scholarly journals Effective uncertainty visualization for aftershock forecast maps

2021 ◽  
Author(s):  
Max Schneider ◽  
Michelle McDowell ◽  
Peter Guttorp ◽  
E. Ashley Steel ◽  
Nadine Fleischhut

Abstract. Earthquake models can produce aftershock forecasts, which have recently been released to lay audiences following large earthquakes. While visualization literature suggests that displaying forecast uncertainty can improve how forecast maps are used, research on uncertainty visualization is missing from earthquake science. We designed a pre-registered online experiment to test the effectiveness of three visualization techniques for displaying aftershock forecast maps and their uncertainty. These maps showed the forecasted number of aftershocks at each location for a week following a hypothetical mainshock, along with the uncertainty around each location’s forecast. Three different uncertainty visualizations were produced: (1) forecast and uncertainty maps adjacent to one another; (2) the forecast map depicted in a color scheme, with the uncertainty shown by the transparency of the color; and (3) two maps that showed the lower and upper bounds of the forecast distribution at each location. Unlike previous experiments, we compared the three uncertainty visualizations using tasks that are systematically designed to address broadly applicable and user-generated communication goals. We compared task responses between participants using uncertainty visualizations and using the forecast map shown without its uncertainty (the current practice). Participants completed two map-reading tasks that targeted several dimensions of the readability of uncertainty visualizations. Participants then performed a comparative judgment task, which demonstrated whether a visualization was successful in reaching two key communication goals: indicating where many aftershocks and no aftershocks are likely (sure bets) and where the forecast is low but the uncertainty is high enough to imply potential risk (surprises). All visualizations performed equally well in the goal of communicating sure bet situations. But the visualization with lower and upper bounds was substantially better than the other designs at communicating surprises. These results have implications for the communication of forecast uncertainty both within and beyond earthquake science.

2021 ◽  
Author(s):  
Max Schneider ◽  
Michelle McDowell ◽  
Peter Guttorp ◽  
E. Ashley Steel ◽  
Nadine Fleischhut

<p>Seismicity rate estimates and the earthquake forecasts they yield vary spatially and are usually represented as heat maps. While visualization literature suggests that displaying forecast uncertainty can improve how forecast maps are used, research on uncertainty visualization (UV) is missing from earthquake science. We present a pre-registered online experiment to test the effectiveness of three UV techniques for displaying aftershock forecasts. These maps show the expected number of aftershocks at each location for a week following a hypothetical mainshock, and we develop maps of the uncertainty around each location’s forecast. Human participants complete experimental tasks using the aftershock forecast displayed with its uncertainty. Three different UVs are producted: (1) forecast and uncertainty maps adjacent to one another; (2) the forecast map depicted in a color scheme, with the uncertainty shown by the transparency of the color; (3) two maps that show the lower and upper bound of the forecast distriubiton at each location. We compare task performance using UVs and using the forecast map shown without its uncertainty (the current practice). Subjects complete two map-reading tasks that target several dimensions of the readability of the three UVs. They then perform a comparative prediction task, which demonstrates whether a UV is successful in reaching two key communication goals: indicating where an aftershock and no aftershocks are likely (“sure bets’’) and where the forecast is low but the uncertainty is high enough to imply potential risk (“potential surprises’’). All UVs perform equally well in the goal of communicating “sure bet’’ situations. But the UV with lower and upper bounds is significantly better than the other UVs at communicating “potential surprises.” We discuss the implications of these results for communication of forecast uncertainty within and beyond earthquake science.</p>


1997 ◽  
Vol 84 (1) ◽  
pp. 176-178
Author(s):  
Frank O'Brien

The author's population density index ( PDI) model is extended to three-dimensional distributions. A derived formula is presented that allows for the calculation of the lower and upper bounds of density in three-dimensional space for any finite lattice.


Author(s):  
S. Yahya Mohamed ◽  
A. Mohamed Ali

In this paper, the notion of energy extended to spherical fuzzy graph. The adjacency matrix of a spherical fuzzy graph is defined and we compute the energy of a spherical fuzzy graph as the sum of absolute values of eigenvalues of the adjacency matrix of the spherical fuzzy graph. Also, the lower and upper bounds for the energy of spherical fuzzy graphs are obtained.


Entropy ◽  
2021 ◽  
Vol 23 (8) ◽  
pp. 940
Author(s):  
Zijing Wang ◽  
Mihai-Alin Badiu ◽  
Justin P. Coon

The age of information (AoI) has been widely used to quantify the information freshness in real-time status update systems. As the AoI is independent of the inherent property of the source data and the context, we introduce a mutual information-based value of information (VoI) framework for hidden Markov models. In this paper, we investigate the VoI and its relationship to the AoI for a noisy Ornstein–Uhlenbeck (OU) process. We explore the effects of correlation and noise on their relationship, and find logarithmic, exponential and linear dependencies between the two in three different regimes. This gives the formal justification for the selection of non-linear AoI functions previously reported in other works. Moreover, we study the statistical properties of the VoI in the example of a queue model, deriving its distribution functions and moments. The lower and upper bounds of the average VoI are also analysed, which can be used for the design and optimisation of freshness-aware networks. Numerical results are presented and further show that, compared with the traditional linear age and some basic non-linear age functions, the proposed VoI framework is more general and suitable for various contexts.


2021 ◽  
Vol 37 (3) ◽  
pp. 919-932
Author(s):  
Byeong Moon Kim ◽  
Byung Chul Song ◽  
Woonjae Hwang

2021 ◽  
Vol 103 (5) ◽  
Author(s):  
Li Peng ◽  
Wen-Bin He ◽  
Stefano Chesi ◽  
Hai-Qing Lin ◽  
Xi-Wen Guan

2021 ◽  
Vol 33 (4) ◽  
pp. 973-986
Author(s):  
Young Jae Sim ◽  
Paweł Zaprawa

Abstract In recent years, the problem of estimating Hankel determinants has attracted the attention of many mathematicians. Their research have been focused mainly on deriving the bounds of H 2 , 2 {H_{2,2}} or H 3 , 1 {H_{3,1}} over different subclasses of 𝒮 {\mathcal{S}} . Only in a few papers third Hankel determinants for non-univalent functions were considered. In this paper, we consider two classes of analytic functions with real coefficients. The first one is the class 𝒯 {\mathcal{T}} of typically real functions. The second object of our interest is 𝒦 ℝ ⁢ ( i ) {\mathcal{K}_{\mathbb{R}}(i)} , the class of functions with real coefficients which are convex in the direction of the imaginary axis. In both classes, we find lower and upper bounds of the third Hankel determinant. The results are sharp.


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