In this paper, to further improve the coverage and performance of unmanned aerial vehicle (UAV) communication systems, we propose a reconfigurable intelligent surface (RIS)-assisted UAV scheme where ...an RIS installed on a building is used to reflect the signals transmitted from the ground source to an UAV, and the UAV is deployed as a relay to forward the decoded signals to the destination. To model the statistical distribution of the RIS-assisted ground-to-air (G2A) links, we develop a tight approximation for the probability density function (PDF) of the instantaneous signal-to-noise ratio (SNR). By using the obtained distribution, analytical expressions for the outage probability, average bit error rate (BER), and average capacity are derived. Results show that the use of RISs can effectively improve the coverage and reliability of UAV communication systems.
Consider an instance
of the Gaussian free field on a simply connected planar domain
with boundary conditions
on one boundary arc and
on the complementary arc, where
is the special constant
. We argue ...that even though
is defined only as a random distribution, and not as a function, it has a well-defined zero level line
connecting the endpoints of these arcs, and the law of
is
. We construct
in two ways: as the limit of the chordal zero contour lines of the projections of
onto certain spaces of piecewise linear functions, and as the only path-valued function on the space of distributions with a natural Markov property. We also show that, as a function of
is “local” (it does not change when
is modified away from
) and derive some general properties of local sets.
We establish existence and uniqueness for Gaussian free field flow lines started at
interior
points of a planar domain. We interpret these as rays of a random geometry with imaginary curvature and ...describe the way distinct rays intersect each other and the boundary. Previous works in this series treat rays started at
boundary
points and use Gaussian free field machinery to determine which chordal
SLE
κ
(
ρ
1
;
ρ
2
)
processes are time-reversible when
κ
<
8
. Here we extend these results to whole-plane
SLE
κ
(
ρ
)
and establish continuity and transience of these paths. In particular, we extend ordinary whole-plane SLE reversibility (established by Zhan for
κ
∈
0
,
4
) to all
κ
∈
0
,
8
. We also show that the rays of a given angle (with variable starting point) form a space-filling planar tree. Each branch is a form of
SLE
κ
for some
κ
∈
(
0
,
4
)
, and the curve that traces the tree in the natural order (hitting
x
before
y
if the branch from
x
is left of the branch from
y
) is a space-filling form of
SLE
κ
′
where
κ
′
:
=
16
/
κ
∈
(
4
,
∞
)
. By varying the boundary data we obtain, for each
κ
′
>
4
, a family of space-filling variants of
SLE
κ
′
(
ρ
)
whose time reversals belong to the same family. When
κ
′
≥
8
, ordinary
SLE
κ
′
belongs to this family, and our result shows that its time-reversal is
SLE
κ
′
(
κ
′
/
2
-
4
;
κ
′
/
2
-
4
)
. As applications of this theory, we obtain the local finiteness of
CLE
κ
′
, for
κ
′
∈
(
4
,
8
)
, and describe the laws of the boundaries of
SLE
κ
′
processes stopped at stopping times.
We establish a general existence and uniqueness of integrable adapted solutions to scalar backward stochastic differential equations with integrable parameters, where the generator g has an ...iteratedlogarithmic uniform continuity in the second unknown variable z. The result improves our previous one in 12.
In theory, two extreme forms of substances exist: the pure form and the single-molecule mixture form. The latter contains a mixture of molecules with molecularly different structures. Inspired by the ...“chemical space” concept, in this paper, I report a study of the single-molecule mixture state that combines model construction and mathematical analysis, obtaining some interesting results. These results provide theoretical evidence that the single-molecule mixture state may indeed exist in realistic synthetic or natural polymer systems.
Choosing a probability distribution to represent daily precipitation depths is important for precipitation frequency analysis, stochastic precipitation modeling and in climate trend assessments. ...Early studies identified the two-parameter gamma (G2) distribution as a suitable distribution for wet-day precipitation based on the traditional goodness-of-fit tests. Here, probability plot correlation coefficients and L-moment diagrams are used to examine distributional alternatives for the wet-day series of daily precipitation for hundreds of stations at the point and catchment scales in the United States. Importantly, both Pearson Type-III (P3) and kappa (KAP) distributions perform very well, particularly for point rainfall. Our analysis indicates that the KAP distribution best describes the distribution of wet-day precipitation at the point scale, whereas the performance of G2 and P3 distributions are comparable for wet-day precipitation at the catchment scale, with P3 generally providing the improved goodness of fit over G2. Since the G2 distribution is currently the most widely used probability density function, our findings could be considerably important, especially within the context of climate change investigations.
Elicitation is a key task for subjectivist Bayesians. Although skeptics hold that elicitation cannot (or perhaps should not) be done, in practice it brings statisticians closer to their clients and ...subject-matter expert colleagues. This article reviews the state of the art, reflecting the experience of statisticians informed by the fruits of a long line of psychological research into how people represent uncertain information cognitively and how they respond to questions about that information. In a discussion of the elicitation process, the first issue to address is what it means for an elicitation to be successful; that is, what criteria should be used. Our answer is that a successful elicitation faithfully represents the opinion of the person being elicited. It is not necessarily "true" in some objectivistic sense, and cannot be judged in that way. We see that elicitation as simply part of the process of statistical modeling. Indeed, in a hierarchical model at which point the likelihood ends and the prior begins is ambiguous. Thus the same kinds of judgment that inform statistical modeling in general also inform elicitation of prior distributions. The psychological literature suggests that people are prone to certain heuristics and biases in how they respond to situations involving uncertainty. As a result, some of the ways of asking questions about uncertain quantities are preferable to others, and appear to be more reliable. However, data are lacking on exactly how well the various methods work, because it is unclear, other than by asking using an elicitation method, just what the person believes. Consequently, one is reduced to indirect means of assessing elicitation methods. The tool chest of methods is growing. Historically, the first methods involved choosing hyperparameters using conjugate prior families, at a time when these were the only families for which posterior distributions could be computed. Modern computational methods, such as Markov chain Monte Carlo, have freed elicitation from this constraint. As a result, now both parametric and nonparametric methods are available for low-dimensional problems. High-dimensional problems are probably best thought of as lacking another hierarchical level, which has the effect of reducing the as-yet-unelicited parameter space. Special considerations apply to the elicitation of group opinions. Informal methods, such as Delphi, encourage the participants to discuss the issue in the hope of reaching consensus. Formal methods, such as weighted averages or logarithmic opinion pools, each have mathematical characteristics that are uncomfortable. Finally, there is the question of what a group opinion even means, because it is not necessarily the opinion of any participant.
We study the convergence of Langevin-Simulated Annealing type algorithms with multiplicative noise, i.e. for V : R d → R V : \mathbb {R}^d \to \mathbb {R} a potential function to minimize, we ...consider the stochastic differential equation d Y t = − σ σ ⊤ ∇ V ( Y t ) dY_t = - \sigma \sigma ^\top \nabla V(Y_t) d t + a ( t ) σ ( Y t ) d W t + a ( t ) 2 Υ ( Y t ) d t dt + a(t)\sigma (Y_t)dW_t + a(t)^2\Upsilon (Y_t)dt , where ( W t ) (W_t) is a Brownian motion, where σ : R d → M d ( R ) \sigma : \mathbb {R}^d \to \mathcal {M}_d(\mathbb {R}) is an adaptive (multiplicative) noise, where a : R + → R + a : \mathbb {R}^+ \to \mathbb {R}^+ is a function decreasing to 0 0 and where Υ \Upsilon is a correction term. This setting can be applied to optimization problems arising in Machine Learning; allowing σ \sigma to depend on the position brings faster convergence in comparison with the classical Langevin equation d Y t = − ∇ V ( Y t ) d t + σ d W t dY_t = -\nabla V(Y_t)dt + \sigma dW_t . The case where σ \sigma is a constant matrix has been extensively studied; however little attention has been paid to the general case. We prove the convergence for the L 1 L^1 -Wasserstein distance of Y t Y_t and of the associated Euler scheme Y ¯ t \bar {Y}_t to some measure ν ⋆ \nu ^\star which is supported by argmin ( V ) \operatorname {argmin}(V) and give rates of convergence to the instantaneous Gibbs measure ν a ( t ) \nu _{a(t)} of density ∝ exp ( − 2 V ( x ) / a ( t ) 2 ) \propto \exp (-2V(x)/a(t)^2) . To do so, we first consider the case where a a is a piecewise constant function. We find again the classical schedule a ( t ) = A log − 1 / 2 ( t ) a(t) = A\log ^{-1/2}(t) . We then prove the convergence for the general case by giving bounds for the Wasserstein distance to the stepwise constant case using ergodicity properties.
Making and Evaluating Point Forecasts Gneiting, Tilmann
Journal of the American Statistical Association,
06/2011, Letnik:
106, Številka:
494
Journal Article
Recenzirano
Odprti dostop
Typically, point forecasting methods are compared and assessed by means of an error measure or scoring function, with the absolute error and the squared error being key examples. The individual ...scores are averaged over forecast cases, to result in a summary measure of the predictive performance, such as the mean absolute error or the mean squared error. I demonstrate that this common practice can lead to grossly misguided inferences, unless the scoring function and the forecasting task are carefully matched. Effective point forecasting requires that the scoring function be specified ex ante, or that the forecaster receives a directive in the form of a statistical functional, such as the mean or a quantile of the predictive distribution. If the scoring function is specified ex ante, the forecaster can issue the optimal point forecast, namely, the Bayes rule. If the forecaster receives a directive in the form of a functional, it is critical that the scoring function be consistent for it, in the sense that the expected score is minimized when following the directive. A functional is elicitable if there exists a scoring function that is strictly consistent for it. Expectations, ratios of expectations and quantiles are elicitable. For example, a scoring function is consistent for the mean functional if and only if it is a Bregman function. It is consistent for a quantile if and only if it is generalized piecewise linear. Similar characterizations apply to ratios of expectations and to expectiles. Weighted scoring functions are consistent for functionals that adapt to the weighting in peculiar ways. Not all functionals are elicitable; for instance, conditional value-at-risk is not, despite its popularity in quantitative finance.