Commit 33974bdb authored by Philipp Arras's avatar Philipp Arras
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Show plots in html docu

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%% Cell type:markdown id: tags:
# Code example: Wiener filter
%% Cell type:markdown id: tags:
## Introduction
IFT starting point:
$$d = Rs+n$$
Typically, $s$ is a continuous field, $d$ a discrete data vector. Particularly, $R$ is not invertible.
IFT aims at **inverting** the above uninvertible problem in the **best possible way** using Bayesian statistics.
NIFTy (Numerical Information Field Theory) is a Python framework in which IFT problems can be tackled easily.
Main Interfaces:
- **Spaces**: Cartesian, 2-Spheres (Healpix, Gauss-Legendre) and their respective harmonic spaces.
- **Fields**: Defined on spaces.
- **Operators**: Acting on fields.
%% Cell type:markdown id: tags:
## Wiener filter on one-dimensional fields
### Assumptions
- $d=Rs+n$, $R$ linear operator.
- $\mathcal P (s) = \mathcal G (s,S)$, $\mathcal P (n) = \mathcal G (n,N)$ where $S, N$ are positive definite matrices.
### Posterior
The Posterior is given by:
$$\mathcal P (s|d) \propto P(s,d) = \mathcal G(d-Rs,N) \,\mathcal G(s,S) \propto \mathcal G (s-m,D) $$
where
$$m = Dj$$
with
$$D = (S^{-1} +R^\dagger N^{-1} R)^{-1} , \quad j = R^\dagger N^{-1} d.$$
Let us implement this in NIFTy!
%% Cell type:markdown id: tags:
### In NIFTy
- We assume statistical homogeneity and isotropy. Therefore the signal covariance $S$ is diagonal in harmonic space, and is described by a one-dimensional power spectrum, assumed here as $$P(k) = P_0\,\left(1+\left(\frac{k}{k_0}\right)^2\right)^{-\gamma /2},$$
with $P_0 = 0.2, k_0 = 5, \gamma = 4$.
- $N = 0.2 \cdot \mathbb{1}$.
- Number of data points $N_{pix} = 512$.
- reconstruction in harmonic space.
- Response operator:
$$R = FFT_{\text{harmonic} \rightarrow \text{position}}$$
%% Cell type:code id: tags:
``` python
N_pixels = 512 # Number of pixels
def pow_spec(k):
P0, k0, gamma = [.2, 5, 4]
return P0 / ((1. + (k/k0)**2)**(gamma / 2))
```
%% Cell type:markdown id: tags:
### Implementation
%% Cell type:markdown id: tags:
#### Import Modules
%% Cell type:code id: tags:
``` python
%matplotlib inline
import numpy as np
import nifty8 as ift
import matplotlib.pyplot as plt
plt.rcParams['figure.dpi'] = 100
plt.style.use("seaborn-notebook")
```
%% Cell type:markdown id: tags:
#### Implement Propagator
%% Cell type:code id: tags:
``` python
def Curvature(R, N, Sh):
IC = ift.GradientNormController(iteration_limit=50000,
tol_abs_gradnorm=0.1)
# WienerFilterCurvature is (R.adjoint*N.inverse*R + Sh.inverse) plus some handy
# helper methods.
return ift.WienerFilterCurvature(R,N,Sh,iteration_controller=IC,iteration_controller_sampling=IC)
```
%% Cell type:markdown id: tags:
#### Conjugate Gradient Preconditioning
- $D$ is defined via:
$$D^{-1} = \mathcal S_h^{-1} + R^\dagger N^{-1} R.$$
In the end, we want to apply $D$ to $j$, i.e. we need the inverse action of $D^{-1}$. This is done numerically (algorithm: *Conjugate Gradient*).
<!--
- One can define the *condition number* of a non-singular and normal matrix $A$:
$$\kappa (A) := \frac{|\lambda_{\text{max}}|}{|\lambda_{\text{min}}|},$$
where $\lambda_{\text{max}}$ and $\lambda_{\text{min}}$ are the largest and smallest eigenvalue of $A$, respectively.
- The larger $\kappa$ the slower Conjugate Gradient.
- By default, conjugate gradient solves: $D^{-1} m = j$ for $m$, where $D^{-1}$ can be badly conditioned. If one knows a non-singular matrix $T$ for which $TD^{-1}$ is better conditioned, one can solve the equivalent problem:
$$\tilde A m = \tilde j,$$
where $\tilde A = T D^{-1}$ and $\tilde j = Tj$.
- In our case $S^{-1}$ is responsible for the bad conditioning of $D$ depending on the chosen power spectrum. Thus, we choose
$$T = \mathcal F^\dagger S_h^{-1} \mathcal F.$$
-->
%% Cell type:markdown id: tags:
#### Generate Mock data
- Generate a field $s$ and $n$ with given covariances.
- Calculate $d$.
%% Cell type:code id: tags:
``` python
s_space = ift.RGSpace(N_pixels)
h_space = s_space.get_default_codomain()
HT = ift.HarmonicTransformOperator(h_space, target=s_space)
# Operators
Sh = ift.create_power_operator(h_space, power_spectrum=pow_spec)
R = HT # @ ift.create_harmonic_smoothing_operator((h_space,), 0, 0.02)
# Fields and data
sh = Sh.draw_sample_with_dtype(dtype=np.float64)
noiseless_data=R(sh)
noise_amplitude = np.sqrt(0.2)
N = ift.ScalingOperator(s_space, noise_amplitude**2)
n = ift.Field.from_random(domain=s_space, random_type='normal',
std=noise_amplitude, mean=0)
d = noiseless_data + n
j = R.adjoint_times(N.inverse_times(d))
curv = Curvature(R=R, N=N, Sh=Sh)
D = curv.inverse
```
%% Cell type:markdown id: tags:
#### Run Wiener Filter
%% Cell type:code id: tags:
``` python
m = D(j)
```
%% Cell type:markdown id: tags:
#### Results
%% Cell type:code id: tags:
``` python
# Get signal data and reconstruction data
s_data = HT(sh).val
m_data = HT(m).val
d_data = d.val
plt.plot(s_data, 'r', label="Signal", linewidth=2)
plt.plot(d_data, 'k.', label="Data")
plt.plot(m_data, 'k', label="Reconstruction",linewidth=2)
plt.title("Reconstruction")
plt.legend()
plt.show()
```
%% Cell type:code id: tags:
``` python
plt.plot(s_data - s_data, 'r', label="Signal", linewidth=2)
plt.plot(d_data - s_data, 'k.', label="Data")
plt.plot(m_data - s_data, 'k', label="Reconstruction",linewidth=2)
plt.axhspan(-noise_amplitude,noise_amplitude, facecolor='0.9', alpha=.5)
plt.title("Residuals")
plt.legend()
plt.show()
```
%% Cell type:markdown id: tags:
#### Power Spectrum
%% Cell type:code id: tags:
``` python
s_power_data = ift.power_analyze(sh).val
m_power_data = ift.power_analyze(m).val
plt.loglog()
plt.xlim(1, int(N_pixels/2))
ymin = min(m_power_data)
plt.ylim(ymin, 1)
xs = np.arange(1,int(N_pixels/2),.1)
plt.plot(xs, pow_spec(xs), label="True Power Spectrum", color='k',alpha=0.5)
plt.plot(s_power_data, 'r', label="Signal")
plt.plot(m_power_data, 'k', label="Reconstruction")
plt.axhline(noise_amplitude**2 / N_pixels, color="k", linestyle='--', label="Noise level", alpha=.5)
plt.axhspan(noise_amplitude**2 / N_pixels, ymin, facecolor='0.9', alpha=.5)
plt.title("Power Spectrum")
plt.legend()
plt.show()
```
%% Cell type:markdown id: tags:
## Wiener Filter on Incomplete Data
%% Cell type:code id: tags:
``` python
# Operators
Sh = ift.create_power_operator(h_space, power_spectrum=pow_spec)
N = ift.ScalingOperator(s_space, noise_amplitude**2)
# R is defined below
# Fields
sh = Sh.draw_sample_with_dtype(dtype=np.float64)
s = HT(sh)
n = ift.Field.from_random(domain=s_space, random_type='normal',
std=noise_amplitude, mean=0)
```
%% Cell type:markdown id: tags:
### Partially Lose Data
%% Cell type:code id: tags:
``` python
l = int(N_pixels * 0.2)
h = int(N_pixels * 0.2 * 2)
mask = np.full(s_space.shape, 1.)
mask[l:h] = 0
mask = ift.Field.from_raw(s_space, mask)
R = ift.DiagonalOperator(mask)(HT)
n = n.val_rw()
n[l:h] = 0
n = ift.Field.from_raw(s_space, n)
d = R(sh) + n
```
%% Cell type:code id: tags:
``` python
curv = Curvature(R=R, N=N, Sh=Sh)
D = curv.inverse
j = R.adjoint_times(N.inverse_times(d))
m = D(j)
```
%% Cell type:markdown id: tags:
### Compute Uncertainty
%% Cell type:code id: tags:
``` python
m_mean, m_var = ift.probe_with_posterior_samples(curv, HT, 200, np.float64)
```
%% Cell type:markdown id: tags:
### Get data
%% Cell type:code id: tags:
``` python
# Get signal data and reconstruction data
s_data = s.val
m_data = HT(m).val
m_var_data = m_var.val
uncertainty = np.sqrt(m_var_data)
d_data = d.val_rw()
# Set lost data to NaN for proper plotting
d_data[d_data == 0] = np.nan
```
%% Cell type:code id: tags:
``` python
plt.axvspan(l, h, facecolor='0.8',alpha=0.5)
plt.fill_between(range(N_pixels), m_data - uncertainty, m_data + uncertainty, facecolor='0.5', alpha=0.5)
plt.plot(s_data, 'r', label="Signal", alpha=1, linewidth=2)
plt.plot(d_data, 'k.', label="Data")
plt.plot(m_data, 'k', label="Reconstruction", linewidth=2)
plt.title("Reconstruction of incomplete data")
plt.legend()
```
%% Cell type:markdown id: tags:
## Wiener filter on two-dimensional field
%% Cell type:code id: tags:
``` python
N_pixels = 256 # Number of pixels
sigma2 = 2. # Noise variance
def pow_spec(k):
P0, k0, gamma = [.2, 2, 4]
return P0 * (1. + (k/k0)**2)**(-gamma/2)
s_space = ift.RGSpace([N_pixels, N_pixels])
```
%% Cell type:code id: tags:
``` python
h_space = s_space.get_default_codomain()
HT = ift.HarmonicTransformOperator(h_space,s_space)
# Operators
Sh = ift.create_power_operator(h_space, power_spectrum=pow_spec)
N = ift.ScalingOperator(s_space, sigma2)
# Fields and data
sh = Sh.draw_sample_with_dtype(dtype=np.float64)
n = ift.Field.from_random(domain=s_space, random_type='normal',
std=np.sqrt(sigma2), mean=0)
# Lose some data
l = int(N_pixels * 0.33)
h = int(N_pixels * 0.33 * 2)
mask = np.full(s_space.shape, 1.)
mask[l:h,l:h] = 0.
mask = ift.Field.from_raw(s_space, mask)
R = ift.DiagonalOperator(mask)(HT)
n = n.val_rw()
n[l:h, l:h] = 0
n = ift.Field.from_raw(s_space, n)
curv = Curvature(R=R, N=N, Sh=Sh)
D = curv.inverse
d = R(sh) + n
j = R.adjoint_times(N.inverse_times(d))
# Run Wiener filter
m = D(j)
# Uncertainty
m_mean, m_var = ift.probe_with_posterior_samples(curv, HT, 20, np.float64)
# Get data
s_data = HT(sh).val
m_data = HT(m).val
m_var_data = m_var.val
d_data = d.val
uncertainty = np.sqrt(np.abs(m_var_data))
```
%% Cell type:code id: tags:
``` python
cmap = ['magma', 'inferno', 'plasma', 'viridis'][1]
mi = np.min(s_data)
ma = np.max(s_data)
fig, axes = plt.subplots(1, 2)
data = [s_data, d_data]
caption = ["Signal", "Data"]
for ax in axes.flat:
im = ax.imshow(data.pop(0), interpolation='nearest', cmap=cmap, vmin=mi,
vmax=ma)
ax.set_title(caption.pop(0))
fig.subplots_adjust(right=0.8)
cbar_ax = fig.add_axes([0.85, 0.15, 0.05, 0.7])
fig.colorbar(im, cax=cbar_ax)
```
%% Cell type:code id: tags:
``` python
mi = np.min(s_data)
ma = np.max(s_data)
fig, axes = plt.subplots(3, 2, figsize=(10, 15))
sample = HT(curv.draw_sample(from_inverse=True)+m).val
post_mean = (m_mean + HT(m)).val
data = [s_data, m_data, post_mean, sample, s_data - m_data, uncertainty]
caption = ["Signal", "Reconstruction", "Posterior mean", "Sample", "Residuals", "Uncertainty Map"]
for ax in axes.flat:
im = ax.imshow(data.pop(0), interpolation='nearest', cmap=cmap, vmin=mi, vmax=ma)
ax.set_title(caption.pop(0))
fig.subplots_adjust(right=0.8)
cbar_ax = fig.add_axes([.85, 0.15, 0.05, 0.7])
fig.colorbar(im, cax=cbar_ax)
```
%% Cell type:markdown id: tags:
### Is the uncertainty map reliable?
%% Cell type:code id: tags:
``` python
precise = (np.abs(s_data-m_data) < uncertainty)
print("Error within uncertainty map bounds: " + str(np.sum(precise) * 100 / N_pixels**2) + "%")
plt.imshow(precise.astype(float), cmap="brg")
plt.colorbar()
```
......
%% Cell type:markdown id: tags:
# Showcasing the Correlated Field model
**Skip to `Parameter Showcases` for the meat/veggies ;)**
The field model roughly works like this:
`f = HT( A * zero_mode * xi ) + offset`
`A` is a spectral power field which is constructed from power spectra that hold on subdomains of the target domain.
It is scaled by a zero mode operator and then pointwise multiplied by a gaussian excitation field, yielding
a representation of the field in harmonic space.
It is then transformed into the target real space and a offset added.
The power spectra `A` is constructed of are in turn constructed as the sum of a power law component
and an integrated Wiener process whose amplitude and roughness can be set.
## Setup code
%% Cell type:code id: tags:
``` python
%matplotlib inline
import nifty8 as ift
import matplotlib.pyplot as plt
plt.rcParams['figure.dpi'] = 100
plt.style.use("seaborn-notebook")
import numpy as np
n_pix = 256
x_space = ift.RGSpace(n_pix)
```
%% Cell type:code id: tags:
``` python
# Plotting routine
def plot(fields, spectra, title=None):
# Plotting preparation is normally handled by nifty8.Plot
# It is done manually here to be able to tweak details
# Fields are assumed to have identical domains
fig = plt.figure(tight_layout=True, figsize=(10, 3))
if title is not None:
fig.suptitle(title, fontsize=14)
# Field
ax1 = fig.add_subplot(1, 2, 1)
ax1.axhline(y=0., color='k', linestyle='--', alpha=0.25)
for field in fields:
dom = field.domain[0]
xcoord = np.arange(dom.shape[0]) * dom.distances[0]
ax1.plot(xcoord, field.val)
ax1.set_xlim(xcoord[0], xcoord[-1])
ax1.set_ylim(-5., 5.)
ax1.set_xlabel('x')
ax1.set_ylabel('f(x)')
ax1.set_title('Field realizations')
# Spectrum
ax2 = fig.add_subplot(1, 2, 2)
for spectrum in spectra:
xcoord = spectrum.domain[0].k_lengths
ycoord = spectrum.val_rw()
ycoord[0] = ycoord[1]
ax2.plot(xcoord, ycoord)
ax2.set_ylim(1e-6, 10.)
ax2.set_xscale('log')
ax2.set_yscale('log')
ax2.set_xlabel('k')
ax2.set_ylabel('p(k)')
ax2.set_title('Power Spectrum')
fig.align_labels()
plt.show()
# Helper: draw main sample
main_sample = None
def init_model(m_pars, fl_pars):
global main_sample
cf = ift.CorrelatedFieldMaker(m_pars["prefix"])
cf.set_amplitude_total_offset(m_pars["offset_mean"], m_pars["offset_std"])
cf.add_fluctuations(**fl_pars)
field = cf.finalize(prior_info=0)
main_sample = ift.from_random(field.domain)
print("model domain keys:", field.domain.keys())
# Helper: field and spectrum from parameter dictionaries + plotting
def eval_model(m_pars, fl_pars, title=None, samples=None):
cf = ift.CorrelatedFieldMaker(m_pars["prefix"])
cf.set_amplitude_total_offset(m_pars["offset_mean"], m_pars["offset_std"])
cf.add_fluctuations(**fl_pars)
field = cf.finalize(prior_info=0)
spectrum = cf.amplitude
if samples is None:
samples = [main_sample]
field_realizations = [field(s) for s in samples]
spectrum_realizations = [spectrum.force(s) for s in samples]
plot(field_realizations, spectrum_realizations, title)
def gen_samples(key_to_vary):
if key_to_vary is None:
return [main_sample]
dct = main_sample.to_dict()
subdom_to_vary = dct.pop(key_to_vary).domain
samples = []
for i in range(8):
d = dct.copy()
d[key_to_vary] = ift.from_random(subdom_to_vary)
samples.append(ift.MultiField.from_dict(d))
return samples
def vary_parameter(parameter_key, values, samples_vary_in=None):
s = gen_samples(samples_vary_in)
for v in values:
if parameter_key in cf_make_pars.keys():
m_pars = {**cf_make_pars, parameter_key: v}
eval_model(m_pars, cf_x_fluct_pars, f"{parameter_key} = {v}", s)
else:
fl_pars = {**cf_x_fluct_pars, parameter_key: v}
eval_model(cf_make_pars, fl_pars, f"{parameter_key} = {v}", s)
```
%% Cell type:markdown id: tags:
## Before the Action: The Moment-Matched Log-Normal Distribution
Many properties of the correlated field are modelled as being lognormally distributed.
The distribution models are parametrized via their means and standard-deviations (first and second position in tuple).
To get a feeling of how the ratio of the `mean` and `stddev` parameters influences the distribution shape,
here are a few example histograms: (observe the x-axis!)
%% Cell type:code id: tags:
``` python
fig = plt.figure(figsize=(13, 3.5))
mean = 1.0
sigmas = [1.0, 0.5, 0.1]
for i in range(3):
op = ift.LognormalTransform(mean=mean, sigma=sigmas[i],
key='foo', N_copies=0)
op_samples = np.array(
[op(s).val for s in [ift.from_random(op.domain) for i in range(10000)]])
ax = fig.add_subplot(1, 3, i + 1)
ax.hist(op_samples, bins=50)
ax.set_title(f"mean = {mean}, sigma = {sigmas[i]}")
ax.set_xlabel('x')
del op_samples
plt.show()
```
%% Cell type:markdown id: tags:
## The Neutral Field
To demonstrate the effect of all parameters, first a 'neutral' set of parameters
is defined which then are varied one by one, showing the effect of the variation
on the generated field realizations and the underlying power spectrum from which
they were drawn.
As a neutral field, a model with a white power spectrum and vanishing spectral power was chosen.
%% Cell type:code id: tags:
``` python
# Neutral model parameters yielding a quasi-constant field
cf_make_pars = {
'offset_mean': 0.,
'offset_std': (1e-3, 1e-16),
'prefix': ''
}
cf_x_fluct_pars = {
'target_subdomain': x_space,
'fluctuations': (1e-3, 1e-16),
'flexibility': (1e-3, 1e-16),
'asperity': (1e-3, 1e-16),
'loglogavgslope': (0., 1e-16)
}
init_model(