Simulation Data#
- class EarlyStopping.SimulationData#
[Source] A collection of static methods for the creation of simmulation data.
Description
Collection of serveral important examples of inverse problems.
References
[Toolbox] The ill-posed inverse problems heat, deriv2, gravity, phillips are based on the Hansen-Matlab toolbox.
Methods
diagonal_data(
sample_size,type)Creation of diagonal design with smooth, supersmooth and rough signals
gravity(
sample_size,a,b,d)Discretised gravity operator with parameters controlling the ill-posedness of the problem
heat(
sample_size,kappa)Discretised heat semigroup with parameter controlling the ill-conditioning of the matrix
deriv2(
sample_size,example)Discretisation of a Fredholm integral equation whose kernel K is the Green’s function
phillips(
sample_size)Create data based on the famous Phillips example
additive_smooth(
sample_size,noise_level)Create data with a smooth additive model using sine, quadratic, linear, and exp.components
additive_step(
sample_size,noise_level)Create data with a stepwise additive model using piecewise constant functions
additive_linear(
sample_size,noise_level)Create data with a linear additive model using piecewise linear interpolation
additive_hills(
sample_size,noise_level)Create data with a hills additive model using sinusoidal components with varying frequencies
- static SimulationData.deriv2(sample_size, example=1)#
Mildly ill-posed inverse problem based on the discretization of a Fredholm integral equation of the first kind whose kernel K is the Green’s function for the second derivative:
\[\begin{split}K(s,t) = \begin{cases}s(t-1) , &s < t, \\ t(s-1) , &s \geq t.\end{cases}\end{split}\]The right hand side \(g\) and the solution \(f\) can be chosen as follows:
\(g(s) = (s^3 - s)/6\) and \(f(t) = t\).
\(g(s) = \exp(s) + (1-e)s - 1\) and \(f(t) = \exp(t)\).
\(g(s) = \begin{cases}(4s^3 - 3s)/24, &s < 0.5,\\(-4s^3 + 12s^2 - 9s + 1)/24, &s \geq 0.5.\end{cases}\) and \(f(t) = \begin{cases}t, &t < 0.5,\\1-t, &t \geq 0.5.\end{cases}\).
Parameters
sample_size:
int. Specifies the size of the design matrix to be generated.example:
int. The example to be used. Must be 1, 2 or 3.Returns
design:
ndarray. The design matrix.response_noiseless:
ndarray. The noiseless response. (Based on discretisation).true_signal:
ndarray. The true signal.
- static SimulationData.heat(sample_size, kappa=1)#
A first kind Volterra integral equation with \([0,1]\) as integration interval. The kernel is \(K(s,t) = k(s-t)\) with the heat kernel
\[k(t) = \frac{t^{-1/2}}{2\kappa \sqrt{\pi}}\exp\left(-\frac{1}{4 \kappa^2 t}\right) .\]Parameters
sample_size:
int. Specifies the size of the design matrix to be generated.kappa:
int. Here, kappa controls the ill-conditioning of the matrix:\(\kappa = 5\) gives a well-conditioned problem.
\(\kappa = 1\) gives an ill-conditioned problem.
Returns
design:
ndarray. The design matrix.response_noiseless:
ndarray. The noiseless response. An exact soltuion is constructed, and the response is produced by applying the design to the signal.true_signal:
ndarray. The true signal.
- static SimulationData.gravity(sample_size, a=0, b=1, d=0.25)#
Discretisation of a 1-D model problem in gravity surveying, in which a mass distribution \(f(t)\) is located at depth d, while the vertical component of the gravity field \(g(s)\) is measured at the surface.
The resulting problem is a first-kind Fredholm integral equation with kernel
\[K(s,t) = d(d^2 + (s-t)^2)^{-3/2},\]with the right-hand side given by
\[f(t) = \sin(\pi t) + 0.5\sin(2\pi t).\]The problem is discretized by means of the midpoint quadrature rule.
Parameters
sample_size:
int. Specifies the size of the design matrix to be generated.a:
int. The lower bound of the integration interval.b:
int. The upper bound of the integration interval.d:
int. The depth at which the magnetic deposit is located.Returns
design:
ndarray. The design matrix.response_noiseless:
ndarray. The noiseless response. The response is produced by applying the design to the signal.true_signal:
ndarray. The true signal.
- static SimulationData.phillips(sample_size)#
Discretization of the famous first-kind Fredholm integral equation deviced by D. L. Phillips.
Define the function
\[\begin{split}\phi(x) = \begin{cases}1 + \cos(x\pi/3), & |x| < 3,\\0, & |x| \geq 3.\end{cases}\end{split}\]The kernel K, the solution f, and the right-hand side g are given by:
\(K(s,t) = \phi(s-t)\)
\(f(t) = \phi(t)\)
\(g(s) = (6-|s|)(1+0.5 \cos(s\pi/3)) + 9/(2\pi)\sin(|s|\pi/3)\)
Both integration intervals are [-6,6].
Parameters
sample_size:
int. Specifies the size of the design matrix to be generated. Must be a multiple of 4.Returns
design:
ndarray. The design matrix.response_noiseless:
ndarray. The noiseless response. (obtained from discretisation).true_signal:
ndarray. The true signal.