I have a time series data and I will be adding more data points in a consistent manner. I want to figure out whether the new data point added is an outlier, in regards to the previously observed data points. I was wondering if I should be using a linear regression method with a moving average method, in order to ensure that I use the same number of data points each time a new data point comes in.
I would like some help on how to approach this problem. Thank you.
The suggsted approach is to use all of the data to form a reasonable XARMAX model which might be a simple weighted average of the past (ARMA model) OR perhaps a deterministic model(X) with possible level shifts /local time trends/seasonal pulses/pulses or some rich combination of these two types of predictors (memory and causals). Now with Intervention Detection schemes one can test for and possibly find a pulse/outlier at the most recent point. If one is found then you can conclude that the last point was not predictable from the past and/or deterministic structure. Essentially this enables one to say that the probability of observing the last values BEFORE it was observed is x%. Now it is very important that the error process from the model have the Gaussian Conditions , one of which is constant variance and of course that the model parameters were invariant over time.
To summarize (from Bacon) ; To do science is to search for repeated patterns.
To detect anomalies is to identify values that do not follow repeated patterns.
For whoever knows the ways of Nature will more easily notice her deviations
and, on the other hand, whoever knows her deviations will more accurately
describe her ways.One learns the rules by observing when the current rules fail. In modern statistical jargon this means you need to have a useful model ! In my opinion this is the essence of the comment from @Xi'an
Another way to restate your question in my words ...
Can you tell me the probability that a single data point (e.g. the latest
reading) came from the distribution represented by all the previous data points?
I have been involved with writing commercial software that will take a pre-existing model and then conduct the test that you require providing an early-warning / heads-up alert. See What predictive models allow me to make new predictions on a series in constant time, without needing to recompute previous ones? for a similar discussion. This of course could be done by those seeking free software solutions but it could require some special purpose programming.
This is another example of "For any complex problem there is a simple solution. And it's always wrong."
