Description C – For Mathematicians,

esp. Probabilists, Applied Mathematicians, Statisticians

HPD: The Capability for “SMAO” & Consequent Implications

What Does SMAO Stand for?

    SMAO (or SM, SA, & SO) = Stochastic Modeling, Analysis, & Optimization. This is in contrast to Deterministic MAO which most in the technical community are familiar with. Since attacking real-world problems mostly requires Stochastics, SMAO is needed. By providing extremely user-friendly software suites for SA** & SO, and a novel methodology for SM, HPD enables one to think & work naturally with Random Variables (RVs) & Stochastic Variables (SVs) with ease, in essence, enabling “Transcendence from the Deterministic to the Stochastic Realm.” This is all due to HPD’s revolutionizing how we model, analyze, & optimize. Note: As important as having the SA & SO software is how to structure a problem and “Model” it Stochastically, taking into account all variables whose Variability must be addressed.
    ** HPD’s SA can handle any relationship z = g(X), for which the distribution for any xi (X ≡ {xi}) 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, SA (just as in Deterministics), thus enabling conducting SA 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 21st 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.