Logistic function - convert values to probabilities
In many home automation scenarios we have an observed value, maybe a sensor reading, maybe a time interval since something happened and we want to convert that value to a probability, e.g. what's the probability that someone left the room given how long it's been since any activity there.
Previously I described using the atan function for that purpose. The logistic function has a similar shape and can be used as an alternative. There's an interesting discussion here on their
differences. In summary: atan has a 'fatter tail'.
I've used both for converting distances or errors to probabilities and most recently I've used the logistic function to compute the probability that two distances belong to the same observation. For example, if something measures as being 5m away, how likely is it that it's in a room that's measured as being 7m away?
Let's take the difference between a measured distance and a recorded distance. Using y = 1 - 2 / (1 + e^((5-x)*1)) we map that difference onto the logisic curve and then values above zero can be used as the probability that the device matches the recording for this distance, and values below zero can be used (in the negative) as the probability that the device does not match the distance. Around 4m we want the probability of either to be about zero - essentially at this error the data isn't telling us anything.
double error = fabs(measured_distance - recording_distance);
// Use a sigmoid function: the logistic curve
double prob = 1 - 1 / (1 + exp((4-error)*1));
if (prob > 0)
// use positive probability to increase probability that
// the device IS here
...
else
// use negative value to increase probability that
// the device is NOT here
...
We can shift the logistic curve with the offset (4) and scale and invert it as necessary to get whatever probability curve we want from an observed value.
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