widehat{\sigma}\left([\mathcal{S}(\left\{v\right\}\u21dd\lambda )s\left(\left\{v\right\}\right)\phantom{\rule{2.84526pt}{0ex}}:\phantom{\rule{2.84526pt}{0ex}}v\in \lambda ]\right)\le \u03f5$, and
- $\widehat{\sigma}([\mathcal{S}(\left\{v\right\}\u21dd\lambda )s\left(\left\{v\right\}\right)\phantom{\rule{2.84526pt}{0ex}}:\phantom{\rule{2.84526pt}{0ex}}v\in \lambda ]\cup \left[s\left(\lambda \right)\right]\backslash \left[s\left(w\right)\right])\le \u03f5$ for all $w\in \lambda $.

**Theorem**

**1.**

**consistency radius**.

**Lemma**

**1.**

**Proof.**

#### 4.3. Maximal Consistent Subcomplexes

**Theorem**

**2.**

- The assignment α is consistent on each ${\Delta}_{{W}_{i}}$, and any subcomplex on which α is consistent has some ${\Delta}_{{W}_{i}}$ as a supercomplex.
- $\bigcup \mathrm{star}\left({\Delta}_{{W}_{i}}\right)$ is a cover of Δ.

**Lemma**

**2.**

- $\gamma \notin {\Delta}_{{W}_{i}}$ for all i.
- Every subcomplex on which α is consistent has some ${\Delta}_{{W}_{i}}$ as a supercomplex.
- $\bigcup \mathrm{star}\left({\Delta}_{{W}_{i}}\right)$ is a cover of Δ.

**Proof.**

**Proof of**

**Theorem 2.**

#### 4.4. Measures on Consistent Subcomplexes

**Definition**

**6.**

**graded**if there exists a rank function $r:P\to \mathbb{N}\phantom{\rule{2.84526pt}{0ex}}\cup \phantom{\rule{2.84526pt}{0ex}}\left\{0\right\}$ such that $r\left(s\right)=0$ if s is a minimal element of P, and $r\left(q\right)=r\left(p\right)+1$ if $p\prec q$ in P. If $r\left(s\right)=i$, we say that s has rank i. The maximum rank, $\underset{p\in P}{max}}\left\{r\left(p\right)\right\$, is called the

**rank**of $\mathcal{P}$.

**Proposition**

**1.**

**full**. Specifically, $\mathcal{A}=\{{a}_{1},{a}_{2},\cdots ,{a}_{k}\}$ is said to be full if ${\bigcup}_{i=1}^{k}{a}_{i}=\{1,2,\cdots ,n\}$. The set of all full ideals $\overline{\mathcal{I}\left({2}^{n}\right)}$ forms an induced subposet $\overline{J\left({2}^{n}\right)}$ of the graded poset $J\left({2}^{n}\right)$. Next, we show that $\overline{J\left({2}^{n}\right)}$ is also graded.

**Proposition**

**2.**

**Proof.**

**Definition**

**7.**

#### 4.5. Consistency Filtrations

**consistency filtration**, of vertex covers corresponding to landmark $\u03f5$ values ${\u03f5}_{0}=0<{\u03f5}_{1}<\cdots <{\u03f5}_{\ell -1}<{\u03f5}_{\ell}={\u03f5}^{*}$ where we recall that ${\u03f5}^{*}$ is the consistency radius, i.e., the smallest value of $\u03f5$ for which $\alpha $ is a $(\Delta ,\mathcal{S},{\mathbf{C}}_{\u03f5})$-pseudosection. The corresponding refinement of vertex covers is ${C}_{0}\le {C}_{1}\le \cdots \le {C}_{\ell -1}\le {C}_{\ell}=U$ where each is a set of subcomplexes whose union is $\Delta $. We can also compute the sequence of cover measures ${p}_{0}<{p}_{1}<\cdots <{p}_{\ell -1}<{p}_{\ell}=1$ for each ${C}_{i}$. (A somewhat different perspective on the consistency filtration is discussed in [51], in which the consistency filtration is shown to be both functorial and robust to perturbations, and is so in both the sheaf and the assignment.)

## 5. Results

#### 5.1. Overall Measurements

#### 5.2. Example: Bear N024

#### 5.3. Example: Minute 5.41

- There is a landmark non-zero consistency value $0<{\u03f5}_{i}\le {\u03f5}^{*}$ which does not exceed the consistency radius;
- There is a prior set of consistent faces ${\mathrm{\Gamma}}_{i-1}$;
- A new consistent face $\gamma \in \Delta $ is added so that ${\mathrm{\Gamma}}_{i}={\mathrm{\Gamma}}_{i-1}\left\{\gamma \right\}$;
- There is a corresponding vertex cover ${\mathrm{\Lambda}}_{i}$, which is a coarsening of the prior ${\mathrm{\Lambda}}_{i-1}$;
- And which has a cover measure $\overline{r}\left(\mathrm{\Lambda}\right)$.

- ${\u03f5}_{0}=0$:
- If we insist that no error be tolerated, that is that all data be consistent, then any nontrivial set in the vertex cover, produced by Theorem 2, cannot contain nontrivial faces of $\Delta $. As such, the set of consistent faces are just the singletons ${\mathrm{\Gamma}}_{0}=\{\left\{V\right\},\left\{R\right\},\left\{T\right\},\left\{G\right\}\}$, and the vertex cover is ${\mathrm{\Lambda}}_{0}=\{\{T,G\},\{V,G\}$, $\left\{R\right\}\}$ with $\overline{r}\left({\mathrm{\Lambda}}_{0}\right)=2/11$.
- ${\u03f5}_{1}=9.48$:
- If we relax our error threshold to the next landmark value, while still well below our consistency radius, the readings on V and R are considered consistent within this tolerance, so that the face $Y=\{V,R\}$ is added, yielding the new set of consistent faces$${\mathrm{\Gamma}}_{1}={\mathrm{\Gamma}}_{0}\left\{Y\right\}=\{\left\{V\right\},\left\{R\right\},\left\{T\right\},\left\{G\right\},\{V,R\}\}.$$The new vertex cover is ${\mathrm{\Lambda}}_{1}=\{\{V,R\},\{V,G\},\{G,T\}\}$, with cover measure $\overline{r}\left({\mathrm{\Lambda}}_{1}\right)=3/11$.
- ${\u03f5}_{2}=15.9$:
- Continuing on, next T and R come into consistency, adding the face $Z=\{R,T\}$, yielding$${\mathrm{\Gamma}}_{2}=\{\left\{V\right\},\left\{R\right\},\left\{T\right\},\left\{G\right\},\{V,R\},\{T,R\}\},$$$${\mathrm{\Lambda}}_{2}=\{\{V,R\},\{V,G\},\{G,T\},\{R,T\}\},\phantom{\rule{1.em}{0ex}}\overline{r}\left({\mathrm{\Lambda}}_{2}\right)=4/11.$$
- ${\u03f5}_{3}=18.42$:
- The next landmark introduces the three-way interaction $H=\{V,T,R\}$ (for notational simplicity just note that ${\mathrm{\Gamma}}_{3}={\mathrm{\Gamma}}_{2}\left\{H\right\}$). However, the vertex cover is unchanged, yielding ${\mathrm{\Lambda}}_{3}={\mathrm{\Lambda}}_{2}$ and $\overline{r}\left({\mathrm{\Lambda}}_{3}\right)=\overline{r}\left({\mathrm{\Lambda}}_{2}\right)=4/11$.
- ${\u03f5}_{4}=20.35$:
- Next T and V are reconciled, adding $X=\{V,T\}$ to ${\mathrm{\Gamma}}_{4}$. Now ${\mathrm{\Lambda}}_{4}=\{\{V,T,R\},\{V,T,G\}\}$, with $\overline{r}\left({\mathrm{\Lambda}}_{4}\right)=7/11$.
- ${\u03f5}_{5}={\u03f5}^{*}=464.5$:
- Finally we arrive at our consistency radius with the bear collar G being reconciled to R adding the face $\{B,G\}$ to ${\mathrm{\Gamma}}_{5}$. Our vertex cover is naturally now the coarsest, i.e., just the set of vertices ${\mathrm{\Lambda}}_{5}=\left\{\{V,T,R,G\}\right\}$ as a whole, with $\overline{r}\left({\mathrm{\Lambda}}_{5}\right)=1$.

## 6. Comparison Statistical Model

- Estimate the locations, with corresponding uncertainties, over time of the bear collar and the human (see Figures 18 and 19 below).
- Estimate accuracy parameters of the involved sensors based on each of the single runs of the experiment (see Table 5 below).
- Combine information across the multiple experimental runs to estimate the accuracy of the sensors (see Equation (5) below).

#### 6.1. High-Level State and Observation Equations - With Examples

#### 6.2. Estimation of Parameters

#### 6.3. Example Outputs

#### 6.4. Comparisons

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**NCSU/NCWRC urban/suburban black bear (Ursus americanus) study area, Asheville, North Carolina, USA

**Figure 4.**Consistency radii ${\u03f5}^{*}$ over time, full system. (

**Left**) Bears. (

**Right**) Dummy collars.

**Figure 5.**Consistency radii ${\u03f5}^{*}$ over time, full system. (

**Left**) Bears. (

**Right**) Dummy collars.

**Figure 10.**Time-integrated sensors streams. Variables $V,T,R$, and B as in the model, with Theta being the offset to the bear on the VHF receiver (see Equation (1)), $x.N$ being northing, and $x.E$ being easting in UTM.

**Figure 14.**(

**Left**) A raw assignment on collar N024. (

**Right**) Processed data and spread measures (variance in m, in blue).

**Figure 15.**(

**Left**) A raw assignment on collar N024. (Consistency vs. cover coarseness for example assignment.

**Figure 17.**Output for the DLM for the human position (red) plotted against the measured position (blue).

**Figure 18.**KF Estimates of locations of the bear for N024. (Estimates of location are shown in green, the GPS data is shown in blue, and the confidence intervals are shown in gray.)

**Figure 19.**KF Estimates of locations of thehuman for N024. (Estimates of location are shown in green, the GPS data is shown in blue, and the confidence intervals are shown in gray.)

$\mathit{H}=$ Human Position | $\mathit{B}=$ Bear Position | |
---|---|---|

$V=$ Vehicle GPS ⟨lat,long,ft⟩ | √ | |

$T=$ Text | √ | |

$R=$ Receiver ⟨UTM N,UTM E,m,deg,m⟩ | √ | √ |

G = GPS on Bear ⟨UTM N, UTM E,m⟩ | √ |

Vertex | Data Format | Description | Stalk |
---|---|---|---|

$G=$Bear Collar | (E, N, m) | Position and elevation of bear from collar | ${\mathbb{R}}^{3}$ |

V=Vehicle GPS | (lat, long, ft) | Position and elevation of human from vehicle | ${\mathbb{R}}^{3}$ |

$T=$Text | string | Text description of human’s location | set of strings |

$R=$Radio VHF Device | (E, N, m, m, deg) | Position and elevation of human and position of bear relative to human | ${\mathbb{R}}^{5}$ |

Distance Code | Distance (m) |
---|---|

2 | 1500 |

3 | 1000 |

4 | 750 |

5 | 500 |

6 | 375 |

$\mathit{\u03f5}$ | New Consistent Face | Vertex Cover | Measure |
---|---|---|---|

0.00 | $\left\{\right\{T,G\},\{V,G\},\{R\left\}\right\}$ | $2/11$ | |

9.48 | $Y=\{V,R\}$ | $\left\{\right\{V,R\},\{V,G\},\{G,T\left\}\right\}$ | $3/11$ |

15.90 | $Z=\{R,T\}$ | $\left\{\right\{V,R\},\{V,G\},\{G,T\},\{R,T\left\}\right\}$ | $4/11$ |

18.42 | $H=\{V,T,R\}$ | $\left\{\right\{V,R\},\{V,G\},\{G,T\},\{R,T\left\}\right\}$ | $4/11$ |

20.35 | $X=\{V,T\}$ | $\left\{\right\{V,T,R\},\{V,T,G\left\}\right\}$ | $7/11$ |

464.50 | $B=\{R,G\}$ | $\left\{\right\{V,T,R,G\left\}\right\}$ | 1 |

**Table 5.**Estimated KF parameters for collar N024. The first two rows are the standard deviation estimates for the $\omega $ parameters. The remaining rows contain standard deviation estimates for the $\u03f5$ parameters.

Parameter Standard Deviation | Value (m) |
---|---|

Bear state update | 0.008 |

Human state update | 26.524 |

Bear GPS Obs. | 16.091 |

VHF GPS Obs. | 0.032 |

Vehicle GPS Obs. | 91.434 |

Street sign Obs. | 27.597 |

VHF Obs. | 663.998 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Joslyn, C.A.; Charles, L.; DePerno, C.; Gould, N.; Nowak, K.; Praggastis, B.; Purvine, E.; Robinson, M.; Strules, J.; Whitney, P.
A Sheaf Theoretical Approach to Uncertainty Quantification of Heterogeneous Geolocation Information. *Sensors* **2020**, *20*, 3418.
https://doi.org/10.3390/s20123418

**AMA Style**

Joslyn CA, Charles L, DePerno C, Gould N, Nowak K, Praggastis B, Purvine E, Robinson M, Strules J, Whitney P.
A Sheaf Theoretical Approach to Uncertainty Quantification of Heterogeneous Geolocation Information. *Sensors*. 2020; 20(12):3418.
https://doi.org/10.3390/s20123418

**Chicago/Turabian Style**

Joslyn, Cliff A., Lauren Charles, Chris DePerno, Nicholas Gould, Kathleen Nowak, Brenda Praggastis, Emilie Purvine, Michael Robinson, Jennifer Strules, and Paul Whitney.
2020. "A Sheaf Theoretical Approach to Uncertainty Quantification of Heterogeneous Geolocation Information" *Sensors* 20, no. 12: 3418.
https://doi.org/10.3390/s20123418