@@ -34,8 +34,8 @@ Discretisation and index notation
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To compute anything numerically, we first need to represent the problem in finite dimensions.
As for stochastic processes, several discretisations of :math:`\mathcal{S}` like collocation methods, expansion into orthogonal polynomials, etc. can be used (see [6]_, [7]_ for an overview and further information about their reliability).
In particular, NIFTy uses the midpoint method as reviewed in section 2.1 in [6]_ and Fourier expansion.
As for stochastic processes, several discretisations of :math:`\mathcal{S}` like collocation methods, expansion into orthogonal polynomials, etc. can be used (see [1]_, [2]_ for an overview and further information about their reliability).
In particular, NIFTy uses the midpoint method as reviewed in section 2.1 of [1]_ and Fourier expansion.
Without going into the details, discretisation methods basically introduce a finite set of basis functions :math:`\{\phi_i\}_{i\in \mathcal{I}}`, where :math:`\mathcal{I}` denotes a generic index set with :math:`|\mathcal{I}| = N` being the chosen discretisation dimension.
Any Riemannian manifold :math:`(\mathcal{M},g)` is equipped with a canonical scalar product given by
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@@ -70,7 +70,7 @@ After projection, any function :math:`f \in \mathcal{S}` is represented by its a
which defines an embedding :math:`\hat{\mathcal{S}} \hookrightarrow \mathcal{S} = \mathcal{F}(\mathcal{M})`.
**Changes of bases** are performed by reapproximating the :math:`\{\phi_i\}_{i\in \mathcal{I}}` in terms of another basis :math:`\{\phi'_i\}_{i\in \mathcal{I'}}`:
**Changes of base** are performed by reapproximating the :math:`\{\phi_i\}_{i\in \mathcal{I}}` in terms of another basis :math:`\{\phi'_i\}_{i\in \mathcal{I'}}`:
.. math::
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@@ -81,7 +81,7 @@ The latter is e.g. true for regular collocation grids on tori and the associated
The discrete Fourier transform then maps between those bases without loss of information.
**Discretisation of operators** works in the same way by expansion.
For illustration purposes, let :math:`A: \mathcal{S} \rightarrow \mathcal{S}` be a not necessarily linear operator.
For illustration purposes, let :math:`A: \mathcal{S} \rightarrow \mathcal{S}` be a (not necessarily linear) operator.
The result of its action on functions :math:`s` is known and may be expanded in :math:`\{\phi_i\}_{i\in \mathcal{I}}`, i.e.
.. math::
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@@ -94,14 +94,14 @@ Integrals can now be written as
.. [6] Bruno Sudret and Armen Der Kiureghian (2000), "Stochastic Finite Element Methods and Reliability: A State-of-the-Art Report"
.. [7] Dongbin Xiu (2010), "Numerical methods for stochastic computations", Princeton University Press.
.. [1] Bruno Sudret and Armen Der Kiureghian (2000), "Stochastic Finite Element Methods and Reliability: A State-of-the-Art Report"
.. [2] Dongbin Xiu (2010), "Numerical methods for stochastic computations", Princeton University Press.
Resolution and self-consistency
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@@ -129,40 +129,40 @@ Apparently, the discretisation and the discretised response need to satisfy a se
.. math::
R = \hat{R} \circ D \, .
An obvious corrollary is that different discretisations :math:`D, D'` with resulting discretised responses :math:`\hat{R}, \hat{R}'` will need to satisfy
An obvious corollary is that different discretisations :math:`D, D'` with resulting discretised responses :math:`\hat{R}, \hat{R}'` will need to satisfy
.. math::
\hat{R} \circ D = \hat{R}' \circ D' \, .
NIFTy is implemented such that in order to change resolution, only the line of code defining the space needs to be altered.
It automatically takes care of depended structures like volume factors, discretised operators and responses.
A visualisation of this can be seen in figure 2 and 3, which displays the MAP inference of a signal at various resolutions.
It automatically takes care of dependent structures like volume factors, discretised operators and responses.
A visualisation of this can be seen in figure 2, which displays the MAP inference of a signal at various resolutions.
.. figure:: images/converging_discretization.png
:scale: 80%
:align: center
Figure 3: Inference result converging at high resolution.
Figure 2: Inference result converging at high resolution.
Implementation in NIFTy
-----------------------
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.. currentmodule:: nifty5
Most codes in NIFTy will contain the description of a measurement process,
or more generally, a log-likelihood.
Most codes in NIFTy will contain the description of a measurement process or,
more generally, a log-likelihood.
This log-likelihood is necessarily a map from the quantity of interest (a field) to a real number.
The likelihood has to be unitless because it is a log-probability and should not scale with resolution.
Often, likelihoods contain integrals over the quantity of interest :math:`s`, which have to be discretized, e.g. by a sum
The log-likelihood has to be unitless because it is a log-probability and should not scale with resolution.
Often, log-likelihoods contain integrals over the quantity of interest :math:`s`, which have to be discretized, e.g. by a sum
Here the domain of the integral :math:`\Omega = \dot{\bigcup_q} \; \Omega_i` is the disjoint union over smaller :math:`\Omega_i`, e.g. the pixels of the space, and :math:`s_i` is the discretized field value on the :math:`i`-th pixel.
This introduces the weighting :math:`V_i=\int_{\Omega_i}\text{d}x\, 1`, also called the volume factor, a property of the space.
NIFTy aids you in constructing your own likelihood by providing methods like :func:`~field.Field.weight`, which weights all pixels of a field with its corresponding volume.
NIFTy aids you in constructing your own log-likelihood by providing methods like :func:`~field.Field.weight`, which weights all pixels of a field with their corresponding volume.
An integral over a :class:`~field.Field` :code:`s` can be performed by calling :code:`s.weight(1).sum()`, which is equivalent to :code:`s.integrate()`.
Volume factors are also applied automatically in the following places:
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@@ -170,32 +170,38 @@ Volume factors are also applied automatically in the following places:
- some response operators, such as the :class:`~library.los_response.LOSResponse`. In this operator a line integral is descritized, so a 1-dimensional volume factor is applied.
- In :class:`~library.correlated_fields.CorrelatedField` as well :class:`~library.correlated_fields.MfCorrelatedField`, the field is multiplied by the square root of the total volume in configuration space. This ensures that the same field reconstructed over a larger domain has the same variance in position space in the limit of infinite resolution. It also ensures that power spectra in NIFTy behave according to the definition of a power spectrum, namely the power of a k-mode is the expectation of the k-mode square, divided by the volume of the space.
Note that in contrast to some older versions of NIFTy, the dot product of fields does not apply a volume factor
Note that in contrast to some older versions of NIFTy, the dot product :code:`s.vdot(t)` of fields does **not** apply a volume factor, but instead just sums over the field components,
.. math::
s^\dagger t = \sum_i s_i^* t_i .
s^\dagger t = \sum_i \overline{s^i}\, t^i \, ,
If this dot product is supposed to be invariant under changes in resolution, then either :math:`s_i` or :math:`t_i` has to decrease as the number of pixels increases, or more specifically, one of the two fields has to be an extensive quantity while the other has to be intensive.
One can make this more explicit by denoting intensive quantities with upper index and extensive quantities with lower index
where the bar denotes complex conjugation.
This dot product is **not** invariant under changes in resolution, as then the number of discretised field components increases.
Upper index components like :math:`s^i`, however, are designed **not** to scale with the volume.
One solution to obtain a resolution independent quantity is to make one of the two factors extensive while the other stays intensive.
This is more explicit when intensive quantities are denoted by an upper index and extensive quantities by a lower index,
.. math::
s^\dagger t = (s^*)^i t_i
s^\dagger t = \overline{s^i} t_i
where we used Einstein sum convention.
This notation connects to the theoretical discussion before.
One of the field has to have the volume metric already incorperated to assure the continouum limit works.
Here, the volume metric is incorporated by lowering one index, i.e. :math:`t_i = v_{ij}\,t^j`.
When building statistical models, all indices will end up matching this upper-lower convention automatically, e.g. for a Gaussian log-likelihood :math:`L` we have
.. math::
L = \frac{1}{2}s^i \left(S^{-1}\right)_{ij} s^j
L = \frac{1}{2}\overline{s^i} \left(S^{-1}\right)_{ij} s^j
