Statistics formula for wifi positioning.

Assuming I have $3$ access point namely: $AC_1$, $AC_2$ and $AC_3$ and I want to know my location using this access point and a mobile device that will get signal from the access points.

First thing that I did is save the signal level for different location in the database.

• In the first location the signal level for $AC_1$ is $-20 \ \rm{dBm}$, the signal level for $AC_2$ is $-30 \ \rm{dBm}$ and signal level for $AC_3$ is $-40 \ \rm{dBm}$.

• In second location the signal level for $AC_1$ is $-10 \ \rm{dBm}$, the signal level for $AC_2$ is $-40 \ \rm{dBm}$ and signal level for $AC_3$ is $-50 \ \rm{dBm}$.

• Finally, for the third location the signal level for $AC_1$ is $-30 \ \rm{dBm}$, the signal level for $AC_2$ is $-50 \ \rm{dBm}$ and the signal level for $AC_3$ is $-60 \ \rm{dBm}$.

Now during my actual location detection phase how will I know my current location ? For example I read this signal : $AC_1$ is $-15 \ \rm{dBm}$, $AC_2$ is $-35 \ \rm{dBm}$ and $AC_3$ is $-45 \ \rm{dBm}$. How will I know my current location? My big problem is that signal level is not the same in one location there are times that it will fluctuate. What do you think is the best thing to do for this ?

• Do not attempt to indent the beginning of each paragraph—that triggers preformatted mode. Commented Jul 18, 2013 at 4:34
• Duplicate from SO: stackoverflow.com/questions/16485370/… Commented Jul 18, 2013 at 5:10

The basic idea is to use an empirical model of signal strength, which looks like: $$P(r)= 10 \alpha P(r_0) \log\left(\frac{r}{r_0}\right) - l \ \text{WAF}$$ where l is the number of walls between the transmitter and the receiver; WAF is the wall attenuation factor; $P(r)$ (measured in $\text{dBm}$) is the power received by a given mobile station whose distance from a given transmitter or access point is $r$ (meters); $r_0$, the reference distance from the transmitter; and $P(r_0)$, the signal power at this reference point. The parameter $\alpha$, called the exponent value, indicates the rate at which the path loss increases as distance $r$ increases. In real life you know practically nothing, because any of these factors can fluctuate, but usually it is assumed (by convention) that you have some information about so-called predictor variables ($r, \ l$). And relying on this knowledge you use statistical methods to estimate the distance to the access point (AP) (and then by triangulation (you have $3$ AP's) find your position).
For example Y. Chen and H. Kobayashi in their paper used linear regression and multiple linear regression to estimate the unknown parameters $P_0, \ \text{WAF}, \ \alpha$. And then used least-square estimation to find how an error in power estimation results in the following estimation of location error. Different researchers suggest other ways of filtering.