In this paper, we shall firstly illustrate why we should discuss the Aumann type set-valued Lebesgue integral of a set-valued stochastic process with respect to time t under the condition that the set-valued stochastic process takes nonempty compact subset of d-dimensional Euclidean space. After recalling some basic results about set-valued stochastic processes, we shall secondly prove that the...

Here in, an application of a new seismic inversion algorithm in one of Iran’s oilfields is described. Stochastic (geostatistical) seismic inversion, as a complementary method to deterministic inversion, is perceived as contribution combination of geostatistics and seismic inversion algorithm. This method integrates information from different data sources with different scales, as prior informat...

In this paper we propose a new method for pairwise rigid point set registration. We pay special attention to noise robustness, outlier resistance and global optimal alignment. The problem of registering two point clouds in space is converted to a minimization of a nonlinear cost function. We propose a cost function that aims to reduce the impact of noise and outliers. Its definition is based on...

Compute the Chernoff bound on P [X ≥ a] where X is a random variable that satisfies the exponential law f X (x) = λe −λx u(x). Recall, from Equation 4.6-4 on page 214 in [3], that the Chernoff bound for a continuous random variable X is given by P [X ≥ a] ≤ argmin t>0 e −at θ X (t) , (1) where θ X (t) is the moment-generating function θ X (t) E[e tX ] = ∞ −∞ e tx f X (x)dx (2) as defined by Equ...

Recall, from Theorem 8.1-2 on page 490 in [4], that a random process X(t) with autocorrelation function RXX(t1, t2) has a m.s. derivative at time t if ∂RXX(t1, t2)/∂t1∂t2 exists at t1 = t2 = t. Furthermore, we recall that the correlation function RXX(t1, t2) is related to the covariance function KXX(t1, t2) as follows. RXX(t1, t2) = KXX(t1, t2) + μX(t1)μX(t2) (1) For this problem we have a cons...

This paper studies Lebesgue integral of a fuzzy closed set-valued stochastic process with respect to the time t. Firstly, a progressively measurable fuzzy closed set-valued stochastic process is discussed and an almost everywhere problem in the former Aumann type Lebesgue integral of the level-set process is pointed out. Secondly, a new definition of the Lebesgue integral by decomposable closur...

Problem 6.25 Consider a wide sense stationary random sequence X[n] input to a linear filter with impulse response h[n] = 1/2, n = {0, 1} 0, otherwise. (1) Write the PSD of the output sequence S Y Y (ω) in terms of the PSD of the input sequence S XX (ω). (a) Show that the PSD is real-valued, even if X[n] is a complex-valued random sequence. (b) Show that if X[n] is real-valued, then S XX (ω) = S...