A new statistical modeling approach developed by Prof. Haim Shore
Response Modeling Methodology (RMM) is now on Wikipedia!! RMM is a general platform for modeling monotone convex relationships, which I have been developing over the last fifteen years (as of May, 2017), applying it to various scientific and engineering disciplines.
A new entry about Response Modeling Methodology (RMM) has now been added to Wikipedia, with a comprehensive literature review::
Response Modeling Methodology – Haim Shore (Wikipedia).
Since studying as an undergraduate student at the Technion (Israel Institute of Technology) and learning, for the first time in my life, that randomness too has its own laws (in the form of statistical distributions, amongst others), I have become extremely appreciative of the ingenuity of the concept of statistical distribution. The sheer combining of randomness with laws, formulated in the language of mathematics not unlike any other branch of the exact sciences, fascinated me considerably, young man that I was at the time.
That admiration has all since evaporated as I have become increasingly aware of the gigantic number of statistical distributions, defined and used within the science of statistics to describe random behavior, either of real-world phenomena or of sample-statistics embedded in statistical-analysis procedures (like hypothesis testing). I realized that unlike with modern-day physics, engaged to this day in the unification of the basic forces of nature, the science of statistics has failed to carry out similar attempts at unification. What the latter implies for me is derivation of a single universal distribution, relative to which all current distributions might be regarded as statistically insignificant random deviations (not unlike a sample average is a random deviation from the population mean). Such unification has never materialized, or even been attempted or debated, within the science of statistics.
Personally, I attribute this failure at unification to the fact that current foundations of statistics, with its basic concepts like probability function, probability density function (pdf) or distribution function (often denoted cumulative density function, or CDF), have been established back in the eighteenth century to derive various early-day distributions. These foundations have not been challenged ever since. Some well-known mathematicians of the time, like Jacob and Daniel Bernoulli, Abraham de Moivre, Carl Friedrich Gauss, Pierre-Simon Laplace and Joseph Louis Lagrange have all used those basic terms of statistics to derive specific distributions. However, the basic tenets underlying formation of those mathematical models of random variation have not been challenged to this day. Central amongst these tenets is the belief that random phenomena, with their associated properly-defined random variables, have each its own specific distribution. That tenet remained intact and unchallenged to this day. Consequently, no serious attempt at unification has ever become the core objective of the science of statistics. Furthermore, no discussion of how to proceed in the pursuit of the “universal distribution” has ever been conducted.
My sentiment about the feasibility of revolutionizing the concept of statistical distribution and deriving a universal distribution, relative to which all current distributions may be regarded as random deviations, has changed dramatically with the introduction of a new non-linear modeling approach, denoted Response Modeling Methodology, RMM). I have developed RMM back in the closing years of the previous century (Shore, 2005, and references therein), and only some years later I realized that the “Continuous Monotone Convexity (CMC)” property, part and parcel of RMM, could serve to derive the universal distribution, in the sense described in the previous paragraph. (Read about the CMC property in another post in this blog).
The results of the new realization are two articles (Shore 2015, 2017), one of which has already been published and the second currently under review (see references here).
In a workshop about recent advances in the application of statistical methods to quality engineering and management, conducted in March of 2013 by the Open University of Israel, I have delivered a presentation (Hebrew) about SPC-based modeling and monitoring of ecological processes. The lecture was based on my recently published article:
Shore, H. (2013), Modeling and Monitoring Ecological Systems—A Statistical Process Control Approach. Qual. Reliab. Engng. Int.. doi: 10.1002/qre.1544
A link to the presentation is given below:
Haim Shore_SPC monitoring of ecological processes_Open University_March 2013
On March, 6, 2014, my research team delivered a seminar (in English) at the Soroka Medical Center, describing our ongoing research project in modeling and monitoring fetal growth. The research team includes, besides myself, Dr. Diamanta Benson-Karhi (from the Open University) and Professor Asher Bashiri (from Soroka and Ben-Gurion University).
Concurrently, our graduate student, Mrs. Maya Malamud, had concluded her thesis and delivered a presentation (in Hebrew) during her final exam session.
Both presentations are now accessible here (in PDF format):
Haim Shore_SPC-based Monitoring of Fetal Growth_Presentation (English)_March 2014
Maya Malamud_Fetal Growth Study_Final Exam Presentation (Hebrew)_March 2014
Article published in Quality Ebgineering:
SPC-based Monitoring of Fetal Growth
What is the linkage between the science of statistics and “Stamp Collecting”? More than you can imagine.. This blog entry (with the linked article and PP presentation) was originally posted, for a restricted time period, on the Community Blog of the American Statistical Association (ASA), where the linked items were visible to members only. The blog entry is now displayed, with the linked items, visible to all.
This is the fourth and last message in this series about the consequences to statistical modeling of the continuous monotone convexity (CMC) property. The new message discusses implications of the CMC property to modeling random variation.
As a departure point for this discussion, some historic perspective about the development of the principle of unification in human perception of nature can be useful.
Our ancestors believed in a multiplicity of gods. All phenomena of nature had their particular gods and various manifestations of same phenomenon were indeed different displays of wishes, desires and emotions of the relevant god. Thus, Prometheus was a deity who gave fire to the human race and for that was punished by Zeus, the king of the gods; Poseidon was the god of the seas; and Eros was the god of desire and attraction.
This convenient “explanation” for the diversity of nature phenomena had all but disappeared with the advent of monotheism. Under the “umbrella” of a single god, ancient gods were “deleted”, to be replaced by a “unified” and “unifying” almighty god, the source of all nature phenomena.
And the three major monotheistic religions had been born.
The “concept” of unification, however, did not stop there. It was migrated to science, where pioneering giants of modern scientific thinking observed diverse phenomena of nature and had attempted to unify them into an all-encompassing mathematics-based theory, from which the separate phenomena could be deduced as special cases. Some of the most well-known representatives of this mammoth shift in human thinking, in those early stages of modern science, were Copernicus (1473-1543), Johannes Kepler (1571-1630), Galileo Galilei (1564-1642) and Isaac Newton (1642-1727).
In particular, the science of physics had been at the forefront of these early attempts to pursue the basic concept of unity in the realm of science. Ernest Rutherford (1871–1937), known as the father of nuclear physics and the discoverer of the proton (in 1919), made the following observation at the time:
The assertion, quoted in Kaku (1994, p. 131), intended to convey a general sentiment that the drive to converge the five fundamental forces of nature into a unifying theory, nowadays a central theme of modern physics, represented science at its best. Furthermore, this is the only correct approach to the scientific investigation of nature. By contrast, at least until recently, most other scientific disciplines have engaged in taxonomy (“bug collecting” or “stamp collecting”). With “stamp collecting” the scientific inquiry is restricted to the discovery and classification of the “objects of enquiry”, particular to that science. However, this never culminates, as in physics, in a unifying theory from which all these objects may be deductively derived as “special cases”.
Is statistics a science of “stamp collecting”?
Observing the abundance of statistical distributions, identified to-date, an unavoidable conclusion is that statistics is indeed a science engaged in “stamp collecting”. Furthermore, serious attempts at unification (partial, at least) are rarely reported in the literature.
In a recent article (Shore, 2015), I have attempted a new paradigm for modeling random variation. The new paradigm, so I believe, may constitute an initial effort to unite all distributions under a unified “umbrella distribution”. In the new paradigm, the “Continuous Monotone Convexity (CMC)” property plays a central role in deriving a general expression to the normal-based quantile function of a generic random variable (assuming a single mode and a non-mixture distribution). Employing numeric fitting to current distributions, the new model has been shown to deliver accurate representation to scores of differently-shaped distributions (including some suggested by anonymous reviewers). Furthermore, negligible deviations from the fitted general model may be attributed to the natural imperfection of the fitting procedure or being perceived as realization of random variation around the fitted general model, not unlike a sample average is a random deviation from the population mean.
In a more recent effort (Shore, 2017), a new paradigm for modeling random variation is introduced and validated via certain predictions about known “statistical facts” (like the Central Limit Theorem), shown to be empirically true, and via distribution fitting, via 5-moment matching procedure, to a sample of known distributions.
These topics and others are addressed extensively in the afore-cited new article. It is my judgment that at present the CMC property constitutes the only possible avenue for achieving in statistics (as in most other modern branches of science) unification of the “objects of enquiry”, as these relate to modeling random variation.
In the affiliated Article #4 , I introduce in a more comprehensive fashion (yet minimally technical) an outline of the new paradigm and elaborate on how the CMC property is employed to arrive at a “general model of random variation”. A related PowerPoint presentation, delivered last summer at a conference in Michigan, is also displayed.
Haim Shore_4_ASA_PP Presentation_Feb 2014
References
[1] Kaku M (1994). Hyperspace- A Scientific Odyssey Through Parallel Universes, Time Warps and the Tenth Dimension. Book. Oxford University Press Inc., NY.
[3] Shore, H. (2017). The Fundamental Process by which Random Variation is Generated. Under review.
This post explains the central role of Continuous Monotone Convexity (CMC) in Response Modeling Methodology (RMM).
In earlier blog entries, the unique effectiveness of the Box-Cox transformation (BCT) was addressed. I concluded that the BCT effectiveness could probably be attributed to the Continuous Monotone Convexity (CMC) property, unique to the inverse BCT (IBCT). Rather than requiring the analyst to specify a model in advance (prior to analysis), the CMC property allows the data, via parameter estimation, determine the final form of the model (linear, power or exponential). This would most likely lead to better fit of the—estimated model, as cumulative reported experience with implementation of IBCT (or BCT) clearly attest to.
In the most recent blog entry in this series, I have introduced the “Ladder of Monotone Convex Functions”, and have demonstrated that IBCT delivers only the first three “steps” of the Ladder. Furthermore, IBCT can be extended so that a single general model can represent all monotone convex functions belonging to the Ladder. This transforms monotone convexity into a continuous spectrum so that the discrete “steps” of the Ladder (the separate models) become mere points on that spectrum.
In this third entry on the subject (and Article #3, linked below), I introduce in a more comprehensive fashion (yet minimally technical) the general model from which all the Ladder functions can be derived as special cases. This model was initially conceived in the last years of the previous century (Shore, 2005, and references therein) and had since been developed into a comprehensive modeling approach, denoted Response Modeling Methodology (RMM). In the affiliated article, an axiomatic derivation of RMM basic model is outlined and specific adaptations of RMM to model systematic variation and to model random variation are addressed. Published evidence for the capability of RMM to replace current published models, previously derived within various scientific and engineering disciplines as either theoretical, empirical or semi-empirical models, is reviewed. Disciplines surveyed include chemical engineering, software quality engineering, process capability analysis, ecology and ultra-sound-based fetal-growth modeling (based on cross-sectional data).
This blog entry (with the linked article given below) was originally posted on the site of the American Statistical Association (ASA), where the linked article was visible to members only.
In a previous post in this series, I have discussed reasons for the effectiveness of the Box-Cox (BC) transformation, particularly when applied to a response variable within linear regression analysis. The final conclusion was that this effectiveness could probably be attributed to the “Continuous Monotone Convexity (CMC)” property, owned by the inverse BC transformation. It was emphasized that the latter, comprising the three most fundamental monotone convex functions, the “linear-power-exponential” trio, delivers only partial representation to a whole host of models of monotone convex relationships, which can be arranged in a hierarchy of monotone convexity. This hierarchy had been denoted the “Ladder of Monotone Convex Functions.”
In this post (and Article #2, linked below), I address in more detail the nature of the CMC property. I specify models included in the Ladder, and show how one can deliver, via a single model, representation to all models belonging to the Ladder (analogously with the inverse BC transformation, a special case of that model). Furthermore, I point to published evidence demonstrating that models of the Ladder may often substitute, with negligible loss in accuracy, published models of monotone convexity, which had been derived from theoretical discipline-specific considerations.
This blog entry (with the linked article given below) was originally posted on the site of the American Statistical Association (ASA), where the linked article was visible to members only.
Haim Shore Blog