Method and system for evaluating a measurement data sample or preparing an evaluation thereof

Abstract

The invention relates to a method and an arrangement for evaluating a measurement data sample generated by measuring a plurality of workpieces, wherein a statistical distribution system exists or is established, which is capable of describing the frequency of measurement data values ​​as a function of the measurement data values, examples of which are distinguishable from each other according to a corresponding moment value of two moments of the corresponding statistical distribution, specifically the skewness and the kurtosis, wherein a set of all statistical distributions capable of describing the frequency of measurement data values ​​in the entire value interval is defined from the statistical distribution system, the value interval being a specified value interval or a value interval of the measurement data actually occurring in the sample, the corresponding moment values ​​of the skewness and the kurtosis are determined based on the measurement data sample corresponding to the first statistical distribution, the determined moment values ​​are used to check whether the defined set contains a statistical distribution with a moment value of the determined skewness and kurtosis, and to generate a corresponding test result. The invention also relates to a method and an arrangement for preparing an evaluation of a measurement data sample.

Description

Technical Field

[0001] The invention relates to a method and an arrangement for evaluating measurement data samples generated by measuring a plurality of workpieces or for preparing an evaluation thereof. Background Art

[0002] The concept of statistical process control is known from the field of quality control of processes for producing and / or machining workpieces and workpiece arrangements. By means of statistical modeling of a process, a statistical distribution can be obtained, based on which, in particular, statements can be made about the frequency of certain measured values ​​generated from the process. For example, the controllability of a process can be a result of the statistical modeling. Due to the statistical modeling, it can also be determined or can alternatively be determined that the quality of the process meets the specifications if the variance of the measured values ​​is small.

[0003] Such statistical models are usually created by using information based on process knowledge and on the evaluation of the acquired measurement data. For example, when measuring the radius of a workpiece, the known process can be described by a normal distribution. In this case, the measured values ​​generated by measuring a plurality of workpieces of the same type are randomly scattered around the expected value. The normal distribution allows, in particular, parameters such as the variance to be determined therefrom.

[0004] It is not possible to clearly describe the result of every process using the normal distribution. Instead, there are other statistical distributions which describe the frequency of the measured values ​​of a measured variable as a function of the measured value. Unlike the case of the normal distribution, these distributions are particularly suitable only for the statistical description of measured values ​​within a value interval that is restricted on one side or on both sides. Such distributions do not assign frequencies to values ​​outside the value interval and / or the frequencies and therefore probabilities of values ​​outside the value interval are set to the value zero.

[0005] For example, the Pearson distribution system is a system with multiple known and frequently used statistical distributions. If the measured value can now use the measured data sample, one of the statistical distributions of the distribution system can be selected and used for statistical modeling. This is based on the following basic concept: the distribution is created only based on the information of the available data. In order to create the statistical distribution, different programs can be selected. In particular, the maximum likelihood principle can be followed, that is, the probability of maximizing the distribution of the statistics of the available data that is correctly described. According to different programs, the so-called moments of the statistical distribution can be determined according to the measured data, and then the distribution that fits these moments can be selected. In the scope of statistical modeling using the distribution based on the measured data or the data derived from the measured data, the situation that may occur is that the distribution describes the frequency of the measured values ​​or the frequency of the values ​​derived from the measured values ​​in most of the occurring or possible value intervals, but the values ​​are also outside the value interval that the distribution can describe or may also be outside the value interval. In fact, this is disadvantageous because the statistical distribution is not optimally modeled, and the user may encounter an unexpected situation, that is, the occurring measured values ​​do not fall into the value interval of the distribution and therefore do not contribute to the distribution or cannot be predicted by the distribution. In this case, there is an obvious modeling error. Summary of the invention

[0006] The object of the invention is to specify a method or an arrangement for evaluating measurement data samples generated by measuring a plurality of workpieces or for preparing an evaluation of measurement data samples, with which method and arrangement a high-quality statistical modeling of the measurement data is possible.

[0007] The solution is based on the basic idea of ​​testing whether a first statistical distribution or a first statistical distribution family is suitable for describing the frequency of measured data values ​​as a function of the measured data values ​​within a specified value interval or within a value interval of the measured data actually occurring in a measured data sample, the sample being statistically described by the first statistical distribution.

[0008] For the test, the measured data sample is first evaluated. Further, the result of the test then helps to evaluate the measured data sample, because a statistical distribution that is not suitable for modeling samples within the entire value interval may be identified as such a distribution. In particular, this allows a more suitable statistical distribution to be determined.

[0009] Further steps of sample evaluation, in particular determining parameters of a distribution that is particularly suitable for statistical modeling of the sample, may be further components of the method for evaluating the sample. A suitable statistical distribution represents a further possible outcome of the sample evaluation. A determination that the first statistical distribution is not suitable for statistical modeling of the sample over the entire specified value or actually occurring value interval of the measurement data also represents a possible outcome of the sample evaluation.

[0010] In particular, the measurement data may be measurement data obtained by measuring the workpiece by one or more coordinate measuring machines. The type of coordinate measuring machine is not limited to any particular type. If the measured values ​​are comparable, different coordinate measuring machines and / or different types of coordinate measuring machines may also be used to generate the measurement data contained in the sample. Examples of types of coordinate measuring machines are portal or gantry coordinate measuring machines, coordinate measuring machines with articulated arms, robots or machine tools comprising at least one measuring sensor and in particular devices comprising optical measuring sensors or other types of measuring sensors, which are fixedly positioned in the measuring space. The arrangement of cameras may also be used, for example, to measure complex workpieces in such a way that the contour of the workpiece surface is determined in a coordinate system of the workpiece or the measuring space. Based on this, in particular the positions of surface points or surface areas of the workpiece can be determined in sequence.

[0011] Different types of measuring sensors are suitable for generating the measurement data. By way of example, tactile sensors and / or optical sensors are particularly used on conventional coordinate measuring machines. Sensors that measure in a contactless manner are, for example, distance sensors and / or imaging sensors, such as cameras. Distance sensors can operate according to different measuring principles, examples of which include capacitive sensors, inductive sensors, time-of-flight sensors (such as TOF cameras (time-of-flight cameras) and / or color confocal sensors. Furthermore, at least one sensor of the coordinate measuring machine can be moved by a drive device, and the position of the measuring sensor determined according to the calibration system of the coordinate measuring machine can be included in the determination of the workpiece coordinates.

[0012] The coordinate measuring machines and measuring sensors described above are only examples. In general, measurement data of a workpiece can be generated in any desired manner. In particular, the measurement data are coordinates of at least a portion of the workpiece surface and / or the interior of the workpiece. The coordinates of, for example, a cavity boundary or a transition of different materials can be determined, for example, by invasive measurement methods, such as passing X-ray radiation or different invasive radiation through the workpiece from different directions and then performing a computer-aided reconstruction (such as a CT method). In particular, the measurement data can also be measurement data such as radius, length, roundness and curvature, which are derived from at least one coordinate and, for example, multiple coordinates of the corresponding workpiece. Such derived data, which in particular correspond to multiple coordinates of the corresponding workpiece, can also be determined directly by a special measuring device without explicitly determining the coordinates. For example, a camera image allows the determination of the roundness or diameter of a cylindrical area of ​​a workpiece. Measuring devices with tactile sensors that directly determine such derived measurement variables are also known.

[0013] Furthermore, the measurement data may be or include measurement data derived from raw data generated by a corresponding measuring device or measuring arrangement. Thus, for example, raw data obtained by measuring a plurality of similar workpieces may be combined to form an average value and / or outliers, i.e. meaningless or unreasonable values ​​may be eliminated or corrected in the raw data.

[0014] The multiple workpieces that form the basis of the measurement data during the measurement can in particular be simple workpieces or complex workpieces assembled from multiple workpieces in succession. All that is required is that comparable measured values ​​or values ​​derived therefrom are available during the evaluation. Workpieces are therefore to be understood as meaning simple components, such as screws or pins, as well as assembled workpieces, or irregularly shaped workpieces, such as doors for motor vehicles and machines or another arrangement of workpieces.

[0015] A measurement data sample is to be understood as meaning any selection of available measurement data and / or any processing result and / or the measurement data as a whole. The measurement data form the basis for statistical description or modeling.

[0016] As mentioned, it is possible and proposed within the scope of the present invention to test a sample as to whether a first statistical distribution is suitable for a statistical description. In particular, the first statistical distribution can be created based solely on the sample. However, it is sufficient for the test to determine only certain properties of the first statistical distribution. As described in more detail below, all that is required for the first statistical distribution are certain moments of the distribution that can be determined. Optionally, there can be a preselection of a distribution type, for example a distribution from a Pearson distribution system. However, the first statistical distribution can also be a distribution that is not part of a statistical distribution system that is used as a basis for an optionally determined second statistical distribution.

[0017] The term "moment of a statistical distribution" is a technical term in statistics. Typically, moments include expected value, variance, skewness and kurtosis. These moments and optionally higher-order moments can be used as features of the corresponding statistical distribution.

[0018] It is now proposed to define a set of all those statistical distributions that can describe the frequency of measured data values ​​in the entire value interval of the statistical distribution system of the above-mentioned value interval of the measured data, which is a value interval or a value interval that specifies the measured data actually occurring in the sample. In this case, the frequency or probability of the measured data value is usually zero or undefined in each manifestation of the statistical distribution and should not be understood as a description of the frequency. The actual or possible occurrence of this measured data value will not be fully described by a probability value or frequency value of zero. At least one exemplary embodiment of a set of statistical distributions suitable for describing the frequency of the entire value interval is discussed below. When the first statistical distribution of the sample is also generated and / or the first statistical distribution is characterized in terms of its suitability for preparing the test, this definition of the set of suitable statistical distributions can appear respectively. However, as long as the value intervals of the measured data of each sample are the same, it is sufficient to define the set once, because the set depends only on the value interval, not on the frequency or probability of the measured data values ​​of the sample. In particular, a set of suitable statistical distributions can be defined before, during and / or after the first statistical distribution of the sample is generated or characterized.

[0019] In particular, when defining a set of all those statistical distributions that are able to describe the frequency of the measured data values ​​in the entire value interval, the boundaries of the set can be determined from the value interval. The set can be uniquely determined from the boundaries.

[0020] It has already been mentioned that even if the first statistical distributions can belong to the statistical distribution system in one configuration, they do not necessarily belong to the statistical distribution system. However, it is necessary that moment values ​​of skewness and kurtosis corresponding to the first statistical distributions can be determined, respectively, in order to characterize that these first statistical distributions or the first statistical distribution family have the same skewness value and the same kurtosis value.

[0021] In order to be able to check whether the set of suitable statistical distributions contains a distribution suitable for the measured data sample, moment values ​​of the skewness and kurtosis corresponding to the first statistical distribution are now determined respectively.

[0022] There are various options for determining moment values ​​of skewness and kurtosis corresponding to the first statistical distribution. Firstly, the skewness and kurtosis values ​​can first be determined directly from the measured data samples. Optionally, the corresponding first statistical distribution can then be specified in more detail, for example by specifying a corresponding probability density function, which has the same skewness and kurtosis values ​​as the features. In particular, this can be done according to a procedure for defining a statistical distribution according to its moments.

[0023] A further option includes determining a first statistical distribution that statistically models the measured data of the sample, for example according to a process that maximizes the probability that the first statistical distribution correctly models the measured data of the sample.The skewness and kurtosis values ​​of the first statistical distribution may then be determined.

[0024] However, if the first statistical distribution is not part of the system of statistical distributions, a statistical distribution having the same skewness and kurtosis values ​​and which statistically models the corresponding measurement data in a suitable manner may be selected or may be present in the system. For simplification purposes, it may therefore be assumed, for example, that within the system a statistical distribution having the skewness and kurtosis values ​​determined from the measurement data sample statistically models the measurement data in a suitable manner. In this regard, "statistically modeling in a suitable manner" does not necessarily mean that the distribution models the measurement data within the entire value interval of the measurement data. This is still checked.

[0025] Now, the determined moment values ​​of skewness and kurtosis are used to check whether the defined set contains a statistical distribution with the determined moment values ​​of skewness and kurtosis. A corresponding test result is generated and optionally output. In particular, the test result may include a statistical distribution in which the determined skewness value and kurtosis value exist within the defined set. In this case, if the first statistical distribution belongs to the distribution system, the first statistical distribution can be used as a suitable statistical distribution for modeling the measured data values ​​within the entire value interval. If the first statistical distribution has not been fully generated by the time the test is performed, it can now be remedied. If the first statistical distribution does not belong to the system, or if the skewness value and kurtosis value are determined directly from the measured data sample, the distribution contained in the system and having the determined skewness value and kurtosis value can be used as a suitable statistical distribution for modeling the measured data values ​​within the entire value interval.

[0026] However, the test result may also include that the determined moment values ​​of 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 first family of statistical distributions is not suitable. In this case, it is not necessary to completely generate the first statistical distribution in all details required for statistical modeling. In particular, it is not necessary to specify a functional equation of the statistical distribution, which 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 to output the test result and / or this triggers the determination of a suitable second statistical distribution.

[0027] When defining a set of all those statistical distributions that are able to describe the frequency of the measured data values ​​in the entire value interval, a value pair of the skewness and the kurtosis of the statistical distribution of the system corresponding to the set can be determined from the value interval. Thus, a relationship is established between the value interval and the skewness and the kurtosis of the statistical distribution of the system. The value pairs determined with information about the boundaries of the value interval specifically form a set. However, for the sake of clarity, the term "set" shall only be used in conjunction with a suitable statistical distribution.

[0028] In particular, when defining a set of all those statistical distributions that are capable of describing the frequencies of the measured data values ​​in the entire value interval, a boundary curve in a plane spanned by the skewness and kurtosis of the statistical distribution of the system can be determined from the value interval. If the distribution system is a Pearson distribution system, this plane can also be referred to as the Pearson plane. The value pairs of skewness and kurtosis on and on one side of the boundary curve correspond to a set of suitable statistical distributions for the system. In particular, this allows a simple determination based on specific values ​​of skewness and kurtosis whether a statistical distribution is suitable for statistical modeling of the measured data in the entire value interval.

[0029] An example of a statistical distribution system is the Pearson distribution system, in which there are variations in the number of distributions belonging to the system. For example, there are variations of distributions having eight types, and there are distributions having more than eight types, such as twelve types, where some of the twelve types are subtypes of the eight types.

[0030] It is preferred that the sample and / or the first statistical distribution are standardized to the situation that the expected value is zero and the variance is one. In the case of a given sample or distribution, by shifting the measured value or distribution to the expected zero value and dividing the independent variable of the measured value or probability density function by the standard deviation, this is made possible. For example, if it has been determined that the statistical distribution suitable for statistical modeling in the entire value interval of the sample is suitable, then the independent variable of the measured value or probability density function can be multiplied by the standard deviation and the displacement that can be reversed to the expected zero value by shifting in the opposite direction. As a result, a statistical distribution suitable for modeling and having the correct expected value and the correct variance is then obtained. However, it is not necessary to carry out such standardization. However, in particular, the expenditure of the set of those statistical distributions for defining all the frequencies of the measured data values ​​that can describe the entire value interval is higher.

[0031] In particular, a method for evaluating a sample of measurement data generated by measuring a plurality of workpieces is proposed, wherein a statistical distribution system exists or is established, which is capable of describing the frequency of measurement data values ​​as a function of the measurement data values, wherein examples of the statistical distribution system are distinguishable from one another according to a respective one of two moments of the respective statistical distribution, in particular the skewness and the kurtosis, and wherein - from the statistical distribution system for a value interval of the measurement data a set of all those statistical distributions capable of describing the frequency of measurement data values ​​in the entire value interval is defined, the value interval being a specified value interval or a value interval of the measurement data actually occurring in the sample,

[0032] - determining corresponding moment values ​​of the skewness and the kurtosis from the measured data samples corresponding to a first statistical distribution,

[0033] The determined moment values ​​are used to test whether the defined set contains a statistical distribution with the determined moment values ​​of skewness and kurtosis, and to generate corresponding test results.

[0034] Furthermore, an arrangement for evaluating measurement data samples generated by measuring a plurality of workpieces is proposed, wherein a statistical distribution system exists or is established, which is capable of describing the frequency of measurement data values ​​as a function of the measurement data values, wherein examples of the statistical distribution system are distinguishable from one another according to a respective one of two moments of the respective statistical distribution, in particular the skewness and the kurtosis, and wherein the arrangement comprises:

[0035] - definition means configured to define, from a system of statistical distributions for value intervals of measurement data, a set of all those statistical distributions capable of describing the frequency of measurement data values ​​in the entire value interval, the value interval being a specified value interval or a value interval of the measurement data actually occurring in the sample,

[0036] - moment determination means configured to determine corresponding moment values ​​of skewness and kurtosis from the measured data samples corresponding to the first statistical distribution,

[0037] - a testing device configured to use the determined moment values ​​to test whether the defined set contains a statistical distribution with moment values ​​of the determined skewness and kurtosis, and to generate a corresponding test result.

[0038] The scope of the invention also includes a method for preparing an evaluation of measurement data samples generated by measuring a plurality of workpieces, wherein a statistical distribution system exists or is established, which is capable of describing the frequency of measurement data values ​​as a function of the measurement data values, wherein instances of the statistical distribution system are distinguishable from one another according to a respective one of two moments of the respective statistical distribution, in particular the skewness and the kurtosis, and wherein

[0039] - defining, from the system of statistical distributions for the value intervals of the measured data, the set of all those statistical distributions which are able to describe the frequency of the measured data values ​​in the entire value interval, the value interval being the specified value interval or a value interval of the sample to be evaluated of the measured data,

[0040] -Determine a statistical distribution from the set.

[0041] The configuration of the arrangement corresponds to the configuration of the method, so that the configurations of the methods described below also correspond to the configuration of the arrangement, respectively.

[0042] In particular, the determined statistical distribution can therefore be used in a further step of the method or during operation of the arrangement for statistical modeling of the measurement data of the sample. However, it is also not necessary for the measurement data sample to be present when preparing the evaluation, and it is also not necessary for the measurement data to be present. For example, a value interval can be specified without any relation to specific measurement data. Alternatively, a value interval for specific but not yet available measurement data can be defined. Naturally, however, the method for preparing the evaluation and the arrangement for preparing the evaluation also relate to the situation where specific measurement data and possibly also the sample to be evaluated already exist.

[0043] The preparation of the evaluation is based on the same idea as the above-described evaluation of the measured data, in which it is checked whether a first statistical distribution is suitable for modeling the measured data samples over the entire value interval. This idea consists in defining, from the system of statistical distributions for the value interval of the measured data, a set of all those statistical distributions which are able to describe the frequency of the measured data values ​​over the entire value interval.

[0044] Preferably, the statistical distribution is uniquely determined from the set according to a given rule, ie the statistical distribution uniquely emerges from the given rule.

[0045] In particular, the specified rule is intended to determine from the distribution system only the statistical distribution that is defined exactly for the specified value interval; that is, in conjunction with the terminology used in the description of the figures, the support set of the distribution is equal to the range of the measured data samples. However, depending on the boundaries of the value interval, there are cases where such a unique determination is impossible. However, if the boundaries of the value interval meet two conditions, the statistical distribution can be uniquely determined. The first condition is that the value of the left boundary of the value interval is less than the value of the right boundary of the value 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 value pair of skewness and kurtosis of the unique result of the rule is located in the "forbidden area", which will be described in more detail in the description of the figures. In that case, the result is not allowed.

[0046] The invention relates to the evaluation and / or preparation for evaluation of measurement data samples generated by measuring a plurality of workpieces. Even if this does not fall within the scope of protection of the appended claims, the invention can also be applied to the evaluation of measurement data samples generated by measuring other measurement objects. This applies both to the method and to the arrangement and the computer program. This also applies to all configurations and exemplary embodiments described in this specification. Thus, for example, humans or animals may be discussed as measurement objects and their evaluation and / or preparation is related to the corresponding measurement data.

[0047] In order to illustrate that skewness and kurtosis can be determined as distributions of moments of a distribution, a two-dimensional diagram can be plotted in which the values ​​of skewness can be plotted in the direction of a first axis (e.g. the x-axis) and the values ​​of kurtosis can be plotted along a second axis (e.g. the y-axis) at an angle, in particular at right angles, to the first axis. Thus, each point in the defined value pair plane corresponds to a family of distributions, the further moments of which, such as expected value and variance, are different in particular for the members of the family. Thus, if a family of distributions corresponding to a particular value pair consisting of a particular value of skewness and a particular value of kurtosis is selected, the distribution within this family that best corresponds to a particular sample, in particular in terms of expected value and variance, can be determined. In the plane of the two-dimensional diagram, the defined set in particular has a boundary line of value pairs of distributions that do not belong to the defined set of suitable distributions, wherein the defined set is located on the side of the boundary line with a larger value of kurtosis.

[0048] According to one configuration of the method, if it follows from the test results that the defined set does not contain a statistical distribution with moment values ​​of the determined skewness and kurtosis, a second statistical distribution of the sample is determined, wherein the second statistical distribution is a statistical distribution contained in the defined set. In the case of a corresponding configuration of the arrangement, the arrangement has means for determining the second statistical distribution. The second statistical distribution is thus suitable for describing the frequency or probability of the measured data values ​​within the entire specified value interval or the value interval of the sample.

[0049] In particular, a rule can be specified which, when followed, determines, for any first statistical distribution or for the moment values ​​of the skewness and kurtosis determined, a second statistical distribution contained in the defined set. In the process, a rule can be specifically specified that allows the second statistical distribution to be determined in a unique way; that is, a second statistical distribution with different moment values ​​of skewness and kurtosis is uniquely derived from the moment values ​​of the skewness and kurtosis determined. For example, such a rule can state that the second statistical distribution has the same skewness value and is a distribution contained in the set whose kurtosis value differs as little as possible from the determined kurtosis value of the first statistical distribution. In conjunction with the aforementioned two-dimensional diagram, this means a displacement along the kurtosis axis until the area of ​​the defined set is reached. It is also possible to specify rules for various situations, and one of these rules is not applicable in some cases, and thus makes the application of a different one of these specified rules necessary.

[0050] Rules are advantageous because reproducible results may be obtained and / or the second statistical distribution may be determined automatically.However, the selection of the second statistical distribution within the defined set may also be left to the user's discretion.

[0051] In particular, a distance measure is defined for two statistical distributions, respectively, which are capable of describing the frequency of measured data values ​​as a function of measured data values. The distance measure describes the distance between the two statistical distributions. In one configuration of the method, the second statistical distribution is determined in a way that the value of the distance measure between the first statistical distribution or the distribution corresponding to the first statistical distribution and the second statistical distribution in the statistical distribution system is the minimum value of the distance measure between the first statistical distribution or the distribution corresponding to the first statistical distribution and the second statistical distribution in the defined set in the statistical distribution system. 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 is not part of the statistical distribution system, then the distance to the corresponding distribution in the system is preferably considered. For example, the first statistical distribution has the same skewness value and kurtosis value as the corresponding distribution in the system. However, in particular if the first statistical distribution is not part of the distribution system, it is also possible to directly determine its distance to the distribution belonging to the system and minimize the distance measure of this distance in order to determine the second statistical distribution belonging to the system.

[0052] The distance measure can be defined in different ways. One option includes defining the distance measure in conjunction with the distance in the plane of the above-mentioned two-dimensional representation, i.e., the skewness-kurtosis value pair. In this case, the distance measure can be, for example, a measure of the Euclidean distance in the value pair plane. Another option includes defining the distance measure in conjunction with the distance of the two probability density functions to be compared in terms of the frequency or probability of the measured data values ​​on the measured data values. For example, the distance measure can therefore be the integral of the power of the absolute value of the frequency (or probability) difference within the interval of the union of the defined ranges of the two statistical distributions or within the value interval specified for the integral. For example, the exponent of the power can be one or preferably two.

[0053] The defined distance measure is an exemplary embodiment of the aforementioned specified rule, which in this case furthermore comprises that the first statistical distribution and the second statistical distribution have a minimum distance from each other or at most a specified maximum value of the distance measure.

[0054] Optionally, the generation of measurement data of the sample can also be a step of the method according to the present invention. As an alternative or in addition thereto, the evaluation of the aforementioned second statistical distribution can be a step of the method according to the present invention. For example, this evaluation step can include determining whether the process of producing and / or processing the measured workpiece (the measurement data has been obtained by measurement) can be carried out and / or whether a specified quality standard is met. For example, the quality standard can include at least one of the moments of the second statistical distribution that meets the specified conditions. For example, in terms of the variance of the statistical distribution (second moment), it can be specified that the variance is not greater than a specified value.

[0055] Determining moment values ​​of skewness and kurtosis based on at least the measurement data samples corresponding to the first statistical distribution and checking whether the defined set contains a statistical distribution with the determined moment values ​​of skewness and kurtosis based on the determined moment values, and producing corresponding test results can be performed by executing a computer program on a computer or a computer network. Preparing an evaluation of measurement data samples generated by measuring a plurality of workpieces, defining a set of statistical distributions and / or determining a statistical distribution from the set can also be performed by executing a computer program on a computer or a computer network.

[0056] In particular, the computer program contains program instructions which, when executed on a computer or on a computer network, prompt the computer or computer network to carry out a method in one of the configurations described in the present description. Thus, in particular when carrying out the method for evaluating a measurement data sample, in particular the moment determination device and the verification device of the arrangement can be implemented using the computer program via a computer or a computer network. The distribution determination device can also be implemented in this way. In preparing an evaluation of a measurement data sample, in particular the definition device and / or the distribution determination device can be implemented using the computer program via a computer or a computer network. BRIEF DESCRIPTION OF THE DRAWINGS

[0057] Exemplary embodiments and background of the present invention will now be described with reference to the accompanying drawings. In the various figures of the accompanying drawings:

[0058] Figure 1 For an exemplary measurement data sample, a frequency distribution of the measurement values ​​and a statistical distribution that is not suitable for describing the frequency distribution over the entire value interval of the sample are shown,

[0059] Figure 2 shows examples of three different statistical distributions that differ from each other in terms of skewness,

[0060] Figure 3 shows an example of two different statistical distributions that differ from each other in terms of kurtosis,

[0061] Figure 4 shows an example of a plane whose points are defined by pairs of values ​​of skewness and kurtosis and is called a Pearson plane, the figure having eight types of plane regions respectively assigned to Pearson distribution systems having eight types of distribution, and a "forbidden region",

[0062] Figure 5 Shows Figure 4 A simplified representation of a section of the Pearson plane region is shown, with two specific points in the Pearson plane marked with crosses,

[0063] Figure 6 shows the statistical distribution, whose skewness and kurtosis values ​​correspond to Figure 5 The lower center point marked with a cross,

[0064] Figure 7 shows the statistical distribution, whose skewness and kurtosis values ​​correspond to Figure 5 The upper right point marked with a cross in the figure, where Figure 6 and Figure 7 The expected value and variance of the statistical distribution shown in have the same value, which is also Figure 4 and Figure 5 The basis of the representation in

[0065] Figure 8 For an exemplary embodiment, a section of the Pearson plane is shown, which contains the set of solutions to the following condition: the corresponding 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,

[0066] Fig. 9 against Figure 8, showing a section of the Pearson plane that contains the set of solutions to the condition that the corresponding statistical distribution should be able to model the frequency or probability of measured data values ​​up to the right edge of the value interval,

[0067] Fig.10 against Figure 8 and Fig. 9 An exemplary embodiment of the invention shows a section from a Pearson plane, the section containing Figure 8 and Fig. 9 The intersection of the solution sets of

[0068] Fig.11 A section from the Pearson plane is shown, the section representing different regions and their boundaries, where skewness varies along the horizontal axis and kurtosis varies along the vertical axis,

[0069] Fig.12 shows portions of the solution set from a segment of the Pearson plane having a left support region (i.e., a support region in terms of the left boundary of the interval of values) for a first parameter value,

[0070] Fig.13 shows a section of the Pearson plane having portions of the solution set of the left support region for a second parameter value,

[0071] Fig.14 Schematically shows an arrangement of a device for evaluating measurement data samples generated by measuring a plurality of workpieces, which can also be interpreted as a flow chart,

[0072] Fig.15 Shows Fig.10 The portion of the segment of the Pearson plane where Fig.15 The shaded area in corresponds to Fig.10 , and wherein two points in the plane corresponding to a first statistical distribution outside the solution set and a second statistical distribution at the edge of the solution set are marked with a cross, and

[0073] Fig.16 against Figure 1 , showing an exemplary sample and frequency distribution of measured values, showing a statistical distribution suitable for describing the frequency distribution over the entire value range of the sample. DETAILED DESCRIPTION

[0074] exist Figure 1 In , the frequency of the measured data values ​​is represented in the form of bars for small local intervals of the value intervals of the measured data values ​​of the measured data samples, respectively. Thus, the measured data values ​​are plotted along the horizontal axis and the frequencies or probabilities are plotted along the vertical axis. Further, Figure 1The function curve of the first statistical distribution for modeling the frequency distribution is shown. Obviously, the first statistical distribution only models the frequencies in the value interval bounded by the left side, which value interval starts at about -1.9 measured data values. However, the sample also contains measured data values ​​less than -1.9. Therefore, the first statistical distribution is not suitable for modeling the frequencies in the entire value interval of the sample.

[0075] For example, you can Figure 1 The sample below specifies a value interval of the measured data values ​​that actually occur or a larger value interval as a value interval in which the statistical distribution should be able to model the frequency or probability of the measured data values. Hereinafter, this specified value interval is also referred to as a range.

[0076] Instead, there is a range of values ​​within which the statistical distribution can model the frequency or probability of the measured data values. In the following, this range of values ​​is also called the support set of the statistical distribution. Figure 1 In the case shown, the range is therefore not contained in the support set. However, this is sought.

[0077] As mentioned above, moments can be assigned as features specifically to the statistical distribution of the Pearson distribution system. The four moments are the expected value v 1 , variance μ 2 , skewness and Kurtosis Combined with the random variable X, the nth moment is obtained by the following formula:

[0078] v n (X) = E(X n )

[0079] The expected value is generated by inserting n = 1. E represents the expected value operator. The nth central moment μ n It is obtained by the following formula:

[0080] μ n (X) = E((Xv 1 (X) n ), n≥1

[0081] According to this, the corresponding equation of variance μ 2 It can be obtained by inserting n = 2. The nth central normalized moment It is obtained by the following formula:

[0082]

[0083] From this, by plugging in n=3 and 4, we can obtain the skewness and Kurtosis The corresponding equations are given by . As has been demonstrated elsewhere, the central standardized moments of skewness and kurtosis are independent of the moments of the expected value and variance, so the test of whether a statistical distribution is suitable for modeling samples over the entire interval of values ​​and the determination of the suitable statistical distribution with respect to skewness and kurtosis, respectively, can be performed for any expected value and any value of the variance, and the results of the test or determination are valid. In the case of standardization described elsewhere, care should be taken to determine the solution set in terms of standardized measurements or standardized statistical distributions.

[0084] Figure 2 Three different statistical distributions are shown, which differ from each other in skewness. Here, the value of the skewness of the probability density function shown in the solid line is zero, because the probability density function is symmetrical. The probability density function shown in the dotted line has a negative skewness; the probability density function shown in the dotted line has a positive skewness. Figure 3 Two different statistical distributions are shown, which differ from each other in kurtosis. The kurtosis only takes positive values. The kurtosis of the probability density function shown by the dashed line is greater than the kurtosis of the probability density function shown by the solid line.

[0085] It is conventional to adjust the first two moments of the distribution, namely the expected value and the variance. However, below we first consider only the skewness and kurtosis; these allow each distribution or family of distributions to be uniquely identified.

[0086] Specifically, a Pearson distribution system with eight types of distributions is considered in the following exemplary embodiments. For this system, Figure 4 The so-called skewness-kurtosis plane is shown, wherein, for an exemplary embodiment, the skewness values ​​are plotted along the axis extending horizontally and the kurtosis values ​​are plotted along the axis extending vertically. The skewness-kurtosis plane is referred to as the Pearson plane hereinafter. Each point of the Pearson plane outside the forbidden area (defined by a coordinate pair including a skewness coordinate and a kurtosis coordinate) is uniquely assigned to a distribution or a distribution family. The members of the family also differ in the values ​​of further moments, such as in particular expected values ​​and variances. There is a continuous relationship between the relative position in the Pearson plane and the form of the associated distribution or distribution family. This is not only directly practical in terms of skewness and kurtosis, but also in terms of further properties, in particular value intervals, within which the distribution family is suitable for statistical modeling of sample or measurement data values ​​and predicting their probabilities. Therefore, if two points are close together on the Pearson plane, when two associated distributions take the same expected value and the same variance, the two distributions are also similar.

[0087] Figure 4 The example of a Pearson plane shown in was created based on a normalization in which the expected value has a value of zero and the variance has a value of one.

[0088] exist Figure 4In the figure, a dotted line approximating a parabola can be seen at the bottom of the shown area of ​​the plane. The value pairs of skewness and kurtosis below this line have no associated distribution. Therefore, the plane area located below this line can be called a non-permitted area or a forbidden area. When the skewness value is zero, the dotted line extends from the aforementioned line parallel to the kurtosis axis to the solid line approximating a parabola. The value pairs on the dotted line belong to the distribution of the Pearson type 2 distribution. Along the extension of the dotted line, the dot-dashed line extends parallel to the kurtosis axis, and its value pairs belong to the distribution of the Pearson type 7 distribution. The value pairs of the aforementioned solid line belong to the Pearson type 3 distribution. A similar parabolic dotted line is shown above the solid line, and the points of the value pairs of the distribution of the Pearson type 5 distribution are located on it. The value pairs of the points between the forbidden area and the solid line belong to the distribution of the Pearson type 1 distribution. The value pairs of the points between the solid line and the upper dotted line belong to the distribution of the Pearson type 6 distribution. The value pairs of the points above the upper dotted line belong to the distribution of the Pearson type 4 distribution. Located at the common intersection or end of all the preceding lines, except the lower dashed line, is a pair of values ​​of a unique point of the distribution, also known as the Pearson's type 0 distribution, which is a normal distribution.

[0089] To highlight two points in the plane, Figure 5 Two crosses are drawn in the region of the Pearson plane shown in . Figure 6 and Figure 7 shows the associated distributions. Both distributions belong to Pearson type 1 distribution. Figure 6 The skewness of the distribution shown is negative and is smaller than Figure 7 The positive skewness of the distribution shown is closer to zero. Therefore, this distribution looks better than Figure 7 The distribution shown in is more symmetrical. Figure 7 The kurtosis of the distribution shown in has a positive value and is greater than Figure 6 Positive values ​​of the distribution shown in .

[0090] The range of the sample represents the interval of measured data values. Now, the distribution that fits the sample needs to have a support set that at least contains this interval range. Therefore, if the range is obtained by the following formula:

[0091] Range = [r min , r max ]

[0092] And the support set of the distribution is obtained by:

[0093] Support set = [v min , v max ],

[0094] Then the following should apply:

[0095]

[0096] That is, the range should be completely contained in the support set. This requirement is equivalent to:

[0097] v min ·≤·r min , (1)

[0098] v max ·≥·r max (2)

[0099] It follows from equation (1) that we seek a distribution whose support set lower bound v min Less than or equal to range r min Any distribution can be visualized as a skewness / kurtosis point in the Pearson plane. Therefore, the set of all distributions that satisfy equation (1) can be represented by a set in the Pearson plane. This yields a solution set, for an exemplary embodiment, Figure 8 The solution set is shaded in . Note that the solution set is bounded below by the convex boundary line, i.e., it lies above the boundary line. Similarly, the solution set can be obtained starting from equation (2) for Fig. 9 The solution set is shown by shading in the exemplary embodiment of FIG. Figure 4 Same as in Figure 8 and Fig. 9 The dashed lines forming the boundaries of the forbidden area are shown in Fig.10 The overall solution set of the exemplary embodiment is shown with cross-hatching in FIG. Figure 8 and Fig. 9 These figures should be understood as schematic with respect to the fact that the partial solution sets in extend into the forbidden region. In reality, the partial solution sets do not extend into the forbidden region.

[0100] exist Fig.10 The intersection of the boundary curves of the partial solution set shown by the dotted line and the dashed line in has a value pair of skewness and kurtosis that corresponds to the statistical distribution within the distribution system with a support set equal to the range. In an exemplary embodiment, this intersection is located at a zero skewness value, but negative or positive skewness values ​​may also be used in other cases.

[0101] M L represents the solution set starting from equation (1), and M R represents the solution set starting from equation (2), which is defined as follows:

[0102] M L ={(s,k):v min (s,k)·≤·r min}as well as

[0103] M R ={(s,k):r max ·≤·vmax (s,k)}.

[0104] Here, (s, k) represents a pair of values ​​of skewness s and kurtosis k.

[0105] Now, the following describes the method for determining the set M L and M R An exemplary embodiment of .

[0106] The Pearson distribution system is based on the following general differential equation for the probability density function. The normalized solution of the general differential equation f(x) is

[0107]

[0108] Polynomial

[0109] a(x)=x+a 0 ·, ...a 0 ∈R (4)

[0110] b(x)=b 2 +b 1 x+b 0 ·,……b 0 , b 1 , b 2 ∈R, (5)

[0111] Wherein, R represents the set of real numbers, which is referred to as the Pearson probability density function hereinafter.

[0112] The coefficients of the polynomials a(x) and b(x) defined in equations (4) and (5) parameterize the still unknown probability density function f(x). The form and definition range of the probability density function depend in particular on the number and location of the zeros of the denominator polynomial b(x).

[0113] In order to relate the probability density function to a statistically significant variable, we first establish the coefficient a = (a 0 , 1) T and b=(b 0 , b 1 , b 2 ) T The relationship between the specified set of moments of the probability density function and the normalized moments (see above) is invariant with respect to scaling and shifting of the random variable X, and its standardized moments are or have been calculated. Therefore, the standardized moments of the random variable X are can be considered to be related to its original moment and central moment v 1 (X) and μ 2(X) (see above). Starting from this, we can define the parameterized moment tuple M of the Pearson probability density function and its assigned random variable X. The tuple M has been defined above. In the process, the aforementioned labels v are introduced for expected value, variance, skewness and kurtosis 1 (X), μ 2 (X) and By using the corresponding index (e.g., M 1：2 (X)=(v 1 (X), μ 2 (X))) to reference a partial set of elements of a tuple M.

[0114] In the following, the original moment The sum tuple M is related to the parameters a and b of the general differential equation. The coupling between moments and coefficients can be written in the form of a matrix equation:

[0115]

[0116] In the following, the aforementioned normalization is performed to simplify the solution. However, the corresponding solution can also be obtained without such normalization. Then, the solution equation becomes correspondingly more complicated. For v 1 =0 and v 2 =1, the solution of equation (6) is as follows:

[0117]

[0118] in, and are used as labels for the moments of skewness and kurtosis, and where Taking into account equations (4) and (6), the Pearson probability density function can now be parameterized as follows:

[0119] Given the proportional and displacement components M * 1：2 (X) = {v * 1 , μ * 2} and the formal component M * 3：4 (X) = {ζ * 1 , * 2 The corresponding Pearson probability density function can be constructed in two steps. In the first step, the coefficient a is calculated using equation (7) * and b * , and solve the Pearson differential equation with these parameters in order to obtain the normalized Pearson probability density function f s(x) and the corresponding random variable X. In the second step, the transformation is set and the required Pearson random variable X is defined as X = τ -1 (X s ) and its probability density function is defined as f(x) = τ′(x)f s (τ(x)). Here, the superscript index -1 indicates the inverse function, and the superscript comma indicates the first derivative of the function. Now, M(X) = M * This shows that the tuple M * 1：2 (X) and M * 3：4 Each of (X), ie the scale and displacement components on the one hand and the form component on the other hand, can be processed independently of each other within the scope of a procedure for fitting the Pearson probability density function to the corresponding moments of the distribution.

[0120] The Pearson plane is defined as the real plane E, 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 with respect to the zero point. The distinction criterion κ(ζ) for the three regions and the corresponding boundary curves is defined as:

[0121]

[0122] The main area of ​​the Pearson plane E can be specified as follows. If x 1 and x 2 are the zeros of the polynomial b(x;ζ), then its coefficients b are elements of the three-dimensional space of real numbers and are calculated using equation (7), and then the region R of the Pearson plane E can be specified as follows 1 , R 4 and R 6 :

[0123]

[0124] The boundary curve C of the specified area i (i=2, 3, 5, 7) (hereinafter referred to as the Pearson curve) can be specified as follows:

[0125]

[0126] Furthermore, the aforementioned forbidden region of the Pearson plane is introduced

[0127]

[0128] , and get its boundary line

[0129]

[0130] First, the forbidden area contains the point pair ζ = (ζ 1 , 2 ), these pairs of points never appear as solutions to the calculations of skewness and kurtosis. Second, the solutions of the conventional Pearson differential equation for points on the forbidden curve are not integrable and for this reason cannot represent a probability density function. Therefore, the region in the Pearson plane with all the points for which solutions can be found can be defined as follows:

[0131]

[0132] In other words, the regions as a whole may be specified as the set of points corresponding to the entire plane minus the set of points in the prohibited region and minus the set of points on the boundary line of the prohibited region. Fig.11 The three regions R are shown 1 , R 4 and R 6 and the two Cs in their aforementioned boundary curves 3 and C 5 . Moreover, with Figure 4 Similar, in Fig.11 The lower part shows the prohibited area and its boundary lines. Fig.11 The point P shown in 0 The relative position of the pair of values ​​representing the skewness and kurtosis of the normal distribution.

[0133] In the following, only the standardized Pearson distribution is considered. Why such consideration is sufficient has been supported above. Now, first according to the skewness ζ 1 and kurtosis ζ 2 Derive an expression for the support interval of the Pearson distribution. Then, derive the curve or boundary line of the aforementioned solution set. Finally, derive the set of solution sets for the right boundary of the support set, and derive the set of solution sets for the left boundary of the support set.

[0134] The zeros x of the polynomial b(x) defined in equation (5) are 1 and x 2 It can be defined as follows:

[0135]

[0136] as well as

[0137]

[0138] If the coefficient b i is specified as a function of skewness and kurtosis, then the zero point x i =x i (ζ) The following equation appears:

[0139]

[0140] as well as

[0141]

[0142] set up The following helper functions can be specified:

[0143]

[0144] Zero point x 1 and x 2 They are complementary in terms of sgn(c), i.e. in terms of sign. This yields the following substitution equation:

[0145]

[0146] as well as

[0147]

[0148] In terms of equations (18) and (19), it should be noted that the argument of the square root in the numerator of the fraction is not defined only for negative expressions. Therefore, the functions according to equations (18) and (19) are not defined for the forbidden area and its boundary lines.

[0149] Combined with the matrix 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 is represented by ξ R Expressed as, support set interval I = (ξ L ,ξ R ) is defined as follows:

[0150]

[0151] as well as

[0152]

[0153] Here, "other" has its conventional meaning. Starting from equations (20) and (21), we can denote the boundary curve c of the solution region L (t) and c R (t)Specify the following implicit equation:

[0154]

[0155] where "for" has its conventional meaning. Before calculating these curves, some preliminary reflections are given below. The equation for the zero point x (1) (ζ)=ξ L and x (2)(ζ)=ξ R The naive solution of solution) produces a curve s(t;ξ) defined by:

[0156]

[0157] For 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, the singularity exists at

[0158]

[0159] For ξ≠0, there are intersections at the following points of the forbidden curve, in particular at the boundary line of the forbidden region:

[0160]

[0161] for Exists on the aforementioned curve or boundary line C 5 The intersection point of , see equation (9):

[0162]

[0163] For ξ≠0, the unique global minimum exists at:

[0164]

[0165] Equation x (1) (ζ)=ξ L and x (2) (ζ)=ξ R Solve by branching the curve s(t;ξ), for x (1) (ζ), the branch is located at The right side of , and for x (2) (ζ), located to its left.

[0166] Although s(t; ξ) is not a solution to Eq. (22), the following considerations show that only the definition range of the curve s(t; ξ) needs to be adjusted to obtain a solution. L (t;ξ L ) and c R (t;ξ R ) has the following solution:

[0167] c L (t;ξ L):=s(t;ξ L )and

[0168] as well as

[0169] c R (t;ξ R ):=s(t;ξ R )and

[0170] Here, "and" and "if" have their conventional meanings. It follows that there is a transformation between the left and right boundaries of the support set. The following relations apply:

[0171] c R (t;ξ R )=c L (-t; -ξ R )

[0172] Combined with equation (22), it is as follows:

[0173] x L (c L (t;ξ L ))=ξ L <0 and x R (c R (t;ξ R ))=ξ R >0

[0174] For the region of the Pearson plane outside the forbidden region, we can now find the left or lower boundary v of the support set as follows min For ξ<0, define the set R L (ξ) and on the right or upper boundary v of the support set max For ξ>0, we define the set R R (ξ), these sets are equivalent to the aforementioned solution set M of equation (1) L and the aforementioned solution set M of equation (2) R :

[0175]

[0176] It follows that if ξ < 0 and if the point with coordinate ζ in the Pearson plane lies in the solution set or support region R at the lower boundary of the support set L (ξ), then the relation a≤ξ applies to the Pearson probability density function with support region (a, b). Similarly, if ξ>0 and if the point with coordinate ζ in the Pearson plane lies in the solution set or support region R at the right boundary of the support R (ξ), then b ≥ ξ applies.

[0177] The left support region R with ξ<0L (ξ) can be represented by the set of points S(ξ) in the Pearson plane, which unifies the following sets S i (ξ), where i = 1, 2, 3:

[0178]

[0179] Here, D ξ represents the left boundary curve c of the support set L , v(ξ) represents the intermediate equation of equation (17), and R represents the area in the Pearson plane without the forbidden area and its boundary line. Fig.12 It is shown that for ξ = -2,5, at the left or lower boundary v of the support set min Solution set R L (ξ). For ξ=-0,5, Fig.13 This solution set R is shown L (ξ). Although Fig.12 All three sets S are shown i , but in Fig.13 In the case of 3 is empty. This applies to the range Because the left boundary curve c L and the aforementioned curve C 5 There is no intersection. Fig.12 and Fig.13 In the set S 2 The boundary curves of are highlighted in each case by showing them with thick solid lines. The dashed lines, as the set S 1 The boundary curve is the boundary line of the prohibited area.

[0180] From the left support region R L Starting with the relationship between (ξ) and ξ, we can illustrate the following: The following applies to two negative values ​​b≤a<0:

[0181]

[0182] If a value ξ is given L <0<ξ R , that is, the solution set that defines the left and right boundaries of the support set, and the following applies to the intersection point:

[0183]

[0184] The above description contains exemplary embodiments for defining a set of all those statistical distributions that are capable of describing the frequencies of the measured data values ​​in the entire value interval (range). Exemplary embodiments of the present invention are now described in conjunction with testing whether the defined set contains a statistical distribution with moment values ​​of skewness and kurtosis determined for a sample.

[0185] Fig.14 The arrangement of a device for evaluating measurement data samples generated by measuring a plurality of workpieces is schematically shown. Fig.14 The illustration in can also be regarded as a flow chart for illustrating an embodiment of a method for evaluating a measurement data sample.

[0186] For example, the coordinate measuring machine 1 measures a plurality of workpieces and transmits the measurement data to the measurement data storage 3, optionally after preprocessing the measurement data. The definition device 5 is configured to define, from a statistical distribution system for a value interval of the measurement data, a set of all those statistical distributions that can describe the frequency of the measurement data values ​​in the entire value interval, which is a specified value interval or a value interval of the measurement data actually occurring in the sample. Fig.14 In the embodiment of the present invention, the input of the definition device 5 is connected to the output of the measurement data memory 3. Thus, the definition device 5 can determine the value interval in particular from the available measurement data. As an alternative or in addition thereto, additional information about the value interval that exceeds the value interval of the measurement data can be obtained, which additional information is stored in the measurement data memory 3. In practice, however, the information about the specified value interval can also be made available to the definition device 5 in any other way, so that the connection between the measurement data memory 3 and the definition device 5 is not mandatory.

[0187] The output of the definition means 5 is connected to the input of a moment determination means 7, which is configured to determine corresponding moment values ​​of skewness and kurtosis from the measured data samples corresponding to the first statistical distribution. During operation or when performing the method, the moment determination means 7 determines the skewness value and the kurtosis value of the sample and transmits these values ​​to the testing means 9, which is configured to use the determined moment values ​​to test whether the defined set contains a statistical distribution with the determined moment values ​​of skewness and kurtosis and produce a corresponding test result.

[0188] It is clear from the test result whether such a statistical distribution exists in the defined set. If it does, the signal output by the test device 9 can, for example, confirm that the first statistical distribution is suitable for the purpose of statistical modeling of samples throughout the specified value interval. If it does not exist, the test device 9 outputs the test result or signal to the distribution determination device 11, which determines the second statistical distribution, which is suitable for the purpose of statistical modeling of samples within the entire specified value interval. In this case, the distribution determination device 11 can, for example, determine the value pair corresponding to the suitable statistical distribution from the Pearson plane. Using information about the value pair, a suitable statistical distribution can then be generated as the second statistical distribution.

[0189] The defining means 5 and the distribution determining means 11 may also be formed without Fig.14The arrangement of the other means shown. Further, the definition means 5 and the distribution determination means 11 can be used to prepare an evaluation of a sample of measured data. In both cases, the definition means 5 is configured to define, from a system of statistical distributions of value intervals for the measured data, a set of all those statistical distributions that are able to describe the frequency of the measured data values ​​in the entire value interval, which is a specified value interval or a value interval of the sample of measured data to be evaluated. The distribution determination means 11 is then configured to determine the statistical distribution from the defined set.

[0190] if Fig.14 If the method is interpreted as a flow chart, then in method step 1, measurement data are generated, which are stored in method step 3. In method step 5, a set of all those statistical distributions is defined which can describe the frequency of the measurement data values ​​in the entire value interval. In method step 7, corresponding moment values ​​of skewness and kurtosis are determined from the measurement data samples corresponding to the first statistical distribution. In method step 9, the determined moment values ​​are used to check whether the defined set contains a statistical distribution with the determined moment values ​​of skewness and kurtosis, and a corresponding test result is generated. The order of these method steps is determined by Fig.14 The arrow in is generated.

[0191] Optional or specific configuration of the device or method steps in Fig.14 In the diagram, reference numerals 1, 3 and 11 are used. If the first statistical distribution is to be tested with regard to its suitability for statistically modeling samples within the entire specified value interval, method step 5 need not necessarily always be performed.

[0192] In particular, based on a distance measure for the distance between two distributions in the Pearson plane, the distribution determination device 11 may determine the distribution whose distance from the first statistical distribution is the smallest as the second statistical distribution. Fig.15 An example of two distributions in the Pearson plane with a minimum distance, in this case the minimum Euclidean distance in the Pearson plane, is shown. The cross located closer to the bottom right represents the first statistical distribution in the case of a lack of suitability for statistical modeling of samples over the entire specified value interval. This lack of suitability is evident from the fact that the cross is outside the shaded area of ​​the solution set of the fitted statistical distribution. The cross represents the distribution by marking a point in the Pearson plane corresponding to the distribution, which in turn corresponds to a pair of values ​​for the skewness and kurtosis of the distribution. According to the rules specified here, the suitable second statistical distribution is determined from the first statistical distribution by determining the point in the solution set of the fitted distribution in the Pearson plane that has the shortest distance to a point of the first statistical distribution. In all cases, this point with the shortest distance is located at the edge of the solution set.

[0193] Similar to Figure 1 In the diagrammatic way, Fig.16 Now shown is a statistical distribution, more precisely a probability density function, which is determined as described above and is therefore located in the solution set of a suitable statistical distribution and is therefore suitable for statistically modeling the measured data values ​​of a sample within the entire specified value interval. Fig.16 This is evident from the fact that Figure 1 In contrast, the probability density function represented by the solid line also takes on positive frequency or probability values ​​in the region of measured data values ​​less than -1.9.

[0194] In the case of preparing an evaluation of the measured data sample, for example, only the method steps of defining a set of all those statistical distributions which are able to describe the frequency of the measured data values ​​within the entire value interval and of determining the statistical distribution belonging to this set can be performed. Naturally, this does not exclude the evaluation of the measured data sample by means of this statistical distribution after the statistical distribution has been determined.

Claims

1. A method for evaluating measurement data samples generated by measuring a plurality of workpieces by one or more coordinate measuring machines, the method being executed by a computer program on a computer or a computer network, in, - from a Pearson-type statistical distribution system for describing the frequencies of the measured data values ​​as a function of the measured data values, a set of all those statistical distributions capable of describing the frequencies of the measured data values ​​in the entire value interval of the sample is defined, wherein the distributions of the system are respectively describable according to the moment values ​​of two moments of the corresponding statistical distribution, in particular the skewness and the kurtosis, said system being present or being established when the method is performed, wherein, Evaluation of the sample identifies that the system is unable to describe the distribution of the frequencies of the measured data values ​​within the entire value interval of the sample, - determining corresponding moment values ​​of the skewness and the kurtosis based on the measured data sample, - the determined moment value is used to test whether the defined set contains a statistical distribution with the determined moment value of skewness and kurtosis, and to produce a corresponding test result, i.e., such a statistical distribution does not exist in the defined set or such a statistical distribution does exist, - if it follows from the test result that the defined set contains a statistical distribution with moment values ​​having the determined skewness and kurtosis, determining that the statistical distribution of the sample is a first statistical distribution, - if it follows from the test result that the defined set does not contain a statistical distribution with moment values ​​having the determined skewness and kurtosis, determining a second statistical distribution of the sample, and wherein, The second statistical distribution is a statistical distribution contained in the defined set, The obtained statistical distribution is used to determine whether the process by which the measured workpiece has been manufactured and / or processed is controllable and / or whether the frequency distribution of the measured data values ​​regarding the sample meets predetermined quality criteria for the process.

2. The method according to claim 1, in, Distance measures are defined for two statistical distributions respectively, the two statistical distributions are capable of describing the frequencies of these measured data values ​​as a function of the measured data values, the distance measures describing the distance between the two statistical distributions, and wherein the value of the distance measure between a first statistical distribution in the statistical distribution system or a distribution corresponding to the first statistical distribution and the second statistical distribution is the minimum value of the distance measure between the first statistical distribution or the corresponding distribution and the statistical distributions in the defined set.

3. The method according to claim 1 or 2, in, When defining the set of all those statistical distributions that are able to describe the frequency of measured data values ​​in the entire value interval of the sample, the boundaries of the set are determined from the value interval.

4. The method according to claim 1 or 2, in, When a set of all those statistical distributions that are able to describe the frequency of measured data values ​​in the entire value interval of the sample is defined, a value pair of skewness and kurtosis of the statistical distribution of the system corresponding to the set is determined from the value interval.

5. The method according to claim 4, in, When the set of all those statistical distributions that are able to describe the frequency of the measured data values ​​in the entire value interval of the sample is defined, the boundary curves in the plane spanned by the skewness and kurtosis of the statistical distribution of the system are determined from the value interval.

6. A system for evaluating measurement data samples generated by measuring a plurality of workpieces by one or more coordinate measuring machines, in, The system includes: - definition means configured to define, from a Pearson-type statistical distribution system for describing the frequencies of the measured data values ​​as a function of the measured data values, a set of all those statistical distributions capable of describing the frequencies of the measured data values ​​in the entire value interval of the sample, wherein the distributions of the system are respectively describable according to the moment values ​​of two moments of the corresponding statistical distribution, in particular the skewness and the kurtosis, wherein the evaluation of the sample identifies the distributions of the system which are not capable of describing the frequencies of the measured data values ​​in the entire value interval of the sample, - moment determination means configured to determine corresponding moment values ​​of skewness and kurtosis from the measured data samples, - a testing device configured to use the determined moment values ​​to test whether the defined set contains a statistical distribution with the determined moment values ​​of skewness and kurtosis and to produce a corresponding test result, i.e., the absence of such a statistical distribution or the presence of such a statistical distribution, - a distribution determination device, which is configured to determine a second statistical distribution of the sample if it is derived from the test result that the defined set does not contain a statistical distribution with moment values ​​of the determined skewness and kurtosis, and wherein the second statistical distribution is a statistical distribution contained in the defined set, and the distribution determination device is configured to determine: if it is derived from the test result that the defined set contains a statistical distribution with moment values ​​of the determined skewness and kurtosis, then the statistical distribution of the sample is determined to be the first statistical distribution, wherein the system is designed to determine based on the obtained statistical distribution whether the process by which the measured workpiece has been manufactured and / or processed is controllable and / or whether the frequency distribution of the measured data values ​​of the sample meets a predetermined quality standard of the process.

7. The system according to claim 6, in, The distribution determination device is configured to determine the second statistical distribution from the defined set in the following way, that is, based on the distance measure of the two statistical distributions defined respectively, the value of the distance measure between the first statistical distribution in the statistical distribution system or the distribution corresponding to the first statistical distribution and the second statistical distribution is the minimum value of the distance measure between the first statistical distribution or the corresponding distribution and the statistical distributions in the defined set, and the distance measure describes in each case the distance between two statistical distributions, which are able to describe the frequency of these measured data values ​​as a function of the measured data values.

8. The system according to claim 6 or 7, in, The defining means is configured to, when defining a set of all those statistical distributions capable of describing the frequency of the measured data values ​​in the entire value interval of the sample, determine the boundaries of the set from the value interval.

9. The system according to claim 6 or 7, in, The definition device is configured to, when defining a set of all those statistical distributions that can describe the frequency of measured data values ​​in the entire value interval of the sample, determine from the value interval a value pair of skewness and kurtosis of the statistical distribution of the system corresponding to the set.

10. The system according to claim 9, in, The definition device is configured to determine the boundary curve in the plane spanned by the skewness and kurtosis of the statistical distribution of the system from the value interval when defining the set of all those statistical distributions that can describe the frequency of the measured data values ​​in the entire value interval of the sample.

11. A method for preparing an evaluation of measurement data samples generated by measuring a plurality of workpieces by means of one or more coordinate measuring machines, the method being executed by means of a computer program on a computer or a computer network, wherein - defining, from a Pearson-type statistical distribution system for describing the frequencies of the measured data values ​​as a function of the measured data values, a set of all those statistical distributions capable of describing the frequencies of the measured data values ​​in the entire value interval of the sample, in, The distributions of the system can be described in accordance with the moment values ​​of two moments of the corresponding statistical distribution, in particular the skewness and the kurtosis, respectively, and the system exists or is established when the method is performed, wherein: Evaluation of the sample identifies that the system is unable to describe the distribution of the frequencies of the measured data values ​​within the entire value interval of the sample, - the statistical distribution is uniquely determined from the set according to the boundaries of the value interval, the boundaries of the value interval satisfying two conditions, the first condition being that the value of the left boundary of the value interval is less than the value of the right boundary of the value interval, the second condition being that the left boundary multiplied by the right boundary is less than -1, and if the second condition is not satisfied, the moment values ​​of the skewness and the kurtosis are located in the forbidden region, The obtained statistical distribution is used to determine whether the process by which the measured workpiece has been manufactured and / or processed is controllable and / or whether the frequency distribution of the measured data values ​​regarding the sample meets predetermined quality criteria for the process.

12. The method according to claim 11, in, When defining the set of all those statistical distributions that are able to describe the frequency of measured data values ​​in the entire value interval of the sample, the boundaries of the set are determined from the value interval.

13. The method according to claim 11 or 12, in, When a set of all those statistical distributions that are able to describe the frequency of measured data values ​​in the entire value interval of the sample is defined, a value pair of skewness and kurtosis of the statistical distribution of the system corresponding to the set is determined from the value interval.

14. The method according to claim 13, in, When the set of all those statistical distributions that are able to describe the frequency of the measured data values ​​in the entire value interval of the sample is defined, the boundary curves in the plane spanned by the skewness and kurtosis of the statistical distribution of the system are determined from the value interval.

15. A system for preparing an evaluation of measurement data samples generated by measuring a plurality of workpieces by one or more coordinate measuring machines, in, The system includes: - definition means configured to define, from a Pearson-type statistical distribution system for describing the frequencies of the measured data values ​​as a function of the measured data values, a set of all those statistical distributions capable of describing the frequencies of the measured data values ​​in the entire value interval of the sample, wherein the distributions of the system are respectively describable according to the moment values ​​of two moments of the corresponding statistical distribution, in particular the skewness and the kurtosis, wherein the evaluation of the sample identifies the distributions of the system which are not capable of describing the frequencies of the measured data values ​​in the entire value interval of the sample, - distribution determination means, the distribution determination means being configured to uniquely determine the statistical distribution from the set according to a boundary of a value interval, the boundary of the value interval satisfying two conditions, a first condition being that the value of the left boundary of the value interval is less than the value of the right boundary of the value interval, a second condition being that the left boundary multiplied by the right boundary is less than -1, and if the second condition is not satisfied, Then the moment values ​​of the skewness and kurtosis are in the forbidden area, Therein, the system is designed to determine based on the obtained statistical distribution whether the process by which the measured workpiece has been manufactured and / or processed is controllable and / or whether the frequency distribution of the measured data values ​​about the sample meets predetermined quality standards for the process.

16. The system according to claim 15, in, The defining means is configured to, when defining a set of all those statistical distributions capable of describing the frequency of the measured data values ​​in the entire value interval of the sample, determine the boundaries of the set from the value interval.

17. The system according to claim 15 or 16, in, The definition device is configured to, when defining a set of all those statistical distributions that can describe the frequency of measured data values ​​in the entire value interval of the sample, determine from the value interval a value pair of skewness and kurtosis of the statistical distribution of the system corresponding to the set.

18. The system according to claim 17, in, The definition device is configured to determine the boundary curve in the plane spanned by the skewness and kurtosis of the statistical distribution of the system from the value interval when defining the set of all those statistical distributions that can describe the frequency of the measured data values ​​in the entire value interval of the sample.

Images