RFC 7459

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)

the Presence Information Data Format Location Object (PIDF-LO)

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.

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.

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.

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.

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

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

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].

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

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.

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].

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

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)".

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

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.

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.

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.

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%.

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%.

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.

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.

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.

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

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.

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]).

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

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

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.

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.

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

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

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.

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

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.

the uncertainty representations from [RFC5491]. That is, the

confidence of the resultant rectangular Polygon or Prism is assumed

to be 95%.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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)".

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.

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

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

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.

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.

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).

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

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).

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.

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.

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%.

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.

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.

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.

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.

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.

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

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.

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

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

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

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)

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.

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)

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.

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.

This section presents some examples of how to apply the methods

described in Section 5.

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.

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.

<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.

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.

<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.

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.

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.

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%.

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%.

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%.

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

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.

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

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>

<?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>

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

<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>

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

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.

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.

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.

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>.

[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>.

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.

[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.

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)

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)

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)

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)

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)

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.

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.

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

Peter Rhodes provided assistance with some of the mathematical

groundwork on this document. Dan Cornford provided a detailed review

and many terminology corrections.

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