volume.rst 11.7 KB
 Julia Stadler committed Mar 05, 2019 1 Discretisation and Volume in NIFTy  Max-Niklas Newrzella committed Jan 29, 2019 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 ================================== .. note:: Some of this discussion is rather technical and may be skipped in a first read-through. Setup ..... IFT employs stochastic processes to model distributions over function spaces, in particular Gaussian processes :math:s \sim \mathcal{G}(s,k) where :math:k denotes the covariance function. The domain of the fields, and hence :math:k, is given by a Riemannian manifold :math:(\mathcal{M},g), where :math:g denotes a Riemannian metric. Fields are defined to be scalar functions on the manifold, living in the function space :math:\mathcal{F}(\mathcal{M}). Unless we find ourselves in the lucky situation that we can solve for the posterior statistics of interest analytically, we need to apply numerical methods. This is where NIFTy comes into play. .. figure:: images/inference.png :width: 80% :align: center Figure 1: Sketch of the various spaces and maps involved in the inference process. A typical setup for inference of such signals using NIFTy is shown in figure 1. We start with a continuous signal :math:s \in \mathcal{S}, defined in some function space :math:\mathcal{S} := \mathcal{F}(\mathcal{M}) over a manifold :math:(\mathcal{M},g) with metric :math:g. This is measured by some instrument, e.g. a telescope. The measurement produces data in an unstructured data space :math:\mathcal{D} via a known response function :math:R : \mathcal{S} \rightarrow \mathcal{D} and involves noise :math:\mathcal{D} \ni n \sim \mathcal{N}(n | 0, N) with known covariance matrix :math:N. In the case of additive noise, the result of the measurement is given by .. math:: d = R(s) + n \, . Discretisation and index notation ................................. To compute anything numerically, we first need to represent the problem in finite dimensions.  Martin Reinecke committed Jan 29, 2019 37 38 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.  Max-Niklas Newrzella committed Jan 29, 2019 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72  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 .. math:: \left< a , b \right>_{\mathcal{M}} = \int_{\mathcal{M}} \mathrm{d}x \, \sqrt{|g|} \, a(x) \, b(x) \, . Projection to the finite basis :math:\{\phi_i\}_{i\in \mathcal{I}} is then given by .. math:: f^i = v^{ij} \, \left< f , \phi_j \right>_{\mathcal{M}} \, where the Einstein summation convention is assumed and we defined the volume metric .. math:: v_{ij} = \left< \phi_i , \phi_j \right>_{\mathcal{M}} \, , along with its inverse, :math:v^{ij}, satisfying :math:v^{ij}v_{jk} = \delta^i_k. Obviously, the basis :math:\{\phi_i\}_{i\in \mathcal{I}} needs to be chosen s.th. the volume metric is invertible, otherwise we run into trouble. Volume factors are encoded into the :math:v_{ij}. For specific choices of the basis :math:\{\phi_i\}_{i\in \mathcal{I}}, e.g. indicator functions in the case of a pixelation, the entries of :math:v_{ij} are indeed just the volumes of the elements. Lowering and raising indices works with :math:v_{ij} and :math:v^{ij} just as usual. After projection, any function :math:f \in \mathcal{S} is represented by its approximation :math:\hat{f} \in \hat{\mathcal{S}} \simeq \mathbb{R}^N, where .. math:: \hat{f} = f^i\,\phi_i \, , which defines an embedding :math:\hat{\mathcal{S}} \hookrightarrow \mathcal{S} = \mathcal{F}(\mathcal{M}).  Martin Reinecke committed Jan 30, 2019 73 **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'}}:  Max-Niklas Newrzella committed Jan 29, 2019 74 75 76 77 78 79 80 81 82 83  .. math:: \phi_i \approx (\phi_i)^j \, \phi'_j \, =: \beta_i^{\,j} \, \phi'_j \, , which in general implies additional loss of information unless the two bases are compatible, i.e. encode the same information. The latter is e.g. true for regular collocation grids on tori and the associated cropped Fourier series. The discrete Fourier transform then maps between those bases without loss of information. **Discretisation of operators** works in the same way by expansion.  Martin Reinecke committed Jan 29, 2019 84 For illustration purposes, let :math:A: \mathcal{S} \rightarrow \mathcal{S} be a (not necessarily linear) operator.  Max-Niklas Newrzella committed Jan 29, 2019 85 86 87 88 89 90 91 92 93 94 95 96 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:: A[s] = (A[s])^k \, \phi_k \, , where the domain of the operator may be restricted to the image of the embedding given above. Integrals can now be written as .. math:: \left< s , A[t] \right>_{\mathcal{M}} \approx s^i \left< \phi_i , \phi_j \right>_{\mathcal{M}} (A[t])^j \equiv s^i \, v_{ij} \, (A[t])^j \, ,  Martin Reinecke committed Jan 29, 2019 97 where the appearance of the volume metric can be hidden by lowering the first index of the operator,  Max-Niklas Newrzella committed Jan 29, 2019 98 99 100 101 102  .. math:: (A[w])_k := v_{km} \, (A[w])^m \, .  Martin Reinecke committed Jan 29, 2019 103 104 Hence, the volume metric need not be carried along if the operators are defined in this fashion right from the start. Linear operators mapping several functions to another function are completely specified by their action on a given basis, and we define  Max-Niklas Newrzella committed Jan 29, 2019 105 106 107 108 109 110 111 112 113 114 115 116 117  .. math:: A^k_{\,\,mn\ldots} := (A[\phi_m,\phi_n,\ldots])^k \, . If :math:A is a (linear) integral operator defined by a kernel :math:\tilde{A}: \mathcal{M} \times \cdots \mathcal{M} \rightarrow \mathbb{R}, its components due to :math:\{\phi_i\}_{i\in \mathcal{I}} are given by .. math:: A^k_{\,\,ij\ldots} &= v^{km} \, \left< \phi_m, A[\phi_i,\phi_j,\ldots] \right>_{\mathcal{M}} \\ &= v^{km} \, \int_{\mathcal{M}} \mathrm{d} x\,\sqrt{|g|}\,\left(\prod_{n}^{|\{ij\ldots\}|}\int_{\mathcal{M}} \mathrm{d} y_n \, \sqrt{|g|}\right) \,\,\phi_m(x)\, \tilde{A}(x,y_1,y_2,\ldots)\, \phi_i(y_1) \, \phi_j(y_2) \cdots \, .  Martin Reinecke committed Jan 29, 2019 118 119 .. [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.  Max-Niklas Newrzella committed Jan 29, 2019 120 121 122 123 124 125 126 127 128 129 130 131  Resolution and self-consistency ............................... Looking at figure 1, we see that the there are two response operators: On the one hand, there is the actual response :math:R: \mathcal{S} \rightarrow \mathcal{D} of the instrument used for measurement, mapping the actual signal to data. On the other hand, there is a discretised response :math:\hat{R}: \hat{\mathcal{S}} \rightarrow \mathcal{D}, mapping from the discretised space to data. Apparently, the discretisation and the discretised response need to satisfy a self-consistency equation, given by .. math:: R = \hat{R} \circ D \, .  Martin Reinecke committed Jan 29, 2019 132 An obvious corollary is that different discretisations :math:D, D' with resulting discretised responses :math:\hat{R}, \hat{R}' will need to satisfy  Max-Niklas Newrzella committed Jan 29, 2019 133 134 135 136 137  .. 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.  Martin Reinecke committed Jan 29, 2019 138 139 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.  Max-Niklas Newrzella committed Jan 29, 2019 140   141 142 .. figure:: images/converging_discretization.png :scale: 80%  Max-Niklas Newrzella committed Jan 29, 2019 143 144  :align: center  Martin Reinecke committed Jan 29, 2019 145  Figure 2: Inference result converging at high resolution.  Max-Niklas Newrzella committed Jan 29, 2019 146 147 148  Implementation in NIFTy  Max-Niklas Newrzella committed Jan 30, 2019 149 .......................  Max-Niklas Newrzella committed Jan 29, 2019 150 151 152  .. currentmodule:: nifty5  Martin Reinecke committed Jan 29, 2019 153 154 Most codes in NIFTy will contain the description of a measurement process or, more generally, a log-likelihood.  Max-Niklas Newrzella committed Jan 29, 2019 155 This log-likelihood is necessarily a map from the quantity of interest (a field) to a real number.  Max-Niklas Newrzella committed Jan 30, 2019 156 157 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  Max-Niklas Newrzella committed Jan 29, 2019 158 159 160  .. math::  Max-Niklas Newrzella committed Jan 30, 2019 161  \int_\Omega \text{d}x\, s(x) \approx \sum_i s^i\int_{\Omega_i}\text{d}x\, 1  Max-Niklas Newrzella committed Jan 29, 2019 162   Julia Stadler committed Mar 05, 2019 163 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 discretised field value on the :math:i-th pixel.  Max-Niklas Newrzella committed Jan 29, 2019 164 This introduces the weighting :math:V_i=\int_{\Omega_i}\text{d}x\, 1, also called the volume factor, a property of the space.  Max-Niklas Newrzella committed Jan 30, 2019 165 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.  Max-Niklas Newrzella committed Jan 29, 2019 166 167 168 169 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: - :class:~operators.harmonic_operators.FFTOperator as well as all other harmonic operators. Here the zero mode of the transformed field is the integral over the original field, thus the whole field is weighted once.  Julia Stadler committed Mar 05, 2019 170 171  - Some response operators, such as the :class:~library.los_response.LOSResponse. In this operator a line integral is discretised, so a 1-dimensional volume factor is applied. - In :class:~library.correlated_fields.CorrelatedField as well :class:~library.correlated_fields.MfCorrelatedField both describing fields with a smooth, a priori unknown correlation structure specified by a power spectrum. The field is multiplied by the square root of the total volume of it domain's harmonic counterpart. This ensures that the same power spectrum can be used regardless of the chosen resolution, provided the total volume of the space remains the same. It also guarantees that the power spectra in NIFTy behave according to their definition, i.e. the power of a mode :math:s_k is the expectation value of that mode squared, divided by the volume of its space :math:P(k) = \left\langle s_k^2 \right\rangle / V_k  Max-Niklas Newrzella committed Jan 29, 2019 172   Max-Niklas Newrzella committed Jan 30, 2019 173 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,  Max-Niklas Newrzella committed Jan 29, 2019 174 175  .. math::  Max-Niklas Newrzella committed Jan 30, 2019 176  s^\dagger t = \sum_i \overline{s^i}\, t^i \, ,  Max-Niklas Newrzella committed Jan 29, 2019 177   Max-Niklas Newrzella committed Jan 30, 2019 178 179 180 181 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.  Martin Reinecke committed Jan 30, 2019 182 One solution to obtain a resolution independent quantity is to make one of the two factors extensive while the other stays intensive.  Max-Niklas Newrzella committed Jan 30, 2019 183 This is more explicit when intensive quantities are denoted by an upper index and extensive quantities by a lower index,  Max-Niklas Newrzella committed Jan 29, 2019 184 185  .. math::  Max-Niklas Newrzella committed Jan 30, 2019 186  s^\dagger t = \overline{s^i} t_i  Max-Niklas Newrzella committed Jan 29, 2019 187 188  where we used Einstein sum convention.  Max-Niklas Newrzella committed Jan 30, 2019 189 Here, the volume metric is incorporated by lowering one index, i.e. :math:t_i = v_{ij}\,t^j.  Max-Niklas Newrzella committed Jan 29, 2019 190 191 192 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::  Max-Niklas Newrzella committed Jan 30, 2019 193  L = \frac{1}{2}\overline{s^i} \left(S^{-1}\right)_{ij} s^j  Max-Niklas Newrzella committed Jan 29, 2019 194   Max-Niklas Newrzella committed Jan 30, 2019 195 with the covariance defined by  Max-Niklas Newrzella committed Jan 29, 2019 196 197  .. math::  Max-Niklas Newrzella committed Jan 30, 2019 198  S^{ij} = \left\ .  Max-Niklas Newrzella committed Jan 29, 2019 199   Martin Reinecke committed Jan 30, 2019 200 Consequently, the inverse covariance operator will automatically have lower indices,  Max-Niklas Newrzella committed Jan 29, 2019 201   Max-Niklas Newrzella committed Jan 30, 2019 202 203 204 205 206 207 .. math:: \left(S^{-1}\right)_{ij} S^{jk} = \delta^{\,\,k}_j\ , ensuring that the whole log-likelihood expression does not scale with resolution. **This upper-lower index convention is not coded into NIFTy**, in order to not reduce user freedom. One should however have this in mind when constructing log-likelihoods in order to ensure resolution independence.