Splines

Question 1

Splines

Answer 1

• piecewise polynomial functions used to approx complex relantionships
• create smooth, flexible curves
• each segment is defined by a polynomial function, with curve combining all segments together

Question 2

Basis Functions

Answer 2

• individual polynomial segments
• building blocks to construct the spline

Question 3

Knots

Answer 3

• points where the polynomial functions or basis functions join
• choice of number of knots affetcs the flexibility and smoothness of the spline
• more knots - more flexible
• less knots - more smooth

Question 4

Penalised Regression Spline

Answer 4

• spline that incorporates a penalty term
• prevents overfitting, ensuring spline captures the underlying trend without being overly sensitive to noise
• tuning parameter - controls trade-off between fit and smoothness
• low lambda - more flexible and wiggly
• high lambda - smoother

Question 5

Smoothness Penalty

Answer 5

• measures the integrated squared second derivatve of function, reflecting the curvature of the function

Question 6

Thin-Plate Regression Basis Function

Answer 6

• basis function
• depends on Euclidean distance between points
• attach points to thin sheet

Question 7

Why Splines

Answer 7

• provide flexible, smooth and interpretable way to capture complex relantionships
• useful for non-linear relantionships to capture the underlying patterns
• clustering - reduce dimension
• spatial - modelling smooth spatial surfaces