Using International Standards to Control the Positional Quality of Spatial Data

F.J. Ariza-López, and J. Rodríguez-Avi

Abstract A positional quality control method based on the application and for this reason, it is proposed to express the results of of the International Standard ISO 2859 is proposed. This entails quality control checks by means of percentiles (Maune, 2007) a common framework for dealing with the control of all other of the observed distribution. The latter has been introduced spatial data quality components (e.g., completeness, consis- by the ASPRS into the Guidelines for reporting vertical accu- tency, etc.). We propose a relationship between the parameters racy for lidar data (ASPRS, 2004). “acceptable quality level” and “limiting quality” of the inter- Another source of criticism is related to the use of points national standard and positional quality by means of observed as control elements. Some researchers (e.g., Joao, 1998) have error models. This proposal does not require any assumption criticized PAAMs for being limited to well-defined points, and for positional errors (e.g., normality), which means that the ap- also for failing to address more complex control elements like plication is universal. It can be applied to any type of positional linear and aerial ones. In the last two decades the positional and geometric controls (points, line-strings), to any dimension assessment (planimetric) of line-strings elements has been an (1D, 2D, 3D, etc.) and with parametric or non-parametric error issue of great interest for researchers, and there are some pro- models (e.g., lidar). This paper introduces ISO 2859, presents posed methods for their positional accuracy assessment. There the statistical bases of the proposal and develops two examples are methods based on distances (e.g., the Hausdorff Distance of application, the first dealing with a lot-by-lot control and the Method by Abbas et al. (1995) and the Mean Distance Method second, isolated lot control. by Skidmore and Turner (1992), and methods based on buf- fers (e.g., the Single Buffer Overlay Method by Goodchild and Hunter (1997), and the Double Buffer Overlay Method by Introduction Tveite and Langaas (1999). In many of these cases the resulting Positional quality is one of the most desirable characteristics of errors follow non-parametric distributions (distribution free). spatial data and is determined by positional accuracy. Positional PAAMs can be classified by means of the statistical ap- quality is assessed by positional accuracy, which is a Deliveredmatter of by Ingentaproach: estimating methods (e.g., NSSDA) and control methods renewed interest because of the capabilitiesIP: 192.168.39.210 offered by Global On: Fri, 01(e.g., Oct NMAS 2021, EMAS 01:31:31, ASLSM). In the first group the estimation is Navigation Satellite SystemsCopyright: (GNSS) and American the need Society for greater for spa Photogrammetry- expressed by and means Remote of a confidence Sensing interval using an un- tial interoperability to support the Spatial Data Infrastructures. derlying statistical model. If PAAMs are going to be used for In a Spatial Data Set (SDS) the position of a real world the acceptance or rejection of products (control approach), entity (feature) is described/recorded with position values of explicit information about user (buyers or consumers) and geometric objects (e.g., points, line-strings, shapes, etc.) in an producer risks are needed. The introduction of such risks is appropriate coordinate system. Positional accuracy represents a new paradigm that has not been considered in any of the the nearness of those values to the entity’s “true” position in traditional PAAMs. Some research has focused on users’ and that system. Positional accuracy has traditionally been evalu- producers’ risks, e.g., for the cases of the EMAS (Ariza-López et ated using control points. Following this idea, there are very al., 2008) and the ASLSM (Ariza-López et al., 2010). many statistical Positional Accuracy Assessment Methodolo- The use of industrial standards for quality control of spatial gies (PAAM), for example: National Map Accuracy Standard data is possible, for instance through the use of ISO 2859 (Sam- (NMAS) (USBB, 1947), Engineering Map Accuracy Standard pling procedures for inspection by attributes) for counting er- (EMAS) (ASCE, 1983), National Standard for Spatial Data Accu- rors (e.g., completeness, consistency, etc.) and ISO 3951 (Sam- racy (NSSDA) (FGDC, 1998), STANAG 2215 (STANAG, 2002), ASPRS pling procedures for inspection by variables) for continuous Accuracy Standards for Large-Scale Maps (ASLSM) (ASPRS, errors (e.g., positional errors). Interest in applying these stan- 1990) and the ASPRS Positional Accuracy Standards for Digital dards to spatial data has been increasing since the inclusion Geospatial Data (ASPRS, 2015). of some guides and examples of use previously in ISO 19114 The majority of PAAMs take as an underlying hypothesis (Geographic Information - Quality Evaluation Procedures) and the Gaussian distribution of positional errors, but several now in ISO 19157 (Geographic Information - Data Quality). For studies indicate that this hypothesis is not true. For instance, instance, ISO 2859 is applied widely throughout the world to in the case of GNSS-error distribution, the Rayleigh distribu- control attributes, e.g., for controlling the Land Parcel Identifi- tion (Logsdon, 1995); and for the case of geocoding errors, a cation System in Europe (Milenov et al., 2010), for geological log-normal distribution (Cayo and Talbot, 2003). For the case data in China (Xie et al., 2008) and for diverse spatial data of vertical errors in digital elevation models there are many quality controls in New Zealand by the National Topographic references (e.g., Bonin and Rousseaux 2005, Oksanen and Sar- Hydrographic Authority (NTHA, 2004). ISO 3951 has a much jakoski 2006) indicating that error distribution is not Normal, lower diffusion and application in the field of spatial data. A

F.J. Ariza-López is with the Universidad de Jaén, Photogrammetric Engineering & Remote Sensing Departamento de Ingeniería Cartográfica, Geodésica y Vol. 81, No. 8, August 2015, pp. 657–668. Fotogrametría, E-23071-Jaén, Spain ([email protected]). 0099-1112/15/657–668 J. Rodríguez-Avi is with the Universidad de Jaén, © 2015 American Society for Photogrammetry Departamento de Estadística e Investigación Operativa, and Remote Sensing E-23071-Jaén, Spain. doi: 10.14358/PERS.81.8.657

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08-15 August Peer Reviewed.indd 657 8/3/2015 12:57:01 PM pioneering reference related to positional controls comes from than the product being controlled (ISO 3951). the Norwegian Mapping Authority, which has been applying In this paper we propose the use of ISO 2859-1 and ISO ISO 3951 for many years (Statens Kartverk, 2007). 2859-2 for positional quality control. This standard has been The lot is a crucial element to both international standards; designed for general quality control when using counting a lot is a set of elements. Quality is fitness for purpose, thus of errors (defects or defective items), which in a spatial data the elements have to be selected taking in consideration the context means the counting of thematic and attribute errors, purpose of the product or control, with a utilitarian focus. inconsistency errors, completeness errors, etc. But positional Therefore, we first have to define what elements make up the errors are a continuous statistical variable. In order to pass population of interest (e.g., an image, an aerial photo, a build- from a continuous variable to a counting variable, we use the ing, a property, a well-defined point, a road, etc.). In some statistical method proposed by Ariza-López and Rodríguez- cases we can consider different elements, for instance, in the Avi (2014). This method offers the possibility to work with case of controlling a road data base: (a) each instance record- parametric and non-parametric models. ed in the dataset, (b) each section of a kilometer in length, We believe that statistical models based on distribution (c) each administrative section, and (d) each arc or segment functions (parametric models) are suitable for situations, such between two nodes of a topological dataset, etc. as those of the past, where few data were available, when data One main feature of a lot is its size. The size is the number collection was expensive, and when computational resources of items in the lot. Therefore, the size depends on the establish- were scarce. The current situation is different; we now talk ment of an element type. For the above example on a road, each about big data, and therefore we consider that statistical para- of the possible elements carries a different lot size. Another main metric models must give way to new methods based on work- feature of a lot is homogeneity. Lots should be homogeneous, ing directly with the populations of data (errors in this case). which means that the units in the lot should have a common The proposal we make in this paper goes in this direction. origin (e.g., the same contractor or surveying company, same After this introduction the paper is organized in five sec- methods, same processes, same machines, same operators, etc.). tions. The following section presents the main ideas about These two international standards (ISO 2859 and ISO 3951) statistical testing in acceptance sampling. The next is cen- are about acceptance sampling (applying sampling plans). Ac- tered on the International Standard ISO 2859-1 because it is ceptance sampling is a form of inspection that is applied to lots the basis of the proposal; the historical origins and its most of items before or after a process. By means of a sample, the important features are presented. The fourth section sum- purpose is to decide whether a lot satisfies a predefined quality marizes how to pass from quantitative values (measured standard (e.g., is the imagery acceptably geo-registered or not). positional errors) to counting values (positional defectives); Lots that satisfy these standards are passed or accepted, and this is achieved by the model proposed by Ariza-López and those that do not are rejected. Acceptance sampling is concerned Rodríguez-Avi (2014). An example of a positional control ap- with inspection and decision-making regarding products, one plied to a sequence of lots is given in the Example of Applica- of the oldest aspects of quality assurance. Here, it is important tion section. Finally, the conclusions are presented aiming to to highlight two aspects of acceptance sampling (Montgomery,Delivered highlightby Ingenta the main features of this proposal. 2001): (a) the purpose is to sentence lots, IP:accept 192.168.39.210 or reject, not On: Fri, 01 Oct 2021 01:31:31 to estimate quality (it is not anCopyright: estimation American process), and Society (b) the for Photogrammetry and Remote Sensing most effective use of acceptance sampling is not to inspect qual- User and Producer Risk in Acceptance Sampling ity in the product, but rather as an auditing tool to ensure that The purpose of quality control is to establish and maintain the output of a process conforms to the requirements. Sampling the conformity of products with design requirements, mainly schemes designated in these standards are applicable, but not expressed as standards or specifications. One of the most im- limited, to inspection of (ISO 2859-1): end items, components portant aspects of statistical quality control is the acceptance and raw materials, operations, materials in process, supplies in control of products. Acceptance sampling is useful in the case storage, maintenance operations, data or records, and admin- of some quality elements of spatial data (e.g., positional ac- istrative procedures. As can be understood, the application of curacy, completeness, etc.) in the following situations: ISO 2859 to control the positional quality, and in general to any • When the cost of 100 percent inspection is extremely high. quantitative aspect of spatial data, allows for a single common • When 100 percent inspection is not technologically framework for all the aspects, quantitative and qualitative, of feasible or would require so much time that production cartographic production (e.g., completion, consistence, theme, would be impacted. records and administrative procedures, services, etc.). This is of • When there are many items to be inspected and the great interest to companies and mapping agencies. inspection error rate is high. The use of these industrial standards for positional accu- • When there is potential product liability risk and con- racy control is possible. Since positional errors are a continu- tinuous monitoring of the product is necessary. ous variable, this can be achieved directly my means of the Positional controls can be understood as industrial ac- ISO 3951 parts 1 (single sampling plans for lot-by-lot inspec- ceptance processes based on sampling in order to control SDS tion for a single quality characteristic) and 2 (single sampling coming from suppliers (internal or external). When accep- plans for lot-by-lot inspection of independent quality charac- tance sampling is performed in industry a sampling plan is teristics). This idea was mentioned by ISO 19114 and now has required. Sampling plans are the key elements of acceptance been extended with new examples by ISO 19157 (see Annex sampling. Given a lot of size N, a sampling plan is no more F of ISO 19157 or Statens Kartverk (2007) for examples). than a plan that specifies the sample sizen and the accep- However, ISO 3951 has a major drawback: it is only applicable tance/rejection criteria (acceptance number c). Thus, follow- when the controlled errors follow a Normal distribution and ing the example of Montgomery (2001), if the lot size is N = is very sensitive to the presence of outliers. The assumption 10,000, the sampling plan n = 89, c = 2 means that from a lot of normality is critical because it is used in all sampling plans of size 10,000 a random sample of n = 89 units is inspected in order to calculate the defective fraction of a lot or process and the number of nonconforming (defective items or defects) (Montgomery, 2001). Small deviations from normality can d is observed. If the number of observed defectives d ≤c = 2, generate great differences in the calculated defective fraction, the lot will be accepted and if d >c the lot will be rejected. generating bad acceptance decisions. Furthermore, it is as- Many of the PAAMs (e.g., NMAS) are fully equivalent to the ac- sumed that measurement error is negligible, so it is necessary ceptance process described above; if the data meet accuracy that the accuracy of the control method be ten times better

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08-15 August Peer Reviewed.indd 658 8/3/2015 12:57:01 PM requirements this fact will be reflected in their metadata (re- randomness, actual OC curves are not step-shaped (Figure 1b). member the sentence “This map complies with NMAS”). But a Thus, an important feature of a sampling plan is how it dis- sampling plan establishes and makes explicit to both producer criminates between lots of high and low quality. The ability of and acquirer (user) clearer than a PAAM the conditions of the a sampling plan to discriminate between lots of high and low process: definition of lot, lot size, sampling method, sample quality is described by its OC curve. A sampling plan cannot size, and producer’s and user’s risk (described below), etc. provide perfect discrimination between good and bad lots be- The decision from an investigated sample on whether cause of the randomness of the sampling, so some low-quality or not a lot satisfies stated requirements can be carried out lots will inevitably be accepted and some high-quality lots through hypothesis testing (achieving a pass/fail decision). will inevitably be rejected. The degree to which a sampling A statistical hypothesis is a statement about a probability plan discriminates between good and bad lots is a function of distribution function or about the values of the parameters the steepness of its OC curve: the steeper the curve, the more of a probability distribution function (parametric case). In discriminating the sampling plan. statistical testing two alternatives are always considered: (a) A common approach to the design of an acceptance sam-

H0: the so-called Null Hypothesis, and (b) H1: the so-called pling plan is to require that the OC curve pass through two Alternative Hypothesis. For example, in relation to PAAMs, the designed points (Montgomery, 2001). Usually the two points EMAS applies two tests together (bias tests and variance tests) used are those corresponding to the user and producer risks. to each positional coordinate (X, Y, and Z). Assuming that a binomial approximation is appropriate, the Since sampling is a random procedure, two kinds of errors sample size n and the acceptance number c are the solution to

may be committed. If the H0 is rejected when it is true, then a the equations Equation 1 and Equation 2: type I error (α) has occurred. If the H0 is not rejected when it is false, then a type II error (β) has occurred. The Type I error is c n! α ppdn()−d called producer’s risk because it denotes the probability that 11−=∑ 11− d=0 dn!!()− d a good lot/product will be rejected, or the probability that a (1) process producing acceptable values of a particular quality c n! dn−d characteristic will be rejected as producing unsatisfactory β = pp22()1− ∑ dn!!− d ones. The Type II error is called user’s risk because it denotes d=0 () (2) the probability of accepting a lot/product of poor quality. Sometimes it is more convenient to work with the power of where α is the Size of Type I error (producer’s risk), β is the the test (Montgomery, 2001), which is the probability of cor- Size of Type II error (user’s risk), n is the Sampling size, c is the Acceptance number, d is the rectly rejecting H0 (Power = – β = P{reject H0}). In relation to the size n of the sample when testing statisti- Number of defectives in the sample,

cal hypotheses, the general procedure is to specify a value for p1 is the Probability of producer risk for the point, and p2 is α and then to design a test procedure so that a small value of the Probability of user risk for the point. β is obtained. Thus the producer’s risk is directly controlledDelivered by IngentaEquation 1 and Equation 2 are nonlinear and there is no or chosen by α; and the user’s risk isIP: generally 192.168.39.210 a function On:of n Fri, 01simple Oct direct2021 solution.01:31:31 Duncan (1986) gives a description of and is controlled indirectly.Copyright: The larger American the size Societyof the sample, for Photogrammetry some techniques and Remotefor solving Sensing this system of equations: the Lar- the smaller the user’s risk (Montgomery, 2001). In general, son binomial nomograph can be used as a graphical approach PAAMs provide a sample size in an attempt to ensure the pro- and Kapusta et al. (2011) propose a mathematical model for de- ducer risk (remember the sentence “at least 20 points”). This signing specific acceptance plans and provide a Matlab® code. suggested sample size is not related to the size and variability of the population and does not ensure that user risk will be kept under a specific threshold. Only recent proposals, such as ISO 2859-1 and ISO 2859-2 the new ASPRS positional accuracy standard (ASPRS, 2015), re- ISO 2859-1 late the required sample size to the size of the project area. In Acceptance sampling was first introduced during World War ISO 2859 the size n is related to lot size and established risks. II to determine whether to accept or reject a given batch of Acceptance plans are summarized by means of Operating military supplies. A cost (and time) saving method was needed Characteristic curves (OC curves) (Figure 1). OC curves plot which would not require 100 percent of elements to be tested, the probability or frequency f of accepting lots in the Y-axis while still maintaining an adequate quality level. Dodge (1969) versus the percent defectives πd on the X-axis. When it comes identifies July 1942 as the birth of this standard under the title to 100 percent control OC curves represent an ideal case and of Standard Inspection Procedures. It achieved the status of De- have a step shape (Figure 1a). Good lots are always accepted, partment of Defense Standard, JAN-STD-105, with the develop- and bad lots are always rejected. Because of the sampling ment by the Statistical Research Group of Columbia University

(a) (b) Figure 1. An operating characteristic curve: (a) operating curve for an ideal situation, and (b) operating curve for a sampling situation.

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08-15 August Peer Reviewed.indd 659 8/3/2015 12:57:28 PM and was superseded by MIL-STD-105A in 1950. The issuance of of lots, and for this reason in order to apply ISO 2859-1, we MIL-STD-105A started a sequence of revisions through to MIL- require, at least, a sequence of ten lots. If there are not ten lots STD-105D until its withdrawal in 1995. This standard was di- we have to analyze lots individually and apply ISO 2859-2, rectly adopted by the International Organization for Standard- detailed below. It has to be noted that the AQL is a property of ization as ISO 2859-1, is still in force, and the latest revision is the production process that is shown in the product present- from 1999. ISO 2859-1 is a well-known method and a widely ed as a lot, which means that the AQL is not a property of the extended standard in all industrial scopes (Juran and Godfrey, sampling plan. The AQL is simply a standard against which 1999). When comparing ISO 2859-1 with other Lot Acceptance to judge the lots, and it is hoped that the production process Sampling plans (e.g., Dodge-Romig and Philips), Banovac et will operate at a fallout level that is considerably better than al. (2012) conclude that the D-Romig plan protects the cus- the AQL (Montgomery, 2001). If the latter is not true, there will tomer and ISO 2859-1 protects the producer, while the Philips be a great rejection in the acceptance process. For this reason, plans are somewhere in between. Zero acceptance number ISO 2859 and the use of AQL are adequate when the aim is to sampling plans (Squeglia, 2008) is another option to take into maintain a quality level as a target. Based on the Technical account, but the basic ideas of the method are the same. There Report of the Geographical Survey Institute of Japan (JGSI, is some criticism of ISO 2859-1’s purpose and formulation and 2002), we can establish the following relationships: its appropriateness to address industrial needs (von Collani, • AQL = 0 % : No error is permitted (an error loses the 2004). But analysis of such criticism clearly shows that it is not value as product.). centered on the statistical basics of the standard but mainly • AQL =5 % : It is desirable that there is no error. on the unclear and confusing writing. The ISO 2859-1 does not • AQL = 10 % : Slight error is permitted. include a statistical demonstration of the standard but there • AQL = 20 % : Error is permitted to some extent. are many sources (e.g., Montgomery, 2001; Stephens, 2001). Recently, Banovac et al. (2012) analyzed the characteristics of Because random sampling cannot identify all lots that lot acceptance sampling plans by attributes and also elaborated contain more than AQL percentage of defectives, users recognize a mathematical base. At present, the interest in sampling plans that some lots that actually contain more defectives than AQL is related to economical aspects, and there are many recent will be accepted. However, there is an upper limit for the per- studies analyzing relations between quality, sample size (cost) centage of defective items that the user is willing to tolerate in and risk requirements in order to determine optimal sampling accepted lots. This value is known as the lot tolerance percent plans (Hsu and Hsu, 2012; Nikolaidis and Nenes, 2009; Stout defective (LTPD). Thus, users desire a quality equal at or better and Hardwick, 2005, Ferrell and Chhoker 2002). than the AQL, and are willing to live with poorer quality, but in A sampling plan is a lot-sentencing procedure in which a this case they do not accept any lots with a defective percentage decision about an entire lot of size N is taken from the result greater or equal to the LTPD. The LTPD should also be set based of a random sample of size n selected from the lot. The deci- on the criticality of the characteristic that is being inspected sion is taken by comparing the accounted number of defects (the more critical the characteristic, the smaller the LTPD). The probability that a bad lot containing defectives equal to that are present in the sample with an acceptance numberDelivered c. by Ingenta Thus, if the lot size is N a sampling plan is defined by n{ , c}. the LTPD will be accepted is known as the consumer’s risk, or IP: 192.168.39.210 On: Fri,beta 01 (β), Oct or the 2021 probability 01:31:31 of making a Type II error. On the other In relation to the size of the Copyright:lot, larger lots American are preferred Society over for Photogrammetry and Remote Sensing smaller ones because statistical sampling is more efficient hand, the probability that a good lot containing defectives equal when inspecting larger populations (lots) than smaller ones. It to the AQL will be rejected is known as the producer’s risk, or is also important that lots be conformable to handling systems alpha (α), or the probability of making a Type I error. The latter used by the producer and the consumer. means that the probability of accepting a good lot with quality better or equal to the AQL is 95 percent. This is because ISO 2859 Another important parameter for the use of ISO sampling series and ISO 3951 series are designed to have a producer’s risk plans (e.g., ISO 2859 and ISO 3951) is the AQL (acceptable of 5 percent and a consumer’s risk of 10 percent, but in the case quality level). The AQL represents the worst or poorest level of quality for the production process that the consumer would of ISO 2859 series because of the nature of counting variables, consider to be acceptable as a process average (Montgomery, these levels are not assured for all sampling plans. 2001). Users (buyers or consumers) are generally willing to The inspection level designates the relative amount of tolerate lots that contain small percentages of defects. This fig- inspection. There are seven inspection levels (S1, S2, S3, S4, I, II, and III) proposed by the standard. Inspection levels I, ure is known as the acceptable quality level. The AQL should be set based on the criticality of the characteristic that is being II, and III are for general use, and unless otherwise specified inspected (the more critical the characteristic, the smaller the level II shall be used. At each inspection level three different severities of in- AQL should be). For example, although there is no direct rela- spection can be applied: tionship an AQL of 5 percent could be related to class I of the • Normal inspection. The acceptance criterion has been ASLSM standard and an AQL of 10 percent with class II of that devised to secure the producer a high probability of same rule. Also, the AQL should be set so that the good incom- acceptance when the process average of the lot is better ing lots are of better quality than the AQL. Otherwise, the sup- plier will be overwhelmed with rejected lots and there may than the AQL. not be enough accepted lots for production to continue. It is • Tightened inspection. The acceptance criterion is tighter than that for the corresponding plan for normal considered that AQL provides a guide for the producer on the level of quality that needs to produced so that the acceptance inspection and is invoked when the inspection results criteria can be satisfied (sampling clause) most of the time for of a predetermined number of consecutive lots indicate that the process average might be poorer than the AQL. a given desired quality. Because AQL is expressed in percent- age values (1 % , 2 % ,...). the greater the value the poorer the • Reduced inspection. The sample size is smaller than that for the corresponding plan for normal inspection quality. Thus, an AQL of 1 % means that 1 percent of the lot and with an acceptance criterion that is comparable to contains defects while an AQL of 2 % means that 2 percent of the lot contains defects. It is obvious that a 2 percent defective that for the corresponding plan for normal inspection. lot is worse than a 1 percent defective lot. This is confusing, Reduced inspection is at the discretion of the respon- sible authority. and for this reason AQL was renamed, from “acceptable qual- ity level” to “acceptance quality limit.” It is the limit, and it is not really “acceptable.” Process average refers to a sequence

660 August 2015 PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING

08-15 August Peer Reviewed.indd 660 8/3/2015 12:57:28 PM Figure 2. Switching rules between severities of inspection (normal, tightened, and reduced) of ISO 2859-1 (ISO 1999).

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Figure 3. General view of the decisions when applying ISO 2859 parts 1 and 2.

There are switching rules that operate to require normal, ISO 2859-2 tightened, or reduced inspection. Figure 2 present the pos- ISO 2859-2 was conceived to supplement ISO 2859-1 for those sible circumstances and rules to apply. situations where the switching rules of ISO 2859-1 are not The choice of inspection level is independent of these applicable, such as when lots are isolated (the application of three severities of inspection. Thus, the inspection level that existing PAAMs can be considered in this way) or we have a has been specified shall be kept unchanged when switching sequence with fewer than ten lots (Figure 4). This standard between normal, tightened, and reduced inspection. Figure 3 provides sampling plans indexed by a series of preferred val- shows a general view of the process described above. ues of limiting quality (LQ). The LQ can be defined as a quality These schemes (inspection levels and severities) are level which for the purposes of sampling inspection is limited intended to be used for a continuing series of lots, that is, a to a low probability of acceptance. LQ does not provide a series long enough to allow the switching rules to be ap- reliable guide for the consumer as to the true quality of the ac- plied. These rules provide: (a) protection for the consumer cepted lots. Acceptance only indicates that the quality of the (by means of a switch to tightened inspection or discontinua- accepted lot is better than the LQ; it does not say how much tion of sampling inspection) should a deterioration in quality better. So AQL is not used directly, but there is a relation be- be detected, and (b) an incentive (at the discretion of the tween LQ and AQL. As stated in the standard, the limiting qual- responsible authority) to reduce inspection costs (by means ity should be chosen realistically at a minimum of three times of a switch to reduced inspection) should consistently good the desired quality (the AQL). The plans present in ISO 2859-2 quality be achieved. have been designed in accordance with single sampling plans ISO 2859-1 allows the use of double and multiple sampling under normal inspection from ISO 2859-1. The consumer’s risk plans. Single-sampling plans involve only a single sample, but is usually below 10 percent and is almost always below 13 the initial sample size is larger than the expected number of percent. ISO 2859-2 offers two procedures: Procedure A, which observations taken under double- or multiple-sampling plans. may be used when producer and user both wish to regard the This stems from the fact that a very good or very poor quality lot lot to be in isolation, and Procedure B, when the producer will often be accepted or rejected early in a multiple-sampling regards the lot as one of a continuing series but the consumer plan, and sampling can therefore be terminated. considers that the lot is received in isolation.

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08-15 August Peer Reviewed.indd 661 8/3/2015 12:57:30 PM Proposal of a Statistical Model for Spatial Data Adequate For ISO population of errors under control. BaMs can be derived for 2859-1 and ISO 2859-2 any dimension and for any kind of error measure based on In order to apply ISO 2859-1 or ISO 2859-2, we need a mecha- any distance definition. For example, Figure 4 shows the nism to convert a continuous variable, such as positional observed planimetric error distribution of the Base Model for errors (1D, 2D, 3D, or nD), into a counting variable and, at the a road dataset measured by means of the Hausdorff distance. same time, allow for the use of the framework proposed by The BaM plots the cumulative frequency distribution of this standard. Here, we propose applying the method pre- positional errors on the Y-axis and the size in meters of such sented by Ariza-López and Rodríguez-Avi (2014a), where a errors on the X-axis. This BaM comes from the control of the complete statistical formulation and demonstration of the road dataset of the product, which is called MTA10v (from behavior of this method is presented when working with “Mapa Topográfico de Andalucía escala 1:10,000 vectorial”). commonly assumed parametric models for positional errors The MTA10v is the official cartography of the Autonomous Re- (Normal, Chi2, and Gamma distributions). In Ariza-López and gion of Andalusia (Spain). It is a topographic-vector database Rodríguez-Avi (2014b), there are examples of its application which includes the geometric axis of all paved roads. The to line strings in 2D and 3D with non-parametric models. BaM is derived from a large control of more than 1,200 km of This method is based on two statistical models, a base the road axis included in this product and a GPS field survey. model and a binomial model, and has been developed to be Table 1 summarizes other main facts of the data sources of the applied to isolated lots. The first is the Base Model BaM( ), BaM (see Ariza-López et al., 2011 for more details). which can be any parametric or non-parametric model, The second model is the Binomial Model (BiM) and is but with the sole condition of adequately representing the applied over the former. By means of the BiM the control is

Table 1. Principal Characteristics of the Data Sources Originating from the Base Model Presented in Figure 4 Characteristics MTA10v Product subset GPS Field Survey Total length 1,210 Km 1,210 Km Total cases 1,254 road segments 1,254 road segments Mean length 965 m 965 m Standard deviation of the length 1671 m 1671 m Total points involved 28,823 points 122,467 points Mean points per road segment 22.98 points/road segment 97.66 points/road segment Mean distance between points 41.98 m 9.88 m Standard deviation of points distance 28.49 m 3.19 m Positional accuracy 10.65 m (95%) 1.41 m (95%)

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Figure 4. Base Model of 2D positional errors measured by Hausdorff distance of a line-string data set representing paved roads of the “Mapa Topográfico de Andalucía.”

Figure 5. General idea of the basis of the proposed method.

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08-15 August Peer Reviewed.indd 662 8/3/2015 12:57:31 PM developed on the population, not on a parameter of the popu- explanation has been developed considering that AQL and π lation. This idea is illustrated in Figure 5. are subrogates of Tol, but the same is also valid if Tol is con- The BaM represents the error behavior in the population, sidered a subrogate of AQL and π. so given a metric tolerance Tol (here being the maximum po- Figure 6 shows these ideas graphically. Let us consider a

sitional error in meters one is willing to accept), the percent- BaM represented by the solid curve labeled with Co; if Tol = age of error cases greater than the desired Tol is π. In other 13 m through the BaM we obtain that πo=12 percent, meaning words, Tol is the value corresponding to the 1-π percentile. In that 12 percent of position errors are greater than 13 m. If we

a control sample of size n, the fact that the error Ei in element want AQL to = πo =12 percent , it means that in order to assure i verifies Ei >Tol is defined as a fail event in a binomial sense a high average of accepted lots, the producer must be able to (we call it a “positional defective: in this case). The BiM con- supply data with some better positional quality, which means sists of counting the number of positional defectives, f. This that the actual BaM of the product must be a curve over the

test follows a Binomial B(n,π) distribution, and the probabil- Co; for example, the curve labeled with CbQ. Because πbQ = 7 ity of positional defectives is: percent <πo = 12 percent, this situation assures a high level of acceptance. But if the data supply is of a poorer quality than n  n k nk− demanded, the situation can be shown graphically by the PF ≥→fF Bn(),ππ = ()1− π (3) π π   ∑  k curve labeled CpQ. Because pQ = 17 percent > o = 12 percent kf= this situation assures rejections and the stopping of the sup- ply by means of applying ISO 2859-1 switching rules. where π = P[Ei >Tol] is the probability (BaM model) that a The above example is based on an empirical BaM (observed), point i has an error value, Ei, greater than the specified toler- but we can also work with parametric models. For example, ance, Tol; F is the random variable “number of defective for a specification of a positional error of 1 m in each coordi- points in a sample of size n”; f is the number of sampling nate (planimetric error of 2.4477 m for 95 percent confidence), points that are defective (that is to say, their ECM is greater we have Tol = 2.4477 m and π = 5 percent, so the AQL should than Tol; n is the sampling size; k is a summation index (we be slightly higher than 5 percent; and it means using tabulated sum all probabilities between k=w and k=n; P[A ≥ b|A(θ)] values of ISO 2859-1 that AQL = 6.5 percent, so that in this case, is the operator “probability”– that is to say, it indicates the the control specification is Tol( = 2.4477 m, AQL = 6.5 percent). value of the probability that the random variable A takes a Also, the presence of bias and outliers is related to the AQL value greater than or equal to b when the random variable by means of the BaM. The situation is very simple; in the case A follows a probability distribution  with parameter θ; and of bias there is a right shift of the entire BaM. This means that: B(n,π) is the binomial distribution with parameters n and π. • For the same initial Tol, the π value increases dramati- The null hypothesis is: cally and as a consequence AQL increases. • For a given AQL the Tol must be increased. H0: The SDS is adequate. Given a signification value α( ) (type In the case of outliers, there is down shift of the right tail I error or producer’s risk), it means that errors are distributedDelivered by Ingenta of the BaM. This right tail is the part of the model that accu- according to the BaM and only π percentIP: 192.168.39.210 of cases are greater On: Fri, 01 Oct 2021 01:31:31 than Tol versus mulates frequency coming from the largest error values. This Copyright: American Society for Photogrammetrydown shift ofand the Remote right tail Sensing is equivalent to a partial right shift H1: The SDS is not adequate. of the base model. In this case we must pay attention to the Now the question is how to use this model in conjunction position of the Tol with respect to the right tail of the model with ISO 2859-1. The answer is relatively easy because this affected by the outliers. If the Tol is not in the affected tail, method is based on counting fail events (we call them here the situation is similar to the prior one; but if the Tol is in the “positional defectives”) and counting of defectives/defects affected tail, the situation is totally equivalent to a general is what we need in order to apply ISO 2859-1. Given a ran- shift of the base model by bias. dom sample taken from a lot, the idea is to consider that an It is relevant to note that the presented method is also valid element is of good quality if its error is less than the toler- when the BaM is unknown. By taking the user’s preferred values ance and that an element is a positional defective if its error {Tol, AQL} and by counting the positional defectives in the is greater than the tolerance. But some additional explana- sample, this procedure can be applied protecting the user com- tions are needed. In positional controls, we are accustomed pletely. Here the problem is for the producer if the actual BaM of to working with root mean squared errors or with standard deviations and with confidence intervals in order to express the positional accuracy of a data set. But in ISO 2859-1 we must use another parameter in order to express the quality of a lot, the previously defined AQL. The preferences of the user are now expressed by the pair {Tol, AQL}: the desired metric Tol and the upper limit of the percentage of cases that are greater than this tolerance and which he is willing to tolerate in accepted lots. Nevertheless, there is a relation between both Tol and AQL and this relation comes from the BaM. The relation is that AQL=f(π), where π is the same as previously defined. Looking at the AQL values (…0.65 % , 1 % , 1.5 % , 2.5 % ...) proposed in ISO 2859-1, it is clear that the tolerance must be situated on the right side of the BaM model and it must be selected in order to allow only small percentages of positional defectives. For the producer to obtain a high acceptance of his lots, he must work with processes that, for the same Tol, ensure that the actual defective percentage is less than the AQL. This can be stated in another way: given a BaM and a tolerance value Figure 6. Relations between metric tolerance (Tol), AQL and π in a Tol, the AQL value should be chosen to be greater than π. This Base Model.

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08-15 August Peer Reviewed.indd 663 8/3/2015 12:57:33 PM the data set is bad in relation to the quality desires of the user. inspection level II. In this way, entering into Table 1 of ISO All the previous discussion in this section is about the rela- 2859-1 (shown here as Figure 7) with the mean lot size (91 tion between the AQL and the BaM, but the use of ISO 2859-2 is items) results in a sample size code letter F. The next step somewhat different because it is indexed by means of the LQ. is going to Table 2.A of ISO 2859-1, which is entitled “single As stated above, the AQL is not used directly but there is a rela- sampling plans for normal inspection (master table).” Enter- tion between LQ and AQL. The proposed relation is to consider ing into this table (here shown as Figure 8) with the sample

LQ = k × AQL, where k ≥3. In order to protect the user using this size code letter F and AQLc1 = 6.5 percent, gives a recom- LQ value, the standard reduces the probability of acceptance mended sample size n = 20,an acceptance value Ac = 3, and a significantly by means of increasing the sample size, for the rejection value Re = 4. As the ISO standard states, the sample same lot size, and maintaining or reducing the acceptance val- of control must be extracted by means of a random procedure. ue. In real terms, this means a greater demand for quality and This is not a problem because current GIS software has tools therefore a lower acceptability for the LQ in order to protect the for this purpose. But, prior to applying this sampling plan, we user. For example, let us consider a Binomial distribution B(n, must agree with the estimated performance of this sampling p) where n = 20 and p is the probability of success in each trial, plan. This can be analyzed by means of the corresponding OC

and two cases: (a) p1 = 10 % (=AQL); (b) p2 = 30 percent (=LQ=3 curve, which is offered by ISO 2859-1 in Table 10-*, where * × AQL). Table 2 shows the results for the same count of cases (f is the code letter, here * = F. Information of risk is offered by = {0,1,…,5}), and we can clearly observe that the probabilities ISO 2859-1 in Table 5.A for producer risks and in Table 6.A for of case two are always fewer than the probabilities of case one. user risks. A summary of the principal values of these tables of the International Standard is shown in Table 3 for AQL = 6.5 % and neighboring values of AQL. Example of Application This section shows two examples of the application of the Table 2. Example of Comparison of Acceptance Probabilities for the Same proposed method jointly with ISO 2859-1 and ISO 2859-2. One Count of Cases f when LQ = 3 × AQL of its major advantages is the possibility of working lot-by-lot when using ISO 2859-1, but remember that for isolated lots or Case Mean f=0 f=1 f=2 f=3 f=4 f=5 when the length of the sequence of lots is less than ten, we 1-B(n=20, n×p1=2 0.1215 0.3917 0.6769 0.8670 0.9568 0.9887 must apply ISO 2859-2. Here for both examples, the BaM being p1=0.1) used is that presented in Figure 4. 2-B(n =20, For the example of application of ISO 2859-1, let us n×p2=6 0.0008 0.0076 0.0354 0.1071 0.2375 0.4164 p2=0.3) consider that we wish to control a road data set supply in Andalucía (Spain), and that the data supply is organized by the map sheet distribution of the National Topographic Map Table 3. User and Producer Risks for Sampling Plan F Obtained from of Spain. In this way the mean accumulated number of road Tables of ISO 2859-1 segments (arcs between topological nodes) per map sheetDelivered is by Ingenta AQL=4% AQL=6.5% AQL=10% 91, with a mean accumulated length equal to 220 km. Because IP: 192.168.39.210 On: Fri,α (producer 01 Oct risk)2021 01:31:31 the BaM is derived for road segments,Copyright: we American must consider Society lots for of Photogrammetry and Remote4.74% Sensing 4.31% 1.66% road segments. In this case the mean lot size is 91 and for the (Table 5.A of ISO 2859-1) sake of simplicity in this example, we are going to consider β (consumer risk) 25.5% 30.4% 41.5% that the lot size is always 91. Now let us consider a supply (Table 6.A of ISO 2859-1) sequence of 15 lots. If we take Tol = 16.3 m the BaM (Figure 4) gives π = 5 %, so we can consider that AQLc1 = 6.5 % (we take If the sampling plan is considered satisfactory we can the next and greater ALQ value proposed by the standard). start its application. Now consider that control samples are We start the application of ISO 2859-1 by considering general realized lot by lot. For each lot (column labeled C1) Table 4

Figure 7. Figure of “Table 1 - Sample Size Code Letters” from ISO 2859-1 (ISO 1999).

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08-15 August Peer Reviewed.indd 664 8/3/2015 12:57:35 PM Table 4 - Example of application of ISO 2859-1 to a 2D positional accuracy control the line-string dataset represented by the Base Model of Figure 4 C1 C2 C3 C4 C5 C6 C7 Lot Data Statistical data Defectives AQL=4% AQL=6.5% AQL=10% µ σ 95% Id Cases of measured errors [m] greater than Tol =16.3 m N Ac-Re D Ac-Re D Ac-Re D [m] [m] percentile [m] 11.41; 1.63; 1; 5.97; 8.13; 9.59; 7.65; 6.39; 2.81; 5.29; 8.69; 1 6.43 3.29 11.24 0 2-3 A 3-4 A 5-6 A 3.05; 2.08; 4.99; 11.23; 9.1; 8.25; 3.25; 7.66; 10.43 5.78; 5.37; 11.61; 4.02; 8.32; 4.56; 6.93; 9.06; 10.28; 7.81; 2 7.77 3.20 14.06 0 2-3 A 3-4 A 5-6 A 9.07; 7.55; 2.78; 6.04; 14.06; 3.02; 14.1; 9.11; 6.72; 9.14 1.74; 6.91; 2.23; 5.73; 1.43; 6.74; 4.06; 7.38; 7.86; 8.25; 3.23; 3 6.57 3.37 10.44 0 2-3 A 3-4 A 5-6 A 7.12; 10.16; 3.8; 8.43; 10.18; 7.08; 8.99; 15.36; 4.73 7.39; 6.8; 7.02; 11.75; 5.89; 15.81; 3.77; 8.17; 11.43; 10.17; 4 8.77 3.72 14.63 0 2-3 A 3-4 A 5-6 A 1.43; 14.56; 3.25; 9.52; 9.05; 10.17; 7.88; 12.49; 12.38; 6.42 13.33; 7.78; 3.07; 3.8; 6.59; 7.82; 7.94; 4.88; 2.84; 3.2; 13.02; 5 6.18 3.39 13.04 0 2-3 A 3-4 A 5-6 A 8.92; 1.47; 3.67; 2.01; 4.71; 7.29; 4.96; 6.06; 10.28 3.61; 5.85; 5.78; 7.29; 9.44; 14.32; 6.91; 2.46; 6.61; 16.75; 6 8.66 5.14 16.92 3 2-3 R 3-4 A 5-6 A 5.1; 11.98; 10.18; 19.99; 4.98; 5.72; 7.08; 4.94; 5.15; 19.12 5.08; 6.75; 5.04; 7.31; 9.69; 7.47; 5.89; 7.78; 6.99; 10.41; 7 6.94 2.61 10.59 0 2-3 A 3-4 A 5-6 A 8.88; 7.29; 4.38; 4.52; 8.17; 6.26; 14.04; 3.08; 2.83; 6.84 18.61; 11.98; 3.07; 6.72; 10.59; 6.26; 12.23; 3.23; 5.08; 3.97; 8 8.16 5.25 18.65 3 2-3 R 3-4 A 5-6 A 19.28; 3.08; 5.21; 7.12; 9.51; 17.38; 4.22; 5.81; 3.08; 6.74 9.34; 4.05; 8.14; 1.59; 9.51; 14.17; 3.02; 10.34; 11.02; 7.5; 9 8.05 3.81 14.25 0 1-2 A 3-4 A 5-6 A 7.63; 6.03; 7.02; 6.8; 15.77; 2.92; 8.86; 10.51; 12.73; 4.09 4.98; 13.68; 10.79; 13.85; 21.92; 4.4; 1.33; 11.28; 4.32; 3.08; 10 8.10 5.39 14.26 1 1-2 A 3-4 A 5-6 A 5.43; 1.86; 12.97; 13.82; 8.88; 8.19; 3.55; 1.47; 9.47; 6.8 5.05; 7.71; 20.11; 2.98; 6.7; 3.32; 6.55; 5.64; 6.74; 5.64; 3.61; 11 7.16 4.55 18.69 2 1-2 R 3-4 A 5-6 A 6.56; 3.5; 8.71; 10.14; 18.61; 4.96; 6.86; 4.55; 5.33 8.73; 8.94; 4.41; 12.62; 3.77; 14.77; 13.11; 11.21; 7.41; 16.89; 12 Delivered8.83 by 4.09Ingenta 14.87 1 1-2 A 3-4 A 5-6 A 11.28; 8.43; 10.31; 5.94; 14.56; 4.74; IP:5.15; 192.168.39.210 6.07; 3.28; 5.06 On: Fri, 01 Oct 2021 01:31:31 12.12; 10.17; 8; 5.15; 2.64;Copyright: 5.63; 3.54; American 14.84; 4.13; Society 11.04; for Photogrammetry and Remote Sensing 13 7.11 4.73 14.98 1 1-2 A 3-4 A 5-6 A 17.57; 0.75; 13.33; 1.92; 5.4; 2.54; 8.53; 3.2; 3.8; 7.82 9.33; 12.97; 3.7; 5.59; 4.76; 4.78; 6.8; 9.87; 1.31; 9.78; 41.72; 14 8.50 8.78 19.18 2 1-2 R 3-4 A 5-6 A 4.13; 9.89; 2.46; 2.72; 3.63; 17.99; 6.31; 4.72; 7.64 14.97; 13.02; 1.88; 4.91; 2.36; 15.46; 2.78; 13.04; 4.65; 10.62; 15 8.47 5.22 15.62 1 1-2 A 3-4 A 5-6 A 2.36; 6.08; 9.75; 10.24; 12.49; 2.36; 18.57; 6.11; 12.71; 4.96

Figure 8. Figure of “Table 2-A Single Sampling Plan for Normal Inspection (Master Table)” from ISO 2859-1 (ISO 1999).

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08-15 August Peer Reviewed.indd 665 8/3/2015 12:57:38 PM shows the errors measured in each of the control samples five consecutive lots accepted under tightened inspection. (column labeled C2), a statistical summary (column labeled It has to be remarked that in the above three examples the C3), and the count of positional defectives (column labeled ISO 2859-1 assures that the producer risk is in the order of 5

C4). Now because we decided that AQLc1 = 6.5 we must pay percent, and user risk in the order of 10 percent by means of attention to the column labeled C6 in Table 4. Here the ac- applying the switching rules. ceptance and rejection values are presented: (“Ac-Re” in the For the example of application of ISO 2859-2 consider now Table 4 under column C6). Now we must think that lots arrive a supply of an isolated lot of characteristics similar to the one by one, and make a decision of acceptance or rejection previous case (same BaM - Figure 4, Tol = 16.3 m, AQL= 6.5 one by one, and apply the switching rules if necessary. This % , size belonging to the interval [91, 150]). This standard is decision is taken comparing the amount of positional defec- indexed by the LQ parameter and the recommendation of the tives with the acceptance value as explained before. So a standard is: “the limiting quality should be chosen realisti- decision is made, and the column labeled D under column cally at a minimum of three times the desired quality.” So we C6 shows “A” when accepted and “R” when the correspond- consider LQ = 20 % because it is the nearest to 3 × 6.5 % of ing lot is rejected. In this case any lot is rejected. In this case the preferred LQ value proposed in the standard. Entering into switching rules are not needed and the complete inspection is Table A of ISO 2859-2 (shown in Figure 9) with LQ = 20 % and run under the normal inspection severity. the lot size, results in the following sampling plan: n = 13, Ac Now let us suppose the same data supply but consider- = 0. The acceptation/rejection decision is taken by compar-

ing [Tol = 16.3 m; AQLc2 = 10 percent ]; because AQLc2> AQLc1> ing the amount of positional defectives with the acceptance π the exigency of positional quality is lesser here. Table 4 value. If we find 0 positional defectives in the sample the lot shows the results for this case. In the column labeled C7 the is accepted. But prior to applying this sampling plan, we must acceptance and rejections values and the final decision are agree with its estimated performance. This can be analyzed by presented. As can be observed, there is no rejection, and all means of the corresponding OC curve, which is offered by ISO lots are accepted. 2859-2 in Table B9 “Single sampling plans for limiting quality Finally let us suppose the same data supply but consider- 20.0 percent.” Additionally, information of risks (user’s and

ing [Tol = 16.3 m; AQLc3 = 4 %]; because AQLc3 <π the exigency producer’s) is offered by ISO 2859-2 in Table D1. The outputs of positional quality is greater here. Table 4 shows the results of the last table are: (a) 4.8 percent probability of acceptance for this case, so in the column labeled C5 the acceptance and at the limiting quality (20.0 percent) (the risk of the user); (b) rejection values and the final decision are presented. As can be 0 percent probability of rejection at 0 percent nonconforming observed, there two rejections (lots numbers 6 and 8). Because (the risk of the producer). If we consider that the sampling 2 of 5 or fewer consecutive lots are not accepted, the Interna- plan performance is satisfactory, we can start its application. tional Standard requires passing to tightened inspection. This Now we analyze the following sample: {5.87; 13.33; 2.05; means greater severity and is achieved by decreasing the Ac 9.82; 9.55; 6.13; 4.38; 4.55; 7.2; 7.71; 2.93; 8.88; 7.92}. In this and Re values (now Ac = 1, Re = 2) while maintaining sample sample there is no case greater than the tolerance, so d = 0 and size. Normal inspection is not re-instated because there Deliveredare not becauseby Ingenta d ≤Ac = 0 the conclusion is that the lot is accepted. IP: 192.168.39.210 On: Fri, 01 Oct 2021 01:31:31 Copyright: American Society for Photogrammetry and Remote Sensing

Figure 9. Figure of “Table A - Single Sampling Plans Indexed by Limiting Quality (LQ) (Procedure A)” from ISO 2859-2 (ISO 1985).

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08-15 August Peer Reviewed.indd 666 8/3/2015 12:57:40 PM Conclusions References This paper presents a method for positional quality control Abbas, L., P. Grussenmeyer, and P. Hottier, 1995. Contrôle de la valid for n-dimensional spatial data (e.g., 1D, 2D, 3D) based planimétrie d´une base de données vectorielles: Une nouvelle on acceptance sampling by means of ISO 2859-1. In order to méthode basée sur la distance de Hausdorff: El méthode du understand this method, we have presented the basics of ac- contrôle linéaire, Bulletin Societé Française de Photogrammétrie et Télédétection, 137:6–11. ceptance by sampling, including the user and producer risks, Ariza-López, F.J, and J. Rodríguez-Avi, 2014a. A statistical and the International Standard ISO 2859-1. The capability for model inspired by the National Map Accuracy Standard, dealing with positional errors within a standard originated for Photogrammetric Engineering & Remote Sensing, 80(3):271–281. non-quantitative values is achieved by the model proposed Ariza-López, F.J, and J. Rodríguez-Avi, 2014b. A method of positional by Ariza-López and Rodríguez-Avi (2014), and this model quality control testing for 2D and 3D line strings, Transactions in have been summarized. An example of its application to a GIS, doi: 10.111/tgis.12117. sequence of lots has been shown for a specific metric toler- Ariza-López, F.J., A.D. Atkinson-Gordo and J. Rodríguez-Avi, 2008. ance Tol and discussing the relation of the AQL parameter in Acceptance curves for the positional control of geographic data relation to the actual quality of the lots. bases, Journal of Surveying Engineering, 134(1):26–32. The main features of the proposed method can be summa- Ariza-López, F.J., A.D. Atkinson-Gordo, J. Rodríguez-Avi, and rized as follows: J.L. García-Balboa, 2010. Analysis of user and producer risk • It is not limited by the requirement of normality of when applying the ASPRS Standards for Large Scale Maps, positional errors, one of the major drawbacks of other Photogrammetric Engineering & Remote Sensing, 76(5):625–632. existing methods (e.g., The NSSDA or ISO 3951). Ariza-López, F.J, A.T. Mozas-Calvache, M.A. Ureña-Cámara, M.V. • No previous underlying assumption is needed for the Alba-Fernández, J.L. García-Balboa, J. Rodríguez-Avi, and J.J. base statistical model of errors, which means that the Ruiz-Lendínez, 2011. Influence of sample size on line-based application is universal (e.g., 1D, 2D, 3D, etc. with positional assessment methods for road data, ISPRS Journal of parametric or non-parametric error models). Photogrammetry and Remote Sensing, 66(5):708–719 • It is based on a statistical contrast process and requires ASCE, 1983. Map Uses, Scales and Accuracies for Engineering and smaller sample sizes than statistical accuracy estima- Associated Purposes, American Society of Civil Engineers, tion processes. Committee on Cartographic Surveying, Surveying and Mapping Division, New York. • This method introduces the user and producer risk into positional accuracy controls, which is a very desirable ASPRS, 1990. ASPRS Accuracy Standards for Large Scale Maps, Photo- grammetric Engineering & Remote Sensing, 56(7):1068–1070. circumstance because it gives transparency to trade relations. ASPRS, 2004. ASPRS Guidelines: Vertical Accuracy Reporting for Lidar Data, American Society for Photogrammetry and Remote • It can be applied to any type of positional and geo- Sensing, Bethesda, Maryland. metric controls (points, line-strings, etc.) because the ASPRS, 2015. ASPRS Positional Accuracy Standards for Digital statistical basis is very simple. Geospatial Data, November 2014, Photogrammetric Engineering • Positional quality is expressed in a very simpleDelivered and by Ingenta& Remote Sensing, Volume 81, No. 3, 53 p., URL: http:// understandable way by the percentageIP: 192.168.39.210 of positional On: Fri, 01 Octwww.asprs.org/Standards-Activities.html 2021 01:31:31 , doi:10.14358/ defectives combinedCopyright: with a metric American tolerance. Society for PhotogrammetryPERS.81.4.281.281. and Remote Sensing • Positional quality is expressed in the same way as other Banovac, E., D. Pavlovic, and N. Vistica, 2012. Analyzing the spatial data quality elements and can be expressed by characteristics of sampling by attributes, Recent Researches in the percentage of defectives or defects. Circuits, Systems, Multimedia and Automatic Control (V. Niola, • The same control framework is valid for other quality Z. Bojkovic, and M.I. Garcia-Planas, editors), WSEAS Press, aspects (e.g., thematic, completeness, logical consis- pp:158–163. tence), which is a very desirable circumstance in order Bonin, O., and F. Rousseaux, 2005. Digital terrain model computation to facilitate quality analysis, management, and reporting. from contour lines: How to derive quality information from • The method can be applied to lot by lot data supplies; artifact analysis, GeoInformatica, 9:253–268. this situation is very new for positional accuracy con- Cayo, M.R., and T.O. Talbot, 2003. Positional error in automated trols and is of great interest for supply contracts. geocoding of residential addresses, International Journal of • It is based on the application of a very well-known Health Geographics, 2:10. international standard (ISO 2859-1), and this can stimu- Dodge, H.F., 1969. Notes on the evolution of the acceptance sampling late the transfer of knowledge and best practices from plans, Part III, Journal of Quality Technology, 1(2):77–88. other sectors of the industry to the geomatic sector. Duncan, A.J., 1986. Quality Control and Industrial Statistics, Fifth edition, Irwin, Homewood, Illinois, 992 p. • The use of a BaM as described opens up the opportunity of applying all the possibilities that the ISO 2859 series Ferrell, W.G., and A. Chhoker, 2002. Design of economically optimal acceptance sampling plans with inspection error, Computers & of standards offers: isolated lot inspection (ISO 2859-2), Operations Research, 29(10):1283–1300. skip-lot sampling (ISO 2859-3), assessment of declared quality levels (2859-4), and sequential sampling plans FGDC, 1998. FGDC-STD-007: Geospatial Positioning Accuracy Standards, Part 3. National Standard for Spatial Data Accuracy, (ISO 2859-5). Federal Geographic Data Committee, Reston, Virginia. Goodchild, M.F., and G. Hunter, 1997. A simple positional accuracy measure for linear features, International Journal of Acknowledgments Geographical Information Science, 11(3):299–306. This work has been funded by the Ministry of Science and Hsu, L., and J. Hsu, 2012. Economic design of acceptance sampling Technology of Spain and the European Regional Develop- plans in two-stage supply chain, Advances in Decision ment Fund under Grant No. BIA2011-23217. The authors also Sciences, Volume 2012, Article ID 359082, 14 p., doi: acknowledge the Regional Government of Andalusia (Spain) 10.1155/2012/359082. for the financial support since 1997 for their research group ISO, 1985. ISO 2859-2:1985 Sampling Procedures for Inspection by (Ingeniería Cartográfica) with code PAIDI-TEP-164. Attributes - Part 2: Sampling Plans Indexed by Limiting Quality (LQ) for Isolated Lot Inspection, International Organization for Standardization, Geneva.

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