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Intuition Sobolev spaces and smoothing splines

0

$begingroup$

With inputs $X_1, dots, X_n$ in a closed interval $[a,b]$ and $a<b$ the smoothing spline estimate $hat{f}$ of a given odd order $k$ is given by minimizing the following penalized residual sum of squares problem with $lambda geq 0$
begin{equation}
sum_{i=1}^{n}(Y_i - f(X_i))^2 + lambda int_{a}^{b}(f^{(r)}(x))^2dx quad text{ with $r=(k+1)/2$} quad (1)
end{equation}
over all functions $f$ in the so-called Sobolev space
$$
mathcal{W}_2^{(r)} = {f: f (r-1) text{-times continuously differentiable and} int_a^b[f^{(r)}(x)]^2 dx < infty }.
$$

Question:

Is there any intuition why $f$ must be taken from this function space? Can I say that this function space guarantees that the regularization term in (1) is well-defined and the estimated function is sufficiently smooth (measured by the continuous differentiability)?

functional-analysis statistics sobolev-spaces spline

share|cite|improve this question

asked Mar 24 at 10:57

user483161user483161

617

$endgroup$

add a comment | 

0

$begingroup$

With inputs $X_1, dots, X_n$ in a closed interval $[a,b]$ and $a<b$ the smoothing spline estimate $hat{f}$ of a given odd order $k$ is given by minimizing the following penalized residual sum of squares problem with $lambda geq 0$
begin{equation}
sum_{i=1}^{n}(Y_i - f(X_i))^2 + lambda int_{a}^{b}(f^{(r)}(x))^2dx quad text{ with $r=(k+1)/2$} quad (1)
end{equation}
over all functions $f$ in the so-called Sobolev space
$$
mathcal{W}_2^{(r)} = {f: f (r-1) text{-times continuously differentiable and} int_a^b[f^{(r)}(x)]^2 dx < infty }.
$$

Question:

Is there any intuition why $f$ must be taken from this function space? Can I say that this function space guarantees that the regularization term in (1) is well-defined and the estimated function is sufficiently smooth (measured by the continuous differentiability)?

functional-analysis statistics sobolev-spaces spline

share|cite|improve this question

asked Mar 24 at 10:57

user483161user483161

617

$endgroup$

add a comment | 

$begingroup$

With inputs $X_1, dots, X_n$ in a closed interval $[a,b]$ and $a<b$ the smoothing spline estimate $hat{f}$ of a given odd order $k$ is given by minimizing the following penalized residual sum of squares problem with $lambda geq 0$
begin{equation}
sum_{i=1}^{n}(Y_i - f(X_i))^2 + lambda int_{a}^{b}(f^{(r)}(x))^2dx quad text{ with $r=(k+1)/2$} quad (1)
end{equation}
over all functions $f$ in the so-called Sobolev space
$$
mathcal{W}_2^{(r)} = {f: f (r-1) text{-times continuously differentiable and} int_a^b[f^{(r)}(x)]^2 dx < infty }.
$$

Question:

Is there any intuition why $f$ must be taken from this function space? Can I say that this function space guarantees that the regularization term in (1) is well-defined and the estimated function is sufficiently smooth (measured by the continuous differentiability)?

functional-analysis statistics sobolev-spaces spline

share|cite|improve this question

asked Mar 24 at 10:57

user483161user483161

617

$endgroup$

share|cite|improve this question

asked Mar 24 at 10:57

user483161user483161

617

share|cite|improve this question

asked Mar 24 at 10:57

user483161user483161

617

asked Mar 24 at 10:57

user483161user483161

617

asked Mar 24 at 10:57

user483161user483161

617