MOdel

MOdel ?

MOdel @filename

MOdel $

For each parameter required by the MOdel command, you will be prompted for four numbers - VAL, SIG, PLO, and PHI - as described below. For each parameter, you should enter an initial value for VAL; but you can usually default on the other three numbers.

VAL: This is the actual value of the parameter. Although CURFIT will often find the the best set of parameters to model the data, it never hurts to start it with parameters near the expected best fit.

SIG: Any value of SIG greater than or equal to 0 will not affect the outcome of Fit. After you fit the model, SIG will contain the one-sigma curvature errors. This number is used by the Uncertainty command to start a formal error determination. If the Uncertainty command fails to converge because the original error estimate is wrong, sometimes you can help the convergence by adjusting SIG to be a better estimate before using Uncertainty. If you set SIG=-1, then the parameter is frozen such that CURFIT is not allowed to change the parameter value while fitting. If you set SIG=-IPAR, the next number (PLO) will default to 1, such that the current parameter value is forced to equal the value of parameter IPAR. (Note: IPAR can not equal 1 or the current parameter number.) If you place a number (N) after SIG, this will force the current parameter to be N times the specified parameter. (N defaults to 1.0.)

Example:

PLT> MOdel GAUS GAUS
  1, GC: VAL( 1.00), SIG( 0.00 ), PLO( 0.00), PHI( 0.00)?
,-4,2
  2, GW: VAL( 1.00), SIG( 0.00 ), PLO( 0.00), PHI( 0.00)?

  3, GN: VAL( 1.00), SIG( 0.00 ), PLO( 0.00), PHI( 0.00)?

  4, GC: VAL( 1.00), SIG( 0.00 ), PLO( 0.00), PHI( 0.00)?
3.
  5, GW: VAL( 1.00), SIG( 0.00 ), PLO( 0.00), PHI( 0.00)?

  6, GN: VAL( 1.00), SIG( 0.00 ), PLO( 0.00), PHI( 0.00)?

defines a model consisting of two gaussians, with the x values of the centers differing by a factor of 2. Although you did not enter a value for parameter 1, it will be set to the value of 6 (2 times value of parameter 4). This relation will be maintained throughout a fit.

PLO, PHI: If SIG greater than or equal to 0 and if PLO<PHI, the parameter value is constrained to lie in the range PLO to PHI. Note: If PLO and PHI are both the same (say equal to zero), then the parameter will not be constrained in any way. In general, if you have difficulty fitting some data, the best thing to do is to freeze some parameters near to their expected values and then fit the reduced parameter set. When a good fit has been found with the reduced set, thaw some of the parameters and refit. If this method does not work, then you may be forced to use PLO and PHI to limit certain parameters to a meaningful range.

QDP/PLT currently supplies the model components defined below, which may be combined into multi-component models:

MOdel CONS

MOdel LINR

MOdel QUAD

MOdel CUBI

MOdel X4

MOdel X5

MOdel POWR

MOdel SIN

MOdel GAUS

MOdel EXP

MOdel AEXP

MOdel BURS

MOdel SBUR

MOdel PEAR

MOdel WIND

MOdel KING

MOdel LORE

MOdel SPLN #

For unconstrained y values, the natural spline condition is imposed, which sets y''= 0 at the boundaries. You may not extrapolate this function outside the interval fitted.

It is possible to impose a periodic boundary condition on the spline curve. To do this, constrain the y position of the last knot to be the same as the first. When this constraint is detected, the program automatically forces the first derivatives to match at the two boundaries. For this case, you are allowed to access the function outside the interval fitted. However, the function is assumed to be periodic, with the period given by the difference in x between the first and last knots.

For example, MOdel SPLN 5 will generate a 5-knot spline (10 parameters). The spline can be added to other models; thus MOdel SPLN 5 GA would add a 5-knot spline to a gaussian. Hence, the spline would model the `background' and the gaussian, a `line'.

It is possible for the x position of two knots to lie between two adjacent data points. This results in a local c² minimum as the lower knot adjusts to fit data below it, the upper knot adjusts to fit data above it. A strong wiggle occurs between the two knots but since there are no data points there, c² is not affected. In this section, two knots very close to each other will be called a collision. If collision occurs during a fit, then convergence will be very slow.

One method to greatly reduce the number of collisions is to first fit the y locations before attempting to fit the x locations. By default, the knots are evenly spaced in the x direction and are not allowed to vary. For the first fit you should leave the x positions frozen, although you can move the knots (using Newpar) to concentrate them where the function is changing rapidly. Once a reasonable set of y positions is determined you can then thaw the x positions and re-fit. You should never thaw the end points: They determine the range over which the spline is to be evaluated.

With the above recipe, collisions can still occur. The straight-forward method to separate the knots is: Use Newpar to re-position the two knots, freeze the x locations, and then re-fit. After this the knots will sometimes stay separated when you thaw their positions and re-fit. The trick is to force the knots far enough apart so that they will not be attracted to the local minimum, but not so far apart as to grossly distort the fit.

Sometimes two knots collide when you are trying to fit the data with too few knots. This case can be easily tested for by increasing the number of knots and re-fitting.

MOdel AKIM #

Like SPLN two different boundary conditions are allowed. If the last y value is unconstrained, then the code uses `virtual' knots outside the boundaries to determine the function at the boundaries. The locations of the virtual knots mirror the location of the knots just inside the boundaries. If the y position of the last knot is constrained to match the y position of the first knot then a periodic boundary condition is imposed.

MOdel DEMO

MOdel $codfile

At the present time only one COD function can be defined in a model, although this function can be referenced more than once. If you wish to combine two COD functions, you will need to write a third function that combines the first two.

COD should be used for all simple components that cannot be expressed by adding together the built-in components. Since a COD function is interpreted, it will run slower than the user-defined component. However, since COD is highly efficient and supports many mathematical functions, it is expected that the interpreter will be good enough for most purposes. For large numbers of points (>10⁴) or models that involve reading a disk file, the user is advised to write a Fortran function using the user component.

Example:

: GAUS ! The file must contain a : followed by a dummy name
X      ! Push current value of X onto the stack
X      ! Push current value of X onto the stack
*      ! Multiply the top two numbers on the stack to get X*X
P1     ! Push the value of parameter 1 onto the stack
*      ! Multiply to get P1*X*X
NEG    ! Negate the number on the top of the stack (-P1*X*X)
EXP    ! Calculate EXP of -P1*X*X
P2     ! Push the value of parameter 2 onto the stack
*      ! Multiply to get P2*EXP(-P1*X*X)
;      ! The function must end with a ; character

: GAUS X X * P1 * NEG EXP P2 * ;