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rfc:rfc7459

Internet Engineering Task Force (IETF) M. Thomson Request for Comments: 7459 Mozilla Updates: 3693, 4119, 5491 J. Winterbottom Category: Standards Track Unaffiliated ISSN: 2070-1721 February 2015

          Representation of Uncertainty and Confidence in
   the Presence Information Data Format Location Object (PIDF-LO)

Abstract

 This document defines key concepts of uncertainty and confidence as
 they pertain to location information.  Methods for the manipulation
 of location estimates that include uncertainty information are
 outlined.
 This document normatively updates the definition of location
 information representations defined in RFCs 4119 and 5491.  It also
 deprecates related terminology defined in RFC 3693.

Status of This Memo

 This is an Internet Standards Track document.
 This document is a product of the Internet Engineering Task Force
 (IETF).  It represents the consensus of the IETF community.  It has
 received public review and has been approved for publication by the
 Internet Engineering Steering Group (IESG).  Further information on
 Internet Standards is available in Section 2 of RFC 5741.
 Information about the current status of this document, any errata,
 and how to provide feedback on it may be obtained at
 http://www.rfc-editor.org/info/rfc7459.

Thomson & Winterbottom Standards Track [Page 1] RFC 7459 Uncertainty & Confidence February 2015

Copyright Notice

 Copyright (c) 2015 IETF Trust and the persons identified as the
 document authors.  All rights reserved.
 This document is subject to BCP 78 and the IETF Trust's Legal
 Provisions Relating to IETF Documents
 (http://trustee.ietf.org/license-info) in effect on the date of
 publication of this document.  Please review these documents
 carefully, as they describe your rights and restrictions with respect
 to this document.  Code Components extracted from this document must
 include Simplified BSD License text as described in Section 4.e of
 the Trust Legal Provisions and are provided without warranty as
 described in the Simplified BSD License.

Thomson & Winterbottom Standards Track [Page 2] RFC 7459 Uncertainty & Confidence February 2015

Table of Contents

 1. Introduction ....................................................4
    1.1. Conventions and Terminology ................................4
 2. A General Definition of Uncertainty .............................5
    2.1. Uncertainty as a Probability Distribution ..................6
    2.2. Deprecation of the Terms "Precision" and "Resolution" ......8
    2.3. Accuracy as a Qualitative Concept ..........................9
 3. Uncertainty in Location .........................................9
    3.1. Targets as Points in Space .................................9
    3.2. Representation of Uncertainty and Confidence in PIDF-LO ...10
    3.3. Uncertainty and Confidence for Civic Addresses ............10
    3.4. DHCP Location Configuration Information and Uncertainty ...11
 4. Representation of Confidence in PIDF-LO ........................12
    4.1. The "confidence" Element ..................................13
    4.2. Generating Locations with Confidence ......................13
    4.3. Consuming and Presenting Confidence .......................13
 5. Manipulation of Uncertainty ....................................14
    5.1. Reduction of a Location Estimate to a Point ...............15
         5.1.1. Centroid Calculation ...............................16
                5.1.1.1. Arc-Band Centroid .........................16
                5.1.1.2. Polygon Centroid ..........................16
    5.2. Conversion to Circle or Sphere ............................19
    5.3. Conversion from Three-Dimensional to Two-Dimensional ......20
    5.4. Increasing and Decreasing Uncertainty and Confidence ......20
         5.4.1. Rectangular Distributions ..........................21
         5.4.2. Normal Distributions ...............................21
    5.5. Determining Whether a Location Is within a Given Region ...22
         5.5.1. Determining the Area of Overlap for Two Circles ....24
         5.5.2. Determining the Area of Overlap for Two Polygons ...25
 6. Examples .......................................................25
    6.1. Reduction to a Point or Circle ............................25
    6.2. Increasing and Decreasing Confidence ......................29
    6.3. Matching Location Estimates to Regions of Interest ........29
    6.4. PIDF-LO with Confidence Example ...........................30
 7. Confidence Schema ..............................................31
 8. IANA Considerations ............................................32
    8.1. URN Sub-Namespace Registration for ........................32
    8.2. XML Schema Registration ...................................33
 9. Security Considerations ........................................33
 10. References ....................................................34
    10.1. Normative References .....................................34
    10.2. Informative References ...................................35

Thomson & Winterbottom Standards Track [Page 3] RFC 7459 Uncertainty & Confidence February 2015

 Appendix A. Conversion between Cartesian and Geodetic
             Coordinates in WGS84 ..................................36
 Appendix B. Calculating the Upward Normal of a Polygon ............37
    B.1. Checking That a Polygon Upward Normal Points Up ...........38
 Acknowledgements ..................................................39
 Authors' Addresses ................................................39

1. Introduction

 Location information represents an estimation of the position of a
 Target [RFC6280].  Under ideal circumstances, a location estimate
 precisely reflects the actual location of the Target.  For automated
 systems that determine location, there are many factors that
 introduce errors into the measurements that are used to determine
 location estimates.
 The process by which measurements are combined to generate a location
 estimate is outside of the scope of work within the IETF.  However,
 the results of such a process are carried in IETF data formats and
 protocols.  This document outlines how uncertainty, and its
 associated datum, confidence, are expressed and interpreted.
 This document provides a common nomenclature for discussing
 uncertainty and confidence as they relate to location information.
 This document also provides guidance on how to manage location
 information that includes uncertainty.  Methods for expanding or
 reducing uncertainty to obtain a required level of confidence are
 described.  Methods for determining the probability that a Target is
 within a specified region based on its location estimate are
 described.  These methods are simplified by making certain
 assumptions about the location estimate and are designed to be
 applicable to location estimates in a relatively small geographic
 area.
 A confidence extension for the Presence Information Data Format -
 Location Object (PIDF-LO) [RFC4119] is described.
 This document describes methods that can be used in combination with
 automatically determined location information.  These are
 statistically based methods.

1.1. Conventions and Terminology

 The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
 "SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this
 document are to be interpreted as described in [RFC2119].

Thomson & Winterbottom Standards Track [Page 4] RFC 7459 Uncertainty & Confidence February 2015

 This document assumes a basic understanding of the principles of
 mathematics, particularly statistics and geometry.
 Some terminology is borrowed from [RFC3693] and [RFC6280], in
 particular "Target".
 Mathematical formulae are presented using the following notation: add
 "+", subtract "-", multiply "*", divide "/", power "^", and absolute
 value "|x|".  Precedence follows established conventions: power
 operations precede multiply and divide, multiply and divide precede
 add and subtract, and parentheses are used to indicate operations
 that are applied together.  Mathematical functions are represented by
 common abbreviations: square root "sqrt(x)", sine "sin(x)", cosine
 "cos(x)", inverse cosine "acos(x)", tangent "tan(x)", inverse tangent
 "atan(x)", two-argument inverse tangent "atan2(y,x)", error function
 "erf(x)", and inverse error function "erfinv(x)".

2. A General Definition of Uncertainty

 Uncertainty results from the limitations of measurement.  In
 measuring any observable quantity, errors from a range of sources
 affect the result.  Uncertainty is a quantification of what is known
 about the observed quantity, either through the limitations of
 measurement or through inherent variability of the quantity.
 Uncertainty is most completely described by a probability
 distribution.  A probability distribution assigns a probability to
 possible values for the quantity.
 A probability distribution describing a measured quantity can be
 arbitrarily complex, so it is desirable to find a simplified model.
 One approach commonly taken is to reduce the probability distribution
 to a confidence interval.  Many alternative models are used in other
 areas, but study of those is not the focus of this document.
 In addition to the central estimate of the observed quantity, a
 confidence interval is succinctly described by two values: an error
 range and a confidence.  The error range describes an interval and
 the confidence describes an estimated upper bound on the probability
 that a "true" value is found within the extents defined by the error.
 In the following example, a measurement result for a length is shown
 as a nominal value with additional information on error range (0.0043
 meters) and confidence (95%).
    e.g., x = 1.00742 +/- 0.0043 meters at 95% confidence

Thomson & Winterbottom Standards Track [Page 5] RFC 7459 Uncertainty & Confidence February 2015

 This measurement result indicates that the value of "x" is between
 1.00312 and 1.01172 meters with 95% probability.  No other assertion
 is made: in particular, this does not assert that x is 1.00742.
 Uncertainty and confidence for location estimates can be derived in a
 number of ways.  This document does not attempt to enumerate the many
 methods for determining uncertainty.  [ISO.GUM] and [NIST.TN1297]
 provide a set of general guidelines for determining and manipulating
 measurement uncertainty.  This document applies that general guidance
 for consumers of location information.
 As a statistical measure, values determined for uncertainty are found
 based on information in the aggregate, across numerous individual
 estimates.  An individual estimate might be determined to be
 "correct" -- for example, by using a survey to validate the result --
 without invalidating the statistical assertion.
 This understanding of estimates in the statistical sense explains why
 asserting a confidence of 100%, which might seem intuitively correct,
 is rarely advisable.

2.1. Uncertainty as a Probability Distribution

 The Probability Density Function (PDF) that is described by
 uncertainty indicates the probability that the "true" value lies at
 any one point.  The shape of the probability distribution can vary
 depending on the method that is used to determine the result.  The
 two probability density functions most generally applicable to
 location information are considered in this document:
 o  The normal PDF (also referred to as a Gaussian PDF) is used where
    a large number of small random factors contribute to errors.  The
    value used for the error range in a normal PDF is related to the
    standard deviation of the distribution.
 o  A rectangular PDF is used where the errors are known to be
    consistent across a limited range.  A rectangular PDF can occur
    where a single error source, such as a rounding error, is
    significantly larger than other errors.  A rectangular PDF is
    often described by the half-width of the distribution; that is,
    half the width of the distribution.
 Each of these probability density functions can be characterized by
 its center point, or mean, and its width.  For a normal distribution,
 uncertainty and confidence together are related to the standard
 deviation of the function (see Section 5.4).  For a rectangular
 distribution, the half-width of the distribution is used.

Thomson & Winterbottom Standards Track [Page 6] RFC 7459 Uncertainty & Confidence February 2015

 Figure 1 shows a normal and rectangular probability density function
 with the mean (m) and standard deviation (s) labeled.  The half-width
 (h) of the rectangular distribution is also indicated.
  • * Normal PDF : — Rectangular PDF : : .———*—————*———. | : | | : | | * ←- s –>: * | | * : : : * | | : | | * : : : * | | * : * | | : : : | : * | : : : | * * | :←—– h ——>| * .——-+…….:………:………:…….+——-*. m Figure 1: Normal and Rectangular Probability Density Functions For a given PDF, the value of the PDF describes the probability that the "true" value is found at that point. Confidence for any given interval is the total probability of the "true" value being in that range, defined as the integral of the PDF over the interval. The probability of the "true" value falling between two points is found by finding the area under the curve between the points (that is, the integral of the curve between the points). For any given PDF, the area under the curve for the entire range from negative infinity to positive infinity is 1 or (100%). Therefore, the confidence over any interval of uncertainty is always less than 100%. Thomson & Winterbottom Standards Track [Page 7] RFC 7459 Uncertainty & Confidence February 2015 Figure 2 shows how confidence is determined for a normal distribution. The area of the shaded region gives the confidence © for the interval between "m-u" and "m+u". * ::::: ::::::::: ::::::::::: *:::::::::::::::* ::::::::::::::: ::::::::::::::::: *:::::::::::::::::::::* *:::::::::::::::::::::::* ::::::::::::::::::::::: *:::::::::::: c ::::::::::::* *:::::::::::::::::::::::::::::* |:::::::::::::::::::::::::::::| |:::::::::::::::::::::::::::::| * |:::::::::::::::::::::::::::::| * * |:::::::::::::::::::::::::::::| * .……….!:::::::::::::::::::::::::::::!……….***.

| | |

               (m-u)            m            (m+u)
             Figure 2: Confidence as the Integral of a PDF
 In Section 5.4, methods are described for manipulating uncertainty if
 the shape of the PDF is known.

2.2. Deprecation of the Terms "Precision" and "Resolution"

 The terms "Precision" and "Resolution" are defined in RFC 3693
 [RFC3693].  These definitions were intended to provide a common
 nomenclature for discussing uncertainty; however, these particular
 terms have many different uses in other fields, and their definitions
 are not sufficient to avoid confusion about their meaning.  These
 terms are unsuitable for use in relation to quantitative concepts
 when discussing uncertainty and confidence in relation to location
 information.

Thomson & Winterbottom Standards Track [Page 8] RFC 7459 Uncertainty & Confidence February 2015

2.3. Accuracy as a Qualitative Concept

 Uncertainty is a quantitative concept.  The term "accuracy" is useful
 in describing, qualitatively, the general concepts of location
 information.  Accuracy is generally useful when describing
 qualitative aspects of location estimates.  Accuracy is not a
 suitable term for use in a quantitative context.
 For instance, it could be appropriate to say that a location estimate
 with uncertainty "X" is more accurate than a location estimate with
 uncertainty "2X" at the same confidence.  It is not appropriate to
 assign a number to "accuracy", nor is it appropriate to refer to any
 component of uncertainty or confidence as "accuracy".  That is,
 saying the "accuracy" for the first location estimate is "X" would be
 an erroneous use of this term.

3. Uncertainty in Location

 A "location estimate" is the result of location determination.  A
 location estimate is subject to uncertainty like any other
 observation.  However, unlike a simple measure of a one dimensional
 property like length, a location estimate is specified in two or
 three dimensions.
 Uncertainty in two- or three-dimensional locations can be described
 using confidence intervals.  The confidence interval for a location
 estimate in two- or three-dimensional space is expressed as a subset
 of that space.  This document uses the term "region of uncertainty"
 to refer to the area or volume that describes the confidence
 interval.
 Areas or volumes that describe regions of uncertainty can be formed
 by the combination of two or three one-dimensional ranges, or more
 complex shapes could be described (for example, the shapes in
 [RFC5491]).

3.1. Targets as Points in Space

 This document makes a simplifying assumption that the Target of the
 PIDF-LO occupies just a single point in space.  While this is clearly
 false in virtually all scenarios with any practical application, it
 is often a reasonable simplifying assumption to make.
 To a large extent, whether this simplification is valid depends on
 the size of the Target relative to the size of the uncertainty
 region.  When locating a personal device using contemporary location
 determination techniques, the space the device occupies relative to

Thomson & Winterbottom Standards Track [Page 9] RFC 7459 Uncertainty & Confidence February 2015

 the uncertainty is proportionally quite small.  Even where that
 device is used as a proxy for a person, the proportions change
 little.
 This assumption is less useful as uncertainty becomes small relative
 to the size of the Target of the PIDF-LO (or conversely, as
 uncertainty becomes small relative to the Target).  For instance,
 describing the location of a football stadium or small country would
 include a region of uncertainty that is only slightly larger than the
 Target itself.  In these cases, much of the guidance in this document
 is not applicable.  Indeed, as the accuracy of location determination
 technology improves, it could be that the advice this document
 contains becomes less relevant by the same measure.

3.2. Representation of Uncertainty and Confidence in PIDF-LO

 A set of shapes suitable for the expression of uncertainty in
 location estimates in the PIDF-LO are described in [GeoShape].  These
 shapes are the recommended form for the representation of uncertainty
 in PIDF-LO [RFC4119] documents.
 The PIDF-LO can contain uncertainty, but it does not include an
 indication of confidence.  [RFC5491] defines a fixed value of 95%.
 Similarly, the PIDF-LO format does not provide an indication of the
 shape of the PDF.  Section 4 defines elements to convey this
 information in PIDF-LO.
 Absence of uncertainty information in a PIDF-LO document does not
 indicate that there is no uncertainty in the location estimate.
 Uncertainty might not have been calculated for the estimate, or it
 may be withheld for privacy purposes.
 If the Point shape is used, confidence and uncertainty are unknown; a
 receiver can either assume a confidence of 0% or infinite
 uncertainty.  The same principle applies on the altitude axis for
 two-dimensional shapes like the Circle.

3.3. Uncertainty and Confidence for Civic Addresses

 Automatically determined civic addresses [RFC5139] inherently include
 uncertainty, based on the area of the most precise element that is
 specified.  In this case, uncertainty is effectively described by the
 presence or absence of elements.  To the recipient of location
 information, elements that are not present are uncertain.
 To apply the concept of uncertainty to civic addresses, it is helpful
 to unify the conceptual models of civic address with geodetic
 location information.  This is particularly useful when considering

Thomson & Winterbottom Standards Track [Page 10] RFC 7459 Uncertainty & Confidence February 2015

 civic addresses that are determined using reverse geocoding (that is,
 the process of translating geodetic information into civic
 addresses).
 In the unified view, a civic address defines a series of (sometimes
 non-orthogonal) spatial partitions.  The first is the implicit
 partition that identifies the surface of the earth and the space near
 the surface.  The second is the country.  Each label that is included
 in a civic address provides information about a different set of
 spatial partitions.  Some partitions require slight adjustments from
 a standard interpretation: for instance, a road includes all
 properties that adjoin the street.  Each label might need to be
 interpreted with other values to provide context.
 As a value at each level is interpreted, one or more spatial
 partitions at that level are selected, and all other partitions of
 that type are excluded.  For non-orthogonal partitions, only the
 portion of the partition that fits within the existing space is
 selected.  This is what distinguishes King Street in Sydney from King
 Street in Melbourne.  Each defined element selects a partition of
 space.  The resulting location is the intersection of all selected
 spaces.
 The resulting spatial partition can be considered as a region of
 uncertainty.
 Note:  This view is a potential perspective on the process of
    geocoding -- the translation of a civic address to a geodetic
    location.
 Uncertainty in civic addresses can be increased by removing elements.
 This does not increase confidence unless additional information is
 used.  Similarly, arbitrarily increasing uncertainty in a geodetic
 location does not increase confidence.

3.4. DHCP Location Configuration Information and Uncertainty

 Location information is often measured in two or three dimensions;
 expressions of uncertainty in one dimension only are rare.  The
 "resolution" parameters in [RFC6225] provide an indication of how
 many bits of a number are valid, which could be interpreted as an
 expression of uncertainty in one dimension.
 [RFC6225] defines a means for representing uncertainty, but a value
 for confidence is not specified.  A default value of 95% confidence
 should be assumed for the combination of the uncertainty on each
 axis.  This is consistent with the transformation of those forms into

Thomson & Winterbottom Standards Track [Page 11] RFC 7459 Uncertainty & Confidence February 2015

 the uncertainty representations from [RFC5491].  That is, the
 confidence of the resultant rectangular Polygon or Prism is assumed
 to be 95%.

4. Representation of Confidence in PIDF-LO

 On the whole, a fixed definition for confidence is preferable,
 primarily because it ensures consistency between implementations.
 Location generators that are aware of this constraint can generate
 location information at the required confidence.  Location recipients
 are able to make sensible assumptions about the quality of the
 information that they receive.
 In some circumstances -- particularly with preexisting systems --
 location generators might be unable to provide location information
 with consistent confidence.  Existing systems sometimes specify
 confidence at 38%, 67%, or 90%.  Existing forms of expressing
 location information, such as that defined in [TS-3GPP-23_032],
 contain elements that express the confidence in the result.
 The addition of a confidence element provides information that was
 previously unavailable to recipients of location information.
 Without this information, a location server or generator that has
 access to location information with a confidence lower than 95% has
 two options:
 o  The location server can scale regions of uncertainty in an attempt
    to achieve 95% confidence.  This scaling process significantly
    degrades the quality of the information, because the location
    server might not have the necessary information to scale
    appropriately; the location server is forced to make assumptions
    that are likely to result in either an overly conservative
    estimate with high uncertainty or an overestimate of confidence.
 o  The location server can ignore the confidence entirely, which
    results in giving the recipient a false impression of its quality.
 Both of these choices degrade the quality of the information
 provided.
 The addition of a confidence element avoids this problem entirely if
 a location recipient supports and understands the element.  A
 recipient that does not understand -- and, hence, ignores -- the
 confidence element is in no worse a position than if the location
 server ignored confidence.

Thomson & Winterbottom Standards Track [Page 12] RFC 7459 Uncertainty & Confidence February 2015

4.1. The "confidence" Element

 The "confidence" element MAY be added to the "location-info" element
 of the PIDF-LO [RFC4119] document.  This element expresses the
 confidence in the associated location information as a percentage.  A
 special "unknown" value is reserved to indicate that confidence is
 supported, but not known to the Location Generator.
 The "confidence" element optionally includes an attribute that
 indicates the shape of the PDF of the associated region of
 uncertainty.  Three values are possible: unknown, normal, and
 rectangular.
 Indicating a particular PDF only indicates that the distribution
 approximately fits the given shape based on the methods used to
 generate the location information.  The PDF is normal if there are a
 large number of small, independent sources of error.  It is
 rectangular if all points within the area have roughly equal
 probability of being the actual location of the Target.  Otherwise,
 the PDF MUST either be set to unknown or omitted.
 If a PIDF-LO does not include the confidence element, the confidence
 of the location estimate is 95%, as defined in [RFC5491].
 A Point shape does not have uncertainty (or it has infinite
 uncertainty), so confidence is meaningless for a Point; therefore,
 this element MUST be omitted if only a Point is provided.

4.2. Generating Locations with Confidence

 Location generators SHOULD attempt to ensure that confidence is equal
 in each dimension when generating location information.  This
 restriction, while not always practical, allows for more accurate
 scaling, if scaling is necessary.
 A confidence element MUST be included with all location information
 that includes uncertainty (that is, all forms other than a Point).  A
 special "unknown" is used if confidence is not known.

4.3. Consuming and Presenting Confidence

 The inclusion of confidence that is anything other than 95% presents
 a potentially difficult usability problem for applications that use
 location information.  Effectively communicating the probability that
 a location is incorrect to a user can be difficult.

Thomson & Winterbottom Standards Track [Page 13] RFC 7459 Uncertainty & Confidence February 2015

 It is inadvisable to simply display locations of any confidence, or
 to display confidence in a separate or non-obvious fashion.  If
 locations with different confidence levels are displayed such that
 the distinction is subtle or easy to overlook -- such as using fine
 graduations of color or transparency for graphical uncertainty
 regions or displaying uncertainty graphically, but providing
 confidence as supplementary text -- a user could fail to notice a
 difference in the quality of the location information that might be
 significant.
 Depending on the circumstances, different ways of handling confidence
 might be appropriate.  Section 5 describes techniques that could be
 appropriate for consumers that use automated processing.
 Providing that the full implications of any choice for the
 application are understood, some amount of automated processing could
 be appropriate.  In a simple example, applications could choose to
 discard or suppress the display of location information if confidence
 does not meet a predetermined threshold.
 In settings where there is an opportunity for user training, some of
 these problems might be mitigated by defining different operational
 procedures for handling location information at different confidence
 levels.

5. Manipulation of Uncertainty

 This section deals with manipulation of location information that
 contains uncertainty.
 The following rules generally apply when manipulating location
 information:
 o  Where calculations are performed on coordinate information, these
    should be performed in Cartesian space and the results converted
    back to latitude, longitude, and altitude.  A method for
    converting to and from Cartesian coordinates is included in
    Appendix A.
       While some approximation methods are useful in simplifying
       calculations, treating latitude and longitude as Cartesian axes
       is never advisable.  The two axes are not orthogonal.  Errors
       can arise from the curvature of the earth and from the
       convergence of longitude lines.

Thomson & Winterbottom Standards Track [Page 14] RFC 7459 Uncertainty & Confidence February 2015

 o  Normal rounding rules do not apply when rounding uncertainty.
    When rounding, the region of uncertainty always increases (that
    is, errors are rounded up) and confidence is always rounded down
    (see [NIST.TN1297]).  This means that any manipulation of
    uncertainty is a non-reversible operation; each manipulation can
    result in the loss of some information.

5.1. Reduction of a Location Estimate to a Point

 Manipulating location estimates that include uncertainty information
 requires additional complexity in systems.  In some cases, systems
 only operate on definitive values, that is, a single point.
 This section describes algorithms for reducing location estimates to
 a simple form without uncertainty information.  Having a consistent
 means for reducing location estimates allows for interaction between
 applications that are able to use uncertainty information and those
 that cannot.
 Note:  Reduction of a location estimate to a point constitutes a
    reduction in information.  Removing uncertainty information can
    degrade results in some applications.  Also, there is a natural
    tendency to misinterpret a Point location as representing a
    location without uncertainty.  This could lead to more serious
    errors.  Therefore, these algorithms should only be applied where
    necessary.
 Several different approaches can be taken when reducing a location
 estimate to a point.  Different methods each make a set of
 assumptions about the properties of the PDF and the selected point;
 no one method is more "correct" than any other.  For any given region
 of uncertainty, selecting an arbitrary point within the area could be
 considered valid; however, given the aforementioned problems with
 Point locations, a more rigorous approach is appropriate.
 Given a result with a known distribution, selecting the point within
 the area that has the highest probability is a more rigorous method.
 Alternatively, a point could be selected that minimizes the overall
 error; that is, it minimizes the expected value of the difference
 between the selected point and the "true" value.
 If a rectangular distribution is assumed, the centroid of the area or
 volume minimizes the overall error.  Minimizing the error for a
 normal distribution is mathematically complex.  Therefore, this
 document opts to select the centroid of the region of uncertainty
 when selecting a point.

Thomson & Winterbottom Standards Track [Page 15] RFC 7459 Uncertainty & Confidence February 2015

5.1.1. Centroid Calculation

 For regular shapes, such as Circle, Sphere, Ellipse, and Ellipsoid,
 this approach equates to the center point of the region.  For regions
 of uncertainty that are expressed as regular Polygons and Prisms, the
 center point is also the most appropriate selection.
 For the Arc-Band shape and non-regular Polygons and Prisms, selecting
 the centroid of the area or volume minimizes the overall error.  This
 assumes that the PDF is rectangular.
 Note:  The centroid of a concave Polygon or Arc-Band shape is not
    necessarily within the region of uncertainty.

5.1.1.1. Arc-Band Centroid

 The centroid of the Arc-Band shape is found along a line that bisects
 the arc.  The centroid can be found at the following distance from
 the starting point of the arc-band (assuming an arc-band with an
 inner radius of "r", outer radius "R", start angle "a", and opening
 angle "o"):
    d = 4 * sin(o/2) * (R*R + R*r + r*r) / (3*o*(R + r))
 This point can be found along the line that bisects the arc; that is,
 the line at an angle of "a + (o/2)".

5.1.1.2. Polygon Centroid

 Calculating a centroid for the Polygon and Prism shapes is more
 complex.  Polygons that are specified using geodetic coordinates are
 not necessarily coplanar.  For Polygons that are specified without an
 altitude, choose a value for altitude before attempting this process;
 an altitude of 0 is acceptable.
    The method described in this section is simplified by assuming
    that the surface of the earth is locally flat.  This method
    degrades as polygons become larger; see [GeoShape] for
    recommendations on polygon size.
 The polygon is translated to a new coordinate system that has an x-y
 plane roughly parallel to the polygon.  This enables the elimination
 of z-axis values and calculating a centroid can be done using only x
 and y coordinates.  This requires that the upward normal for the
 polygon be known.

Thomson & Winterbottom Standards Track [Page 16] RFC 7459 Uncertainty & Confidence February 2015

 To translate the polygon coordinates, apply the process described in
 Appendix B to find the normal vector "N = [Nx,Ny,Nz]".  This value
 should be made a unit vector to ensure that the transformation matrix
 is a special orthogonal matrix.  From this vector, select two vectors
 that are perpendicular to this vector and combine these into a
 transformation matrix.
 If "Nx" and "Ny" are non-zero, the matrices in Figure 3 can be used,
 given "p = sqrt(Nx^2 + Ny^2)".  More transformations are provided
 later in this section for cases where "Nx" or "Ny" are zero.
        [   -Ny/p     Nx/p     0  ]         [ -Ny/p  -Nx*Nz/p  Nx ]
    T = [ -Nx*Nz/p  -Ny*Nz/p   p  ]    T' = [  Nx/p  -Ny*Nz/p  Ny ]
        [    Nx        Ny      Nz ]         [   0      p       Nz ]
               (Transform)                    (Reverse Transform)
             Figure 3: Recommended Transformation Matrices
 To apply a transform to each point in the polygon, form a matrix from
 the Cartesian Earth-Centered, Earth-Fixed (ECEF) coordinates and use
 matrix multiplication to determine the translated coordinates.
    [   -Ny/p     Nx/p     0  ]   [ x[1]  x[2]  x[3]  ...  x[n] ]
    [ -Nx*Nz/p  -Ny*Nz/p   p  ] * [ y[1]  y[2]  y[3]  ...  y[n] ]
    [    Nx        Ny      Nz ]   [ z[1]  z[2]  z[3]  ...  z[n] ]
        [ x'[1]  x'[2]  x'[3]  ... x'[n] ]
      = [ y'[1]  y'[2]  y'[3]  ... y'[n] ]
        [ z'[1]  z'[2]  z'[3]  ... z'[n] ]
                       Figure 4: Transformation
 Alternatively, direct multiplication can be used to achieve the same
 result:
    x'[i] = -Ny * x[i] / p + Nx * y[i] / p
    y'[i] = -Nx * Nz * x[i] / p - Ny * Nz * y[i] / p + p * z[i]
    z'[i] = Nx * x[i] + Ny * y[i] + Nz * z[i]
 The first and second rows of this matrix ("x'" and "y'") contain the
 values that are used to calculate the centroid of the polygon.  To
 find the centroid of this polygon, first find the area using:
    A = sum from i=1..n of (x'[i]*y'[i+1]-x'[i+1]*y'[i]) / 2

Thomson & Winterbottom Standards Track [Page 17] RFC 7459 Uncertainty & Confidence February 2015

 For these formulae, treat each set of coordinates as circular, that
 is "x'[0] == x'[n]" and "x'[n+1] == x'[1]".  Based on the area, the
 centroid along each axis can be determined by:
    Cx' = sum (x'[i]+x'[i+1]) * (x'[i]*y'[i+1]-x'[i+1]*y'[i]) / (6*A)
    Cy' = sum (y'[i]+y'[i+1]) * (x'[i]*y'[i+1]-x'[i+1]*y'[i]) / (6*A)
 Note:  The formula for the area of a polygon will return a negative
    value if the polygon is specified in a clockwise direction.  This
    can be used to determine the orientation of the polygon.
 The third row contains a distance from a plane parallel to the
 polygon.  If the polygon is coplanar, then the values for "z'" are
 identical; however, the constraints recommended in [RFC5491] mean
 that this is rarely the case.  To determine "Cz'", average these
 values:
    Cz' = sum z'[i] / n
 Once the centroid is known in the transformed coordinates, these can
 be transformed back to the original coordinate system.  The reverse
 transformation is shown in Figure 5.
    [ -Ny/p  -Nx*Nz/p  Nx ]     [       Cx'        ]   [ Cx ]
    [  Nx/p  -Ny*Nz/p  Ny ]  *  [       Cy'        ] = [ Cy ]
    [   0        p     Nz ]     [ sum of z'[i] / n ]   [ Cz ]
                   Figure 5: Reverse Transformation
 The reverse transformation can be applied directly as follows:
    Cx = -Ny * Cx' / p - Nx * Nz * Cy' / p + Nx * Cz'
    Cy = Nx * Cx' / p - Ny * Nz * Cy' / p + Ny * Cz'
    Cz = p * Cy' + Nz * Cz'
 The ECEF value "[Cx,Cy,Cz]" can then be converted back to geodetic
 coordinates.  Given a polygon that is defined with no altitude or
 equal altitudes for each point, the altitude of the result can be
 either ignored or reset after converting back to a geodetic value.

Thomson & Winterbottom Standards Track [Page 18] RFC 7459 Uncertainty & Confidence February 2015

 The centroid of the Prism shape is found by finding the centroid of
 the base polygon and raising the point by half the height of the
 prism.  This can be added to altitude of the final result;
 alternatively, this can be added to "Cz'", which ensures that
 negative height is correctly applied to polygons that are defined in
 a clockwise direction.
 The recommended transforms only apply if "Nx" and "Ny" are non-zero.
 If the normal vector is "[0,0,1]" (that is, along the z-axis), then
 no transform is necessary.  Similarly, if the normal vector is
 "[0,1,0]" or "[1,0,0]", avoid the transformation and use the x and z
 coordinates or y and z coordinates (respectively) in the centroid
 calculation phase.  If either "Nx" or "Ny" are zero, the alternative
 transform matrices in Figure 6 can be used.  The reverse transform is
 the transpose of this matrix.
  if Nx == 0:                              | if Ny == 0:
      [ 0  -Nz  Ny ]       [  0   1  0  ]  |            [ -Nz  0  Nx ]
  T = [ 1   0   0  ]  T' = [ -Nz  0  Ny ]  |   T = T' = [  0   1  0  ]
      [ 0   Ny  Nz ]       [  Ny  0  Nz ]  |            [  Nx  0  Nz ]
             Figure 6: Alternative Transformation Matrices

5.2. Conversion to Circle or Sphere

 The circle or sphere are simple shapes that suit a range of
 applications.  A circle or sphere contains fewer units of data to
 manipulate, which simplifies operations on location estimates.
 The simplest method for converting a location estimate to a Circle or
 Sphere shape is to determine the centroid and then find the longest
 distance to any point in the region of uncertainty to that point.
 This distance can be determined based on the shape type:
 Circle/Sphere:  No conversion necessary.
 Ellipse/Ellipsoid:  The greater of either semi-major axis or altitude
    uncertainty.
 Polygon/Prism:  The distance to the farthest vertex of the Polygon
    (for a Prism, it is only necessary to check points on the base).

Thomson & Winterbottom Standards Track [Page 19] RFC 7459 Uncertainty & Confidence February 2015

 Arc-Band:  The farthest length from the centroid to the points where
    the inner and outer arc end.  This distance can be calculated by
    finding the larger of the two following formulae:
       X = sqrt( d*d + R*R - 2*d*R*cos(o/2) )
       x = sqrt( d*d + r*r - 2*d*r*cos(o/2) )
 Once the Circle or Sphere shape is found, the associated confidence
 can be increased if the result is known to follow a normal
 distribution.  However, this is a complicated process and provides
 limited benefit.  In many cases, it also violates the constraint that
 confidence in each dimension be the same.  Confidence should be
 unchanged when performing this conversion.
 Two-dimensional shapes are converted to a Circle; three-dimensional
 shapes are converted to a Sphere.

5.3. Conversion from Three-Dimensional to Two-Dimensional

 A three-dimensional shape can be easily converted to a two-
 dimensional shape by removing the altitude component.  A Sphere
 becomes a Circle; a Prism becomes a Polygon; an Ellipsoid becomes an
 Ellipse.  Each conversion is simple, requiring only the removal of
 those elements relating to altitude.
 The altitude is unspecified for a two-dimensional shape and therefore
 has unlimited uncertainty along the vertical axis.  The confidence
 for the two-dimensional shape is thus higher than the three-
 dimensional shape.  Assuming equal confidence on each axis, the
 confidence of the Circle can be increased using the following
 approximate formula:
    C[2d] >= C[3d] ^ (2/3)
 "C[2d]" is the confidence of the two-dimensional shape and "C[3d]" is
 the confidence of the three-dimensional shape.  For example, a Sphere
 with a confidence of 95% can be simplified to a Circle of equal
 radius with confidence of 96.6%.

5.4. Increasing and Decreasing Uncertainty and Confidence

 The combination of uncertainty and confidence provide a great deal of
 information about the nature of the data that is being measured.  If
 uncertainty, confidence, and PDF are known, certain information can
 be extrapolated.  In particular, the uncertainty can be scaled to
 meet a desired confidence or the confidence for a particular region
 of uncertainty can be found.

Thomson & Winterbottom Standards Track [Page 20] RFC 7459 Uncertainty & Confidence February 2015

 In general, confidence decreases as the region of uncertainty
 decreases in size, and confidence increases as the region of
 uncertainty increases in size.  However, this depends on the PDF;
 expanding the region of uncertainty for a rectangular distribution
 has no effect on confidence without additional information.  If the
 region of uncertainty is increased during the process of obfuscation
 (see [RFC6772]), then the confidence cannot be increased.
 A region of uncertainty that is reduced in size always has a lower
 confidence.
 A region of uncertainty that has an unknown PDF shape cannot be
 reduced in size reliably.  The region of uncertainty can be expanded,
 but only if confidence is not increased.
 This section makes the simplifying assumption that location
 information is symmetrically and evenly distributed in each
 dimension.  This is not necessarily true in practice.  If better
 information is available, alternative methods might produce better
 results.

5.4.1. Rectangular Distributions

 Uncertainty that follows a rectangular distribution can only be
 decreased in size.  Increasing uncertainty has no value, since it has
 no effect on confidence.  Since the PDF is constant over the region
 of uncertainty, the resulting confidence is determined by the
 following formula:
    Cr = Co * Ur / Uo
 Where "Uo" and "Ur" are the sizes of the original and reduced regions
 of uncertainty (either the area or the volume of the region); "Co"
 and "Cr" are the confidence values associated with each region.
 Information is lost by decreasing the region of uncertainty for a
 rectangular distribution.  Once reduced in size, the uncertainty
 region cannot subsequently be increased in size.

5.4.2. Normal Distributions

 Uncertainty and confidence can be both increased and decreased for a
 normal distribution.  This calculation depends on the number of
 dimensions of the uncertainty region.

Thomson & Winterbottom Standards Track [Page 21] RFC 7459 Uncertainty & Confidence February 2015

 For a normal distribution, uncertainty and confidence are related to
 the standard deviation of the function.  The following function
 defines the relationship between standard deviation, uncertainty, and
 confidence along a single axis:
    S[x] = U[x] / ( sqrt(2) * erfinv(C[x]) )
 Where "S[x]" is the standard deviation, "U[x]" is the uncertainty,
 and "C[x]" is the confidence along a single axis.  "erfinv" is the
 inverse error function.
 Scaling a normal distribution in two dimensions requires several
 assumptions.  Firstly, it is assumed that the distribution along each
 axis is independent.  Secondly, the confidence for each axis is
 assumed to be the same.  Therefore, the confidence along each axis
 can be assumed to be:
    C[x] = Co ^ (1/n)
 Where "C[x]" is the confidence along a single axis and "Co" is the
 overall confidence and "n" is the number of dimensions in the
 uncertainty.
 Therefore, to find the uncertainty for each axis at a desired
 confidence, "Cd", apply the following formula:
    Ud[x] <= U[x] * (erfinv(Cd ^ (1/n)) / erfinv(Co ^ (1/n)))
 For regular shapes, this formula can be applied as a scaling factor
 in each dimension to reach a required confidence.

5.5. Determining Whether a Location Is within a Given Region

 A number of applications require that a judgment be made about
 whether a Target is within a given region of interest.  Given a
 location estimate with uncertainty, this judgment can be difficult.
 A location estimate represents a probability distribution, and the
 true location of the Target cannot be definitively known.  Therefore,
 the judgment relies on determining the probability that the Target is
 within the region.
 The probability that the Target is within a particular region is
 found by integrating the PDF over the region.  For a normal
 distribution, there are no analytical methods that can be used to
 determine the integral of the two- or three-dimensional PDF over an
 arbitrary region.  The complexity of numerical methods is also too
 great to be useful in many applications; for example, finding the
 integral of the PDF in two or three dimensions across the overlap

Thomson & Winterbottom Standards Track [Page 22] RFC 7459 Uncertainty & Confidence February 2015

 between the uncertainty region and the target region.  If the PDF is
 unknown, no determination can be made without a simplifying
 assumption.
 When judging whether a location is within a given region, this
 document assumes that uncertainties are rectangular.  This introduces
 errors, but simplifies the calculations significantly.  Prior to
 applying this assumption, confidence should be scaled to 95%.
 Note:  The selection of confidence has a significant impact on the
    final result.  Only use a different confidence if an uncertainty
    value for 95% confidence cannot be found.
 Given the assumption of a rectangular distribution, the probability
 that a Target is found within a given region is found by first
 finding the area (or volume) of overlap between the uncertainty
 region and the region of interest.  This is multiplied by the
 confidence of the location estimate to determine the probability.
 Figure 7 shows an example of finding the area of overlap between the
 region of uncertainty and the region of interest.
                  _.-""""-._
                .'          `.    _ Region of
               /              \  /  Uncertainty
            ..+-"""--..        |
         .-'  | :::::: `-.     |
       ,'     | :: Ao ::: `.   |
      /        \ :::::::::: \ /
     /          `._ :::::: _.X
    |              `-....-'   |
    |                         |
    |                         |
     \                       /
      `.                   .'  \_ Region of
        `._             _.'       Interest
           `--..___..--'
        Figure 7: Area of Overlap between Two Circular Regions

Thomson & Winterbottom Standards Track [Page 23] RFC 7459 Uncertainty & Confidence February 2015

 Once the area of overlap, "Ao", is known, the probability that the
 Target is within the region of interest, "Pi", is:
    Pi = Co * Ao / Au
 Given that the area of the region of uncertainty is "Au" and the
 confidence is "Co".
 This probability is often input to a decision process that has a
 limited set of outcomes; therefore, a threshold value needs to be
 selected.  Depending on the application, different threshold
 probabilities might be selected.  A probability of 50% or greater is
 recommended before deciding that an uncertain value is within a given
 region.  If the decision process selects between two or more regions,
 as is required by [RFC5222], then the region with the highest
 probability can be selected.

5.5.1. Determining the Area of Overlap for Two Circles

 Determining the area of overlap between two arbitrary shapes is a
 non-trivial process.  Reducing areas to circles (see Section 5.2)
 enables the application of the following process.
 Given the radius of the first circle "r", the radius of the second
 circle "R", and the distance between their center points "d", the
 following set of formulae provide the area of overlap "Ao".
 o  If the circles don't overlap, that is "d >= r+R", "Ao" is zero.
 o  If one of the two circles is entirely within the other, that is
    "d <= |r-R|", the area of overlap is the area of the smaller
    circle.
 o  Otherwise, if the circles partially overlap, that is "d < r+R" and
    "d > |r-R|", find "Ao" using:
       a = (r^2 - R^2 + d^2)/(2*d)
       Ao = r^2*acos(a/r) + R^2*acos((d - a)/R) - d*sqrt(r^2 - a^2)
 A value for "d" can be determined by converting the center points to
 Cartesian coordinates and calculating the distance between the two
 center points:
    d = sqrt((x1-x2)^2 + (y1-y2)^2 + (z1-z2)^2)

Thomson & Winterbottom Standards Track [Page 24] RFC 7459 Uncertainty & Confidence February 2015

5.5.2. Determining the Area of Overlap for Two Polygons

 A calculation of overlap based on polygons can give better results
 than the circle-based method.  However, efficient calculation of
 overlapping area is non-trivial.  Algorithms such as Vatti's clipping
 algorithm [Vatti92] can be used.
 For large polygonal areas, it might be that geodesic interpolation is
 used.  In these cases, altitude is also frequently omitted in
 describing the polygon.  For such shapes, a planar projection can
 still give a good approximation of the area of overlap if the larger
 area polygon is projected onto the local tangent plane of the
 smaller.  This is only possible if the only area of interest is that
 contained within the smaller polygon.  Where the entire area of the
 larger polygon is of interest, geodesic interpolation is necessary.

6. Examples

 This section presents some examples of how to apply the methods
 described in Section 5.

6.1. Reduction to a Point or Circle

 Alice receives a location estimate from her Location Information
 Server (LIS) that contains an ellipsoidal region of uncertainty.
 This information is provided at 19% confidence with a normal PDF.  A
 PIDF-LO extract for this information is shown in Figure 8.

Thomson & Winterbottom Standards Track [Page 25] RFC 7459 Uncertainty & Confidence February 2015

   <gp:geopriv>
     <gp:location-info>
       <gs:Ellipsoid srsName="urn:ogc:def:crs:EPSG::4979">
         <gml:pos>-34.407242 150.882518 34</gml:pos>
         <gs:semiMajorAxis uom="urn:ogc:def:uom:EPSG::9001">
           7.7156
         </gs:semiMajorAxis>
         <gs:semiMinorAxis uom="urn:ogc:def:uom:EPSG::9001">
           3.31
         </gs:semiMinorAxis>
         <gs:verticalAxis uom="urn:ogc:def:uom:EPSG::9001">
           28.7
         </gs:verticalAxis>
         <gs:orientation uom="urn:ogc:def:uom:EPSG::9102">
           43
         </gs:orientation>
       </gs:Ellipsoid>
       <con:confidence pdf="normal">95</con:confidence>
     </gp:location-info>
     <gp:usage-rules/>
   </gp:geopriv>
                 Figure 8: Alice's Ellipsoid Location
 This information can be reduced to a point simply by extracting the
 center point, that is [-34.407242, 150.882518, 34].
 If some limited uncertainty were required, the estimate could be
 converted into a circle or sphere.  To convert to a sphere, the
 radius is the largest of the semi-major, semi-minor and vertical
 axes; in this case, 28.7 meters.
 However, if only a circle is required, the altitude can be dropped as
 can the altitude uncertainty (the vertical axis of the ellipsoid),
 resulting in a circle at [-34.407242, 150.882518] of radius 7.7156
 meters.
 Bob receives a location estimate with a Polygon shape (which roughly
 corresponds to the location of the Sydney Opera House).  This
 information is shown in Figure 9.

Thomson & Winterbottom Standards Track [Page 26] RFC 7459 Uncertainty & Confidence February 2015

   <gml:Polygon srsName="urn:ogc:def:crs:EPSG::4326">
     <gml:exterior>
       <gml:LinearRing>
         <gml:posList>
           -33.856625 151.215906 -33.856299 151.215343
           -33.856326 151.214731 -33.857533 151.214495
           -33.857720 151.214613 -33.857369 151.215375
           -33.856625 151.215906
         </gml:posList>
       </gml:LinearRing>
     </gml:exterior>
   </gml:Polygon>
                   Figure 9: Bob's Polygon Location
 To convert this to a polygon, each point is firstly assigned an
 altitude of zero and converted to ECEF coordinates (see Appendix A).
 Then, a normal vector for this polygon is found (see Appendix B).
 The result of each of these stages is shown in Figure 10.  Note that
 the numbers shown in this document are rounded only for formatting
 reasons; the actual calculations do not include rounding, which would
 generate significant errors in the final values.

Thomson & Winterbottom Standards Track [Page 27] RFC 7459 Uncertainty & Confidence February 2015

 Polygon in ECEF coordinate space
    (repeated point omitted and transposed to fit):
          [ -4.6470e+06  2.5530e+06  -3.5333e+06 ]
          [ -4.6470e+06  2.5531e+06  -3.5332e+06 ]
  pecef = [ -4.6470e+06  2.5531e+06  -3.5332e+06 ]
          [ -4.6469e+06  2.5531e+06  -3.5333e+06 ]
          [ -4.6469e+06  2.5531e+06  -3.5334e+06 ]
          [ -4.6469e+06  2.5531e+06  -3.5333e+06 ]
 Normal Vector: n = [ -0.72782  0.39987  -0.55712 ]
 Transformation Matrix:
      [ -0.48152  -0.87643   0.00000 ]
  t = [ -0.48828   0.26827   0.83043 ]
      [ -0.72782   0.39987  -0.55712 ]
 Transformed Coordinates:
           [  8.3206e+01  1.9809e+04  6.3715e+06 ]
           [  3.1107e+01  1.9845e+04  6.3715e+06 ]
  pecef' = [ -2.5528e+01  1.9842e+04  6.3715e+06 ]
           [ -4.7367e+01  1.9708e+04  6.3715e+06 ]
           [ -3.6447e+01  1.9687e+04  6.3715e+06 ]
           [  3.4068e+01  1.9726e+04  6.3715e+06 ]
 Two dimensional polygon area: A = 12600 m^2
 Two-dimensional polygon centroid: C' = [ 8.8184e+00  1.9775e+04 ]
 Average of pecef' z coordinates: 6.3715e+06
 Reverse Transformation Matrix:
       [ -0.48152  -0.48828  -0.72782 ]
  t' = [ -0.87643   0.26827   0.39987 ]
       [  0.00000   0.83043  -0.55712 ]
 Polygon centroid (ECEF): C = [ -4.6470e+06  2.5531e+06  -3.5333e+06 ]
 Polygon centroid (Geo): Cg = [ -33.856926  151.215102  -4.9537e-04 ]
              Figure 10: Calculation of Polygon Centroid
 The point conversion for the polygon uses the final result, "Cg",
 ignoring the altitude since the original shape did not include
 altitude.
 To convert this to a circle, take the maximum distance in ECEF
 coordinates from the center point to each of the points.  This
 results in a radius of 99.1 meters.  Confidence is unchanged.

Thomson & Winterbottom Standards Track [Page 28] RFC 7459 Uncertainty & Confidence February 2015

6.2. Increasing and Decreasing Confidence

 Assume that confidence is known to be 19% for Alice's location
 information.  This is a typical value for a three-dimensional
 ellipsoid uncertainty of normal distribution where the standard
 deviation is used directly for uncertainty in each dimension.  The
 confidence associated with Alice's location estimate is quite low for
 many applications.  Since the estimate is known to follow a normal
 distribution, the method in Section 5.4.2 can be used.  Each axis can
 be scaled by:
    scale = erfinv(0.95^(1/3)) / erfinv(0.19^(1/3)) = 2.9937
 Ensuring that rounding always increases uncertainty, the location
 estimate at 95% includes a semi-major axis of 23.1, a semi-minor axis
 of 10 and a vertical axis of 86.
 Bob's location estimate (from the previous example) covers an area of
 approximately 12600 square meters.  If the estimate follows a
 rectangular distribution, the region of uncertainty can be reduced in
 size.  Here we find the confidence that Bob is within the smaller
 area of the Concert Hall.  For the Concert Hall, the polygon
 [-33.856473, 151.215257; -33.856322, 151.214973;
 -33.856424, 151.21471; -33.857248, 151.214753;
 -33.857413, 151.214941; -33.857311, 151.215128] is used.  To use this
 new region of uncertainty, find its area using the same translation
 method described in Section 5.1.1.2, which produces 4566.2 square
 meters.  Given that the Concert Hall is entirely within Bob's
 original location estimate, the confidence associated with the
 smaller area is therefore 95% * 4566.2 / 12600 = 34%.

6.3. Matching Location Estimates to Regions of Interest

 Suppose that a circular area is defined centered at
 [-33.872754, 151.20683] with a radius of 1950 meters.  To determine
 whether Bob is found within this area -- given that Bob is at
 [-34.407242, 150.882518] with an uncertainty radius 7.7156 meters --
 we apply the method in Section 5.5.  Using the converted Circle shape
 for Bob's location, the distance between these points is found to be
 1915.26 meters.  The area of overlap between Bob's location estimate
 and the region of interest is therefore 2209 square meters and the
 area of Bob's location estimate is 30853 square meters.  This gives
 the estimated probability that Bob is less than 1950 meters from the
 selected point as 67.8%.

Thomson & Winterbottom Standards Track [Page 29] RFC 7459 Uncertainty & Confidence February 2015

 Note that if 1920 meters were chosen for the distance from the
 selected point, the area of overlap is only 16196 square meters and
 the confidence is 49.8%.  Therefore, it is marginally more likely
 that Bob is outside the region of interest, despite the center point
 of his location estimate being within the region.

6.4. PIDF-LO with Confidence Example

 The PIDF-LO document in Figure 11 includes a representation of
 uncertainty as a circular area.  The confidence element (on the line
 marked with a comment) indicates that the confidence is 67% and that
 it follows a normal distribution.
   <pidf:presence
       xmlns:pidf="urn:ietf:params:xml:ns:pidf"
       xmlns:dm="urn:ietf:params:xml:ns:pidf:data-model"
       xmlns:gp="urn:ietf:params:xml:ns:pidf:geopriv10"
       xmlns:gs="http://www.opengis.net/pidflo/1.0"
       xmlns:gml="http://www.opengis.net/gml"
       xmlns:con="urn:ietf:params:xml:ns:geopriv:conf"
       entity="pres:alice@example.com">
     <dm:device id="sg89ab">
       <gp:geopriv>
         <gp:location-info>
           <gs:Circle srsName="urn:ogc:def:crs:EPSG::4326">
             <gml:pos>42.5463 -73.2512</gml:pos>
             <gs:radius uom="urn:ogc:def:uom:EPSG::9001">
               850.24
             </gs:radius>
           </gs:Circle>
 <!--c--> <con:confidence pdf="normal">67</con:confidence>
         </gp:location-info>
         <gp:usage-rules/>
       </gp:geopriv>
      <dm:deviceID>mac:010203040506</dm:deviceID>
    </dm:device>
  </pidf:presence>
              Figure 11: Example PIDF-LO with Confidence

Thomson & Winterbottom Standards Track [Page 30] RFC 7459 Uncertainty & Confidence February 2015

7. Confidence Schema

 <?xml version="1.0"?>
 <xs:schema
     xmlns:conf="urn:ietf:params:xml:ns:geopriv:conf"
     xmlns:xs="http://www.w3.org/2001/XMLSchema"
     targetNamespace="urn:ietf:params:xml:ns:geopriv:conf"
     elementFormDefault="qualified"
     attributeFormDefault="unqualified">
   <xs:annotation>
     <xs:appinfo
         source="urn:ietf:params:xml:schema:geopriv:conf">
       PIDF-LO Confidence
     </xs:appinfo>
     <xs:documentation
         source="http://www.rfc-editor.org/rfc/rfc7459.txt">
       This schema defines an element that is used for indicating
       confidence in PIDF-LO documents.
     </xs:documentation>
   </xs:annotation>
   <xs:element name="confidence" type="conf:confidenceType"/>
   <xs:complexType name="confidenceType">
     <xs:simpleContent>
       <xs:extension base="conf:confidenceBase">
         <xs:attribute name="pdf" type="conf:pdfType"
                       default="unknown"/>
       </xs:extension>
     </xs:simpleContent>
   </xs:complexType>
   <xs:simpleType name="confidenceBase">
     <xs:union>
       <xs:simpleType>
         <xs:restriction base="xs:decimal">
           <xs:minExclusive value="0.0"/>
           <xs:maxExclusive value="100.0"/>
         </xs:restriction>
       </xs:simpleType>
       <xs:simpleType>
         <xs:restriction base="xs:token">
           <xs:enumeration value="unknown"/>
         </xs:restriction>
       </xs:simpleType>
     </xs:union>
   </xs:simpleType>

Thomson & Winterbottom Standards Track [Page 31] RFC 7459 Uncertainty & Confidence February 2015

   <xs:simpleType name="pdfType">
     <xs:restriction base="xs:token">
       <xs:enumeration value="unknown"/>
       <xs:enumeration value="normal"/>
       <xs:enumeration value="rectangular"/>
     </xs:restriction>
   </xs:simpleType>
 </xs:schema>

8. IANA Considerations

8.1. URN Sub-Namespace Registration for

    urn:ietf:params:xml:ns:geopriv:conf
 A new XML namespace, "urn:ietf:params:xml:ns:geopriv:conf", has been
 registered, as per the guidelines in [RFC3688].
 URI:  urn:ietf:params:xml:ns:geopriv:conf
 Registrant Contact:  IETF GEOPRIV working group (geopriv@ietf.org),
    Martin Thomson (martin.thomson@gmail.com).
 XML:
     BEGIN
       <?xml version="1.0"?>
       <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
         "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
       <html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
         <head>
           <title>PIDF-LO Confidence Attribute</title>
         </head>
         <body>
           <h1>Namespace for PIDF-LO Confidence Attribute</h1>
           <h2>urn:ietf:params:xml:ns:geopriv:conf</h2>
           <p>See <a href="http://www.rfc-editor.org/rfc/rfc7459.txt">
              RFC 7459</a>.</p>
         </body>
       </html>
     END

Thomson & Winterbottom Standards Track [Page 32] RFC 7459 Uncertainty & Confidence February 2015

8.2. XML Schema Registration

 An XML schema has been registered, as per the guidelines in
 [RFC3688].
 URI:  urn:ietf:params:xml:schema:geopriv:conf
 Registrant Contact:  IETF GEOPRIV working group (geopriv@ietf.org),
    Martin Thomson (martin.thomson@gmail.com).
 Schema:  The XML for this schema can be found as the entirety of
    Section 7 of this document.

9. Security Considerations

 This document describes methods for managing and manipulating
 uncertainty in location.  No specific security concerns arise from
 most of the information provided.  The considerations of [RFC4119]
 all apply.
 A thorough treatment of the privacy implications of describing
 location information are discussed in [RFC6280].  Including
 uncertainty information increases the amount of information
 available; and altering uncertainty is not an effective privacy
 mechanism.
 Providing uncertainty and confidence information can reveal
 information about the process by which location information is
 generated.  For instance, it might reveal information that could be
 used to infer that a user is using a mobile device with a GPS, or
 that a user is acquiring location information from a particular
 network-based service.  A Rule Maker might choose to remove
 uncertainty-related fields from a location object in order to protect
 this information.  Note however that information might not be
 perfectly protected due to difficulties associated with location
 obfuscation, as described in Section 13.5 of [RFC6772].  In
 particular, increasing uncertainty does not necessarily result in a
 reduction of the information conveyed by the location object.
 Adding confidence to location information risks misinterpretation by
 consumers of location that do not understand the element.  This could
 be exploited, particularly when reducing confidence, since the
 resulting uncertainty region might include locations that are less
 likely to contain the Target than the recipient expects.  Since this
 sort of error is always a possibility, the impact of this is low.

Thomson & Winterbottom Standards Track [Page 33] RFC 7459 Uncertainty & Confidence February 2015

10. References

10.1. Normative References

 [RFC2119]  Bradner, S., "Key words for use in RFCs to Indicate
            Requirement Levels", BCP 14, RFC 2119, March 1997,
            <http://www.rfc-editor.org/info/rfc2119>.
 [RFC3688]  Mealling, M., "The IETF XML Registry", BCP 81, RFC 3688,
            January 2004, <http://www.rfc-editor.org/info/rfc3688>.
 [RFC3693]  Cuellar, J., Morris, J., Mulligan, D., Peterson, J., and
            J. Polk, "Geopriv Requirements", RFC 3693, February 2004,
            <http://www.rfc-editor.org/info/rfc3693>.
 [RFC4119]  Peterson, J., "A Presence-based GEOPRIV Location Object
            Format", RFC 4119, December 2005,
            <http://www.rfc-editor.org/info/rfc4119>.
 [RFC5139]  Thomson, M. and J. Winterbottom, "Revised Civic Location
            Format for Presence Information Data Format Location
            Object (PIDF-LO)", RFC 5139, February 2008,
            <http://www.rfc-editor.org/info/rfc5139>.
 [RFC5491]  Winterbottom, J., Thomson, M., and H. Tschofenig, "GEOPRIV
            Presence Information Data Format Location Object (PIDF-LO)
            Usage Clarification, Considerations, and Recommendations",
            RFC 5491, March 2009,
            <http://www.rfc-editor.org/info/rfc5491>.
 [RFC6225]  Polk, J., Linsner, M., Thomson, M., and B. Aboba, Ed.,
            "Dynamic Host Configuration Protocol Options for
            Coordinate-Based Location Configuration Information", RFC
            6225, July 2011, <http://www.rfc-editor.org/info/rfc6225>.
 [RFC6280]  Barnes, R., Lepinski, M., Cooper, A., Morris, J.,
            Tschofenig, H., and H. Schulzrinne, "An Architecture for
            Location and Location Privacy in Internet Applications",
            BCP 160, RFC 6280, July 2011,
            <http://www.rfc-editor.org/info/rfc6280>.

Thomson & Winterbottom Standards Track [Page 34] RFC 7459 Uncertainty & Confidence February 2015

10.2. Informative References

 [Convert]  Burtch, R., "A Comparison of Methods Used in Rectangular
            to Geodetic Coordinate Transformations", April 2006.
 [GeoShape] Thomson, M. and C. Reed, "GML 3.1.1 PIDF-LO Shape
            Application Schema for use by the Internet Engineering
            Task Force (IETF)", Candidate OpenGIS Implementation
            Specification 06-142r1, Version: 1.0, April 2007.
 [ISO.GUM]  ISO/IEC, "Guide to the expression of uncertainty in
            measurement (GUM)", Guide 98:1995, 1995.
 [NIST.TN1297]
            Taylor, B. and C. Kuyatt, "Guidelines for Evaluating and
            Expressing the Uncertainty of NIST Measurement Results",
            Technical Note 1297, September 1994.
 [RFC5222]  Hardie, T., Newton, A., Schulzrinne, H., and H.
            Tschofenig, "LoST: A Location-to-Service Translation
            Protocol", RFC 5222, August 2008,
            <http://www.rfc-editor.org/info/rfc5222>.
 [RFC6772]  Schulzrinne, H., Ed., Tschofenig, H., Ed., Cuellar, J.,
            Polk, J., Morris, J., and M. Thomson, "Geolocation Policy:
            A Document Format for Expressing Privacy Preferences for
            Location Information", RFC 6772, January 2013,
            <http://www.rfc-editor.org/info/rfc6772>.
 [Sunday02] Sunday, D., "Fast polygon area and Newell normal
            computation", Journal of Graphics Tools JGT, 7(2):9-13,
            2002.
 [TS-3GPP-23_032]
            3GPP, "Universal Geographical Area Description (GAD)",
            3GPP TS 23.032 12.0.0, September 2014.
 [Vatti92]  Vatti, B., "A generic solution to polygon clipping",
            Communications of the ACM Volume 35, Issue 7, pages 56-63,
            July 1992,
            <http://portal.acm.org/citation.cfm?id=129906>.
 [WGS84]    US National Imagery and Mapping Agency, "Department of
            Defense (DoD) World Geodetic System 1984 (WGS 84), Third
            Edition", NIMA TR8350.2, January 2000.

Thomson & Winterbottom Standards Track [Page 35] RFC 7459 Uncertainty & Confidence February 2015

Appendix A. Conversion between Cartesian and Geodetic Coordinates in

           WGS84
 The process of conversion from geodetic (latitude, longitude, and
 altitude) to ECEF Cartesian coordinates is relatively simple.
 In this appendix, the following constants and derived values are used
 from the definition of WGS84 [WGS84]:
    {radius of ellipsoid} R = 6378137 meters
    {inverse flattening} 1/f = 298.257223563
    {first eccentricity squared} e^2 = f * (2 - f)
    {second eccentricity squared} e'^2 = e^2 * (1 - e^2)
 To convert geodetic coordinates (latitude, longitude, altitude) to
 ECEF coordinates (X, Y, Z), use the following relationships:
    N = R / sqrt(1 - e^2 * sin(latitude)^2)
    X = (N + altitude) * cos(latitude) * cos(longitude)
    Y = (N + altitude) * cos(latitude) * sin(longitude)
    Z = (N*(1 - e^2) + altitude) * sin(latitude)
 The reverse conversion requires more complex computation, and most
 methods introduce some error in latitude and altitude.  A range of
 techniques are described in [Convert].  A variant on the method
 originally proposed by Bowring, which results in an acceptably small
 error, is described by the following:
    p = sqrt(X^2 + Y^2)
    r = sqrt(X^2 + Y^2 + Z^2)
    u = atan((1-f) * Z * (1 + e'^2 * (1-f) * R / r) / p)
    latitude = atan((Z + e'^2 * (1-f) * R * sin(u)^3)
    / (p - e^2 * R * cos(u)^3))
    longitude = atan2(Y, X)
    altitude = sqrt((p - R * cos(u))^2 + (Z - (1-f) * R * sin(u))^2)

Thomson & Winterbottom Standards Track [Page 36] RFC 7459 Uncertainty & Confidence February 2015

 If the point is near the poles, that is, "p < 1", the value for
 altitude that this method produces is unstable.  A simpler method for
 determining the altitude of a point near the poles is:
    altitude = |Z| - R * (1 - f)

Appendix B. Calculating the Upward Normal of a Polygon

 For a polygon that is guaranteed to be convex and coplanar, the
 upward normal can be found by finding the vector cross product of
 adjacent edges.
 For more general cases, the Newell method of approximation described
 in [Sunday02] may be applied.  In particular, this method can be used
 if the points are only approximately coplanar, and for non-convex
 polygons.
 This process requires a Cartesian coordinate system.  Therefore,
 convert the geodetic coordinates of the polygon to Cartesian, ECEF
 coordinates (Appendix A).  If no altitude is specified, assume an
 altitude of zero.
 This method can be condensed to the following set of equations:
    Nx = sum from i=1..n of (y[i] * (z[i+1] - z[i-1]))
    Ny = sum from i=1..n of (z[i] * (x[i+1] - x[i-1]))
    Nz = sum from i=1..n of (x[i] * (y[i+1] - y[i-1]))
 For these formulae, the polygon is made of points
 "(x[1], y[1], z[1])" through "(x[n], y[n], x[n])".  Each array is
 treated as circular, that is, "x[0] == x[n]" and "x[n+1] == x[1]".
 To translate this into a unit-vector; divide each component by the
 length of the vector:
    Nx' = Nx / sqrt(Nx^2 + Ny^2 + Nz^2)
    Ny' = Ny / sqrt(Nx^2 + Ny^2 + Nz^2)
    Nz' = Nz / sqrt(Nx^2 + Ny^2 + Nz^2)

Thomson & Winterbottom Standards Track [Page 37] RFC 7459 Uncertainty & Confidence February 2015

B.1. Checking That a Polygon Upward Normal Points Up

 RFC 5491 [RFC5491] stipulates that the Polygon shape be presented in
 counterclockwise direction so that the upward normal is in an upward
 direction.  Accidental reversal of points can invert this vector.
 This error can be hard to detect just by looking at the series of
 coordinates that form the polygon.
 Calculate the dot product of the upward normal of the polygon
 (Appendix B) and any vector that points away from the center of the
 earth from the location of polygon.  If this product is positive,
 then the polygon upward normal also points away from the center of
 the earth.
    The inverse cosine of this value indicates the angle between the
    horizontal plane and the approximate plane of the polygon.
 A unit vector for the upward direction at any point can be found
 based on the latitude (lat) and longitude (lng) of the point, as
 follows:
    Up = [ cos(lat) * cos(lng) ; cos(lat) * sin(lng) ; sin(lat) ]
 For polygons that span less than half the globe, any point in the
 polygon -- including the centroid -- can be selected to generate an
 approximate up vector for comparison with the upward normal.

Thomson & Winterbottom Standards Track [Page 38] RFC 7459 Uncertainty & Confidence February 2015

Acknowledgements

 Peter Rhodes provided assistance with some of the mathematical
 groundwork on this document.  Dan Cornford provided a detailed review
 and many terminology corrections.

Authors' Addresses

 Martin Thomson
 Mozilla
 331 E Evelyn Street
 Mountain View, CA  94041
 United States
 EMail: martin.thomson@gmail.com
 James Winterbottom
 Unaffiliated
 Australia
 EMail: a.james.winterbottom@gmail.com

Thomson & Winterbottom Standards Track [Page 39]

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