02. Forcasting

HaiyueAugust 16, 2023About 2 min

1 Principles of statistical forecasting

The main item refers not only to forecasting but any type of statistical modelling.
It is that one builds the model one part (usually the majority) of the data, and then tests the results on the remainder.
These sets are termed the Training and Testing sets.
In standard statistical modelling, often the sets are sampled from the data at random.
In time series forecasting, this is not possible, as the model building set has to have consecutive data, as does the testing set.
This is further complicated with data that includes seasonality, as at least the training set has to contain multiples of the fundamental period.
For instance, for hourly solar radiation, the training set would comprise a number of years. We also have to cater for leap years in some way. The easiest way is to delete Feb 29.

In keeping with the idea of training and testing sets, I took the first 75 years as the training set, reserving 22 years for testing.
I fitted a line through those 75 data points.
I then took the residuals of that fit, and developed an $AR(2)$ model for them.
Let $L(t)$ be the level of the lake, and $f(t)$ be the linear fit.
Then $r(t) = L(t) − f(t)$ will be the residuals.
The models are

$f(t) = 10.64 − 0.041t$
$r(t) = 0.962r(t − 1) − 0.318r(t − 2)$
$L(t) = 10.64 − 0.41t + 0.962r(t − 1) − 0.318r(t − 2)$

The solar forecasting models are given by