METHOD AND ARRANGEMENT FOR EVALUATING OR PREPARING AN EVALUATION OF A SAMPLE OF MEASUREMENT DATA FROM A MEASUREMENT OF A MULTIPLE OF WORKPIECES

Description

[0001] The invention relates to a method executed on a computer or computer network by a computer program and an arrangement for evaluating a sample of measurement data from a measurement of a plurality of workpieces by one or more coordinate measuring machines.

[0002] The concept of statistical process control is well-known in the field of quality control for processes involved in the manufacture and / or processing of workpieces and workpiece arrangements. Statistical modeling of a process yields a statistical distribution, which allows for statements to be made, in particular, about the frequency of specific measured values ​​generated by the process. For example, according to the patent claims, the result of statistical modeling can be that the process is controllable. Especially when the spread of the measured values ​​is small, the result of statistical modeling can also, or alternatively, indicate that its quality meets the specifications.

[0003] Typically, such a statistical model is created by using information from both the process itself and the analysis of measurement data. For example, it is known that when measuring the radius of a workpiece, the process can be described by a normal distribution. In this case, the measured values ​​from a large number of identical workpieces scatter randomly around an expected value. The normal distribution allows parameters such as, in particular, the spread to be determined from it.

[0004] Not every process can be meaningfully described by a normal distribution. Instead, other statistical distributions exist that describe the frequency of measured values ​​of a quantity as a function of the measured value. Unlike a normal distribution, these distributions may only be suitable for the statistical description of measured values ​​within a one-sided or two-sided bounded interval. Such a distribution assigns no frequency to values ​​outside the interval, and / or the frequency, and thus the probability, of values ​​outside the interval is set to zero.

[0005] For example, the Pearson distribution system is a system of multiple well-known and widely used statistical distributions. If a sample of measurement data is available for a given measurement, one of the statistical distributions from the distribution system can be selected and used for statistical modeling. This is based on the fundamental idea that a distribution is created solely from information contained in the available data. Various approaches can be used to create the statistical distribution. In particular, one can proceed according to the principle of maximum likelihood, i.e., the probability that the distribution correctly describes the statistics of the available data is maximized. Another approach involves determining the so-called moments of the statistical distribution from the measurement data and then selecting a distribution that matches these moments.When statistically modeling with a distribution based on measurement data or derived data, it can happen that the distribution accurately describes the frequency of the measured values ​​or the derived values ​​over a large part of the occurring or possible value interval, but values ​​also lie, or can lie, outside the value interval described by the distribution. In practice, this has the disadvantage that the statistical distribution is not optimally modeled, and users may encounter the undesirable case where occurring measured values ​​do not fall within the value interval of the distribution and therefore do not contribute to the distribution or cannot be predicted by it. In this case, there is an obvious modeling error.

[0006] WO 99 / 42811 A1 describes a method for identifying the corrosion of a metal object, comprising the steps of providing electrode means capable of generating electrochemical noise signals and analyzing the statistical distribution of the noise signals to provide an indication of corrosion parameters, wherein the parameters include the extent and / or nature of the corrosion.

[0007] T. Segreto et al., "Feature Extraction and Pattern Recognition in Acoustic Emission Monitoring of Robot Assisted Polishing," PROCEDIA CIRP, Volume 28, December 1, 2015, pages 22-27, describes the use of sensors that emit an acoustic signal for monitoring a surface polishing process. A statistical analysis of the received signals is performed.

[0008] US 2016 / 225389 A1 describes a method for detecting an abnormal noise, wherein a standard pattern vector is generated with a feature set of a standard tone as a component and an original input pattern vector with a feature set of an input sound as a component.

[0009] CN 107 220 216 A concerns the calculation of spare parts requirements and in particular an approximate calculation method for spare parts requirements of the Weibull type using key figures.

[0010] None of the four documents mentioned above concerns the evaluation of a sample of measurement data from a survey of a large number of workpieces by one or more coordinate measuring machines.

[0011] It is an object of the present invention to provide a method and an arrangement for evaluating a sample of measurement data from a measurement of a plurality of workpieces by one or more coordinate measuring machines, with which measurement data for the quality control of processes for the manufacture and / or machining of workpieces and arrangements of workpieces can be statistically modeled with high quality.

[0012] The attached patent claims define the scope of protection. Claim 1 defines a method executed on a computer or computer network by a computer program. Claim 8 defines an arrangement. Claim 15 defines a computer program for executing the method.

[0013] The solution is based on the fundamental idea that for a first statistical distribution or for a group of first statistical distributions with which a sample of measurement data can be statistically described, it is checked whether these are suitable to describe the frequency of measurement data values ​​as a function of the measurement data values ​​within a given interval of values ​​or within an interval of values ​​of the measurement data actually occurring in the sample.

[0014] To conduct the test, an initial evaluation of the sample measurement data takes place. Furthermore, the result of this test directly informs the evaluation of the measurement data sample, as a statistical distribution that is unsuitable for modeling the sample across the entire interval of values ​​can be identified as such. This, in particular, makes it possible to determine a more suitable statistical distribution.

[0015] Further steps in the sample analysis, in particular the determination of parameters for the distribution suitable for statistical modeling of the sample, can be additional components of the sample analysis procedure. The appropriate statistical distribution represents one possible outcome of the sample analysis. The finding that the initial statistical distribution is not suitable for statistically modeling the sample across the entire specified or actual interval of measured values ​​represents another possible outcome of the sample analysis.

[0016] The measurement data are obtained by measuring the workpieces using one or more coordinate measuring machines (CMMs). The type of CMM is not limited to a specific type. As long as the measured values ​​are comparable, different CMMs and / or CMMs of different types can be used to generate the measurement data contained in the sample. Examples of CMM types include portal or gantry-type CMMs, CMMs with an articulated arm, robots or machine tools with at least one measuring sensor, and devices with a measuring sensor, particularly an optical one, or of another type, where the measuring sensor is fixed in the measuring space. For example,Even a complex workpiece can be measured using an array of cameras in such a way that the contours of the workpiece's surface are determined within a coordinate system of the workpiece or the measuring space. From this, the positions of surface points or surface areas of the workpiece can be determined.

[0017] Various types of measuring sensors are suitable for generating measurement data. For example, tactile and / or optical sensors are used, particularly on classic coordinate measuring machines. Non-contact measuring sensors include, for example, distance sensors and / or imaging sensors such as cameras. Distance sensors can operate according to different measuring principles, such as capacitive sensors, inductive sensors, time-of-flight sensors like TOF cameras, and / or chromatic confocal sensors. Furthermore, at least one sensor of the coordinate measuring machine can be moved by means of a drive device and can incorporate the position of the measuring sensor, determined from a scale system of the coordinate measuring machine, into the determination of the workpiece coordinates.

[0018] The coordinate measuring machines and sensors described above are merely examples. In general, workpiece measurement data can be generated in any way by one or more coordinate measuring machines. This measurement data consists, in particular, of coordinates of at least a portion of the workpiece surface and / or the workpiece interior. Coordinates of, for example, cavity boundaries or transitions between different materials can be determined using invasive measurement methods such as X-ray or other invasive radiation scanning of the workpiece from different directions, followed by computer-aided reconstruction (e.g., CT scanning). In particular, the measurement data can also be derived from at least one coordinate and, for example, from several coordinates of the respective workpiece, including parameters such as radii, lengths, roundness, and curvature.Such derived data, which in particular correspond to several coordinates of the respective workpiece, can also be determined directly using special measuring instruments without explicitly determining the coordinates. For example, camera images make it possible to determine the roundness or diameter of a cylindrical section of a workpiece. Measuring instruments with tactile sensors are also known that directly determine such derived measured quantities.

[0019] Furthermore, the measurement data can be derived from, or contain, the raw data generated by the respective measuring device or measuring setup. For example, the raw data obtained from measuring a number of identical workpieces can be averaged, and / or outliers, i.e., nonsensical or implausible values, can be eliminated or corrected from the raw data.

[0020] Given the multitude of workpieces whose measurements form the basis for the measurement data, these can be simple or complex, often composed of multiple parts. The only requirement is that comparable measured values ​​or derived values ​​are available for evaluation. Therefore, a workpiece can be anything from a simple component like a screw or a pin to a composite or irregularly shaped workpiece like a car door, as well as a machine or any other arrangement of workpieces.

[0021] A sample of measurement data refers to any selection and / or processing result from available measurement data of the type mentioned above, and / or to the complete measurement data set. The measurement data forms the basis for statistical description and / or modeling.

[0022] As mentioned, it is possible, and proposed within the scope of the invention, to test whether a first statistical distribution is suitable for statistical description of the sample. In particular, the first statistical distribution can be created solely on the basis of the sample. For the test, however, it is sufficient to determine only certain properties of the first statistical distribution. As will be described in more detail below, it is only necessary for the first statistical distribution that certain moments of the distribution can be determined. Optionally, a preselection of a distribution type, such as a distribution from the Pearson distribution system, can take place. However, the first statistical distribution can also be a distribution that is not part of the system of statistical distributions that serves as the basis for a second statistical distribution that may be determined.

[0023] The term "moments" of a statistical distribution is a statistical term. These moments typically include the expected value, the variance, the skewness, and the kurtosis. These moments, and optionally higher-order moments, can be used as characteristic values ​​for the respective statistical distribution.

[0024] It is now proposed that, for the aforementioned interval of measurement data, which is either a predetermined interval or an interval of the measurement data actually occurring in the sample, a set of all statistical distributions capable of describing the frequency of measurement data values ​​within the entire interval is defined from a system of statistical distributions. It is not considered a description of frequency if the frequency or probability for a measurement data value is generally zero or undefined for every value of the statistical distribution. Actual or possible occurrences of this measurement data value would not be adequately described by a probability or frequency value of zero. At least one embodiment of how the set of statistical distributions suitable for describing the frequency within the entire interval is determined is presented.The process of defining the set of suitable statistical distributions will be discussed later. This definition can occur whenever an initial statistical distribution is generated for a sample and / or the initial statistical distribution is characterized in preparation for testing its suitability. However, as long as the interval of values ​​of the measurement data is the same for different samples, it suffices to define the set only once, since the set depends solely on the interval of values ​​and not on the frequency or probability of the measurement data values ​​in the sample. In particular, the definition of the set of suitable statistical distributions can take place before, simultaneously with, and / or after the generation or characterization of the initial statistical distribution for the sample.

[0025] In particular, when defining the set of all statistical distributions capable of describing the frequency of measured data values ​​within the entire interval of values, a boundary of the set can be determined from the interval of values. The set can then be uniquely determined from this boundary.

[0026] It has already been mentioned that the first statistical distribution does not necessarily have to belong to the system of statistical distributions, although it can be included in certain configurations. However, it is necessary that a moment value for skewness and kurtosis corresponding to the first statistical distribution can be determined in order to characterize it, or a group of first statistical distributions with the same skewness and kurtosis values.

[0027] In order to check whether the set of suitable statistical distributions contains a distribution suitable for the sample of measurement data, a moment value of the skewness and the kurtosis is now determined according to the first statistical distribution.

[0028] There are several ways to determine the moment values ​​of skewness and kurtosis according to the first statistical distribution. Firstly, skewness and kurtosis values ​​can be directly determined from a sample of measurement data. Optionally, a corresponding first statistical distribution can then be specified in more detail, for example, by providing the relevant probability density function that exhibits the same skewness and kurtosis values ​​as parameters. In particular, this can be done using the method of defining the statistical distribution according to its moments.

[0029] Another possibility is to determine a first statistical distribution that statistically models the sample measurement data, for example, using the method of maximizing the probability that the first statistical distribution correctly models the sample measurement data. Then, the skewness and kurtosis values ​​of the first statistical distribution can be determined.

[0030] If the first statistical distribution does not belong to the system of statistical distributions, then a statistical distribution can be chosen or may already exist within the system that has the same skewness and kurtosis values ​​and that adequately models the respective measurement data. For the sake of simplicity, it can therefore be assumed, for example, that the statistical distribution within the system that has the skewness and kurtosis values ​​determined from the sample of measurement data adequately models the measurement data. At this point, "adequately models" does not necessarily mean that the distribution models the measurement data across the entire interval of values. This will be examined later.

[0031] Based on the determined moment values ​​for skewness and kurtosis, the system checks whether a statistical distribution exists within the defined set that exhibits the determined moment values ​​for skewness and kurtosis. A corresponding test result is generated and optionally output. The test result may, in particular, confirm the existence of a statistical distribution with the determined values ​​for skewness and kurtosis within the defined set. In this case, the first statistical distribution can be used as a suitable statistical distribution for modeling the measured data values ​​across the entire value interval, provided it belongs to the system of distributions. If the first statistical distribution has not yet been fully generated before the test is performed, this can now be done.If the first statistical distribution does not belong to the system, or if the skewness and kurtosis values ​​were determined directly from the sample of measurement data, the distribution contained in the system with the determined skewness and kurtosis values ​​can be used as a suitable statistical distribution for modeling the measurement data values ​​in the entire interval of values.

[0032] The test result may also be that the determined moment values ​​for skewness and kurtosis do not belong to any statistical distribution within the defined set of suitable distributions. In this case, the test result indicates that the first statistical distribution, or the group of first statistical distributions, is unsuitable. In this case, it is not necessary to generate the first statistical distribution completely in all the details required for statistical modeling. In particular, it is not necessary to specify a function equation for the statistical distribution that describes the frequency or probability of occurrence of the measured data values ​​as a function of the measured data values. In this case, it is preferred that the test result is output and / or that the process of determining a suitable second statistical distribution is initiated.

[0033] When defining the set of all statistical distributions capable of describing the frequency of measured data values ​​within the entire interval of values, pairs of values ​​representing the skewness and kurtosis of statistical distributions of the system can be derived from the interval of values. These pairs correspond to the set. Thus, a relationship is established between the interval of values ​​and the skewness and kurtosis of statistical distributions of the system. The pairs of values ​​determined using information, particularly about the boundaries of the interval of values, also constitute a set. However, for the sake of clarity, the term "set" will be used in the following only in relation to the appropriate statistical distributions.

[0034] In particular, when defining the set of all statistical distributions capable of describing the frequency of measurement data values ​​across the entire interval, a boundary curve can be derived from the interval in a plane spanned by the skewness and kurtosis of statistical distributions of the system. This plane can also be called the Pearson plane if the system of distributions is a Pearson distribution system. The pairs of values ​​for skewness and kurtosis on the boundary curve and on one side of the boundary curve then correspond to the set of suitable statistical distributions of the system. This makes it possible, in a simple way, to determine, based on specific values ​​of skewness and kurtosis, whether a statistical distribution is suitable for statistically modeling measurement data across the entire interval.

[0035] The system of statistical distributions is the Pearson distribution system, although there are variants with regard to the number of distributions belonging to the system. For example, there is a variant with eight types of distributions, and there are variants with more than eight, for example twelve, types of distributions, where some of the twelve types are subtypes of the eight types.

[0036] It is preferred that the sample and / or the initial statistical distribution be standardized to the case where the expected value is zero and the variance is one. For a given sample or distribution, this is possible by shifting the measured values ​​or the distribution to the expected value of zero and dividing the measured values ​​or the argument of the probability density function by the standard deviation. Once, for example, a statistical distribution suitable for modeling over the entire interval of values ​​in the sample has been determined, the measured values ​​or the argument of the probability density function can be multiplied by the standard deviation, and the shift to the expected value of zero can be reversed by shifting in the opposite direction. The result is a statistical distribution suitable for modeling with the correct expected value and the correct variance.However, such standardization is not necessary. In particular, the effort required to define the set of all statistical distributions capable of describing the frequency of measurement values ​​within the entire interval of values ​​is then higher.

[0037] A method for evaluating a sample of measurement data from a measurement of a plurality of workpieces according to claim 1 has, among other things, the following features and further features: A system of statistical distributions exists or is established which is capable of describing a frequency of measurement data values ​​as a function of the measurement data values, wherein instances of the system of statistical distributions are distinguishable from one another by a moment value of two moments, namely a skewness and a kurtosis, of the respective statistical distribution, and wherein For a value interval of the measurement data, which is a predefined value interval or a value interval of the measurement data actually occurring in the sample, a set of all those statistical distributions is defined from the system of statistical distributions which are able to describe a frequency of measurement data values ​​in the entire value interval, a moment value of skewness and kurtosis is determined from the sample of measurement data according to a first statistical distribution, it is checked on the basis of the determined moment values ​​whether a statistical distribution exists in the defined set that exhibits the determined moment values ​​of skewness and kurtosis, and a corresponding test result is generated.

[0038] According to claim 3, an arrangement for evaluating a sample of measurement data from a measurement of a plurality of workpieces has, among other things, the following features and further features: A system of statistical distributions exists or is established which is capable of describing a frequency of measurement data values ​​as a function of the measurement data values, wherein instances of the system of statistical distributions are distinguishable from one another by a moment value of two moments, namely a skewness and a kurtosis, of the respective statistical distribution, and wherein the arrangement has: a definition device designed to define, for a value interval of the measurement data, which is a predetermined value interval or a value interval of the measurement data actually occurring in the sample, a set of all those statistical distributions from the system of statistical distributions which are able to describe a frequency of measurement data values ​​in the entire value interval; a moment determination device designed to determine, from the sample of measurement data according to a first statistical distribution, a moment value of skewness and kurtosis each; a testing device designed to check, on the basis of the determined moment values, whether a statistical distribution exists in the defined set which exhibits the determined moment values ​​of skewness and kurtosis, and to generate a corresponding test result.

[0039] Although more general than defined by the claims, a method for preparing an evaluation of a sample of measurement data from a measurement of a large number of workpieces is described here, wherein a system of statistical distributions exists or is established which is capable of describing a frequency of measurement data values ​​as a function of the measurement data values, wherein instances of the system of statistical distributions are distinguishable from one another by a moment value of two moments, namely a skewness and a kurtosis, of the respective statistical distribution, and wherein For a value interval of measurement data, which is a predetermined value interval or a value interval of a sample of measurement data to be evaluated, a set of all those statistical distributions is defined from the system of statistical distributions which are able to describe a frequency of measurement data values ​​in the entire value interval, and a statistical distribution is determined from this set.

[0040] The specific features of the order correspond to specific features of the procedure, so that in particular each of the specific features of the procedure described below also corresponds to a specific feature of the order.

[0041] According to the claims, the statistical distribution thus determined is used in a further step of the method or during the operation of the arrangement for the statistical modeling of the measurement data of a sample. However, no sample of measurement data, nor even any measurement data itself, needs to be available during the preparation of the evaluation. The value interval can, for example, be specified without any reference to specific measurement data. Alternatively, the value interval can be defined for specific, but not yet available, measurement data. Of course, the method for preparing the evaluation and the arrangement for preparing the evaluation also apply to the case where specific measurement data and, if applicable, the sample to be evaluated are already available.

[0042] The preparation for the evaluation is based on the same principle as the evaluation of the measurement data described above, namely a check to see if an initial statistical distribution is suitable for modeling a sample of measurement data across the entire interval of values. This principle is based on defining, from the system of statistical distributions, a set of all statistical distributions capable of describing the frequency of measurement data values ​​within the entire interval.

[0043] Preferably, a statistical distribution is uniquely determined from the set according to a given rule, i.e., the statistical distribution results uniquely from the given rule.

[0044] Specifically, the given rule is simply to determine the statistical distribution from the system of distributions that is precisely defined for the given interval of values; that is, in terms of the concepts used in the figure description, the support of the distribution is equal to the range of the sample of measurement data. However, depending on the boundaries of the interval of values, there are cases in which such a unique determination is not possible. If, however, the boundaries of the interval of values ​​satisfy two conditions, such a unique determination of a statistical distribution is possible. The first condition is that the left boundary of the interval of values ​​is smaller than the right boundary of the interval. The second condition is that the left boundary multiplied by the right boundary is less than -1.If the second condition is not met, the pair of values ​​for skewness and kurtosis of the unique result of the rule lies in the "forbidden range," which is described in more detail in the figure description. In this case, the result is invalid.

[0045] The invention relates to the evaluation of a sample of measurement data from a survey of a large number of workpieces and, optionally, also to the preparation of the evaluation. Although this is not within the scope of the attached claims, the invention can also be applied to the evaluation of a sample of measurement data from a survey of other objects. This applies to the method, the arrangement, and the computer program. It also applies to all embodiments and exemplary embodiments described in this description. Therefore, for example, persons or animals can also be considered as objects of measurement, and the evaluation and / or its preparation relates to corresponding measurement data.

[0046] To illustrate distributions for which skewness and kurtosis can be determined as moments of the distribution, a two-dimensional diagram can be drawn in which skewness values ​​can be plotted along a first axis (e.g., x-axis) and kurtosis values ​​along a second axis (e.g., y-axis) that is perpendicular to the first axis. Each point in the plane of value pairs defined in this way corresponds to a group of distributions, where the other moments of this group of distributions, such as the expected value and the variance, are different for the members of the group.Therefore, if a group of distributions is selected that corresponds to a specific pair of values ​​consisting of a specific skewness value and a specific kurtosis value, then a distribution can be determined within this group that best matches a specific sample, particularly with respect to its expected value and variance. In the plane of the two-dimensional diagram, the defined set has a boundary line separating it from pairs of values ​​of distributions that do not belong to the defined set of suitable distributions, with the defined set lying on the side of larger kurtosis values ​​at the boundary line.

[0047] According to the claims, a second statistical distribution is determined for the sample if the test result shows that no statistical distribution exists in the defined set that exhibits the determined moment values ​​of skewness and kurtosis, wherein the second statistical distribution is a statistical distribution contained in the defined set. In a corresponding configuration of the arrangement, it includes a device for determining the second statistical distribution. The second statistical distribution is thus suitable for describing the frequency or probability of the measured data values ​​over the entire predefined interval of values ​​or the interval of values ​​of the sample.

[0048] In particular, a rule can be specified, the observance of which, for any first statistical distribution or for the determined moment values ​​of skewness and kurtosis, yields a second statistical distribution contained within the defined set. It is especially possible to specify a rule by which the second statistical distribution can be determined uniquely; that is, a second statistical distribution with different moment values ​​of skewness and kurtosis follows unambiguously from the determined moment values ​​of skewness and kurtosis. Such a rule could, for example, state that the second statistical distribution has the same skewness value and is the distribution within the set whose kurtosis value differs as little as possible from the determined kurtosis value of the first statistical distribution.Referring to the two-dimensional diagram mentioned above, this means a shift along the kurtosis axis until the range of the defined quantity is reached. It is also possible that rules are / will be specified for different cases, and that in some cases one of these rules is not applicable, and therefore another of the specified rules must be applied.

[0049] A rule-based approach has the advantage of ensuring reproducible results and / or automatically determining the second statistical distribution. However, it is also possible to leave it to the user to select a second statistical distribution within the defined set.

[0050] In particular, a distance measure is defined for each pair of statistical distributions capable of describing the frequency of measured data values ​​as a function of those values. The distance measure describes the difference between the two statistical distributions. In one embodiment of the procedure, the second statistical distribution is determined such that the value of the distance measure between the first statistical distribution, or between a distribution corresponding to the first statistical distribution in the system of statistical distributions, and the second statistical distribution is a minimum of the distance measure between the first statistical distribution, or between a distribution corresponding to the first statistical distribution, and the statistical distributions in the defined set.In simplified terms, the second statistical distribution is the distribution in the defined set that is closest to the first statistical distribution or the corresponding distribution. If the first statistical distribution does not belong to the system of statistical distributions, then preferably the distance to the corresponding distribution in the system is considered. For example, the first statistical distribution has the same skewness and kurtosis values ​​as the corresponding distribution in the system. However, particularly when the first statistical distribution is not part of the system of distributions, it is also possible to directly determine its distance to a distribution belonging to the system and to minimize the measure of this distance in order to determine the second statistical distribution belonging to the system.

[0051] The distance measure can be defined in different ways. One possibility is to define the distance measure in terms of the distance in the aforementioned two-dimensional representation, i.e., in the plane of skewness-kurtosis value pairs. In this case, the distance measure could, for example, be the measure of the Euclidean distance in the plane of value pairs. Another possibility is to define the distance measure in terms of the difference between the two probability density functions being compared with respect to the functional dependence of the frequency or probability of the measured data values ​​on the measured data values. For example, the distance measure could therefore be the integral of a power of the absolute value of the difference in frequencies (or probabilities) over the interval of the union of the domains of the two statistical distributions or over a predetermined interval of values ​​for integration.The exponent of the power can be, for example, one or preferably two.

[0052] The defined distance measure is an embodiment of the aforementioned predefined rule, wherein the rule in this case also stipulates that the first and second statistical distributions have the minimum distance to each other or at most a predefined maximum value of the distance measure.

[0053] Optionally, generating the measurement data of the sample can also be a step in the method according to the invention. According to the claims, evaluating the aforementioned second statistical distribution is a step in the method according to the invention. This evaluation step includes determining whether the process in which the measured workpieces (from whose measurement the measurement data were obtained) were manufactured and / or processed is capable and / or whether a predetermined quality criterion is met or not. For example, the quality criterion may consist of at least one of the moments of the second statistical distribution fulfilling a predetermined condition. For example, with regard to the spread of the statistical distribution (second moment), it may be specified that the spread is not greater than a predetermined value.

[0054] At least the determination of the moment values ​​of skewness and kurtosis from the sample of measurement data according to an initial statistical distribution, and the verification, based on the determined moment values, of whether a statistical distribution exists in the defined set that exhibits the determined moment values ​​of skewness and kurtosis, and the generation of a corresponding test result, can be carried out by executing a computer program on a computer or computer network. Likewise, the preparation of an evaluation of a sample of measurement data from a measurement of a large number of workpieces, the definition of the set of statistical distributions, and / or the determination of a statistical distribution from the set can be carried out by executing a computer program on a computer or computer network.

[0055] In particular, the computer program contains program instructions that, when executed on the computer or computer network, cause the computer or computer network to execute the method in one of the configurations described in this description. Therefore, when executing the method for evaluating a sample of measurement data, it is particularly possible that the torque determination device and the test device of the arrangement are implemented by a computer or a computer network using the computer program. The distribution determination device can also be implemented in this way. In the case of preparing an evaluation of a sample of measurement data, it is particularly possible that the definition device and / or the distribution determination device are implemented by a computer or a computer network using the computer program.

[0056] Exemplary embodiments and background of the invention will now be described with reference to the accompanying drawing. The individual figures in the drawing show: Fig. 1 shows a frequency distribution of the measured values ​​for an example sample of measurement data and a statistical distribution not suitable for describing the frequency distribution over the entire interval of values ​​of the sample. Fig. 2 shows examples of three different statistical distributions that differ from each other in terms of their skewness. Fig. 3 shows examples of two different statistical distributions that differ from each other in terms of their kurtosis. Fig. 4 shows an example of a plane whose points are defined by pairs of values ​​of skewness and kurtosis and which is called the Pearson plane, where the figure shows regions of the plane that are each assigned to the eight types of a Pearson distribution system with eight distribution types, as well as the "forbidden region". Fig. 5 shows a simplified representation with a section of the Fig. 4 The depicted area of ​​the Pearson plane, where two specific points in the Pearson plane are marked by a cross, Fig. 6 shows a statistical distribution whose skewness and kurtosis values ​​correspond to the lower central point marked by a cross in Fig. 5 corresponding to Fig. 7, a statistical distribution whose skewness and kurtosis values ​​correspond to the upper right point marked by a cross in Fig. 5 correspond to the expected value and variance of the in Fig. 6 and Fig. 7 The statistical distributions shown have the same value as those shown in the representation. Fig. 4 and Fig. 5 The underlying principle is shown in Fig. 8, which, for an exemplary embodiment, shows a section of the Pearson plane containing a solution set for the condition that the respective statistical distribution should be able to model the frequency or probability of the measured data values ​​up to the left edge of the value interval. Fig. 9 shows the exemplary embodiment of the Fig. 8 a section of 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 measured data values ​​up to the right edge of the value interval, Fig. 10 for the embodiment of the Fig. 8 and Fig. 9 a section of the Pearson plane that represents the intersection of the solution sets from Fig. 8 and Fig. 9 Fig. 11 contains a section of the Pearson plane showing different regions and their boundaries, with the skewness varying along the horizontal axis and the kurtosis along the vertical axis; Fig. 12 shows a section of the Pearson plane with subregions of a solution set for the left support region, i.e., the support region with respect to the left boundary of the interval of values, for a first parameter value; Fig. 13 shows a section of the Pearson plane with subregions of a solution set for the left support region for a second parameter value; Fig. 14 schematically shows an arrangement of devices for evaluating a sample of measurement data from a survey of a large number of workpieces, which can also be interpreted as a flowchart; Fig. 15 shows a section of the Pearson plane. Fig. 10 , where the hatched area in Fig. 15 the cross-hatched area in Fig. 10 corresponds and where two points in the plane are marked by crosses, representing a first statistical distribution outside the solution set and a second statistical distribution at the boundary of the solution set, and Fig. 16 for the example sample and the frequency distribution of the measured values ​​from Fig. 1 a statistical distribution suitable for describing the frequency distribution over the entire interval of values ​​of the sample.

[0057] In Fig. 1 For each small sub-interval of the value range of measurement data from a sample of measurement data, the frequencies of the measurement data values ​​are represented as bars. Accordingly, the measurement data values ​​are plotted on the horizontal axis and the frequencies or probabilities on the vertical axis. Furthermore, in Fig. 1 The graph shows the function curve of a first statistical distribution that models the frequency distribution. It is evident that the first statistical distribution only models frequencies within a left-bounded interval of values, where the interval begins approximately at the measurement value -1.9. However, the sample also contains measurement values ​​less than -1.9. Therefore, the first statistical distribution is not suitable for modeling frequencies across the entire interval of values ​​in the sample.

[0058] For example, for the sample that the Fig. 1 The underlying principle is that the interval of values ​​of the actually occurring measurement data, or a larger interval than that within which a statistical distribution should be able to model the frequency or probability of the measurement data values, is specified. This specified interval of values ​​is also referred to as the range in the following.

[0059] In contrast, there is the interval of values ​​within which a statistical distribution can model the frequency or probability of the measured data values. This interval of values ​​is also referred to as the support of the statistical distribution. With regard to the in Fig. 1 In the case described, the range is therefore not included in the support. However, this is the goal.

[0060] In particular, the statistical distributions of the Pearson distribution system can be assigned moments as characteristic parameters, as mentioned above. Four of these moments are the expected value. v 1, the variance µ 2. The skewness µ̃ 3 and the kurtosis µ̃ 4. With respect to a random variable X, the nth moment is given by: ν n X = E X n

[0061] By inserting n E = 1 yields the expected value. E denotes the expected value operator. The nth central moment µ n is given by: μ n X = E X − ν 1 X n , n ≥ 1

[0062] This can be derived by inserting n = 2 the corresponding equation for the variance µ 2 will be obtained. The nth central, standardized moment µ̃ n is given by: μ ˜ n = μ n X μ 2 X n / 2 , n ≥ 2

[0063] From this, by inserting n = 3 and 4 the corresponding equations for the skewness µ̃ 3 and the kurtosis µ̃ 4. As will be explained elsewhere, the central, standardized moments of skewness and kurtosis are independent of the moments of expectation and variance. Therefore, the suitability of a statistical distribution for modeling a sample across its entire interval of values, and the determination of a suitable statistical distribution, can each be performed with respect to skewness and kurtosis for any expected value and any variance value, and the results of the test or determination are valid. In the case of the standardization described elsewhere, it is important to ensure that the solution set is determined with respect to the standardized measurements or the standardized statistical distribution.

[0064] Fig. 2 The diagram shows three different statistical distributions that differ in their skewness. The probability density function represented by a solid line has a skewness of zero, as it is symmetrical. The probability density function represented by a dashed line has a negative skewness value, and the probability density function represented by a dotted line has a positive skewness value. Fig. 3 The figure shows two different statistical distributions that differ in their kurtosis. Kurtosis only takes on positive values. The kurtosis of the probability density function represented by a dashed line is greater than the kurtosis of the probability density function represented by a solid line.

[0065] It is common practice to fit the first two moments of a distribution: the expected value and the variance. However, in the following discussion, we will initially focus only on the skewness and kurtosis, which can uniquely identify any distribution or group of distributions.

[0066] Specifically, the following example considers a Pearson distribution system with eight types of distributions. Fig. 4 The diagram shows the so-called skew-kurtosis plane for this system, where, in one embodiment, skewness values ​​are plotted along the horizontal axis and kurtosis values ​​along the vertical axis. The skew-kurtosis plane is subsequently referred to as the Pearson plane. Every point on the Pearson plane outside the forbidden region, defined by a coordinate pair consisting of a skewness coordinate and a kurtosis coordinate, is uniquely assigned to a distribution or a group of distributions. The members of the group are further distinguished by the values ​​of additional moments, such as the expected value and the variance. There is a continuous relationship between the position in the Pearson plane and the shape of the corresponding distribution or group of distributions.This applies not only directly to skewness and kurtosis but also to other properties, especially the interval of values ​​over which the group of distributions is suitable for statistically modeling sample or measurement data values ​​and predicting their probability. Thus, if two points in the Pearson plane are close to each other, the corresponding two distributions will also be similar, assuming the same expected value and variance for both distributions.

[0067] The example of the Pearson plane, which in Fig. 4 The representation shown was created on the basis of a standardization where the expected value has the value zero and the variance has the value one.

[0068] In Fig. 4 A roughly parabolic dashed line is visible at the bottom of the depicted area of ​​the plane. For pairs of skewness and kurtosis values ​​below this line, there are no corresponding distributions. The area of ​​the plane below this line can therefore be described as an inadmissible or forbidden region. At a skewness value of zero, a dotted line extends from this line, parallel to the kurtosis axis, to a roughly parabolic solid line. The pairs of values ​​on the dotted line belong to distributions of Pearson distribution type 2. Extending from the dotted line, parallel to the kurtosis axis, is a dash-dotted line, whose pairs of values ​​belong to distributions of Pearson distribution type 7. The pairs of values ​​on the aforementioned solid line belong to Pearson distribution type 3.Above the solid line is a parabolic dashed line, on which the points of the value pairs of the distributions of Pearson distribution type 5 lie. The value pairs of the points between the forbidden area and the solid line belong to distributions of Pearson distribution type 1. The value pairs of the points between the solid line and the upper dashed line belong to distributions of Pearson distribution type 6. The value pairs of the points above the upper dashed line belong to distributions of Pearson distribution type 4. At the common intersection or endpoint of all the aforementioned lines, with the exception of the lower dashed line, lies the value pair of the single point of the distribution of Pearson distribution type 0, also known as the normal distribution.

[0069] In the Fig. 5 Within the depicted area of ​​the Pearson plane, two crosses are drawn to highlight two points in the plane. The corresponding distributions are shown in Fig. 6 and Fig. 7 shown. These two distributions belong to Pearson distribution type 1. The skewness of the in Fig. 6 The distribution shown is negative and closer to zero than the positive skewness of the one in Fig. 7 The distribution shown is therefore more symmetrical than the distribution shown in Fig. 7 is shown. The kurtosis of the in Fig. 7 The distribution shown has a positive value with a larger amount than the positive value of the one in Fig. 6 distribution shown.

[0070] The range of a sample represents an interval of the measured data values. It is now required that the distribution fitted to the sample has a support that is at least equal to the range of the interval. Therefore, if the range is given by Range = r min r max and the support of the distribution by Support = v min v max , then the following shall apply: Range ⊆ Support , That is, the entire range should be included in the support. This requirement is equivalent to: v min ≤ r min , v max ≥ r max .

[0071] Equation (1) shows that a distribution is sought whose lower support limit v min less than or equal to the minimum of the range r min Each distribution can be visualized as a skewness / kurtosis point in the Pearson plane. Thus, the set of all distributions that satisfy equation (1) can be represented as a set in the Pearson plane. This results in a solution set, which, for one example, is shaded in Fig. 8 is shown. It is worth noting that the solution set below is bounded by a convex boundary line, i.e., lies above the boundary line. Similarly, starting from equation (2), a solution set can be obtained which, for the embodiment shown in Fig. 9 is shown hatched. Fig. 8 and Fig. 9 is as a guide, as in Fig. 4 The dashed line represents the boundary of the forbidden area. The total solution set for this example is given in Fig. 10 shown cross-hatched. Regarding the fact that the partial solution sets are in Fig. 8 and Fig. 9 These figures are to be understood schematically, as they do not extend into the forbidden area. In reality, the partial solution sets do not extend into the forbidden area.

[0072] The intersection point of the limit curves of the partial solution sets that are in Fig. 10 The pair of values ​​for skewness and kurtosis, represented by a dotted and a dashed line, corresponds to the statistical distribution within the system of distributions for which the support equals the range. In this example, this intersection point lies at a skewness value of zero, but in other cases, it can also assume negative or positive skewness values.

[0073] M L denotes the solution set starting from equation (1) and M R denotes the solution set starting from equation (2), which can be defined as follows: M L = s k : v min s k ≤ r min and M R = s k : r max ≤ v max s k .

[0074] This refers to (s, k) a pair of values ​​for skewness s and kurtosis k.

[0075] The following is an example of how to determine the quantities. M L and M R described.

[0076] The Pearson distribution system is based on the following ordinary differential equation for a probability density function. A normalized solution f ( x ) the ordinary differential equation f ′ x = a x b x f x with the polynomials a x = x + a 0 , a 0 ∈ R b x = b 2 + b 1 x + b 0 , b 0 , b 1 , b 2 ∈ R , where R denotes the set of real numbers, it will henceforth be called the Pearson probability density function.

[0077] The coefficients of the polynomials a ( x ) and b ( x ), as defined in equations (4) and (5), parameterize the still unknown probability density function f(x). In particular, the shape and domain of the probability density function depend strongly on the number and location of the zeros of the denominator polynomial. b(x) away.

[0078] To relate the probability density function to quantities that have statistical significance, a relationship between the coefficients is established. a = ( a 0, 1) T< and b = ( b 0 , b 1 , b 2) T< on the one hand, and a specific set of moments of the probability density function. The standardized moments µ̃ n The moments shown above are invariant with respect to scaling and shifting transformations of the random variable X for which they are or have been calculated. Therefore, the standardized moments can µ̃ n the random variable X as independent of its raw and central moments v 1 ( X ) and µ 2 ( X) (see above). Based on this, a definition of a parameterized tuple M of moments for a Pearson probability density function and its associated random variable X can be given. The definition of the tuple M has already been given above. The aforementioned notations were used in this context. v 1 ( X ), µ 2 ( X ), µ̃ 3 and µ̃ The expected value, variance, skewness, and kurtosis are introduced in four parts. Reference is made to a subset of the elements of the tuple M by using the corresponding indices, e.g. M 1:2 ( X ) = ( v 1 ( X ), µ 2 ( X )).

[0079] The raw moments are described below. ν i X i = 1 4 and the tuple M in relation to the parameters a and bThe coupling between the moments and the coefficients can be written in the form of a matrix equation as: 0 1 2 ν 1 1 1 2 ν 1 3 ν 2 ν 1 2 ν 1 3 ν 2 4 ν 3 ν 2 3 ν 2 4 ν 3 5 ν 4 ν 3 b 0 b 1 b 2 a 0 = − ν 1 ν 2 ν 3 ν 4 .

[0080] The standardization mentioned above is applied below to simplify the solution. However, a corresponding solution can also be derived without such standardization. The solution equations will then be correspondingly more complex. For v 1 = 0 and v 2 = 1 is the solution to equation (6): b = 1 c 4 ζ 2 − 3 ζ 1 2 3 + ζ 2 ζ 1 2 ζ 2 − 3 ζ 1 2 − 6 , a = − b 1 1 , where µ̃ 3 ( X ) = ζ 1 ( X ) and µ̃ 4 ( X ) = ζ 2 ( X ) are used as terms for the moments of skewness and kurtosis, and where c = 2 9 + 6 ζ 1 2 − 5 ζ 2 Taking into account equation (4) and equation (6), the parameterization of the Pearson probability density function can now be carried out as follows: If a parameterization moment tuple M* with a scaling and shifting component M* 1:2 ( X ) = { v* 1 , µ* 2} and a form component M* 3:4 ( X Given ) = {ζ* 1 , ζ* 2}, the corresponding Pearson probability density function can be constructed in two steps. In a first step, the coefficients are a * and b* are calculated using equation (7) and the Pearson differential equation problem is solved with these parameters to obtain a standardized Pearson probability density function. f s ( x ) and obtain the corresponding random variable X. In a second step, a transformation is performed. τ x = x − ν * 1 / μ * 2 set up and the required Pearson random variable X as X= τ -1< ( X s ) and its probability density function as f(x) = τ'( x ) f s ( τ ( x )) is defined. The superscript -1 denotes the inverse function, and the comma denotes the first derivative of the function. Now, M(X) = M*. This shows that each of the tuples M* 1:2 ( X ) and M* 3:4 ( X ) , This means that the scaling and shifting component on the one hand and the shape component on the other hand can be handled independently of each other in the procedure for fitting the Pearson probability density function to the corresponding moments of the distribution.

[0081] The Pearson plane is defined as the plane E of real numbers, where the first coordinate is the skewness and the second coordinate is the kurtosis. The Pearson plane E is then divided into three regions, characterized by their behavior with respect to the zeros. The distinguishing criterion κ (ζ) for the three regions and the corresponding boundary curves is defined as: κ ζ : = ζ 1 2 ζ 2 + 3 2 4 2 ζ 2 − 3 ζ 1 2 − 6 4 ζ 2 − 3 ζ 1 2 .

[0082] The main areas of Pearson Level E can be specified as follows. x 1 and x 2 the zeros of the polynomial b(x; ζ) are, whose coefficients b are elements of the three-dimensional space of real numbers and are calculated using equation (7), then the areas can be R 1 , R 4 and R 6 The Pearson level E can be specified as follows: R 1 : = ζ ∈ E : x 1 , x 2 ∈ ℝ ∧ x 1 x 2 < 0 = ζ ∈ E : κ ζ < 0 , R 4 : = ζ ∈ E : x 1 , x 2 ∈ ℂ = ζ ∈ E : κ ζ ∈ 0 1 , R 6 : = ζ ∈ E : x 1 , x 2 ∈ ℝ ∧ x 1 x 2 > 0 = ζ ∈ E : κ ζ > 1 .

[0083] The border curves C i with i = 2, 3, 5, 7 of the aforementioned ranges, hereinafter referred to as Pearson curves, can be specified as follows: C 2 : = ζ ∈ E : ζ 1 = 0 ∧ ζ 2 ∈ 1 3 , C 3 : = ζ ∈ E : 2 ζ 2 − 3 ζ 1 2 − 6 = 0 = ζ ∈ E : κ ζ ∈ − ∞ , ∞ ; C 5 : = ζ ∈ E : κ ζ = 1 , C 7 : = ζ ∈ E : ζ 1 = 0 ∧ ζ 2 > 3 .

[0084] Furthermore, the aforementioned prohibited area R ⊗ : = ζ ∈ E : ζ 2 < ζ 1 2 + 1 introduced the Pearson Plain, whose boundary line C ⊗ : = ζ ∈ E : ζ 2 = ζ 1 2 + 1 Given the forbidden region, the region contains, on the one hand, pairs of points ζ = (ζ₁, ζ₂) that can never be solutions to the calculations of skewness and kurtosis. On the other hand, the solutions of the ordinary Pearson differential equations for points on the forbidden curve are not integrable and therefore do not represent a probability density distribution. Therefore, the regions containing all points in the Pearson plane for which a solution can be found can be defined as follows: R : = E \ C ⊗ ∪ R ⊗ .

[0085] In words, the totality of these areas can be specified as the set of points corresponding to the entire plane less the set of points in the forbidden area and less the set of points on the boundary line of the forbidden area. Fig. 11 For an exemplary embodiment, the above-mentioned three areas are shown. R 1 , R 4 , R 6 and two of their aforementioned boundary curves C 3 and C 5 Furthermore, in the lower part of the Fig. 11 similar to in Fig. 4 The prohibited area and its boundary line are shown. The one in Fig. 11 depicted point P 0 represents the position of the pair of values ​​for skewness and kurtosis of the normal distribution.

[0086] In the following, only standardized Pearson distributions are considered. The reasons for the sufficiency of such a consideration have already been explained above. First, an expression for the support interval of Pearson distributions is derived as a function of the skewness ζ₁ and the kurtosis ζ₂. Then, curves or boundary lines for the solution sets mentioned above are derived. Finally, sets of solution sets for the right and left support boundaries are derived.

[0087] The zeros x 1 and x 2 of the polynomial defined in equation (5) b(x) can be specified as follows: x 1 = − b 1 − b 1 2 − 4 b 0 b 2 2 b 2 and x 2 = − b 1 + b 1 2 − 4 b 0 b 2 2 b 2 .

[0088] If the coefficients b i When expressed as functions of skewness and kurtosis, the following equations result for the zeros. x i = x i ( ζ ): x 1 ζ = u ζ + sgn c υ ζ w ζ and x 2 ζ = u ζ − sgn c υ ζ w ζ .

[0089] With c = 2 9 + 6 ζ 1 2 − 5 ζ 2 The following helper functions can be specified: u ζ : = ζ 1 ζ 2 + 3 , υ ζ : = − 36 ζ 1 4 + ζ 1 2 ζ 2 2 + 78 ζ 2 − 63 − 32 ζ 2 − 3 ζ 2 , w ζ : = 6 ζ 1 2 − 4 ζ 2 + 12 .

[0090] The zeros x 1 and x The two equations are complementary with respect to sgn(c), i.e., with respect to sign. This leads to the following alternative equations: x 1 ζ : = u ζ − v ζ w ζ and x 2 ζ : = u ζ + v ζ w ζ .

[0091] Referring to equations (18) and (19), it should be noted that the argument of the square root in the numerator of the fraction is only undefined for negative expressions. Therefore, the functions according to equations (18) and (19) are not defined for the forbidden region and its boundary.

[0092] The support interval I = ( ξ L , ξ R ) with respect to a standardized Pearson distribution with a moment tuple M* 3:4 = ζ, where the left boundary of the support interval is given by ξ L and the right boundary of the support interval by ξ R The term is defined as follows: ξ L = x L ζ : = x 1 ζ , ζ ∈ R 1 ∪ C 2 ∨ ζ 1 > 0 ∧ ζ ∈ R 6 ∪ C 5 , − 2 ζ 1 , ζ 1 > 0 ∧ ζ ∈ C 3 , − ∞ else and ξ R = x R ζ : = x 2 ζ , ζ ∈ R 1 ∪ C 2 ∨ ζ 1 < 0 ∧ ζ ∈ R 6 ∪ C 5 , − 2 ζ 1 , ζ 1 < 0 ∧ ζ ∈ C 3 , ∞ , else .

[0093] Here, "else" means otherwise. Starting from equations (20) and (21), limit curves can now be... c L ( t ) and c R ( t ) the solution domains are given the following implicit equations: x L c L t = ξ L for ξ L < 0 , x R c R t = ξ R for ξ R > 0 where "for" means for. Before these curves are calculated, preliminary considerations are made. The naive solution of the equations x (1) (ζ) = ξ L and x (2) (ζ) = ξ R for the zeros leads to the curve s ( t ; ξ), which is defined by: s t ξ : = t 6 ξ 2 − 3 ξt + 3 t 2 + 3 ξ 2 t 2 4 + 2 ξ 2 + ξt für die x (1) ( s ( t ; ξ L )) = ξ L and x (2) ( s ( t; ξ R )) = ξ R The curve s(t; ξ) has the following properties: For ξ ≠ 0, a singularity exists at ζ 1 ∞ ξ : = − 4 + 2 ξ 2 / ξ ;

[0094] For ξ ≠ 0, there exists an intersection point at the following point of the forbidden curve, namely the boundary line of the forbidden region: ζ 1 ⊗ ξ : = ξ 2 − 1 / ξ

[0095] For ξ > 2 There exists an intersection point with the aforementioned curve or boundary line. C 5 , see equation (9): ζ 1 5 ξ : = − 4 ξ / ξ 2 − 1

[0096] For ξ ≠ 0, there exists a unique global minimum at: ζ 1 min : = ζ 1 ∞ − 2 ξ 2 + 4 / ξ 2 + 5 = 2 ξ + 2 / ξ − ξ 2 + 4 / ξ 2 + 5

[0097] The equations x (1) (ζ) = ξ L and x (2) (ζ) = ξ R are through the branch of the curve s ( t; ξ ) solved, which for x (1) (ζ) to the right and for x (2) (ζ) to the left of ζ 1 ∞ ξ lies.

[0098] Although s ( t; ξ) not the solution of equations (22), but the following consideration shows that only the domain of the curve s ( t; ξ) must be adjusted to obtain the solution. The implicit equations (22) for the support boundaries C L ( t; ξ L ) and c R ( t; ξ R ) have the solutions c L t ξ L : = s t ξ L with D c L : = ζ 1 ⊗ ξ L , ζ 1 5 ξ L , if ξ L ∈ − ∞ , − 2 , ζ 1 ⊗ ξ L , ζ 1 ∞ ξ L , if ξ L ∈ − 2 , 0 and c R t ξ R : = s t ξ R with D c R : = ζ 1 5 ξ R , ζ 1 ⊗ ξ R , if ξ R ∈ 2 ∞ . ζ 1 ∞ ξ R , ζ 1 ⊗ ξ R , if ξ R ∈ 0 2 .

[0099] Here, "with" means "with" and "if" means "if". It follows that there is a transformation between the right and left support boundaries. The following relationship holds: c R t ; ξ R = c L − t ; − ξ R

[0100] With regard to equations (22), it follows that: x L c L t ; ξ L = ξ L < 0 und x R c R t ; ξ R = ξ R > 0

[0101] For the region of the Pearson plane outside the forbidden region, the solution sets mentioned above can now be used. M L of equation (1) and M R of equation (2) equivalent sets R L (ξ) for ξ < 0 with respect to the left or lower support limit v min and R R (ξ) for ξ > 0 with respect to the right or upper support limit v max as follows: R L ξ = ζ ∈ R : x L ζ ≤ ξ , ξ < 0 , R R ξ = ζ ∈ R : x R ζ ≥ ξ , ξ > 0 .

[0102] It follows that for a Pearson probability density function with a support region ( a, b ) the relationship a≤ ξ holds if ξ < 0 and if the point in the Pearson plane with coordinates ζ lies in the solution set or support area R. L (ξ) lies at the left support boundary. Similarly, b ≥ ξ if ξ > 0 and if the point in the Pearson plane with coordinates ζ lies in the solution set or support region R. R (ξ) lies at the right support boundary.

[0103] The left support area R L (ξ) for ξ < 0 can be expressed by the set S (ξ) can be expressed by points in the Pearson plane, which represent the following individual sets S i (ξ) unites, where i = 1, 2, 3: S 1 ξ : = ζ ∈ R : ζ 1 ≤ D ξ , S 2 ξ : = ζ ∈ R : ζ 1 ∈ D ξ ∧ ζ 2 ≥ c L ⋅ ; ξ , S 3 ξ : = ζ ∈ R : ζ 1 ≥ D ξ ∧ υ ξ < 0 ,

[0104] This refers to D ξ the domain of definition of the left support boundary curve c L , itself v (ξ) to the middle of equations (17) and denotes R the region in the Pearson plane excluding the forbidden region and its boundary line. Fig. 12 shows the solution set R L (ξ) with respect to the left or lower support limit v min for ξ = -2.5. Fig. 13 This solution set R L (ξ) for ξ = -0.5. While Fig. 12 all three quantities S i shows is the amount S 3 in the case of the Fig. 13 empty. This applies to the area − 2 ≤ ξ < 0 , since there is no intersection point for the left boundary curve c L and the aforementioned curve C 5 exists. In Fig. 12 und Fig. 13 is in each case the limit curve of the set S 2 is highlighted by representing the boundary curve with a broad, solid line. The dashed line represents the boundary curve of the set. S 1 is the boundary line of the prohibited area.

[0105] Based on the connection between the left support area R L (ξ) and ξ, the following can be stated: For two negative values ​​b ≤ a < 0 applies: R L b ⊆ R L a , ζ ∈ R L a \ R L b ⇒ x L ζ , 0 ⊂ b 0 .

[0106] If values ​​ξ L < 0 < ξ R Given that solution sets are defined for the right and left support boundaries, the following applies to the intersection: R L ∩ R ξ L ξ R : = R L ξ L ∩ R R ξ R = ζ ∈ R : ξ L ξ R ⊆ supp f ⋅ ; ζ .

[0107] The preceding description contains an exemplary embodiment for defining the set of all statistical distributions capable of describing the frequency of measured data values ​​within the entire interval (the range). An exemplary embodiment of the invention is now described with respect to verifying whether a statistical distribution exists within the defined set that exhibits the moment values ​​of skewness and kurtosis determined for a sample.

[0108] Fig. 14 The diagram schematically shows an arrangement of devices for evaluating a sample of measurement data from the measurement of a large number of workpieces. The representation in the Fig. 14 It can also be interpreted as a flowchart to explain one embodiment of a method for evaluating a sample of measurement data.

[0109] For example, a coordinate measuring machine 1 measures the multitude of workpieces and transmits the measurement data, optionally after preprocessing, to a measurement data storage device 3. A definition device 5 is designed to define, for a value interval of the measurement data, which is either a predefined value interval or a value interval of the measurement data actually occurring in the sample, from the system of statistical distributions a set of all those statistical distributions that are capable of describing a frequency of measurement data values ​​in the entire value interval. Fig. 14 An input of the definition device 5 is connected to an output of the measurement data storage device 3. Therefore, the definition device 5 can determine the value interval, particularly from the available measurement data. Alternatively or additionally, it can receive supplementary information stored in the measurement data storage device 3 regarding a value interval extending beyond the value interval of the measurement data. In practice, however, the information about the specified value interval can also be provided to the definition device 5 in other ways, so a connection between the measurement data storage device 3 and the definition device 5 is not strictly necessary.

[0110] An output of the definition device 5 is connected to an input of a moment determination device 7, which is configured to determine a moment value for skewness and kurtosis from the sample of measurement data according to a first statistical distribution. During operation or during the execution of the procedure, the moment determination device 7 determines values ​​for skewness and kurtosis for the sample and transmits the values ​​to a test device 9, which is configured to check, based on the determined moment values, whether a statistical distribution exists in the defined quantity that exhibits the determined moment values ​​for skewness and kurtosis, and to generate a corresponding test result.

[0111] The test result determines whether such a statistical distribution exists in the defined set or not. If so, then, for example, a signal issued by test unit 9 can confirm that the first statistical distribution is suitable for the purpose of statistically modeling the sample over the entire specified interval of values. If not, then test unit 9 outputs the test result or a signal to a distribution determination unit 11, which determines a second statistical distribution suitable for the purpose of statistically modeling the sample over the entire specified interval of values. Distribution determination unit 11 can, for example, identify the pair of values ​​from the Pearson plane that corresponds to a suitable statistical distribution.Using the information about the pair of values, the appropriate statistical distribution can then be generated as a second statistical distribution.

[0112] The definition device 5 and the distribution determination device 11 can also be used without the others in Fig. 14 The depicted devices form an arrangement. Furthermore, the definition device 5 and the distribution determination device 11 can serve to prepare the evaluation of a sample of measurement data. In both cases, the definition device 5 is designed to define, for a given interval of measurement data, either a predefined interval or an interval of a sample of measurement data to be evaluated, a set of all statistical distributions from the system of statistical distributions that are capable of describing a frequency of measurement data values ​​within the entire interval. The distribution determination device 11 is then designed to determine a statistical distribution from the defined set.

[0113] If Fig. 14 Interpreted as a flowchart, the process begins with the generation of measurement data in step 1, which is then stored in step 3. In step 5, the set of all statistical distributions capable of describing the frequency of measurement values ​​within the entire interval is defined. In step 7, a moment value for skewness and kurtosis is determined from the sample of measurement data according to a first statistical distribution. In step 9, the determined moment values ​​are used to check whether a statistical distribution exists within the defined set that exhibits the determined moment values ​​for skewness and kurtosis, and a corresponding test result is generated. The sequence of the process steps is indicated by the arrows in the diagram. Fig. 14 .

[0114] Optional facilities or procedural steps belonging to specific configurations are described in Fig. 14 Designated with reference numbers 1, 3 and 11. Procedure step 5 does not necessarily have to be carried out every time a first statistical distribution is to be tested for its suitability for statistically modeling a sample over the entire given interval of values.

[0115] In particular, the distribution determination device 11 can, based on a distance measure for the distance between two distributions in the Pearson plane, identify as the second statistical distribution the one whose distance to the first statistical distribution is minimal. Fig. 15 The diagram shows an example of two distributions in the Pearson plane that have a minimal distance, in this case the minimum Euclidean distance in the Pearson plane. The cross located further to the right and below represents the first statistical distribution in a case where it is not suitable for statistically modeling the sample over the entire given interval of values. This lack of suitability is evident because the cross lies outside the shaded area of ​​the solution set of suitable statistical distributions. The cross represents the distribution by marking a point in the Pearson plane corresponding to the distribution, which in turn corresponds to the pair of values ​​for the skewness and kurtosis of the distribution.According to the procedure given here, the appropriate second statistical distribution is determined from the first statistical distribution by identifying the point in the solution set of suitable distributions lying in the Pearson plane that has the smallest distance to the point of the first statistical distribution. In all cases, this point of smallest distance lies on the boundary of the solution set.

[0116] Fig. 16 now shows analogous to the representation in Fig. 1 A statistical distribution, more precisely a probability density function, determined in the manner described above, which therefore lies in the solution set of suitable statistical distributions and is thus suitable to statistically model the sample measurement values ​​over the entire given interval of values. Fig. 16 This can be seen from the fact that, in contrast to Fig. 1The distribution density function represented by the unbroken line also assumes positive frequency values ​​or probability values ​​in the range of measurement data values ​​smaller than -1.9.

[0117] In the case of preparing the analysis of a sample of measurement data, for example, only the procedural steps of defining the set of all those statistical distributions capable of describing a frequency of measurement values ​​within the entire interval of values, and determining the statistical distribution corresponding to this set, can be carried out. This does not, of course, preclude the possibility of analyzing a sample of measurement data using this statistical distribution after its determination.

Claims

1. Method executed by a computer program on a computer or computer network, for evaluating a sample of measurement data from measuring a multiplicity of workpieces by one or more coordinate measuring machines, wherein - a set of all those statistical distributions that are able to describe a frequency of measurement data values in the entire value interval of the sample is defined from a system of Pearson-type statistical distributions for describing a frequency of measurement data values as a function of the measurement data values, wherein the distributions of the system are in each case describable by a moment value of two moments, specifically a skewness and a kurtosis, of the respective statistical distribution, said system being in existence or set up upon execution of the method, wherein evaluation of the sample identifies distributions of the system which are unable to describe the frequency of measurement data values in the entire value interval of the sample, - a respective moment value of the skewness and the kurtosis is ascertained from the sample of measurement data corresponding to a first statistical distribution, - 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, specifically that either no such statistical distribution exists in the defined set or such a statistical distribution exists in said defined set, - a statistical distribution which is such an existing statistical distribution is obtained or, if no such statistical distribution exists in the defined set, a second statistical distribution is ascertained and obtained for the sample, and this second statistical distribution is a statistical distribution contained in the defined set, and - on the basis of the statistical distribution obtained, it is determined whether a process in which the measured workpieces were produced and / or processed can be controlled and / or whether a specified quality criterion for the process with regard to the frequency distribution of the measurement data values of the sample is fulfilled or not.

2. Method according to Claim 1, 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, said measure of distance describing a distance between the two statistical distributions, and wherein 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.

3. Method according to Claim 1 or 2, wherein a boundary of the set 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 of the sample.

4. Method according to Claim 1, 2 or 3, 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 of the sample.

5. Method according to Claim 3 and according to Claim 4, wherein 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 of the sample.

6. Method according to any of Claims 1 to 5, wherein the second statistical distribution is ascertained by following a specified rule for the ascertained moment values of the skewness and the kurtosis and wherein the second statistical distribution follows uniquely from the ascertained moment values of the skewness and the kurtosis and has different moment values of the skewness and the kurtosis.

7. Method according to any of Claims 1 to 6, wherein the measurement data in each case are coordinates of at least some of a workpiece surface and / or of a workpiece interior or are measurement data derived from at least one coordinate of the respective workpiece.

8. Arrangement for evaluating a sample of measurement data from measuring a multiplicity of workpieces by one or more coordinate measuring machines, wherein the arrangement comprises: - a definition device (5) 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 of the sample from a system of Pearson-type statistical distributions for describing a frequency of measurement data values as a function of the measurement data values, wherein the distributions of the system are in each case describable by a moment value of two moments, specifically a skewness and a kurtosis, of the respective statistical distribution, wherein evaluation of the sample identifies distributions of the system which are unable to describe the frequency of measurement data values in the entire value interval of the sample, - a moment ascertainment device (7) 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, - a checking device (9) 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, specifically that either no such statistical distribution exists or such a statistical distribution exists, - a distribution ascertainment device (11) which is configured to ascertain a second statistical distribution for the sample if no such statistical distribution exists, wherein the second statistical distribution is a statistical distribution contained in the defined set, wherein the arrangement is configured to obtain a statistical distribution which is such an existing statistical distribution in the defined set or, if no such statistical distribution exists in the defined set, which is the second statistical distribution contained in the defined set, and to determine, on the basis of the statistical distribution obtained, whether a process in which the measured workpieces were produced and / or processed can be controlled and / or whether a specified quality criterion for the process with regard to the frequency distribution of the measurement data values of the sample is fulfilled or not.

9. Arrangement according to Claim 8, wherein the distribution ascertainment device (11) is configured to ascertain the second statistical distribution from the defined set in such a way that, on the basis of 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, the measure of distance describing 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.

10. Arrangement according to Claim 8 or 9, wherein the definition device (5) 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 of the sample.

11. Arrangement according to Claim 8, 9 or 10, wherein the definition device (5) 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 of the sample.

12. Arrangement according to Claim 10 and according to Claim 11, wherein the definition device (5) 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 of the sample.

13. Arrangement according to any of Claims 8 to 12, wherein the arrangement is configured to ascertain the second statistical distribution by following a specified rule for the ascertained moment values of the skewness and the kurtosis, wherein the second statistical distribution follows uniquely from the ascertained moment values of the skewness and the kurtosis and has different moment values of the skewness and the kurtosis.

14. Arrangement according to any of Claims 8 to 13, wherein the measurement data in each case are coordinates of at least some of a workpiece surface and / or of a workpiece interior or are measurement data derived from at least one coordinate of the respective workpiece.

15. Computer program for carrying out the method according to any of Claims 1 to 7.