Description Usage Arguments Details Value See Also Examples

A linear differential operator of order $m$ is defined, usually to specify a roughness penalty.

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`nderiv` |
a nonnegative integer specifying the order $m$ of the highest order derivative in the operator |

`bwtlist` |
a list of length $m$. Each member contains a functional data object that acts as a weight function for a derivative. The first member weights the function, the second the first derivative, and so on up to order $m-1$. |

To check that an object is of this class, use functions `is.Lfd`

or `int2Lfd`

.

Linear differential operator objects are often used to define roughness penalties for smoothing towards a "hypersmooth" function that is annihilated by the operator. For example, the harmonic acceleration operator used in the analysis of the Canadian daily weather data annihilates linear combinations of $1, sin(2 pi t/365)$ and $cos(2 pi t/365)$, and the larger the smoothing parameter, the closer the smooth function will be to a function of this shape.

Function `pda.fd`

estimates a linear differential operator object
that comes as close as possible to annihilating a functional data
object.

A linear differential operator of order $m$ is a linear combination of the derivatives of a functional data object up to order $m$. The derivatives of orders 0, 1, ..., $m-1$ can each be multiplied by a weight function $b(t)$ that may or may not vary with argument $t$.

If the notation $D^j$ is taken to mean "take the derivative of order $j$", then a linear differental operator $L$ applied to function $x$ has the expression

$Lx(t) = b_0(t) x(t) + b_1(t)Dx(t) + ... + b_{m-1}(t) D^{m-1} x(t) + D^mx(t)$

There are `print`

, `summary`

, and `plot`

methods for
objects of class `Lfd`

.

a linear differential operator object

`int2Lfd`

,
`vec2Lfd`

,
`fdPar`

,
`pda.fd`

`plot.Lfd`

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