Action: SIZESHAPE_POSITION_LINEAR_PROJ

Module	sizeshape
Description	Usage
Calculates a linear projection in the space of a given reference configurational distribution in size-and-shape space.
output value	type
the linear projection	scalar

Details and examples

Calculates a linear projection in the space of a given reference configurational distribution in size-and-shape space.

The linear projection is given by:

$$l(\mathbf{x}) = \mathbf{v}\cdot(\mathbf{R}\cdot(\mathbf{x}(t) - \vec{{\zeta}}(t)) - \mathbf{\mu}),$$

Where $\mathbf{v}$ is a vector of linear coefficients, $\mathbf{x}(t)$ is the configuration at time t, $\vec{\zeta}(t)$ is the difference in the geometric mean of the current configuration and that of the reference configuration $\mathbf{\mu}$. $\vec{\zeta}(t) = \frac{1}{N} \sum_{i=1}^N \vec{x_{i}}(t) - \frac{1}{N} \sum_{i=1}^N \vec{\mu_{i}}(t)$, for N atoms.

$\mathbf{R}$ is an optimal rotation matrix that minimizes the Mahalanobis distance between the current configuration and reference. $\mathbf{R}$ is obtained by using Kabsch algorithm within the code. The Mahalanobis distance is given as:

$$d(\mathbf{x}, \mathbf{\mu}, \mathbf{\Sigma}) = \sqrt{(\mathbf{x}-\mathbf{\mu})^T \mathbf{\Sigma}^{-1} (\mathbf{x}-\mathbf{\mu})}$$

where, $\mathbf{\Sigma}^{-1}$ is the $N\times N$ precision matrix. See also SIZESHAPE_POSITION_MAHA_DIST for information about calculating Mahalanobis distance in size-and-shape space.

Size-and-shape Gaussian Mixture Model (shapeGMM) \cite Heidi-shapeGMM-2022 is a probabilistic clustering technique that is used to perform structural clusteing on ensemble of molecular configurations and to obtain reference $(\mathbf{\mu})$ and precision $(\mathbf{\Sigma}^{-1})$ corresponding to each of the cluster centers. Please chcek out shapeGMMTorch-GitHub and shapeGMMTorch-PyPI for examples and informations on preforming shapeGMM clustering.

Examples

In the following example, a group is defined with atom indices of selected atoms and then linear projection is calculated for the given reference, precision and coefficients. Each file is a space separated list of 3N floating point numbers.

Click on the labels of the actions for more information on what each action computes

#SETTINGS INPUTFILES=regtest/sizeshape/rt-sizeshape/global_avg.txt
#SETTINGS INPUTFILES=regtest/sizeshape/rt-sizeshape/global_precision.txt
#SETTINGS INPUTFILES=regtest/sizeshape/rt-sizeshape/ld1_scalings.txt

UNITSThis command sets the internal units for the code. More details LENGTHthe units of lengths=A TIMEthe units of time=ps ENERGYthe units of energy=kcal/mol
The UNITS action with label  calculates somethingGROUPDefine a group of atoms so that a particular list of atoms can be referenced with a single label in definitions of CVs or virtual atoms. More details ATOMSthe numerical indexes for the set of atoms in the group=18,20,22,31,33,35,44,46,48,57,59,61,70,72,74,83,85,87,96,98,100,109,111 LABELa label for the action so that its output can be referenced in the input to other actions=ga_listThe GROUP action with label ga_list calculates the following quantities:
 Quantity   
 Type   
 Description  
ga_list
atoms
indices of atoms specified in GROUP
projThe SIZESHAPE_POSITION_LINEAR_PROJ action with label proj calculates the following quantities:
 Quantity   
 Type   
 Description  
proj
scalar
the linear projection: SIZESHAPE_POSITION_LINEAR_PROJCalculates a linear projection in the space of a given reference configurational distribution in size-and-shape space. More details ...
   REFERENCECoordinates of the reference structure=
regtest/sizeshape/rt-sizeshape/global_avg.txt
 Click here to see an extract from this file.
  
  ×  
FILE: regtest/sizeshape/rt-sizeshape/global_avg.txt
    
4.265825962982551012e+00 2.336690414907679203e-01 3.662809312682036023e+00
4.214057267340745483e+00 1.072049513718947578e-01 4.314804999781733486e+00
3.671228981308725103e+00 1.274792340073149510e-01 3.977231205572313577e+00
2.343698812761148886e+00 -2.841411661146072132e-01 3.478414401687389024e+00
1.686436487941495699e+00 -8.665088225218358087e-01 4.100811861077803044e+00
4.360230108844224128e-01 -1.347015411672896512e+00 3.844097514796287030e+00
3.117457174800847969e-02 -1.024227536411313366e+00 2.585940217022081988e+00
6.808808862178362675e-01 -5.870202775664047357e-01 2.290245252111865248e+00
4.586351396567848737e-01 -2.433497671270518359e-01 1.192357902655695634e+00
4.931328549364416558e-01 2.515992483355074527e-01 2.956934483804412803e-02
1.406321673638208702e+00 6.682431861735423917e-01 7.908513055476143472e-02
1.646908036369386430e+00 1.186975749838515615e+00 -8.956858825186609918e-01
5.912530303131902087e-01 8.617407774506029572e-01 -1.344660953846545715e+00
8.530666038486142688e-02 2.536604398394486926e-01 -4.506541385680229994e-01
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-3.499742687863982127e+00 -9.747491888605476218e-01 -1.653484266030961436e+00
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-2.499045339091049645e+00 2.342840846507300567e-01 -3.442979964675248805e+00
-2.331926055743990922e+00 8.142912721692037170e-01 -4.695739761431558357e+00
-2.568879301577981611e+00 1.091951813009595762e+00 -5.625053366993444826e+00
-2.213980543775061705e+00 1.112069908442180788e+00 -5.229306740929514419e+00
  
   PRECISIONPrecision Matrix (inverse of covariance)=
regtest/sizeshape/rt-sizeshape/global_precision.txt
 Click here to see an extract from this file.
  
  ×  
FILE: regtest/sizeshape/rt-sizeshape/global_precision.txt
    
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3.260580287172311076e-01 -9.512588298321154046e-01 -4.436445385268852210e-02 2.418471544653607364e+00 -2.003420141528640397e-01 -1.061985799126625452e+01 -6.462280445941964935e+00 1.861784076616177952e+00 -2.619725312876233669e+00 8.332814408542049689e-01 1.435784250738115198e+01 -5.288020624787915125e+01 1.412525470264524756e+02 -1.209752078734303211e+02 2.834210966776290253e+00 6.872035320031325512e+01 -3.713437396884358321e+01 -1.200889435095781153e+02 1.505009903985806829e+02 -2.360284378048359315e+02 1.984952390410417138e+02 -1.726639859088383133e+01 2.367061880986858213e+01
-2.407079734553001227e-01 4.764495021104730443e-01 -9.273163347354730357e-01 1.735086438573440581e+00 -1.571162986767073333e+00 -5.387113217480632876e+00 -1.842331321362554952e+00 -1.086057555055923052e+00 6.554258795806483517e-01 2.913611038688055821e+00 5.897940047881299996e+00 -3.703333668761484176e+01 8.807560198206526536e+01 -6.791623844252907816e+01 -5.661727157052321324e+00 3.215537051004270808e+01 -1.732989764142075018e+01 -6.822817263798721399e+01 7.459360068624346241e+01 -3.788478747503720001e+01 -1.726639859088383133e+01 8.664655681400094522e+01 -3.077439487873002122e+01
7.276273416339069655e-01 -8.160183386215708046e-01 3.336243824700811378e-01 5.801075102971581243e-01 -9.332852800342261723e-02 -8.134892698422866442e-01 -2.065992221782699811e+00 1.511386856157765468e+00 -7.910864037361950984e-01 -2.121436706560488439e+00 2.999410141876153624e+00 -1.054810568719527941e+00 1.554174029562684467e+01 -1.351901672842707747e+01 -1.161697304761181604e+00 -3.184204072141046726e+00 5.018169342462520000e+00 -1.558766901584039388e+01 -9.689475113469024903e-01 -3.966409304890950338e+00 2.367061880986858213e+01 -3.077439487873002122e+01 2.653581617274004500e+01
  
   COEFFSVector of linear coefficients=
regtest/sizeshape/rt-sizeshape/ld1_scalings.txt
 Click here to see an extract from this file.
  
  ×  
FILE: regtest/sizeshape/rt-sizeshape/ld1_scalings.txt
    
-1.008123761803866980e+00
1.626510375189665680e+00
4.491793750288705644e-01
-6.586147643691930842e-01
8.235017815364878979e-01
3.478980458875061688e-01
1.481264411141690829e+00
-3.038312014911420000e+00
-1.267599717071444143e+00
-1.585188804951326025e-01
-5.817395001067517946e-01
-6.260861234524053298e-01
-3.387375174876071648e-01
9.372083330539093815e-01
2.554555712335253581e-01
2.167666577399882111e-02
-6.051555596404274429e-01
-1.380047808122754072e-01
4.218768793816384255e-01
-1.281803232189904440e+00
-9.491404066182579369e-01
-1.339507084104512580e-01
-8.935651980303346842e-02
-8.508449794582281789e-02
-2.148524234383437648e+00
4.226170448258242729e+00
1.677729266602153713e+00
-1.357153299731779228e-01
2.085836564032336471e-01
-7.003628642152187611e-02
5.765462050962143897e-02
-9.802581922121870184e-01
-5.445968486733449021e-01
-4.132029558223072052e-01
6.261097028889341942e-01
2.257791572712143091e-01
-3.200628724867338182e-01
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-1.977133624004201451e-01
-4.769408025188121014e-01
8.678530538596880284e-01
3.176011660686111626e-01
1.944520543315805039e+00
-4.508400989736514042e+00
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2.386745619994993017e+00
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6.025099475452838860e+00
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9.449132130777106175e-02
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5.344767556698636346e-01
2.778728406758145253e-01
5.657267029451286122e-01
-1.164396506907206197e+00
-7.932515522656270468e-01
-3.322057545545195278e+00
6.199518797075218579e+00
2.565033662356621669e+00
-6.500590254132574408e-01
8.874930491073512728e-01
1.671230651011225421e-01
2.608378228283110278e+00
-5.709980861399765573e+00
-2.296414790623169999e+00
  
   GROUPGroup of atoms being used=ga_list
...

PRINTPrint quantities to a file. More details ARGthe labels of the values that you would like to print to the file=proj STRIDE the frequency with which the quantities of interest should be output=1 FILEthe name of the file on which to output these quantities=COLVAR FMT the format that should be used to output real numbers=%8.8f

Input

The atoms that serve as the input for this action are specified using one or more of the keywords in the following table.

Keyword	Type	Description
GROUP	atoms	Group of atoms being used

Full list of keywords

The following table describes the keywords and options that can be used with this action

Keyword	Type	Default	Description
GROUP	input	none	Group of atoms being used
PRECISION	compulsory	none	Precision Matrix (inverse of covariance)
REFERENCE	compulsory	none	Coordinates of the reference structure
COEFFS	compulsory	none	Vector of linear coefficients
NUMERICAL_DERIVATIVESThis keyword do not have examples	optional	false	calculate the derivatives for these quantities numerically
NOPBCThis keyword do not have examples	optional	false	ignore the periodic boundary conditions when calculating distances
SERIALThis keyword do not have examples	optional	false	Perform the calculation in serial, for debug purposes only