The most usual are the mean and variance. Search on Wikipedia for more details.

No, there is nothing precisely like that.

"Random" is what we can not explain or predict (for whatever reason; it does not matter if there is no such possible explanation or if we are just not aware of one).

The model uses some predictors (known to us; like the time of the day, the wether conditions including the day in the year, etc.) and makes a prediction of the response (the energy consumption) - the response value we should expect, given the corresponding values of the predictors. You can see the model as a mathematical formula of the predictor values. The formula contains parameters that make the model flexible and adjustable to observed data (think of an intercept and a slope of a regression line, or the frequency and amplitude of a sinusoidal wave).

The deviation of observations from these expected values are called residuals. They are not explained by the model and are thus considered "random". This randomness is mathematically handled by a probability distribution: we don't say that a particular resudual will be this or that large; instead we give a probability distribution (more correctly, we give the probability distribution of the response, conditional on the predictors). Using this probability model allows us to find the probability of the observed data (what is called the likelihood) given any combination of chosen values of the model parameters. Usually, we "fit" these parameters to maximize this likelihood (-> maximum likelihood estimates).

Thus, given a fitted model (on a given set of observations), we have a (maximized) likelihood (which depends on the data and on the functional model and on the probability model).

This can be used to compare different models. One might just see which of the models has the largest (maximized) likelihood. There are a few practical problems, because models with more parameters can get higher likelihood s just because they are more flexible - not more "correct". This is tried to be accounted for in by giving penalties for the model flexibility. This leads to the formulation of different information criteria (AIC, BIC, DIC and alikes, that all differ in the way the penalties are counted).

So, after that long post, you may look for such ICs to compare different models. The limitation remains that the models are all compared only on the data that was used to fit them, without guarantee that they will behave similar for new data. So if you have enough data it might be wise to fit the models using only a subset of the available data and then check how well these models predict the rest of the data. It does not really matter how you quantify this; I would plot the differences of the models side-by-side in a boxplot or a scatterplot.

Dear Mohammad,

In agreement with Jochen Wilhelm , but from a different point of view I recommend you to build simple but interpretable models, and use the standard deviation/variance of the residuals as a measure of randomness.

HTH,

Ákos

Thank you very much indeed Ulrich, Jochen and Ákos.

Dear Ákos (and Ulrich):

I want to pose the same question to you as I have done to Jochen:

Does more variance/St Dev across my residual values indicate (i) more randomness in my observed values or (ii) a weak model that doesn’t fit the observed data very well or (iii) both?

Dear Mohammad,

I think, both. If the model is good (not in statistical terms but in terms of the theory), variance may indicate randomness, but you can never be sure that your model is good enough...

HTH,

Ákos

Fantastic,

I suppose what is essentially at the bottom of it all is that nature is to complex for our mathematical/statistical models, and as Jochen said whatever we don't understand fully we just call 'random'.

Thanks a lot again Ákos.