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scalation/scalation/scalation.calculus/DB_Spline

DB_Spline

scalation.calculus.DB_Spline
class DB_Spline(ττ: VectorD, mMax: Int, clamp: Boolean) extends B_Spline, DBasisFunction

The DB_Spline class provides B-Spline basis functions with derivatives for various orders 'm', where the order is one more than the degree. A spline function is a piecewise polynomial function where the pieces are stitched together at knots with the goal of maintaining continuity and differentability. B-Spline basis functions form a popular form of basis functions used in Functional Data Analysis.

Value parameters

clamp

whether or not to clamp the ends of the knot vector using B_Spline.clamp

mMax

the maximum order, allowing splines orders from 1 to mMax

ττ

the time-points of the original knots in the time dimension

Attributes

See also
Graph
Supertypes
class B_Spline
class Object
trait Matchable
class Any
Show all

Members list

Value members

Concrete methods

def d1bb(m: Int)(j: Int)(t: Double): Double

First derivatives of order 'm' B-Spline basis functions (general recurrence).

First derivatives of order 'm' B-Spline basis functions (general recurrence).

Value parameters

j

indicates which spline function

m

the order of the spline function (degree = order - 1)

t

the time parameter

Attributes

override def d1bf(m: Int)(j: Int)(t: Double): Double

First derivatives of order 'm' B-Spline basis functions (dynamic programming apporach)

First derivatives of order 'm' B-Spline basis functions (dynamic programming apporach)

Value parameters

j

indicates which spline function

m

the order of the spline function (degree = order - 1)

t

the time parameter

Attributes

Definition Classes
def d1bfr(m: Int)(j: Int)(t: Double): Double

Adjusted derivatives of order 'm' B-Spline basis functions (general recurrence). These are adjusted so that the first "usable" spline is at j = 0. The valid range of usable splines is defined in range.

Adjusted derivatives of order 'm' B-Spline basis functions (general recurrence). These are adjusted so that the first "usable" spline is at j = 0. The valid range of usable splines is defined in range.

Value parameters

j

indicates which spline function

m

the order of the spline function (degree = order - 1)

t

the time parameter

Attributes

def d2bb(m: Int)(j: Int)(t: Double): Double

Second derivatives of order 'm' B-Spline basis functions (general recurrence).

Second derivatives of order 'm' B-Spline basis functions (general recurrence).

Value parameters

j

indicates which spline function

m

the order of the spline function (degree = order - 1)

t

the time parameter

Attributes

override def d2bf(m: Int)(j: Int)(t: Double): Double

Obtain the value of 2nd derivative of the m-th order 'j'-th basis function at time 't'. Or alternatively, obtain the 2nd derivative basis function by calling d2bf(m)(j) only. Ex: val x = d2bf(m)(j)(t) retrieves the 2nd derivative value of the j-th basis function at 't'. val f = d2bf(m)(j) retrieves the 2nd derivative of the j-th basis function.

Obtain the value of 2nd derivative of the m-th order 'j'-th basis function at time 't'. Or alternatively, obtain the 2nd derivative basis function by calling d2bf(m)(j) only. Ex: val x = d2bf(m)(j)(t) retrieves the 2nd derivative value of the j-th basis function at 't'. val f = d2bf(m)(j) retrieves the 2nd derivative of the j-th basis function.

Value parameters

j

indicates which basis function

m

the order of the basis function

t

the time parameter

Attributes

Definition Classes
def d2bfr(m: Int)(j: Int)(t: Double): Double

Adjusted second derivatives of order 'm' B-Spline basis functions (general recurrence). These are adjusted so that the first "usable" spline is at j = 0. The valid range of usable splines is defined in range.

Adjusted second derivatives of order 'm' B-Spline basis functions (general recurrence). These are adjusted so that the first "usable" spline is at j = 0. The valid range of usable splines is defined in range.

Value parameters

j

indicates which spline function

m

the order of the spline function (degree = order - 1)

t

the time parameter

Attributes

override def dnabf_(n: Int)(m: Int)(t: Double): VectorD

Obtain the value of nth derivative of the m-th order basis functions (all) at time 't'. Or alternatively, obtain the basis function by calling dnabf(m)(j) only. Ex: val x = dnabf(m)(t) retrieves the nth derivative value of the value of all the basis functions at 't'. val f = dnabf(m) retrieves the nth derivative value of all the basis functions.

Obtain the value of nth derivative of the m-th order basis functions (all) at time 't'. Or alternatively, obtain the basis function by calling dnabf(m)(j) only. Ex: val x = dnabf(m)(t) retrieves the nth derivative value of the value of all the basis functions at 't'. val f = dnabf(m) retrieves the nth derivative value of all the basis functions.

Value parameters

m

the order of all the basis function

n

the order of the derivative

t

the time parameter

Attributes

Definition Classes
def dnabfr(n: Int)(m: Int)(t: VectorD): MatrixD

Obtain the value of nth derivative of the m-th order basis functions (all) at time 't'. Or alternatively, obtain the basis function by calling dnabf(m)(j) only. Ex: val x = dnabfr(n)(m)(t) retrieves the nth derivative value of the value of all the basis functions at 't'. val f = dnabfr(n)(m) retrieves the nth derivative value of all the basis functions. Note that this is the recursive approach as opposed to the dynamic programming approach.

Obtain the value of nth derivative of the m-th order basis functions (all) at time 't'. Or alternatively, obtain the basis function by calling dnabf(m)(j) only. Ex: val x = dnabfr(n)(m)(t) retrieves the nth derivative value of the value of all the basis functions at 't'. val f = dnabfr(n)(m) retrieves the nth derivative value of all the basis functions. Note that this is the recursive approach as opposed to the dynamic programming approach.

Value parameters

m

the order of all the basis function

n

the order of the derivative

t

the time parameter

Attributes

def dnabfr_(n: Int)(m: Int)(t: Double): VectorD

Obtain the value of nth derivative of the m-th order basis functions (all) at time 't'. Or alternatively, obtain the basis function by calling dnabf(m)(j) only. Ex: val x = dnabfr(m)(t) retrieves the nth derivative value of the value of all the basis functions at 't'. val f = dnabfr(m) retrieves the nth derivative value of all the basis functions. Note that this is the recursive approach as opposed to the dynamic programming approach.

Obtain the value of nth derivative of the m-th order basis functions (all) at time 't'. Or alternatively, obtain the basis function by calling dnabf(m)(j) only. Ex: val x = dnabfr(m)(t) retrieves the nth derivative value of the value of all the basis functions at 't'. val f = dnabfr(m) retrieves the nth derivative value of all the basis functions. Note that this is the recursive approach as opposed to the dynamic programming approach.

Value parameters

m

the order of all the basis function

n

the order of the derivative

t

the time parameter

Attributes

def dnbb(n: Int)(m: Int)(j: Int)(t: Double): Double

N-th derivatives of order 'm' B-Spline basis functions (general recurrence).

N-th derivatives of order 'm' B-Spline basis functions (general recurrence).

Value parameters

j

indicates which spline function

m

the order of the spline function (degree = order - 1)

n

the n-th derivative to be computed

t

the time parameter

Attributes

def dnbf(n: Int)(m: Int)(j: Int)(t: Double): Double

Adjusted n-th derivative of order 'm' B-Spline basis functions (general recurrence). These are adjusted so that the first "usable" spline is at j = 0. The valid range of usable splines is defined in range.

Adjusted n-th derivative of order 'm' B-Spline basis functions (general recurrence). These are adjusted so that the first "usable" spline is at j = 0. The valid range of usable splines is defined in range.

Value parameters

j

indicates which spline function

m

the order of the spline function (degree = order - 1)

n

the n-th derivative to be computed

t

the time parameter

Attributes

def dnbfr(n: Int)(m: Int)(j: Int)(t: Double): Double

Adjusted n-th derivative of order 'm' B-Spline basis functions (general recurrence). These are adjusted so that the first "usable" spline is at j = 0. The valid range of usable splines is defined in range.

Adjusted n-th derivative of order 'm' B-Spline basis functions (general recurrence). These are adjusted so that the first "usable" spline is at j = 0. The valid range of usable splines is defined in range.

Value parameters

j

indicates which spline function

m

the order of the spline function (degree = order - 1)

n

the n-th derivative to be computed

t

the time parameter

Attributes

Inherited methods

def abf(m: Int)(t: VectorD): MatrixD

Obtain the value of the m-th order basis functions (all) at time 't'. Or alternatively, obtain the basis function by calling bf(m)(j) only. Ex: val x = bf(m)(t) retrieves the value of all the basis functions at 't'. val f = bf(m) retrieves all the basis functions.

Obtain the value of the m-th order basis functions (all) at time 't'. Or alternatively, obtain the basis function by calling bf(m)(j) only. Ex: val x = bf(m)(t) retrieves the value of all the basis functions at 't'. val f = bf(m) retrieves all the basis functions.

Value parameters

m

the order of all the basis function

t

the time parameter

Attributes

Inherited from:
BasisFunction
override def abf_(m: Int)(t: Double): VectorD

Evaluate each of the τ.dim-1-many order m B-spline basis functions at t non-recursively, using an efficient dynamic programming approach. ==Example==

Evaluate each of the τ.dim-1-many order m B-spline basis functions at t non-recursively, using an efficient dynamic programming approach. ==Example==

val m = 4                       // order m (degree m-1)
val t = VectorD (1, 2, 3, 4)    // original time points
val N = B_Spline (t, k)         // B_Spline instance
val z = N.abf_ (m)(t)           // vector of evaluations
println ("order k basis functions at t = z")

==Implementation Details== Let N(m)(i) denote the i-th order k basis function evaluated at t. Then each N(m)(i) depends on the evaluation of N(m-1)(i) and N(m-1)(i+1). Here is an example, given a set of order m=4 B-spline basis functions and knot vector τ of length n:

N(1)(0), N(1)(1), N(1)(2), N(1)(3), ..., N(1)(n-1)
      |/       |/       |/            |/
N(2)(0), N(2)(1), N(2)(2), ..., N(2)(n-2)
      |/       |/            |/
N(3)(0), N(3)(1), ..., N(3)(n-3)
      |/            |/
N(4)(0), ..., N(4)(n-4)

This algorithm applies the procedure described above using O(n) storage and O(k*n) floating point operations.

Value parameters

t

point to evaluate

Attributes

See also

Carl de Boor (1978). A Practical Guide to Splines. Springer-Verlag. ISBN 3-540-90356-9.

Definition Classes
Inherited from:
B_Spline
def apply(m: Int)(j: Int)(t: Double): Double

Obtain the value of the m-th order 'j'-th basis function at time 't'. Or alternatively, obtain the basis function by calling bf(m)(j) only. Ex: val x = bf(m)(j)(t) retrieves the value of the j-th basis function at 't'. val f = bf(m)(j) retrieves the j-th basis function.

Obtain the value of the m-th order 'j'-th basis function at time 't'. Or alternatively, obtain the basis function by calling bf(m)(j) only. Ex: val x = bf(m)(j)(t) retrieves the value of the j-th basis function at 't'. val f = bf(m)(j) retrieves the j-th basis function.

Value parameters

j

indicates which basis function

m

the order of the basis function

t

the time parameter

Attributes

Inherited from:
BasisFunction
def bb(m: Int)(j: Int)(t: Double): Double

Order 'm' B-Spline basis functions (general recurrence).

Order 'm' B-Spline basis functions (general recurrence).

Value parameters

j

indicates which spline function

m

the order of the spline function (degree = order - 1)

t

the time parameter

Attributes

Inherited from:
B_Spline
def bf(m: Int)(j: Int)(t: Double): Double

Evaluate the 'j+1'-th order m B-spline basis function at t non-recursively, using an efficient dynamic programming approach. ==Example==

Evaluate the 'j+1'-th order m B-spline basis function at t non-recursively, using an efficient dynamic programming approach. ==Example==

val m = 4                       // order m (degree m-1)
val t = VectorD (1, 2, 3, 4)    // original time points
val N = B_Spline (t, k)         // B_Spline instance
val j = 0                       // spline j in N.range(m)
val z = N.bf (m)(j)(t(2))       // evaluate at t(2)
println ("order k basis function j at t(2) = z")

==Implementation Details== Let N(m)(i) denote the i-th order k basis function evaluated at t. Then each N(m)(i) depends on the evaluation of N(m-1)(i) and N(m-1)(i+1). Here is an example, given a set of order m=4 B-spline basis functions and knot vector τ of length n:

N(1)(0), N(1)(1), N(1)(2), N(1)(3)
      |/       |/       |/
N(2)(0), N(2)(1), N(2)(2)
      |/       |/
N(3)(0), N(3)(1)
      |/
N(4)(0)

This algorithm applies the procedure described above using O(k) storage and O(k*k) floating point operations.

Value parameters

t

point to evaluate

Attributes

See also

Carl de Boor (1978). A Practical Guide to Splines. Springer-Verlag. ISBN 3-540-90356-9.

Inherited from:
B_Spline
def bf1(j: Int)(t: Double): Double

Obtain the value of the 1st order 'j'-th basis function at time 't'. Or alternatively, obtain the basis function by calling bf1(j) only. Ex: val x = bf1(j)(t) retrieves the value of the j-th basis function at 't'. val f = bf1(j) retrieves the j-th basis function.

Obtain the value of the 1st order 'j'-th basis function at time 't'. Or alternatively, obtain the basis function by calling bf1(j) only. Ex: val x = bf1(j)(t) retrieves the value of the j-th basis function at 't'. val f = bf1(j) retrieves the j-th basis function.

Value parameters

j

indicates which basis function

t

the time parameter

Attributes

Inherited from:
BasisFunction
def bf2(j: Int)(t: Double): Double

Obtain the value of the 2nd order 'j'-th basis function at time 't'. Or alternatively, obtain the basis function by calling bf2(j) only. Ex: val x = bf2(j)(t) retrieves the value of the j-th basis function at 't'. val f = bf2(j) retrieves the j-th basis function.

Obtain the value of the 2nd order 'j'-th basis function at time 't'. Or alternatively, obtain the basis function by calling bf2(j) only. Ex: val x = bf2(j)(t) retrieves the value of the j-th basis function at 't'. val f = bf2(j) retrieves the j-th basis function.

Value parameters

j

indicates which basis function

t

the time parameter

Attributes

Inherited from:
BasisFunction
def bf3(j: Int)(t: Double): Double

Obtain the value of the 3rd order 'j'-th basis function at time 't'. Or alternatively, obtain the basis function by calling bf3(j) only. Ex: val x = bf3(j)(t) retrieves the value of the j-th basis function at 't'. val f = bf3(j) retrieves the j-th basis function.

Obtain the value of the 3rd order 'j'-th basis function at time 't'. Or alternatively, obtain the basis function by calling bf3(j) only. Ex: val x = bf3(j)(t) retrieves the value of the j-th basis function at 't'. val f = bf3(j) retrieves the j-th basis function.

Value parameters

j

indicates which basis function

t

the time parameter

Attributes

Inherited from:
BasisFunction
def bf4(j: Int)(t: Double): Double

Obtain the value of the 4th order 'j'-th basis function at time 't'. Or alternatively, obtain the basis function by calling bf4(j) only. Ex: val x = bf4(j)(t) retrieves the value of the j-th basis function at 't'. val f = bf4(j) retrieves the j-th basis function.

Obtain the value of the 4th order 'j'-th basis function at time 't'. Or alternatively, obtain the basis function by calling bf4(j) only. Ex: val x = bf4(j)(t) retrieves the value of the j-th basis function at 't'. val f = bf4(j) retrieves the j-th basis function.

Value parameters

j

indicates which basis function

t

the time parameter

Attributes

Inherited from:
BasisFunction
def bfr(m: Int)(j: Int)(t: Double): Double

Adjusted order 'm' B-Spline basis functions (general recurrence). These are adjusted so that the first "usable" spline is at j = 0. The valid range of usable splines is defined in range.

Adjusted order 'm' B-Spline basis functions (general recurrence). These are adjusted so that the first "usable" spline is at j = 0. The valid range of usable splines is defined in range.

Value parameters

j

indicates which spline function

m

the order of the spline function (degree = order - 1)

t

the time parameter

Attributes

Inherited from:
B_Spline
def count(m: Int): Int

The number of basis functions for a specified order.

The number of basis functions for a specified order.

Value parameters

m

the order of the basis function

Attributes

Inherited from:
BasisFunction
def dnabf(n: Int)(m: Int)(t: VectorD): MatrixD

Obtain the value of nth derivative of the m-th order basis functions (all) at time 't'. Or alternatively, obtain the basis function by calling dnabf(m)(j) only. Ex: val x = dnabf(n)(m)(t) retrieves the nth derivative value of the value of all the basis functions at 't'. val f = dnabf(n)(m) retrieves the nth derivative value of all the basis functions.

Obtain the value of nth derivative of the m-th order basis functions (all) at time 't'. Or alternatively, obtain the basis function by calling dnabf(m)(j) only. Ex: val x = dnabf(n)(m)(t) retrieves the nth derivative value of the value of all the basis functions at 't'. val f = dnabf(n)(m) retrieves the nth derivative value of all the basis functions.

Value parameters

m

the order of all the basis function

n

the order of the derivative

t

the time parameter

Attributes

Inherited from:
DBasisFunction
def dot_(n: Int)(m: Int)(i: Int, j: Int)(g: DBasisFunction, a: Double, b: Double): Double

Compute the dot/inner product of nth derivative of 'this' basis function and that of basis function 'g'.

Compute the dot/inner product of nth derivative of 'this' basis function and that of basis function 'g'.

Value parameters

a

the start of the interval

b

the end of the interval

g

the other function

j

indicates which basis function

m

the order of the basis function

n

the order of the derivative

Attributes

Inherited from:
DBasisFunction
def dot_(m: Int)(i: Int, j: Int)(g: BasisFunction, a: Double, b: Double): Double

Compute the dot/inner product of 'this' basis function object and basis function 'g'.

Compute the dot/inner product of 'this' basis function object and basis function 'g'.

Value parameters

a

the start of the interval

b

the end of the interval

g

the other function

i

indicates which basis function of 'this'

j

indicates which basis function of 'g'

m

the order of the basis function

Attributes

Inherited from:
BasisFunction
override def getCache(m: Int, t: VectorD): Array[MatrixD]

Retrieves the cached design matrices and penalty matrices

Retrieves the cached design matrices and penalty matrices

Value parameters

m

the order of all the basis function

t

the time parameter

Attributes

Definition Classes
Inherited from:
DBasisFunction
def getOrder: Int

Retrieve the order of the this B_Spline.

Retrieve the order of the this B_Spline.

Attributes

Inherited from:
B_Spline
def range(m: Int): Range

Range of "usable" splines when using the bs function. This is needed, since extra splines may be generated by the general B-spline recurrence.

Range of "usable" splines when using the bs function. This is needed, since extra splines may be generated by the general B-spline recurrence.

Value parameters

m

the order of the spline

Attributes

Inherited from:
B_Spline
def recomputeCache: Unit

Recompute cached matrices.

Recompute cached matrices.

Attributes

Inherited from:
BasisFunction
def size(m: Int): Int

The number of usable spline basis functions for a specified order, given the configured knot vector.

The number of usable spline basis functions for a specified order, given the configured knot vector.

Value parameters

m

the order of the spline

Attributes

Inherited from:
B_Spline
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