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. * Menu: * 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

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 ] END ! return to main command parser

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.

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. ----------------------------------------------------------------------------- READ unit-number [CARD] [edit-spec] 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.

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.

Editing a time series The EDIT command allows the time series specifications to be modified directly. WARNING:: This command gives the user direct access to most time 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).

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

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>]

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]

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 TRAJ ... (or READ ...) 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 category recognition codes for analog input patterns. Appl. Optics 26:4919- 4930. 2) Pao, Y.-H. 1989. Adaptive Pattern Recognition and Neural Networks, Addison- 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)

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. 2) The READ command 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.

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

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Modified, updated and generalized by C.L. Brooks, III

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