DE102021212157A1 - Process for improving the statistical description of measured values - Google Patents

Abstract

A method is proposed for improving the statistical description of measured values of parts or devices with the same characteristic, comprising:- providing measured values from a plurality of the parts or devices, the measured values relating to the same characteristic;- establishing a density function of a Gaussian Mixture distribution based on the measured values;- Fitting the density function of the Gaussian mixture distribution using an iterative application of the Expectation Maximization algorithm; wherein the density function is changed in such a way that a component of the density function is removed as soon as the variance of this component falls below a predefined threshold value during the iterative application of the expectation maximization algorithm.

Description

In the statistical description of measured values - especially in statistical process control and the prediction of expected measured value distributions with statistical means - the adaptation of statistical distributions to the measured values is a fundamental step. Measured values can be, for example, dimensions, radii or other geometric or structural or physical properties of parts or devices. Parts can be, for example, components or semi-finished products. When “parts” are spoken of, they can mean structurally identical parts or parts that have at least one structurally identical feature.

In the simplest case, measured values of a certain (structurally identical) feature of the parts follow a unimodal distribution and can typically be represented by a Gaussian distribution (normal distribution). In practice, however, it is often the case that the measured values follow distributions that have multiple peaks. A multi-peak can, for example, be due to the fact that parts on which the measured values have been measured are subject to differences in production. Differences can arise, for example, between production during early shifts and late shifts or due to different batches of basic materials or basic components of the parts. Parts that have the same properties, for example parts from the early shift or parts that are based on the same batch of a basic ingredient, can each follow a unicodal distribution. If, however, parts from different layers or parts that have basic components from different batches are considered together, more complex, in particular multi-peak distributions can result, since the unimodal distributions are superimposed.

• • Messwerte von Teilen aus Frühschichten folgen einer Normalverteilung $N_0\left({\mu }_0,{\sigma }_0^2\right);$
  
• • Messwerte von Teilen aus Spätschichten folgen einer Normalverteilung $N_1\left({\mu }_1,{\sigma }_1^2\right);$
  

wobei 2/3 der Teile in Frühschichten hergestellt werden. Die Dichtefunktion der Gaußschen Mischverteilung ergibt sich zu: $$ƒ\left(x\right)=\frac{2}{3}\cdot {ƒ}_{N\left({\mu }_0,{\sigma }_0^2\right)}\left(x\right)+\frac{1}{3}\cdot {ƒ}_{N\left({\mu }_1,{\sigma }_1^2\right)}\left(x\right),$$

wobei ${ƒ}_{N\left(\mu ,{\sigma }^2\right)}$

eine Dichtefunktion einer Gauß-Verteilung (Normalverteilung) mit den Parametern µ, σ² ist. µ bezeichnet den jeweiligen Erwartungswert, σ² die jeweilige Varianz. Die einzelnen Dichtefunktionen ${ƒ}_N$

der Gauß-Verteilungen (Normalverteilungen) werden als Komponenten der Dichtefunktion der Gaußschen Mischverteilung bezeichnet. Das Funktionsargument x ist eine Eingangsgröße und stellt praktisch einen Messwert dar, für den die Dichtefunktion der Gaußschen Mischverteilung ausgeführt werden soll. Die Dichtefunktion der Gaußschen Mischverteilung ergibt eine Kurve. Für ein vorgegebenes Intervall ergibt die Fläche unter der Kurve (das Integral) in diesem Intervall eine Wahrscheinlichkeit, dass ein Messwert in diesem Intervall liegt.A frequently used model for such multi-peak distributions is the Gaussian mixed distribution. It consists of a number n of normal distributions that are superimposed and each have weightings. An example is given below to illustrate this:

• • Measured values from parts from early shifts follow a normal distribution $N_0\left({\mathrm{<figure-callout\; id="0"\; label="\mu "\; filenames="DE102021212157A1\_0030.png"\; state="\{\{state\}\}">\mu </figure-callout>}}_0,{\mathrm{<figure-callout\; id="0"\; label="\sigma "\; filenames="DE102021212157A1\_0030.png"\; state="\{\{state\}\}">\sigma </figure-callout>}}_0^2\right);$
  
• • Measured values from parts from late shifts follow a normal distribution $N_1\left({\mu }_1,{\sigma }_1^2\right);$
  

2/3 of the parts are produced in early shifts. The density function of the Gaussian mixed distribution results from: $$ƒ\left(x\right)=\frac{2}{3}\cdot {ƒ}_{N\left({\mathrm{<figure-callout\; id="0"\; label="\mu "\; filenames="DE102021212157A1\_0030.png"\; state="\{\{state\}\}">\mu </figure-callout>}}_0,{\mathrm{<figure-callout\; id="0"\; label="\sigma "\; filenames="DE102021212157A1\_0030.png"\; state="\{\{state\}\}">\sigma </figure-callout>}}_0^2\right)}\left(x\right)+\frac{1}{3}\cdot {ƒ}_{N\left({\mu }_1,{\sigma }_1^2\right)}\left(x\right),$$

whereby ${ƒ}_{N\left(\mu ,{\sigma }^2\right)}$

is a density function of a Gaussian distribution (normal distribution) with the parameters µ, σ ² . µ denotes the respective expected value, σ ² the respective variance. The individual density functions ${ƒ}_N$

of the Gaussian distributions (normal distributions) are called components of the density function of the Gaussian mixed distribution. The function argument x is an input variable and practically represents a measured value for which the density function of the Gaussian mixed distribution is to be carried out. The density function of the Gaussian mixed distribution gives a curve. For a given interval, the area under the curve (the integral) in this interval gives a probability that a measured value lies in this interval.

mit den Gewichtungsfaktoren α_i. Im obigen Beispiel ist α₀ = 2/3 und α₁ = 1/3. Der Metaparameter N gibt die Anzahl der Komponenten vor. Im obigen Beispiel ist N = 2.In general, the formula for a density function of a Gaussian mixed distribution is: $$f\left(x\right)={\displaystyle \sum _{i=1}^N\left({\alpha }_i\cdot {ƒ}_{N\left({\mu }_i,{\sigma }_i^2\right)}\left(x\right)\right)},$$

with the weighting factors α _i . In the example above, α ₀ = 2/3 and α ₁ = 1/3. The meta parameter N specifies the number of components. In the example above, N = 2.

For a certain, fixed N, i.e. for a certain number of components, a parameter Θ can be set up with $$\Theta =\left[{\mu }_1,...{\mu }_N,{\sigma }_1^2,...,{\mathrm{<figure-callout\; id="2"\; label="\sigma \; N"\; filenames="DE102021212157A1\_0027.png,DE102021212157A1\_0029.png"\; state="\{\{state\}\}">\sigma </figure-callout>}}_N^{},{\alpha }_1,...,{\alpha }_N\right].$$

The density function of the Gaussian mixed distribution can now be formulated particularly simply as ƒ (x; Θ).

In order to describe the measured values in a statistically accurate way - for example for the purpose of process control or the prediction of processes and expected measured values - a suitable value of the metaparameter N (number of components) must be found and a suitable parameter Θ must be determined. "Suitable" means that the density function of the Gaussian mixed distribution ƒ (x; Θ) reflects the real measured values as realistically as possible, the density function of the Gaussian mixed distribution ƒ (x; Θ) is therefore well adapted to the real measured values and their frequencies.

In the following, let S = {s ₁ , ..., s _K } be a set of measured values (total of K measured values) to which a density function of the Gaussian mixed distribution is to be adapted. First, a criterion is defined to quantify how well a determined density function of the Gaussian mixed distribution fits the set of measured values. A so-called likelihood function (sometimes also called plausibility function) is usually used for this purpose. It is: $$\mathrm{L.}\left(\mathrm{S.};\Theta \right)={\displaystyle \prod _{i=1}^Kƒ\left(s_i;\Theta \right)}.$$

A higher value of the likelihood function is interpreted as a better fit. For numerical reasons, the log-likelihood is often used in practice. It has the same property that a higher value of the log-likelihood function is associated with a better fit.

A common procedure for adapting a density function of a Gaussian mixed distribution to a set of measured values (the set of measured values can also be referred to as a “random sample”, if applicable) is an iterative method. The starting point is a parameter for iteration step P: $${\Theta }^{\left(\text{P.}\right)}=\left[{\mu }_1^{\left(\mathrm{P.}\right)},...,{\mu }_N^{\left(\mathrm{P.}\right)},{\sigma }_1^{2\left(\mathrm{P.}\right)},...,{\sigma }_N^{2\left(\mathrm{P.}\right)},{\alpha }_1^{\left(\mathrm{P.}\right)},...,{\alpha }_N^{\left(\mathrm{P.}\right)}\right]$$

It should be noted that an estimate and / or an application of known heuristics can be used to determine a starting value of the parameter - as the basis of a first iterative step.

First, information about the expected frequencies of the measured values is calculated from the current parameter. They calculate from: $$H_j\left(t\right)=\frac{{\alpha }_j\cdot {ƒ}_{N\left(st;{\mu }_j,{\sigma }_j^2\right)}}{{\displaystyle {\sum }_{\text{u}=1}^N\left({\alpha }_u\cdot {ƒ}_{N\left(st;{\mu }_j,{\sigma }_j^2\right)}\right)}},t=\mathrm{1,...,}K;j=\mathrm{1,...,}N$$

$${\mu }_j^{\left(P+1\right)}=\frac{{\displaystyle {\sum }_{t=1}^K\left(h_j^{\left(P\right)}\left(t\right)\cdot x_t\right)}}{{\displaystyle {\sum }_{t=1}^K{h}_j^{\left(P\right)}\left(t\right)}},$$

$${\sigma }_j^{2\left(P+1\right)}=\frac{{\displaystyle {\sum }_{t=1}^K{h}_j^{\left(P\right)}\left(t\right){\left(x_t-{\mu }_j^{\left(P\right)}\right)}^2}}{{\displaystyle {\sum }_{t=1}^N{h}_j^{\left(P\right)}\left(t\right)}},$$

woraus sich der neue Parameter ergibt: $${\Theta }^{\left(\text{P}+1\right)}=\left[{\mu }_1^{\left(P+1\right)},\dots ,{\mu }_N^{\left(P+1\right)},{\sigma }_1^{2\left(P+1\right)},\dots ,{\sigma }_N^{2\left(P+1\right)},{\alpha }_1^{\left(P+1\right)},\dots ,{\alpha }_N^{\left(P+1\right)}\right]$$

With the help of formula (6), the components of the parameter are adjusted below in such a way that the value of the likelihood function increases. In iteration step P + 1, based on the expected frequencies in relation to iteration step P: $${\alpha }_j^{\left(\mathrm{P.}+1\right)}=\frac{1}{K}{\displaystyle \sum _{t=1}^K{H}_j^{\left(\mathrm{P.}\right)}\left(t\right)},$$

$${\mu }_j^{\left(\mathrm{P.}+1\right)}=\frac{{\displaystyle {\sum }_{t=1}^K\left(H_j^{\left(\mathrm{P.}\right)}\left(t\right)\cdot x_t\right)}}{{\displaystyle {\sum }_{t=1}^K{H}_j^{\left(\mathrm{P.}\right)}\left(t\right)}},$$

$${\sigma }_j^{2\left(\mathrm{P.}+1\right)}=\frac{{\displaystyle {\sum }_{t=1}^K{H}_j^{\left(\mathrm{P.}\right)}\left(t\right){\left(x_t-{\mu }_j^{\left(\mathrm{P.}\right)}\right)}^2}}{{\displaystyle {\sum }_{t=1}^N{H}_j^{\left(\mathrm{P.}\right)}\left(t\right)}},$$

from which the new parameter results: $${\Theta }^{\left(\text{P.}+1\right)}=\left[{\mu }_1^{\left(\mathrm{P.}+1\right)},...,{\mu }_N^{\left(\mathrm{P.}+1\right)},{\sigma }_1^{2\left(\mathrm{P.}+1\right)},...,{\sigma }_N^{2\left(\mathrm{P.}+1\right)},{\alpha }_1^{\left(\mathrm{P.}+1\right)},...,{\alpha }_N^{\left(\mathrm{P.}+1\right)}\right]$$

In summary, the number N of components is selected first. A specification or an estimated value can be used for this purpose. A start value for the parameter Θ is then initialized according to known methods (estimation, heuristics). Then Θ is changed iteratively in order to improve the value of the associated likelihood function, i.e. to increase it.

This procedure described and shown here in abbreviated form is called the expectation maximization algorithm (“EM algorithm”). The iteration rules shown generally have the effect that the value of the associated likelihood function becomes better or at least remains the same in each iteration.

A prerequisite for the correct functioning of the EM algorithm is, however, that the number of components N of the density function of the Gaussian mixed distribution has been selected to match the (real) measured values. This can mean in particular that the number of components N corresponds to the number of peaks of the real distribution, that is to say that the peaks can be correctly represented by the density function of the Gaussian mixed distribution. If too few components were used, the density function of the Gaussian mixed distribution does not have the necessary flexibility to be able to represent a distribution of the measured values well and realistically.

If, on the other hand, too many components are used, the EM algorithm can lead to so-called degenerate components. When adapting the density function of the Gaussian mixed distribution, it can happen, for example, that if the real measured values correspond to a distribution that could be easily represented by two components, two components of the density function of the Gaussian mixed distribution emulate the (real) distribution well, and a third one Component is "free". If the EM algorithm maximizes a value of the likelihood function (or the log-likelihood function), such a free component is adapted to a single measured value or to a set of very few measured values. Specifically, this means that the variance of the free component continues to approach 0 with each iteration step. Thus, an assigned probability for a range around the measured value approaches infinity. This leads, on the one hand, to an unrealistic course of the density function of the Gaussian mixed distribution and, on the other hand, to numerical problems and ultimately to a non-evaluability and thus non-usability of the density function of the Gaussian mixed distribution.

It is an object of the present invention to avoid the described problem of degenerating components so that the described consequences - an unrealistic course and non-evaluability and non-usability of the density function of the Gaussian mixed distribution - can be avoided.

• - Bereitstellen von Messwerten von einer Mehrzahl der Teile oder Vorrichtungen, wobei sich die Messwerte auf das gleiche Merkmal beziehen;
• - Aufstellen einer Dichtefunktion einer Gaußschen Mischverteilung auf Basis der Messwerte;
• - Anpassung der Dichtefunktion der Gaußschen Mischverteilung mithilfe einer iterativen Anwendung des Expectation-Maximization-Algorithmus;

gekennzeichnet dadurch, dass
die Dichtefunktion derart verändert wird, dass eine Komponente der Dichtefunktion entfernt wird, sobald bei der iterativen Anwendung des Expectation-Maximization-Algorithmus die Varianz dieser Komponente unter einen vordefinierten Schwellwert fällt.A method is proposed for improving the statistical description of measured values of parts or devices with the same characteristic, having:

• Providing measured values from a plurality of the parts or devices, the measured values relating to the same feature;
• - Establishing a density function of a Gaussian mixed distribution on the basis of the measured values;
• - Adaptation of the density function of the Gaussian mixed distribution with the aid of an iterative application of the expectation maximization algorithm;

characterized in that
the density function is changed in such a way that a component of the density function is removed as soon as the variance of this component falls below a predefined threshold value during the iterative application of the expectation maximization algorithm.

The measured values can in particular be dimensions (for example in m, cm, mm, μm or nm) or radii (for example in m, cm, mm) or angles (for example in degrees or rad). The measured values can have been determined in advance using measurement technology. The measured values can also include temperatures (e.g. in ° Celsius or Kelvin), chemical and / or material composition of the parts (e.g. a metal content within an alloy) or other structural properties (e.g. a surface roughness), others physical properties (e.g. density; electrical or magnetic properties, e.g. resistance, conductivity, inductance, capacitance; thermodynamic properties, e.g. heat capacity; speeds; accelerations; optical properties).

The parts can be, for example, workpieces, components or semi-finished products. The parts have the same (e.g. structurally identical) feature, for example a dimension, a radius or an angle. As an alternative or in addition, identical or structurally identical parts can be involved. The measured values refer to the structurally identical feature. This can mean, for example, that the measured values are determined at the same point of the individual parts or at the same structural and / or geometric feature, so that - if the parts were exactly the same - an identical value would result.

The devices can be, for example, devices, for example medical devices, optical devices (microscopes, cameras, binoculars), optoelectronic devices, metrological devices (e.g. coordinate measuring devices), semiconductor devices (e.g. devices used in semiconductor production).

It should be mentioned that the method described can also be used in other applications of the expectation maximization algorithm. Other applications need not relate to measurements from parts or devices. Other applications can also relate to measured values from parts or devices that are, however, not structurally identical.

The described method can be carried out in an automated manner, in particular with the aid of a computing device, for example a computer (in particular having a processor, a power supply, a main memory, an input and an output device and a storage medium). The claimed steps can be carried out in the order: "Providing the measured values ..."; "Setting up the density function ..."; "Adaptation of the density function ..." are carried out, in particular carried out one after the other, so that the next step begins after the end of the previous step.

Changing the density function by removing a component means in practice that a density function of a Gaussian distribution (normal distribution), i.e. a summand, is removed from formula (2).

A density function with too many components can be compared to a mechanical system that has one degree of freedom too much - that is, a free degree of freedom that, for example, enables uncontrolled movement or displacement of a part of the mechanical system.

The predefined threshold value can in particular be an empirical value. For example, it can correspond to a variance that would cause a probability to increase too sharply, for example an increase to more than one and a half times or more than double a peak value of the measured values. Other methods are conceivable. For the present invention it is mainly and essentially relevant that a component of the density function is removed at all. In particular, the removal can be carried out as soon as the first signs of degeneration of the component appear, but the degeneration has not yet progressed so far that numerical problems and / or a completely unrealistic course of the density function results.

The variances can be checked for all components before or after each application (that is, before or after each iteration step) of the EM algorithm.

The method presented represents a simple way of avoiding the problem of degenerating components in applications of the expectation maximization algorithm. In this way, the reliability, quality and realism of statistical descriptions of the measured values can be improved, in particular with the aid of a density function of a Gaussian mixed distribution. The computing effort and the programming effort are low. Predictions about the properties of the measured values can be improved, statistical process control is improved.

In one embodiment of the method according to the invention, after the component has been removed, the iterative application of the expectation maximization algorithm to the changed density function is continued or carried out again.

This means that the density function, which has now been reduced in its number of components, forms the basis for further applications of the expectation maximization algorithm. This refinement allows the advantages of the method according to the invention to be used easily. A program code that allows an iterative application of the EM algorithm with the original number of components can be easily changed or does not require any change in order to implement an application of the EM algorithm for the reduced number of components.

In one embodiment of the method according to the invention, the iterative application of the expectation maximization algorithm is continued in such a way that a counting of the iterative steps of the iterative application of the expectation maximization algorithm is continued, a predefined maximum number of iteration steps being maintained.

To limit the computational effort and complexity of the method, a predefined maximum number of allowed iteration steps can be provided. The predefined maximum number can be retained if the EM algorithm is continued with the modified density function. The continuation can then be carried out in such a way that only the number of allowed iteration steps remaining after the change in the density function remains. If, for example, the predefined maximum number of iteration steps is 15 and the change in the density function (i.e. the removal of a component) takes place after the 7th iteration step, there are still 8 iteration steps left in order to adapt the now changed density function using the EM algorithm.

In this embodiment, the changed density function is based on the density function in the state before the removal of the one component. This means in particular that the partial parameters in formula (3) (μ, σ, α) that are not affected by the removal can be at least partially retained.

The design presented can be implemented quickly in terms of programming effort and is low in terms of computing effort.

In one embodiment of the method according to the invention, the iterative application of the expectation maximization algorithm is continued in such a way that a count of the iterative steps of the iterative application of the expectation maximization algorithm is restarted, a predefined maximum number of iteration steps being maintained.

In this embodiment, a continuation can be undertaken in such a way that the number of allowed iteration steps remaining after the change in the density function is set again to the predefined maximum number of iteration steps. If, for example, the predefined maximum number of iteration steps is 15 and changing the density function (i.e. removing a component) after the 9th

Iteration step takes place, there are then again 15 iteration steps in order to achieve an adaptation of the now changed density function with the aid of the EM algorithm.

In this embodiment, the changed density function is based on the density function in the state before the removal of the one component. This means in particular that the partial parameters in formula (3) (μ, σ, α) that are not affected by the removal can be at least partially retained.

The presented embodiment can be implemented quickly in terms of programming effort and is still low in terms of computing effort due to the restriction to the predefined maximum number of iteration steps and offers good adaptation of the changed density function.

In one embodiment of the method according to the invention, the iterative application of the expectation maximization algorithm is carried out again in such a way that a counting of the iterative steps of the iterative application of the expectation maximization algorithm is restarted, a predefined maximum number of iteration steps being maintained.

In this embodiment, the changed density function is not based on the density function in the state before the removal of the one component. That means in particular that all partial parameters in formula (3) (µ, σ, α) can be determined again, whereby starting values can be specified using known methods (estimation, application of known heuristics).

The presented embodiment can be implemented quickly in terms of programming effort and is still low in terms of computing effort due to the restriction to the predefined maximum number of iteration steps and offers good adaptation of the changed density function.

In one embodiment of the method according to the invention, the measured values are standardized in such a way that the mean value of the measured values is / are equal to 0 and / or the variance of the measured values is / are equal to 1.

$${\sigma }_s=\sqrt{\frac{1}{K-1}{\displaystyle {\sum }_{i=1}^K{\left(s_i-{\mu }_s\right)}^2}},$$

$$S_{std.}:=\frac{\left(S-{\mu }_s\right)}{{\sigma }_s},$$

Such a standardization can be done using the following formulas: $${\mu }_s=\frac{1}{K}{\displaystyle \sum _{i=1}^K{s}_i},$$

$${\sigma }_s=\sqrt{\frac{1}{K-1}{\displaystyle {\sum }_{i=1}^K{\left(s_i-{\mu }_s\right)}^2}},$$

$${\mathrm{S.}}_{std.}:=\frac{\left(\mathrm{S.}-{\mu }_s\right)}{{\sigma }_s},$$

Here S corresponds to _std. the set of measured values S (see above), now in standardized form. Standardization makes sense so that the variances of the components of the density function are always on a matching scale during applications of the EM algorithm, regardless of a scaling or an order of magnitude of the measured values. It is thus possible to compare the (then standardized) measured values and also to compare the predefined threshold value in several applications of the method according to the invention for different measured values. If the measured values of several applications of the method according to the invention are always in the same order of magnitude, then a standardization can possibly be dispensed with.

In one embodiment of the method according to the invention, the standardization is reversed after the application of the expectation maximization algorithm.

The following formula in particular can be used for this purpose: $$ƒ\left(x\right)=ƒ\left(\frac{x-{\mu }_s}{{\sigma }_s};\Theta \right)\cdot \frac{1}{{\sigma }_s}.$$

Undoing allows the density function to be used with direct reference to the measured values on a real scale.

Furthermore, a computing device is proposed which is designed to carry out the method according to the invention (in particular in one of the embodiments presented). The steps of the method can in particular be in the form of a program (for example program code) in a programming environment and / or in compiled form on the computing device. The measured values can also be available, for example as a table or csv. File.

The computing device can in particular have a computer (in particular having a processor, a power supply, a working memory, an input and an output device and a storage medium).

With regard to configurations of the computing device according to the invention, reference is made in full to configurations of the method according to the invention in its configurations.

• 1 eine schematische Ansicht einer Ausgestaltung des erfindungsgemäßen Verfahrens;
• 2a beispielartige Verteilungen zweier Komponenten (zwei Normalverteilungen) und zugehörige Messwerte;
• 2b beispielartige überlagerte Verteilungen der beiden in 2a dargestellten Komponenten (zwei überlagerte Normalverteilungen), die eine Dichtefunktion einer Gaußschen Mischverteilung (häufig auch einfach „Dichtefunktion“ genannt) ergeben, und zugehörige Messwerte;
• 3a eine beispielartige Dichtefunktion mit geringem Wert einer zugehörigen Likelihood-Funktion (erst wenige Iterationsschritte des EM-Algorithmus sind vorgenommen worden) und zugehörige Messwerte;
• 3b eine beispielartige Dichtefunktion mit höherem Wert einer zugehörigen Likelihood-Funktion, basierend auf der Dichtefunktion, die in 3a dargestellt wird, und zugehörige Messwerte, die den in 3a dargestellten Messwerten entsprechen;
• 4a eine beispielartige Dichtefunktion zu Beginn einer Anwendung des EM-Algorithmus, und zugehörige Messwerte, die den in 3a dargestellten Messwerten entsprechen;
• 4b beispielartige Dichtefunktion bei einer fortgeschrittenen Anwendung des EM-Algorithmus, basierend auf der Dichtefunktion, die in 4a dargestellt wird, und zugehörige Messwerte, die den in 3a dargestellten Messwerten entsprechen;
• 4c eine beispielartige Dichtefunktion nach einer abgeschlossenen Anwendung des EM-Algorithmus, basierend auf der Dichtefunktion, die in 4b dargestellt wird, und zugehörige Messwerte, die den in 3a dargestellten Messwerten entsprechen;
• 5a eine beispielartige Dichtefunktion zu Beginn einer Anwendung des EM-Algorithmus, und zugehörige Messwerte,
• 5b eine beispielartige Dichtefunktion bei einer fortgeschrittenen Anwendung des EM-Algorithmus, basierend auf der Dichtefunktion, die in 5a dargestellt wird, und zugehörige Messwerte, die den in 5a dargestellten Messwerten entsprechen;
• 5c eine beispielartige Dichtefunktion nach einer abgeschlossenen Anwendung des EM-Algorithmus, basierend auf der Dichtefunktion, die in 5a dargestellt wird, und zugehörige Messwerte, die den in 5a dargestellten Messwerten entsprechen, wobei eine Degenerierung (siehe linker Bereich des Diagramms) einer Komponente der Dichtefunktion stattgefunden hat;
• 6a eine beispielartige Dichtefunktion bei einer fortgeschrittenen Anwendung des EM-Algorithmus, wobei die Anzahl der Komponenten gegenüber den Dichtefunktionen in 5a-5c erfindungsgemäß um eine Komponente reduziert worden ist, und zugehörige Messwerte, die den in 5a dargestellten Messwerten entsprechen;
• 6b eine beispielartige Dichtefunktion bei einer abgeschlossenen Anwendung des EM-Algorithmus, basierend auf der Dichtefunktion, die in 6a dargestellt wird, wobei die Anzahl der Komponenten gegenüber den Dichtefunktionen in 5a-5c erfindungsgemäß um eine Komponente reduziert worden ist, und zugehörige Messwerte, die den in 5a dargestellten Messwerten entsprechen;
• 7a eine beispielartige Dichtefunktion, während einer fortgeschrittenen Anwendung des EM-Algorithmus, wobei eine Komponente der Dichtefunktion sich (im linken Bereich des Diagramms) in einem Vorstadium einer Degenerierung befindet, und zugehörige Messwerte, die den in 5a dargestellten Messwerten entsprechen;
• 7b eine beispielartige Dichtefunktion, während einer weiter fortgeschrittenen Anwendung des EM-Algorithmus, basierend auf der Dichtefunktion, die in 7a dargestellt ist, wobei die Komponente der Dichtefunktion, die sich in 7a in einem Vorstadium einer Degenerierung befand, erfindungsgemäß entfernt worden ist, und zugehörige Messwerte, die den in 5a dargestellten Messwerten entsprechen;
• 7c eine beispielartige Dichtefunktion, während einer noch weiter fortgeschrittenen Anwendung des EM-Algorithmus, basierend auf der Dichtefunktion, die in 7b dargestellt ist, wobei die Komponente der Dichtefunktion, die sich in 7a in einem Vorstadium einer Degenerierung befand, nach wie vor entfernt ist, und zugehörige Messwerte, die den in 5a dargestellten Messwerten entsprechen.

Refinements of the invention will now be presented with reference to the accompanying drawing. The individual figures in the attached drawing show:

• 1 a schematic view of an embodiment of the method according to the invention;
• 2a example distributions of two components (two normal distributions) and associated measured values;
• 2 B exemplary superimposed distributions of the two in 2a shown components (two superimposed normal distributions), which result in a density function of a Gaussian mixed distribution (often also simply called “density function”), and associated measured values;
• 3a an exemplary density function with a low value of an associated likelihood function (only a few iteration steps of the EM algorithm have been carried out) and associated measured values;
• 3b an exemplary density function with a higher value of an associated likelihood function based on the density function shown in 3a is displayed, and associated measured values that correspond to the in 3a correspond to the measured values shown;
• 4a an exemplary density function at the beginning of an application of the EM algorithm, and associated measured values that correspond to the in 3a correspond to the measured values shown;
• 4b exemplary density function in an advanced application of the EM algorithm, based on the density function described in 4a is displayed, and associated measured values that correspond to the in 3a correspond to the measured values shown;
• 4c an exemplary density function after a completed application of the EM algorithm, based on the density function described in 4b is displayed, and associated measured values that correspond to the in 3a correspond to the measured values shown;
• 5a an exemplary density function at the beginning of an application of the EM algorithm, and associated measured values,
• 5b an exemplary density function in an advanced application of the EM algorithm, based on the density function described in 5a is displayed, and associated measured values that correspond to the in 5a correspond to the measured values shown;
• 5c an exemplary density function after a completed application of the EM algorithm, based on the density function described in 5a is displayed, and associated measured values that correspond to the in 5a the measured values shown correspond, with a degeneration (see left-hand area of the diagram) of a component of the density function having taken place;
• 6a an exemplary density function in an advanced application of the EM algorithm, where the number of components versus the density functions in 5a-5c has been reduced by one component according to the invention, and associated measured values that correspond to the in 5a correspond to the measured values shown;
• 6b an exemplary density function in a completed application of the EM algorithm, based on the density function described in 6a is shown, with the number of components versus the density functions in 5a-5c has been reduced by one component according to the invention, and associated measured values that correspond to the in 5a correspond to the measured values shown;
• 7a an exemplary density function, during an advanced application of the EM algorithm, with a component of the density function (in the left-hand area of the diagram) being in a preliminary stage of degeneration, and associated measured values corresponding to the in 5a correspond to the measured values shown;
• 7b an exemplary density function, during a more advanced application of the EM algorithm based on the density function presented in 7a is shown, where the component of the density function, which is in 7a was in a preliminary stage of degeneration, has been removed according to the invention, and associated measured values that correspond to the in 5a correspond to the measured values shown;
• 7c an exemplary density function, during an even more advanced application of the EM algorithm, based on the density function presented in 7b is shown, where the component of the density function, which is in 7a was in a preliminary stage of degeneration, is still removed, and associated readings that correspond to the in 5a correspond to the measured values shown.

The diagrams included in the 2a until 7c are shown have standardized measured values on the abscissa (x-axis). The ordinate (y-axis) shows standardized frequencies or probabilities.

1 shows a schematic view of an embodiment of the method according to the invention. In a first step S1 measured values are provided. For example, it can each be a dimension between two edges. For example, there are 100 measured values for identical parts. The measured values can optionally be standardized, for example using forms 11-13. This is in 1 indicated with a bracketed "S".

In a second step S2 a density function of a Gaussian density function is set up on the basis of the standardized measured values (formula 2). For example, the measured values can have a three-peak curve if they are shown in a diagram (abscissa: measured values, ordinate: frequencies). This is due, for example, to the fact that the parts, although they are structurally identical, come from three different plants. 50 parts come from a first work, 30 parts from a second work, 20 parts from a third work. It would be possible (for illustration) that this results in three normal distributions with weighting factors of 0.5, 0.3 and 0.2. However, the density function initially has four components. In the following course it will be shown that one component is superfluous and behaves in a degenerative manner.

In a third step S3 an iterative adaptation of the density function is carried out using the EM algorithm (formulas 3-10). The known likelihood function is used here (formula 4). After each iteration step, there is a check in step 4 as to whether a component has a variance that is below a predefined threshold value ε. The predefined threshold value can be rigidly specified from the start or, alternatively, can be predefined separately before one or each check. The steps S3 and S4 are executed iteratively several times in sequence, as indicated by round arrows in the 3 and 4th is symbolized.

For example, after the 13th iteration step (the maximum number of iteration steps is 30, for example) a variance of a component falls below the threshold value ε. In this case, the component is removed (indicated by an “X” in 1 ), so that there are only three components left, and the iteration process with the third step S3 continued with the then changed density function until the 25th iteration step is reached (i.e. 12 times). Alternatively, 25 iteration steps could follow on the basis of the changed density function. Alternatively, based on a new density function with only three components, 25 iteration steps could follow.

The standardization is then reversed, e.g. B. Using Formula 14, in 1 indicated by an "S" in brackets. The density function now obtained is the result of the method according to the invention (indicated by a tick in 1 ).

Represented as an algorithm, an embodiment of the method according to the invention could be as follows:

erster Iterationsschritt: Θ = Θ⁽⁰⁾ Standardisierung: ${\mu }_s=\frac{1}{K}{\displaystyle {\sum }_{i=1}^K{s}_i},{\sigma }_s=\sqrt{\frac{1}{K-1}{\displaystyle {\sum }_{i=1}^K{\left(s_i-{\mu }_s\right)}^2}},S:=\left(S-{\mu }_s\right)/{\sigma }_s,$

while (maximale Anzahl Iterationsschritte oder sonstige Endbedingung nicht erreicht):
for j = 1, ..., N do:
H_j = [h_j(1; Θ), ..., h_j(K; Θ)]
$${\mu }_j^{\left(P+1\right)}=\text{update}\_\mu \left(H_j\right)$$

$${\sigma }_j^{2\left(P+1\right)}=\text{update}\_{\sigma }^2\left(H_j,{\mu }^{\left(k\right)}\right)$$

$${\alpha }_j^{\left(P+1\right)}=\text{update}\_\alpha \left(H_j\right)$$

end for
for j = 1, ..., N do:
if ${\sigma }_j^{2\left(P+1\right)}<\epsilon$

then
Entferne Komponente $j:\left({\mu }_j^{\left(P+1\right)},{\sigma }_j^{2\left(P+1\right)},{\alpha }_j^{\left(P+1\right)}\right)$

end if
end for $$\Theta =\left[{\mu }_1^{\left(P+1\right)},\dots ,{\mu }_n^{\left(P+1\right)},{\sigma }_1^{2\left(P+1\right)},\dots ,{\sigma }_n^{2\left(P+1\right)},{\alpha }_1^{\left(P+1\right)},\dots ,{\alpha }_n^{\left(P+1\right)}\right]$$

(wobei die entfernte Komponente fehlt)
end while
return ΘNecessary: $\mathrm{S.}=\left[s_1,...,s_k\right],\epsilon ,{\Theta }^{\left(0\right)}=\left[{\mu }_1^{\left(0\right)},...,{\mu }_n^{\left(0\right)},{\sigma }_1^{2\left(0\right)},...,{\sigma }_n^{2\left(0\right)},{\alpha }_1^{\left(0\right)},...,{\alpha }_n^{\left(0\right)}\right]$

first iteration step: Θ = Θ ⁽⁰⁾ standardization: ${\mu }_s=\frac{1}{K}{\displaystyle {\sum }_{i=1}^K{s}_i},{\sigma }_s=\sqrt{\frac{1}{K-1}{\displaystyle {\sum }_{i=1}^K{\left(s_i-{\mu }_s\right)}^2}},\mathrm{S.}:=\left(\mathrm{S.}-{\mu }_s\right)/{\sigma }_s,$

while (maximum number of iteration steps or other end condition not reached):
for j = 1, ..., N do:
H _j = [h _j (1; Θ), ..., h _j (K; Θ)]
$${\mu }_j^{\left(\mathrm{P.}+1\right)}=\text{update}\_\mu \left(H_j\right)$$

$${\sigma }_j^{2\left(\mathrm{P.}+1\right)}=\text{update}\_{\sigma }^2\left(H_j,{\mu }^{\left(k\right)}\right)$$

$${\alpha }_j^{\left(\mathrm{P.}+1\right)}=\text{update}\_\alpha \left(H_j\right)$$

end for
for j = 1, ..., N do:
if ${\sigma }_j^{2\left(\mathrm{P.}+1\right)}<\epsilon$

then
Remove component $j:\left({\mu }_j^{\left(\mathrm{P.}+1\right)},{\sigma }_j^{2\left(\mathrm{P.}+1\right)},{\alpha }_j^{\left(\mathrm{P.}+1\right)}\right)$

end if
end for $$\Theta =\left[{\mu }_1^{\left(\mathrm{P.}+1\right)},...,{\mu }_n^{\left(\mathrm{P.}+1\right)},{\sigma }_1^{2\left(\mathrm{P.}+1\right)},...,{\sigma }_n^{2\left(\mathrm{P.}+1\right)},{\alpha }_1^{\left(\mathrm{P.}+1\right)},...,{\alpha }_n^{\left(\mathrm{P.}+1\right)}\right]$$

(with the removed component missing)
end while
return Θ

A computing device according to the invention which is designed to carry out the method according to the invention can have a program code in a memory which has such an algorithm.

In 2a two normal distributions are shown as gradients (solid line), which are based on standardized measured values that are also shown (bar-like). In 2 B becomes a density function from the in 2a shown, the weighting factor of the in 2a The normal distribution arranged further to the left has a weighting factor of 2/3, and the normal distribution arranged further to the right has a weighting factor of 1/3.

In 3a an exemplary, three-peaked density function is shown in which only a small number of iteration steps of the EM algorithm for adaptation have been carried out. 3b shows a state of the density function in which some iteration steps have already been carried out. The value of the associated likelihood function is increased. In the 3a and 3b the underlying standardized measured values are also shown.

In 4a an exemplary, three-peaked density function is shown in which no iteration step of the EM algorithm for adaptation has yet been carried out. 4b shows a state of the density function in which some iteration steps have already been carried out. 4c shows the state of the density function when all iteration steps have been carried out.

In the 4a-4c the underlying standardized measured values are also shown.

In 5a an exemplary density function is shown in which very few iteration steps of the EM algorithm for adaptation have been carried out. 5b shows a state of the density function in which some iteration steps have already been carried out. In the left-hand area of the diagram, due to just a single measured value, an incipient degeneration of a component is developing. 5c shows the state of the density function when further iteration steps have been carried out. A degenerate component can be seen in the left-hand area of the diagram, which shows a very strongly delimited, abrupt course towards very high function values. The variance of this component is very high. The method according to the invention was not used here. In the 5a-5c the underlying standardized measured values are also shown.

On the other hand, as in the 6a and 6b shown, a density function reduced by one component is used from the outset in relation to the previously discussed measured values, this occurs in 5c shown problem does not occur. In the 6a and 6b the underlying standardized measured values are also shown. If the method according to the invention is applied to the situation as described in 5b is shown, applied, at most an incipient degeneration will develop, as in 7a shown. Due to the removal of a component according to the invention, that is to say the change in the density function according to the invention, however, as in FIG 7b shown an avoidance of degenerative effects of the density function. Another application of the EM algorithm leads to the in 7c shown course. In the 7a-7c the underlying standardized measured values are also shown.

List of reference symbols

S1

first step

S2

second step

S3

Third step

S4

fourth step

Claims (8)

Method for improving the statistical description of measured values of parts or devices with the same characteristic, comprising: Providing measured values from a plurality of the parts or devices, the measured values relating to the same feature; - Establishing a density function of a Gaussian mixed distribution on the basis of the measured values; - Adaptation of the density function of the Gaussian mixed distribution with the aid of an iterative application of the expectation maximization algorithm; characterized in that the density function is changed in such a way that a component of the density function is removed as soon as the variance of this component falls below a predefined threshold value during the iterative application of the expectation maximization algorithm. Procedure according to Claim 1 , characterized in that after the component has been removed, the iterative application of the expectation maximization algorithm to the changed density function is continued or carried out again. Procedure according to Claim 2 , characterized in that the iterative application of the expectation maximization algorithm is continued in such a way that a counting of the iterative steps of the iterative application of the expectation maximization algorithm is continued, a predefined maximum number of iteration steps being maintained. Procedure according to Claim 2 , characterized in that the iterative application of the expectation maximization algorithm is continued in such a way that a count of the iterative steps of the iterative application of the expectation maximization algorithm is restarted, a predefined maximum number of iteration steps being maintained. Procedure according to Claim 2 , characterized in that the iterative application of the expectation maximization algorithm is carried out again in such a way that a count of the iterative steps of the iterative application of the expectation maximization algorithm is restarted, a predefined maximum number of iteration steps being maintained. Method according to one of the Claims 1 - 5 , characterized in that the measured values are standardized in such a way that the mean value of the measured values is / are equal to 0 and / or the variance of the measured values is / are equal to 1. Procedure according to Claim 6 , characterized in that the standardization is reversed after the application of the expectation maximization algorithm. Computing device designed to carry out the method according to one of the Claims 1 - 7th .

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