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+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.
+
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+Thomson & Winterbottom       Standards Track                    [Page 1]
+
+RFC 7459                Uncertainty & Confidence           February 2015
+
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+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.
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+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
+
+
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+
+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
+
+
+
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+
+Thomson & Winterbottom       Standards Track                    [Page 5]
+
+RFC 7459                Uncertainty & Confidence           February 2015
+
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+   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%.
+
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+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 (c)
+   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.
+
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+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
+
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+Thomson & Winterbottom       Standards Track                   [Page 39]
+