**LASER beam diffraction is an example of Binary Classification. They do not exist exclusively logical classifications: they all are Physical**

*Classification* is subject to Statistics and Applied Probability Theory studies. It is concerned with the investigation of sets of objects in order to establish if they can be summarized in terms of a small number of classes of similar objects. In the following, an introduction to the basic concepts of *Binary Classification* and their observed consequences in the industrial automated Quality Assurance, namely the Electronic Inspection. A subject vital to understand what the analytic, automated quality assurance equipment are and how they operate. The optoelectronic devices are electrical-to-optical or optical-to-electrical transducers, or instruments that use such devices in their operation.

**Information Theory is the only key to reject defective products without to reject the correct. An highly interdisciplinary field. The sectors directly related to the Electronic Inspection in the Food and Beverage Bottling Lines are: Physics, Statistics, Applied Probability Theory, Communication Theory and Computer Science**

An ensemble of these devices, gating circuits, A/D converters, amplifiers, comparators, HMI, CPUs, memories and digital I/O circuits receiving inputs (i.e., a Trigger) and controlling a rejection device. Optoelectronics and Theory of Information are the two main keys to understand all Electronic Inspectors, so to get the best of them, Quality of the products, safety of the final Customers and minimum Production losses. All Electronic Inspection equipments are a subset of the family of the Binary Classifiers. Six basic terms, whose relation is graphically displayed below:

**TP True Positive**Defective**TN True Negative**Actually non-defective**FP False Positive**Actually non-defective evaluated defective**FN False Negative**Actually defective objects, erroneously evaluated as non-defective**P Positive**Evaluated as defective**N Negative**Evaluated as non-defective

**P****rocess a statistically significative sample of containers by mean of any Inspection. In the meantime, check one-by-one all rejects' attribution. On base of the Sensitivity threshold set, they’ll arise their respective statistical distributions of the hits and of the misses, True Positives, False Positives, True Negative and False Negatives. Repeating the same test after a 2-fold increase of the containers’ speed, you’d see that the same Sensitivity of the former test is now corresponding to higher rejects TP + FP (** **abridged by Jutta234/CC BY-SA 3.0/2006)**

We will now recall the most fundamental concepts of Binary Classifiers:

Accuracy

* *

* “Degree of proximity of the measurements to the true value”*

Imagine to know that a PET container format has a specific nominal content of 1500 ml, then implying impossibility to sell bottles’ whose content is:

< ( target content – 1.5 % ).

- at a product filling temperature of 8 ºC, create 2 test bottles, whose difference on filling height equals 2 mm: 2 mm around the priorly established level,
- pass the underfilled bottle 200 consecutive times thru the fill level inspection and remark in a graphic all measurements’ results: adimensional numbers. Observable a familiar gaussian profile similar to those in the figure before,
- calculate its average and write it in the graphic, where it appears remarked as:
*Reference value*, - later, at a product filling temperature of 25 °C repeat the same identical tests, discovering a newly formed gaussian shape, similar to the prior but whose average sensibly differs by the
*Reference value*, *accuracy*is that difference, hinting to systematic differences in between the two sets of measurements: in the case considered, we know the ambient temperature created the difference. The*accuracy*of the measurements is influenced by systematic errors.

Reproducibility

* *

* “Variation arising using the same measurement process among *

* different instruments and Operators, and over longer time periods”*

In the Food, Beverage and Pharma Packaging Lines, changeovers to the same format (also, same inspection program) after a while are performed by different Production Operators. Here *Instrument* is the same electronic inspection equipment made up by several inspections, each of them differentially correlated to all what is their own *Environment. * As an example, correlated to:

- ambient temperature,
- ambient relative humidity,
- beverage temperature,
- containers’ temperature,
- static electric charge (i.e., in the Labeller Machines),
- mechanical setup,
- cleaning of the detectors,
- wearing of the mechanical parts,
- wearing of the Conveyors’ guides,
- wearing of the illumination sources (i.e., strobo flashers),
- …...

Also, Operators which considering the changeovers comprised in an extended period of time generally differ. If the electronic inspection equipment design, installation or commissioning do not allow to reproduce the conditions met in the past for the same format, false rejects shall arise. False rejects when trying to process and produce in the presently altered conditions the equipment whom parameterisation account for different conditions.

False Positives

Actually non-defective evaluated defective (i.e., false rejects).

Sensitivity

* “Test's ability to identify positive results”*

In this framework, *Positive* and *Negative* results are not meant as actual defects or actual correct objects, because also including False Positives and Negatives:

** Sensitivity quantifies the capability of an automated measurement equipment, to limit the amount of defective containers (i.e., underfilled, mislabelled, malcapped, leaking, etc.) mixed in the FMCG Packaging Line Production (** **RIA Novosti****/2015)**

Specificity

“Capability to identify as non-defective what really is not defective”

**Specificity measures the capability of our measurement equipment to specify the initial conditions of the correlation between a Detector and an object. As an example, the capability to correlate the greatest amount of histories where cap+bottle and Detector of the measurement equipment coexis****t**

In the language of a Packaging Company and its Quality Assurance and Control Depts., the capability to limit false-rejects. A definition not clearing how this may be obtained. Due to the relevance of the Specificity we’ll go much deeper than the standard statistical depth, til the limits of what is today known: the Quantum level of the measurement process. It is conceived that each collection of particle, for example a common plastic cap or a bottle in the figure on right side, comes with its own base in the *Hilbert* *space*. In the case of particles, the total space is calculated by taking tensor products between copies of the space associated to each particle. * Spaces which cannot be inferred by the Measurements’ outcomes. After considering that similar outcomes would also be obtained after a rotation of the state vector in the **state space,* *by a unitary transformation.* *We are referring to the outcomes of the Measurements sourcing the Information deriving by the physical computation whose result is named “Binary Classification”.*

The Information about the position basis and the tensor product structure is not encoded in the outcomes of our Measurements. Also (further detail here) we can only ask whether the system is in a small subset of a possible state. Those particular states for which the property we measure is well-defined. Definition meaning stability of the measured signal and amplitude over the Detector’s or Observer’s own threshold Sensitivity. Knowingly, if the initial conditions of a measurement or observation could be specified completely with infinite precision, the solution should be unique. But, the Detector (or, Observer) of a measurememt equipment *“questions the Object”* about only one of its physical properties. After establishing the correlation of the Detector of a measurement equipment and object (i.e., cap+bottle), the “answer” is encoded in their resulting superposition:

*positive answer*, i.e., an Information state “1“ or “yes”, if a specific physical property results well-defined in a small subset of a possible state;*negative answer*, i.e., an Information state “0“ or “no”, on the opposite.

on practice, … bits of Information. See figure below: at any instant of Time, there are coexisting realities compatible with those answers. Looking the fine-details, the measurement selects a subset among all solutions of the Schrödinger equation. But the amount of the remained solutions limits the possible answers to the question. Our measurements allow us to specify only partially the initial conditions. That’s why there are later more possible solutions, say more possible measurement outcomes.

Each measurement selects a subset among all solutions of the Schrödinger equation. But the possible answers to the question are limited to the remained solutions. Our measurements allow us to specify only partially the initial conditions, i.e., those at time *t*0. That’s why they coexist additional possible solutions at time *t*2 ( **Stoica, O. G. in ****eds.** **Aguirre, A., Foster, B., and Merali, Z./2015)**

With reference to the figure before, the observed fluctuations of a measurement started at time *t*0, shall show a dispersion progressively sharpening at *t*1, *t*2,… What, from a:

*Classic perspective,*was tentatively explained in terms of increased measurement*reproducibility.*Circular explanation, without any added explanatory power;*Quantum perspective,*is the effect of the selection of a subset of all the coexisting solutions of the Schrödinger equation. A selection of*histories*.

Correlation Time acts asymptotically selecting a single history where object and Detector of a measurement equipment coexist, for *t* → ∞. No measurement technology may proceed to a total retrieval of Information by macroscopic objects. *Specificity measures the capability of our measurement equipment to specify the initial conditions of the correlation between a Detector and an object. * What has a counterpart in the capability to correlate the greatest amount of histories where object and measurement equipment and its Detector coexist.

Receiver Operating Characteristic

Receiver Operating Characteristic (ROC) curve, is a plot that illustrating the performance of a Binary Classifier as its sensitivity threshold is being varied. The curve (see figure below) is created by plotting the True Positive against the False Positive rate (FP), at several sensitivity threshold settings. In the industrial automated QA Optoelectronics’ measurement and analytic applications, an Inspection is a Binary Classifier.

It may always identify positive results. Being these a superposition of the True Positives and False Positives, an excessive Sensitivity results dagnine. If forced it stops completely the Production Line if, in the meantime, another fundamental feature of the Detector or Receiver named *Specificity*, is not sufficient. Refer to the figure on side showing two gaussian distributions. Here TN, TP, FN, FP indicate in what a way Binary Classifiers separate the infeeding objects whose properties are random variables:

- right side, the normal distribution characteristic of the measurement device,
- left side, objects featuring a property which is a random variable,
*sensitivity*threshold, the vertical black colour line, being the way left us to control how-many of the objects to reject.

The individual inspections’ Technical Guarantees are typically provided by Vendors by mean of three amounts:

where the Defects' Detection ratio is

and False Rejection ratio is

Example 1

Is it Possible to Detect All Defects?

“*Signal Theory explains why the only way to eliminate 100 % of the defects implies a Sensitivity setup so high to result later in the rejection of the 100 % of the actually non-defective containers. What means to stop completely Production”*

Is it possible to reject all defects? At a first sight, setting a sebsitivity high enough, we’d be always rejecting a bottle without a cap, mixed in a flow of capped. The correct answer to the question is dual and subte:

- practically no,
- theoretically yes.

As an example, in the figure below the flow of 500 ml Coca-Cola™ Regular PET bottles includes defective containers also.

**To detect 100 % of the defects means a Sensitivity setup so high to imply rejection of 100 % of the actually non-defective containers. What implies impossibility to produce**

Defective because of any or more of the following conditions:

- filling content lower than the legal requirements for the minimum,
- cap too-high or inclined,
- cap’s tamper-evident ring broken,
- “flagging” rather than wrap-around label,
- no wrap-around label,
- not LASER- nor ink-marked,
- no Best-Before-Date (BBD) marking,
- etc.

The key point to understand is that only an improvement to a superior technology assures a decrease in the presence of these defects. Just a decrease. Vital to comprehend that Signal Theory explains why the only way to eliminate 100 % of the defects implies a Sensitivity setup so high to result later in the rejection of the 100 % of the actually non-defective containers. What means to stop completely Production. A much deeper answer is affirmative: it is possible to reject 100.000…% of the defects in the *State Space. * Say where each container has a complete description, including all *thinkable and unthinkable measurements* we have still not performed (further details here). A possibility requiring infinite Time available to establish the correlation between Detector (or Receiver) and Object.

**Mixed in this flow of bottles there are surely also those defective. Underfilled, mal positioned caps, flagging labels, etc. How can we reject all defects? Signal Theory theorems’ show that detection of 100 % of the defects means a Sensitivity setup so high to imply rejection of 100 % of the actually non-defective containers. What implies impossible production**

Example 2

Guarantees of a X-Ray Fill Level Inspection

Inspections’ technological guarantees are always referred to at least three amounts. As a practical example, we’ll cite in the following the technological guaranteed values for the X-ray Fill Level inspection declared for a Canning Line by a Vendor of Electronic Inspection equipments:

* 1.0 mm under filling level** **99.9 % 0.01 %*

*defect size and physical property defects’ detection ratio % false rejection ratio %*

Sometimes the Vendors specify a few additional informations which are decisive to compare the performances actually guaranteed. Namely, the statistical confidence intervals measured in units of one standard deviation (1σ ), that the classification shall separate two Objects differing in one of their physical properties for the indicate defect size. Clearly, to guarantee 1.0 mm under or over filling detected and rejected with 1σ or 3σ, implies performance of the equipments extremely different. As an example:

*Change in a Physical property to classify*: underfilling of 1.0 mm

*Defects’ Detection ratio*: classified to 3σ = 99.730 0204 %, meaning that less than 1 bottle each 370398 may be out of the symmetric interval, then a*False Negative*in Signal Theory language or a non-rejected defect in Bottling’s layman language

*False reject ratio: 4*σ = 99.993 666 %, meaning that less than 1 bottle each 15187 may be out of the symmetric interval, then a*False Positive*in Signal Theory language, a*False Reject*in the Bottling’s laymen language.