# CHARMM c28a3 correl.doc

```
```

#### File: Correl ]-[ Node: Top Up: (commands.doc) -=- Next: Syntax

```

Correlation Functions

The CORREL commands may be used to obtain a set of time series
for a given property from a trajectory. Once obtained, the time series
may be manipulated as required, saved or plotted, or to generate
correlation functions  ( C(tau) = <A(t).A(t+tau)> ). The correlation
functions may be manipulated, saved, plotted, and transformed to find
spectral density (Fourier transform of C(tau)), etc and determine the
correlation times.

Alternately, a covariance matrix may be computed for a collection
of time series. This option will compute the full matrix for use
in entropy calculations or for other applications.

Reorienting a coordinate trajectory is possible using the
COMPARE command. For details see *note reorient:(dynamc.doc)Merge.

* Syntax::              The syntax of the correlation command
* General::             General information regarding the correlation section
* Enter::               How to specify time series
* Trajectory::          How to reference to trajectory files
* Edit::                How the edit the time series specifications
* Mantime::             How to manipulate time series
* Corfun::              How to generate correlation functions.
* Spectrum::            How to get a spectrum from a correlation function
* Cluster::             How to cluster time series data into similar groups
* IO::                  Input/output guide to correlation functions and series
* Examples::            Just what it says

```

#### File: Correl ]-[ Node: Syntax Up: Top -=- Previous: Top -=- Next: General

```
Syntax for the CORREL command and subcommands

[SYNTAX CORRelation functions]

Syntax:

CORREL  [ MAXTimesteps int ]  [ MAXSeries int ]  [ MAXAtoms ] [ COVAriance]  -
default 512         default 2        default 100

[ nonbond-spec ] [ hbond-spec ] [ image-spec ] [NOUPdate]
[  INBFrq 0    ] [  IHBFrq 0  ] [  IMGFrq 0  ]

hbond-spec        *note Hbonds:(hbonds.doc).
nonbond-spec      *note Nbonds:(nbonds.doc).
image-spec        *note Images:(images.doc)Update.

Subcommands:

miscellaneous-commands

COOR coordinate-manipulation-command

{   DUPLicate  time-series-name                         }
{                                                       }
ENTEr  name     { [ BONDs   repeat(2x(atom-spec))        ] [ GEOMetry ] } c
{ [ ANGLe   repeat(3x(atom-spec))        ] [ ENERgy   ] } c
{ [ DIHEd   repeat(4x(atom-spec)) [NOTR] ]              } c
{ [ IMPRo   repeat(4x(atom-spec))        ]              } c
{                                                       }
{ [ ATOM ] [ X   ]  repeat(atom-spec) [ MASS ]          } e
{ [ FLUC ] [ Y   ]                                      } c
{          [ Z   ]                                      }
{          [ R   ]                                      }
{          [ XYZ ]                                      }
{                                                       }
{   VECT   [ X   ]  repeat(2x(atom-spec))               } e
{          [ Y   ]                                      }
{          [ Z   ]                                      }
{          [ R   ]                                      }
{          [ XYZ ]                                      }
{                                                       }
{  ATOM DOTProduct repeat(2x(atom-spec)) [NORMal] [MASS]} e
{  FLUC DOTProduct repeat(2x(atom-spec)) [NORMal] [MASS]} e
{  VECT DOTProduct repeat(4x(atom-spec)) [NORMal]       } e
{                                                       }
{  ATOM CROSsproduct rep.(2x(atom-spec)) [NORMal] [MASS]} e
{  FLUC CROSsproduct rep.(2x(atom-spec)) [NORMal] [MASS]} e
{  VECT CROSsproduct rep.(4x(atom-spec)) [NORMal]       } e
{                                                       }
{  HBONd [4x(atom-spec)]*++       [ ENERgy  ]           } c
{                                 [ DISTance]           }
{                                 [ HANGle  ]           }
{                                 [ AANGle  ]           }
{                                                       }
{  DISTance repeat(2x(atom-spec))                       } c
{                                                       }
{ [ GYRAtion ] [ CUT real ] [ MASS ]       s**          } c
{ [ DENSity  ]                             s**          } c
{                                                       }
{   RMS  [ MASS ] [ ORIEnt ]               s**          } c
{   MODE mode-number                       s**          } c**
{   TEMPerature  [ NDEGF int ]             s**          } v
{   ENERGY                                              } c
{   HELIx [ SELE atom-selection END ]                   } c
{   INERtia  [ SELE atom-selection END ]   [ TRACe ]    } c
{   PUCK  RESI <resid> [SEGI <segid>]                   } c
{   USER user-value [ repeat(atom-spec) ]  s**          } e
{                                                       }
{   CELL cell-spec                                      }
{   TIME [ AKMA ]                                       }
{   ZERO                                                }
{                                                       }

( code: c-coordinates, v-velocities, e-either )
c**  MODE time series is allowed only if CORREL is invoked from VIBRAN.

s**  these utilize the first atom selection in the next TRAJ command.

*++  Hydrogen bond atom order is one of:
Donor,Hydrogen,Acceptor,Acceptor-antecedent
Donor,Hydrogen,Acceptor
Donor,Acceptor

cell-spec::= one of { A B C ALPHa BETA GAMMa ALL SHAPe }

atom-spec::= {residue-number atom-name}
{ segid  resid atom-name }
{ BYNUm  atom-number     }
{ SELE atom-selection END} ***

atom-selection::= see *note Selection:(select.doc)
*** Note: If an atom-selection is used for atom-spec's, then
all atom-spec's must be contained within one atom-selection

TRAJectory [ FIRStu int ] [ NUNIt int ] [ BEGIn int ] [ STOP int ]
[ SKIP int ] [ VELOcity ]  [first-atom-selection]
[ ORIEnt  [MASS]  second-atom-selection  ]

{ ALL                     } [P2] [UNIT int]
SHOW    { time-series-name        }
{ CORRelation-function    }        (defines ?P2, ?AVER, ?FLUC)

{ ALL                     }
EDIT    { time-series-name        }  edit-spec
{ CORRelation-function    }

edit-spec::=  [INDEx int] [VECCod int] [CLASs int] [SECOnd int]
[TOTAl int] [SKIP int] [DELTa real]
[VALUe real] [NAME new-name] [OFFSet real]

READ  { time-series-name  } unit-spec edit-spec { [FILE]              }
{ CORRelation-funct }                     { CARD                }
{ DUMB  [COLUmn int]  }

{ ALL                     }             { [FILE]              }
WRITe   { time-series-name        }  unit-spec  { CARD                }
{ CORRelation-function    }             { PLOT                }
{ DUMB [ TIME ]       }

MANTIME time-series-name
{ DAVErage            } ! Q(t) = Q(t) - <Q(t)>, <> implies time average
{ NORMal              } ! Q(t) = Q(t) / |Q(t)|
{ SQUAre              } ! Q(t) = Q(t) ** 2
{ COS                 } ! Q(t) = COS(Q(t))  (in degrees)
{ ACOS                } ! Q(t) = ACOS(Q(t)) (in degrees)
{ COS2                } ! Q(t) = 3*COS(Q(t))**2 - 1 (in degrees)
{ AVERage integer     } ! Q(t) = < Q(ti) >(ti=t-NUTIL+1,t)
{ SQRT                } ! Q(t) = SQRT(Q(t))
{ FLUCt name2         } ! print zero time fluctuations
{ DINItial            } ! Q(t) = Q(t) - Q(1)
{ DELN integer        } ! Q(t) = Q(t) - <Q(ti)>(ti=t-NUTIL+1,t)
{ OSC                 } ! print oscillations
{ COPY name2          } ! Q(t) = Q2(t)
{ ADD  name2          } ! Q(t) = Q(t) + Q2(t)
{ RATIo name2         } ! Q(t) = Q(t) / Q2(t)
{ DOTProdcut name2    } ! Q(T) x-comp=Q(T).Q2(T)
Q2(T)x-comp=angle Q(T) vs Q2(T) degrees
{ CROSproduct name2   } ! Q(T) = Q(T)xQ2(T)
{ KMULt name2         } ! Q(t) = Q(t) * Q2(t)
{ PROB integer        } ! Q(t) = PROB(Q(t))
{ HIST min max nbins  } ! Q(ibin) = Fraction of Q(t) values in ibin
{ POLY integer        } ! fit time series to polynomial (0-10)
[REPLace] [WEIGh name]
{ CONTinuous [real]   } ! make a (dihedral) time series continuous
Q(t) = Q(t)+ n(t)*2*real, n(t)=integer
(default real is 180.0)
{ LOG                 } ! Q(t) = LOG(Q(t))
{ EXP                 } ! Q(t) = EXP(Q(t))
{ IPOWer integer      } ! Q(t) = Q(t) ** integer
{ MULT   real         } ! Q(t) = real * Q(t)
{ DIVIde real         } ! Q(t) = Q(t) / real
{ SHIFt  real         } ! Q(t) = Q(t) + real
{ DMIN                } ! Q(t) = Q(t) - QMIN
{ ABS                 } ! Q(t) = ABS(Q(t))
{ DIVFirst            } ! Q(t) = Q(t) / Q(1)
{ DIVMaximum          } ! Q(t) = Q(t) / ABS(Q(MAX))
{ INTEgrate           } ! Q(t) = Integral(0 to t) (Q(t)dt)
{ TEST  real          } ! Q(t) = COS(2*PI*t*real/TTOT)
{ ZERO                } ! Q(t) = 0.0
{ DERIvative          } ! Q(t) = (Q(t+dt)-Q(t))/dt

CORFUN 2x(time-series-name)
{ [ PRODuct ]  [ FFT   ] [ LTC  ] [ P0 ] [ NONOrm ] } [ TOTAl int ]
{              [ DIREct] [ NLTC ] [ P1 ]            }
{                                 [ P2 ]            }
{                                                   }
{  DIFFerence                                       }

SPECtrum  [FOLD] [RAMP] [SWITch] [SIZE integer]

CLUSter time-series-name RADIus <real> [ MAXCluster <int> ] -
[ MAXIteration <int> ] [ MAXError <real> ] -
[ NFEAture <int> ] [ UNICluster <int> ] -
[ UNIMember <int> ] [ UNIInitial <int>] -
[ CSTEP <int> ] [ BEGIn <int> ] -
[ STOP <int> ] [ ANGLE ]

```

#### File: Correl ]-[ Node: General Up: Top -=- Next: Enter -=- Previous: Syntax

```
General discussion regarding time series and correlation functions

Discussion:

The CORREL command invokes the CORREL subcommand parser.
The keyword values MAXTimesteps, MAXSeries, and MAXAtoms may be
specified for space allocation greater than the default options.
If there in insufficient virtual address memory for the space request,
it may be possible to achive the desired results by removing the
nonbond lists before running the CORREL command.

The MAXTimesteps value is the largest number of steps any
time series will contain. The MAXSeries keyword is the largest number
of timeseries that will be contained at any time within CORREL.
A vector time series will counts as 3 time series in allocating space.
The MAXAtoms keyword allocates space for the atoms that are specified
in the ENTER commands (also duplicating a time series requires more space
for atoms). For bonds, angles, dihedrals, and improper dihedral
specifications, one extra value is needed for each entry to hold the
CODES value (so each bond uses 3 atom entries, 4 for angles...).

If the COVAriance keyword is given, no time series will be
computed, but instead, a complete equal time covariance matrix will
be computed. For this option, only one TRAJectory command is allowed.
The covariance matrix is then obtained by writing the time series, where
the elements are covariant with other time series.

The ENTER defines a time series. Many time series may be specified.
A time series is defined by the following items;

Name              - Each time series must have a unique (4 character) name.
Class code        - The type of time series (BOND, USER, ATOM,...)
Number of steps   - The number of time steps currently valid
Velocity code     - Was the time series read from velocities?
Skip value        - What multiple of delta do the time steps represent?
Delta             - Integration time step
Offset            - Time of first element
Secondary code    - Depends on Class code (Geometry/Energy)(X/Y/Z...)
Vector code       - 1=simple time series, 3=vector, 0=Y or Z part of vector
Value             - Utility series value, depends on Class code
Mass weighting    - Are the elements to be mass weighted (only for ATOM)
Average           - Time series average
Fluctuation       - Time series fluctuation about the average
Atom pointer      - Pointer into first specified atom in atom list
Atom count        - Number atom entries given in the ENTER command
Time series       - Series values from (1,NTOT)

The TRAJectory command processes all of the time series which
have a NTOT (number of steps) count of zero. For this process,
the main coordinates are used for reading the trajectory. If flutucations
are requested, the comparison coordinates MUST be filled with the
reference (or average) coordinates before invoking the TRAJectory
command. Allowing multiple TRAJectory commands separated by enter
commands make it possible to compute correlation function between
positions and velocities, or even for different trajectories.

The EDIT command allows the user to directly modify the time
series specifications.

The MANTIME command allows the user to manipulate the time
series values (and sometimes some of the specifications).

The SHOW command will display the specification data for all
of the time series.

```

#### File: Correl ]-[ Node: Enter Up: Top -=- Next: Trajectory -=- Previous: General

```
Specifying time series

The ENTER command defines a new time series. Each time series
specified by different enter commands must have a unique name (up to
4 characters). With this command, a time series may be defined and
then must be later filled with a TRAJectory command (or a MANTIME COPY,
or a READ time-series command). Alternativly, a time series may be retrieved
from an existing file, or duplicated from another time series that
currently exists.

The time series names "ALL" and "CORR" may not be used, and
are reserved for selecting all of the time series or the correlation
function respectivly.

The ENTER options are;

-----------------------------------------------------------------------------
DUPLicate  time-series-name
This causes an exact copy of an existing time series to be
created (except with a different name). This may be useful where
several different type of manipulations are required on a single
time series.

-----------------------------------------------------------------------------
This causes a time series to be created and all data then
read in from an existing time series file. All time series (up to the
maximum allowed) will be read with this command.

-----------------------------------------------------------------------------
[ BONDS   repeat(2x(atom-spec))        ] [ GEOMETRY ]
[ ANGLE   repeat(3x(atom-spec))        ] [ ENERGY   ]
[ DIHEd   repeat(4x(atom-spec)) [NOTR] ]
[ IMPRo   repeat(4x(atom-spec))        ]
These specifications cause a particular internal coordinate
(or an average of several) to define the time series. It is not necessary
that the specified atoms have a corresponding PSF entry, but if ENERGY is
requested, the specified atoms must be able to produce a valid parameter
code. The default is GEOMETRY. With geometry, any 4 atoms may be specified.
A velocity trajectory should not be used to fill these types of time series.
The NOTR option for dihedral prevents the analysis of dihedral transitions.

-----------------------------------------------------------------------------
[ ATOM ] [ X   ]  repeat(atom-spec) [ MASS ]
[ FLUC ] [ Y   ]
[ Z   ]
[ R   ]
[ XYZ ]
These ENTER commands define a time series, Q(t), based on atom
positions or velocities. The ATOM option uses the (X,Y,Z,R,or XYZ) values
directly.  The FLUCtuation option subtracts off the reference values
(contained in the comparison coordinates). For example, if the average
structure is desired as the reference value, then the command:
COOR DYNA COMP trajectory-spec
would be required before invoking the TRAJECTORY command.
If more than one atom is specified, then Q(t) values are
averaged over atoms.  If MASS is specified, then mass weighting is used in
this averaging of Q(t) values.  The properties X,Y,Z, and R cause a scalar
time series to be created with the requested property. The XYZ option causes
a vector time series to be created.
ATOM:  Q(t) = X(t)
FLUC:  Q(t) = X(t) - Xref

-----------------------------------------------------------------------------
VECT   [ X   ]  repeat(2x(atom-spec))
[ Y   ]
[ Z   ]
[ R   ]
[ XYZ ]
The VECTor command is similar to the ATOM and FLUCuation
commands listed above, except the values are given by the difference
in position or velocity of 2 atoms. If more than one pair of atoms
is specified, then the values for each vector are averaged.
Q(t) = X1(t) - X2(t)
-----------------------------------------------------------------------------
ATOM DOTProduct  repeat(2x(atom-spec))
FLUC DOTProduct  repeat(2x(atom-spec))
VECT DOTProduct  repeat(4x(atom-spec))

ATOM CROSsproduct  repeat(2x(atom-spec))
FLUC CROSsproduct  repeat(2x(atom-spec))
VECT CROSsproduct  repeat(4x(atom-spec))

These ENTER commands produce a scalar time series for
velocities or positions with the following definitions;

ATOM DOTP:  Q(t) =  ( r1(t) | r2(t) )
FLUC DOTP:  Q(t) =  ( (r1(t)-r1(ref)) | (r2(t)-r2(ref)) )
VECT DOTP:  Q(t) =  ( (r1(t)-r2(t)) | (r3(t)-r4(4)) )

If more than one set of atoms is specified, then the vector values
are averaged.  The dotproduct is then computed from the
averaged vectors.  NOTE: the vectors are averaged, NOT the resultant
dotproducts or crossproducts.   For the FLUC option, the reference
coordinates must be in the comparison coordinate set.

-----------------------------------------------------------------------------
[ GYRAtion ] [ CUT real ]
[ DENSity  ]
These commands define a scalar time series for a coordinate
trajectory. The density calculation is based about the origin on all
atoms within the CUT value; the radius of gyration is for all atoms
within distance CUT of the geometric center of the molecule, and no
mass weighting is applied.

-----------------------------------------------------------------------------
MODE mode-number
This option generates a scalar time series which is obtained
by projecting the velocities onto the specified normal mode, or to
project the coordinate diplacement from the reference strucure. The
result is given by;
velocity:  Q(t) = < root(mass)*v(t) | q >
position:  Q(t) = < root(mass(i))*(r(t)-r(ref)) | q >
-----------------------------------------------------------------------------
TEMPerature
The time series is the temperature at each point.
If NDEFG is specified as a positive value, then this is used instead of
the NDEGF values from the trajectory file.  If a negative NDEGF value
is specified, then NDEGF will be set to 3 times the number of selected
atoms in the trajectory associated trajectory command.
---------------------------------------------------------------------
HELIx atom-selection
The x,y, and z components of the normalized vector defining the
axis af a cylindrical surface best fitting the selected atoms.
So you end up with a three-dimesnional vector series.
Intended for say alpha helices where the selection would be something
like: SELE ATOM * * CA .AND. RESID 23:36 END, to give the axis of
an alpha helix running from residue 23 to residue 36.
---------------------------------------------------------------------
INERtia atom-selection [ TRACe ]
The x,y, and z components of the normalized vector defining
the principal axis obtained from diagonalizing the moment of inertia
tensor for the selected atoms at each time point.  The eigenvector
corresponding to the smallest eigenvalue is returned, and 180 deg flips
of the axis are explicitly prohibited (nonphysical).
The optional TRACe keyword returns the sorted eigenvalues as a
three column time series, instead of the principal axis vector.
---------------------------------------------------------------------
CELL  cell-spec
If the cell-spec is one of the 6 unit cell parameters A, B, C,
ALPHA, BETA, or GAMMA, then a single time series corresponding to that
component is return.  The keyword ALL returns a 6 element time series,
with the columns in the order given above.  The SHAPE keyword returns
the shape matrix for the unit cell at each time point, in lower diagonal
form.  The shape matrix has the angles as cosines, while ALPHA, BETA, and
GAMMA are in degrees.
---------------------------------------------------------------------
RMS  [ORIE]
The RMS deviation from the COMPARISON coordinate set is
computed, with a rotation to obtain a best fit if ORIEnt is specified.
---------------------------------------------------------------------
PUCK RESI <resid> [SEGI <segid>]
The sugar pucker phase and amplitude are calculated for
the (deoxy)ribose of the specified residue; the first segment is
the default. This gives a two-dimensional vector, with component 1
being the phase (degrees) and component 2 the pucker amplitude
(Angstroms), as defined by Cremer&Pople (JACS 1975).
-----------------------------------------------------------------------------
USER user-value [ repeat(atom-spec) ]
The USRTIM routine is called for each coordinate or velocity
set. The user value and atom list is also passed along. See the
description in (USERSB.FLX)USRTIM for more details.
Q(t) = Whatever you want!
-----------------------------------------------------------------------------
TIME [ AKMA ]
The time is returned in picoseconds unless AKMA is specified.
Q(t) = t
-----------------------------------------------------------------------------
ZERO
A zero time series is specified ( Q(t)=0 ).
This option is useful for cases where time series will be read with
the DUMB option. For these cases, the EDIT command may also be needed
to get desired results.

```

#### File: Correl ]-[ Node: Trajectory Up: Top -=- Next: Edit -=- Previous: Enter

```
Specification of the Trajectory Files

The TRAJectory command reads a number of trajectory files whose
Fortran unit numbers are specified sequentially. The first unit is given
by the FIRSTU keyword and must be specified. NUNIT gives the number of
units to be scanned, and defaults to 1.

BEGIN, STOP, and SKIP are used to specify which steps in the
trajectory are actually used. BEGIN specifies the first step number to
be used. STOP specifies the last. SKIP is used to select steps
periodically as follows: only those steps whose step number is evenly
divisible by STEP are selected. The default value for BEGIN is the first
step in the trajectory; for STOP, it is the last step in the trajectory;
and for SKIP, the default is 1.

The first atom selection in the TRAJectory command is meaningful
only for those time series that require an atom selection.  These are
time series defined by the following ENTER commands: GYRAtion, DENSity,
RMS, MODE, TEMPerature, and optionally USER.

General reorienting of a coordinate trajectory is possible using the
MERGE command. For details see *note reorient:(dynamc.doc)Merge.
It is also possible to perform a simple rms best fit of each frame with the
reference coordinates (comparison set) using the ORIEnt option.  For this
option a second atom selection MUST be provided and a MASS keyword is an
option that allows for a mass weighting of the best fit.

If VELOcity is specified, a velocity trajectory will be looked
for. Otherwise, a coordinate trajectory is expected.

Any time series that has a zero count (NTOT=0) will be
filled by this comand. The time series count will then be filled
with the total number of steps processed for each of these series.
Any time series with a nonzero count (NTOT>0) will not be affected
by this command. The count may be set to zero for a time series with
the EDIT command.

Upon conclusion, the average and flucutation as well as some
other data is presented on each of the processed time series.

If any of the time series to be filled require a reference
coordinate set, then the comparison coordinates MUST be filled with the
reference (or average) coordinates before invoking the TRAJectory
command. Upon completion, the main coordinates contain the last coordinate
set read from the trajectory, and the comparison coordinates are unaffected.

```

#### File: Correl ]-[ Node: Edit Up: Top -=- Next: Mantime -=- Previous: Trajectory

```
Editing a time series

The EDIT command allows the time series specifications
to be modified directly.

series specification. There is NO checking to see if what is being done
makes sense. As such, this command is versitile and dangerous.

The EDIT command must be followed by a valid time series name.
All subsequent keywords will be based on that time series.
The series name "ALL" will cause the edit spec to operate on all
the time series. The name "CORR" will cause the edit to occur on the
correlation function.

The following may be specified for a time series;

INDEx integer   - May be specified to modify X,Y, or Z (1,2,3 resp)
of a vector timeseries. Otherwise, all are modified.
The index number is in fact an offset from the specified
time series, where a value of 1 represents the selected
time series. A value of 5 will cause the edit operation
to modify the fourth time series from the specified.

CLASs integer   - May be used to specify a class code (consult source).

TOTAl integer   - The total number of valid steps may be altered, but
none of the values are modified. By setting this
value to zero, the time series is then ready again
for the next TRAJectory command.

SKIP integer    - May be specified to reset the SKIP value. This may be
useful after reading an external time series.

DELTa real      - May be specified to modify the basic time step. The
actual time step for a series is (SKIP*DELTA).

OFFSet real     - The time of the first element in the time series.

VECCod integer  - User may specify a vector code. This may be useful
in merging 3 separate time series into a vector
time series (or the reverse). In fact any number of
time series may be grouped together with this option.
For example, if a table with 5 time series is desired,
setting VECCOD to 5 for the first one and the writing
this time series will output all 5.

VALUe real      - This allows the user to modify the series utility
value. Its function depends on the Class code.
This value is currently used for (USER, GYRAtion,
DENSity, MODE, and TIME)

SECOndary int   - The secondary class code may be modified (consult source).

```

#### File: Correl ]-[ Node: Mantime Up: Top -=- Next: Corfun -=- Previous: Edit

```
Manipulating the Time Series

The MANTIME command allows the user to manipulate selected
time series, Q(t), and performs the operation requested by the option
and leaves the resultant time series as the active time series.
This helps in performing various permuations of manipulations to increase
the options without increasing the number of ENTER commands.

The keyword ordering must be followed exactly.

DAVErage        subtracts the average of the time series from all elements.

NORMal          normalises the vectorial time series.
(i.e. creates the unit vector by dividing all elements for
a given value of t by r(t) = sqrt(x**2 + y**2 + z**2) ).

SQUAre          squares all the elements

COS             obtains the cosine of all elements.

ACOS            obtains the arc-cosine of all elements.

COS2            calculates 3*cos**2 - 1 for all elements.

AVERage integer calculates the average for every <integer> consecutive
points and increases the time interval by a factor of
<integer>. Note: NTOT is divided by <integer>.

SQRT            obtains square root for all elements.
Negative elements are set to -SQRT(-q(t)).

FLUCt name      The Q(t) remains unchanged.
A second (b) timeseries must be specified.
The zero time fluctuations are computed and printed
out.  The following variables are computed:
A = <Qa(t) . Qb(t)>
B = sqrt <Qa(t)**2>
C = sqrt <Qb(t)**2>
D = A/(B*C)

DINItial        subtracts the value of the first element from all elements.
Q(t) = Q(t) - Q(1)

DELN integer    Q(I) = Q(I) - <Q(I)> I FROM 1 TO N, FROM N+1 TO N+N ETC.
(untested).

OSC             counts the number of oscillations in Q(t) / unit time step.
The Q(t) remains unchanged.

COPY name       This copies the second time series to the first. NTOT
of the first is set to that of the second.

ADD name        Q(t) = Q(t) + Q2(t); add the second time series to the first

RATIo name      Q(t) = Q(t) / Q2(t)

CROSsprod name  Q(T) = Q(T) x Q2(T); the 3D crossproduct of the two
3D vectors formed by the selected and named timeseries
DOTProd name    Q(T) = x-comp of Q(T)= Q(T) . Q2(T)
x-comp of Q2(T) angle in degrees between the two vectors
NOTE! Modifies Q2 as well as Q
to get just the x-comp you may then edit the selected series:
EDIT series VECCOD 1

KMULt name      Q(t) = Q(t) * Q2(t)

PROB integer    give the probability to find a specific value of the
time series. <integer> subdivisions of the time series
are considered so that there are integer+1 values.

HIST min max nbins
Q(ibin) = Fraction of Q(t) values within ibin
This command replaces a time series with a
histogram of the time series divided into "nbins" with
a range from "min" to "max".  The histogram values sum to 1.

POLY integer [REPLace] [WEIGh name]
fit time series to polynomial. The order should
be in the range of 0 to 10.
Order 0 will provide just the average,
Order 1 will fit the time series to a stright line.
Order 2 will fit to a quadratic function.
The REPLace option will replace the time series with
fitted one.  The WEIGht option will wait all data
by the values in a second time series.

CONTinuous real Q(t) = Q(t) + n(t) , where n(t) is an integer such that
the ABS(Q(t)-Q(t-1))<=real
The default value is 180.0, which is appropriate for
making a dihedral time series continuous.  A different
positive value may be selected (such as a box size...).

LOG             Q(t) = LOG(Q(t))

EXP             Q(t) = EXP(Q(t))

IPOWer integer  Q(t) = Q(t) ** integer

MULT real       Q(t) = Q(t) * <real>

DIVI real       Q(t) = Q(t) / <real>

SHIFt real      Q(t) = Q(t) + <real>

DMIN            Q(t) = Q(t) - QMIN, QMIN being the minimum of the time series.

ABS             Q(t) = ABS(Q(t))

DIVFirst        Q(t) = Q(t) / Q(1)

DIVMax          Q(t) = Q(t)/ ABS(Q(t) with max norm)

INTEgrate       Q(t) = Integral(0-t) [ Q(t) dt ]

TEST real       Q(t) = COS ( 2 * PI * <real> / NTOT )

ZERO            Q(t) = 0
This option zeroes the specified time series.

DERIvative      Q(t) = (Q(t+dt)-Q(t))/dt, the last point is set to the one
before last

```

#### File: Correl ]-[ Node: Corfun Up: Top -=- Next: Spectrum -=- Previous: Mantime

```
Calculating a Correlation Function

CORFUN: This option takes the specified time series and calculates the
desired correlation function from it.  The resultant correlation function
is saved in a time series named "CORR" which may then be used in subsequent
CORREL manipulation or write commands.  If multiple CORFUN commands are
requested, then the "CORR" time series is overwritten.

In the following, Qa and Qb refer to the time series that were
extracted using the CORREL command.

PRODuct This option (default) generates a correlation function that is the
product of the time series elements.
C(tau) = < Q1(t)*Q2(t+tau) >

DIFFerence
The difference option is an alternative of the product option
and it generates a function that is useful in calculating
diffusion constants (slope at long tau).
C(tau) = < ( Q1(t) - Q2(t+tau) ) **2 >

FFT     This option is to calculate the correlation function using the FFT
method.  There are certain limitations on the prime factors
in the total number of points.

DIRECT  This option is to calculate the correlation function using the
direct multiplication method.

P1      This option gives the direct correlation function, <Qa(0).Qb(t)>.
If Qa and Qb are unit vectors, then this is also the first
order Legendre Polynomial

P2      This is to obtain the correlation function of second order Legendre
Polynomial, (3 <[Qa(0).Qb(t)]**2> - 1)/2.  For all applications
that I can think of, Qa and Qb will be unit vectors. For P2, LTC = 0
and NORM = 1

NLTC    no long tail correction.

LTC     long tail correction (subtracts <Qa>**2 if autocorrelation,
<Qa>*<Qb> if cross correlation.  There is no LTC for P2
so NLTC and LTC give same result.)
This feature is to be used with care.  If the Qa and Qb are
fluctuations from the mean (i.e. FLCT or MANTIME DELTA), then
this can serve as a correction for roundoff error.  Otherwise,
they are not centered about the mean, this correction causes
the C.F. to be a less accurate calculation of fluctuations from
the mean, i.e.
<Qa(0).Qb(t)> - LTC = <Qa(0).Qb(t)> - <Qa>*<Qb>
= <delta Qa(0) . delta Qb(t)>

NONORM  Correlations are not normalized. This is useful for adding
correlations computed in different trajectories.
(P2 is not normalized)
The correlation functions are normalized unless NONORM is specified.

TOTAL integer
The TOTAL value determines the number of points to keep in
the correlation function. The number of points may not be
grater than the number of points in the time series. A reasonable
value is about 1/4 to 1/3 the length of the time series.
Correlation function values near the end have little weight.
The default value is the nearest power of two less than half of
the time series length.

The defaults are FFT, P0, NLTC.

Note:  The correlation time which is given by the program is calculated
by an exponential fit to the first NTOT/8 points or up to the
first crossing of the time axis.  This value should be considered
a (poor) estimate, it is meaningful only for correlation functions
which decay exponentially to zero with no oscillations.

For P0,
C(t) = (c(t) - ltc)/N
ltc and Normalization factors, N, are:
LTC, autocorrelation:
ltc = <Qa>**2    for P0
= 0          for P2
N = C(0) - ltc
= <Qa**2> - ltc
LTC, cross-correlations:
ltc = <Qa>*<Qb>
N = sqrt[ (<Qa**2> - <Qa>**2) * (<Qb**2> - <Qb>**2) ]
NLTC, autocorrelation:
ltc = 0
N = C(0)
NLTC, cross-correlations:
ltc = 0
N = sqrt [<Qa**2>*<Qb**2>]

```

#### File: Correl ]-[ Node: Spectrum Up: Top -=- Next: Cluster -=- Previous: Corfun

```
Generating a Spectrum from Correlation Functions

There is a command, SPECtral-density, which may be used to generate
a spectrum from a correlation function. The synatax is;

SPECtrum [SIZE integer] [FOLD] [RAMP] [SWITch]

```

#### File: Correl ]-[ Node: Cluster Up: Top -=- Next: IO -=- Previous: Spectrum

```
Clustering Time Series Data

This command clusters time series data obtained within the CORREL
facility.  The time series must first be defined using CORREL's ENTEr
command and the data read in via TRAJ or READ.  The CLUSter command
clusters these data into groups with similar time series values, with
each cluster being defined by a "cluster center".  The cluster centers are
output to UNICluster, and a list of time points and assigned clusters is
given in the cluster membership file (UNIMember).

For example, if you want to find similar conformations of a peptide
using dihedral angles, you would first define the set of dihedral angles to
be considered, say angle(1) -> angle(M), as M time series.  If the time series
were each N time steps long, then you would be clustering N "patterns", with
each pattern M "features" long.

Consecutive time series are clustered.  If the first time series
is, for example, "ts1" then the "veccod" of this time series can be
changed to the number of time series to be clustered:
CORREL ...
ENTE ts1 ...
ENTE ts2 ...
...
ENTE tsM ...
EDIT ts1 veccod M
CLUSTER ts1 ...
END
Alternatively, NFEAture M can be specified in the CLUSter command line.
Note that vector time series count as three features.

The Clustering Algorithm

ART-2' is a step-wise optimal clustering algorithm based on a
self-organizing neural net (Carpenter & Grossberg, 1987; Pao, 1989;
Karpen et al., 1993).  The algorithm optimizes cluster assignment subject
to a constraint on cluster radius, such that no member of a cluster is more
than a specified distance from the cluster center.  This optimization is
carried out as an iterative minimization procedure that minimizes the
Euclidean distance between the cluster center and its members.

A self-organizing net is created with each output node representing
a cluster.  The number of pattern features is equal to the number of input
nodes.  The weights of the connections between the input layer (layer i)
and the output layer (layer j) are denoted by b(j,i).  For each cluster j,
b(j,i), i = 1, nfeature, is the cluster center.  To create the net (which is
synonomous to learning the classification scheme or cluster centers) the
following algorithm is implemented:

1. To initialize the network, assign b(1,i) equal to the first
pattern tq(1,i) for i = 1, nfeature.

2. For each pattern number k, calculate the Euclidean distance (rms)
between the pattern tq(k,i) and all cluster centers b(j,i), where
j is the cluster index.

rms(j,k) = sqrt[sum [(b(j,i)-tq(k,i))**2] for i = 1, nfeature]

3. Find cluster j such that rms(j,k) < rms(i,k) for all i<>j.  If
rms(j,k) <= Threshold, then update b(j,i):

b(j,i) = ((m-1)*b(j,i) + tq(k,i))/m,

where m is the number of prior updates of b(j,i).  Note that
b(j,i) is the average of feature i for all patterns currently
assigned to cluster j.

4. If rms > Threshold for all prior cluster centers (j=1,numclusters),
then create a new cluster center by increasing the number of
output nodes by one, and assign the weights b(numclusters,i) of
this node the value of the pattern tq(k,i).

5. Repeat 2.-4. until all patterns have been input.

6. Compare the new set of cluster centers with the last set.  If
the difference between them is less than MAXError, then halt
clustering.

7. If the difference between the sets of cluster centers is greater
than MAXError, then use the new set of cluster centers as the
starting cluster centers, and repeat steps 2.-6.  Else, clustering
is complete.

Note that the cluster centers currently being calculated in step 3
are only used for the comparison in step 2 during the first
iteration with no initial cluster centers.  Otherwise, the centers
calculated in the previous iteration (or read from UNIInit) are
used in the comparison in step 2.  Hence, in the initial "learning"
phase, cluster centers are recalculated as each new member is added.
In subsequent "refining" phases, cluster centers are not updated
until all conformations are read in and assigned.

References:

1) Carpenter, G. A., & Grossberg, S. 1987. ART 2: Self-organization of stable
4930.

Wesley, New York.

3) Karpen, M. E., Tobias, D. T., & Brooks III, C. L. 1993. Statistical
clustering techniques for analysis of long molecular dynamics trajectories.
I: Analysis of 2.2 ns trajectories of YPGDV. Biochemistry  32:412-420.

CLUSter Parameters

CLUSter time-series-name RADIus <real> [ MAXCluster <int> ] -
[ MAXIteration <int> ] [ MAXError <real> ] -
[ NFEAture <int> ] [ UNICluster <int> ] -
[ UNIMember <int> ] [ UNIInitial <int>] -
[ CSTEP <int> ] [ BEGIn <int> ] -
[ STOP <int> ] [ ANGLE ]

1. time-series-name: The name of the first time series (as defined by
the ENTE command) to be clustered.

2. RADIus: Maximum radius of cluster.  The rms cutoff or threshold for
assignment to a cluster.

3. MAXCluster:  Maximum number of clusters (default = 50).

4. MAXIteration:  The maximum number of iterations allowed.  If the
clustering has not converged by this number of iterations, all
clusters are output (default = 20).

5. MAXError:  If the rms difference between the position of the cluster
centers for the last two iterations is less than maxerror, the system
is considered converged and the clustering is halted (default = 0.001).

6. NFEAture:  This variable gives the number of features in the input
pattern, that is, the number of time series to be clustered at a time.
The default is the veccod parameter associated with 'time-series-name'.
NFEATure time series are clustered, starting with 'time-series-name'
and continuing with the next nfeature-1 series specified in subsequent
'ENTE' commands (default = veccod of time-series-name).

7. UNICluster:  The unit number of the output cluster file.  If UNIC = -1
(the default), the cluster parameters are output to the standard output.

8. UNIMember:  The unit number of the output membership file.  This file
lists each time point and the cluster(s) associated with the specified
time series at that time point.  If UNIM = -1 (the default), the
membership list is not output.

9. UNIInit:  The unit number of the file with the initial cluster centers.
If UNII = -1 (the default), no initial cluster centers are specified.

10. CSTEp:  This variable gives the spacing between time series in the
input vector.  For each timepoint k, the set of patterns clustered is
tq(k,1) -> tq(k,nfeature), tq(k,1 + cstep) -> tq(k,nfeature + cstep),
...,tq(k,nserie - nfeature + 1) -> tq(k,nserie) (default = nfeature).

11. BEGIn:  Indicates frame in time series where clustering begins
(default = 1).

12. STOP:  Indicates the frame in the time series where clustering ends
(default = minimum length (TOTAl in SHOW) of time series).

13. ANGLe:  A logical flag which when true specifies angle data is to be
clustered, taking angle periodicity into account (default = .FALSE.).

Caveats

The clustering algorithm is initial-guess dependent, i.e., it is
dependent on the input order of the patterns.  The order of presentation
in CLUSter is simply the consecutive frames of the time series.  To check
for stable clustering, cluster centers can be calculated from time series
with the time frames randomized.  This is not currently implemented in
CHARMM, so the user will have to write a set of time series to a file
and then randomize row position outside of CHARMM.

It is relatively straight forward to compare features derived from
similar measures (i.e., time series with the same "class codes", for
example all DIHE/GEOM).  In some applications it may be desired to "mix"
units in the pattern, for example, cluster a set of time series derived
from both atomic positions and energies.  How best to compare "apples &
oranges" is a problem from measurement theory, and is application-specific.
Normalizing the variables such that they have unit variance is one
possibility, and this can be done by 1) determining the standard deviation
of the time series (FLUC given by the SHOW command), and 2) using this
value in the MANTim DIVI command.  Since only differences between features
are used in the clustering algorithm, shifting the time series to zero
mean is not necessary.

Duda & Hart have a good discussion of the issues involved in
clustering and normalization:

Duda, R. O., & Hart, P. E., Pattern Classification and Scene Analysis,
Wiley, New York, pp. (1973).

Cluster Output

The following data are output to UNIC for each cluster:

Cluster Index - The clusters are numbered starting with 1.
No. of Members - Number of patterns assigned to the cluster.
Cumulative No. of Members - The total number of patterns within the
cluster radius.  This can be higher than the No. of Members due
to patterns being within the maximum radius of more than one cluster.
Standard Deviation of Patterns within Cluster -
For cluster j with the number of features = Nfeature, this is
sqrt(sum((tq(k,i) - b(j,i))**2)/Nfeature*N(j)) where the sum is
over i = 1, Nfeature and over all k such that tq(k) is a member
of j.  N(j) = the number of members in cluster j.  Note that
b(j,i) = <tq(k,i)> (averaged over k in cluster j).
Maximum Distance - the longest distance between the cluster center and
an assigned pattern, normalized by sqrt(Nfeature).
Cluster Centers - (b(j,i), i = 1, Nfeature)

The following data are output to UNIM:

Cluster index of the assigned cluster
Time series time step
Time series index of first time series in pattern
Distance of pattern from cluster center, normalized by sqrt(Nfeature)

```

#### File: Correl ]-[ Node: IO Up: Top -=- Next: Examples -=- Previous: Cluster

```
Input/Output of time series and correlation functions.

1) The SHOW command

{ ALL                     }
SHOW    { time-series-name        }
{ CORRelation-function    }

The SHOW command displays to print unit various data regarding
the specified time series. This command is automatically run after the
ENTER and EDIT commands as a verification of the last action.

READ  { time-series-name  } unit-spec edit-spec { [FILE]              }
{ CORRelation-funct }                     { CARD                }
{ DUMB  [COLUmn int]  }

The READ command allows a time series or correlation function to
be directly read. The file formats for time series and correlation
functions is identical. There are three basic methods by which time
series may be read: FILE (default), CARD, and DUMB. The FILE and CARD
options expect a file of specific type generated by the corresponding
WRITE command. The DUMB option will read a free field card file with
NO title or other header. The COLUmn option (default 1) may be specified
to start reading the time series from any specified column. The DUMB
option will usually include some edit specifications to properly set
the time steps (etc.).

3) The WRITe command

{ ALL                     }              { [FILE]        }
WRITe   { time-series-name        }  unit-spec   { CARD          }
{ CORRelation-function    }              { PLOT          }
{ DUMB [ TIME ] }

The WRITe command will write out time series or a correlation function.
All of the write options expect a title to follow this command.
There are several file formats; FILE (default), CARD, PLOT, and DUMB.
The FILE and CARD options will write out all data regarding the specified
time series with the expectation for later retrival by Charmm or another
program. The PLOT option will create a BINARY file for plotting by PLT2.
The first line of the title is used as the plot title, but this may be
reset in PLT2.
The DUMB options will simply write out the values with no title
or header to a card file, one value to a line. If the TIME option is
specified, the time value will preceed the time series values (as needed
for an X-Y plot). If the time series is a vector type, then 3 values will
be given on each line. The DUMB option is useful for making plot files,
or for feeding the data to other programs.
With the EDIT command, a user may merge 3 separate sequential
time series into a vector time series (or the reverse). In fact any number
of time series may be grouped together with this option.  For example,
if a table with 5 time series is desired, setting VECCOD to 5 for the
first one and the writing this time series will output all 5.

```

#### File: Correl ]-[ Node: Examples Up: Top -=- Previous: IO

```
Examples

These examples are meant to be a partial guide in setting up
input files for CORREL. The test cases may be examined for a wider
set of applications.

Example (1)

CORREL MAXSERIES 1 MAXTIMESTEPS 500 MAXATOMS 5
ENTER AAAA  TORSION MAIN 28 N MAIN 28 CA MAIN 28 C MAIN 29 N    GEOMETRY
TRAJECTORY FIRSTU 51 NUNIT 5 BEGIN 26000 STOP 31000 SKIP 10
MANTIME AAAA DAVER
WRITE AAAA UNIT 20 DUMB TIME
* title
*
WRITE AAAA CARD UNIT 10
* title for card
* file containing the time series
*
CORFUN AAAA AAAA   FFT NLTC P0
WRITE CORREL  UNIT 21 DUMB TIME
* title
*
WRITE CORREL FILE UNIT 11
* title for binary correlation function
*

Extracts the time series, PHI(t), for phi dihedral of residue 28.
Makes the time series the fluctuation from the mean, delta PHI(t).
Makes a plot file of delta PHI(t) vs. time.
Makes binary file of delta PHI(t).
Calculates C(t) = <delta PHI(0) . delta PHI(t)> / <PHI**2> by FFT
with no long tail correction.
Makes a plot file of C(t) vs. t.
Makes a binary file of C(t).

Example (2)

CORREL MAXSERIES 2 MAXTIMESTEPS 500 MAXATOMS 10
ENTER PHI  TORSION MAIN 27 C  MAIN 28 N  MAIN 28 CA  MAIN 28 C  GEOMETRY
ENTER PSI  TORSION MAIN 28 N  MAIN 28 CA MAIN 28 C   MAIN 29 N  GEOMETRY
TRAJECTORY FIRSTU 51 NUNIT 5 BEGIN 26000 STOP 31000 SKIP 10
MANTIME PHI DAVER
MANTIME PSI DAVER
CORFUN  PHI PSI  FFT NLTC P0 NONORM
WRITE CORREL FILE UNIT 11
* title for cross correlation binary file
*
WRITE CORREL PLOT UNIT 12
* plot title
*

Extracts the time series PHI(t), for phi dihedral, and PSI(t), for
the psi dihedral, of residue 28.
Makes the time series the fluctuation from the mean.
Calculates C(t) = <delta PHI(0) . delta PSI(t)> by FFT with no
long tail correction.
Makes a binary file of C(t).
Makes a binary PLT2 file for plotting

Example (3) Fluorescence Depolarization, for example

CORREL MAXSERIES 6 MAXTIMESTEPS 500 MAXATOMS 8
ENTER V1  VECTOR XYZ  MAIN 28 NE1 MAIN 28 CZ3 MAIN 28  NE1 MAIN 28 CE3
ENTER V2  VECTOR XYZ  MAIN 28  CD1 MAIN 28 CH2 MAIN 28 CD1 MAIN 28 CZ3
TRAJECTORY FIRSTU 51 NUNIT 5 BEGIN 26000 STOP 31000 SKIP 10
MANTIME V1 NORMAL
MANTIME V2 NORMAL
SHOW ALL
CORFUN  V1 V2   FFT P2
WRITE CORREL PLOT UNIT 21
* title for plot
*

Extracts the time series, consisting of the average of the vectors
NE1 - CZ3 and NE1 - CE3 == V1(t) and of the average of CD1 - CH2 and
CD1 - CZ3 == V2(t).
Makes V1(t) and V2(t) unit vectors.
Displays data regarding both time series
Calculates P2(t) = (3< (V1(0)*V2(t))**2 > - 1) / 2
Makes a binary plot file for PLT2
```

#### CHARMM .doc Homepage

Information and HTML Formatting Courtesy of:

NIH/DCRT/Laboratory for Structural Biology
FDA/CBER/OVRR Biophysics Laboratory
Modified, updated and generalized by C.L. Brooks, III
The Scripps Research Institute