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I have a $100\times100$ grid. I have a transmitter on each corner, $4$ in total.

$$\begin{array}{rl}\text{Transmitter (a) is at}&(0,0);\\ \text{(b) is at}&(100,0);\\ \text{(c) is at}&(0,100);\\ \text{(d) is at}&(100,100).\end{array}$$

The user is able to measure the signal power from each transmitter.

I need to determine roughly where the user is on the grid knowing that the power/distance from the user to... $$\begin{array}{rl}\text{transmitter (b) is}&53\\ \text{transmitter (c) is}&51\\ \text{transmitter (d) is}&72\end{array}$$

There is a margin for error on the transmitter power/distance as I translate itf from decibels into distance on the grid: there is about a $20\%$ margin for error.

It's not important, but I translate the signal power to roughly $1$ unit of distance for argument's sake.

What is the equation I need to apply to these values to get a rough $(x,y)$ coordinate for the user?

Trying to remember back to when I was in school, I vaguely remember getting the points where circles intersect and averaging them out or something like that.

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Is this a hard question or have I not provided enough information in the question? Many Thanks, Code –  Code Dec 18 '11 at 19:16
It's not an easy question once you introduce the issue of margin of error. It is possible to do (GPS systems may do something along these lines, though in 3 dimensions), but I don't have a reference offhand for just how to solve the problem with the margin of error. –  Isaac Jan 6 '12 at 1:24
If the transmitters are at the same signal level, the measured power at the user should be proportional to the inverse square of the distance (or maybe the inverse distance if the signal follows the ground, or maybe something in between). The signal levels might be measured in dBm instead of V/m. Either of these may corrupt your data-you need to understand what it is telling you. –  Ross Millikan Jul 7 '12 at 16:26

1 Answer 1

One approach is to consider this as a probabilistic inverse problem. It may be overkill for purely theoretical purposes where there is a geometric formula for the exact solution, but this is how I would do it in a real-world situation where there is uncertainty. The following framework is probably how it's actually done for real GPS devices.

Probabilistic/Bayesian theory:

Define things as follows:

  • Call the unknown point location $p=(x,y)$,
  • the measurement locations $m_1,m_2,m_3$, ($m_1=(x_{m1},y_{m1})$, etc)
  • the measured distances $d=(d_1,d_2,d_3)$,
  • and the measurement noise $\eta = (\eta_1,\eta_2,\eta_3)$.

Then the situation can be encapsulated via the following equation: $$d=g(p)+\eta,$$

where $g(p):=(||m_1-p||,||m_2-p||,||m_3-p||)$. Based on measurements of $d$, we want to estimate $p$. The continuous density version of Bayes' theorem tells us that $$f_P(p|d)=\frac{f_D(d|p)f_P(p)}{f_D(d)}.$$ Let's break down what this means piece by piece:

  • $f_P(p|d)$ is the posterior probability density for your reciever to be at location $p$, taking into account that you measured the distances to be $d$.
    • Our goal is to find either the whole distribution $f_P(p|d)$, or maybe just certain properties of it like the expected value, maximum liklihood point, variance, etc.
    • The expected value and maximum liklihood point of the posterior $f_P(p|d)$ would be good "best guesses" for where the reciever is, and the variances would say how certain you are of that guess.
  • $f_D(d|p)$ is the likelihood that you would measure a set of distances $d$ if the true reciever point was $p$, taking into account the noise.
    • Considering a slight rearrangement of the model equation, $\eta=d-g(p)$, we can express the liklihood as $$f_D(d|p)=\rho_{\eta}(d-g(p))$$ where $\rho_{\eta}$ is the probability distribution of the noise $\eta$.
    • The noise distribution $\rho_\eta$ has to be modeled based on your knowledge of the measurement device - more on this later.
  • $f_P(p)$ is the prior probability distribution on the reciever location - how likely you think it is to be in certain location before you even measure the distances.
    • For example, if any location is as likely as any other, $f_P(p)$ would be a uniform distribution on the box.
  • $f_D(d)$ is the prior probability distribution on the distance measurements $d$.
    • Since $f_D(d)$ doesn't depend on $p$, it's basically just a normalization constant.
    • Properties of $f_D(d)$ are important in theory to show that the bayesian framework is legitimate, but in practice you can just ignore it. It won't effect the mean, maximum liklihood point, variance, etc.

Now lets address the modeling issues mentioned above. It might be reasonable to say that the measurements are mean-zero normally distributed random variables with standard deviations $\sigma_1, \sigma_2,\sigma_3$ for the distance measurements $d_1, d_2$, and $d_3$ respectively, and the prior distribution is uniform on the box. Then the liklihood is, $$f_D(d|p)=e^{-\sigma_1^{-2}(d_1-||m_1-p||)^2-\sigma_2^{-2}(d_2-||m_2-p||)^2-\sigma_3^{-2}(d_3-||m_3-p||)^2},$$ so the posterior is, $$f_P(p|d)=N\begin{cases}e^{-\sigma_1^{-2}(d_1-||m_1-p||)^2-\sigma_2^{-2}(d_2-||m_2-p||)^2-\sigma_3^{-2}(d_3-||m_3-p||)^2}, & p \text{ is in the box} \\ 0 & p \text{ not in the box}\end{cases}$$ where $N$ is some fixed normalization constant.

Extracting information from the posterior:

Given the posterior distribution, there are a few things you might want to do.

Drawing samples:

First, you might want to draw samples from it. Markov-chain monte carlo is a good way to do this since it works without you knowing the normalization constant. MCMC also has advantages for high-dimensional problems, but that doesn't matter since this problem is small. However if you have thousands of measurement devices and thousands of points then such considerations will be important.

Best guess for reciever location:

Second, you might want to find the maximum liklihood point - in other words the point $p$ where $f_P(p|d)$ is maximized. To do that, solve the constrained convex optimization problem, \begin{align}min_p & ~ \sigma_1^{-2}(d_1-||m_1-p||)^2+\sigma_2^{-2}(d_2-||m_2-p||)^2+\sigma_3^{-2}(d_3-||m_3-p||)^2 \\ \text{such that } & p \text{ is in the box}\end{align}

Uncertainty quantification:

Third, you might want to find the uncertainty. A quick and dirty way to do this is to draw a few, say 5, samples from the posterior and then take their variance. A more sophisticated way is to take the trace of the inverse of the hessian of the optization functional. It turns out that the inverse of the hessian is the covariance of a gaussian approximation to the posterior, though a proof of this is somewhat involved.

For an explanation on Bayesian inversion for a similar but easier example, you might read the excellent explanatory paper, "Estimating parameters in physical models through Bayesian inversion: a complete example" by Allmaras et. al.

Summary: The situation can be framed as a probabilistic inverse problem, where noise-corrupted measured distances are known and the recievers' position is unknown. Under this framework, Bayes' theorem provides a formula for the posterior probability density for the receiver position, based on (1) the prior probability densities, (2) the noise distribution, and (3) the measured data. If the prior is modeled as a uniform distribution and the noise is gaussian, then the posterior takes a particularly simple form. Samples from the posterior can be drawn with Markov chain monte carlo, and the posterior maximum liklihood point can be found by solving a computationally easy convex optimization problem.

Edit: I did a quick code up of the above maximum likelihood calculation in python using an off-the-shelf optimization routine from scipy, and it seems to work well.

Code: (I release this to the public domain if you want to use it. Doesn't use gradient information)

import numpy as np
from scipy.optimize import minimize
from math import sqrt
from math import pow
import matplotlib.pyplot as plt

def recieverMAPpoint(d1,d2,d3):
    #coordinates of constraints box
    m1x=100.; m1y=0.
    m2x=0.; m2y=100.
    m3x=100.; m3y=100.
    #measurement undertainty=20%
    s1=.2*d1; s2=.2*d2; s3=.2*d3 
    #optimization functional J
    def J(p):
        x=p[0]; y=p[1]
        return pow(r1/s1,2) + pow(r2/s2,2) + pow(r3/s3,2)
    #constraints box B
    B = ({'type': 'ineq',
      'fun' : lambda p: np.array([p[0], p[1]])},
     {'type': 'ineq',
      'fun' : lambda p: np.array([100-p[0], 100-p[1]])})
    #minimize J over B
    res = minimize(J,np.array([50,50]),constraints=B,options={'disp': True},method='SLSQP')
    #plot constraints box
    #plot distance circles
    plt.gca().add_patch(c1); plt.gca().add_patch(c2); plt.gca().add_patch(c3)
    #plot optimal point
    return res.x

Test Cases:

Example from original post:

>>> recieverMAPpoint(53,51,72)
Optimization terminated successfully.    (Exit mode 0)
            Current function value: 6.47506375609
            Iterations: 9
            Function evaluations: 36
            Gradient evaluations: 9
array([ 48.3354663 ,  50.77404997])

enter image description here

Measurements error overestimates distance:

>>> recieverMAPpoint(70,100,40)
Optimization terminated successfully.    (Exit mode 0)
            Current function value: 0.241068422167
            Iterations: 12
            Function evaluations: 48
            Gradient evaluations: 12

enter image description here

Measurement error underestimates distance:

>>> recieverMAPpoint(30,100,70)
Optimization terminated successfully.    (Exit mode 0)
            Current function value: 0.472436673945
            Iterations: 13
            Function evaluations: 52
            Gradient evaluations: 13
array([ 85.47617804,  27.93369774])

enter image description here

Distance measurements exact:

>>> recieverMAPpoint(80.6,67.1,92.2)
Optimization terminated successfully.    (Exit mode 0)
            Current function value: 1.20656973089e-07
            Iterations: 11
            Function evaluations: 44
            Gradient evaluations: 11
array([ 30.012965  ,  39.98444119])

enter image description here

Optimal point on edge of box (constraint active):

>>> recieverMAPpoint(60,120,130)
Optimization terminated successfully.    (Exit mode 0)
            Current function value: 0.526683755534
            Iterations: 13
            Function evaluations: 52
            Gradient evaluations: 13
array([  3.96901773e+01,  -1.76150977e-13])

enter image description here

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