13 Lecture 04
13.1 Standardizing variables
(x - mean(x)) / sd(x) or scale(x)
Result = mean of 0, sd of 1
Helps software fit
Value = 1 is equal to 1 SD
13.2 Plotting uncertainty - sample from posterior
(if multivariate normal)
- Approximate posterior from mean, standard deviation
- Sample from multivariate normal distribution of parameters
- Use samples to generate predictions that integrate over uncertainty
extract_samples returns a, b, sigma, … and you can plot each
13.3 Polynomials
Polynomials have bad behavior especially at the boundaries of the data
They don’t fit locally, and are not actually flexibly.
Eg. a polynomial of 3rd degree will necessarily have two turns - this has to happen irrespective of the data
13.4 Splines
Locally wiggly functions, combined by interpolation
Geocentric - describing relationships - not exploring them
13.5 Basis splines
Bayesian B-splines = P-splines
Similar to linear models but with synthetic variables
\(\mu = \alpha + w_{1} \beta_{1} + + w_{2} \beta_{2} + + w_{3} \beta_{3} + + w_{4} \beta_{4} + ...\)
Knots are often picked at equal intervals in data, though strategies vary
At each knot, the knot’s function is at 100%, moving away from it, the neighboring functions turn on
Parameters always have more uncertainty than predictions
Caution: over fitting