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I have a set of devices each with list of times T that represent when a device turns on e.g.

Device A : [Mon 16:03, Mon 15:59, Wed 16:05, ... n]

I am detecting patterns of usage for example the next day a person turns the switch on at average T+/-5 minutes there is likely to be a strong link between that time and the average T value. We can say there is a pattern and it can be built up as the days go on. If there is a day without the value (switch wasn't turned on) ie. miss then the confidence can be reduced. One problem being is that a days missing data would need to be accounted for. We can say if confidence goes below a threshold then a pattern doesn't exist.

I have created a simple working version (not taking into account the misses) but I'm more interested in what greater minds would consider the best way to evaluate and detect if there is a daily occurrence of an event. I thought this is the best place for this since I'm interested in an elegant and beautiful way of approaching this. Are there better statistic models that exist to work out patterns like this? Thank-you

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one fairly obvious thing to try is to generate power spectra of the timing data (using an fft) and look for significant peaks. if you have a signal of period 1 day then you know something is happening daily; if you also have a signal of period 7 days then you know that there is a weekly component - maybe they don't have the same behaviour at weekends.

that's a classical, quantitative approach. you could also try playing with more modern, unstructured approaches - perhaps training a neural net to recognise patterns in some way? and there's no reason why you shouldn't combine these - the power spectrum might provide parameters (periods) that are used to present the data to the net in a more structured form (eg by taking times modulo the appropriate periods).

finally, i would also do a literature search and see what others have done. playing around with google, it seems like "temporal pattern detection" would be one suitable phase.

ps also, i would separate the detection of variations from the detection of patterns. first, i would work on detecting patterns. only once that is working well would i think about extending it to "5min earlier each day" etc. partly because it is better to start simple, but also because i am not convinced such corrections are important. most people don't consistently do something 5min earlier every day, or they would end up being busy at night... we are naturally cyclical, so i would look for cycles first and only consider linear variations if there was evidence that they were important.

pps statistics only comes into this in assessing the evidence for particular hypotheses. it's more about pattern matching / pattern detection and you might get more replies by adding those tags.

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According to your definition, a pattern exist if an event e occurs within a 10 minutes interval each 24 hours, with a probability that is higher than a given threshold. This is similar to the sunrise problem. In your case the Boolean daily event is not the sunrise, but the occurrence of event e within the expected interval.

The probability for event e to occur tomorrow, can be calculated according to the rule of succession. According to your definition, if this value is higher than the given threshold - a pattern exist.

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