Splines Flashcards Preview

MATH5824M Generalised Linear and Additive Models > Splines > Flashcards

Flashcards in Splines Deck (22)
Loading flashcards...
1
Q

y

A

dependent variable

2
Q

t

A

explanatory variable (1-dimensional)

3
Q

Underlying Function Assumption

A

-have yi observations at each ti for i=1,..,n where observation times ti are ordered : t1 < t2 < … < tn
-assume that observations yi are noisy versions of an underlying smooth function:
yi = f(ti) + εi

4
Q

f

A
  • we want to estimate f
  • can do this using:
  • -parametric models e.g. f(t) = α + βt
  • -non-parametric models i.e. f is a smooth function of t
5
Q

Basic Spline Theory

Word Definition

A
  • let t1 < t2 < … < tn be a fixed set of ‘knots’

- a spline of order p≥1 is a piecewise polynomial of order p that is (p-1) times differentiable at the knots

6
Q

Basic Spline Theory

Equation Definition

A

-let t1 < t2 < … < tn be a fixed set of ‘knots’
f(t) = Σ aij t^j
-for ti ≤ t < ti+1, i=0,…,n and j=0,…,p
-sum from j=0 to j=p
-where to=-∞ and tn+1=+∞

7
Q

Smoothness Constraint

A

f^(l) (ti-) = f^(l)(ti+)

  • for l=0,…,p-1 and i=1,…,n
  • i.e. the limit of the lth differential of f(ti) as t approaches ti from below equals the same differential as t approaches ti from above
8
Q

Linear Spline

A

-consider the sequence t1 < … < tn
-define gi(t) = |t-ti|, i.e. we shift g to be centred at each knot
-then:
f(t) = a0 + a1*t + Σ bi gi(t)
-sum from i=1 to i=n is a linear spline
-at each knot ti, the function gi(t) is continuous

9
Q

Basis for the Vector Space of Linear Splines

A

B = {1, t, g1(t), …, gn(t)} forms a basis for the vector space of linear splines

10
Q

Natural Spline

Definition

A

-natural splines are a special case of polynomial splines of odd order
-a spline is said to be natural if beyond the boundary knots t1 & tn its (p+1)/2 higher-order derivatives are zero, i.e.:
f^(j)(t) = 0
for j=(p+1)/2, …, p AND t≤t1 or t≥tn

11
Q

How many constraints does an order-p natural spline have?

A

-an order-p natural spline has p+1 constraints
f^(j)(t1-) = f^(j)(tn+) = 0
-for (p+1)/2, …, p

12
Q

Natural Linear Splines

Constraints

A

-a natural Iinear spline has p+1=2 extra constraints:
f’(t1-) = f’(tn+) = 0
-so f(t) is constant in the outer intervals

13
Q

Natural Cubic Splines

Constraints

A

-a natural cubic spline has p+1=4 extra constraints:
f’‘(t1-) = f’‘(tn+) = 0
f’’‘(t1-) = f’’‘(tn+) = 0
=> f(t) is linear in the outer intervals

14
Q

Total Degrees of Freedom of a Natural Spline

A

n + p + 1 - (p + 1) = n

-dimensional space of natural splines = n regardless of p

15
Q

Linear Natural Splines

Representation

A

f(t) = ao + Σ bi |t-ti|

  • with Σbi=0
  • sum form i=1 to i=n
16
Q

Cubic Natural Splines

Representation

A

f(t) = a0 + a1*t + Σ bi |t-ti|³

  • with Σ bi = Σ bi*t = 0
  • sum from i=1 to i=n
17
Q

Roughness

Definition

A

J^(ν)(f) = ∫ [f^(ν)(t)]² dt , ν≥1

-integrate from -∞ to +∞

18
Q

Roughness of a Natural Linear Spline

Direct Calculation Formula

A

J^(1)(f) = -2 Σ Σ bi bk |ti-tk|

-sum from i=1 to i=n & k=1 to k=n

19
Q

Roughness of a Natural Linear Spline

Matrix Form

A

J^(1)(f) = c^(1) b^T K^(1) b

  • where c^(1) = -2
  • and b is the nx1 vector of spline coefficients b1,…,bn
  • and K^(1) is the nxn matrix whose (i,k)th element is |ti-tk|^p
  • where p=2ν-1
20
Q

Roughness of a Natural Cubic Spline

Direct Calculation Formula

A

J^(1)(f) = 12 Σ Σ bi bk |ti-tk|³

-sum from i=1 to i=n & k=1 to k=n

21
Q

Roughness of a Natural Cubic Spline

Matrix Form

A

J^(2)(f) = c^(2) b^T K^(2) b

  • where c^(1) = 12
  • and b is the nx1 vector of spline coefficients b1,…,bn
  • and K^(1) is the nxn matrix whose (i,k)th element is |ti-tk|^p
  • where p=2ν-1
22
Q

The Interpolation Problem and the Optimal Interpolating Function

A

-for fixed integer ν≥1 representing an order of smoothness, the objective is to find the smoothest function which passes through a set of data points
-i.e. we want to find f to mimimise:
J^(ν)(f) = ∫ [f^(ν)(t)]² dt
-subject to f(ti)=yi for i=1,…,n
-the solution f is the optimal interpolating function