\(\renewcommand{\AA}{\text{Å}}\)

# compute slcsa/atom command

## Syntax

    compute ID group-ID slcsa/atom twojmax nclasses db_mean_descriptor_file lda_file lr_decision_file lr_bias_file maha_file value

- ID, group-ID are documented in compute command

- slcsa/atom = style name of this compute command

- twojmax = band limit for bispectrum components (non-negative integer)

- nclasses = number of crystal structures used in the database for the classifier SL-CSA

- db_mean_descriptor_file = file name of file containing the database mean descriptor

- lda_file = file name of file containing the linear discriminant analysis matrix for dimension reduction

- lr_decision_file = file name of file containing the scaling matrix for logistic regression classification

- lr_bias_file = file name of file containing the bias vector for logistic regression classification

- maha_file = file name of file containing for each crystal structure: the Mahalanobis distance threshold for sanity check purposes, the average reduced descriptor and the inverse of the corresponding covariance matrix

- c_ID[1] = compute ID and output data column of previously defined compute sna/atom command

## Examples

    compute b1 all sna/atom 9.0 0.99363 8 0.5 1.0 rmin0 0.0 nnn 24 wmode 1 delta 0.3
    compute b2 all slcsa/atom 8 4 mean_descriptors.dat lda_scalings.dat lr_decision.dat lr_bias.dat maha_thresholds.dat c_b1[1]

## Description

Added in version 7Feb2024.

Define a computation that performs the Supervised Learning Crystal Structure Analysis (SL-CSA) from (Lafourcade) for each atom in the group. The SL-CSA tool takes as an input a per-atom descriptor (bispectrum) that is computed through the compute sna/atom command and then proceeds to a dimension reduction step followed by a logistic regression in order to assign a probable crystal structure to each atom in the group. The SL-CSA tool is pre-trained on a database containing \(C\) distinct crystal structures from which a crystal structure classifier is derived and a tutorial to build such a tool is available at SL-CSA.

The first step of the SL-CSA tool consists in performing a dimension reduction of the per-atom descriptor \(\mathbf{B}^i \in \mathbb{R}^{D}\) through the Linear Discriminant Analysis (LDA) method, leading to a new projected descriptor \(\mathbf{x}^i=\mathrm{P}_\mathrm{LDA}(\mathbf{B}^i):\mathbb{R}^D \rightarrow \mathbb{R}^{d=C-1}\):

\[\mathbf{x}^i = \mathbf{C}^T_\mathrm{LDA} \cdot (\mathbf{B}^i - \mu^\mathbf{B}_\mathrm{db})\]

where \(\mathbf{C}^T_\mathrm{LDA} \in \mathbb{R}^{D \times d}\) is the reduction coefficients matrix of the LDA model read in file lda_file, \(\mathbf{B}^i \in \mathbb{R}^{D}\) is the bispectrum of atom \(i\) and \(\mu^\mathbf{B}_\mathrm{db} \in \mathbb{R}^{D}\) is the average descriptor of the entire database. The latter is computed from the average descriptors of each crystal structure read from the file mean_descriptors_file.

The new projected descriptor with dimension \(d=C-1\) allows for a good separation of different crystal structures fingerprints in the latent space.

Once the dimension reduction step is performed by means of LDA, the new descriptor \(\mathbf{x}^i \in \mathbb{R}^{d=C-1}\) is taken as an input for performing a multinomial logistic regression (LR) which provides a score vector \(\mathbf{s}^i=\mathrm{P}_\mathrm{LR}(\mathbf{x}^i):\mathbb{R}^d \rightarrow \mathbb{R}^C\) defined as:

\[\mathbf{s}^i = \mathbf{b}_\mathrm{LR} + \mathbf{D}_\mathrm{LR} \cdot {\mathbf{x}^i}^T\]

with \(\mathbf{b}_\mathrm{LR} \in \mathbb{R}^C\) and \(\mathbf{D}_\mathrm{LR} \in \mathbb{R}^{C \times d}\) the bias vector and decision matrix of the LR model after training both read in files lr_fil1 and lr_file2 respectively.

Finally, a probability vector \(\mathbf{p}^i=\mathrm{P}_\mathrm{LR}(\mathbf{x}^i):\mathbb{R}^d \rightarrow \mathbb{R}^C\) is defined as:

\[\mathbf{p}^i = \frac{\mathrm{exp}(\mathbf{s}^i)}{\sum\limits_{j} \mathrm{exp}(s^i_j) }\]

from which the crystal structure assigned to each atom with descriptor \(\mathbf{B}^i\) and projected descriptor \(\mathbf{x}^i\) is computed as the argmax of the probability vector \(\mathbf{p}^i\). Since the logistic regression step systematically attributes a crystal structure to each atom, a sanity check is needed to avoid misclassification. To this end, a per-atom Mahalanobis distance to each crystal structure CS present in the database is computed:

\[d_\mathrm{Mahalanobis}^{i \rightarrow \mathrm{CS}} = \sqrt{(\mathbf{x}^i - \mathbf{\mu}^\mathbf{x}_\mathrm{CS})^\mathrm{T} \cdot \mathbf{\Sigma}^{-1}_\mathrm{CS} \cdot (\mathbf{x}^i - \mathbf{\mu}^\mathbf{x}_\mathrm{CS}) }\]

where \(\mathbf{\mu}^\mathbf{x}_\mathrm{CS} \in \mathbb{R}^{d}\) is the average projected descriptor of crystal structure CS in the database and where \(\mathbf{\Sigma}_\mathrm{CS} \in \mathbb{R}^{d \times d}\) is the corresponding covariance matrix. Finally, if the Mahalanobis distance to crystal structure CS for atom i is greater than the pre-determined threshold, no crystal structure is assigned to atom i. The Mahalanobis distance thresholds are read in file maha_file while the covariance matrices are read in file covmat_file.

The SL-CSA framework provides an automatic computation of the different matrices and thresholds required for a proper classification and writes down all the required files for calling the compute slcsa/atom command.

The compute slcsa/atom command requires that the compute sna/atom command is called before as it takes the resulting per-atom bispectrum as an input. In addition, it is crucial that the value twojmax is set to the same value of the value twojmax used in the compute sna/atom command, as well as that the value nclasses is set to the number of crystal structures used in the database to train the SL-CSA tool.

## Output info

By default, this compute computes the Mahalanobis distances to the different crystal structures present in the database in addition to assigning a crystal structure for each atom as a per-atom vector, which can be accessed by any command that uses per-atom values from a compute as input. See the Howto output page for an overview of LAMMPS output options.

## Restrictions

This compute is part of the EXTRA-COMPUTE package. It is only enabled if LAMMPS was built with that package. See the Build package page for more info.

## Related commands

compute sna/atom

## Default

none

(Lafourcade) Lafourcade, Maillet, Denoual, Duval, Allera, Goryaeva, and Marinica, Comp. Mat. Science, 230, 112534 (2023)