Statistical Methods

Question 1

How to eg see if there is a statistically significant decrease in a time series for a period?

Answer 1

Eg using ordinary least square linear regression (and see sign level of slope coefficient?)

Question 2

How can one asses the goodness of a prediction?

Answer 2

1. Mean error in prediction, which can be further disentangled to mean when there is a positive difference, and mean when there is a negative difference: eg sum of ind errors divided by long-term mean value of what is predicted times number of predictions -> hence a summary measure of prediction….
2. Calculate likelihood that a prediction is within a desired error margin.

Question 3

CCA- Canonical correlation analysis

Answer 3

Multivariate statistical model
Eg multiple regression predict a single dependent variable from multiple indep variables, whereas CCA simultaneously predicts multiple dependent variables from multiple indep variables.
CCA derives a set of weights for each set variables so the resulting composite variables (pairs of canonical variables) are maximally correlated (canonical correlation). This gives first canonical mode. Procedure can then be repeated, attaining successive canonical modes derived from residual or leftover variance from earlier modes. I.e. The second pair of canonical variables is derived so that they exhibit the max corr, but are derived based on variance not accounted for by the first canonical mode. Since successive pairs of canonical variables are based on residual variance, each of the pairs of variables is orthogonal and independent of all other variables derived from the same set of data.

Question 4

What does the canonical correlation squared mean?

Answer 4

Then it represents the amount of variance in one canonical variable accounted for by the other canonical variable. This may also be called the amount of shared variance between the two canonical variables.

Question 5

PCA-principal component analysis

Answer 5

A method that allows reduction of the initial data set dimensions into a few representative variables (modes).

The new variables called principal components and often referred to as modes of variability, are obtained as linear combinations of the initial variables.

These combinations are obtained such that the new variables account for the maximum fraction of the variance contained in the original data set.

The number of modes retained in the PCA analysis is a compromise between the need to retain as much as possible of the signal and the requirement of noise reduction.