# Chi square figures of merit

Within the NNPDF methodology various figures of merit are used, each of which can be used in different situations. To avoid confusion, it is important to understand the differences between the various figures of merit, and to understand which definition we are referring to in a given context. In particular, it is worth stressing that whenever a figure of merit is discussed, the $$t_0$$ method (discussed below) applies.

Here we we provide an overview of the different figures of merit, and discuss when each of them is used.

## The basis of the loss functions: 𝜒²

The $$\chi^2$$ figures of merit used in the NNPDF methodology are all based on the chi square statistic:

$\chi^{2}=\sum_{i, j}^{N_{\text {dat }}}(D-P)_{i} C_{i j}^{-1}(D-P)_{j},$

where $$D_i$$ is the $$i$$-th datapoint, $$P_i$$ is the prediction of the corresponding datapoint calculated from the convolution product between the FastKernel tables for point $$i$$ and the PDF model, and $$C_{ij}$$ is the covariance between datapoints $$i$$ and $$j$$.

The covariance matrix accounts for correlated systematic uncertainties, normalization uncertainties, and statistical uncertainties as provided by the experimental collaborations.

We refer to this figure of merit as experimental $$\chi^2$$.

Note

This definition of $$\chi^2$$ is not used as a figure of merit anywhere in NNDPF fits. Instead, variations discussed below are used.

### Avoiding bias: t₀ method

The $$t_0$$ method introduced in [BDDF+10] aims to remove systematic biases as a result of a naive treatment of multiplicative uncertainties. This is done by redefining the covariance matrix in the definition of $$\chi^2$$, resulting in a $$t_0$$ covariance matrix $$C_{t_0}$$ and a corresponding figure of merit sometimes denoted by $$\chi^2_{t_0}$$. The new covariance matrix is constructed by replacing the central value of the data (which is used as reference for multiplicative uncertainties) with the theory predictions computed using some existing PDF set, which needs to be specified.

Note

From NNPDF2.0 onwards the t₀ formalism has been used to define the figure of merit used during the fitting of the PDFs.

Note

The $$t_0$$ method is not used by default in other validphys applications, and instead the default is to compute the experimental $$\chi^2$$. To compute $$\chi^2_{t_0}$$, users need to specify

use_t0: True
t0pdfset: <Some LHAPDF set>


in the relevant namespace. This will instruct actions such as validphys.results.dataset_chi2_table() to compute the $$t_0$$ estimator.

## Missing higher order uncertainties

Another source of uncertainties that we may want to include in the covariance matrix are theoretical uncertainties, particularly missing higher order uncertainties estimated through scale variations. These uncertainties can be considered in the figure of merit through the implementation of a ‘theory covariance matrix’. A paper discussing the formalism can be found here: [AK+19a]. For a tutorial see How to include a theory covariance matrix in a fit.

## Future test: including PDF errors

To test the generalization power of the NNPDF fitting framework in the region where PDFs are not constrained by data, the ‘future test’ has been developed. The figure of merit considered in a future test is again the $$\chi^2$$, however, in this case the covariance matrix is not only the covariance matrix corresponding to the datasets, but it is instead the sum of the covariance matrix describing the data uncertainties and the covariance matrix describing the PDF uncertainties.

For a more detailed discussion of the future test formalism see e.g. [CMFN21], or learn How to run a Future Test

## Regularized covariance matrices

Information about the accuracy of the experimental uncertainty is generally not available, nevertheless inaccuracies in an experimental covariance matrix can lead to problems during optimization. Simply making a conservative estimate of the correlations does not always guarantee this problem is avoided and this is where the regularized covariance matrix comes in: it aims to provide a matrix which is closely related to the original experimental covariance matrix while avoiding the problems during optimization.

The stability characteristic for a given dataset can be computed using the validphys.covmats.covmat_stability_characteristic(). All the dataset covariance matrices can be altered so that their stability characteristic is less than a given value by specifying such value as a norm_threshold parameter in the runcard. Adding it in an analysis results in computing a regularized $$\chi^2$$ that is less sensitive to inaccuracies in the correlation model. Adding it in a fit runcard results in a fit with regularized covariance matrices.

Note

There is currently no support for displaying regularized $$\chi^2$$ values in vp-comparefits

A more detailed discussion of regularization procedure, and how it is used within NNPDF can be found in sections 4.2 and 8.7 of the NNPDF4.0 paper .

## The weighted fit method

To determine whether a specific dataset shows inconsistencies with the global dataset, one can produce a PDF determination in which that measurement is given an increased weight (usually equal to the combined weight of the other datasets). The idea being that if – in oder to accommodate the dataset under investigation – the agreement to the other datasets deteriorates, this dataset is likely inconsistent with the global dataset.

When performing a weighted fit the figure of merit is hence redefined as

$\chi^{2}=\frac{1}{N_{\text {dat }}-N_{\text {dat }}^{(j)}} \sum_{i \neq j}^{n_{\text {exp }}}N_{\text {dat }}^{(i)}\chi_{i}^{2} +\omega^{(j)} \chi_{j}^{2}$

with $$w^{(j)}=N_{\rm dat}/N^{(j)}_{\rm dat}$$.

A dataset can be given an additional weight by explictitly writing a weight key for a given dataset in the n3fit runcard. For example, while the default weight is 1, one can set the weight of the HERACOMB_SIGMARED_C dataset to 100 by adding the following to the runcard:

dataset_inputs:
- {dataset: HERACOMB_SIGMARED_C, frac: 0.75, weight: 100}


## Experimental, validation, and training 𝜒²

When performing a PDF fit we generally distinguish three different definitions of the $$\chi^2$$ loss function, namely the experimental loss $$\chi^2_{\rm exp}$$, the training loss $$\chi^2_{rm tr}$$ and the validation loss $$\chi^2_{val}$$, all of which are defined using the $$t_0$$ method. Here the experimental loss is calculated with respect to the experimental covariance matrix and corresponding central values, while the training and validation losses are defined with respect to the central values of the psuedodata replicas.

The training and validation losses are used for cross-correlation in the early stopping algorithm, and can further be adjusted to ensure positivity and integrability of the resulting PDFs after the fit by adding a component to the loss function (see below).

More details of these loss functions and the role they play within the training of the neural network can be found in the methodology overview.

### Positivity and integrability: Lagrange multipliers

Generally in an NNPDF fit we will want to ensure positivity and integrability of the resulting PDFs. This is enforced by means of Lagrange multipliers, which provide an additional contribution to the definition of the chi squared loss function.

For an discussion of how exactly the loss function is adjusted upon including the Lagrange multipliers, see sections 3.1.3 and 3.1.4 of the NNPDF4.0 paper .

An explanation of how the runcard should be adjusted to include the additional positivity Lagrange multiplier can be found here, while the analogous information for integrability can be found here.

## Hyperoptimized figure of merit

To test the generalization power of a given methodology (a specific set of hyperparameter values), we employ hyperoptimization, specifically we use K-folds cross-validation. The idea of K-folds cross-validation is to create subsets of data representative of the global dataset, and then perform a fit to $$K-1$$ subsets while using the $$K^{\rm th}$$ subset as a test set to check the generalization performance after the neural network has been trained. The figure of merit that is minimized during the hyperoptimization routine is obtained by summing over all $$K$$ test losses that are obtained after performing $$K$$ fits to each possible combination of $$K-1$$ datasets.

For a more detailed description of the hyperoptimization loss see the documentation of the hyperoptimization algorithm.