Links to related topics:

### Case StudiesA collection of case studies in Root Cause Analysis, not only by the Author. They are mainly referred to the Food and Beverage Machinery, Devices and Processes there in. Also included a few from Nuclear Power Production technology, Astronautics and Biology, of special technical interest because referred to Industrial Design, Instrumentation and Machinery.…

### Contamination of bottled beveragesThere are scores of Food and Beverage Safety violations terminated in the: hospitals before,tribunals later, worldwide Press in the meantime. The Food and Beverage Contaminations deriving by such violations, a…

### Product LiabilityIn the Food and Beverage Bottling Lines of all of the World, the majority of the packaged beers and part of the mineral waters are notoriously glass bottled. Glass really means Quality when compared to PET but, …it introduces a risk non existing with plastic packagings. …

Introduction

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Root Cause Analysis (RCA) is much more than a technique of investigation of Industrial Machinery's inefficiencies or malfunctions. Part of it are the pillars of the Scientific Method used to encounter the causes for an observed effect. Yet over three centuries ago scientists were using some of the tools of Root Cause Analysis, like the search for changes in the properties of a system, after following one of the conditions. Search for changes when, as an example, trying to figure out the reasons why:

- electromagnetic induction is related to determinate effects, but not others,
- objects follow a curved parabolic trajectory,
- sky is blue under some conditions, but not others.

From a modern physical point of view, to root cause analyse the problems affecting a system, i.e. an Industrial Machinery, its subsystems and components, means to determine *who* or *what*, *when*, *where* and in *what extent* established the:

- Topology of the system,
- Information Flow between its sub-systems,
- Information Flow between the system and its Environment.

Reason for this are the categories *what, who, when, where, what extent used to *label objects, times, spaces and amounts. They have* *been* *proved derivate concepts. As an example: space and time. Since decades physics’ M-Theory does no more consider any them “fundamentals”. Fundamental is only the quantum field. Space and time just the way we name some of the (apparent) properties of that “fundamental”, filtered by our limited natural sensorial and computational capabilities.

Why to Look for Changes

“It looks to me that we are venting something,” came Lovell's report from Apollo 13.

*“We're venting something out into space. It's a gas of some sort.”*

“Everyone started looking at data” remembered Wendt,

*“and saying Where could it be? How could it be? and What can we do?”*

Some glorious and famous texts of Root Cause Analysis and Problem Solving published over twenty years ago, set its start sixty years ago. A start related to the improved solution of problems of Astronautics or minimisation of the rejects during production of Motorola radios. Texts mainly focused on the strategies to identify a Root Cause’s pattern. The most ancient method to identify a Root Cause for an Effect, looks for *changes*. Searching for *what/who, when, where *and in *what extent* changed in a system. Limiting the domain of this presentation to our own field, Industrial Machinery and Instrumentation, we’ll be referring to the *changes* of the properties happened to an “object”.

Changes of:

*physical properties*, like dimensions, temperatures, electric currents, etc. represented by mean of numbers part of the set of the Complex numbers ℂ (i.e., the vector potential of a high frequency alternate voltage applied to an electronic component), including Real numbers ℝ or symbols;*chemical properties*, like the concentration of reagents in a solution;*non chemical-physics statuses*, and qualities, represented by Real numbers, symbols (Yes, No, 0, 1, Up, Down, High, Low, True, False, etc.) like the mechanical or production efficiency of a Filler or Blower Machine.

*“What really a change encompasses is information, measured in bits”*

What really a change encompasses is information, measured in bits. No change in a physical property means the minimum possible information corresponding to that property. Then, a *root cause analyst looks for changes simply because that is the way the Informations are encoded*.

Objects

We’ll proceed on a step-by-step basis, not giving anything for granted. We front immediately the problem to define what is an “object”. The most modern and general answer (see figure below) is a mathematical one. It conceives an object like:

- what exists in a space, named
*state space,*encompassing all possible object’s states, - remaining itself after all possible transformations (see figure below).

** **What is an “object”? Today's only non-circular logic answer describes the object as an entity existing in the state space *M***. ***M*** includes all possible states of an object. The rules ***f**α* **and** ** ***f**β* **assign logical predicates to the representations ***fα (M)* **and** fβ (M) **for the state** x. **The transformation ***fβ . f -1α ***is a map pointing to the “objective” state*** x. *It is the invariance under transformations among all representations which confirming that *x* is an “object”. Invariance implying that *x* abstracts, say remains itself, by all possible transformations* ***(image credit S. Y. Auyang, 1995)**

Objects’ Distinguishability Condition

*“The interaction of objects, together with the process of making distinctions, results in the transfer of a quantity named *Information”

The objects whose properties are studied by the methods of Root Cause Analysis are always those which can be distinguished. At different times, the objects have the capacity to distinguish themselves from other objects and from themselves. The interaction of objects, together with the process of making distinctions, results in the transfer of a quantity named *Information. * Some objects are capable of distinguishing themselves in more ways than others. These objects have a greater Information capacity. *Objects’ distinguishability condition* is* *so obvious, to result frequently overlooked. May a rigorous application of Root Cause Analysis’ methods assure us success to understand the causes for a change intervened in a system of objects, like a Machine ? The answer is negative. This should be the case if we could know all the data inherent to that system of objects. To know all data means to have all information about each one of the objects. In the figure below we are indicating as a black colour flow the impressive amount of interactions and related flow of information relating an object to an elementary example of *inspection, *an* *Information Gathering and Using System (IGUS). Black colour flow continous, to represent the fact that the amount of informations and interactions exchanged between an object and an inspection (IGUS) is much more dense than what later really apprehended by the inspection. The detection phase where they start the interactions between the object and the inspection, implies a filtering reducing the information an inspection may derive by an object. The figure below let us understand that Root Cause Analysis has limits in its possibility to describe the reality of the facts which caused an actual .

One and the same same macroscopic object is always composed of particles, directly entering in the signal detection process relevant for the Food and Beverage Automated Quality Controls and machinery. Some of the families of particles, i.e. the electrons, are undistiguishable. *Distinguishability* is defined as the extrinsic quality of an object which permits us to say that:

- it is one specific entity, and not another;

or:

- that the object is in one particular
*state*, and not another.

The subject, at a first sight purely theoretic or philosophical, is in the reality extremely relevant when root cause analysing the causes for a malfunction in an Electronic Inspector, industrial equipment or machinery, in a Food and Beverage Packaging Line. And results specially relevant for the category of Inspectors operating with sensors *in-the-Machine*. Those where the “Machine” is a Labeller, Filler, Closer, Capper, Seamer or Blowformer. For, if we cannot differentiate between two or more object (i.e., containers) or states of the objects (i.e., *where* they are in a Shifting-Register, *what* defects they have), it is inconceivable that the object can later produce on its own any reduction of such indeterminacy. More directly, if one cannot in principle identify properties which determine an object’s *state,* then the object is incapable of carrying information.

State Space

Arena of all Dynamical Systems (e.g. Machinery)

The 4-dimensional space-time we perceive and conceive by mean of our 100 billion of neurons, frequently named *physical space,* is not the *State Space,* rather a subset. All *objects* we perceive directly, show us a mere instantaneous and constantly changing projection of their complex existence in an higher dimensional space (> 9 dimensions). Normally, we relate ourselves with systems of objects. A scholar definition summarises the *state* *of a system* like: *“Present value of some of the inner elements of the system, that change separately, but not completely unrelated, with respect to the output of the system”*. In other words, the state of a system is an explicit account of the values of the system inner components. Then, what is *State Space*? To start to grab this concept, think to the drum in a gun used as Russian-roulette. Russian-roulette emulates quite well in the collective imaginary all of the named “Montecarlo-devices”. *Montecarlo-devices* are man-made systems trying to reproduce *Nature’s constant launch of a dice over all of its existing faces.* A constantly happening process, confirmed by theory and experiments, also if happening in spaces, dimensions and energy levels we cannot directly perceive with our eyes. When the drum is charged with *n* bullets the gun is riskier than when:

- charged with
*n - 1*bullets, - the drum has
*m**- 1*slots*,*rather than*m.*

**Imagine the drum in a gun used to play Russian-roulette. The assessment of its ****intuitively different Risks when charged with ***n*** or ***n - 1*** bullets, is strictly due to the different ***State Spaces*** occupied by the differently charged drum. The intuitively different Risk's assessment, tells openly we are constantly using ***State Space***'s idea, also if many are not aware of this**

The intuitively different assessment for the risk involved when shooting one or the other gun, is strictly due to the different measures of the state space in each particular case. Through all this, the bullets, *n* and the slots *m* are *conditions *which have to be present in the Past, to induce a risk to mortally shot oneselves in their Future. As seen from the point of view of our own macroscopic geometric sizes, time intervals, masses and energies, it exists a definite time-orientation. Time-arrow sequencing all processes as:

cause ⇒ effect

Effect in the Future that we’ll be representing below as a vertex (an inverted “ V ”). In this specific example regarding a Russian-roulette gun, the Effect is better named as a plurality of Effects. A spectrum of different Futures, each one associated to a certain measure of negative outcome. What in layman language is named *Risk*. Shown in the following the vertices formed in such a way in the state space representing a gun differently configured with n = (1, 2) of the complete set existing for:

*m*(1, 2, … 6) slots in the drum;*n*(1, 2, … 6) bullets;

** ****When Root Cause Analysis (RCA) asks what did not happened it proceeds toward the truth of what happened, which changed the Past to the Present status. The deepest **Analysis are those conceived in the

**, where all of the possible states have full existence, or in its modern variant, the**

*State Space***Hilbert Space**. Thus, allowing to create efficient mathematical models where different sets of Causes and Conditions determines Effects of different extent. Around fifteen years ago, they started to be represented in the Hilbert Space the properties of objects as macroscopic as the drum visible above and also much bigger. Since then, it is no more a space limited to the atomic and subatomic scales

**The State Space of an Automatic Teller Machine (ATM) (image credit M. Hammer****, ****2014) **

risk = linear function (n, m)

∧

bullets, *n* slots, *m*

100 % 50 % 33 % 25 % 20 % 16 %

∧ ∧ ∧ ∧ ∧ ∧

1 1 1 2 1 3 1 4 1 5 1 6

100 % 100 % 67 % 50 % 40 % 33 %

∧ ∧ ∧ ∧ ∧ ∧

2 1 2 2 2 3 2 4 2 5 2 6

** . . . . .**

*“A complete answer to the question set above: “*what is *state space* *?” (…) requires us thinking in a space wider than that commonly encountered in the most technological applications. *

Such a point of view makes sense of the fact that yet seven years ago the Lloyd’s (at London, UK) were financing scientific conferences rejoining physicists. Out of the strictly scientific circles, are the Insurers and scholars of Actuarial Science and of the Financial Sciences, those who has truly understood that the physicists' *State Space* (and, its subset, the Hilbert space) encompasses all possible *unfavourable** outcomes* of a given present *state*, attributing to each one of them a measure named **risk**. A complete answer to the question set above: “what is *state space* ?”, involving causes and effects, energy, material bodies’ masses, space and time, is not intuitive. Less intuitive than that purely mathematical suggested centuries ago by the combinatorial calculations of Probability. It requires us thinking in a space much wider than that commonly encountered in the technological applications.

A space so wide to have room for all the possible *states *(temperatures, inclinations, introductions, constructive or destructive superposition's amplitude of the multitude of their atomic substructures, etc.)* *in which the bullets can occupy the drum. Thus including in some cases an infinity of *states* which are uncomprehensible and simply impossible to imagine for a human being. States which however have a precise mathematical quantitative definition. An example ? States like: “occupation at a certain time of a definite slot, by a well definite bullet lying ...1 meter far from the drum”. All of the Industrial Machinery's *systems* used for the Food and Beverage Packaging, are *dynamical systems*. *Dynamical systems* are all those evolving with Time. The term “evolve” is a synonimous of “change” and “vary”. The *variables* completely describing the state of a *dynamical system* are named the *state variables*. And the set of all their possible values is the *state space *whose* *points* *can be:

- continous, i.e., the real
*ℝ*or complex numbers*ℂ*; - discrete, just isolated points, i.e. the natural numbers
*ℕ*; - infinite- or finite-dimensional.

** ** **The set of all numbers includes the subsets of the ****> 3- dimensional**** Hypercomplex numbers ***ℍ***, including the 2- dimensional C****omplex numbers set ***ℂ*, **of special interest for all ***dynamical systems,*** included the Industrial Machinery object of Root Cause Analysis. **** Complex numbers ***ℂ*** include the familiar ****Real numbers ***ℝ*** (image credit stratocaster47, 2014)**

State Space as Events’ arena

*“In the world of the extremely complex dynamical systems, to know precisely what it happened in the Past, does not allow to know the Future. *

*This is the rationale for Root Cause Analysis. *

*It is not sufficient to create some Alexandria-like bibliotheque of the Past machinery's or devices' failures, to infer the Root Cause for a Present fault”*

In the next sections we’ll introduce other spaces, particularly relevant for the technological application, namely *sample, configuration and phase space:*

*Sample space*is where the observations are made.*Configuration**p**hase**spaces*are where the*dynamical systems*are studied;

all of them subsets of the infinitely wider *State Space. *Their comprehension opens the way to understand the deeper reasons why the *State Space* is closely focused on the Machinery's, Devices’ and Processes’ inefficiencies or malfunctions studied by Root Cause Analysis. Aren’t Machinery, Devices and Processes themselves *dynamical systems* of Mechanics, Industrial Chemistry, Thermodynamics, Electronics or Electromechanical Automation ? *State space* is the most general thinkable evolutionary arena. This is true also for sectors of the human activities completely different than the dynamical systems like Machinery and Devices. Two examples below:

Example 1

State Space and Investment Theory

As an example of this intrinsic universality, consider a field of application for Root Cause Analysis’ methods completely different than Machinery or Devices. Namely, the important part of the Financial Science named Investment Theory. Here, to decide if a certain investment is worth the **risk** associated, means much more than recording and analysing the past historical series. A banal integration of Past economic results carries to Future huge losses for Present investments.

Example 2

State Space and Positional Astronomy

Another example, from an (apparently) different discipline, in the ancient Aztec civilization. Their predictions of the Future events (e.g., celestial bodies' rise and set times along all of the year), relevant for their mainly agricultural economy, was based on the empiric registration of extremely long historical series of Past positions for the same celestial bodies. Not having developed a 3-dimensional Geometry, Trigonometry and Analystic Geometry, they were compelled to massive and pure integration of observational values. Imagine an integration of data, made without Gauss’ integral nor Linear Regression. But, the mere knowledge of the Past (in the reality, just some of the many branches of the entire tree-like evolution in the *State Space*), does not encompasses the Future states. And because of this reason their predictions never reached that precision reached yet one thousand five hundred years before by their Greek counterparts.

Sample Space

*“What macroscopically looks like *“tricked dices” is the effect of the reversibility of all processes happening at the Events’ spatial and temporal scales”

As we have yet introduced, *State Space* is the widest thinkable existing. It includes all others and, between these, a couple relevant for the Industrial measurements. *Random Variable* is a recurrent word when measuring illumination, weight, induction, power, voltage, current intensity, distance, time, etc. Of these physical properties of an object, the information we deduce is in the form of constantly floating numbers. A *Random Variable* X: Ω → E is a measurable function from the set of the possible outcomes Ω to some set E. With reference to the figure below, all *random variables* are points in the *sample Space. Sample space *may be thinked as one of the possible projections of the *state space. *

*Random variables* are projections in the *Sample SpaceState Space* of objects’ properties having their full existence in the . The function manifests the relation existing between these two spaces. On side are shown in red and blue color two different functions. As an example: red could represent the transfer function of the Energy of a X-ray photon and blue the transfer function of the Polarisation of the same photon. Distinct physical properties of a single object, whose values appear like two random variables when measured by mean of two measurement devices (e.g., a X-ray phototransistor and a X-ray polarimeter) in their own Sample Spaces. Part of the individual points in the *Sample Space* are simultaneously related to several points in the State Space (image credit Dong, Hong Kong University, **2010)**

The word *projection* implying a certain function* *necessary to transfer the amounts in the *State Space* to the *Sample Space. *Our present ideas are based over the outcomes of actual measurements and their comparisons with precedently recorded data, then: *what is the maximum resolving power to focus different statuses (directions), say to discriminate problems, selecting just the one we really are in front ?* An easier answer, after considering that different problems differ for *at least* 1 bit. Considering the amount of variables represented by Real numbers and the fact we measure just an outcome of the many, it is comprehensible whoever's difficulty to focus what really is the problem.

Then, it is definetely better to start with a visual idea of what is an *“expected result”*.

** An example of measurement in two Sample Spaces of the properties (Energy, Polarization) of an object. Yellow and gray colour are a couple of X-ray Silicon Imaging Detectors. ****A Compton scattering Event in the upper detector implies a first measurement of Energy. By the measured value, it is determined the maximum scattering angle and thus drawn a hollow cone in which a coincidence Event can occur. If a photon absorbed in the lower detector corresponds to a scattering in the upper both in time and space, its polarization can be found. Two measurements of Energy, both random variables, result in X-ray's total Energy and Polarization angle (image credit Muleri, et al., 2012)**

Resulys Expected in the Sample Space

*“...a dice 3-axis' simmetry shape has stable positions in a gravitational field only over one of these sides, as seen by a human perspective”*

The scholar texts of Probability Theory makes wide use of the dices to represent and calculate the expected results. Imagine to toss a single dice. Probability Theory is assuming fair (a synonimous of “ideal”) dices. A fair dice could only be an ideal geometric object, like:

- perfect cube,
- with same weight of the 6 sides,
- zero friction coefficient,
- ...dot-like ! Dot-like, because the mere presence of a center of mass disjoint by surface of the object, under the action of an external gravitational field, creates tidal forces acting differentially on its various sections.

We commonly assume as *“expected results”* of the toss of a dice just the symbols:

because the dice's 3-axis simmetry shape has stable positions in a gravitational field only over one of these sides, as seen by a human perspective. An apparently trivial remark, whose consequences are non-trivial for all physical concepts that depend on *“incomplete information”. *An example of them in Statistical Mechanics, where we regard the position and shape of a solid body as physically given even when we do not know them, while we describe its molecules objectively by a distribution of possible states characterized by a certain temperature parameter. They are arguments of *dynamical stability* in contrast to changes rapid and controllable only be mean of instruments, the reason why we assume to be the six values seeen above as the only possible results of the tossings of a dice. Doing this, we forget that all variables are equally real. Then, any such distinction must not be based on the vaguely defined difference between what we can easily observe and what would require a certain instrumental effort to find out. The diagram at right side illustrates one the basic geometric definitions of the Stability Theory of a *dynamical system*, due to its main developer Aleksandr Mikhailovich Lyapunov. Showed a 2-dimensional case where *x1* and *x2* could represent e.g., a couple of spatial coordinates, momenta, etc. We see that however narrow a cylinder of radius *ε* with the* 0**t* axis may be, there is a *δ**-*neighbourhood of the point *(0, 0, t0)* in the plane *t = t0* such that all integral curves:* *

* x1 = x1 (t), x2 = x2 (t)*

** **
**Stability of a dynamical system following ****Lyapunov**

emanating from the neighbourhood will remain inside the cylinder for all *t > t0*. This is a reasonable base for the definition of relatively *stable Events.* As you can see, nothing is said about the existence of the integral curves deviating out of the circle whose radius is *ε. * Also if unstable, they can exist. But, in the laymen's language the criterion of *dynamical stability is still erroneously synonimous of reality or existence*.

** A common LED may be enlightened along times so short that no human eye shall ever be capable to perceive. Nonetheless, the instrumentation confirms the existence of what we alone should erroneously label ***nonexistent*

Example 1

Expected result with a LED illuminated along 20 ns

Refer to the case shown above where a LED pulse illuminates our eye’s retina. Following the light intensity and other factors, the majority of us detects LED light pulses as short as 20 *μ*s*. *But, what about if the pulse is one thousand times shorter, lasting just 20 ns ? The fact we are naturally equipped to detect some Signals (associated to Events) and not others, does not mean that the undetected Signals do not exist.

Example 2

Expected results for the forces making electron's stability

Another example the *charge*. By mean of the common term *charge*, one century ago it was exclusively meant the source of the electromagnetic force, say the electron (electric charge). Today the electron is also associated to another kind of “charge” which is that one guaranteeing the stability of the matter. Today are known four charges guaranteeing the stability of the fundamental forces and twelve guaranteeing the stability of all what exists. “Exists” just in the sense that pops-up during the experiments along time intervals long enough to be detected.

* *Detector of the B-bar experiment at Stanford Linear Accelerator Laboratory. One of those capable to record Events of extremely brief duration (credit SLAC, Stanford University, 2012)

Example 3

Expected Results Tossing a Real Dice in the State Space

Returning to the expected results when tossing a single dice, the reality known today to the Science makes justice of the fact that in the *State Space* the dice admits many more configurations. Many more “expected results” than the six listed above. Paraphrasing the facts, we can say that* a dice has six sides, six points of stability, when the clock of our observations, a relative of the refresh time of the monitor screen you are using to read this texts, lies in the order of magnitude of our natural senses. * But try to look at the same dice illuminating it with extremely brief pulses of light, filming the scene with equipment emulating an extremely fast camera. One clocking time periods 10^{-18 }seconds or smaller. You’ll see a quite different shape for the same “object”, not less real that the former. An example of just one of the many “expected results” in the *State Space* above, in the ternary Event (3, 5, 6). One of the many Events, before the dice stabilizes itself over one the sides.

Example 4

Expected Results Tossing Two Dices

Consider now the multiple arrangements of two common dices, visible at left side. They represent all their possible tossings, based under the unreal assumption that these dices are *fair*. Assumption since decades contradicted by the certainty that, on the opposite, a gambling system truly *fair* is impossible to build. The entire *Sample Space* on side, comprising 36 Events, is an example of “expected result”. Quantum optics' experimental results (i.e., Mach-Zehnder interferometer) show *“unexpected results”*. As an example, imagine to beam a single photon originating by a laser source of light ( a particularly monochromatic one) toward one of the six faces of a cube made of quartz.

** Try to toss ****two ideal fair dices and you’ll have established a Sample Space comprising 36 Events. “Event” is a combination of the outcomes of the tossing of the couple of dices. They do not exist “fair dices” nor it exist a way to manufacture them that ideal way, perfectly symmetric, perfectly balanced over all their sides. That’s why the true number of Events results higher, coherently with the potentially infinite**

*State Space’s*dimensionality. In other words, in a case of infinite dimension for the

*State Space's*, we can know the*State*only by mean of an infinite number of observations in infinite*Sample Spaces,*along an infinite TimeWe’d expect the photon proceeding out coming by any of the other five faces. But, repeating the test many times you’ll discover unexpected cases when the photon gets out of that face you normally shine it into, ...before you shine it. Clearly, an Effect like this is a violation of causality. The Time-ordered Cause-Effect succession of Events we are accustomed to consider a canon for all what happens in our life and activity. Observations like these are not predicted by all theories of light. The theory since decades encasing these facts is Quantum Field Theory (QFT). Violations of physical laws, yet prefetched by theories including the laws we commonly use as a special case.

Other results, expected in the State Space

Representing in the figure below one of the dices having a surface blank, rather than dot-marked, we are hinting to the unexpected cases. It has been understood that, contrary to what Albert Einstein imagined, *“God does play dices …and they are tricked !” *What macroscopically looks like *“tricked dices”* is the effect of the reversibility of all processes happening at the Events’ spatial and temporal scales.

**In 1970 it had been discovered why, at our own scales, all ****processes ****look us irreversible. Why the glass panes’ tendential state, their future expected evolution, is that of a broken glass. A reason different than the Thermodynamic assumed until a few decades ago and still presented at least at the High Schools level courses of Science**

Reversible in the sense that the value of a variable, after an infinitesimal time may assume a new value and, immediately later, also reaquire the former value. Such a behaviour, should look us definetely random or, chaotic. We name *random* the behaviour describing observed states *unrelated* to precedent others. No relation of Cause and Effect. With reference to the figure below, at left side the real behaviour and, at right side, the evolution we observe because of Decoherence. Decoherence process defined as such an* irreversible transformation of a controllable Superposition into an uncontrollable** **entanglement with the Environment*. Why *“uncontrollable”* ?

Quite banally, because the Environment includes a mind boggling amount of entangled elements, differentially related with each one of the controllable elements of the Superposition. A Polar-star may guide each root cause Analyst in the navigation between so many scenarios, so many Events. Mechanical or electronics failures, Quality pitfalls, Process’ sudden and unexpected stops: all Events existing in the *Sample Space. Sample Space* associated to a macroscopic dynamical system, like a Packaging Machine, one of its assembles or an apparently simple Detector device.

** ****When closely looked, a****ll physical and technological processes are reversible. With the discovery of Decoherence in 1970, it had been understood why, on the opposite, they appear us irreversible. The short-duration reversible processes look us like Noise (image credit **

**Klimenko, Maas/2014)**

Yet the Binomial Distribution of Statistics, origin of the Normal (or, Gaussian) Distribution, also if in an implicit way, indicates a quatitative method to measure the **explanatory power of different Root Causes**. The probababilities of different numbers obtained by the throw of two dice offer a good introduction to the ideas of measurement or observation of distinguishable outcomes. The key concept is reflected in a famous observation made by Erwin Schroedinger in 1927, one of the founders of Quantum Mechanics, about the different meaning of the wave function when applied to a space hosting:

**When tossing two dices there are six ways to get a total of 7, but only one way to get a total of 2. Implying different “measures” for the occurrences of 7 and 2. The outcome 7 related to many Causes (dices’ tossings) is 6 times more frequent than the outcome 2.**

**Binomial distribution, and its derivated Normal, implicitly indicate how a present scenario or, outcome, may be quantitatively related to a certain Cause and differently to another**

- a single existing object;
- a couple of existing objects.

Returning to the familiar macroscopic dice, throwing of a single dice, all outcomes are equally probable. But in the throw of two dice, the different possibilities for the total of the two dice are not equally probable because there are more ways to get some numbers than others. There exist six ways to get a total of 7, but only one way to get 2. After tossing the dices, the cases (or scenarios, or branches, themselves *states*) where a 7 is observed result six times those for getting a 2. Throwing a 3 is twice as likely as throwing a 2 because there are two distinguishable ways to get a 3. The probability of getting a given value for the total on the dice may be calculated by taking the total number of ways that value can be produced and dividing it by the total number of distinguishable outcomes. So the probability of a 7 on the dice is 1/6 because it can be produced in 6 ways out of a total of 36 possible outcomes.

Phase Space

** **

The subset of all the points* *of the* **state space* forming a set continous and finite-dimensional, is named *phase space. *The figure at right side shows the *phase spac*e in its most general meaning: a symplectic manifold M. Its central concept originates by the Langrangian function *L*, introduced as the kinetic energy *T* of the system subtracted of the potential energy *V*:

*L** = T - V*

Now consider a *dynamical system* composed of *i* = 1, 2, …, N particles, then *q*i are 3N position coordinates and *p**i *are the related 3N generalised momentum coordinates. Applying Calculus' methods, it can be demonstrated that the generalised momentum coordinates *p**i* can be obtained from the position coordinates and the Lagrangian, using the partial derivatives equation:

*p**i* *= ∂**L** / ∂q**i*

where *∂**q**i* are the partial derivatives of the time-derived position coordinates*, *say the sequence:

* dq1/dt, dq2/dt, …, dqi/dt, …, dqN-1/dt…, dqN/dt*

of the position coordinates *q**i*. The space of the *q* and *p* coordinates specifying a physical system is named *phase space*. Each one point in the phase space is a possible *dynamical state* of the system and has a corresponding vector, which determines how the system will evolve from that state. *Phase space *provides a straightforward introduction to the derived concept of *c*onfiguration space, which is the object of the next section.

** ****How many kettles front of our eyes? A single kettle occupies three visibly different sets of points, three different subspaces of a common State Space, following its energetic level measured by the temperature**

Configuration Space

*Configuration space *can be thought of as *the half of the* *phase space* that contains the position coordinates *q*. Here, the number of state variables is the dimension of the dynamical system. In the *phase space*, every degree of freedom or parameter of the system is represented as an axis of a multidimensional space. Then, a one-dimensional system is called a phase line, while a two-dimensional system is called a phase plane. For every possible state of the system, or allowed combination of values of the system's parameters, a point is included in the multidimensional space.

Root Cause Analysis’ Rationale

In the world of the extremely complex *dynamical systems*, to know precisely what it happened in the Past, does not allow to know the Future. This is the *r**ationale* for Root Cause Analysis applied to the industrial Machinery. It is not sufficient to create some Alexandria-like bibliotheque of Past machinery’s-, assembles’- or devices’-failures, to infer the correct Root Cause for a Present fault. Why? A clear answer is provided in the following by a pendulum, the basic **nonlinear oscillators**. We all have observed that the period of oscillation increases with increasing amplitude of oscillation. Starting near the upside-down position, we'will find that the period becomes much larger than for small-angle oscillations. And, as a matter of fact, the period really approaches infinity, an outcome we'll however never reach in our ordinary conditions. The video below shows how this is evident yet for a relatively simple non-linear oscillator. In the example, the dynamical evolution of a pendulum whose initial value is ~2 and initial amplitude 0.85. The successive amplitudes span in an apparently chaotic way ranging (1.0 - 1.5), associated to values ranging (2.2 - 0.5).

An Ill-Fated Strategy

In ancient times, it was generally considered that a precise knowledge of the Past allowed the knowledge of the Present and Future. Mainly because of the efforts of the French physicist and mathematician Henri Poincare’, today we know it is not true. With reference to the video above, consider that Root Cause Analysis is made in one of the terminal branches developed at the right side of the graphics. Being there and trying to figure* *from what time-ordered chains of Past *Conditions, Agents *and* choices* or, *decisions*, it is possible to recreate the Present incident. As an example, a failure, malfunction or inefficiency status. But, this is possible only having amassed data of high quality about the variables, sampled at time intervals coherent with the process. Today, a few Machines, Equipments or Processes log so many data. Then, as implicitly hinted by the video above, to try to use a record of past Events as a reference when root cause analysing a system, is deemed to fail. Dynamical systems’ complexity and other factors imply a complex and differentiated future evolution. How complex kind of evolution may be inferred looking below an application of Sturm-Liouville Theorem to a simple bidimensional volume Γ(t) in the *Phase Space*, as* **initial* *Condition *at Time t = 1:

A root cause Analyst convinced that a by a precise knowledge of the Past Events it is surely possible to derive the Present status (Incident), has to look carefully the graphics above. Being him at Time t = 3, should he really be capable to imagine that the volume Γ(t) had that circular shape at Time t = 1 ? In principle, considering he has data about that volume at Time t = 2 he’d answer positively. But, the problem is that at Time t = 1 are given *initial Conditions, *and we enter the domain lying in the gray zone between Theoretical Physics and Cosmology. In brief, the *initial Conditions *of what today and temporarely looks shaped as a photoelectric switch or solenoid valve, are and possibly shall remain forever unknown. To set as *initial Conditions, *e.g.* *a property of the photoelectric switch or solenoid valve, the epoch of fabrication means to ignore that these objects are superpositions of elementary objects. Objects existing well before. Also, the temporal cutoff at the fabrication epoch is arbitrary. At this point, the *Positive root cause Analyst* could counter this objection limiting to the Time t = 2 his exam of the recorded data. And soon he’d understand the Root Cause of his own problem is Epistemologic. The *evolutionary paths joining the state at Time t = 1 to the state at the following Time t = 2 are not infinite however too many.* Then, he’d immediately figure that if this is true, then the evolutionary paths joining a state at Time t = 1 to a state at the following Time t = 3 (the Incident Time), can only be many more. We are speaking of amounts of evolutionary paths which in general, for an initial Condition quite simple and regular, may have orders of magnitude with hundredths of zeroes.

The diagram above resumes a simple example in which they happen three changes of a variable or, decisions. Three time-ordered Events. Visibly four histories derive by three Events. Reader could however think to increase the data logging capabilities of its system, what in principle is surely possible, as a strategy guaranteeing success. As an example, storing the process’ data in a gigantic data bank like those owned by Google®, Inc. (see figure below). But, also this strategy is deemed to failure.

Two different problems, since long time conceived by physicists:

*missing knowledge about the initial conditions*of the system;*Nature’s clock is too fast.*Imagine a clock ticking at time intervals >10^{-43 }seconds. All Google, Inc. present Data Centres added to all others existing in the entire World, are not enough to record all the values assumed by the variables of a common equipment (as an example, a PLC and its analog I/Os) during just one second of time.

The approach based over the comparison of the Present Incident with what registered in the Past, is exactly that **Past Events**:

- are known at too wide time scales. We could be missing the Root Cause banally because not sampling what we consider the
*variables*with a frequency adequate to detect the change which caused the Incident. Just an example of these incidents are the Machinery, Equipments and Devices failures due to unexpected wide and temporary over voltages affecting Electronics; - that
**they are recorded with unavoidable errors.**Trying to extrapolate by many variables (e.g., temperature, voltage, current, frequency, weight, energy, power, density, etc.), each one of them known with a wide variance, creates a confusing riddle of alternative possible causes; - that
**few***variables***of the process are monitored and recorded**. Trying to solve a hard problem, one which resisted the attack of others, possibly we are fronting the effects of a*hidden variable.*One before assumed unrelated to the Present condition of Incident. How to know if the case 3. is the actual case? With special reference to the Root Cause Analysis of industrial incidents affecting Machinery, Equipments or Productive processes, the Writer indicates the methods to discriminate what affect by what cannot affect (or, only minimally affect) in those offered by Mathematics, Physics, and Engineering in all its developments and branches inherent to what lies in the focus of the Incident. The Incident's Environment and Past. Why Mathematics, Physics, and Engineering? Because all technological applications are made of that.

Root Cause Analysis conceives this and avoid to fall in the time-consuming trap of the comparison with all other known Cases of incidents presenting some similarity to the one we have in front. The comparison exists as a method, just one of the darts available and used by the root cause Analyst.

Counterintuitive Causes Does Matter

Direct comparison of the *State* and *Phase Spaces*, makes sense of our statement, deepened in the following, that also *causes* particularly counterintuitive or contradictional, have to be accounted for. Accounted when imagining *what* *Causes, *by mean of what *Agents,* originate an *Effect.* *Causes, *exactly like their* Effects *reside into *state space*. Then spaces which may be continous or discrete.

It is presently being studied if the dimensionality of these spaces is a number finite (however, extremely great) or infinite. Graphics at left side, implicit in Liouville's theorem picture, is a visual aid when trying to understand, at its deepest level, what is the task of the root cause Analysts. Shown two successive 2-dimensional sections of a 3-dimensional *Phase Space. * As we have seen, itself a subset of the wider *State Space*. Both regions A0 and A1 at time t0, lie in the past of the space B at time t1. Visibly, the points in the *Phase Space* into A0 and A1 at time t0, keeping apart their simultaneity, are totally unrelated. A graphic way to understand how an *Effect* in the future derive by the sum, or *superposition*, of related and unrelated *Causes *in the past of the problem*. *And now ask yourself: in what a way they’ll be perceived in B at time t1 the Effects of the Causes at A1, with respect to those originated at A0 ? *Noise* is the answer.

**An undesired Effect, e.g. the states constituting a “problem” affecting at time t1 a Machinery, a Device or Productive Process, is the sum or superposition, of strictly and less strictly related Causes. Visible how both regions A0 and A1, subsets of the phase space at time t0, lie in the past of the space B at time t1 (image credit **

**Klimenko, Maas/2014)**

What the figure above is showing us is a *Cause **Ai** for Problems, *existing but unnoticed in the past time t0, amplified until occupy many more states in the set B at the future time t1. Abstracting by the theorem graphically represented above, it can be realized the root cause Analyst's task as *d**epiction of the entire spectrum of the states existing into the sets A**0, A1, A2, …, Ai, …, An-1, An in the past, giving origin to the problem B at present time t1. *What accounts to:

**create**categories**measure**the sets of states A0, A1, A2, …, Ai, …, An in the past, giving origin to the problem B at present time t1. On practice, the attribution of a weight to each one set of states in the past, typically expressed as a probability percentile.

Degrees of Freedom

Measuring an Object’s Complexity in the Hilbert Space

At this point, a relevant observation regarding the list of individual cases composing the state space of the gun’s drum, could be that *all of these combinations of Causes and Effects pre-exists the rotary action* on the gun’s drum. Before to start to move anything, we yet know, on the base of mere computation, what should look the results when playing Russian-roulette. The *State Space* is hinting to a place where all of the possible future *results of an action or measurement, exist before the action or measurement: the Hilbert space*. The definition of *degrees of freedom* we’ll be using is more general and recent (one century ago) than that adopted by Classic Mechanics. Since primary schools we all are familiar with the idea of the 3-dimensional Euclidean space, conceived twenty-five centuries ago. One century ago it was discovered that the Euclidean space is just a special case of the wider Hilbert space.

*Hilbert Space*'s (further infos here) most basic properties:

- abstract vectorial space,
- finite- or infinite-dimensionality;
- possesses the structure of an inner product;
- allows a measure for its length and angle;

Let us define the number of degrees of freedom N of the system to be the natural logarithm of the dimension *N** *of its Hilbert space *H*:

N* =** ln** **N **= ln dim( **H** )*

The *number of degrees of freedom is equal to the number of bits of information needed to characterize a state*. Applying the drum with 6 slots visible above, has *N** = *2^{6 }states, then N* *= 64 ln 2 degrees of freedom. Meaning that this Russian-roulette gun:

- has a state completely specified by 64 bits of information;
*can be used to store 64 bits of information*.

The computation above is just an abstraction: it is ignoring that, in reality, the drum has plenty of substructures made of atoms of different metals. A multitude of atoms of Iron, Carbonium and Silicium, themselves made of electrons, pions, kaons, different kinds of mesons, neutrinos, etc. Whoever agrees that richness of an “object”, evaluated as amount (a *measure*) of structures and substructures, is an indicator of *object’s complexity*. We are understanding that, when using those 64 bits to estimate the *complexity* of the drum, it carries us to underestimate by many orders of magnitude its true complexity. All components of each one metallic atom have their own complexity, their own degrees of freedom *N* calculated by their dimension *N* in the Hilbert space *H*. And it is starting to appear transparent also to non-specialists Nature’s favourite arrangement of the matter: structures composed of other smaller structures. In other words, a Superposition.

Happened and non-happened Events

We expect some of the Readers of this page yet having practised, at least one time, as part of their own duties in an Industrial factory this research for *what/who, when, where *and in* what extent changed* in system.

**Chronologic ordering of different states of Physics ? No, a tree-like structure of Phylogenetics, valid for natural phenomena. ****Since thousands of years, well before Science and Technology were born, changes are conceived in the framework of Cause-Effect relations (image credit Yale University, Peabody Museum of Natural History, 2014)**

In this, nothing truly new or technologic. Since many centuries the studies of Natural Science (i.e., botany or genetics) had that goal. Two figures, at right side and below, published by the Yale University (Peabody Museum of Natural Science) show this classic point of view about *Evolution. *We are (intentionally) naming an educational institution, a Museum of Natural History. An institution without any necessity for know-how in Root Cause Analysis. Nor applications to the solution of technological Problems, like Machinery’s inefficiency. But, also from that uncommon point of view, the centrality of an idea, *Evolution, *stands up. A point of view including a multitude of *causality* relations, where a Root Cause is always displayed in a clear graphical way. The figure above at right side shows three states A, B, C deriving by two different Times 1, 2. A representation considered obvious by Rene’ Descartes, Wilhelm Leibniz and Benjamin Franklin yet centuries ago. The figure below shows one further step. The *branchings* at left and right side of the “ = ” logic symbol, are *topologically* one and the same “object”. Only difference, the transformation deriving by two rotations of 180º, inherent to the mirroring and the exchange of the B and C labels. No difference at all, as seen by a topologic perspective. Comparing this figure with the one above in the section titled “Objects”, it is possible to recognise that meaning of “Object”. An entity existing in the state space M where M includes all possible states of an object. The rules *fα *and *fβ *assign logical predicates to the representations* fα (M)* and *fβ* (M) for the state x. ** **The transformation** *** f _{β } . f ^{-1}_{α }*is a map pointing to the “objective” state

*x.*

** ****Since many centuries it was being considered at least a reasonable conjecture the equivalence of causal relationship and Topology. In 2001 the correctness of this classic idea had been upgraded to the rank of theorem (image credit Yale University, Peabody Museum of Natural History)**

It is the invariance under transformations among all representations which is confirming that *x* is an “object”. Invariance implying that *x* abstracts, say remains itself, by all possible transformations. Also implying that, in a probably inconscious way, the three researchers named above were yet correctly defining an “object” in the *state space,* centuries before this concept was coined. Yet part of their own established mentality, exactly what today are considered the most modern points of view about:

- what is a
*Cause*, - what is its
*Effect,* - how to correctly establish if an observed status of an “object” is an Effect of a certain Cause.

Phylogenetics’ point of view, shown in the figures above, can be reasonably be named “historical”. *The difference between different states (alias, change) of an “object”, or of different objects, chronologically-ordered and topologically-different due to the action of Information. *

Searching for Happened Events

We’ll see in the following that this ancient perspective remained close to an universally accepted *conjecture* until 2001. In that year, it had been demonstrated by mean of theorems the correctness of the Logic underlying these ancient ideas. In the couple of figures above the Events are the timed-vertices where a branching arises. Passing from Phylogenetics to the Relativistic perspective of Physics, the Events are spatial 3-dimensional leaves (or slices, or sheets) of a 4-dimensional foliation. In its most general aspect, a space-time manifold M. Each leaf contains all existing objects, some of them so extended in the space to occupy simultaneously several leaves. (Objects of a detailed presentation here).

** ***Changes* are w**hat the root cause Analysts look for . Changes happened before, during and after an instant of Time. An Event, made critical by the fact that the state of a system after that changed, and changed negatively. The changes, whatever their nature, physical or logical, are due to the action of Information. Diagram shows the history of a computation, as seen by the classic macroscopic perspective, visibly an history of changes. Here **

*b*

**is a state function whose value is referred to the Time parameter**

*t*

**, and whose initial state is**

*b*

**( 0 ) =**

*β*

**. The history of the state of the variable parameter**

*b*

**, corresponds to the time-ordered serie of functions f2( f1( f0(**

*β*

**)))**

**.**

**Leaving**

**aside the formalism, the graph show common analog values expected infeeding the analog inputs of Programmable Logic Computers (PLCs). And the PLCs, originally named “automata”, are the computation devices over which an amount of fundamental theorems were originally tailored. Finally, we see that a bidimensional graph represents correctly the computation happening along the serie of Events 0, 1, 2, 3, …**

**(image credit Deutsch, 2001)**

It means that several adiacent leaves have quite similar material content, the majority of them difficult or impossible to distinguish. Signalling, all signalling electromagnetic or gravitational, happens always and only intra-leaves. Meaning that all interactions are always and only happening between different, however very similar, 3-dimensional spaces. A view where 3-dimensional “leaf” and “universe” are just different names for the same thing. *Changes* happened before, during and after an instant of Time, an Event, made critical by the fact that the state of a system after that changed, and changed negatively. The graph at left side shows the history of a computation, as seen by the classic macroscopic perspective, visibly an history of changes. Here *b* is a parameter whose value is referred to the Time parameter t, and whose initial state is *b*( 0 ) = *β*. The history of the state of the variable parameter *b* here represented, corresponds to the time-ordered serie of functions f2( f1( f0( *β* ))). Leaving aside the formalism, the graph show common analog values expected infeeding the analog inputs of Programmable Logic Computers (PLCs). And the PLCs, originally named “automata”, are exactly the computation devices for which an amount of fundamental theorems were originally tailored. Finally, we see that a bidimensional graph represents correctly the computation the serie of the Events 0, 1, 2, 3, …

Searching for Non-Happened Events

*“...all Events are labelled by Time. When we look for the Time something happened or did not happened, strictly means we are looking for the happened or non-happened Event”*

The research for the complementary *not happened Events*, the states or physico-chemical values which *did not changed* along the period of Time does matter. What did not changed in correspondance with an initially undefined, *change*. Visibly, the most frequent words are *change *and* Time. Time *because Root Cause Analysis is always applied to the solution of “problems” affecting a productive process or device, happening in the space-time. Also, problems non existent until a certain Time. This when, knowingly, all Events are labelled (or, tagged) by Time. Event made critical by the fact that the following instants (and, related Events) are those when the “problem” let us feel its negative effects in terms of reduction on production, quality, safety, etc. The study of the facts and statuses which did not happened at first sight may appear unuseful, an action not deternining what/who, when, where and in what extent *happened*.

A deeper analysis guided by logic, physical and philosophical principles and laws, shows easily that *non-happened Events does matter, as much as happened Events. *Non-happened Events are since nearly one century the majority of the information content of the theories and experiments looking toward the finest details, the smallest spaces, times and energies. Finest details which have not to be considered out of the area of interest of the Industry and of the engineer, rather the best thinkable description today available of what let the systems and processes behave as visibly they are behaving. We refer to all those Events non-registered as outcome of a certain experiment, but however existing as one of the eigenvalues of characterising a system. Each one of them associated to a certain value of Probability of outcome. As an example, after having searched and listed:

*what*happened,- to
*whom*it happened something, *when*something happened,*where*something happened,- in
*what extent*something happened,

it definetely makes sense to search and list also the *complementary* informations:

*what did not*happened,*to whom it did not happened*something,*when*something did not happened,*where*something did not happened,- in
*what extent*something did not happened,

because of reasons considered well founded as seen by completely different perspectives from the:

**layman's point of view**, to cross-check the entire list of what/who-, when-, where-, what extent-happened. The layman point of view assumes implicitly that something (an Effect, a Cause, a Condition) cannot simultaneously happens and non-happens;**forensic point of view**, to prevent a biased analysis which should carry to aberrant judgements. As an example, as early as 1946, the US Supreme Court held in the case of Hickman versus Taylor, 329 U.S. 496 that*“mutual knowledge of all the relevant facts gathered by both parties is essential to proper litigation”***naturalist's point of view**, yet since thousands of years (correctly) hinting to a tree-like structure for the entire genealogy of Life;**physicist's point of view**, to define the boundaries implicit in the underlying Topology of the system (or, of the process) affected by the “problem”. And to define the Information Flow structuring that Topology. An idea based on the assumption that all systems, whatever their size and energy content, can appear shaped following different Geometries, but cannot abide by the fact to be characterised by their own individual Topology itself structured by the Information Flow. Topology and Information are today considered the essence of whatever, wherever, whenever. In this modern interpretation of what is an Event causing a (problematic) Effect, the Information Flow defines, for each one particle, what eigenvalues shall be measured through all of the space. A result made complex by its multiplicity, where different outcomes associated to the same spatial position are part of the matrix of results expected when making a serie of measurements. Also, a result implicitly hinting to the fact that they do not exist two identical Events, say two identical leaves of the manifold*M*. “Leaves” hinting to manifestly tree-like topologic structures.

**The history of a classic computation, as seen zooming until detection of its finest subatomic details, cannot be displayed by a 2-dimensional graph. It is here replaced by a 3-dimensional space including the 2-dimensional slice of the classic perspective. The superimposed Probability of the outcomes of each branch for each one value of the time parameter ***t*** is always 1 (100 %). It is a single value, a main trunk, before the start of the change from state (at ***t*** = -1 ) to superposition ( ***t*** ≤ -1 ) where it splits in 4 branches. Each one of them later interfering with the others until a final interference ( ***t*** ≤ 4 ) where they converge in a single trunk. It is the information flow what structures, following physical laws, the branches’ Topology. In this perspective, the non-happened Events are the sum of all those happened in other branches (image credit Deutsch, 2001) **

Normal Distribution & Modern Principle of Superposition

We are engineers, and that’s why the last ( 4. ) is our own pespective. Consider the graph above showing the history of the classic computation precedently examined and now zoomed until detection of the finest subatomic details which can be registered today. Zooming that much the sequential story of states cannot be any more displayed by mean of the bidimensional graph presented before. That’s why it is here replaced by a 3-dimensional space, including the 2-dimensional slice of the classic perspective. The graph above is showing the time-ordered evolution of the state of a physical system, corresponding to the time-ordered serie of functions f2( f1( f0( *β* ))), where *b*:

- is a parameter whose value is referred to the Time parameter
*t*, - whose initial state is
*b*( -1 ) =*β*, - whose final value is
*b*( 4 ) =*g*(*β*)*≠ β*, - with
*g*(*β*) being*b*, after the computation.

Evident how in the time interval corresponding to the Events happening:

-1 < *t * < 4

the superimposed Probability of all the outcomes is always 1, say 100 %. What precedes can only be interpreted thinking that there is a single value, a main trunk, before the start of the change from state (*t* = -1) to superposition (*t* ≤ -1) where it splits in 4 branches. Each one of the branches later interfering with the others for -1 < *t * < 4, until a final interference happening when *t* < 4, where the branches *superimpose* themselves in a single trunk with the same initial Probability amplitude 1, when *t* = 4.

** The computations parallel-happening at all scales, witness an initial splitting of the pre-existing mix of superimposed terms, followed by an interference. What happens during the initial reduction, when ****-1 <** *t* **< 0, is mirrored by a gaussian shape**** identical to that observed during the final interference when 3**** <** *t* **< 4**** after subtracting a constant. The processes U and its inverse ****U**^{-1 }**correspond to a time-ordered sequence of Events where and when the Information Flow structured four coexisting pathways. Each one of them to a certain extent, the Probability of each one of the Events Time-parameterised at **

*t*

*=*-1, 0, 1, 2, 3, 4. Finally, the parameter

*b*is

*what***, from the state**

*changes**β*

**to its future state**

*g*(

*β*)

**The normal function with a 2-dimensional domain here shown corresponds to a 360º revolution of the profile corresponding to the splitting and following superposition phases. Its special shape was yet partially hinted three centuries ago by Galileo. Two centuries ago it was rigorously and independently derived by de Moivre, Gauss, Adrian and Laplace (image credit Kaushik Ghose/en.wikipedia/2006/CC BY-SA 3.0) **** **
** **

Since now, we warn the Reader about the capital relevance of the process underlined by the word “*superimpose”. *All staff with special expertise in the technological disciplines (Electronics, Electrotechnics, Electromechanics and Automation, Telecommunications and Signals, Industrial Management and Industrial Chemistry) masters the gaussian or normal function. Its common probabilistic interpretation was first empirically observed by Galileo Galilei. Later, over two centuries ago, independently derived by de Moivre, Laplace, Gauss and Adrian. That meaning has been made *classic* by the huge accumulation of experimental and theoretical discoveries of the past one hundred years. Looking again the graph above with a higher zoom, after having pink-coloured two of its details, something relevant arises.

The pink-coloured areas similar to the couple of halves composing the normal distribution of the physical measurements. One of the most commonly observed universal laws. Similarity full of meaning with far reaching explanatory power. The computations parallel-happening at all superimposed scales, witness an initial reduction of the pre-existing mix of superimposed terms, followed by an interference phenomena. Until around sixty years ago gaussian distribution's physical interpretation was still the probabilistic classic one dated 1808. The figure above is showing the paradigmatic revolution happened around the change from Second to Third Millennium. As a matter of fact, if we let any of those two light pink-coloured shapes, visible when:

*splitting*the state function, passing from state to superposition;*the components**interfere*, passing from superposition to state;

revolve 360º around the vertical Probability axe, we'll obtain the normal function in the figure at right side.

That's the origin of the universality of the gaussian distribution in the macroscopic phenomena, observations and measurements. Different superimposed components interfere, changing their topology from that of a superposition to that of a single state. The single state, for which we’ll measure (at Time *t* = 4 ) a value *each one time* different. And, along a repeated serie of measurements, respectful of the density prescribed by the normal distribution law.

A new meaning for *what*, *where*, *when* and *in what extent*

With reference to the figures above, we have now an adequate amount of elements backed by theorems, to express a new meaning for the basic terms of the discipline named Root Cause Analysis:

- the parameter
*b*is precisely*what*, from the state initial value*changes**β*to its future state*g*(*β*); - what happens during the initial reduction,
-1 <*when**t***< 0, is mirrored by a gaussian shape**identical to that observed during the final interference happening3 <*when**t*< 4 after subtracting a constant; - the processes, U and its inverse
^{-1 }correspond to a time-ordered sequence of Events*where*the Information Flow structured four coexisting pathways;*when* - each one of them to a certain
which is the Probability associated to each one of the Events labelled by*extent**t*= -1, 0, 1, 2, 3, 4.

What let the fine-details of the process look that way ? **It is the Information Flow what, following physical laws, structures the branches’ Topology. **By the graphs ot is evident the mechanism which let what we name **n****on-happened Events **simply be** those happened in other branches.** The graphs above display the most important point: the *non-happened Events are as relevant as those happened* *to define that Probability 100 % of outcomes* or, observations, or measurements hinting to the state of a system or process.

*Non-happened* Events are those considered *happened,* as seen by Observers, Machinery, Processes and Devices existing in other histories and observing what happens. Each one of these other histories may be similar but not identical to the others, and their difference is lower bounded to 1 bit.

Focusing the Action Determining a Change

*“...difficulty to focus what is the problem”*

We conclude this section with a remark against the widespread idea following which experience, an alias of amount of data recorded in a memory, helps to root cause analyse a problem. If an *incident* and the related *case history* are mirrored in many others couples *(incident, case history*) existing in the memory of the root cause Analyst, then no application of Root Cause Analysis is necessary at all. On the opposite, Root Cause Analysis is applied to the incidents which:

- do not have known counterparts in the Analyst memory;
- have only some similar aspects and contradictional facts for other aspectes. Then, syllogistic reasoning, one of the most ancient and constantly used method of Logic, does not apply.

The key point of Root Cause Analysis is really the focusing over *what/who, when, where* and in *what extent changed* and their complementary existing branches (or, histories) *where,* *what/who, when *and in *what extent did not changed*. How to explain some truly excellent root cause analysis made by staff never:

- trained in Root Cause Analysis ?
- requested to propose the cause for an effect observed in a discipline they are totally unaware ?

*Intuition* is the only answer.

“Intuition” truly meaning that the straightest way toward the solution of technical problems is open to whoever. Definetely something which is only slightly related to the amount of text books and scholar articles of Root Cause Analysis and Problem Solving red before. And this, in the opinion of who yet red really many of these.

What differentiates a Root Cause by a Condition ?

Other texts of Root Cause Analysis suggest ways to separate conditions and Root Causes. Unfortunately nearly always missing the key point: the *exclusively physical rationale* for Root Cause Analysis and its strategies and techniques. RCA is not a modern method: really modern is its application to the technological problems affecting all processes and systems, industrial and non-industrial. As a matter of fact, to *know the cause (or, causes) for an observed effect*, means the* capability to change the present status of a system. *In these pages we’ll just brief what we can offer in terms of solutions made immediately factual by the accelerated comprehension of the causes, and of their eventual relations with external factors. Subjects thoroughly deepened here.

**In Root Cause Analysis, one of the most difficult activities is to distinguish one or a few Root Causes, floating in the middle of a multitude of Conditions. A Root Cause is the Event Triggering the Effect. The term “Trigger” fully corresponds to the standard weapons' use of that word. In the example below the fuse of a German artillery projectile. Root Cause of the detonation is the percussion of its lower ellipsoidic part and all other visible parts, screws, spring, explosive, etc. separately condition the fuzing action. Limiting our perspective to the projectile fuzing, the process seems fully described. But this is just an effect of the boundaries we arbitrarily introduced. The process of explosion of the projectile does not terminate in its fuzing. Fuzing starts another chained serie of processes. Other processes typically related to other conditions. As an example, the humidity of the chemical explosive conditions its function, impeding it (image abridged by Pascal Casanova) **

One of the examples we’ll consider elsewhere in this web site is the Case Study shows step-by-step *how to correctly differentiate Causes by Conditions.* *T**he Root Cause is: *

*one in a multitude of Conditions;**the one closer to the Effect: immediately before the Effect;**no interposed Condition between the Root Cause and the onset of the Effect.*

Problem Solving Strategies

Our approach to Problem Solving is guided by a Polar Star: modern ideas developed by different physicists, published in 2000 and 2001, conceiving the Information Flows structuring Topologic features. The spatio-temporal and energetic features distinctly perceived and measured by all of us, just the visible tip of an iceberg. A point of view having room for all thinkable assumptions, patterns and thesis, reduced to coexisting, actual and non-alternative scenarios. Scenarios more properly named “branches”. A single observed technological “problem”, an effect, is the superposition of a non-infinite however mind-boggling amount of causes, measured by mean of conditions. Each one *cause* originates in a different Event, say the content of an entire 3-dimensional leaf (or, sheet) part of the 4-dimensional foliation. In that period different paths, relativistic and quantomechanical, converged to the same result (further details here).

** ****They exist innumerable curves joining a Cause Event at p, and ****an Effect Event at q. No one privileged or more real than the others**

*“No observed problem subject to Root Cause Analysis is never completely external or internal to a Machine, device, process or procedure”*

The relativistic is illustrated by the figure above, showing three of the infinite and actual (not potential) worldlines. *There are innumerable curves joining two Events p and q, and none is privileged*. The point q is an Event in the Future of one of the infinite world lines (histories) crossing the Event p. Due to its relevance we remark that there are several world lines connecting them, and not one. Several ways to go from some state in the past to some other state in its future. The dark-violet coloured circle represents a 3-dimensional sphere. **To evaluate the phase difference and the coupling between the Events p and q, we have to account for the contribution from all paths. **Imagine that** **at** p lies a Root Cause **and that at** q an observed Effect. ***Phase difference* and *coupling* quantify the relation much more precisely than a Cause-Effect correlation coefficient.

The concept graphically represented by the figure above says **everything between two Events contributes to their Cause-Effect coupling.** All this means that no observed problem subject to Root Cause Analysis is never completely *external* or* internal* to a Machine, device, process or procedure. This, in bold evident contradiction with the seasoned point of view about relation between causes and effects. That point of view dividing between external or internal the causes for an observed malfunction or inefficiency. The Effect typically an undesired status for a Machine or process, in laymen language named: *“problem”*. One time the most modern ideas are fully digested, the following step toward the search of the root causes simply list the longest possible row of scenarios or branches.

**When a root cause Analyst investigates a “Problem”, he is observing the outcomes of a multitude of physical Events superimposed along precedent Times, spread over a vast volume of Space. The upper points of the branches hint to different starting values for properties (e.g., voltages, temperatures, positions, etc.) in the history of an observation we consider a “problem”. These sets of values coexist with a myriad of other tree-like structures. Each one of them “a problem” itself correlated to the one we are focusing. All around the Analyst slightly different values for the causes, emulating what we are focusing as a problem. A myriad of ways, differing exactly 1 bit, to go from a single Cause at p to the same identical Effect at q. Also, like in common streets, they always exist crossings. Filally, each one of the vertexes tilted upside-down, is the place where a couple of superimposed states interferes, becoming a single physical state**

Biasing: its influence over Root Cause Analysis

Unbiased analysis: *Ω*_{RCA }= 4π steradians

Imagine to be in the centre of a sphere. The *causes* for an observed *effect* (the “problem”) originate in different measure, from wherever all around us. But, just one in the multitude of actions is determining the problematic state at the centre. Like one in the infinity of radiuses in a spherical solid angle *Ω* visible in the figure on side. A winning strategy, part the cartesian idea of the scientific method, implies not to prioritize what we consider our favourite scenarios. Following the French scientist, philosopher and military Rene’ Descartes, *we are not allowed any pre-concept idea equivalent to a preferential scenario.* No functional and successful knowledge can be reached that way. Scientific point of view also meaning the Analyst’s duty to keep a skeptic posture with respect to all interpretations of the facts. Exactly the opposite than what is accomplished by too many of the Analysts forgetting *RCA is an investigational activity*.

** ****They exist innumerable curves joining a Cause Event at p and an Effect Event at q. No one privileged or more real than the others. Also because of this reason, Root Cause Analysis have always to be unbiased. “Unbiased” meaning in all directions and without any preference for some directions. Only unbiased analysis reach the Truth and the success implied in the determination of what created the Problem (image credit Haade, 2007, CC BY-SA 3.0)**

**Beechcraft’s oxygen tank improvements, in the weeks after Apollo 13 incident and the thorough comprehension of its causes thanks to Root Cause Analysis (image courtesy NASA, 1970)**** **
** **

RCA cannot abide by the present rules and practices defining when and in what extent the results of an investigation can be trusted. With reference to the figure above, they exists some relevant questions, along the boundary comprised between Science (and our application, Technology) and Ethics:

- since when the many fields of Engineering are allowed to concentrate the controls just in a thin solid angle
*Ω*RCA ? - Who decides that the domain where to search for the cause or causes of a Machinery malfunction, Design error, Production defects or contractual Commercial intentional falsification has to be limited into the angle
*Ω*RCA ? - Is it in the interests of the investigation of the truth to limit
*Ω*RCA when search for Problems'*causes*?

Trying to answer on your own each one of these three questions we make a giant step forward. Following the steps of Galileo, Newton, Edison or Einstein, who answered these questions before, persons whose Ethical stature was not inferior to their Mathematical.

Apollo 13 Incident:

an Unbiased Root Cause Analysis

Notoriously, one of the first applications of the Root Cause Analysis was the study of the causes for some incidents related to Astronautics at NASA. As an example, studying the famous Apollo 13’s incident happened on April 13, 1970, it gets out that it was a relatively banal error in the exchange of communications between who established a Design modification and one of the NASA’s Contractors, Beechcraft Corp., who did not received an information. In 1965, eight years before Apollo 13 mission it was decided to increase the bus voltage of some electric devices in the Apollo family, from 24 to 65 VDC.

Further conditions let the overheated wiring into one the spherical Oxygen tanks necessary to power generating fuel-cells be exposed to the fluid, provoking an explosion. What exploded was out of the reach of the eyes of the crew. Then, thanks to Root Cause Analysis, from a distance* *over* *320 000 km in a few hours it was understood *what* happened, *where *and *when*.* * The study started by the exam of all *what did not happened*, what systems were correctly functioning. Also, Root Cause Analysis was later applied to understand why the incident happened, until understanding that at least a thermostat rating was incoherent with the new Design. The figure at right side shows how many modifications were made by NASA and its Contractors in the weeks after the incident, to prevent its repetition in the following missions. This Case ranks between the best examples of the Problem Solving power of Root Cause Analysis.

Logically Erroneous Analysis

*“It’s impossible to let a Packaging Machine function as prescribed by a Contract and related Technical Guarantees, thanks to Retorics camouflaging contradictions, tautologies and lies”*

*“The core problem of our technological applications, like the Food and Beverage Packaging Machinery and Devices, is that they all are exclusively driven by the principle of maximum profit”*

In the opposite direction, imagine someone alleging his capability to *simultaneously show the veridicity and falsity of the same assertion*. Deepening, you’ll discover he is not truly *showing*, demonstrating or proving anything, rather simply *argumenting*. *Argumenting* replacing tautologies, contradictions,and other verbal tricks like the intentional omissions to the dialectic rules of Logic demonstration coined twenty five centuries ago. All disguised by mean of a brilliant Retoric. An example of erroneous logic is that of the analyst when considering arguments as correct if they somehow lead to the expected or empirically known result. Facts we know, unfortunately, have the negative tendence to drive us toward them, banally because we are trying to encounter what is causing the Effect we name Problem. Imagine of how many facts we are unaware, and ask yourself:

- is a fact
*less factual*because we are not aware of it ? - is a fact
*more factual*, because we were perceived its existence ?

This kind of erroneous logic does not correspond to a biased analysis: it is a true error, where the Analyst has no guilt. The same its consequence is negative because we’ll terminate to define the wrong and not the true Root Cause for an observed Effect.

Biased Analysis: *Ω*RCA ≪ 4π steradians

Unfortunately, a great share of all the root cause analysis are, as seen from a scientific point of view, *biased. *As a consequence, are not analysis rather the equivalent of a commercial brochure trying to sell something. On practice, trying to sell an explanation for some observed facts deviating the attention of the Readers far from the true Causes. In Physics, Chemistry, Biology or Medicine, a *biased investigation* is close to an intentional falsification. Root cause analysing technical failures, Production inefficiencies or Quality pitfalls, you'll discover that causes for the incidents are a cocktail of: * *

*incompetence,*when designing, commissioning, upgrading, servicing, etc.*written promises*, impossible to fulfill when offering to a prospected Customer,*technical untested convincements*, hitting against basic physical laws,*private interests,**conflicting with those of the Company,*as an example the staff who did not knew how to openly say to their own Boss or Customer they were not capable to reach the goals they were paid to,*etc.*

The links to the Case Studies below offer a variate panorama about what get out when extending in all directions, then to 4π steradians, a technological investigational activity:

Links to Root Cause Analysis' Case Studies:

Analysis and Experts’ “pet theories”

### IntroductionIn the following, we’ll introduce the study of a pattern observed worldwide in the period 2005-2014 in the operative reality of around one-hundred Electronic Inspectors sold to be operated in-the-Machine (Labeller, Filler, Capper, Closer or Seamer). …

### Case Study Sampling Table always jammedIntroductionTypical problems in a Food and Beverage Bottling Line derive by missing materials like motors, Frequency Converters, detectors, etc. In the following we’ll briefly study a Case where the “material” was much more than necessary. …

### Case Study The Brewery which considered the TechnicalGuarantees part of a Contract to be honoured ￼ ROC-curve of a Fill Level inspection. Imagine to pass 1000 times a test container through one and the same Electronic Inspector equipped with a dual Fill Level Inspection: X-ray and High Frequency (or, Optic with camera). …

### IntroductionPowerade® is a famous energy beverage, the first in a row of energy drinks with great success launched in the World, after 1988. In July 2002, The Coca-Cola Company® updated the bottles of the standard Powerade to a new sport-grip bottle. …

### Case study 27% rejects stopping all days a Beverage Bottling Line ￼ The rejects' accumulation table of an Electronic Inspector in a Beverage Bottling Line, as it appeared to the eyes of the Operators and Management along 6 consecutive months after the day of the Line startup. …

### Case study 1 year of Production lost ￼ ￼ ￼In the following a case where the Machine blamed by the Bottler never functioned correctly since the first day. 18 months later, and after two different analysis made by elements acting in conditions of …

### Case study Cap presence control of sensitive beverages in 2001“X” is a great dairy filling and packaging plant. Here were installed, started and commissioned two standalone Full Bottle Inspectors in a new Aseptic Line for PET-bottled dairy products. …

### Case study Cap Presence control of sensitive beverages in 2010Introduction ￼“Y” is a great dairy filling and packaging plant. Here it has been installed a standalone Full Bottle Inspector in a new Aseptic filling Line for PET-bottled dairy products. …

### Case study99.5% of the non-labelled bottles sent to the Market￼ Non labelled and flagging-labelled bottles, manually rejected at the outfeed of a Labeller Machine. Manually rather than automatically rejected by the inspection system, because of a design-error in its configuration, amount of tracking Triggers photocells and Encoders. …

### Case StudyCanning Line’s Layout Relevance ￼ ￼ ￼IntroductionNo one of us has X-Ray's radiographic eyes. Because of this obvious reason, the cans’ content cannot be visually verified if, due to any reason, the Full Can Inspector is not active. …

### Case study Electronic inspection of Pharmaceuticalsin-the-Machine-Cycle When speaking of guarantees of asepticity, Pharmaceutical Industry is since many decades what really and only retains the maximum know-how. …

*“Let the Root Cause Analysis to an Expert of that Machine…..” *

A fundamental case for using expert opinions is when dealing with uncertainty in technical issues:

- with significant uncertainty,
- controversial or contentious,
- complex,
- with limited objective information,
- that can have a significant effect on risk,

are the most suited for expert opinion. But, in the reality, the Experts are alike double-edged swords. They bring in a deep knowledge base and thoughts, but also they frequently infuse biases and, worse, pet theories. The selection of experts should be handled carefully, recognizing uncertainties associated with this type of information, and sometimes with skepticism.

Biased analysis and Packaging Lines' efficiency

We are implicitly identifying the core problem of our technological applications, like the Food and Beverage Packaging Machinery and Devices, in the fact they are exclusively driven by the principle of maximum profit. Increase on the profits, without cuts on the Food and Beverage Safety or other qualitative aspects, really is due and possible. But, it implies a price to pay. To pay before in terms of Industrial Design and Process' improvements, which have to be invested to profit later of their beneficial long-term effects. It’s impossible to let a Packaging Machine function as prescribed by a Contract and related Technical Guarantees, thanks to Retorics camouflaging contradictions, tautologies and lies. That’s why no one root cause Analyst is allowed to straightly go to his/her favourite explanation. Explanation corresponding to just one:

** A biased ****root cause Analysis corresponds to imagine just one ****of the infinite trajectories p-q joining the point p in the Past of the Machinery, Equipment, Device or Process affected at the Present point and time q. What about all other causal explanations for the actually observed facts ?**

- of the trajectories p-q visible at right side;
- small solid angle
*Ω*RCA of the 4π truly existing.

In conclusion, looking far from the area where our intuition seems to indicate in a Root Cause the way-in to the solution of the problem, provides a *measure of the explanatory power of our own intuition oriented toward another direction*. What differentiates the immediately successful analysis made at NASA in 1970 in the aftermaths of the Apollo 13 incident briefed above, by the multitude of the other cases where the publication about what actions caused what incidents had been delayed years, is the different *Ethical value* of the Actors. And not a presumed technical *superiority* of those at NASA in 1970. The *rationale* is the sequence:

- the running state of the Industrial Machinery and Equipments is
*unnatural:*if we leave them alone, soon they’ll stop themselves; - then, if the Packaging Factories are running is also because they are maintained in that state thanks to the solutions given by highly skilled technical Staff on-site;
- as a consequence, if later a technical Problem resists months or years to a final, complete and permanent solution, the reason cannot be technical.

When root cause analysing technological issues, Graphene strives to achieve the following objectives:

- To provide a systems framework for the analysis and modeling of the issue;
- To provide and illustrate methods for issue synthesis in terms of
*what, who, where, when, in what extent*an issue manifest itself and also the conjugates:*what did not, who did, not where did not, when did not*and in*what extent did not;* - To examine and illustrate methods to recreate the same issue in systems apparently not affected;
- To guide the readers to mature a critical thinking-based opinion about what, precedentely experts declared were the causes of the issues;
- To provide methods for visualizing the causal relation existing between causes and effects;
- To provide examples of other practical applications in the same area, where the same issue is not felt.

Links to other topics:

### Links to related topics: ￼Introduction￼Root Cause Analysis (RCA) is much more than a technique of investigation of Industrial Machinery's inefficiencies or malfunctions. Part of it are the pillars of the…

### “Project: Middle English projecte, from Medieval Latin projectum, from Latin, neuter of projectus, past participle of proicere to throw forward, from pro-jacere to throw” (Merriam Webster dictionary definition of “Project”, an Encyclopedia Britannica Co., 2014)

### “False rejects’ volume, implicit in the adoption of the camera-technology, conjures to let the Risk [of sensitive beverages ingested by Consumers] be ranging (5 - 10)%, rather than the supposed <0.1%”

### Graphene's staff played since 1990 on side of the Vendors' of machinery in the Technical Design and in the Field Engineering Departments. Designing before, later commissioning packaging and textile machinery, automated quality assurance electronic inspection equipment and related Conveyors. …

### Field Service EngineeringGraphen is built over an experience of almost thirty years in the industry of the automatic machinery and copntrols. Last twenty serving global Food and Beverage Manufacturers and Logistics. …

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