On Fundamental and Future Thinking due to Being Able to Compute Resulting Distributions:
- Distributions are liberated (thus is Applied Probability): The currently known types (~ 93) of continuous distributions (except for a few very simple ones such as Uniform & Exponential) are but mathematical transformations from the Normal; they are just a micro set of discrete points in the infinite-dimensional space of distributions!! But, resulting distributions depend on causal relationships which can be anything!
Thus, for example: For Design Engineering, the distribution of any level of parameter, y, cannot be Lognormal since the hierarchical relationships between the lowest level Parameter and y do not logistically compound to a lognormal function!!!
- For futurists, perhaps we’ll all be able to think stochastically?: For any z = g(X), since computing z is the same as computing its distribution, we could think of computing stochastically as a transcendence from the Deterministic Realm! This could mean we have historically been thinking limitedly, but will have the capacity to think stochastically in the future!!
On Design Engineering (DE) of Complex Electro-Mechanical Systems
- HPD enable practitioners to properly handle Variability and reach Design decisions. Since addressing Variability comprises most of DE’s work content in many manufacturing industries (especially of complex Electro-mechanical products), this enables significantly reducing (perhaps by 2/3!!) resources, development time, and # of hardware & prototype builds!
- Our advancements in the following significantly contribute to enlightening fundamental concepts and enhancing DE practices:
- Rigorizes Latitude; clarifies distinction between Latitude & Robustness
- Defines Cpks at any level; enables Multi-Level Cpk propagation (e.g., parts to system) (Management asked for this!)
- Clarifies Analysis, Synthesis, & Optimization for handling simple to complex problems
- Finds Stochastic Operating Windows (º Feasible Design Regions) and the most Robust points within them, comprehending all internal & external noises and performance targets
- As importantly, engineers would understand SO much more technical info about their systems (local or global) with HPD definitions, work processes, and results!
For CAE, PLM, MBSE Systems and Digital Twins Initiatives
Since Designs must specify Nominals and Tolerancesfor every part feature and operating setpoint (= some of the critical parameters), both Nominals & Tolerances must be carried in those systems. Likewise, a Digital Twin must consist of both such values.
On All Other Application Areas where Variability Needs to Be Addressed
- For areas such as Material Science, Chemical Engineering, parts manufacturing industries, and ALL technical fields (except computer science) the same HPD Software & Methodology capabilities apply as for DE. Importantly, HPD makes it all easier, correct, and comprehensive.
- For Operations Research Needs, terminologies differ from DE’s, but examples such as Risk Analysis & Stochastic Optimization for simple to complex situations are handled just as for DE
- For Miscellaneous Other Areas: Class projects conducted by Univ. of Rochester students who took HPD-based courses covered a wide range of problem types of their own choosing. Examples included many that relate to their Master’s theses in other disciplines (e.g., “Arsenic in Bangladesh’s Groundwater” or “Air Induction Rate of an Anti-Swirl Stator Mixing Tank Design”) and some to their interests (e.g., to “Variability in Tuba Playing” and “Quality of Calculus Teaching”). All were complex and required FoVs (Flow of Variabilities) as the first step of their structuring a total Stochastic Model.
On Mathematics, especially Probability, Applied Mathematics, & Statistics
For specifics, see HPD/Description C. Here are some major summary points:
- For Probability, liberated the concept of “Types” of distributions and proved inadequacy of certain assumptions and many other concepts
- Liberated Applied Probability thus enabling Transcendence to the Stochastic Realm (think & work naturally with RVs & SVs)
- For Mathematics in general, enabled the concept that Deterministic Realm is but a sub-Realm of the Stochastic Realm
- Enabled unifying Applied Mathematics by providing a comprehensive capability for SMAO (Stochastic “Modeling,” Analysis, & Optimization) and addressing when are Deterministics & Stochastics appropriate.
On Course Contents in Academia
Revolutionizing Stochastics will certainly alter how we teach Probability, Applied Mathematics (see 4th bullet above), Statistics and Engineering. It will also imply many needed additions regarding Stochastic treatments in other disciplines where Variability is important (e.g., Business, the Sciences, . . .)