Description C – For Mathematicians,

esp. Probabilists, Applied Mathematicians, Statisticians

HPD: The Capability for “SMAO” & Consequent Implications

What Does SMAO Stand for?

SMAO

SM, SA

, & SO

Stochastic Modeling, Analysis, & Optimization

Deterministic

MAO

with. Since

attacking real-world problems

mostly requires

Stochastics

SMAO is needed

. By providing

extremely user-friendly

software suites for

** &

, and a

novel

methodology

HPD

think & work

naturally

with ease

revolutionizing

model

analyze,

optimize.

As important

structure a problem

Model” it Stochastically

all

can handle

any

relationship

= g(

), for which the distribution for any

(

X ≡

{

}) does

not

need to be of

any

known type

! It can be

anything described by the user

or pre-computed! That it can compute “exact” output distributions means an

output

can be used as an

input

distribution for the

next level

of, or any other,

(

just as

in Deterministics), thus enabling conducting

for a

Flow of Models

(often needed for addressing a complex problem) to yield a

Flow of Distributions

(which has especially important implications on application to Systems).

Important Implications on Probability

The capability described in ** above means that HPD has liberated the concept of “Types” of distributions since now there is no limit on Types (i.e., there is an infinitude of Types, or no types at all; the ~93 currently “named” types of continuous distribution types are just discreet points in the continuum of distribution-space!)
Comparing results from the above ** capability to a secondary tool in the HPD’s SA suite has uncovered the inadequacy in distribution-fitting with, say, Pearson Systems, thus disproving previously assumed adequacy of fitting with 4 moments
Clarifies the difference between an RV & SV (the latter is needed for Stochastic Optimization & Stochastic Processes)
HPD_VA enables significantly simplifying the understanding & teaching of Probability (HPD way ≡ the revolutionized 21^st century way!)
HPD’s Methodology shows how to apply one of its tools to Stochastic Initial Value Problems (SIVPs); a macro can be programmed for it.

Important Implications on Applied Probability (including Statistics)

Liberates Applied Probability & distributions
HPD’s Sensitivity/Contribution Analysis, based on rigorous mathematics, far supersedes existing techniques (which have gross omissions/errors), thus can correctly reduce the size of Stochastic problems
HPD’s Stochastic Optimization significantly supersedes existing techniques (that for Design Engineering has severe limitations and a fundamental fallacy, and that for Operations Research is in nascent stage)
On Statistics: Renders Hypothesis Testing meaningless since a distribution can be of any of an infinitude of “types” or no “types”
On Statistics: One of HPD’s “Miscellaneous” tools (from our HPD_M suite) fits data/moments to the complete set of Pearson System distribution types (i.e., it does not default to a Normal distribution fit for some situations as existing tools do).

Important Implications on Mathematics/Applied Mathematics

On Mathematics: HPD enables rigorously considering the Deterministic Realm as a sub-realm of the Stochastic Realm. Let {DM} & {SM} stand for the classes of Deterministic & Stochastic Models, respectively. One may consider that {DM} ⊂ SM} since a DM is a degenerate SM with a delta function for its distribution. Thus, HPD enables perceiving Stochastics as unifying Probabilistics & Deterministics; a liberating concept, indeed. (Note: We use Realm because it is much “larger” than the concept of “Space” which was adequate for Deterministics, e.g., in the “space” of Reals.)
On Applied Mathematics: In the same sense as above, but more specifically pertaining to (a) HPD’s capabilities, (b) at what point in a product development process (or decision-process) are Deterministics & Stochastics more appropriately needed, and (c) handling both the Forward & the Inverse problems, HPD enables unifying Applied Mathematics.