Linear Model

This model was presented in Ref. CMMU25.

Model Repository: https://github.com/HEP-PBSP/wmin-model .

What is this model for?

This model is especially suitable for running bayesian fits. It can be used to:

  1. Construct a Proper Orthogonal Decomposition (POD) basis (see Ref. CMMU25 for details on what this is).

  2. Run Bayesian PDF Fits with the POD Parametrisation.

Model description

This model parametrises PDFs as linear combinations of basis functions:

\[f_w(x) = \phi_0(x) + \sum_{k=1}^{N} w_k \phi_k(x),\]

where \(w = (w_1, ..., w_N)\) are the parameters to be inferred, and \(\phi_k(x)\) are carefully chosen basis functions, which in practice are constructed by applying Proper Orthogonal Decomposition (POD) to a basis set of samples of the randomly initialised n3fit Neural Network.

For details on the motivation behind this choice of model see Ref. Ref. CMMU25.

How to use this model

You can find installation instructions in the model repository.

Constructing a POD basis

The following is an example runcard that can be used to construct a POD basis:

meta:
title: POD basis
author: Lazy Person
keywords: ["POD basis", "wmin"]

# NNPDF Neural Net Architecture settings
replica_range_settings:
min_replica: 1
max_replica: 1000   # generate replicas numbered 1 to 1000


impose_sumrule: true
filter_sr_outliers: false   # whether to filter sum rules outliers

fitbasis: EVOL

nodes: [25, 20, 8]

activations: ["tanh", "tanh", "linear"]

initializer_name: "glorot_normal"
layer_type: "dense"


# Number of components to keep
Neig: 10

# theoryid used after SVD to evolve fit
theoryid: 40_000_000

actions_:
- write_pod_basis

This will generate max_replica - min_replica random initialisations of the n3fit Neural Network, that will then be reduced to Neig eigenvectors, which will be the basis elements. It can be run with the command:

wmin runcard.yaml

where wmin is the model-specific executable.

This basis should then be evolved, and the basis elements then need to be shifted by running:

python shift_lhapdf_members.py evolved_directory/postfit/evolved_directory

where the shift_lhapdf_members.py script can be found in the directory wmin-model/wmin/runcards and evolved_directory is the fit or POD basis directory that should have previously been evolved.

Running fits

You can follow Colibri’s analytic and bayesian workflows to run fits with this model. There are, however, a few points to note that are specific to this model.

Analytic fits

Analytic fits are only appropriate for linear models that also have a linear relationship with the data, so should be run with DIS data only.

wmin_settings

This model has specific settings that need to be specified in the runcard in order to run a fit:

wmin_settings:
    wminpdfset: 250503_pod_basis_40k
    n_basis: 10     # number of parameters/weights to be fitted

  • wminpdfset is the POD basis set you should have constructed before running a fit.

  • n_basis is the number of parameters or weights to be fitted (minimised). It should be less than or equal to the number of replicas in wminpdfset.