Flashcards in Splines Deck (22)

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1

## y

### dependent variable

2

## t

### explanatory variable (1-dimensional)

3

## Underlying Function Assumption

###
-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

## f

###
-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

##
Basic Spline Theory

Word Definition

###
-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

##
Basic Spline Theory

Equation Definition

###
-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

## Smoothness Constraint

###
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

## Linear Spline

###
-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

## Basis for the Vector Space of Linear Splines

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

10

##
Natural Spline

Definition

###
-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

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

###
-an order-p natural spline has p+1 constraints

f^(j)(t1-) = f^(j)(tn+) = 0

-for (p+1)/2, ..., p

12

##
Natural Linear Splines

Constraints

###
-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

##
Natural Cubic Splines

Constraints

###
-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

## Total Degrees of Freedom of a Natural Spline

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

-dimensional space of natural splines = n regardless of p

15

##
Linear Natural Splines

Representation

###
f(t) = ao + Σ bi |t-ti|

-with Σbi=0

-sum form i=1 to i=n

16

##
Cubic Natural Splines

Representation

###
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

##
Roughness

Definition

###
J^(ν)(f) = ∫ [f^(ν)(t)]² dt , ν≥1

-integrate from -∞ to +∞

18

##
Roughness of a Natural Linear Spline

Direct Calculation Formula

###
J^(1)(f) = -2 Σ Σ bi bk |ti-tk|

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

19

##
Roughness of a Natural Linear Spline

Matrix Form

###
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

##
Roughness of a Natural Cubic Spline

Direct Calculation Formula

###
J^(1)(f) = 12 Σ Σ bi bk |ti-tk|³

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

21

##
Roughness of a Natural Cubic Spline

Matrix Form

###
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