Introduction - Motivation

Device Calibration - Device Diversity - Manual Calibration

Self-Calibration

Device Signal Strength Self-Calibration using Histograms

- RSS Histograms - Self-Calibration Method

Performance Evaluation - Measurement Setup - Experimental Results

Conclusions

Christos Laoudias∗ , Robert Pich´e† and Christos Panayiotou∗ ∗

KIOS Research Center for Intelligent Systems and Networks, University of Cyprus † Tampere University of Technology, Tampere, Finland

- Concluding Remarks

International Conference on Indoor Positioning and Indoor Navigation, Sydney, Australia

15 November 2012

Outline

Introduction - Motivation

Introduction

Device Calibration - Device Diversity - Manual Calibration

Device Calibration

Self-Calibration - RSS Histograms - Self-Calibration Method

Self-Calibration

Performance Evaluation - Measurement Setup - Experimental Results

Conclusions - Concluding Remarks

Performance Evaluation Conclusions

International Conference on Indoor Positioning and Indoor Navigation, Sydney, Australia

15 November 2012

Motivation of our work I

RSS is intended for determining the signal quality and not for positioning purposes

Introduction - Motivation

Device Calibration - Device Diversity - Manual Calibration

Self-Calibration - RSS Histograms - Self-Calibration Method

Performance Evaluation - Measurement Setup - Experimental Results

Conclusions - Concluding Remarks

International Conference on Indoor Positioning and Indoor Navigation, Sydney, Australia

15 November 2012

Motivation of our work I

RSS is intended for determining the signal quality and not for positioning purposes

I

Different devices do not report RSS values in the same way

Introduction - Motivation

Device Calibration - Device Diversity - Manual Calibration

I

Self-Calibration

I

- RSS Histograms - Self-Calibration Method

I

Performance Evaluation - Measurement Setup - Experimental Results

I

The WiFi standard (IEEE 802.11) defines the RSS Indicator (1 byte integer) for measuring RSS in [0 255] Each vendor’s implementation is limited up to RSSImax RSSI is mapped to power values in dBm internally by the device driver (proprietary information) Even worse: same chipsets may not report the same RSS values due to different antennas or packaging

Conclusions - Concluding Remarks

International Conference on Indoor Positioning and Indoor Navigation, Sydney, Australia

15 November 2012

Motivation of our work I

RSS is intended for determining the signal quality and not for positioning purposes

I

Different devices do not report RSS values in the same way

Introduction - Motivation

Device Calibration

I

- Device Diversity - Manual Calibration

Self-Calibration

I

- RSS Histograms - Self-Calibration Method

I

Performance Evaluation

I

- Measurement Setup - Experimental Results

Conclusions - Concluding Remarks

I

The WiFi standard (IEEE 802.11) defines the RSS Indicator (1 byte integer) for measuring RSS in [0 255] Each vendor’s implementation is limited up to RSSImax RSSI is mapped to power values in dBm internally by the device driver (proprietary information) Even worse: same chipsets may not report the same RSS values due to different antennas or packaging

Using a new device for positioning is feasible, but the RSS values are not compatible with the radiomap

International Conference on Indoor Positioning and Indoor Navigation, Sydney, Australia

15 November 2012

Motivation of our work I

RSS is intended for determining the signal quality and not for positioning purposes

I

Different devices do not report RSS values in the same way

Introduction - Motivation

Device Calibration

I

- Device Diversity - Manual Calibration

Self-Calibration

I

- RSS Histograms - Self-Calibration Method

I

Performance Evaluation

I

- Measurement Setup - Experimental Results

Conclusions

The WiFi standard (IEEE 802.11) defines the RSS Indicator (1 byte integer) for measuring RSS in [0 255] Each vendor’s implementation is limited up to RSSImax RSSI is mapped to power values in dBm internally by the device driver (proprietary information) Even worse: same chipsets may not report the same RSS values due to different antennas or packaging

I

Using a new device for positioning is feasible, but the RSS values are not compatible with the radiomap

I

Best accuracy is guaranteed only if the user carries the same device during positioning, otherwise calibration is required

- Concluding Remarks

International Conference on Indoor Positioning and Indoor Navigation, Sydney, Australia

15 November 2012

Motivation of our work I

RSS is intended for determining the signal quality and not for positioning purposes

I

Different devices do not report RSS values in the same way

Introduction - Motivation

Device Calibration

I

- Device Diversity - Manual Calibration

Self-Calibration

I

- RSS Histograms - Self-Calibration Method

I

Performance Evaluation

I

- Measurement Setup - Experimental Results

Conclusions

The WiFi standard (IEEE 802.11) defines the RSS Indicator (1 byte integer) for measuring RSS in [0 255] Each vendor’s implementation is limited up to RSSImax RSSI is mapped to power values in dBm internally by the device driver (proprietary information) Even worse: same chipsets may not report the same RSS values due to different antennas or packaging

I

Using a new device for positioning is feasible, but the RSS values are not compatible with the radiomap

I

Best accuracy is guaranteed only if the user carries the same device during positioning, otherwise calibration is required

I

Existing calibration methods do not fit well in real-time positioning scenarios

- Concluding Remarks

International Conference on Indoor Positioning and Indoor Navigation, Sydney, Australia

15 November 2012

Device Diversity

Introduction - Motivation

Device Calibration - Device Diversity - Manual Calibration

Self-Calibration - RSS Histograms - Self-Calibration Method

Performance Evaluation - Measurement Setup - Experimental Results

Conclusions - Concluding Remarks

Source: K. Kaemarungsi (2006) International Conference on Indoor Positioning and Indoor Navigation, Sydney, Australia

15 November 2012

Introduction - Motivation

Device Calibration - Device Diversity - Manual Calibration

Self-Calibration - RSS Histograms - Self-Calibration Method

Conclusions - Concluding Remarks

−20

−30

−30

−40 −50 −60 −70 −80 RSS pairs Cloud center Least−squares Fit

−90 −100 −100

Performance Evaluation - Measurement Setup - Experimental Results

−20

Mean RSS from Asus eeePC [dBm]

Mean RSS from HP iPAQ [dBm]

Good News: Linearity between RSS values

I

−90

−80 −70 −60 −50 −40 Mean RSS from HTC Flyer [dBm]

−30

−20

−40 −50 −60 −70 −80 RSS pairs Cloud center Least−squares Fit

−90 −100 −100

−90

−80 −70 −60 −50 −40 −30 Mean RSS from Samsung Nexus S [dBm]

−20

Manual Calibration: Collect several colocated RSS pairs at known locations and estimate the linear coefficients through least squares (2) (1) ¯rij = α12¯rij + β12

International Conference on Indoor Positioning and Indoor Navigation, Sydney, Australia

15 November 2012

Introduction - Motivation

Device Calibration - Device Diversity - Manual Calibration

Self-Calibration - RSS Histograms - Self-Calibration Method

Conclusions

−20

−30

−30

−40 −50 −60 −70 −80 RSS pairs Cloud center Least−squares Fit

−90 −100 −100

Performance Evaluation - Measurement Setup - Experimental Results

−20

Mean RSS from Asus eeePC [dBm]

Mean RSS from HP iPAQ [dBm]

Good News: Linearity between RSS values

−90

−80 −70 −60 −50 −40 Mean RSS from HTC Flyer [dBm]

−30

−20

−40 −50 −60 −70 −80 RSS pairs Cloud center Least−squares Fit

−90 −100 −100

−90

−80 −70 −60 −50 −40 −30 Mean RSS from Samsung Nexus S [dBm]

−20

I

Manual Calibration: Collect several colocated RSS pairs at known locations and estimate the linear coefficients through least squares (2) (1) ¯rij = α12¯rij + β12

I

Limited Applicability: (i) User needs to be familiar with the indoor area and (ii) a considerable data collection effort is required

- Concluding Remarks

International Conference on Indoor Positioning and Indoor Navigation, Sydney, Australia

15 November 2012

Can we do it more efficiently?

Introduction - Motivation

Device Calibration - Device Diversity - Manual Calibration

Self-Calibration

Objectives I

Fully automatic approach with short calibration time

I

Runs concurrently with positioning while the user walks around

I

No user intervention or tedious data collection

- RSS Histograms - Self-Calibration Method

Performance Evaluation - Measurement Setup - Experimental Results

Idea

Conclusions - Concluding Remarks

I

Perform device self-calibration on-the-fly using histograms of RSS values observed simultaneously with positioning

International Conference on Indoor Positioning and Indoor Navigation, Sydney, Australia

15 November 2012

- Motivation

Device Calibration - Device Diversity - Manual Calibration

0.05

0.05

0.045

0.045

0.04

0.04

0.035

0.035

0.03

Probability

Introduction

Probability

RSS Histograms

0.025

0.03 0.025

0.02

0.02

0.015

0.015

0.01

0.01

0.005

0.005

Self-Calibration

0 −100

- RSS Histograms - Self-Calibration Method

−90

−80

−70

−60 −50 −40 Mean RSS Value

−30

−20

0 −100

−10

−90

−80

−70

−60 −50 −40 Mean RSS Value

−30

−20

−10

Figure: HP iPAQ (left) and Asus eeePC (right)

Performance Evaluation

Conclusions Probability

- Concluding Remarks

0.05

1

0.045

0.9

0.04

0.8

0.035

0.7

0.03

0.6

Probability

- Measurement Setup - Experimental Results

0.025 0.02

0.5 0.4

0.015

0.3

0.01

0.2 0.1

0.005 0 −100

HP iPAQ Asus eeePC HTC Flyer

−90

−80

−70

−60 −50 −40 Mean RSS Value

−30

−20

−10

0 −100

−90

−80

−70

−60 −50 −40 Mean RSS Value

−30

−20

−10

Figure: HTC Flyer (left) and Empirical cdfs (right) International Conference on Indoor Positioning and Indoor Navigation, Sydney, Australia

15 November 2012

Self-Calibration Method Reference Device ecdf

Device Calibration

(α , β )

Introduction - Motivation

Device Calibration

Transformation

- Device Diversity - Manual Calibration

s% (k )

Self-Calibration - RSS Histograms - Self-Calibration Method

Performance Evaluation

User Device ecdf

ri

Positioning Algorithm

s(k )

ˆl(k )

Radiomap

- Measurement Setup - Experimental Results

Conclusions - Concluding Remarks

International Conference on Indoor Positioning and Indoor Navigation, Sydney, Australia

15 November 2012

Self-Calibration Method Reference Device ecdf

Device Calibration

(α , β )

Introduction - Motivation

Device Calibration

Transformation

- Device Diversity - Manual Calibration

s% (k )

Self-Calibration - RSS Histograms - Self-Calibration Method

Performance Evaluation

User Device ecdf

ri

Positioning Algorithm

s(k )

ˆl(k )

Radiomap

- Measurement Setup - Experimental Results

Conclusions

1. Create the ecdf of the reference device from the radiomap

- Concluding Remarks

International Conference on Indoor Positioning and Indoor Navigation, Sydney, Australia

15 November 2012

Self-Calibration Method Reference Device ecdf

Device Calibration

(α , β )

Introduction - Motivation

Device Calibration

Transformation

- Device Diversity - Manual Calibration

s% (k )

Self-Calibration - RSS Histograms - Self-Calibration Method

Performance Evaluation

User Device ecdf

ri

Positioning Algorithm

s(k )

ˆl(k )

Radiomap

- Measurement Setup - Experimental Results

Conclusions - Concluding Remarks

1. Create the ecdf of the reference device from the radiomap 2. Create and update the ecdf of the new device by using s(k)

International Conference on Indoor Positioning and Indoor Navigation, Sydney, Australia

15 November 2012

Self-Calibration Method Reference Device ecdf

Device Calibration

(α , β )

Introduction - Motivation

Device Calibration

Transformation

- Device Diversity - Manual Calibration

s% (k )

Self-Calibration - RSS Histograms - Self-Calibration Method

Performance Evaluation

User Device ecdf

ri

Positioning Algorithm

s(k )

ˆl(k )

Radiomap

- Measurement Setup - Experimental Results

Conclusions - Concluding Remarks

1. Create the ecdf of the reference device from the radiomap 2. Create and update the ecdf of the new device by using s(k) 3. Fit a linear mapping between the reference and new device to obtain (α, β) by using “representative” ecdf values

International Conference on Indoor Positioning and Indoor Navigation, Sydney, Australia

15 November 2012

Self-Calibration Method Reference Device ecdf

Device Calibration

(α , β )

Introduction - Motivation

Device Calibration

Transformation

- Device Diversity - Manual Calibration

s% (k )

Self-Calibration - RSS Histograms - Self-Calibration Method

Performance Evaluation

User Device ecdf

ri

Positioning Algorithm

s(k )

ˆl(k )

Radiomap

- Measurement Setup - Experimental Results

Conclusions - Concluding Remarks

1. Create the ecdf of the reference device from the radiomap 2. Create and update the ecdf of the new device by using s(k) 3. Fit a linear mapping between the reference and new device to obtain (α, β) by using “representative” ecdf values 4. Transform the observed RSS values with ˜sj (k) = αsj (k) + β

International Conference on Indoor Positioning and Indoor Navigation, Sydney, Australia

15 November 2012

Self-Calibration Method Reference Device ecdf

Device Calibration

(α , β )

Introduction - Motivation

Device Calibration

Transformation

- Device Diversity - Manual Calibration

s% (k )

Self-Calibration - RSS Histograms - Self-Calibration Method

Performance Evaluation

User Device ecdf

ri

Positioning Algorithm

s(k )

ˆl(k )

Radiomap

- Measurement Setup - Experimental Results

Conclusions - Concluding Remarks

1. Create the ecdf of the reference device from the radiomap 2. Create and update the ecdf of the new device by using s(k) 3. Fit a linear mapping between the reference and new device to obtain (α, β) by using “representative” ecdf values 4. Transform the observed RSS values with ˜sj (k) = αsj (k) + β ˆ 5. Estimate location `(k) with any fingerprint-based algorithm

International Conference on Indoor Positioning and Indoor Navigation, Sydney, Australia

15 November 2012

Inverse ecdf Linear fitting 1 0.9 0.8

HP iPAQ HTC Flyer

0.7

Introduction 0.6 Probability

- Motivation

Device Calibration - Device Diversity - Manual Calibration

0.4 0.3

Self-Calibration

0.2

- RSS Histograms - Self-Calibration Method

0.1 0 -100

Performance Evaluation - Measurement Setup - Experimental Results

0.5

-90

-80

-70

-60 -50 -40 Mean RSS Value

-30

-20

-10

I

F (x) gives the probability that the RSS value is less than x, F −1 (y ) returns the RSS value that corresponds to the y -th cdf percentile

I

Fr (x) and Fu (x) are the ecdfs of the reference and user device

I

Fr−1 (y ) = αFu−1 (y ) + β, y ∈ {0.1, 0.2, . . . , 0.9}

I

(α, β) are initialized to (1, 0) and updated periodically (e.g. every 10 sec) thereafter, while the user is walking

Conclusions - Concluding Remarks

International Conference on Indoor Positioning and Indoor Navigation, Sydney, Australia

15 November 2012

Inverse ecdf Linear fitting 1 0.9 0.8

HP iPAQ HTC Flyer

0.7

Introduction 0.6 Probability

- Motivation

Device Calibration - Device Diversity - Manual Calibration

0.4 0.3

Self-Calibration

0.2

- RSS Histograms - Self-Calibration Method

0.1 0 -100

Performance Evaluation - Measurement Setup - Experimental Results

0.5

-90

-80

-70

-60 -50 -40 Mean RSS Value

-30

-20

-10

I

F (x) gives the probability that the RSS value is less than x, F −1 (y ) returns the RSS value that corresponds to the y -th cdf percentile

I

Fr (x) and Fu (x) are the ecdfs of the reference and user device

I

Fr−1 (y ) = αFu−1 (y ) + β, y ∈ {0.1, 0.2, . . . , 0.9}

I

(α, β) are initialized to (1, 0) and updated periodically (e.g. every 10 sec) thereafter, while the user is walking

Conclusions - Concluding Remarks

International Conference on Indoor Positioning and Indoor Navigation, Sydney, Australia

15 November 2012

Inverse ecdf Linear fitting 1 0.9 0.8

HP iPAQ HTC Flyer

0.7

Introduction 0.6 Probability

- Motivation

Device Calibration - Device Diversity - Manual Calibration

0.4 0.3

Self-Calibration

-87 dBm

0.2

- RSS Histograms - Self-Calibration Method

-66 dBm

0.1 0 -100

Performance Evaluation - Measurement Setup - Experimental Results

0.5

-90

-80

-70

-60 -50 -40 Mean RSS Value

-30

-20

-10

I

F (x) gives the probability that the RSS value is less than x, F −1 (y ) returns the RSS value that corresponds to the y -th cdf percentile

I

Fr (x) and Fu (x) are the ecdfs of the reference and user device

I

Fr−1 (y ) = αFu−1 (y ) + β, y ∈ {0.1, 0.2, . . . , 0.9}

I

(α, β) are initialized to (1, 0) and updated periodically (e.g. every 10 sec) thereafter, while the user is walking

Conclusions - Concluding Remarks

International Conference on Indoor Positioning and Indoor Navigation, Sydney, Australia

15 November 2012

Experimental Setup

Introduction - Motivation

Device Calibration - Device Diversity - Manual Calibration START

Self-Calibration - RSS Histograms - Self-Calibration Method

END

Performance Evaluation - Measurement Setup - Experimental Results

Conclusions

I

560 m2 office, 9 WiFi APs, 5 devices (1 HP iPAQ PDA, 1 Asus eeePC laptop, 1 HTC Flyer Android tablet, 2 Android smartphones)

I

Training Data: 105 reference locations, 20 fingerprints per location (2100 in total) with each device for comparison

I

Testing Data: Route with 2 segments, 96 test locations, 1 fingerprint per location, route sampled 10 times

- Concluding Remarks

International Conference on Indoor Positioning and Indoor Navigation, Sydney, Australia

15 November 2012

Experimental Results – 10 Routes 9

Introduction - Motivation

Device Calibration - Device Diversity - Manual Calibration

Self-Calibration - RSS Histograms - Self-Calibration Method

Performance Evaluation - Measurement Setup - Experimental Results

Mean Positioning Error Per Route [m]

8 7 6 5 4 3 2 1 0

No Calibration

Conclusions

Self−Calibration

Manual Calibration

Device−Specific

1

- Concluding Remarks

0.9 0.8

HP iPAQ HTC Flyer

0.7

Probability

0.6 0.5 0.4 0.3 0.2 0.1 0 -100

-90

-80

-70

-60 -50 -40 Mean RSS Value

-30

-20

-10

Figure: HTC Flyer user with HP iPAQ radiomap International Conference on Indoor Positioning and Indoor Navigation, Sydney, Australia

15 November 2012

Experimental Results – 10 Routes 4

- Motivation

Device Calibration - Device Diversity - Manual Calibration

Self-Calibration - RSS Histograms - Self-Calibration Method

Performance Evaluation

Mean Positioning Error Per Route [m]

3.5

Introduction

3 2.5 2 1.5 1 0.5 0

No Calibration

Self−Calibration Manual Calibration Device−Specific

- Measurement Setup - Experimental Results

Conclusions

1 0.9

- Concluding Remarks

0.8 0.7

Probability

0.6 0.5 0.4 0.3 0.2 Asus eeePC HTC Flyer

0.1 0 -100

-90

-80

-70 -60 -50 Mean RSS Value

-40

-30

-20

Figure: HTC Flyer user with Asus eeePC radiomap International Conference on Indoor Positioning and Indoor Navigation, Sydney, Australia

15 November 2012

Experimental Results – Single Route 18 No Calibration Self−Calibration Device−Specific

16 14

Introduction Positioning Error [m]

- Motivation

Device Calibration - Device Diversity - Manual Calibration

Self-Calibration - RSS Histograms - Self-Calibration Method

12 10 8 6 4

Performance Evaluation

2 0

- Measurement Setup - Experimental Results

0

10

20

30

40

50 Samples

60

70

80

90

100

Conclusions - Concluding Remarks

I

iPAQ radiomap with Flyer user-carried device

I

For the first 10 sec the device is uncalibrated and accuracy is not adequate

I

Beyond that point, the device is automatically calibrated and accuracy is greatly improved

International Conference on Indoor Positioning and Indoor Navigation, Sydney, Australia

15 November 2012

Results with pairwise device combinations Table: Median of the mean error ¯ [m], with and without calibration. Introduction - Motivation

Device Calibration - Device Diversity - Manual Calibration

Self-Calibration - RSS Histograms - Self-Calibration Method

Performance Evaluation

iPAQ eeePC Flyer Desire Nexus S

iPAQ

eeePC

Flyer

Desire

Nexus S

2.7 2.8 (4.4) 3.2 (5.9) 3.4 (6.1) 3.0 (6.2)

2.8 (6.6) 2.3 2.6 (3.0) 2.8 (3.2) 2.6 (2.8)

3.0 (7.5) 2.3 (2.8) 1.9 2.5 (2.5) 2.7 (2.7)

2.9 (8.4) 2.6 (3.5) 2.1 (2.3) 2.4 2.4 (2.5)

2.6 (7.7) 2.5 (2.9) 2.6 (2.7) 2.5 (2.6) 2.3

- Measurement Setup - Experimental Results

Conclusions - Concluding Remarks

I

All 5 devices used as a reference (row) and test device (column)

I

Mean positioning error using device self-calibration (results without calibration shown in parentheses)

I

The diagonal cells report the accuracy when the reference and test devices are the same (i.e. device-specific radiomap is used)

I

Self-calibration improves the accuracy for all device pairs

International Conference on Indoor Positioning and Indoor Navigation, Sydney, Australia

15 November 2012

Concluding Remarks

Introduction - Motivation

Device Calibration - Device Diversity - Manual Calibration

Self-Calibration - RSS Histograms - Self-Calibration Method

Device diversity is one of the reasons that hinders the proliferation of RSS-based positioning systems. Our Contributions I

Low-complexity, yet effective method that allows any mobile device to be self-calibrated

I

Automatic calibration is attained shortly after the user has started positioning

Performance Evaluation - Measurement Setup - Experimental Results

Conclusions - Concluding Remarks

Future Work I

Application in larger scale setups featuring non uniform WiFi AP layouts (possible skewness of the RSS histograms)

I

Integrate with our Airplace indoor positioning platform developed for Android smartphones

http://www2.ucy.ac.cy/~laoudias/pages/platform.html International Conference on Indoor Positioning and Indoor Navigation, Sydney, Australia

15 November 2012

Introduction - Motivation

Device Calibration

Thank you for your attention

- Device Diversity - Manual Calibration

Self-Calibration - RSS Histograms - Self-Calibration Method

Performance Evaluation - Measurement Setup - Experimental Results

Conclusions - Concluding Remarks

Contact Christos Laoudias KIOS Research Center for Intelligent Systems and Networks Department of Electrical & Computer Engineering University of Cyprus Email: [email protected]

International Conference on Indoor Positioning and Indoor Navigation, Sydney, Australia

15 November 2012

Introduction - Motivation

Device Calibration - Device Diversity - Manual Calibration

Self-Calibration - RSS Histograms - Self-Calibration Method

Performance Evaluation

Extra Slides

- Measurement Setup - Experimental Results

Conclusions - Concluding Remarks

International Conference on Indoor Positioning and Indoor Navigation, Sydney, Australia

15 November 2012

RSS Difference Approach

Introduction - Motivation

Device Calibration - Device Diversity - Manual Calibration

Self-Calibration - RSS Histograms - Self-Calibration Method

Performance Evaluation - Measurement Setup - Experimental Results

Conclusions - Concluding Remarks

Assume that a mobile device resides at a location `, which is covered by 2 WiFi APs, namely AP1 and AP2 . The RSS values recorded by the device are given by RSS1

= A − 10γ log10 d1 + X1

RSS2

= A − 10γ log10 d2 + X2

where di , i = 1, 2 is the distance from the i-th AP, while X1 , X2 ∼ N (0, σn2 ) are independent Gaussian noise components disturbing the RSS values. Taking the difference of these RSS values, denoted as RSSD12 , gives d2 + X0 RSSD12 = RSS1 − RSS2 = 10γ log10 d1 where X 0 ∼ N (0, 2σn2 ) is the linear combination of X1 , X2 .

International Conference on Indoor Positioning and Indoor Navigation, Sydney, Australia

15 November 2012

Inverse ecdf Least Squares Fitting

Introduction

If u is a continuous random variable and y = f (u) with monotonically increasing f then f = Fy−1 ◦ Fu . In particular, the inverse cdf ordered pairs

- Motivation

Device Calibration - Device Diversity - Manual Calibration

Self-Calibration - RSS Histograms - Self-Calibration Method

Performance Evaluation - Measurement Setup - Experimental Results

Conclusions - Concluding Remarks

{(ui , yi ) = (Fu−1 (qi ), Fy−1 (qi )) : qi ∈ {0.1, . . . , 0.9}} lie on the curve y = f (u). Proof: We have Fu (u)

=

P(u ≤ u) = P(f (u) ≤ f (u)) =

= P(y ≤ f (u)) = Fy (f (u)). Applying Fy−1 to both sides gives the identity f = Fy−1 ◦ Fu . Also, the components of the inverse cdf ordered pairs satisfy yi = Fy−1 (qi ) = Fy−1 (Fu (ui )) = f (ui ).

International Conference on Indoor Positioning and Indoor Navigation, Sydney, Australia

15 November 2012