This book offers a much needed critical introduction to data mining and 'big data'. Supported by multiple case studies and examples, the authors provide everything needed to explore, evaluate and ...review big data concepts and techniques.
We explore the simultaneous exact controllability of mean and variance of an insurance policy by utilizing the benefit St and premium Pt as control inputs to manage the policy value tV and the ...variance 2σt of future losses. The goal is to determine whether there exist control inputs that can steer the mean and variance from a prescribed initial state at t=0 to a prescribed final state at t=T, where the initial–terminal pair of states (0V,TV) and (2σ0,2σT) represent the mean and variance of future losses at times t=0 and t=T, respectively. The mean tV and variance 2σt are governed by Thiele’s and Hattendorff’s differential equations in continuous time and recursive equations in discrete time. Our study focuses on solving the problem of exact controllability in both continuous and discrete time. We show that our result can be used to devise control inputs St,Pt in the interval 0,T so that the mean and variance partially track a specified curve tV=a(t) and 2σt=b(t), respectively, i.e., at a fine sampling of points in the time interval 0,T.
We derive a Hattendorff differential equation and a recursion governing the evolution of continuous and discrete time evolution respectively of the variance of the loss at time t random variable ...given that the state at time t is j, for a multistate Markov insurance model (denoted by 2σt(j)). We also show using matrix notation that both models can be easily adapted for use in MATLAB for numerical computations.
Abstract
The literature on diversity measures, regardless of the metric used (e.g., Gini-Simpson index, Shannon entropy) has a notable gap: not much has been done to connect these measures back to ...the shape of the original distribution, or to use them to compare the diversity of parts of a given distribution and their relationship to the diversity of the whole distribution. As such, the precise quantification of the relationship between the probability of each type
p
i
and the diversity
D
in non-uniform distributions, both among parts of a distribution as well as the whole, remains unresolved. This is particularly true for Hill numbers, despite their usefulness as ‘effective numbers’. This gap is problematic as most real-world systems (e.g., income distributions, economic complexity indices, rankings, ecological systems) have unequal distributions, varying frequencies, and comprise multiple diversity types with unknown frequencies that can change. To address this issue, we connect case-based entropy, an approach to diversity we developed, to the shape of a probability distribution; allowing us to show that the original probability distribution
g
1
, the case-based entropy curve
g
2
and the
c
{1,
k
}
versus the
c
{
1
,
k
}
*
ln
A
{
1
,
k
}
curve
g
3
, which we call the
slope of diversity
, are one-to-one (or injective), i.e., a different probability distribution
g
1
gives a different curve for
g
2
and
g
3
. Hence, a different permutation of the original probability distribution
g
1
(that leads to a different shape) will uniquely determine the graphs
g
2
and
g
3
. By proving the injective nature of our approach, we will have established a unique way to measure the degree of uniformity of parts as measured by
D
P
/
c
P
for a given part
P
of the original probability distribution, and also have shown a unique way to compute the
D
P
/
c
P
for various shapes of the original distribution and (in terms of comparison) for different curves.
We use a payment pattern of the type {1k,2k,3k,…} to generalize the standard level payment and increasing annuity to polynomial payment patterns. We derive explicit formulas for the present value of ...an n-year polynomial annuity, the present value of an m-monthly n-year polynomial annuity, and the present value of an n-year continuous polynomial annuity. We also use the idea to extend the annuities to payment patterns derived from analytic functions, as well as to payment patterns of the type {1r,2r,3r,…}, with r being an arbitrary real number. In the process, we develop possible approximations to k! and for the gamma function evaluated at real numbers.
We use the representation of a continuous time Hattendorff differential equation and Matlab to compute 2σt(j), the solution of a backwards in time differential equation that describes the evolution ...of the variance of Lt(j), the loss at time t random variable for a multi-state Markovian process, given that the state at time t is j. We demonstrate this process by solving examples of several instances of a multi-state model which a practitioner can use as a guide to solve and analyze specific multi-state models. Numerical solutions to compute the variance 2σt(j) enable practitioners and academic researchers to test and simulate various state-space scenarios, with possible transitions to and from temporary disabilities, to permanent disabilities, to and from good health, and eventually to a deceased state. The solution method presented in this paper allows researchers and practitioners to easily compute the evolution of the variance of loss without having to resort to detailed programming.
The Rao–Nakra model of a three layer sandwich beam is analyzed for exact boundary controllability. The damped and undamped cases are considered. The multiplier method is used to obtain required ...observability inequalities that imply controllability. It is shown that if the control time
T is large enough, under certain other parametric restrictions, the system is exactly controllable.
We state and prove a robustness result for the concept of almost everywhere uniform stability of an invariant attractor for a nonlinear autonomous differential equation. We also show that the concept ...of almost everywhere uniform stability rectifies the drawback of vanishing density on stable manifolds of hyperbolic equilibrium points which exists for a weaker concept of almost everywhere stability.
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