diff --git a/docs/source/volume.rst b/docs/source/volume.rst index a00a850234efbe5fb1e1679f835eb22ce838f14c..cad1c510229ef1196bd76f4145b291719f49a79b 100644 --- a/docs/source/volume.rst +++ b/docs/source/volume.rst @@ -1,4 +1,4 @@ -Discretization and Volume in NIFTy +Discretisation and Volume in NIFTy ================================== .. note:: Some of this discussion is rather technical and may be skipped in a first read-through. @@ -160,15 +160,15 @@ Often, log-likelihoods contain integrals over the quantity of interest :math:s \int_\Omega \text{d}x\, s(x) \approx \sum_i s^i\int_{\Omega_i}\text{d}x\, 1 -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. +Here the domain of the integral :math:\Omega = \dot{\discretised_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. 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 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: - :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. - - 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. + - 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 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,