Coordinate Measuring Machine Measurement and Analysis of Multiple Workpieces

Abstract

A method evaluates a sample of measurement data from measuring multiple workpieces by at least one coordinate measuring machine. A system of statistical distributions describes a frequency of measurement data values. The distributions are distinguishable based on skewness and kurtosis. The method includes defining a set of statistical distributions that are able to describe a frequency of measurement data values in the entire value interval from the system of statistical distributions for a value interval of the measurement data, which is a specified value interval or a value interval of the measurement data actually arising in the sample. The method includes ascertaining the skewness and the kurtosis from the sample of measurement data corresponding to a first statistical distribution. The method includes checking, using the ascertained moment values, whether the defined set contains a statistical distribution that has the ascertained skewness and kurtosis, and producing a corresponding test result.

Claims

1. A computer-executed method for evaluating a sample of measurement data from measuring a plurality of workpieces by at least one coordinate measuring machine, a system of statistical distributions being configured to describe a frequency of measurement data values as a function of the measurement data values, examples of the system of statistical distributions being distinguishable from one another by respectively one moment value of two moments of the respective statistical distribution, the two moments being a skewness and a kurtosis, the method comprising: defining a set of all those statistical distributions that are able to describe a frequency of measurement data values in the entire value interval from the system of statistical distributions for a value interval of the measurement data, which is a specified value interval or a value interval of the measurement data actually arising in the sample; ascertaining a respective moment value of the skewness and the kurtosis from the sample of measurement data corresponding to a first statistical distribution; checking, using the ascertained moment values, whether the defined set contains a statistical distribution that has the ascertained moment values of the skewness and the kurtosis; and producing a corresponding test result based on the checking.

2. The method of claim 1 further comprising: controlling a process for at least one of producing workpieces, processing workpieces, producing arrangements of workpieces, and processing arrangements of workpieces based on a contained statistical distribution, wherein the contained statistical distribution is contained by the defined set of all those statistical distributions that are able to describe a frequency of measurements data values in the entire value interval of the sample of measurement data, and wherein the contained statistical distribution has the ascertained moment values of the skewness and the kurtosis.

3. The method of claim 1 further comprising: determining a quality of a process for at least one of producing workpieces, processing workpieces, producing arrangements of workpieces, and processing arrangements of workpieces based on a contained statistical distribution, wherein the contained statistical distribution is contained by the defined set of all those statistical distributions that are able to describe a frequency of measurements data values in the entire value interval of the sample of measurement data, and wherein the contained statistical distribution has the ascertained moment values of the skewness and the kurtosis.

4. The method of claim 1 wherein: the method further comprises ascertaining a second statistical distribution for the sample in response to the test result indicating that the defined set does not contain a statistical distribution having the ascertained moment values of the skewness and the kurtosis; and the second statistical distribution is a statistical distribution contained in the defined set.

5. The method of claim 4 further comprising controlling a process for at least one of producing workpieces, processing workpieces, producing arrangements of workpieces, and processing arrangements of workpieces based on the second statistical distribution.

6. The method of claim 4 further comprising determining a quality of a process for at least one of producing workpieces, processing workpieces, producing arrangements of workpieces and processing arrangements of workpieces based on the second statistical distribution.

7. The method of claim 4 wherein: a measure of distance has been or is defined for two statistical distributions in each case, which are able to describe a frequency of measurement data values as a function of the measurement data values; the measure of distance describes a distance between the two statistical distributions; and the value of the measure of distance of the first statistical distribution or a distribution corresponding to the first statistical distribution in the system of statistical distributions from the second statistical distribution is a minimum of the measure of distance of the distance of the first statistical distribution or the corresponding distribution from the statistical distributions in the defined set.

8. The method of claim 7 further comprising controlling a process for at least one of producing workpieces, processing workpieces, producing arrangements of workpieces, and processing arrangements of workpieces based on the second statistical distribution.

9. The method of claim 7 further comprising determining a quality of a process for at least one of producing workpieces, processing workpieces, producing arrangements of workpieces, and processing arrangements of workpieces based on the second statistical distribution.

10. The method of claim 1 further comprising ascertaining a boundary of the set from the value interval when defining the set of all those statistical distributions that are able to describe a frequency of measurement data values in the entire value interval.

11. The method of claim 10 wherein: value pairs of the skewness and the kurtosis of statistical distributions of the system, which correspond to the set, are ascertained from the value interval when defining the set of all those statistical distributions that are able to describe a frequency of measurement data values in the entire value interval; and a boundary curve in a plane spanned by the skewness and the kurtosis of statistical distributions of the system is ascertained from the value interval when defining the set of all those statistical distributions that are able to describe a frequency of measurement data values in the entire value interval.

12. The method of claim 1 wherein value pairs of the skewness and the kurtosis of statistical distributions of the system, which correspond to the set, are ascertained from the value interval when defining the set of all those statistical distributions that are able to describe a frequency of measurement data values in the entire value interval.

13. An arrangement for evaluating a sample of measurement data from measuring a plurality of workpieces by at least one coordinate measuring machine, a system of statistical distributions being configured to describe a frequency of measurement data values as a function of the measurement data values, examples of the system of statistical distributions being distinguishable from one another by respectively one moment value of two moments of the respective statistical distribution, the two moments being a skewness and a kurtosis, the arrangement comprising: a definition device configured to define a set of all those statistical distributions that are able to describe a frequency of measurement data values in the entire value interval from the system of statistical distributions for a value interval of the measurement data, which is a specified value interval or a value interval of the measurement data actually arising in the sample; a moment ascertainment device configured to ascertain a respective moment value of the skewness and the kurtosis from the sample of measurement data corresponding to a first statistical distribution; and a checking device configured to (i) use the ascertained moment values to check whether the defined set contains a statistical distribution that has the ascertained moment values of the skewness and the kurtosis and (ii) produce a corresponding test result.

14. The arrangement of claim 13 wherein: the arrangement comprises a distribution ascertainment device configured to ascertain a second statistical distribution for the sample in response to the test result indicating that the defined set does not contain a statistical distribution having the ascertained moment values of the skewness and the kurtosis; and the second statistical distribution is a statistical distribution contained in the defined set.

15. The arrangement of claim 14 wherein: the distribution ascertainment device is configured to ascertain the second statistical distribution from the defined set in such a way that, based on a measure of distance defined for two statistical distributions in each case, the value of the measure of distance of the first statistical distribution or a distribution corresponding to the first statistical distribution in the system of statistical distributions from the second statistical distribution is a minimum of the measure of distance of the distance of the first statistical distribution or the corresponding distribution from the statistical distributions in the defined set; and the measure of distance describes a distance between two statistical distributions in each case, which are able to describe a frequency of measurement data values as a function of the measurement data values.

16. The arrangement of claim 13 wherein the definition device is configured to ascertain a boundary of the set from the value interval when defining the set of all those statistical distributions that are able to describe a frequency of measurement data values in the entire value interval.

17. The arrangement of claim 16 wherein: the definition device is configured to ascertain value pairs of the skewness and the kurtosis of statistical distributions of the system, which correspond to the set, from the value interval when defining the set of all those statistical distributions that are able to describe a frequency of measurement data values in the entire value interval; and the definition device is configured to ascertain a boundary curve in a plane spanned by the skewness and the kurtosis of statistical distributions of the system from the value interval when defining the set of all those statistical distributions that are able to describe a frequency of measurement data values in the entire value interval.

18. The arrangement of claim 13 wherein the definition device is configured to ascertain value pairs of the skewness and the kurtosis of statistical distributions of the system, which correspond to the set, from the value interval when defining the set of all those statistical distributions that are able to describe a frequency of measurement data values in the entire value interval.

19. A computer-executed method for preparing an evaluation of a sample of measurement data from measuring a plurality of workpieces by at least one coordinate measuring machine, a system of statistical distributions being configured to describe a frequency of measurement data values as a function of the measurement data values, examples of the system of statistical distributions being distinguishable from one another by respectively one moment value of two moments of the respective statistical distribution, the two moments being a skewness and a kurtosis, the method comprising: defining a set of all those statistical distributions that are able to describe a frequency of measurement data values in the entire value interval from the system of statistical distributions for a value interval of measurement data, which is a specified value interval or a value interval of a sample, to be evaluated, of measurement data; and ascertaining a statistical distribution from the set.

20. The method of claim 19 further comprising ascertaining a boundary of the set from the value interval when defining the set of all those statistical distributions that are able to describe a frequency of measurement data values in the entire value interval.

21. The method of claim 20 further comprising: ascertaining value pairs of the skewness and the kurtosis of statistical distributions of the system, which correspond to the set, from the value interval when defining the set of all those statistical distributions that are able to describe a frequency of measurement data values in the entire value interval; and ascertaining a boundary curve in a plane spanned by the skewness and the kurtosis of statistical distributions of the system from the value interval when defining the set of all those statistical distributions that are able to describe a frequency of measurement data values in the entire value interval.

22. The method of claim 19 further comprising ascertaining value pairs of the skewness and the kurtosis of statistical distributions of the system, which correspond to the set, from the value interval when defining the set of all those statistical distributions that are able to describe a frequency of measurement data values in the entire value interval.

23. An arrangement for preparing an evaluation of a sample of measurement data from measuring a plurality of workpieces, a system of statistical distributions being configured to describe a frequency of measurement data values as a function of the measurement data values, examples of the system of statistical distributions being distinguishable from one another by respectively one moment value of two moments of the respective statistical distribution, the two moments being a skewness and a kurtosis, the arrangement comprising: a definition device configured to define a set of all those statistical distributions that are able to describe a frequency of measurement data values in the entire value interval from the system of statistical distributions for a value interval of measurement data, which is a specified value interval or a value interval of a sample, to be evaluated, of measurement data; and a distribution ascertainment device configured to ascertain a statistical distribution from the set.

24. The arrangement of claim 23 wherein the definition device is configured to ascertain a boundary of the set from the value interval when defining the set of all those statistical distributions that are able to describe a frequency of measurement data values in the entire value interval.

25. The arrangement of claim 24 wherein: the definition device is configured to ascertain value pairs of the skewness and the kurtosis of statistical distributions of the system, which correspond to the set, from the value interval when defining the set of all those statistical distributions that are able to describe a frequency of measurement data values in the entire value interval; and the definition device is configured to ascertain a boundary curve in a plane spanned by the skewness and the kurtosis of statistical distributions of the system from the value interval when defining the set of all those statistical distributions that are able to describe a frequency of measurement data values in the entire value interval.

26. The arrangement of claim 23 wherein the definition device is configured to ascertain value pairs of the skewness and the kurtosis of statistical distributions of the system, which correspond to the set, from the value interval when defining the set of all those statistical distributions that are able to describe a frequency of measurement data values in the entire value interval.

27. A non-transitory computer-readable medium storing processor-executable instructions that embody the method of claim 1.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0061] The present disclosure will become more fully understood from the detailed description and the accompanying drawings.

[0062] FIG. 1 shows, for an example sample of measurement data, a frequency distribution of the measurement values and a statistical distribution not suitable for describing the frequency distribution over the entire value interval of the sample.

[0063] FIG. 2 shows examples of three different statistical distributions that differ from one another in respect of their skewness.

[0064] FIG. 3 shows examples of two different statistical distributions that differ from one another in respect of their kurtosis.

[0065] FIG. 4 shows an example of a plane whose points are defined by value pairs of the skewness and the kurtosis and which is referred to as Pearson plane, with the FIG. having regions of the plane which are respectively assigned to the eight types of a Pearson distribution system with eight distribution types, and the “forbidden region”.

[0066] FIG. 5 shows a simplified representation with a section of the region of the Pearson plane illustrated in FIG. 4, with two specific points in the Pearson plane being labeled by a cross.

[0067] FIG. 6 shows a statistical distribution whose values of the skewness and the kurtosis correspond to the lower central point in FIG. 5 that is labeled by a cross.

[0068] FIG. 7 shows a statistical distribution whose values of the skewness and the kurtosis correspond to the upper right point in FIG. 5 that is labeled by a cross, wherein the expected value and the variance of the statistical distributions illustrated in FIG. 6 and FIG. 7 have the same value, which also underlie the representation in FIG. 4 and FIG. 5.

[0069] FIG. 8 shows, for one example embodiment, a section from the Pearson plane which contains a solution set for the condition that the respective statistical distribution should be able to model the frequency or probability of the measurement data values up to the left edge of the value interval.

[0070] FIG. 9 shows, for the example embodiment of FIG. 8, a section from the Pearson plane which contains a solution set for the condition that the respective statistical distribution should be able to model the frequency or probability of the measurement data values up to the right edge of the value interval.

[0071] FIG. 10 shows, for the example embodiment of FIG. 8 and FIG. 9, a section from the Pearson plane which contains the intersection of the solution sets of FIG. 8 and FIG. 9.

[0072] FIG. 11 shows a section from the Pearson plane which indicates different regions and their boundary lines, wherein the skewness varies along the horizontal axis and the kurtosis varies along the vertical axis.

[0073] FIG. 12 shows a section from the Pearson plane with portions of a solution set for the left support region, i.e., the support region in respect of the left boundary of the value interval, for a first parameter value,

[0074] FIG. 13 shows a section from the Pearson plane with portions of a solution set for the left support region for a second parameter value.

[0075] FIG. 14 schematically shows an arrangement of devices for evaluating a sample of measurement data from measuring a multiplicity of workpieces, which can also be interpreted as a flowchart.

[0076] FIG. 15 shows a portion of the section from the Pearson plane of FIG. 10, wherein the hatched region in FIG. 15 corresponds to the cross-hatched region in FIG. 10 and wherein crosses are used to mark two points in the plane, said points corresponding to a first statistical distribution outside of the solution set and a second statistical distribution at the edge of the solution set.

[0077] FIG. 16 shows, for the example sample and the frequency distribution of the measurement values from FIG. 1, a statistical distribution suitable for describing the frequency distribution over the entire value interval of the sample.

DETAILED DESCRIPTION

[0078] In FIG. 1, the frequencies of measurement data values are represented in the form of bars, in each case for small partial intervals of the value interval of measurement data values of a sample of measurement data. Accordingly, the measurement data values are plotted along the horizontal axis and the frequencies or probabilities are plotted along the vertical axis. Further, FIG. 1 illustrates the function curve of a first statistical distribution which models the frequency distribution. It is evident that the first statistical distribution only models frequencies in a value interval bounded on the left side, the value interval starting at approximately the measurement data value of −1.9. However, the sample also contains measurement data values that are less than −1.9. Therefore, the first statistical distribution is not suitable for modeling the frequencies over the entire value interval of the sample.

[0079] By way of example, the value interval of the actually arising measurement data values or a larger value interval can be specified for the sample underlying FIG. 1 as the value interval in which a statistical distribution should be able to model the frequency or probability of the measurement data values. Below, this specified value interval is also referred to as the range.

[0080] By contrast, there is the value interval in which a statistical distribution can model the frequency or the probability of the measurement data values. Below, this value interval is also referred to as the support of the statistical distribution. In relation to the case illustrated in FIG. 1, the range is therefore not contained within the support. However, this is sought after.

[0081] As already mentioned above, moments can be assigned as characteristics to the statistical distributions of the Pearson distribution system in particular. Four of the moments are the expected value v.sub.1, the variance μ.sub.2, the skewness {tilde over (μ)}.sub.3, and the kurtosis {tilde over (μ)}.sub.4. In relation to a random variable X, the n-th moment is given by:

v.sub.n(X)=E(X.sup.n)

The expected value arises by inserting. E denotes the expected value operator. The n-th central moment μ.sub.n is given by:
From this, the corresponding equation for the variance μ.sub.2 can be obtained by inserting n=2. The n-th central, standardized moment {tilde over (μ)}.sub.n is given by:

[00001]$\begin{array}{cc}{\tilde{\mu }}_n=\frac{{\mu }_n\left(X\right)}{{{\mu }_2\left(X\right)}^{n/2}},n\ge 2& \end{array}$

[0082] From this, by inserting n=3 and 4, it is possible to obtain the corresponding equations for the skewness {tilde over (μ)}.sub.3 and the kurtosis {tilde over (μ)}.sub.4. As will still be justified elsewhere, the central, standardized moments of the skewness and the kurtosis are independent of the moments of the expected value and the variance, and so the check of the suitability of a statistical distribution for modeling a sample over the entire value interval and the ascertainment of a suitable statistical distribution, in each case with respect to the skewness and the kurtosis, can be undertaken for any expected values and any values of the variance and the results of the check or the ascertainment are valid. In the case of the standardization described elsewhere, care should be taken that the solution set is ascertained in respect of the standardized measurement values or the standardized statistical distribution.

[0083] FIG. 2 shows three different statistical distributions that differ from one another in respect of their skewness. Here, the skewness of the probability density function illustrated by a solid line has a value of zero since the probability density function is symmetrical. The probability density function illustrated by a dashed line has a negative value of the skewness; the probability density function illustrated by a dotted line has a positive value of the skewness. FIG. 3 shows two different statistical distributions that differ from one another in respect of their kurtosis. The kurtosis only assumes positive values. The kurtosis of the probability density function illustrated by a dashed line is greater than the kurtosis of the probability density function illustrated by a solid line.

[0084] It is conventional to adapt the first two moments, the expected value and the variance, of a distribution. However, only the skewness and kurtosis are initially considered below; these allow each distribution or family of distributions to be identified uniquely.

[0085] Specifically, a Pearson distribution system with eight types of distributions is considered in the example embodiment described below. For this system, FIG. 4 shows the so-called skewness-kurtosis plane, wherein, for one example embodiment, values of the skewness are plotted along the axis extending horizontally and values of the kurtosis are plotted along the axis extending vertically. The skewness-kurtosis plane is referred to as Pearson plane below. Each point of the Pearson plane outside of the forbidden region, which is defined by a coordinate pair consisting of a skewness coordinate and a kurtosis coordinate, is uniquely assigned to a distribution or a family of distributions. The members of the family still differ by the values of the further moments, such as the expected value and the variance in particular. There is a continuous relationship between the relative position in the Pearson plane and the form of the associated distribution or family of distributions. This applies not only directly in respect of the skewness and the kurtosis but also in respect of further properties, in particular of the value interval over which the family of distributions is suitable for statistically modeling samples or measurement data values and predicting their probability. Thus, if two points are close together in the Pearson plane, the associated two distributions are also similar if the same expected value and the same variance are assumed for the two distributions.

[0086] The example of the Pearson plane illustrated in FIG. 4 was created on the basis of a standardization in which the expected value has a value of zero and the variance has a value of one.

[0087] In FIG. 4, an approximately parabolic dashed line is evident at the bottom of the illustrated region of the plane. There are no associated distributions for value pairs of skewness and kurtosis lying below this line. The region of the plane lying below this line can therefore be referred to as inadmissible or forbidden region. In the case of the skewness value of zero, a dotted line extends parallel to the kurtosis axis from the aforementioned line to an approximately parabolic solid line. The value pairs on the dotted line belong to distributions of the Pearson type 2 distribution. Along the extension of the dotted line, a dash-dotted line extends parallel to the kurtosis axis and the value pairs thereof belong to distributions of the Pearson type 7 distribution. The value pairs of the aforementioned solid line belong to the Pearson type 3 distribution. A likewise parabolic dashed line is illustrated above the solid line, the points of the value pairs of the distributions of the Pearson type 5 distribution lying thereon. The value pairs of the points between the forbidden region and the solid line belong to distributions of the Pearson type 1 distribution. The value pairs of the points between the solid line and the upper dashed line belong to distributions of the Pearson type 6 distribution. The value pairs of the points above the upper dashed line belong to distributions of the Pearson type 4 distribution. Located at the common intersection or end of all of the aforementioned lines with the exception of the lower dashed line is the value pair of the only point of the distribution of the Pearson type 0 distribution, which is also known as the normal distribution.

[0088] In order to highlight two points in the plane, two crosses are plotted in the region of the Pearson plane illustrated in FIG. 5. The associated distributions are illustrated in FIG. 6 and FIG. 7. These two distributions belong to the Pearson type 1 distribution. The skewness of the distribution illustrated in FIG. 6 is negative and closer to the value of zero than the positive skewness of the distribution illustrated in FIG. 7. Therefore, this distribution looks more symmetrical than the distribution illustrated in FIG. 7. The kurtosis of the distribution illustrated in FIG. 7 has a positive value with a greater magnitude than the positive value of the distribution illustrated in FIG. 6.

[0089] The range of a sample represents an interval of the measurement data values. The distribution fitted to the sample is now required to have a support which at least contains the range of the interval. Thus, if the range is given by

Range=[r.sub.min,r.sub.max]

and the support of the distribution is given by

Support=[v.sub.min,v.sub.max],

then the following should apply:

Range.Math.Support,

i.e., the range should be completely contained within the support. This demand is equivalent to:

v.sub.min≤r.sub.min,  (1)

v.sub.max≥r.sub.max.  (2)

[0090] What emerges from Equation (1) is that a distribution is sought after, the lower support limit v.sub.min of which is less than or equal to the minimum of the range r.sub.min. Any distribution can be visualized as skewness/kurtosis point in the Pearson plane. Thus, the set of all distributions that satisfy Equation (1) can be represented by a set in the Pearson plane. This yields a solution set which is illustrated by hatching in FIG. 8 for one example embodiment. It is worth noting that the solution set is bounded below by a convex boundary line, i.e., located above the boundary line. Equally, a solution set can be obtained proceeding from Equation (2), said solution set being illustrated by hatching for the example embodiment in FIG. 9. To aid orientation, the dashed line forming the boundary to the forbidden region is illustrated in FIG. 8 and FIG. 9, like in FIG. 4. The overall solution set for the example embodiment is illustrated with crosshatching in FIG. 10. These FIGS. should be understood to be schematic in respect of the fact that the partial solution sets in FIG. 8 and FIG. 9 extend into the forbidden region. In reality, the partial solution sets do not extend into the forbidden region.

[0091] The point of intersection of the boundary curves of the partial solution sets, illustrated in FIG. 10 by a dotted and a dashed line, has the value pair of the skewness and the kurtosis which corresponds to that statistical distribution within the system of distributions for which the support equals the range. In the example embodiment, this point of intersection lies at the skewness value of zero but can also adopt negative or positive skewness values in other cases.

[0092] M.sub.L denotes the solution set proceeding from Equation (1) and M.sub.R denotes the solution set proceeding from Equation (2), which can be defined as follows:

M.sub.L={(s,k):v.sub.min(s,k)≤r.sub.min}

and

M.sub.R={(s,k):r.sub.max≤v.sub.max(s,k)}.

Here, (s, k) denotes a value pair of the skewness s and the kurtosis k.

[0093] An example embodiment for determining the sets M.sub.L and M.sub.R is now described below.

[0094] The Pearson distribution system is based on the following conventional differential equation for a probability density function. A normalized solution ƒ(x) of the conventional differential equation

[00002]$\begin{array}{cc}{f}^{\prime }\left(x\right)=\frac{a\left(x\right)}{b\left(x\right)}f\left(x\right)& \left(3\right)\end{array}$

with the polynomials

a(x)=x+a.sub.0, a.sub.0∈R  (4)

b(x)=b.sub.2+b.sub.1x+b.sub.0, b.sub.0,b.sub.1,b.sub.2∈R,  (5)

where R denotes the set of real numbers is referred to below as Pearson probability density function.

[0095] The coefficients of the polynomials a(x) and b(x), as defined in Equations (4) and (5), parameterize the still unknown probability density function ƒ(x). The form and the definition range of the probability density function, in particular, depend strongly on the number and the location of the zeros of the denominator polynomial b(x).

[0096] In order to relate the probability density function to variables with statistical significance, a relationship is established between, firstly, the coefficients a=(a.sub.0,1).sup.T and b=(b.sub.0, b.sub.1, b.sub.2).sup.T and, secondly, a specific set of moments of the probability density function. The standardized moments {tilde over (μ)}.sub.n, see above, are invariant in respect of scaling and displacement transformations of the random variable X, for which they are or have been calculated. Therefore, the standardized moments {tilde over (μ)}.sub.n of the random variable X can be considered to be independent of their raw and central moments v.sub.1(X) and μ.sub.2(X) (see above). Proceeding therefrom, it is possible to define a parameterized tuple M of moments for a Pearson probability density function and its assigned random variables X. The tuple M has already been defined above. In the process, the aforementioned labels v.sub.1(X), μ.sub.2(X), {tilde over (μ)}.sub.3, and {tilde over (μ)}.sub.4 were introduced for the expected value, the variance, the skewness, and the kurtosis. Reference is made to a partial set of the elements of the tuple M by virtue of using the corresponding indices, e.g., M.sub.1:2(X)=(v.sub.1(X), μ.sub.2(X)).

[0097] Below, the raw moments {ν.sub.i(X)}.sub.i=1.sup.4 and the tuple M are related to the parameters a and b of the conventional differential equations. The coupling between the moments and the coefficients can be written in the form of a matrix equation as:

[00003]$\begin{array}{cc}\left[\begin{array}{cccc}0& 1& 2{\nu }_1& 1\\ 1& 2{\nu }_1& 3{\nu }_2& {\nu }_1\\ 2{\nu }_1& 3{\nu }_2& 4{\nu }_3& {\nu }_2\\ 3v_2& 4{\nu }_3& 5v_4& v_3\end{array}\right]\left[\begin{array}{c}{b}_0\\ b_1\\ b_2\\ a_0\end{array}\right]=-\left[\begin{array}{c}{v}_1\\ v_2\\ v_3\\ {\nu }_4\end{array}\right].& \left(6\right)\end{array}$

[0098] Below, the aforementioned standardization is undertaken in order to simplify the solution. However, a corresponding solution can also be derived without such a standardization. Then, the solution equations become correspondingly more complex. For v.sub.1=0 and v.sub.2=1, the solution of Equation (6) is as follows:

[00004]$\begin{array}{cc}b=\frac{1}{c}\left[\begin{array}{c}4{\zeta }_2-3{\zeta }_1^2\\ \left(3+{\zeta }_2\right){\zeta }_1\\ 2{\zeta }_2-3{\zeta }_1^2-6\end{array}\right],a=\left[\begin{array}{c}-b_1\\ 1\end{array}\right],& \left(7\right)\end{array}$

where {tilde over (μ)}.sub.3(X)=ζ.sub.1(X) and {tilde over (μ)}.sub.4(X)=ζ.sub.2(X) are used as labels for the moments of the skewness and the kurtosis and where c=2(9+6ζ.sub.1.sup.2−5ζ.sub.2). Taking account of Equation (4) and Equation (6), the Pearson probability density function can now be parameterized as follows.

[0099] Given a parameterization moment tuple M* with a scaling and displacement component M*.sub.1:2(X)={v*.sub.1,μ*.sub.2} and a form component M*.sub.3:4(X)={ζ*.sub.1,ζ*.sub.2,} the corresponding Pearson probability density function can be constructed in two steps. In a first step, the coefficients a* and b* are calculated using Equation (7) and the problem of the Pearson differential equation is solved with these parameters in order to obtain a standardized Pearson probability density function ƒ.sub.s(x) and the corresponding random variable X. In a second step, a transformation τ(x)=(x−v*.sub.1)/√{square root over (μ.sub.2)} is set up and the required Pearson random variable X is defined as X=τ.sup.−1 (X.sub.s) and its probability density function is defined as ƒ(x)=τ′(x)ƒ.sub.s(τ(x)).

[0100] Here, the superscript index −1 denotes the inverse function and the superscript comma denotes the first derivative of the function. Now, M(X)=M*. This shows that each of the tuples M*.sub.1:2(X) and M*.sub.3:4(X), i.e., the scaling and displacement component on the one hand and the form component on the other hand, can be handled independently of one another within the scope of the procedure of fitting the Pearson probability density function to the corresponding moments of the distribution.

[0101] The Pearson plane is defined as a plane E of real numbers, where the first coordinate is the skewness and the second coordinate is the kurtosis. The Pearson plane E is now subdivided into three regions which are characterized by their behavior in respect of the zeros. The distinguishing criterion κ(ζ) for the three regions and corresponding boundary curves is defined as:

[00005]$\kappa \left(\zeta \right)\text{:=}\frac{{{\zeta }_1^2\left({\zeta }_2+3\right)}^2}{4\left(2{\zeta }_2-3{\zeta }_1^2-6\right)\left(4{\zeta }_2-3{\zeta }_1^2\right)}.$

[0102] The main regions of the Pearson plane E can be specified as set forth below. If x.sub.1 and x.sub.2 are the zeros of the polynomial b(x;ζ), the coefficients b of which are elements of the three-dimensional space of real numbers and are calculated using Equation (7), then the regions R.sub.1, R.sub.4, and R.sub.6 of the Pearson plane E can be specified as follows:

.sub.1:={ζ∈ε:x.sub.1,x.sub.2∈∧x.sub.1x.sub.2<0}={ζ∈ε:κ(ζ)<0},

.sub.4:={ζ∈ε:x.sub.1,x.sub.2∈}={ζ∈ε:κ(ζ)∈(0,1)},

.sub.6:={ζ∈ε:x.sub.1,x.sub.2∈∧x.sub.1x.sub.2>0}={ζ∈ε:κ(ζ)>0}.   (8)

[0103] The boundary curves C.sub.i with i=2, 3, 5, 7 of the specified regions, referred to as Pearson curves below, can be specified as follows:

.sub.2:={ζ∈ε:ζ.sub.1=0∧ζ.sub.2∈(1,3)},

.sub.3:={ζ∈ε:2ζ.sub.2−3ζ.sub.1.sup.2−6=0}={ζ∈ε:κ(ζ)∈{−∞,∞}},

.sub.5:={ζ∈ε:κ(ζ)=1},

.sub.7:={ζ∈ε:ζ.sub.1=0∧ζ.sub.2>3}.   (9)

[0104] Further, the aforementioned forbidden region

.sub..Math.=:{ζ∈ε:ζ.sub.2<ζ.sub.1.sup.2+1}   (10)

of the Pearson plane is introduced, the boundary line of which

.sub..Math.:={ζ∈ε:ζ.sub.2=ζ.sub.1.sup.2+1}   (11)

is given. Firstly, the forbidden region contains pairs of points (ζ.sub.1,ζ.sub.2), which can never occur as a solution to the calculation of the skewness and the kurtosis. Secondly, the solutions of the conventional Pearson differential equations for points on the forbidden curve are not integrable and do not represent a probability density function for this reason. Therefore, the regions with all points in the Pearson plane for which a solution can be found can be defined as follows:

:=ε\(C.sub..Math.∪.sub..Math.).   (12)

[0105] Expressed differently, the totality of these regions can be specified as the set of points which corresponds to the entire plane minus the set of the points in the forbidden region and minus the set of the points on the boundary line of the forbidden region. For one example embodiment, FIG. 11 shows the aforementioned three regions R.sub.1, R.sub.4, R.sub.6 and two of their aforementioned boundary curves C.sub.3 and C.sub.5. Moreover, like in FIG. 4, the forbidden region and its boundary line are illustrated in the lower part of FIG. 11. The point Po illustrated in FIG. 11 represents the relative position of the value pair of the skewness and the kurtosis of the normal distribution.

[0106] Only standardized Pearson distributions are considered below. Why such a consideration suffices was already justified above. Now, an expression for the support interval of Pearson distributions is initially derived as a function of the skewness ζ.sub.1 and the kurtosis ζ.sub.2. Then, curves or boundary lines are derived for the aforementioned solution sets. Finally, sets of the solution sets are derived for the right and the left support boundary are derived.

[0107] The zeros x.sub.1 and x.sub.2 of the polynomial b(x) defined in Equation (5) can be specified as follows:

[00006]$\begin{array}{cc}{x}_1=\frac{-b_1-\sqrt{b_1^2-4b_0{b}_2}}{2b_2}\mathrm{and}& \left(13\right)\\ x_2=\frac{-b_1-\sqrt{b_1^2-4b_0{b}_2}}{2b_2}.& \left(14\right)\end{array}$

[0108] If the coefficients b.sub.i are specified as functions of the skewness and the kurtosis, the following equations arise for the zeros x.sub.i=x.sub.i(ζ):

[00007]$\begin{array}{cc}{x}_{\left(1\right)}\left(\zeta \right)=\frac{u\left(\zeta \right)+\mathrm{sgn}\left(c\right)\sqrt{v\left(\zeta \right)}}{w\left(\zeta \right)}\mathrm{and}& \left(15\right)\\ x_{\left(2\right)}\left(\zeta \right)=\frac{u\left(\zeta \right)-\mathrm{sgn}\left(c\right)\sqrt{v\left(\zeta \right)}}{w\left(\zeta \right)}.& \left(16\right)\end{array}$

[0109] Setting c=2(9+6ζ.sub.1.sup.2−5ζ.sub.2), it is possible to specify the following auxiliary functions:

u(ζ):=ζ.sub.1(ζ.sub.2+3),

v(ζ)=−36ζ.sub.1.sup.4+ζ.sub.1.sup.2(ζ.sub.2.sup.2+78ζ.sub.2−63)−32(ζ.sub.2−3)ζ.sub.2,

w(ζ):=6ζ.sub.1.sup.2−4ζ.sub.2+12.   (17)

[0110] The zeros x.sub.1 and x.sub.2 are complementary in respect of sgn(c), i.e., in respect of the sign. This leads to the following alternative equations:

[00008]$\begin{array}{cc}{x}_{\left(1\right)}\left(\zeta \right)\text{:=}\frac{u\left(\zeta \right)-\sqrt{v\left(\zeta \right)}}{\left(w\right)\left(\zeta \right)}\mathrm{and}& \left(18\right)\\ x_{\left(2\right)}\left(\zeta \right)\text{:=}\frac{u\left(\zeta \right)+\sqrt{v\left(\zeta \right)}}{w\left(\zeta \right)}.& \left(19\right)\end{array}$

[0111] In respect of Equations (18) and (19), it is noted that the argument of the square root in the numerator of the fraction is only not defined for negative expressions. Therefore, the functions as per Equations (18) and (19) are not defined for the forbidden region and its boundary line.

[0112] In relation to a standardized Pearson distribution with a moment tuple M*.sub.3:4=ζ, where the left boundary of the support interval is denoted by ξ.sub.L and the right boundary of the support interval is denoted by R, the support interval I=(ξ.sub.L,ξ.sub.R) is defined as follows:

[00009]$\begin{array}{cc}{\xi }_L=x_L\left(\zeta \right):=\{\begin{array}{cc}{x}_{\left(1\right)}\left(\zeta \right),& \zeta \in 1_{}.Math.{\mathrm{��}}_2\bigvee \left({\zeta }_1>0⩓\zeta \in 6_{}.Math.{\mathrm{��}}_5\right),\\ -\frac{2}{{\zeta }_1},& {\zeta }_1>0⩓\zeta \in {\mathrm{��}}_3,\\ -\infty ,& \mathrm{else}\end{array}{\xi }_R=x_R\left(\zeta \right)\text{:=}\{\begin{array}{cc}{x}_{\left(2\right)}\left(\zeta \right),& \zeta \in 1_{}.Math.{\mathrm{��}}_2\bigvee \left({\zeta }_1<0⩓\zeta \in 6_{}.Math.{\mathrm{��}}_5\right),\\ -\frac{2}{{\zeta }_1},& {\zeta }_1<0⩓\zeta \in {\mathrm{��}}_3,\\ \infty ,& \mathrm{else}.\end{array}& \left(20\right)\end{array}$

[0113] Here, “else” has its conventional meaning. Proceeding from Equations (20) and (21), the following implicit equations can be specified for boundary curves c.sub.L(t) and c.sub.R(t) of the solution regions:

x.sub.L(c.sub.L(t))=ξ.sub.L for ξ.sub.L<0,

x.sub.R(c.sub.R(t))=ξ.sub.R for ξ.sub.R>0,   (22)

[0114] where “for” has its conventional meaning. A few preliminary reflections are made below before these curves are calculated. The naïve solution of the equations x.sub.(1)(ζ)=ξ.sub.L and x.sub.(2)(ζ)=ξ.sub.R for the zeros leads to the curves(t;ξ), which is defined by:

[00010]$\begin{array}{cc}s\left(t;\xi \right):=\left[\begin{array}{c}t\\ \frac{6{\xi }^2-3\mathrm{\xi t}+3t^2+3{\xi }^2{t}^2}{4+2{\xi }^2+\mathrm{\xi t}}\end{array}\right]& \left(23\right)\end{array}$

for x.sub.(1)(s(t;ξ.sub.L))=ξ.sub.L and x.sub.(2)(s(t;ξ.sub.R))=ξ.sub.R The curve s(t;ξ) has the following properties: For ξ≠0, there is a singularity at

ζ.sub.1.sup.∞(ξ):=−(4+2ξ.sup.2)/ξ;

[0115] For ξ≠0, there is a point of intersection at the following point of the forbidden curve, specifically the boundary line of the forbidden region:

ζ.sub.1.sup..Math.(ξ):=(ξ.sup.2−1)/ξ

[0116] For |ξ|>√{square root over (2)}, there is a point of intersection with the aforementioned curve or boundary line C.sub.5, see Equation (9):

ζ.sub.1.sup.5(ξ):=−4ξ/(ξ.sup.2−1)

[0117] For ξ≠0, there is a unique global minimum at:

[00011]${\zeta }_1^{\min }:={\zeta }_1^{\infty }-2\sqrt{{\xi }^2+4/{\xi }^2+5}=\frac{2}{\xi +2/\xi -\sqrt{{\xi }^2+4/\xi +5}}$

[0118] The equations x.sub.(1)(ζ)=ξ.sub.L and x.sub.(2)(ζ)=ξ.sub.R are solved by the branch of the curve s(t;ξ) which, for x.sub.(1)(ζ), is located to the right and, for x.sub.(2)(ζ), to the left of ζ.sub.1.sup.∞(ξ).

[0119] Although s(t;ξ) is not the solution to Equations (22), the following deliberation shows that only the range of definition of the curve s(t;ξ) needs to be adapted in order to obtain the solution. Implicit Equations (22) for the support boundary lines c.sub.L(t;ξ.sub.L) and c.sub.R(t;ξ.sub.R) have the following solutions:

[00012]$\begin{array}{cc}{c}_L\left(t;{\xi }_L\right):=s\left(t;{\xi }_L\right)\mathrm{with}\text{.Math.}\left(c_L\right):=\{\begin{array}{cc}\left({\zeta }_1^{.Math.}\left({\xi }_L\right),{\zeta }_1^5\left({\xi }_L\right)\right),& \mathrm{if}{\xi }_L\in \left(-\infty ,-\sqrt{2}\right),\\ \left({\zeta }_1^{.Math.}\left({\xi }_L\right),{\zeta }_1^{\infty }\left({\xi }_L\right)\right),& \mathrm{if}{\xi }_L\in \left[-\sqrt{2},0\right)\end{array}\mathrm{and}{c}_R\left(t;{\xi }_R\right):=s\left(t;{\xi }_R\right)\mathrm{with}\left(c_R\right):=\{\begin{array}{cc}\left({\zeta }_1^5\left({\xi }_R\right),{\zeta }_1^{.Math.}\left({\xi }_R\right)\right),& \mathrm{if}{\xi }_R\in \left(\sqrt{2},\infty \right).\\ \left({\zeta }_1^{\infty }\left({\xi }_R\right),{\zeta }_1^{.Math.}\left({\xi }_R\right)\right),& \mathrm{if}{\xi }_R\in \left(0,\sqrt{2}\right].\end{array}& \end{array}$

[0120] Here, “with” and “if” have their conventional meanings. What follows therefrom is that there is a transformation between the right and the left support boundary line. The following relationship applies:

c.sub.R(t;ξ.sub.R)=c.sub.L(−t;−ξ.sub.R)

[0121] In relation to Equations (22), the following follows:

x.sub.L(c.sub.L(t;ξ.sub.L)=ξ.sub.L<0 and x.sub.R(c.sub.R(t;ξ.sub.R))=ξ.sub.R>0

[0122] For the region of the Pearson plane outside of the forbidden region, it is now possible to define the sets R.sub.L(ξ) for ξ<0 in respect of the left or lower support boundary v.sub.min and R.sub.R(ξ) for >0 in respect of the right or upper support boundary v.sub.max, which are equivalent to the aforementioned solution sets M.sub.L of Equation (1) and M.sub.R of Equation (2), as follows:

.sub.L(ξ)={ζ∈:x.sub.L(ζ)≤ξ}, ξ<0,

.sub.R(ξ)={ζ∈:x.sub.R(ζ)≥ξ}, ξ>0,

[0123] What follows therefrom is that the relationship a≤ξ (applies to a Pearson probability density function with a support region (a, b) if ξ<0 and if the point of the Pearson plane with the coordinates ζ is located in the solution set or the support region R.sub.L(ξ) for the left support boundary. Analogously, b≥ξ applies if ξ>0 and if the point of the Pearson plane with the coordinates ζ is located in the solution set or the support region R.sub.R(ξ) for the right support boundary.

[0124] The left support region R.sub.L(ξ) for ξ<0 can be expressed by the set S(ξ) of points in the Pearson plane, which unifies the following individual sets S.sub.i(ξ), where i=1, 2, 3:

.sub.1(ξ):={ζ∈:ζ.sub.1≤.sub.ξ},

.sub.2:={ζ∈:ζ.sub.1∈.sub.ξ∧ζ.sub.2≥c.sub.L(⋅;ξ)},

.sub.3(ξ):={ζ∈:ζ.sub.1≥.sub.ξ∧v(ξ)<0},

[0125] Here, D.sub.ξ denotes the range of definition of the left support boundary curve c.sub.L, v(ξ) denotes the middle one of Equations (17) and R denotes the region in the Pearson plane without the forbidden region and its boundary line. FIG. 12 shows the solution set R.sub.L(ξ) in respect of the left or lower support boundary v.sub.min for ξ=−2.5. FIG. 13 shows this solution set R.sub.L(ξ) for ξ=−0.5. While FIG. 12 shows all three sets S.sub.i, the set S.sub.3 is empty in the case of FIG. 13. This applies to the range −√{square root over (2)}≤ξ<0, since there is no point of intersection for the left boundary curve c.sub.L and the aforementioned curve C.sub.5. In FIG. 12 and FIG. 13, the boundary curve of the set S.sub.2 is highlighted in each case by virtue of the boundary curve being represented by a wide solid line. The dashed line, as boundary curve of the set S.sub.1, is the boundary line of the forbidden region.

[0126] Proceeding from the relationship between the left support region R.sub.L(ξ) and ξ, it is possible to state the following: The following applies to two negative values b≤a<0:

.sub.L(b).Math..sub.L(a),

ζ∈.sub.L(a)\.sub.L(b)(x.sub.L(ζ),0)⊂(b,0).

[0127] If values ξ.sub.L<0<ξ.sub.R are given, i.e., solution sets are defined for the right and left support boundary, the following applies to the intersection:

[00013]$\begin{array}{c}{L.Math.R}_{}\left({\xi }_L,{\xi }_R\right):=L_{}\left({\xi }_L\right)R_{}\left({\xi }_R\right)\\ =\left\{\zeta \in \text{:}\left({\xi }_L,{\xi }_R\right).Math.\mathrm{supp}\left(f\left(.Math.;\zeta \right)\right)\right\}.\end{array}$

[0128] The description above contains an example embodiment for defining the set of all those statistical distributions that are able to describe a frequency of measurement data values in the entire value interval (the range). There now is a description of an example embodiment of the invention in relation to the check whether the defined set contains a statistical distribution which has the moment values of the skewness and the kurtosis that were ascertained for a sample.

[0129] FIG. 14 schematically shows an arrangement of devices for evaluating a sample of measurement data from measuring a multiplicity of workpieces. The illustration in FIG. 14 can also be considered to be a flowchart for explaining an embodiment of a method for evaluating the sample of measurement data.

[0130] By way of example, a coordinate measuring machine 1 measures the plurality of workpieces and transfers the measurement data, optionally following preprocessing of the measurement data, to a measurement data memory 3. A definition device 5 is configured to define a set of all those statistical distributions that are able to describe a frequency of measurement data values in the entire value interval from the system of statistical distributions for a value interval of the measurement data, which is a specified value interval or a value interval of the measurement data actually arising in the sample. In FIG. 14, an input of the definition device 5 is connected to an output of the measurement data memory 3. Therefore, the definition device 5 can ascertain the value interval, in particular from the measurement data available. As an alternative or in addition thereto, it can obtain additional information about a value interval going beyond the value interval of the measurement data, said additional information being stored in the measurement data memory 3. However, in practice the information about the specified value interval can also be made available to the definition device 5 in any other way, and so a link between the measurement data memory 3 and the definition device 5 is not mandatory.

[0131] An output of the definition device 5 is connected to an input of a moment ascertainment device 7 which is configured to ascertain a respective moment value of the skewness and the kurtosis from the sample of measurement data corresponding to a first statistical distribution. During the operation or while the method is carried out, the moment ascertainment device 7 ascertains values of the skewness and the kurtosis for the sample and transmits the values to a checking device 9 which is configured to use the ascertained moment values to check whether the defined set contains a statistical distribution which has the ascertained moment values of the skewness and the kurtosis, and to produce a corresponding test result.

[0132] From the test result, it is clear whether or not such a statistical distribution exists in the defined set. Should this be the case, a signal output by the checking device 9, for example, can confirm that the first statistical distribution is suitable for the purpose of statistical modeling of the sample over the entire specified value interval. Should this not be the case, the checking device 9 outputs the test result or a signal to a distribution ascertainment device 11 which ascertains a second statistical distribution that is suitable for the purpose of statistical modeling of the sample over the entire specified value interval. In this case, the distribution ascertainment device 11 can ascertain, e.g., that value pair from the Pearson plane which corresponds to a suitable statistical distribution. Using the information about the value pair, it is possible, in turn, to produce the suitable statistical distribution as second statistical distribution.

[0133] The definition device 5 and the distribution ascertainment device 11 can also form an arrangement without the other devices illustrated in FIG. 14. Further, the definition device 5 and the distribution ascertainment device 11 can serve to prepare the evaluation of a sample of measurement data. In both cases, the definition device 5 is configured to define a set of all those statistical distributions that are able to describe a frequency of measurement data values in the entire value interval from the system of statistical distributions for a value interval of measurement data, which is a specified value interval or a value interval of a sample, to be evaluated, of measurement data. Then, the distribution ascertainment device 11 is configured to ascertain a statistical distribution from the defined set.

[0134] If FIG. 14 is interpreted as a flowchart then measurement data are produced in a method step 1, said measurement data being stored in a method step 3. In a method step 5, the set of all those statistical distributions that are able to describe a frequency of measurement data values in the entire value interval is defined. In a method step 7, a respective moment value of the skewness and the kurtosis is ascertained from the sample of measurement data corresponding to a first statistical distribution. In a method step 9, the ascertained moment values are used to check whether the defined set contains a statistical distribution which has the ascertained moment values of the skewness and the kurtosis, and a corresponding test result is produced. The sequence of the method steps arises from the arrows in FIG. 14.

[0135] Devices or method steps that are optional or belong to specific configurations are denoted by reference signs 1, 3 and 11 in FIG. 14. Method step 5 need not necessarily be always carried out when a first statistical distribution should be checked in respect of its suitability for statistical modeling of a sample over the entire specified value interval.

[0136] In particular, on the basis of a measure of distance for the distance between two distributions in the Pearson plane, the distribution ascertainment device 11 can ascertain that distribution whose distance from the first statistical distribution is minimal as second statistical distribution. FIG. 15 illustrates an example of two distributions in the Pearson plane which have a minimal distance, in this case the minimal Euclidean distance in the Pearson plane. The cross situated more to the bottom right represents the first statistical distribution in a case where it is not suitable for statistically modeling the sample over the entire specified value interval. This lack of suitability is evident from the fact that the cross is located outside of the hatched area of the solution set of suitable statistical distributions. The cross represents the distribution by virtue of marking a point in the Pearson plane which corresponds to the distribution, said point in turn corresponding to the value pair of the skewness and the kurtosis of the distribution. In accordance with the rules specified here, the suitable second statistical distribution is ascertained from the first statistical distribution by virtue of ascertaining the point in the solution set of suitable distributions in the Pearson plane which has the shortest distance from the point of the first statistical distribution. In all cases, this point of shortest distance is located on the edge of the solution set.

[0137] In a manner analogous to the illustration in FIG. 1, FIG. 16 now shows a statistical distribution, more precisely a probability density function, which was ascertained as described above and which is consequently located in the solution set of the suitable statistical distributions and is therefore suitable for statistically modeling the measurement data values of the sample over the entire specified value interval. In FIG. 16, this is evident from the fact that, in contrast to FIG. 1, the probability density function represented by the solid line also adopts positive frequency values or probability values in the region of the measurement data values smaller than −1.9.

[0138] In the case where the evaluation of a sample of measurement data is prepared, it is possible, for example, to only carry out the method steps of defining the set of all those statistical distributions that are able to describe a frequency of measurement data values over the entire value interval and of ascertaining the statistical distribution belonging to the set. Naturally, this does not preclude the ascertainment of the statistical distribution being followed by an evaluation of a sample of measurement data by means of this statistical distribution.

[0139] The term non-transitory computer-readable medium does not encompass transitory electrical or electromagnetic signals propagating through a medium (such as on a carrier wave). Non-limiting examples of a non-transitory computer-readable medium are nonvolatile memory circuits (such as a flash memory circuit, an erasable programmable read-only memory circuit, or a mask read-only memory circuit), volatile memory circuits (such as a static random access memory circuit or a dynamic random access memory circuit), magnetic storage media (such as an analog or digital magnetic tape or a hard disk drive), and optical storage media (such as a CD, a DVD, or a Blu-ray Disc). The phrase “at least one of A, B, and C” should be construed to mean a logical (A OR B OR C), using a non-exclusive logical OR, and should not be construed to mean “at least one of A, at least one of B, and at least one of C.”