GEOPRIV M. Thomson
Internet-Draft Microsoft
Intended status: Informational J. Winterbottom
Expires: September 30, 2012 Andrew Corporation
March 29, 2012
Representation of Uncertainty and Confidence in PIDF-LO
draft-thomson-geopriv-uncertainty-07
Abstract
The key concepts of uncertainty and confidence as they pertain to
location information are defined. Methods for the manipulation of
location estimates that include uncertainty information are outlined.
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Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1. Conventions and Terminology . . . . . . . . . . . . . . . 3
2. A General Definition of Uncertainty . . . . . . . . . . . . . 3
2.1. Uncertainty as a Probability Distribution . . . . . . . . 4
2.2. Deprecation of the Terms Precision and Resolution . . . . 6
2.3. Accuracy as a Qualitative Concept . . . . . . . . . . . . 7
3. Uncertainty in Location . . . . . . . . . . . . . . . . . . . 7
3.1. Representation of Uncertainty and Confidence in PIDF-LO . 7
3.2. Uncertainty and Confidence for Civic Addresses . . . . . . 8
3.3. DHCP Location Configuration Information and Uncertainty . 9
4. Manipulation of Uncertainty . . . . . . . . . . . . . . . . . 9
4.1. Reduction of a Location Estimate to a Point . . . . . . . 10
4.1.1. Centroid Calculation . . . . . . . . . . . . . . . . . 10
4.1.1.1. Arc-Band Centroid . . . . . . . . . . . . . . . . 11
4.1.1.2. Polygon Centroid . . . . . . . . . . . . . . . . . 11
4.2. Conversion to Circle or Sphere . . . . . . . . . . . . . . 14
4.3. Three-Dimensional to Two-Dimensional Conversion . . . . . 15
4.4. Increasing and Decreasing Uncertainty and Confidence . . . 15
4.4.1. Rectangular Distributions . . . . . . . . . . . . . . 16
4.4.2. Normal Distributions . . . . . . . . . . . . . . . . . 16
4.5. Determining Whether a Location is Within a Given Region . 17
4.5.1. Determining the Area of Overlap for Two Circles . . . 18
4.5.2. Determining the Area of Overlap for Two Polygons . . . 19
5. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5.1. Reduction to a Point or Circle . . . . . . . . . . . . . . 19
5.2. Increasing and Decreasing Confidence . . . . . . . . . . . 23
5.3. Matching Location Estimates to Regions of Interest . . . . 23
6. Security Considerations . . . . . . . . . . . . . . . . . . . 24
7. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . 24
8. Informative References . . . . . . . . . . . . . . . . . . . . 24
Appendix A. Conversion Between Cartesian and Geodetic
Coordinates in WGS84 . . . . . . . . . . . . . . . . 25
Appendix B. Calculating the Upward Normal of a Polygon . . . . . 27
B.1. Checking that a Polygon Upward Normal Points Up . . . . . 27
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . . 28
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1. Introduction
Location information represents an estimation of the position of a
Target. Under ideal circumstances, a location estimate precisely
reflects the actual location of the Target. In reality, 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 their 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 area.
1.1. Conventions and Terminology
This document assumes a basic understanding of the principles of
mathematics, particularly statistics and geometry.
Some terminology is borrowed from [RFC3693] and
[I-D.ietf-geopriv-arch].
Mathematical formulae are presented using the following notation: add
"+", subtract "-", multiply "*", divide "/", power "^" and absolute
value "|x|". Precedence is indicated using parentheses.
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)", 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
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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 and 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 succintly 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 result indicates that the measurement indicates that the value
of "x" 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.
This document uses the term _uncertainty_ to refer in general to the
concept as well as more specifically to refer to the error increment.
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.
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 most
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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 (see Section 4.4). For a rectangular distribution, half of
the width of the distribution is used.
Figure 1 shows a normal and rectangular probability density function
with the mean (m) and standard deviation (s) labelled. 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
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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 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 4.4, 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
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information.
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, to
say that 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 2- or 3-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.
3.1. Representation of Uncertainty and Confidence in PIDF-LO
A set of shapes suitable for the expression of uncertainty in
location estimates in the Presence Information Data Format - Location
Object (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 does not include an indication of confidence, but that
confidence is 95%, by definition in [RFC5491]. Similarly, the
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PIDF-LO format does not provide an indication of the shape of the
PDF.
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-dimension shapes like the Circle.
3.2. Uncertainty and Confidence for Civic Addresses
Civic addresses [RFC5139] inherently include uncertainty, based on
the area of the most precise element that is specified. Uncertainty
is effectively defined by the presence or absence of elements --
elements that are not present are deemed to be 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.
Note: This view is one perspective on the process of geo-coding -
the translation of a civic address to a geodetic location.
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 partions 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 to represent a
region of uncertainty. At no stage does this process select a point;
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although, as spaces get smaller this distinction might have no
practical significance and an approximation if a point could be used.
Uncertainty in civic addresses can be increased by removing elements.
This doesn't necessarily improve confidence in the same way that
arbitrarily increasing uncertainty in a geodetic location doesn't
increase confidence.
3.3. 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 [RFC3825] provide an indication of
uncertainty in one dimension.
[RFC3825] defines a means for representing uncertainty, but a value
for confidence is not specified. A default value of 95% confidence
can be assumed for the combination of the uncertainty on each axis.
That is, the confidence of the resultant rectangular polygon or prism
is 95%.
4. 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.
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.
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4.1. 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 minimises 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.
4.1.1. 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
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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.
4.1.1.1. 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)". Negative values are possible if
the angle of opening is greater than 180 degrees; negative values
indicate that the centroid is found along the angle
"a + (o/2) + 180".
4.1.1.2. 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 is known.
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.
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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 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
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)
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Note: The formula for the area of a polygon will return a negative
value if the polygon is specified in 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 either
be ignored or reset after converting back to a geodetic value.
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
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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
4.2. 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 furthest vertex of the polygon
(for a Prism, it is only necessary to check points on the base).
Arc-Band: The furthest 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.
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4.3. Three-Dimensional to Two-Dimensional Conversion
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%.
4.4. 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
both 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.
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 [I-D.thomson-geopriv-location-obscuring]), 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
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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.
4.4.1. Rectangular Distributions
Uncertainty that follows a rectangular distribution can only be
decreased in size. 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 "Cb" 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.
4.4.2. Normal Distributions
Uncertainty and confidence can be both increased and decreased for a
normal distribution. However, the process is more complicated.
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 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
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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.
4.5. Determining Whether a Location is Within a Given Region
A number of applications require that a judgement be made about
whether a Target is within a given region of interest. Given a
location estimate with uncertainty, this judgement can be difficult.
A location estimate represents a probability distribution, and the
true location of the Target cannot be definitively known. Therefore,
the judgement 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
between the uncertainty region and the target region. If the PDF is
unknown, no determination can be made. When judging whether a
location is within a given region, uncertainties using these PDFs can
be assumed to be rectangular. If this assumption is made, the
confidence should be scaled to 95%, if possible.
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.
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_.-""""-._
.' `. _ Region of
/ \ / Uncertainty
..+-"""--.. |
.-' | :::::: `-. |
,' | :: Ao ::: `. |
/ \ :::::::::: \ /
/ `._ :::::: _.X
| `-....-' |
| |
| |
\ /
`. .' \_ Region of
`._ _.' Interest
`--..___..--'
Figure 7: Area of Overlap Between Two Circular Regions
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. In the absence of specific
recommendations, this document suggests that the probability be
greater than 50% before a decision is made. 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.
4.5.1. 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 4.2)
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 formulas provide the area of overlap "Ao".
o If the circles don't overlap, that is "d >= r+R", "Ao" is zero.
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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)
4.5.2. 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.
5. Examples
This section presents some examples of how to apply the methods
described in Section 4.
5.1. Reduction to a Point or Circle
Alice receives a location estimate from her LIS that contains a
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.
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-34.407242 150.882518 34
7.7156
3.31
28.7
43
Figure 8
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. This
information is shown in Figure 9.
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-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
Figure 9
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
results of each of these stages is shown in Figure 10. Note that the
numbers shown are all rounded; no rounding is possible during this
process since rounding would contribute significant errors.
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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
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.
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5.2. Increasing and Decreasing Confidence
Assuming that confidence is known to be 19% for Alice's location
information. This is typical value for a three-dimensional ellipsoid
uncertainty of normal distribution where the standard deviation is
supplied 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 4.4.2 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 covers an area of approximately 12600 square
meters. If the estimate follows a rectangular distribution, the
region of uncertainty can be reduced in size. To find the confidence
that he is within the smaller area of the concert hall, given by 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]. To use this new
region of uncertainty, find its area using the same translation
method described in Section 4.1.1.2, which is 4566.2 square meters.
The confidence associated with the smaller area is therefore 95% *
4566.2 / 12600 = 34%.
5.3. Matching Location Estimates to Regions of Interest
Suppose than 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, we apply the method in
Section 4.5. 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 probability
that Bob is less than 1950 meters from the selected point as 67.8%.
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.
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6. Security Considerations
This document describes methods for managing and manipulating
uncertainty in location. No specific security concerns arise from
most of the information provided.
7. Acknowledgements
Peter Rhodes provided assistance with some of the mathematical
groundwork on this document. Dan Cornford provided a detailed review
and many terminology corrections.
8. 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.
[I-D.ietf-geopriv-arch]
Barnes, R., Lepinski, M., Cooper, A., Morris, J.,
Tschofenig, H., and H. Schulzrinne, "An Architecture for
Location and Location Privacy in Internet Applications",
draft-ietf-geopriv-arch-03 (work in progress),
October 2010.
[I-D.ietf-geopriv-policy]
Cuellar, J., Tschofenig, H., Schulzrinne, H., Polk, J.,
Morris, J., and M. Thomson, "Geolocation Policy: A
Document Format for Expressing Privacy Preferences for
Location Information", draft-ietf-geopriv-policy-25 (work
in progress), October 2011.
[I-D.thomson-geopriv-location-obscuring]
Thomson, M., "Obscuring Location",
draft-thomson-geopriv-location-obscuring-03 (work in
progress), June 2011.
[ISO.GUM] ISO/IEC, "Guide to the expression of uncertainty in
measurement (GUM)", Guide 98:1995, 1995.
[NIST.TN1297]
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Taylor, B. and C. Kuyatt, "Guidelines for Evaluating and
Expressing the Uncertainty of NIST Measurement Results",
Technical Note 1297, Sep 1994.
[RFC3693] Cuellar, J., Morris, J., Mulligan, D., Peterson, J., and
J. Polk, "Geopriv Requirements", RFC 3693, February 2004.
[RFC3694] Danley, M., Mulligan, D., Morris, J., and J. Peterson,
"Threat Analysis of the Geopriv Protocol", RFC 3694,
February 2004.
[RFC3825] Polk, J., Schnizlein, J., and M. Linsner, "Dynamic Host
Configuration Protocol Option for Coordinate-based
Location Configuration Information", RFC 3825, July 2004.
[RFC4119] Peterson, J., "A Presence-based GEOPRIV Location Object
Format", RFC 4119, December 2005.
[RFC5139] Thomson, M. and J. Winterbottom, "Revised Civic Location
Format for Presence Information Data Format Location
Object (PIDF-LO)", RFC 5139, February 2008.
[RFC5222] Hardie, T., Newton, A., Schulzrinne, H., and H.
Tschofenig, "LoST: A Location-to-Service Translation
Protocol", RFC 5222, August 2008.
[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.
[Sunday02]
Sunday, D., "Fast polygon area and Newell normal
computation", Journal of Graphics Tools JGT, 7(2):9-
13,2002, 2002, .
[Vatti92] Vatti, B., "A generic solution to polygon clipping",
Communications of the ACM Vol35, Issue7, pp56-63, 1992,
.
[WGS84] US National Imagery and Mapping Agency, "Department of
Defense (DoD) World Geodetic System 1984 (WGS 84), Third
Edition", NIMA TR8350.2, January 2000.
Appendix A. Conversion Between Cartesian and Geodetic Coordinates in
WGS84
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The process of conversion from geodetic (latitude, longitude and
altitude) to earth-centered, earth-fixed (ECEF) Cartesian coordinates
is relatively simple.
In this section, 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 = atan(Y / X)
altitude = sqrt((p - R * cos(u))^2 + (Z - (1-f) * R * sin(u))^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:
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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)
B.1. Checking that a Polygon Upward Normal Points Up
RFC 5491 [RFC5491] mandates that polygons be presented in anti-
clockwise 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.
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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.
Authors' Addresses
Martin Thomson
Microsoft
3210 Porter Drive
Palo Alto, CA 94304
US
Email: martin.thomson@gmail.com
James Winterbottom
Andrew Corporation
Andrew Building (39)
Wollongong University Campus
Northfields Avenue
Wollongong, NSW 2522
AU
Email: james.winterbottom@andrew.com
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