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: How to structure a problem and Model it Stochastically, taking into account all variables whose Variability must be addressed, is as important as having the SA & SO software.
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). Thus the ~93 currently "named" types of continuous distribution types are just discreet points in the (infinite) space of distribution types.
- Comparing results from the ** capability to another tool in the HPD's SA suite has uncovered the inadequacy in distribution-fitting with Pearson Systems, thus in 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 21st century way).
Important Implications on Applied Probability(including Statistics)
- Liberates Applied Probability (there are exclusions, e.g., Bayesian probability - which instead can be handled with our modeling technique).
- 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)
- HPD’s Methodology shows how to apply one of its tools to Stochastic Initial Value Problems (SIVPs).
- On Statistics: Renders Hypothesis Testing meaningless since a distribution can be of any type.
- 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 delta functions for distributions of the model's variables. 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.