- Bayesian modelling of appliance detections using smoothness and confidence levels
- Time instant combinatorial optimisation

__Bayesian modelling of fridge detections using smoothness and confidence levels__

I'm going start by applying a threshold to the smoothness values calculate after subtracting the fridge's signature from the aggregate signal at each possible interval. This will look something like this:

For each value above the threshold, I will lookup the confidence value as is given by the confidence plot:

So, let:

X = smoothness peak corresponds to fridge cycle

t = time since last cycle

This confidence value will form the prior probability that the fridge is operating during this time period, X

We can form a likelihood too given the distribution of time between cycles, t

Therefore, using Bayes:

P(X | t) proportional to P(t | X) * P(X)

Therefore, the probability of each smoothness peak corresponding to the next fridge's cycle can be calculated, given the previous cycle. The optimum cycle will be then be selected using maximum likelihood. This process will be repeated sequentially for each cycle over the course of the day.

I believe this approach is similar to the hidden Markov model of this problem, although I haven't read much into the underlying theory of this area yet.

__Time instant combinatorial optimisation__

This was explained in detail in the previous post. For convenience, I've repeated it below.

Minimise the error function f:

f =

where:

- t is a time instant in the range 1...T
- n is an appliance the range 1...N
- P agg is the aggregate power
- I is a vector of binary values indicating the state of an appliance. Only one value will be non-zero
- P is a vector of continuous values indicating the power values for each state of an appliance

This approach has been frequently mentioned and dismissed within the NALM literature (initially by Hart in 1992) for a number of valid reasons. I want to investigate how well this works for increasingly complex data:

- Small number of appliances (5) for which the complete set of states for all appliances is known
- As in 1 but with some noise
- As in 1 but for a realistic number of appliances (20)
- As in 3 but with an incomplete set of appliances

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