final def !=(arg0: Any): Boolean

Definition Classes: AnyRef → Any

final def ##(): Int

Definition Classes: AnyRef → Any

final def ==(arg0: Any): Boolean

Definition Classes: AnyRef → Any

val DEBUG: Boolean

Attributes: protected
Definition Classes: B_Spline

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

Obtain the value of the m-th order basis functions (all) at time 't'.

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.

m: the order of all the basis function
t: the time parameter

Definition Classes: BasisFunction

def abf_(m: Int = mMax)(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

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.

m: the order of all the basis function
t: point to evaluate

Definition Classes: B_Spline → BasisFunction
See also: Carl de Boor (1978). A Practical Guide to Splines. Springer-Verlag. ISBN 3-540-90356-9.

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

Obtain the value of the m-th order 'j'-th basis function at time 't'.

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.

m: the order of the basis function
j: indicates which basis function
t: the time parameter

Definition Classes: BasisFunction

final def asInstanceOf[T0]: T0

Definition Classes: Any

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

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

m: the order of the spline function (degree = order - 1)
j: indicates which spline function
t: the time parameter

Definition Classes: 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

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.

m: the order of the basis function
j: indicates which basis function
t: point to evaluate

Definition Classes: B_Spline → BasisFunction
See also: Carl de Boor (1978). A Practical Guide to Splines. Springer-Verlag. ISBN 3-540-90356-9.

def bf1(j: Int)(t: Double): Double

Obtain the value of the 1st order 'j'-th basis function at time 't'.

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.

j: indicates which basis function
t: the time parameter

Definition Classes: BasisFunction

def bf2(j: Int)(t: Double): Double

Obtain the value of the 2nd order 'j'-th basis function at time 't'.

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.

j: indicates which basis function
t: the time parameter

Definition Classes: BasisFunction

def bf3(j: Int)(t: Double): Double

Obtain the value of the 3rd order 'j'-th basis function at time 't'.

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.

j: indicates which basis function
t: the time parameter

Definition Classes: BasisFunction

def bf4(j: Int)(t: Double): Double

Obtain the value of the 4th order 'j'-th basis function at time 't'.

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.

j: indicates which basis function
t: the time parameter

Definition Classes: BasisFunction

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

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

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.

m: the order of the spline function (degree = order - 1)
j: indicates which spline function
t: the time parameter

Definition Classes: B_Spline

def clone(): AnyRef

Attributes: protected[java.lang]
Definition Classes: AnyRef
Annotations: @native() @throws( ... )

def count(m: Int): Int

The number of basis functions for a specified order.

m: the order of the basis function

Definition Classes: BasisFunction

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

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

m: the order of the spline function (degree = order - 1)
j: indicates which spline function
t: the time parameter

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

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

m: the order of the spline function (degree = order - 1)
j: indicates which spline function
t: the time parameter

Definition Classes: DB_Spline → DBasisFunction

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

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

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.

m: the order of the spline function (degree = order - 1)
j: indicates which spline function
t: the time parameter

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

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

m: the order of the spline function (degree = order - 1)
j: indicates which spline function
t: the time parameter

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'.

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.

m: the order of the basis function
j: indicates which basis function
t: the time parameter

Definition Classes: DB_Spline → DBasisFunction

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

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

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.

m: the order of the spline function (degree = order - 1)
j: indicates which spline function
t: the time parameter

def dnabf(n: Int)(m: Int)(t: VectoD): MatrixD

Obtain the value of nth derivative of the m-th order basis functions (all) at time 't'.

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.

n: the order of the derivative
m: the order of all the basis function
t: the time parameter

Definition Classes: DBasisFunction

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'.

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.

n: the order of the derivative
m: the order of all the basis function
t: the time parameter

Definition Classes: DB_Spline → DBasisFunction

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'.

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.

n: the order of the derivative
m: the order of all the basis function
t: the time parameter

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'.

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.

n: the order of the derivative
m: the order of all the basis function
t: the time parameter

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

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

n: the n-th derivative to be computed
m: the order of the spline function (degree = order - 1)
j: indicates which spline function
t: the time parameter

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

Adjusted n-th derivative of order 'm' B-Spline basis functions (general recurrence).

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.

n: the n-th derivative to be computed
m: the order of the spline function (degree = order - 1)
j: indicates which spline function
t: the time parameter

Definition Classes: DB_Spline → DBasisFunction

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

Adjusted n-th derivative of order 'm' B-Spline basis functions (general recurrence).

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.

n: the n-th derivative to be computed
m: the order of the spline function (degree = order - 1)
j: indicates which spline function
t: the time parameter

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'.

n: the order of the derivative
m: the order of the basis function
j: indicates which basis function
g: the other function
a: the start of the interval
b: the end of the interval

Definition Classes: DBasisFunction

def dot_(m: Int)(i: Int, j: Int)(g: BasisFunction, a: Double = 0.0, b: Double = 1.0): Double

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

m: the order of the basis function
i: indicates which basis function of 'this'
j: indicates which basis function of 'g'
g: the other function
a: the start of the interval
b: the end of the interval

Definition Classes: BasisFunction

final def eq(arg0: AnyRef): Boolean

Definition Classes: AnyRef

def equals(arg0: Any): Boolean

Definition Classes: AnyRef → Any

def finalize(): Unit

Attributes: protected[java.lang]
Definition Classes: AnyRef
Annotations: @throws( classOf[java.lang.Throwable] )

final def flaw(method: String, message: String): Unit

Definition Classes: Error

def getCache(m: Int, t: VectoD): Array[MatrixD]

Retrieves the cached design matrices and penalty matrices

m: the order of all the basis function
t: the time parameter

Definition Classes: DBasisFunction → BasisFunction

final def getClass(): Class[_]

Definition Classes: AnyRef → Any
Annotations: @native()

def getOrder: Int

Retrieve the order of the this B_Spline.

Definition Classes: B_Spline → BasisFunction

def hashCode(): Int

Definition Classes: AnyRef → Any
Annotations: @native()

val head: Double

Attributes: protected
Definition Classes: B_Spline

final def isInstanceOf[T0]: Boolean

Definition Classes: Any

val l: Int

Attributes: protected
Definition Classes: B_Spline

final def ne(arg0: AnyRef): Boolean

Definition Classes: AnyRef

val needCompute: Boolean

Attributes: protected
Definition Classes: BasisFunction

final def notify(): Unit

Definition Classes: AnyRef
Annotations: @native()

final def notifyAll(): Unit

Definition Classes: AnyRef
Annotations: @native()

def range(m: Int = mMax): Inclusive

Range of "usable" splines when using the bs function.

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

m: the order of the spline

Definition Classes: B_Spline → BasisFunction

def recomputeCache: Unit

Recompute cached matrices

Definition Classes: BasisFunction

def size(m: Int = mMax): Int

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

m: the order of the spline

Definition Classes: B_Spline → BasisFunction

final def synchronized[T0](arg0: ⇒ T0): T0

Definition Classes: AnyRef

val tail: Double

Attributes: protected
Definition Classes: B_Spline

def toString(): String

Definition Classes: AnyRef → Any

final def wait(): Unit

Definition Classes: AnyRef
Annotations: @throws( ... )

final def wait(arg0: Long, arg1: Int): Unit

Definition Classes: AnyRef
Annotations: @throws( ... )

final def wait(arg0: Long): Unit

Definition Classes: AnyRef
Annotations: @native() @throws( ... )

val Σ: MatrixD

Attributes: protected
Definition Classes: DBasisFunction

val Φ: MatrixD

Attributes: protected
Definition Classes: BasisFunction

val Φt: MatrixD

Attributes: protected
Definition Classes: BasisFunction

val ΦtΦ: MatrixD

Attributes: protected
Definition Classes: BasisFunction

val τ: VectoD

Attributes: protected
Definition Classes: B_Spline

Packages

DB_Spline

class DB_Spline extends B_Spline with DBasisFunction

Instance Constructors

Value Members

Example

Implementation Details

Example

Implementation Details

Inherited from DBasisFunction

Inherited from B_Spline

Inherited from Error

Inherited from BasisFunction

Inherited from AnyRef

Inherited from Any

Ungrouped

Packages

DB_Spline 

class DB_Spline extends B_Spline with DBasisFunction

Instance Constructors

Value Members

Example

Implementation Details

Example

Implementation Details

Inherited from DBasisFunction

Inherited from B_Spline

Inherited from Error

Inherited from BasisFunction

Inherited from AnyRef

Inherited from Any

Ungrouped

DB_Spline