__Time instant combinatorial optimisation__

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

__Time interval combinatorial optimisation__

Extend the previous optimisation problem from a 1-dimensional to a 2-dimensional problem. This is achieved by minimising the error function over an interval of time, instead of for each time instant. The following constraints would be included for each appliance:

- Appliance operation model (possibility and probability of transitions between states)
- Length of state (minimum and maximum duration of states)

__Factorial hidden Markov models__

Represent the problem as a factorial hidden Markov model, where each observation (aggregate power reading) is a function of many state variables (appliance power values). This model is particularly suitable as the observations and state variables are sequential samples over time, and therefore depend on previous samples. If this dependence factorises along the chain of dependence, we can say that the value of a state variable at time t depends entirely on the the value of the same state variable at time t-1.

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