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Root Cause Analysis Whitepaper

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Root Cause Analysis, alike Mathematics or Physics, is a strange area where only verified truth counts, where not to make any favours is the greatest favour. Root Cause Analysis (RCA) and Failure Investigation are important tools in: 

  • Product Complaint, 
  • mitigation activities, 
  • Failure Investigation,  
  • non-conformance, 
  • Risk Management,
  • hazard analysis,

and the basic foundation of a satisfactory Corrective And/or Preventive Actions (CAPA) system. Root Cause Analysis is necessary to verify the issues including data outliers. These are often improperly and arbitrarily dismissed. RCA is required in order to close the logic loops existing on:

  • corrective actions,
  • preventive actions, 

thus allowing a proper impact analysis before to decide any action. Meaning that failure investigation and Root Cause Analysis are an important element of Current Good Manufacturing Practice (cGMP) regulations compliance, enforced by the US Food and Drug Administration.  When non adequately performed it becomes a key source of problems with respect to the norms of the Regulatory Agencies (as an example, FDA in the US, EFSA in the EU, etc.).  Companies’ resources are frequently scarce exactly in those activities that have the greatest positive effect to the Quality and Safety of their Product.  RCA is meant to reduce fire fighting, thus minimizing compliance problems also.  The expectations for meaningful Corrective and/or Preventative Actions (i.e., maintenance, modifications, etc.) supported by: 

  • Root Cause Analysis,
  • results-driven Failure Investigation, 

addressing and resolving the hidden product problems, are really growing among the Regulatory Agencies.  Two significative examples are the FDA's QSIT, referred to equipments and the EU’s ISO 14971 thinked for the Device Risk Management. A valid Corrective And/or Preventive Actions system requires defined failure investigation. This includes a systemic Root Cause Analysis devoted to the true problem resolution. Root Cause Analysis 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.   

Root Cause Analysis (RCA) in a nutshell: from the decision to apply the method to the solution of a problem

Also, 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 changes observed  in 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 the modern Physics 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 present: 

  • 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 String Theory does not consider any more  them fundamentals of Physics. 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. Deeper details about what causal relations underline, here.  

“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?”

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

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”.  They all convey Information the changes of:

  • physical properties, like dimensions, temperatures, electric currents, polarizations, momenta, etc. represented by mean of numbers part of the set of the Complex numbers  (i.e., the vector potential of a high frequency alternate voltage applied to an electronic component), including Real numbers ℝ or symbols;
  • chemical properties, i.e., 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. No change in a physical property means the minimum possible information corresponding to that property.  That is why our brains, Machinery Control Devices are “wired to look for changes”.  Root Cause Analysis, as an investigational activity, makes of the search for changes one of its main techniques to acquire knowledge Then, a Root Cause Analyst looks for changes because that is the way the Informations are encoded.  


We’ll proceed the cartesian way 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, syntesized in the figure below, is a mathematical one.  

 What is an object?  Today's only non-circular logic answer describes an entity existing in the state space M.   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  fβ . f-1α  is a map pointing to the “objective” state x. 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 (  S. Auyang/1995)

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.

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 object.   

  Whatever Detector, Inspection or Analyst filters the dense flow of information encoded in the changes implicit in the interactions between object and Detector. Filtering in the end imposing an upper bound to our possibilities to establish the entire reality about what caused the change from the former to the present state of an object (  Muller/2007) 

In the figure before 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 that an inspection may derive by an object.  The figure above let us understand that Root Cause Analysis has limits in its possibility to describe the reality of the facts which caused an actual Effect.  One and the same same macroscopic object is always composed of particles, directly entering in the signal detection process relevant for the Food, Beverage or Pharma 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; 


  • 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 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.  

 Outfeed area of a Labeller Machine controlled by an Electronic Inspection equipment with Detectors in-the-Labeller Machine and at-the-Conveyor.  The distinguishability condition is partially met by mean of 6 Triggers, 3 inductive into the Labeller + 3 LASER optic at the Conveyor, attributing before and tracking later each container's most important Information: its Identity

An example, the outfeed area of one of these, equipped with Photodetectors in-the-Labeller Machine, in the figure before.  Here the distinguishability condition is partially met by mean of 6 Triggers (3 inductive into the Labeller + 3 LASER optic at the visible Conveyor) attributing before and tracking later each container most important Information: its Identity.  For, if we cannot differentiate between two or more objects (i.e., containers) or states of the objects (i.e., where they are in a Shifting-Register, what defective property they carry), 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 the state of an object, then the object is incapable of carrying information.  

State Space

State in System Theory

We normally relate ourselves with systems of objects. Following the classic Physics concepts encased in System Theory, the state of a system is defined [M.D. Mesarovic, Y. Takahara, 1989] textually as:

“The state enables the determination of a future output solely on the basis of the future input and the state the system is in.  In other words, the state enables a “decoupling” of the past from the present and future.  The state embodies all past history of a system. Knowing the state “supplants” knowledge of the past....  Apparently, for this role to be meaningful, the notion of past and future must be relevant for the system considered”

System Theory, also if applied to modern IT technology and supercomputers, is clearly referring itself to the dated relativistic point of view.  When stating that: “The state embodies all past history of a system” it mimics the definition of Event given by Minkowski in 1907.  A state is then the superposition of all material and radiative influences present in an equitemp spatial hypersurface base of the past lightcone of an Event.  Another 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 and again, the state of a system is an explicit account of the values of the system inner components.  

State Space 

Arena of all Dynamical Phenomena 

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 definition given today by System Theory is not simply obsolete.  It is since several decades falsified by an impressive mass of experiments and newer theories. Between them, the Quantum Field Theory.  Around 75 years ago the British nobelist Paul Dirac encased its nealy created Quantum Field Theory (QFT) in a new space: the State Space. Still today, the widest existing concept of space. The 4D space-time we perceive by mean of our < 100 billion neurons, frequently named physical space, is not the State Space, rather an extremely smaller but not infinitesimal subset.  All objects we perceive directly, show us a mere instantaneous, constantly changing projection of their true existence in a higher dimensional space (≥ 10D).  Then, what is the State Space?  To start to grab this concept, think to the drum in a gun used as Russian-roulette. Russian-roulette emulates 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.  

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 gun is riskier when its drum:

  • is charged with n - 1 bullets, rather than n bullets
  • has - 1 slots, rather than slots.

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:


                          cause  ⇒  effect   

Effect in the Future represented below as a vertex (an inverted “ V ”).  In this specific example regarding a Russian-roulette gun, the Effect is a plurality of Effects.  

                                                                 risk   =   linear function (n, m) 

 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 State Space, where all of the possible states have full existence, or in its modern subset, the Hilbert SpaceThus, 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 a planet. Since then, it is no more a space limited to the objects of atomic or subatomic scale 

The State Space of an Automatic Teller Machine (ATM) (  Hammer/ 2014)  


                                             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

                               .  .  .  .  .  

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

“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”

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; 

Such a point of view makes sense of the fact that yet seven years ago the Lloyd’s (at London, United Kingdom) were financing scientific conferences rejoining theoretical physicists frequently naming Hilbert space properties and applications. 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 in the Packaging Lines, 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  or complex numbers 
  • discrete, just isolated points, i.e. the natural numbers ;
  • infinite- or finite-dimensional.
dynamical_variables. The set of all numbers includes the subsets of the > 3- dimensional Hypercomplex numbers ℍ, including the 2- dimensional Complex 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)

 The set of all numbers includes the subsets of the > 3D Hypercomplex numbers , including the 2D Complex numbers set of special interest for all dynamical systems, included the Industrial Machinery object of Root Cause Analysis.  Complex numbers  include the familiar Real numbers  (   stratocaster47/2014)

State Space Evolution is a Transformation

The laws of evolution are expressed by a transformation. Transformation that we’ll denote N in the following (for “Nature”) of the state space of every quasi-isolated physical system into itself. A mapping giving as the image of any state, the final state evolving naturally from the object state as initial state.  

The evolution of the state space is expressed by a mapping. A transformation of the state space of every quasi-isolated physical system into itself (  abridged by J. Rosen/2008)

Thus exhibiting the causal relation between initial state at time t0 and the final states at time t1, where tt0  say in the future of t0.  Not all states are obtainable as final states.   How are such states obtained to serve as initial states of processes of natural evolution ?   They are a subset of all the states, and precisely those final states of processes during which the system is not quasi-isolated.  If u is the initial state of any physical system, then N(u) is the final state resulting by its historical evolution from the initial state u.  


Aseptic Filling Process: a Transformation in the State Space

Aseptic Bloc 720x541@1x.  Aseptic Bloc. Imagine a Machine Vision system: 

by whatever Vendor, 
devoted to final inspection,
equipped with at least two cameras, 
operating standalone: no sensors in-the-Filler- and Closer-Machines.  
It shall not be possible to reject the bottle below at right side whose cap is malpositioned, with a rejection ratio over >92 % when the associated false reject ratio is <0.01 %.   Meaning 8 dangerous open bottles each 100, identically opened and consecutively present in the Inspector infeed, reaching people …not a Market !  

Contaminated bottles terminating to final Customers, whose visual detection by mean of Operators is nearly impossible, because they are: 

in fast motion, where controlled front of illuminated white panels on single-lane Conveyors;
mixed between thousands of others correctly capped, where they are checked in slow motion on multi-lane Conveyors.  Key-concept is the false rejection ratio (False Positives %).   Detection ratio and false rejection ratio are amounts whose relation grossly resembles the known reciprocal ( x y  = constant).   That <0.01 % are the inspection process' losses whoever should desire to have.   Those 99.99 % of detection ratios shown by some Vendors have full meaning only when multiplied by the conjugate amount, the false rejection ratio.  Then, to make a practical example: adopting the World’s best double camera inspection systems, it’d be necessary to sacrify <1.0 % of the production (one bottle each two hundred) as falsely rejected containers.    Losses  <1.0 % to guarantee that  >99.9 % of the leaking bottles are rejected: less than one each one thousand undetected and passing to the Market.  

Aseptic Bloc. Imagine a Machine Vision system: 

by whatever Vendor, 
devoted to final inspection,
equipped with at least two cameras, 
operating standalone: no sensors in-the-Filler- and Closer-Machines.  
It shall not be possible to reject the bottle below at right side whose cap is malpositioned, with a rejection ratio over >92 % when the associated false reject ratio is <0.01 %.   Meaning 8 dangerous open bottles each 100, identically opened and consecutively present in the Inspector infeed, reaching people …not a Market !  

Contaminated bottles terminating to final Customers, whose visual detection by mean of Operators is nearly impossible, because they are: 

in fast motion, where controlled front of illuminated white panels on single-lane Conveyors;
mixed between thousands of others correctly capped, where they are checked in slow motion on multi-lane Conveyors.
Vendors’ technical guarantees show different digits ?  

The Writer of these notes can surely fulfill and maintain the Electronic Inspector in those Technical Guarantees’ digits along the 15 years of Expected Working Life (EWL).  But, Food and Beverage Factories rarely have staff exclusively specialised and devoted to operate on the Electronic Inspectors.  Meaning that all Electronic Inspectors (exactly like the Formula1 race cars), run set at their maximum performances only during the Acceptance Test freeing successive payments.  And what about the 10-15 years following those payments ?    In brief: performances will depend much more on the training and peculiar skills of the Bottler’s Maintenance staff, than on the brand of the Electronic Inspector.

  Aseptic Filling, Capping and Inspection system in a European Dairy

“The initial states of the dynamical processes involving physical systems, are the final states of processes during which the system is not quasi-isolated”

To make an example, if the process u → N(u) is a filling process in aseptic conditions, u denotes the entire set of initial conditions: parametric recipe, the beverage temperature, the production speed determining the time available to fill each container, the containers’ capacity, etc.  Now, imagine to transform the entire physical system built up by valves, electronics, pneumatics, mechanics, beverage, environment, services, operators, etc.  

Transformation Θ may be any or more combinations of the: 

  1. spatial displacement of a component in a filling valve, 
  2. spatial displacement of a container, 
  3. deformation of a container, 
  4. spatial displacement of a cap, 
  5. deformation of a cap, 
  6. change of the environmental temperature,
  7. change of the beverage temperature,
  8. change of the pressure of the infeeding water,
  9. change of the temperature of the infeeding water,
  10. human parametric error in the Filler recipe,
  11. human error in the syrup room,
  12. human error in the choice of caps,
  13. change of the pressure of the infeeding CO2
  14. change of the mains power frequency,
  15. change of the mains power voltage,
  16. induction of intense electromagnetic interferences (emi) in the signals' low-voltage bus,
  17. jam in the Capper machine outfeed, blocking the flow of containers,
  18. manual opening of one of the machine guards, assuring machine's positive pressure (aka, aseptic process), 
  19. infeed pressure of the compressed air assuring machine’s positive pressure (aka, aseptic process),
  20. ……….

The image of the state u of a physical system is the state Θ(u), or: 

            u  →  Θ(u)

In our example, the final state resulting by its evolution from the initial state of the entire filling and packaging process: 

               u  →  N(u) 

could be the complete sequence of states (moulding-pre-heating-blowing-inspection- rinse-filling-capping-marking-inspection) experienced by a preform, when being transformed in a moulded-pre-heated-blowed-inspected-rinsed-filled-capped-marked-inspected bottle. Transforming the result N(u) by mean of Θ, gives the image state ΘN(u), or:

             N(u)  →  ΘN(u)

Symmetries in the State Space

The graph in the figure above shows an initial setup obtained by mean of a transformation Θ of the state u of a physical system, in the state Θ(u).  In this case, starting from the initial state Θ(u), the transformation taking place yields the result ΘN(u), or:

             Θ(u)   NΘ(u)

Concluding it is possible to deduce a definition for the symmetry transformations of the physical laws, as a transformation Θ for which for all experiments the transformed result equals the result of the final state reached after the transformations due to the process:

             Θ(u)  =  NΘ(u)

for all states u.  Identity hinting to a symmetry in the laws of nature. Symmetry which, in its deepest essence, can be resumed as indifference of the physical laws.  A transformation shall be a symmetry transformation of the laws of nature, if the:

  • latter ignores some aspect of the physical states, 
  • transformation affects only the ignored aspects. 

And if the initial states are a couple, rather than one?  In this case, the couple of initial states related by such a transformation shall be treated impartially by the laws of nature, so that they'll evolve into a couple of final states that are related by precisely the same transformation. The laws of nature ignores the difference between the two states, which is then preserved during the following evolution, thus re-emerging later as the difference between the two final states.  As an example, consider the fundamental spatial-displacement symmetry of the laws of nature, also named isotropy, stating that the laws of nature are the wherever identical.  Performing two experiments that are the same, with the only exception that one is made here and the other there, they'll yield identical outcomes.  Difference reduced to the location.  The set of invertible symmetry transformations of the laws of nature, i.e., the set of all invertible transformations commuting with the evolution transformation, forms a group: the symmetry group of the laws of nature.  

“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”

Considering the fact that symmetry in the state space is the fundamental concept underlying all causal relations, we’ll summarise below following Joe Rosen (2008) its nine fundamental principles: 

  1. Symmetry is immunity to a possible change;
  2. Symmetry implies asymmetry.  For every symmetry there is somewhere an asymmetry;
  3. Undifferentiability of degrees of freedom means their physical identity;
  4. Equivalent states of a cause equivalent states of its effect;
  5. The symmetry group of the cause is a subgroup of the symmetry group of the effect (Curie’s principle);
  6. Equivalent states, as initial states, evolve into equivalent states, as final states, while inequivalent states may evolve into equivalent states;
  7. The symmetry group of the cause is a subgroup of the symmetry group of the effect;
  8. For a quasi-isolated physical system the degree of symmetry cannot decrease as the system evolves, but either remains constant or increases;
  9. As a quasi-isolated system evolves, the populations of the equivalence subspaces of the sequence of states through which it passes cannot decrease, but either remain constant or increase. Also implying that the degree of symmetry of the state of a quasi-isolated system cannot decrease during evolution, but either remains constant or increases.

State Space as Events’ Arena 

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

  1. Sample space is where the observations are made;
  2. Configuration and phase 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 Analytic 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 and Dimension

“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.  Including 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.    

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)

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 (  Dong, Hong Kong University/2010)

  Knots disappear, resolved as unentangled open lines, when observed by a higher dimensional vantage point (K M.W. Evans, et al./2001)

Sample space may be thinked as one of the possible projections of the state space. 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.  An inverse approach to the difference between State and Sample Space is hinted in the graphics below.  Here represented the transition functions between two charts on a manifold.  The graphic at left side shows a space in the set of the Real numbers n.  Implicitly meaning that the amount of mutually orthogonal coordinate axes may be potentially infinite.  At right side a 2D surface whose points pertain to the set 2.  Here, two dark-grey sets Uα, Uβ have a point in a common light-grey area where the sets Uα and Uβ superimpose each other: their intersection set Uα  Uβ.   Visibly, that common point in the intersection set, when viewed from the vantage point of a higher dimensional space, reveals all its fine-structures.   gαβ is the relation existing in the higher dimensional space between the transition functions α and β.  What precedes and follows makes sense of a relatively famous theorem of Knots Theory.  Theorem stating that knots existing in a space with dimension d = i  unentangle and disappear when looked in a space with dimension D > i.  The light-grey intersection set may represents the outcome of a measurement fully compatible with two different, however superimposed, spaces.  As an example, the intersected sets of the Positive and of the Negative outcomes of the measurements (de-facto, an Information Retrieval action) performed by a Binary Classifier, like an Electronic Inspection equipment (see graphics below, at right side) or the widely used Google Search™ algorithm.  In all these applications, the core problem is that we are always front of outcomes coherent with more than just one scenario.  Worse, mutually excluding or nearly-mutually excluding scenarios.  

 Transition functions between two charts on a manifold. Mapping a set into a higher dimensional space reveals details otherwise invisible

“the automated measurement equipments always detect values in low-dimensional spaces”

Hence, the key to understand the deep signification of the observed appearance of False Positive and False Negative (FP and FN in the figure on side), corresponding in the Electronic Inspection Binary Classifications to actually non-defective rejected objects (aka, Production losses) and actually defective non-rejected dangerous objects.  The figure below on right side clearly shows the Superposition from which this enigma takes its nearly full origin.  Only “nearly full” rather than “full” origin, in light of the most modern conceptions of Theoretical Physics.  Ascribing also to Entanglement and Initial Conditions (i.e., amount of degrees of freedom) a causal role for what we observe today wherever, and also in the technological processes.  As an example pertinent to our special metrologic cases and technologies, a sequence where: 

  1. the black point in the graphics above is a group of numbers representing the outcome of the measurement of the observable: 

                            pixel (x, y, t, voltage)

  1. corresponding to the illumination of a certain pixel in a CMOS imaging sensor, after the A/D conversion reducing it to a Natural number normalized in a range depending on the CMOS characteristics;
  2. at a certain Time t or, more efficiently, associated ina Shifting-Register to the value simultaneously counted for the pulses outcoming by an Encoder referred to a Trigger; 
  3. voltage digitised value in neighbourhood of one of the grey range boundaries, the programmed upper or lower limit of a range of grey levels.  So close to result (i.e., included in a certain grey-range with probability 55 % and in the adiacent other with probability 45 %) to make it a difficult to decide. Uncertainty which is a close relative of the False Positives (FP) and False Negatives (FN) in the graphics at right side.

The key point is that mapping a process defined in a space with a certain dimension to a space of superior dimension allows a quantitative definition of a the relation gαβ existing between the transition functions α and β.  In nowadays technological applications, we are rarely or never measuring all State Space's fine-structure and details.  Nearly or never measuring coordinates and momentums at all its points. Meaning that the available automated measurement equipments always detect values in low-dimensional spaces like that at right side in the graphics above.  A further example for this extremely relevant fact in the way the Information about the state of a physical property (i.e. capping, sealing, fill level) is retrieved today by the Machine VIsion imaging equipments based on cameras.  Part of the electromagnetic energy of the photons collected by a each pixel along a certain time, is transformed in an electric potential, later digitalised and associated as a fourth Euclidean coordinate to the couple of coordinate axes x, y identifying that pixel in the matrix.  Having this way a basic representation of the property illumination of a certain pixel at a certain time t, by a vector: 

                                  ( x, y, t, voltage )

whose elements are always Natural or Rational numbers. Just an infinitesimal subset of the Real numbers set .  But, the subspace of the State Space where that pixel is fully representated, where models and computations have been applied during its design phase, as well as the 4-vector representative of each photon collected by that pixel, is in the Hilbert Space H.  And the Hilbert Space  is a complex-valued vector space.  Here each point includes different Real numbers .  

Expected Results 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”

Considering the amount of variables represented by Real numbers  and the fact we measure just an outcome in the multitude, 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”.  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, 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 3-axis' simmetry shape, as seen by a human perspective, has stable positions in a gravitational field only over one of these sides.  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 rapid changes controllable only be mean of instruments, the reason why we assume to be the six values seen 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 2D case where x1  and  x2  could represent for example, a couple of spatial coordinates, momenta, etc.   We see that however narrow a cylinder of radius ε with the 0t  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.  Visibly, 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.   

Example 1 

Expected Result with a LED Illuminated 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 μs.  But, what about if the pulse should be 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.

led brief pulses exist as m

 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 2 

Expected Results for the Forces Making Electron's Stability

detection of brief events med hr

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” 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 Lab. One of those capable to record Events of extremely brief duration (  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.  You’ll see a quite different shape for the same “object”.  Different but not less real that the former.  An example of just one of the many results expected 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  2 Dices in the State Space

sample space of two dices med

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., those of the Mach-Zehnder interferometer) show “unexpected results”.  As an example, imagine to beam a single photon originating by a laser  

  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 exists a way to manufacture them that ideal way, perfectly symmetric, perfectly balanced over all their sides.  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 Time

dices and sample space in med

source of light (a particularly monochromatic one) toward one of the six faces of a cube made of quartz.  We’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 !”   

     “God plays dices ...and they are tricked !”  Also counterintuitive results, with an apparently impossible blank side, have to be accounted for 

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

 In 1970 the great German physicist Dieter Zeh discovered why, at our scales, all processes look irreversible. Why the glass panes’ tendential state, their future expected evolution, is that of a broken glass.  A reason different than the Thermodynamic teached us a few decades ago

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?  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.  


 When closely looked, all 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 (  Klimenko, Maas/2014)

Binomial Distribution and RCA.  In the example referred to the tossing of 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.   Binomial distribution, and its derivated Normal, implicitly indicate how a present scenario or, outcome, may be quantitatively related to a certain Cause

  When tossing 2 dices there are 6 ways to get a total of 7, but only 1 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 or probable, 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 

Sample space associated to a macroscopic dynamical system, like a Packaging Machine, one of its assembles or an apparently simple Detector device.  Yet the Binomial Distribution of Statistics, origin of the Normal (or, Gaussian) Distribution, also if in an implicit way, indicates a quantitative 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 in 1927 by Erwin Schroedinger, one of the founders of Quantum Mechanics, about the different meaning of the wave function when applied to a space hosting: 

  • 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 dices, the different possibilities for the total of the two dices 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

Geometric meaning of the Phase Space (  Thiemann/2007) 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 space 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:

  Geometric meaning of the Phase Space


        L  =  T  -  V

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

                                        pi  =   ∂L / ∂qi 

state space

where ∂qi  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 qi.  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 configuration 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

  Geometric interpretation of the phase space (  abridged by Thiemann/2007) 


Phase space is typically teached as a continous and finite-dimensional space.  But, back in the year 1900 what really Max Planck's analysis of the black-body radiation's spectrum established was in effect a discreteness. Discreteness of the phase space which should be directly detected if the actual measurement systems could allow us to examine spatial separations of around the Planck length (~10-35 meter) and temporal separations of around the Planck time (~10-43 second).  Phase space is a high dimensional mathematical space.  One where each spatial degree of freedom, in a many-particle system, is accompanied by a corresponding momentum degree of freedom.  The figure before shows the phase space in its most general and continous 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 qi  are 3N position coordinates and pi  are the related 3N generalised momentum coordinates.   Applying Calculus' methods, it can be demonstrated that the generalised momentum coordinates  pi  can be obtained from the position coordinates and the Lagrangian, using the partial derivatives equation:                                     

state space

                             pi  =   L / ∂qi 

where qi  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 qi.  The space of the q and p coordinates specifying a physical system is named phase space.  Each point in the phase space represents 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 configuration space, which is the object of the next section.

  How many kettles are there?  A single kettle occupies three different sets of points, three different subspaces of a common State Spacefollowing 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 1D system is called a phase line, while a 2D 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.   

  Composition of the Configuration Space C  (  abridged by R. Penrose/2010)

Root Cause Analysis’ Rationale

After this introduction we start to have all elements to conceive the rationale of Root Cause Analysis. In the world of the extremely complex dynamical systems, to know with a certain precision what it happened in the Past, does not allow to know the Future.  That is why Root Cause Analysis methods have to be applied to the industrial Machinery.   Simply, it is not sufficient to create some Alexandria-like bibliotheque of Past machinery’s-, assembles’- or devices’ known operation and 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 oscillator.  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).   

   Following the dynamical evolution of a simple nonlinear oscillator, allows to understand much of Root Cause Analysis' rationale.  Root Cause Analysis is made in one of the terminal branches developed at the right side of the graphics.   Being there and trying to “imagine” from what time-ordered chain 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 Machinerys, Equipments or Processes log so many data.  By the knowledge about a few Past branchings, then it is impossible to predict what should be the Present couple:  (Value,  Amplitude).   As an example, the actual status of failure 

A Strategy Deemed to Failure

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:

   Evolution of a defined volume Γ(t)  in the phase space.  The region Γ(t) represents the information we have about a system at three distinct and successive times t = 1, 2, 3.   Visibly, the information we have does not increase.  Sturm-Liouville’s Theorem holds its full validity, included those mesoscopic and macroscopic space-time scales where the Equipments, Machinery and Devices operate ( abridged by Susskind/2005)

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 they started to be observed and their properties registered.  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.

  A branching diagram ( abridged by Welch, Morgan/2014) illustrates how 4 histories derive by 3 time-ordered Events.  What in principle seems to suggest that we can know a Past Event, e.g., the Root Cause for a Present failure.  But, we do not know the initial Conditions.  Also, each one second of Time can correspond up to nearly 1043    3D spatial time-slices.  Implying an amount of random variables associated to each one of the Events making the histories above, potentially occupying at least as many locations of memory as 1043 

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:

  1. missing knowledge about the initial conditions of the system;
  2. 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.     

    Servers in a Google, Inc. Data Centre.  All actual Google’s Data Centres added to all others existing in the World, are not enough to record all the values assumed by the variables of a common equipment, i.e. a PLC and its analog and digital I/Os, during one second of time ( Google, Inc./2014)

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

  1. 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; 
  2. 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;
  3. have just few variables of the process 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.   

phase space increase med

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 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 A?

  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 (  Klimenko, Maas/2014)

Noise is one of the correct answers.  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 depiction of the entire spectrum of the states existing into the sets A0, A1, A2, …, Ai, …, An-1, An  in the past, giving origin to the problem B at present time t1.    What accounts to:

  1. create categories of states, following criterias of homogeneity and relevance with respect to the problem B,
  2. 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, co-exist before the action or measurement.  Namely, 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 can be resumed as:

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

The amount of linearly independent vectors gives the dimension N of the Hilbert space H.   The number N of degrees of freedom of the system is the natural logarithm of the dimension N of its Hilbert space H, where: 

                                     N  =  ln N  =  ln dim(H )

The number N of degrees of freedom is equal to the number of bits of information needed to characterize a state.   Then, the 6-slots drum visible before accounts for  N = 2states, say  N = 64 ln 2 degrees of freedom.   Meaning that this Russian-roulette gun: 

  • state is completely specified by 64 bits of information; 
  • can be used to store 64 bits of information.   

The computation presented here is an abstraction 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 elementary components like electrons, pions, kaons, different kinds of mesons, neutrinos, etc.   Whoever agrees that richness of an object, evaluated as amount (then, a measure) of structures and substructures, is an indicator of the object’s complexity.   As an example, the Information content embodied by that drum, and measuring drum's complexity by the amount of the microstates of all its atomic and subatomic components, ranges ~ 1024.   We are understanding that, when using those 64 bits to estimate the 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


phylogenetic tree med hr

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 (  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 or Darwin 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  fβ  .  f -1α  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 ( Yale University, Peabody Museum of Natural History/2014)

phylogenetic rules

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 extended in the space and simultaneously occupying several leaves. (Objects of a detailed presentation here).  

 Changes are what 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( f0β ))).   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, … ( 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( f0β ))).   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:

  1. 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;
  2. 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”;
  3. naturalist's point of view, yet since thousands of years (correctly) hinting to a tree-like structure for the entire genealogy of Life;
  4. 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 first entered in the physical sciences by the mathematical door of Graph Theory, a branch of Discrete Mathematics.

quantum-computation med

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 (  Deutsch/2001)  

Normal Distribution: New Meaning of a Classic idea

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 3D space, including the 2D 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( f0β ))), 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 )  =  gβ )  ≠   β,  
  • with gβ ) being the measured value of the state of 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 4 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 changes, from the state β to its future state gβ(  Deutsch/2001) 


gaussian function 4096x3140@1x

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 on side.  

 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 independently derived by de Moivre, Gauss, Adrian and Laplace (  Kaushik Ghose/en.wikipedia/2006/CC BY-SA 3.0) 

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.  See figure below.  At a macroscopic and classically available scale of dimension, we hint the Reader to simply imagine how many superimposed z(x,y) points lie into each white coloured circle.  Points sharing a same z coordinate corresponding to different couples of measurements (x,y).  The new perspective, is an abstraction of the observation following which a single outcome is expected by several sums or superpositions of the outcomes of dices (see section “Expected results tossing two dices” above).  Abstraction to physical systems simultaneously processed each of them differently correlated with all the other systems (Environment). 

 Processing many couples of measurements (x,y) results in several identical outcomes z contained in the same circle.  A behaviour paralleled when throwing a couple of dices, knowingly resulting in many couple of outcomes whose sum (or, superposition) is a single value.  Same way, each measurement involves many different correlations between the object and its Environment, later converging to a single value

New Meaning for What, Where, When, 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 changes, from the state initial value β to its future state gβ );
  • what happens during the initial reduction, when  -1 < t  < 0,  is mirrored by a gaussian shape identical to that observed during the final interference happening when< 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 which is the Probability associated to each one of the Events labelled by  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 non-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 otherwise 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 matured by who yet red 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.   Modern is its application to the solution of 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 a few Root Cause, 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 (  abridged by Pascal Casanova)  

One of the examples we’ll consider elsewhere in this web site, between the Case Studies shows step-by-step how to correctly differentiate Causes by Conditions.   In general, the Root Cause is: 

  • one in a multitude of Conditions;
  • the one closer to the Effect: immediately before the Effect.

No Condition, interposed between the Root Cause and the onset of the Effect, may exist.

Problem Solving Strategies


Root Cause Analysis, alike Mathematics or Physics, is a strange area where only verified truth counts, where not to make any favours is the greatest favour.  And 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, 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 3D leaf (or, sheet) part of the 4D foliation.  

    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 completely external or internal to a Machine, device, process or procedure”

In that period different paths, relativistic and quantomechanical, converged to the same result (further details here).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 meaning that no observed problem is 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 cause and effect.  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. 

rca 5 med hr

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.  Finally, each one of the vertexes tilted upside-down, is the place where a couple of superimposed states interferes, becoming a single physical state

Bias’ Influence over Root Cause Analysis 

Unbiased analysis:  ΩRCA = 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 we observe from 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 (  Haade/CC BY-SA 3.0/2007)

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: 


  1. since when the many fields of Engineering are allowed to concentrate the controls just in a thin solid angle ΩRCA ?  
  2. 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 ΩRCA ?  
  3. Is it in the interests of the investigation of the truth to limit ΩRCA 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: its Unbiased Root Cause Analysis

Beechcraft’s oxygen tank improvements applied to Apollo 14 in the weeks after Apollo 13 incident and thorough comprehension of its causes thanks to Root Cause Analysis (  NASA/1970)   

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, five 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, starting by Apollo 14.  

Logically Flawed Analysis

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

The NASA example seen before ranks between the best examples of the Problem Solving power of Root Cause Analysis.  Rather the opposite of false or fallacious, for Greeks truth is a-letheia (α-λήθηεια).   Namely what is unveiled, revealed, not-ignored.   Continuously searching for minimal traces that would allow us to open doors on new realities.   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 if is it a fact:

  • less factual because we are not aware of it?   
  • more factual, because we perceived its existence?  

This kind of erroneous logic does not correspond to a biased analysis: it is a true logic error.   Its consequences are negative because we’ll terminate to define the wrong and not the true Root Cause for an observed Effect.

Biased Analysis   ΩRCA  ≪ 4π steradians

“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”

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 gets 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”

               “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 long-term beneficial effects.   It’s impossible to let a Packaging Machine function as prescribed by a Contract and related Technical Guarantees, thanks to Retorics disparaging contradictions, tautologies and lies.   That’s why no one root cause Analyst is allowed to straightly go to his or 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 ΩRCA of the 4π  sterad 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:

  1. the running state of the Industrial Machinery and Equipments is unnatural: if we leave them alone, soon they’ll stop themselves;
  2. 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;   
  3. 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:

  1. To provide a systems framework for the analysis and modeling of the issue;
  2. 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;
  3. To examine and illustrate methods to recreate the same issue in systems apparently not affected;
  4. To guide the readers to mature a critical thinking-based opinion about what, precedentely, Experts declared were the causes of the issues;
  5. To provide methods for visualizing the causal relation existing between causes and effects;
  6. To provide examples of other practical applications in the same area, where the same issue is not felt.

Links to other subjects:

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