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2 GEOPRIV M. Thomson
3 Internet-Draft Mozilla
4 Updates: 3693,4119,5491 (if approved) J. Winterbottom
5 Intended status: Standards Track Unaffiliated
6 Expires: March 20, 2015 September 16, 2014
8 Representation of Uncertainty and Confidence in PIDF-LO
9 draft-ietf-geopriv-uncertainty-03
11 Abstract
13 The key concepts of uncertainty and confidence as they pertain to
14 location information are defined. Methods for the manipulation of
15 location estimates that include uncertainty information are outlined.
17 This draft normatively updates the definition of location information
18 representations defined in RFC 4119 and RFC 5491. It also deprecates
19 related terminology defined in RFC 3693.
21 Status of This Memo
23 This Internet-Draft is submitted in full conformance with the
24 provisions of BCP 78 and BCP 79.
26 Internet-Drafts are working documents of the Internet Engineering
27 Task Force (IETF). Note that other groups may also distribute
28 working documents as Internet-Drafts. The list of current Internet-
29 Drafts is at http://datatracker.ietf.org/drafts/current/.
31 Internet-Drafts are draft documents valid for a maximum of six months
32 and may be updated, replaced, or obsoleted by other documents at any
33 time. It is inappropriate to use Internet-Drafts as reference
34 material or to cite them other than as "work in progress."
36 This Internet-Draft will expire on March 20, 2015.
38 Copyright Notice
40 Copyright (c) 2014 IETF Trust and the persons identified as the
41 document authors. All rights reserved.
43 This document is subject to BCP 78 and the IETF Trust's Legal
44 Provisions Relating to IETF Documents
45 (http://trustee.ietf.org/license-info) in effect on the date of
46 publication of this document. Please review these documents
47 carefully, as they describe your rights and restrictions with respect
48 to this document. Code Components extracted from this document must
49 include Simplified BSD License text as described in Section 4.e of
50 the Trust Legal Provisions and are provided without warranty as
51 described in the Simplified BSD License.
53 Table of Contents
55 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 3
56 1.1. Conventions and Terminology . . . . . . . . . . . . . . . 3
57 2. A General Definition of Uncertainty . . . . . . . . . . . . . 4
58 2.1. Uncertainty as a Probability Distribution . . . . . . . . 5
59 2.2. Deprecation of the Terms Precision and Resolution . . . . 7
60 2.3. Accuracy as a Qualitative Concept . . . . . . . . . . . . 7
61 3. Uncertainty in Location . . . . . . . . . . . . . . . . . . . 8
62 3.1. Targets as Points in Space . . . . . . . . . . . . . . . 8
63 3.2. Representation of Uncertainty and Confidence in PIDF-LO . 9
64 3.3. Uncertainty and Confidence for Civic Addresses . . . . . 9
65 3.4. DHCP Location Configuration Information and Uncertainty . 10
66 4. Representation of Confidence in PIDF-LO . . . . . . . . . . . 10
67 4.1. The "confidence" Element . . . . . . . . . . . . . . . . 11
68 4.2. Generating Locations with Confidence . . . . . . . . . . 12
69 4.3. Consuming and Presenting Confidence . . . . . . . . . . . 12
70 5. Manipulation of Uncertainty . . . . . . . . . . . . . . . . . 13
71 5.1. Reduction of a Location Estimate to a Point . . . . . . . 13
72 5.1.1. Centroid Calculation . . . . . . . . . . . . . . . . 14
73 5.1.1.1. Arc-Band Centroid . . . . . . . . . . . . . . . . 14
74 5.1.1.2. Polygon Centroid . . . . . . . . . . . . . . . . 15
75 5.2. Conversion to Circle or Sphere . . . . . . . . . . . . . 17
76 5.3. Three-Dimensional to Two-Dimensional Conversion . . . . . 18
77 5.4. Increasing and Decreasing Uncertainty and Confidence . . 19
78 5.4.1. Rectangular Distributions . . . . . . . . . . . . . . 19
79 5.4.2. Normal Distributions . . . . . . . . . . . . . . . . 20
80 5.5. Determining Whether a Location is Within a Given Region . 20
81 5.5.1. Determining the Area of Overlap for Two Circles . . . 22
82 5.5.2. Determining the Area of Overlap for Two Polygons . . 23
83 6. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 23
84 6.1. Reduction to a Point or Circle . . . . . . . . . . . . . 23
85 6.2. Increasing and Decreasing Confidence . . . . . . . . . . 27
86 6.3. Matching Location Estimates to Regions of Interest . . . 27
87 6.4. PIDF-LO With Confidence Example . . . . . . . . . . . . . 28
88 7. Confidence Schema . . . . . . . . . . . . . . . . . . . . . . 28
89 8. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 30
90 8.1. URN Sub-Namespace Registration for
91 urn:ietf:params:xml:ns:geopriv:conf . . . . . . . . . . . 30
92 8.2. XML Schema Registration . . . . . . . . . . . . . . . . . 30
93 9. Security Considerations . . . . . . . . . . . . . . . . . . . 31
94 10. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . 31
95 11. References . . . . . . . . . . . . . . . . . . . . . . . . . 31
96 11.1. Normative References . . . . . . . . . . . . . . . . . . 31
97 11.2. Informative References . . . . . . . . . . . . . . . . . 32
98 Appendix A. Conversion Between Cartesian and Geodetic
99 Coordinates in WGS84 . . . . . . . . . . . . . . . . 33
100 Appendix B. Calculating the Upward Normal of a Polygon . . . . . 34
101 B.1. Checking that a Polygon Upward Normal Points Up . . . . . 35
102 Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 35
104 1. Introduction
106 Location information represents an estimation of the position of a
107 Target [RFC6280]. Under ideal circumstances, a location estimate
108 precisely reflects the actual location of the Target. For automated
109 systems that determine location, there are many factors that
110 introduce errors into the measurements that are used to determine
111 location estimates.
113 The process by which measurements are combined to generate a location
114 estimate is outside of the scope of work within the IETF. However,
115 the results of such a process are carried in IETF data formats and
116 protocols. This document outlines how uncertainty, and its
117 associated datum, confidence, are expressed and interpreted.
119 This document provides a common nomenclature for discussing
120 uncertainty and confidence as they relate to location information.
122 This document also provides guidance on how to manage location
123 information that includes uncertainty. Methods for expanding or
124 reducing uncertainty to obtain a required level of confidence are
125 described. Methods for determining the probability that a Target is
126 within a specified region based on its location estimate are
127 described. These methods are simplified by making certain
128 assumptions about the location estimate and are designed to be
129 applicable to location estimates in a relatively small geographic
130 area.
132 A confidence extension for the Presence Information Data Format -
133 Location Object (PIDF-LO) [RFC4119] is described.
135 This document describes methods that can be used in combination with
136 automatically determined location information. These are
137 statistically-based methods.
139 1.1. Conventions and Terminology
141 The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
142 "SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this
143 document are to be interpreted as described in [RFC2119].
145 This document assumes a basic understanding of the principles of
146 mathematics, particularly statistics and geometry.
148 Some terminology is borrowed from [RFC3693] and [RFC6280], in
149 particular Target.
151 Mathematical formulae are presented using the following notation: add
152 "+", subtract "-", multiply "*", divide "/", power "^" and absolute
153 value "|x|". Precedence is indicated using parentheses.
154 Mathematical functions are represented by common abbreviations:
155 square root "sqrt(x)", sine "sin(x)", cosine "cos(x)", inverse cosine
156 "acos(x)", tangent "tan(x)", inverse tangent "atan(x)", two-argument
157 inverse tangent "atan2(y,x)", error function "erf(x)", and inverse
158 error function "erfinv(x)".
160 2. A General Definition of Uncertainty
162 Uncertainty results from the limitations of measurement. In
163 measuring any observable quantity, errors from a range of sources
164 affect the result. Uncertainty is a quantification of what is known
165 about the observed quantity, either through the limitations of
166 measurement or through inherent variability of the quantity.
168 Uncertainty is most completely described by a probability
169 distribution. A probability distribution assigns a probability to
170 possible values for the quantity.
172 A probability distribution describing a measured quantity can be
173 arbitrarily complex and so it is desirable to find a simplified
174 model. One approach commonly taken is to reduce the probability
175 distribution to a confidence interval. Many alternative models are
176 used in other areas, but study of those is not the focus of this
177 document.
179 In addition to the central estimate of the observed quantity, a
180 confidence interval is succinctly described by two values: an error
181 range and a confidence. The error range describes an interval and
182 the confidence describes an estimated upper bound on the probability
183 that a "true" value is found within the extents defined by the error.
185 In the following example, a measurement result for a length is shown
186 as a nominal value with additional information on error range (0.0043
187 meters) and confidence (95%).
189 e.g. x = 1.00742 +/- 0.0043 meters at 95% confidence
191 This result indicates that the measurement indicates that the value
192 of "x" between 1.00312 and 1.01172 meters with 95% probability. No
193 other assertion is made: in particular, this does not assert that x
194 is 1.00742.
196 Uncertainty and confidence for location estimates can be derived in a
197 number of ways. This document does not attempt to enumerate the many
198 methods for determining uncertainty. [ISO.GUM] and [NIST.TN1297]
199 provide a set of general guidelines for determining and manipulating
200 measurement uncertainty. This document applies that general guidance
201 for consumers of location information.
203 As a statistical measure, values determined for uncertainty are
204 determined based on information in the aggregate, across numerous
205 individual estimates. An individual estimate might be determined to
206 be "correct" - by using a survey to validate the result, for example
207 - without invalidating the statistical assertion.
209 This understanding of estimates in the statistical sense explains why
210 asserting a confidence of 100%, which might seem intuitively correct,
211 is rarely advisable.
213 2.1. Uncertainty as a Probability Distribution
215 The Probability Density Function (PDF) that is described by
216 uncertainty indicates the probability that the "true" value lies at
217 any one point. The shape of the probability distribution can vary
218 depending on the method that is used to determine the result. The
219 two probability density functions most generally applicable to
220 location information are considered in this document:
222 o The normal PDF (also referred to as a Gaussian PDF) is used where
223 a large number of small random factors contribute to errors. The
224 value used for the error range in a normal PDF is related to the
225 standard deviation of the distribution.
227 o A rectangular PDF is used where the errors are known to be
228 consistent across a limited range. A rectangular PDF can occur
229 where a single error source, such as a rounding error, is
230 significantly larger than other errors. A rectangular PDF is
231 often described by the half-width of the distribution; that is,
232 half the width of the distribution.
234 Each of these probability density functions can be characterized by
235 its center point, or mean, and its width. For a normal distribution,
236 uncertainty and confidence together are related to the standard
237 deviation of the function (see Section 5.4). For a rectangular
238 distribution, the half-width of the distribution is used.
240 Figure 1 shows a normal and rectangular probability density function
241 with the mean (m) and standard deviation (s) labelled. The half-
242 width (h) of the rectangular distribution is also indicated.
244 ***** *** Normal PDF
245 ** : ** --- Rectangular PDF
246 ** : **
247 ** : **
248 .---------*---------------*---------.
249 | ** : ** |
250 | ** : ** |
251 | * <-- s -->: * |
252 | * : : : * |
253 | ** : ** |
254 | * : : : * |
255 | * : * |
256 |** : : : **|
257 ** : **
258 *** | : : : | ***
259 ***** | :<------ h ------>| *****
260 .****-------+.......:.........:.........:.......+-------*****.
261 m
263 Figure 1: Normal and Rectangular Probability Density Functions
265 For a given PDF, the value of the PDF describes the probability that
266 the "true" value is found at that point. Confidence for any given
267 interval is the total probability of the "true" value being in that
268 range, defined as the integral of the PDF over the interval.
270 The probability of the "true" value falling between two points is
271 found by finding the area under the curve between the points (that
272 is, the integral of the curve between the points). For any given
273 PDF, the area under the curve for the entire range from negative
274 infinity to positive infinity is 1 or (100%). Therefore, the
275 confidence over any interval of uncertainty is always less than
276 100%.
278 Figure 2 shows how confidence is determined for a normal
279 distribution. The area of the shaded region gives the confidence (c)
280 for the interval between "m-u" and "m+u".
282 *****
283 **:::::**
284 **:::::::::**
285 **:::::::::::**
286 *:::::::::::::::*
287 **:::::::::::::::**
288 **:::::::::::::::::**
289 *:::::::::::::::::::::*
290 *:::::::::::::::::::::::*
291 **:::::::::::::::::::::::**
292 *:::::::::::: c ::::::::::::*
293 *:::::::::::::::::::::::::::::*
294 **|:::::::::::::::::::::::::::::|**
295 ** |:::::::::::::::::::::::::::::| **
296 *** |:::::::::::::::::::::::::::::| ***
297 ***** |:::::::::::::::::::::::::::::| *****
298 .****..........!:::::::::::::::::::::::::::::!..........*****.
299 | | |
300 (m-u) m (m+u)
302 Figure 2: Confidence as the Integral of a PDF
304 In Section 5.4, methods are described for manipulating uncertainty if
305 the shape of the PDF is known.
307 2.2. Deprecation of the Terms Precision and Resolution
309 The terms _Precision_ and _Resolution_ are defined in RFC 3693
310 [RFC3693]. These definitions were intended to provide a common
311 nomenclature for discussing uncertainty; however, these particular
312 terms have many different uses in other fields and their definitions
313 are not sufficient to avoid confusion about their meaning. These
314 terms are unsuitable for use in relation to quantitative concepts
315 when discussing uncertainty and confidence in relation to location
316 information.
318 2.3. Accuracy as a Qualitative Concept
320 Uncertainty is a quantitative concept. The term _accuracy_ is useful
321 in describing, qualitatively, the general concepts of location
322 information. Accuracy is generally useful when describing
323 qualitative aspects of location estimates. Accuracy is not a
324 suitable term for use in a quantitative context.
326 For instance, it could be appropriate to say that a location estimate
327 with uncertainty "X" is more accurate than a location estimate with
328 uncertainty "2X" at the same confidence. It is not appropriate to
329 assign a number to "accuracy", nor is it appropriate to refer to any
330 component of uncertainty or confidence as "accuracy". That is, to
331 say that the "accuracy" for the first location estimate is "X" would
332 be an erroneous use of this term.
334 3. Uncertainty in Location
336 A _location estimate_ is the result of location determination. A
337 location estimate is subject to uncertainty like any other
338 observation. However, unlike a simple measure of a one dimensional
339 property like length, a location estimate is specified in two or
340 three dimensions.
342 Uncertainty in two or three dimensional locations can be described
343 using confidence intervals. The confidence interval for a location
344 estimate in two or three dimensional space is expressed as a subset
345 of that space. This document uses the term _region of uncertainty_
346 to refer to the area or volume that describes the confidence
347 interval.
349 Areas or volumes that describe regions of uncertainty can be formed
350 by the combination of two or three one-dimensional ranges, or more
351 complex shapes could be described (for example, the shapes in
352 [RFC5491]).
354 3.1. Targets as Points in Space
356 This document makes a simplifying assumption that the Target of the
357 PIDF-LO occupies just a single point in space. While this is clearly
358 false in virtually all scenarios with any practical application, it
359 is often a reasonable simplifying assumption to make.
361 To a large extent, whether this simplification is valid depends on
362 the size of the target relative to the size of the uncertainty
363 region. When locating a personal device using contemporary location
364 determination techniques, the space the device occupies relative to
365 the uncertainty is proportionally quite small. Even where that
366 device is used as a proxy for a person, the proportions change
367 little.
369 This assumption is less useful as uncertainty becomes small relative
370 to the size of the Target of the PIDF-LO (or conversely, as
371 uncertainty becomes small relative to the Target). For instance,
372 describing the location of a football stadium or small country would
373 include a region of uncertainty that is infinitesimally larger than
374 the Target itself. In these cases, much of the guidance in this
375 document is not applicable. Indeed, as the accuracy of location
376 determination technology improves, it could be that the advice this
377 document contains becomes less relevant by the same measure.
379 3.2. Representation of Uncertainty and Confidence in PIDF-LO
381 A set of shapes suitable for the expression of uncertainty in
382 location estimates in the Presence Information Data Format - Location
383 Object (PIDF-LO) are described in [GeoShape]. These shapes are the
384 recommended form for the representation of uncertainty in PIDF-LO
385 [RFC4119] documents.
387 The PIDF-LO can contain uncertainty, but does not include an
388 indication of confidence. [RFC5491] defines a fixed value of 95%.
389 Similarly, the PIDF-LO format does not provide an indication of the
390 shape of the PDF. Section 4 defines elements to convey this
391 information in PIDF-LO.
393 Absence of uncertainty information in a PIDF-LO document does not
394 indicate that there is no uncertainty in the location estimate.
395 Uncertainty might not have been calculated for the estimate, or it
396 may be withheld for privacy purposes.
398 If the Point shape is used, confidence and uncertainty are unknown; a
399 receiver can either assume a confidence of 0% or infinite
400 uncertainty. The same principle applies on the altitude axis for
401 two-dimension shapes like the Circle.
403 3.3. Uncertainty and Confidence for Civic Addresses
405 Automatically determined civic addresses [RFC5139] inherently include
406 uncertainty, based on the area of the most precise element that is
407 specified. In this case, uncertainty is effectively described by the
408 presence or absence of elements. To the recipient of location
409 information, elements that are not present are uncertain.
411 To apply the concept of uncertainty to civic addresses, it is helpful
412 to unify the conceptual models of civic address with geodetic
413 location information. This is particularly useful when considering
414 civic addresses that are determined using reverse geocoding (that is,
415 the process of translating geodetic information into civic
416 addresses).
418 In the unified view, a civic address defines a series of (sometimes
419 non-orthogonal) spatial partitions. The first is the implicit
420 partition that identifies the surface of the earth and the space near
421 the surface. The second is the country. Each label that is included
422 in a civic address provides information about a different set of
423 spatial partitions. Some partitions require slight adjustments from
424 a standard interpretation: for instance, a road includes all
425 properties that adjoin the street. Each label might need to be
426 interpreted with other values to provide context.
428 As a value at each level is interpreted, one or more spatial
429 partitions at that level are selected, and all other partitions of
430 that type are excluded. For non-orthogonal partitions, only the
431 portion of the partition that fits within the existing space is
432 selected. This is what distinguishes King Street in Sydney from King
433 Street in Melbourne. Each defined element selects a partition of
434 space. The resulting location is the intersection of all selected
435 spaces.
437 The resulting spatial partition can be considered as a region of
438 uncertainty.
440 Note: This view is a potential perspective on the process of geo-
441 coding - the translation of a civic address to a geodetic
442 location.
444 Uncertainty in civic addresses can be increased by removing elements.
445 This does not increase confidence unless additional information is
446 used. Similarly, arbitrarily increasing uncertainty in a geodetic
447 location does not increase confidence.
449 3.4. DHCP Location Configuration Information and Uncertainty
451 Location information is often measured in two or three dimensions;
452 expressions of uncertainty in one dimension only are rare. The
453 "resolution" parameters in [RFC6225] provide an indication of how
454 many bits of a number are valid, which could be interpreted as an
455 expression of uncertainty in one dimension.
457 [RFC6225] defines a means for representing uncertainty, but a value
458 for confidence is not specified. A default value of 95% confidence
459 should be assumed for the combination of the uncertainty on each
460 axis. This is consistent with the transformation of those forms into
461 the uncertainty representations from [RFC5491]. That is, the
462 confidence of the resultant rectangular polygon or prism is assumed
463 to be 95%.
465 4. Representation of Confidence in PIDF-LO
467 On the whole, a fixed definition for confidence is preferable,
468 primarily because it ensures consistency between implementations.
469 Location generators that are aware of this constraint can generate
470 location information at the required confidence. Location recipients
471 are able to make sensible assumptions about the quality of the
472 information that they receive.
474 In some circumstances - particularly with pre-existing systems -
475 location generators might unable to provide location information with
476 consistent confidence. Existing systems sometimes specify confidence
477 at 38%, 67% or 90%. Existing forms of expressing location
478 information, such as that defined in [TS-3GPP-23_032], contain
479 elements that express the confidence in the result.
481 The addition of a confidence element provides information that was
482 previously unavailable to recipients of location information.
483 Without this information, a location server or generator that has
484 access to location information with a confidence lower than 95% has
485 two options:
487 o The location server can scale regions of uncertainty in an attempt
488 to acheive 95% confidence. This scaling process significantly
489 degrades the quality of the information, because the location
490 server might not have the necessary information to scale
491 appropriately; the location server is forced to make assumptions
492 that are likely to result in either an overly conservative
493 estimate with high uncertainty or a overestimate of confidence.
495 o The location server can ignore the confidence entirely, which
496 results in giving the recipient a false impression of its quality.
498 Both of these choices degrade the quality of the information
499 provided.
501 The addition of a confidence element avoids this problem entirely if
502 a location recipient supports and understands the element. A
503 recipient that does not understand - and hence ignores - the
504 confidence element is in no worse a position than if the location
505 server ignored confidence.
507 4.1. The "confidence" Element
509 The confidence element MAY be added to the "location-info" element of
510 the Presence Information Data Format - Location Object (PIDF-LO)
511 [RFC4119] document. This element expresses the confidence in the
512 associated location information as a percentage. A special "unknown"
513 value is reserved to indicate that confidence is supported, but not
514 known to the Location Generator.
516 The confidence element optionally includes an attribute that
517 indicates the shape of the probability density function (PDF) of the
518 associated region of uncertainty. Three values are possible:
519 unknown, normal and rectangular.
521 Indicating a particular PDF only indicates that the distribution
522 approximately fits the given shape based on the methods used to
523 generate the location information. The PDF is normal if there are a
524 large number of small, independent sources of error; rectangular if
525 all points within the area have roughly equal probability of being
526 the actual location of the Target; otherwise, the PDF MUST either be
527 set to unknown or omitted.
529 If a PIDF-LO does not include the confidence element, the confidence
530 of the location estimate is 95%, as defined in [RFC5491].
532 A Point shape does not have uncertainty (or it has infinite
533 uncertainty), so confidence is meaningless for a point; therefore,
534 this element MUST be omitted if only a point is provided.
536 4.2. Generating Locations with Confidence
538 Location generators SHOULD attempt to ensure that confidence is equal
539 in each dimension when generating location information. This
540 restriction, while not always practical, allows for more accurate
541 scaling, if scaling is necessary.
543 A confidence element MUST be included with all location information
544 that includes uncertainty (that is, all forms other than a point). A
545 special "unknown" MAY be used if confidence is not known.
547 4.3. Consuming and Presenting Confidence
549 The inclusion of confidence that is anything other than 95% presents
550 a potentially difficult usability problem for applications that use
551 location information. Effectively communicating the probability that
552 a location is incorrect to a user can be difficult.
554 It is inadvisable to simply display locations of any confidence, or
555 to display confidence in a separate or non-obvious fashion. If
556 locations with different confidence levels are displayed such that
557 the distinction is subtle or easy to overlook - such as using fine
558 graduations of color or transparency for graphical uncertainty
559 regions, or displaying uncertainty graphically, but providing
560 confidence as supplementary text - a user could fail to notice a
561 difference in the quality of the location information that might be
562 significant.
564 Depending on the circumstances, different ways of handling confidence
565 might be appropriate. Section 5 describes techniques that could be
566 appropriate for consumers that use automated processing.
568 Providing that the full implications of any choice for the
569 application are understood, some amount of automated processing could
570 be appropriate. In a simple example, applications could choose to
571 discard or suppress the display of location information if confidence
572 does not meet a pre-determined threshold.
574 In settings where there is an opportunity for user training, some of
575 these problems might be mitigated by defining different operational
576 procedures for handling location information at different confidence
577 levels.
579 5. Manipulation of Uncertainty
581 This section deals with manipulation of location information that
582 contains uncertainty.
584 The following rules generally apply when manipulating location
585 information:
587 o Where calculations are performed on coordinate information, these
588 should be performed in Cartesian space and the results converted
589 back to latitude, longitude and altitude. A method for converting
590 to and from Cartesian coordinates is included in Appendix A.
592 While some approximation methods are useful in simplifying
593 calculations, treating latitude and longitude as Cartesian axes
594 is never advisable. The two axes are not orthogonal. Errors
595 can arise from the curvature of the earth and from the
596 convergence of longitude lines.
598 o Normal rounding rules do not apply when rounding uncertainty.
599 When rounding, the region of uncertainty always increases (that
600 is, errors are rounded up) and confidence is always rounded down
601 (see [NIST.TN1297]). This means that any manipulation of
602 uncertainty is a non-reversible operation; each manipulation can
603 result in the loss of some information.
605 5.1. Reduction of a Location Estimate to a Point
607 Manipulating location estimates that include uncertainty information
608 requires additional complexity in systems. In some cases, systems
609 only operate on definitive values, that is, a single point.
611 This section describes algorithms for reducing location estimates to
612 a simple form without uncertainty information. Having a consistent
613 means for reducing location estimates allows for interaction between
614 applications that are able to use uncertainty information and those
615 that cannot.
617 Note: Reduction of a location estimate to a point constitutes a
618 reduction in information. Removing uncertainty information can
619 degrade results in some applications. Also, there is a natural
620 tendency to misinterpret a point location as representing a
621 location without uncertainty. This could lead to more serious
622 errors. Therefore, these algorithms should only be applied where
623 necessary.
625 Several different approaches can be taken when reducing a location
626 estimate to a point. Different methods each make a set of
627 assumptions about the properties of the PDF and the selected point;
628 no one method is more "correct" than any other. For any given region
629 of uncertainty, selecting an arbitrary point within the area could be
630 considered valid; however, given the aforementioned problems with
631 point locations, a more rigorous approach is appropriate.
633 Given a result with a known distribution, selecting the point within
634 the area that has the highest probability is a more rigorous method.
635 Alternatively, a point could be selected that minimizes the overall
636 error; that is, it minimizes the expected value of the difference
637 between the selected point and the "true" value.
639 If a rectangular distribution is assumed, the centroid of the area or
640 volume minimizes the overall error. Minimizing the error for a
641 normal distribution is mathematically complex. Therefore, this
642 document opts to select the centroid of the region of uncertainty
643 when selecting a point.
645 5.1.1. Centroid Calculation
647 For regular shapes, such as Circle, Sphere, Ellipse and Ellipsoid,
648 this approach equates to the center point of the region. For regions
649 of uncertainty that are expressed as regular Polygons and Prisms the
650 center point is also the most appropriate selection.
652 For the Arc-Band shape and non-regular Polygons and Prisms, selecting
653 the centroid of the area or volume minimizes the overall error. This
654 assumes that the PDF is rectangular.
656 Note: The centroid of a concave Polygon or Arc-Band shape is not
657 necessarily within the region of uncertainty.
659 5.1.1.1. Arc-Band Centroid
661 The centroid of the Arc-Band shape is found along a line that bisects
662 the arc. The centroid can be found at the following distance from
663 the starting point of the arc-band (assuming an arc-band with an
664 inner radius of "r", outer radius "R", start angle "a", and opening
665 angle "o"):
667 d = 4 * sin(o/2) * (R*R + R*r + r*r) / (3*o*(R + r))
669 This point can be found along the line that bisects the arc; that is,
670 the line at an angle of "a + (o/2)".
672 5.1.1.2. Polygon Centroid
674 Calculating a centroid for the Polygon and Prism shapes is more
675 complex. Polygons that are specified using geodetic coordinates are
676 not necessarily coplanar. For Polygons that are specified without an
677 altitude, choose a value for altitude before attempting this process;
678 an altitude of 0 is acceptable.
680 The method described in this section is simplified by assuming
681 that the surface of the earth is locally flat. This method
682 degrades as polygons become larger; see [GeoShape] for
683 recommendations on polygon size.
685 The polygon is translated to a new coordinate system that has an x-y
686 plane roughly parallel to the polygon. This enables the elimination
687 of z-axis values and calculating a centroid can be done using only x
688 and y coordinates. This requires that the upward normal for the
689 polygon is known.
691 To translate the polygon coordinates, apply the process described in
692 Appendix B to find the normal vector "N = [Nx,Ny,Nz]". This value
693 should be made a unit vector to ensure that the transformation matrix
694 is a special orthogonal matrix. From this vector, select two vectors
695 that are perpendicular to this vector and combine these into a
696 transformation matrix.
698 If "Nx" and "Ny" are non-zero, the matrices in Figure 3 can be used,
699 given "p = sqrt(Nx^2 + Ny^2)". More transformations are provided
700 later in this section for cases where "Nx" or "Ny" are zero.
702 [ -Ny/p Nx/p 0 ] [ -Ny/p -Nx*Nz/p Nx ]
703 T = [ -Nx*Nz/p -Ny*Nz/p p ] T' = [ Nx/p -Ny*Nz/p Ny ]
704 [ Nx Ny Nz ] [ 0 p Nz ]
705 (Transform) (Reverse Transform)
707 Figure 3: Recommended Transformation Matrices
709 To apply a transform to each point in the polygon, form a matrix from
710 the ECEF coordinates and use matrix multiplication to determine the
711 translated coordinates.
713 [ -Ny/p Nx/p 0 ] [ x[1] x[2] x[3] ... x[n] ]
714 [ -Nx*Nz/p -Ny*Nz/p p ] * [ y[1] y[2] y[3] ... y[n] ]
715 [ Nx Ny Nz ] [ z[1] z[2] z[3] ... z[n] ]
717 [ x'[1] x'[2] x'[3] ... x'[n] ]
718 = [ y'[1] y'[2] y'[3] ... y'[n] ]
719 [ z'[1] z'[2] z'[3] ... z'[n] ]
721 Figure 4: Transformation
723 Alternatively, direct multiplication can be used to achieve the same
724 result:
726 x'[i] = -Ny * x[i] / p + Nx * y[i] / p
728 y'[i] = -Nx * Nz * x[i] / p - Ny * Nz * y[i] / p + p * z[i]
730 z'[i] = Nx * x[i] + Ny * y[i] + Nz * z[i]
732 The first and second rows of this matrix ("x'" and "y'") contain the
733 values that are used to calculate the centroid of the polygon. To
734 find the centroid of this polygon, first find the area using:
736 A = sum from i=1..n of (x'[i]*y'[i+1]-x'[i+1]*y'[i]) / 2
738 For these formulae, treat each set of coordinates as circular, that
739 is "x'[0] == x'[n]" and "x'[n+1] == x'[1]". Based on the area, the
740 centroid along each axis can be determined by:
742 Cx' = sum (x'[i]+x'[i+1]) * (x'[i]*y'[i+1]-x'[i+1]*y'[i]) / (6*A)
744 Cy' = sum (y'[i]+y'[i+1]) * (x'[i]*y'[i+1]-x'[i+1]*y'[i]) / (6*A)
746 Note: The formula for the area of a polygon will return a negative
747 value if the polygon is specified in clockwise direction. This
748 can be used to determine the orientation of the polygon.
750 The third row contains a distance from a plane parallel to the
751 polygon. If the polygon is coplanar, then the values for "z'" are
752 identical; however, the constraints recommended in [RFC5491] mean
753 that this is rarely the case. To determine "Cz'", average these
754 values:
756 Cz' = sum z'[i] / n
758 Once the centroid is known in the transformed coordinates, these can
759 be transformed back to the original coordinate system. The reverse
760 transformation is shown in Figure 5.
762 [ -Ny/p -Nx*Nz/p Nx ] [ Cx' ] [ Cx ]
763 [ Nx/p -Ny*Nz/p Ny ] * [ Cy' ] = [ Cy ]
764 [ 0 p Nz ] [ sum of z'[i] / n ] [ Cz ]
766 Figure 5: Reverse Transformation
768 The reverse transformation can be applied directly as follows:
770 Cx = -Ny * Cx' / p - Nx * Nz * Cy' / p + Nx * Cz'
772 Cy = Nx * Cx' / p - Ny * Nz * Cy' / p + Ny * Cz'
774 Cz = p * Cy' + Nz * Cz'
776 The ECEF value "[Cx,Cy,Cz]" can then be converted back to geodetic
777 coordinates. Given a polygon that is defined with no altitude or
778 equal altitudes for each point, the altitude of the result can either
779 be ignored or reset after converting back to a geodetic value.
781 The centroid of the Prism shape is found by finding the centroid of
782 the base polygon and raising the point by half the height of the
783 prism. This can be added to altitude of the final result;
784 alternatively, this can be added to "Cz'", which ensures that
785 negative height is correctly applied to polygons that are defined in
786 a "clockwise" direction.
788 The recommended transforms only apply if "Nx" and "Ny" are non-zero.
789 If the normal vector is "[0,0,1]" (that is, along the z-axis), then
790 no transform is necessary. Similarly, if the normal vector is
791 "[0,1,0]" or "[1,0,0]", avoid the transformation and use the x and z
792 coordinates or y and z coordinates (respectively) in the centroid
793 calculation phase. If either "Nx" or "Ny" are zero, the alternative
794 transform matrices in Figure 6 can be used. The reverse transform is
795 the transpose of this matrix.
797 if Nx == 0: | if Ny == 0:
798 [ 0 -Nz Ny ] [ 0 1 0 ] | [ -Nz 0 Nx ]
799 T = [ 1 0 0 ] T' = [ -Nz 0 Ny ] | T = T' = [ 0 1 0 ]
800 [ 0 Ny Nz ] [ Ny 0 Nz ] | [ Nx 0 Nz ]
802 Figure 6: Alternative Transformation Matrices
804 5.2. Conversion to Circle or Sphere
806 The Circle or Sphere are simple shapes that suit a range of
807 applications. A circle or sphere contains fewer units of data to
808 manipulate, which simplifies operations on location estimates.
810 The simplest method for converting a location estimate to a Circle or
811 Sphere shape is to determine the centroid and then find the longest
812 distance to any point in the region of uncertainty to that point.
813 This distance can be determined based on the shape type:
815 Circle/Sphere: No conversion necessary.
817 Ellipse/Ellipsoid: The greater of either semi-major axis or altitude
818 uncertainty.
820 Polygon/Prism: The distance to the furthest vertex of the polygon
821 (for a Prism, it is only necessary to check points on the base).
823 Arc-Band: The furthest length from the centroid to the points where
824 the inner and outer arc end. This distance can be calculated by
825 finding the larger of the two following formulae:
827 X = sqrt( d*d + R*R - 2*d*R*cos(o/2) )
829 x = sqrt( d*d + r*r - 2*d*r*cos(o/2) )
831 Once the Circle or Sphere shape is found, the associated confidence
832 can be increased if the result is known to follow a normal
833 distribution. However, this is a complicated process and provides
834 limited benefit. In many cases it also violates the constraint that
835 confidence in each dimension be the same. Confidence should be
836 unchanged when performing this conversion.
838 Two dimensional shapes are converted to a Circle; three dimensional
839 shapes are converted to a Sphere.
841 5.3. Three-Dimensional to Two-Dimensional Conversion
843 A three-dimensional shape can be easily converted to a two-
844 dimensional shape by removing the altitude component. A sphere
845 becomes a circle; a prism becomes a polygon; an ellipsoid becomes an
846 ellipse. Each conversion is simple, requiring only the removal of
847 those elements relating to altitude.
849 The altitude is unspecified for a two-dimensional shape and therefore
850 has unlimited uncertainty along the vertical axis. The confidence
851 for the two-dimensional shape is thus higher than the three-
852 dimensional shape. Assuming equal confidence on each axis, the
853 confidence of the circle can be increased using the following
854 approximate formula:
856 C[2d] >= C[3d] ^ (2/3)
858 "C[2d]" is the confidence of the two-dimensional shape and "C[3d]" is
859 the confidence of the three-dimensional shape. For example, a Sphere
860 with a confidence of 95% can be simplified to a Circle of equal
861 radius with confidence of 96.6%.
863 5.4. Increasing and Decreasing Uncertainty and Confidence
865 The combination of uncertainty and confidence provide a great deal of
866 information about the nature of the data that is being measured. If
867 uncertainty, confidence and PDF are known, certain information can be
868 extrapolated. In particular, the uncertainty can be scaled to meet a
869 desired confidence or the confidence for a particular region of
870 uncertainty can be found.
872 In general, confidence decreases as the region of uncertainty
873 decreases in size and confidence increases as the region of
874 uncertainty increases in size. However, this depends on the PDF;
875 expanding the region of uncertainty for a rectangular distribution
876 has no effect on confidence without additional information. If the
877 region of uncertainty is increased during the process of obfuscation
878 (see [RFC6772]), then the confidence cannot be increased.
880 A region of uncertainty that is reduced in size always has a lower
881 confidence.
883 A region of uncertainty that has an unknown PDF shape cannot be
884 reduced in size reliably. The region of uncertainty can be expanded,
885 but only if confidence is not increased.
887 This section makes the simplifying assumption that location
888 information is symmetrically and evenly distributed in each
889 dimension. This is not necessarily true in practice. If better
890 information is available, alternative methods might produce better
891 results.
893 5.4.1. Rectangular Distributions
895 Uncertainty that follows a rectangular distribution can only be
896 decreased in size. Increasing uncertainty has no value, since it has
897 no effect on confidence. Since the PDF is constant over the region
898 of uncertainty, the resulting confidence is determined by the
899 following formula:
901 Cr = Co * Ur / Uo
903 Where "Uo" and "Ur" are the sizes of the original and reduced regions
904 of uncertainty (either the area or the volume of the region); "Co"
905 and "Cr" are the confidence values associated with each region.
907 Information is lost by decreasing the region of uncertainty for a
908 rectangular distribution. Once reduced in size, the uncertainty
909 region cannot subsequently be increased in size.
911 5.4.2. Normal Distributions
913 Uncertainty and confidence can be both increased and decreased for a
914 normal distribution. This calculation depends on the number of
915 dimensions of the uncertainty region.
917 For a normal distribution, uncertainty and confidence are related to
918 the standard deviation of the function. The following function
919 defines the relationship between standard deviation, uncertainty, and
920 confidence along a single axis:
922 S[x] = U[x] / ( sqrt(2) * erfinv(C[x]) )
924 Where "S[x]" is the standard deviation, "U[x]" is the uncertainty,
925 and "C[x]" is the confidence along a single axis. "erfinv" is the
926 inverse error function.
928 Scaling a normal distribution in two dimensions requires several
929 assumptions. Firstly, it is assumed that the distribution along each
930 axis is independent. Secondly, the confidence for each axis is
931 assumed to be the same. Therefore, the confidence along each axis
932 can be assumed to be:
934 C[x] = Co ^ (1/n)
936 Where "C[x]" is the confidence along a single axis and "Co" is the
937 overall confidence and "n" is the number of dimensions in the
938 uncertainty.
940 Therefore, to find the uncertainty for each axis at a desired
941 confidence, "Cd", apply the following formula:
943 Ud[x] <= U[x] * (erfinv(Cd ^ (1/n)) / erfinv(Co ^ (1/n)))
945 For regular shapes, this formula can be applied as a scaling factor
946 in each dimension to reach a required confidence.
948 5.5. Determining Whether a Location is Within a Given Region
950 A number of applications require that a judgment be made about
951 whether a Target is within a given region of interest. Given a
952 location estimate with uncertainty, this judgment can be difficult.
953 A location estimate represents a probability distribution, and the
954 true location of the Target cannot be definitively known. Therefore,
955 the judgment relies on determining the probability that the Target is
956 within the region.
958 The probability that the Target is within a particular region is
959 found by integrating the PDF over the region. For a normal
960 distribution, there are no analytical methods that can be used to
961 determine the integral of the two or three dimensional PDF over an
962 arbitrary region. The complexity of numerical methods is also too
963 great to be useful in many applications; for example, finding the
964 integral of the PDF in two or three dimensions across the overlap
965 between the uncertainty region and the target region. If the PDF is
966 unknown, no determination can be made without a simplifying
967 assumption.
969 When judging whether a location is within a given region, this
970 document assumes that uncertainties are rectangular. This introduces
971 errors, but simplifies the calculations significantly. Prior to
972 applying this assumption, confidence should be scaled to 95%.
974 Note: The selection of confidence has a significant impact on the
975 final result. Only use a different confidence if an uncertainty
976 value for 95% confidence cannot be found.
978 Given the assumption of a rectangular distribution, the probability
979 that a Target is found within a given region is found by first
980 finding the area (or volume) of overlap between the uncertainty
981 region and the region of interest. This is multiplied by the
982 confidence of the location estimate to determine the probability.
983 Figure 7 shows an example of finding the area of overlap between the
984 region of uncertainty and the region of interest.
986 _.-""""-._
987 .' `. _ Region of
988 / \ / Uncertainty
989 ..+-"""--.. |
990 .-' | :::::: `-. |
991 ,' | :: Ao ::: `. |
992 / \ :::::::::: \ /
993 / `._ :::::: _.X
994 | `-....-' |
995 | |
996 | |
997 \ /
998 `. .' \_ Region of
999 `._ _.' Interest
1000 `--..___..--'
1002 Figure 7: Area of Overlap Between Two Circular Regions
1004 Once the area of overlap, "Ao", is known, the probability that the
1005 Target is within the region of interest, "Pi", is:
1007 Pi = Co * Ao / Au
1009 Given that the area of the region of uncertainty is "Au" and the
1010 confidence is "Co".
1012 This probability is often input to a decision process that has a
1013 limited set of outcomes; therefore, a threshold value needs to be
1014 selected. Depending on the application, different threshold
1015 probabilities might be selected. In the absence of specific
1016 recommendations, this document suggests that the probability be
1017 greater than 50% before a decision is made. If the decision process
1018 selects between two or more regions, as is required by [RFC5222],
1019 then the region with the highest probability can be selected.
1021 5.5.1. Determining the Area of Overlap for Two Circles
1023 Determining the area of overlap between two arbitrary shapes is a
1024 non-trivial process. Reducing areas to circles (see Section 5.2)
1025 enables the application of the following process.
1027 Given the radius of the first circle "r", the radius of the second
1028 circle "R" and the distance between their center points "d", the
1029 following set of formulas provide the area of overlap "Ao".
1031 o If the circles don't overlap, that is "d >= r+R", "Ao" is zero.
1033 o If one of the two circles is entirely within the other, that is
1034 "d <= |r-R|", the area of overlap is the area of the smaller
1035 circle.
1037 o Otherwise, if the circles partially overlap, that is "d < r+R" and
1038 "d > |r-R|", find "Ao" using:
1040 a = (r^2 - R^2 + d^2)/(2*d)
1042 Ao = r^2*acos(a/r) + R^2*acos((d - a)/R) - d*sqrt(r^2 - a^2)
1044 A value for "d" can be determined by converting the center points to
1045 Cartesian coordinates and calculating the distance between the two
1046 center points:
1048 d = sqrt((x1-x2)^2 + (y1-y2)^2 + (z1-z2)^2)
1050 5.5.2. Determining the Area of Overlap for Two Polygons
1052 A calculation of overlap based on polygons can give better results
1053 than the circle-based method. However, efficient calculation of
1054 overlapping area is non-trivial. Algorithms such as Vatti's clipping
1055 algorithm [Vatti92] can be used.
1057 For large polygonal areas, it might be that geodesic interpolation is
1058 used. In these cases, altitude is also frequently omitted in
1059 describing the polygon. For such shapes, a planar projection can
1060 still give a good approximation of the area of overlap if the larger
1061 area polygon is projected onto the local tangent plane of the
1062 smaller. This is only possible if the only area of interest is that
1063 contained within the smaller polygon. Where the entire area of the
1064 larger polygon is of interest, geodesic interpolation is necessary.
1066 6. Examples
1068 This section presents some examples of how to apply the methods
1069 described in Section 5.
1071 6.1. Reduction to a Point or Circle
1073 Alice receives a location estimate from her LIS that contains an
1074 ellipsoidal region of uncertainty. This information is provided at
1075 19% confidence with a normal PDF. A PIDF-LO extract for this
1076 information is shown in Figure 8.
1078
See RFCXXXX.
1366 1367 1368 END 1370 8.2. XML Schema Registration 1372 This section registers an XML schema as per the guidelines in 1373 [RFC3688]. 1375 URI: urn:ietf:params:xml:schema:geopriv:conf 1377 Registrant Contact: IETF, GEOPRIV working group, (geopriv@ietf.org), 1378 Martin Thomson (martin.thomson@gmail.com). 1380 Schema: The XML for this schema can be found as the entirety of 1381 Section 7 of this document. 1383 9. Security Considerations 1385 This document describes methods for managing and manipulating 1386 uncertainty in location. No specific security concerns arise from 1387 most of the information provided. The considerations of [RFC4119] 1388 all apply. 1390 Providing uncertainty and confidence information can reveal 1391 information about the process by which location information is 1392 generated. For instance, it might reveal information that could be 1393 used to infer that a user is using a mobile device with a GPS, or 1394 that a user is acquiring location information from a particular 1395 network-based service. A Rule Maker might choose to remove 1396 uncertainty-related fields from a location object in order to protect 1397 this information; though it is noted that this information might not 1398 be perfectly protected due to difficulties associated with location 1399 obfuscation, as described in Section 13.5 of [RFC6772]. 1401 Adding confidence to location information risks misinterpretation by 1402 consumers of location that do not understand the element. This could 1403 be exploited, particularly when reducing confidence, since the 1404 resulting uncertainty region might include locations that are less 1405 likely to contain the target than the recipient expects. Since this 1406 sort of error is always a possibility, the impact of this is low. 1408 10. Acknowledgements 1410 Peter Rhodes provided assistance with some of the mathematical 1411 groundwork on this document. Dan Cornford provided a detailed review 1412 and many terminology corrections. 1414 11. References 1416 11.1. Normative References 1418 [RFC2119] Bradner, S., "Key words for use in RFCs to Indicate 1419 Requirement Levels", BCP 14, RFC 2119, March 1997. 1421 [RFC3688] Mealling, M., "The IETF XML Registry", BCP 81, RFC 3688, 1422 January 2004. 1424 [RFC3693] Cuellar, J., Morris, J., Mulligan, D., Peterson, J., and 1425 J. Polk, "Geopriv Requirements", RFC 3693, February 2004. 1427 [RFC4119] Peterson, J., "A Presence-based GEOPRIV Location Object 1428 Format", RFC 4119, December 2005. 1430 [RFC5139] Thomson, M. and J. Winterbottom, "Revised Civic Location 1431 Format for Presence Information Data Format Location 1432 Object (PIDF-LO)", RFC 5139, February 2008. 1434 [RFC5491] Winterbottom, J., Thomson, M., and H. Tschofenig, "GEOPRIV 1435 Presence Information Data Format Location Object (PIDF-LO) 1436 Usage Clarification, Considerations, and Recommendations", 1437 RFC 5491, March 2009. 1439 [RFC6225] Polk, J., Linsner, M., Thomson, M., and B. Aboba, "Dynamic 1440 Host Configuration Protocol Options for Coordinate-Based 1441 Location Configuration Information", RFC 6225, July 2011. 1443 [RFC6280] Barnes, R., Lepinski, M., Cooper, A., Morris, J., 1444 Tschofenig, H., and H. Schulzrinne, "An Architecture for 1445 Location and Location Privacy in Internet Applications", 1446 BCP 160, RFC 6280, July 2011. 1448 11.2. Informative References 1450 [Convert] Burtch, R., "A Comparison of Methods Used in Rectangular 1451 to Geodetic Coordinate Transformations", April 2006. 1453 [GeoShape] 1454 Thomson, M. and C. Reed, "GML 3.1.1 PIDF-LO Shape 1455 Application Schema for use by the Internet Engineering 1456 Task Force (IETF)", Candidate OpenGIS Implementation 1457 Specification 06-142r1, Version: 1.0, April 2007. 1459 [ISO.GUM] ISO/IEC, "Guide to the expression of uncertainty in 1460 measurement (GUM)", Guide 98:1995, 1995. 1462 [NIST.TN1297] 1463 Taylor, B. and C. Kuyatt, "Guidelines for Evaluating and 1464 Expressing the Uncertainty of NIST Measurement Results", 1465 Technical Note 1297, Sep 1994. 1467 [RFC5222] Hardie, T., Newton, A., Schulzrinne, H., and H. 1468 Tschofenig, "LoST: A Location-to-Service Translation 1469 Protocol", RFC 5222, August 2008. 1471 [RFC6772] Schulzrinne, H., Tschofenig, H., Cuellar, J., Polk, J., 1472 Morris, J., and M. Thomson, "Geolocation Policy: A 1473 Document Format for Expressing Privacy Preferences for 1474 Location Information", RFC 6772, January 2013. 1476 [Sunday02] 1477 Sunday, D., "Fast polygon area and Newell normal 1478 computation", Journal of Graphics Tools JGT, 1479 7(2):9-13,2002, 2002, 1480