Module functionals

Introduction

A functional is a function with parameters, defined as f(p;x), where p are the parameters, and x the arguments. Methods are available to calculate the value of a function for a series of argument values for the given set of parameters, and for the automatic calculation of the derivatives with respect to the parameters.

The created functionals can be used for fitiing as provided by casacore.fitting.

A functional has a mask associated with it, to indicate if certain parameters have to be solved for. See masks for details.

To access the functionals module import casacore.functionals.

Functionals are created in a variety of ways, in general by specifying the name of the functional, together with some necessary information like e.g. the order of a polynomial, or the code needed to compile your privately d efined function. Parameters can be set at creation time or later:

>>> from casacore.functionals import functional, gaussian1d
>>> a = gaussian1d()            # creates a 1D Gaussian, default arguments
>>> b = functional('gaussian1') # creates the same one
>>> print a.f(1)                # the value at x=1
[0.062500000000000028]
>>> print a(1)
[0.062500000000000028]
>>> print a.fdf([0,0.5]);                  # value and derivatives
>>> print a([0, 0.5], derivative=True)

In some cases an order can be specified as well (e.g. for polynomials):

>>> from casacore.functionals import functional, poly
>>> a = poly(3)               # creates a 3rd order polynomial
>>> print a
{'ndim': 1, 'masks': array([ True,  True,  True,  True], dtype=bool),
 'params': array([ 1.,  1.,  1.,  1.]), 'npar': 4, 'type': 5, 'order': 3}

An extremely valuable aspect of the Functionals module is the ability to create a functional from a compiled string specifying an arbitrary function. For example, let us make our own polynomial 1 + 2*x + 3*x2 and evaluate it at a few abcissa locations:

>>> from casacore.functionals import compiled
>>> a = compiled('p0 + p1*x + p2*x*x', [1,2,3])   # Define
>>> a([0,10,20])                                  # Evaluate at x=[0,10,20]
[1.0, 321.0, 1241.0]

The functions created can also be used to specify the function to be fitted in a least squares fit.

Class functionals.functional

class casacore.functionals.functional(name=None, order=-1, params=None, mode=None, dtype=0)
f(x)

Calculate the value of the functional for the specified arguments (taking any specified mask into account).

Parameters:x – the value(s) to evaluate at

Example:

a = gaussian1d()
a.f(0.0)
# equivalent
a(0.0)
fdf(x)
Calculate the value of the functional for the specified arguments,
and the derivatives with respect to the parameters (taking any specified mask into account).
Parameters:x –

the value(s) to evaluate at

Example:

a = gaussian1d()
a.fdf(0.0)
# equivalent
a(0.0, derivative=True)
ndim()

Return the dimensionality of the functional

Retval:int
npar()

Return the numper of parameters of the functional :retval: int

Equivalent:

>>> p = poly(2)
>>> print p.npar()
3
>>> print len(p)
3

Class functionals.gaussian1d

class casacore.functionals.gaussian1d(params=None, dtype=0)

Create a 1-dimensional Gaussian with the specified height, width and center. :param params: the [height, center, width] as a list

f(x)

Calculate the value of the functional for the specified arguments (taking any specified mask into account).

Parameters:x – the value(s) to evaluate at

Example:

a = gaussian1d()
a.f(0.0)
# equivalent
a(0.0)
fdf(x)
Calculate the value of the functional for the specified arguments,
and the derivatives with respect to the parameters (taking any specified mask into account).
Parameters:x –

the value(s) to evaluate at

Example:

a = gaussian1d()
a.fdf(0.0)
# equivalent
a(0.0, derivative=True)
ndim()

Return the dimensionality of the functional

Retval:int
npar()

Return the numper of parameters of the functional :retval: int

Equivalent:

>>> p = poly(2)
>>> print p.npar()
3
>>> print len(p)
3

Class functionals.gaussian2d

class casacore.functionals.gaussian2d(params=None, dtype=0)

Create a two-dimensional gaussian.

Parameters:
  • params – list [amplitude, centers, major width, ratio, angle] of Gaussian default is [1, 0, 0, 1, 1, 0]
  • dtype – The data type. One of ‘real’ or 0, or ‘complex’ or 1
f(x)

Calculate the value of the functional for the specified arguments (taking any specified mask into account).

Parameters:x – the value(s) to evaluate at

Example:

a = gaussian1d()
a.f(0.0)
# equivalent
a(0.0)
fdf(x)
Calculate the value of the functional for the specified arguments,
and the derivatives with respect to the parameters (taking any specified mask into account).
Parameters:x –

the value(s) to evaluate at

Example:

a = gaussian1d()
a.fdf(0.0)
# equivalent
a(0.0, derivative=True)
ndim()

Return the dimensionality of the functional

Retval:int
npar()

Return the numper of parameters of the functional :retval: int

Equivalent:

>>> p = poly(2)
>>> print p.npar()
3
>>> print len(p)
3

Class functionals.poly

class casacore.functionals.poly(order, params=None, dtype=0)

Create a polynomial of specified degree. The default parameters are all 1. (Note that using the generic functional function the parameters are all set to 0).

Parameters:
  • order – the order of the polynomial (number of parameters -1)
  • params – the values of the parameters as a list.
  • dtype – the optional data type. Default is float, but will be auto-detected from params. Can be set to ‘complex’.
f(x)

Calculate the value of the functional for the specified arguments (taking any specified mask into account).

Parameters:x – the value(s) to evaluate at

Example:

a = gaussian1d()
a.f(0.0)
# equivalent
a(0.0)
fdf(x)
Calculate the value of the functional for the specified arguments,
and the derivatives with respect to the parameters (taking any specified mask into account).
Parameters:x –

the value(s) to evaluate at

Example:

a = gaussian1d()
a.fdf(0.0)
# equivalent
a(0.0, derivative=True)
ndim()

Return the dimensionality of the functional

Retval:int
npar()

Return the numper of parameters of the functional :retval: int

Equivalent:

>>> p = poly(2)
>>> print p.npar()
3
>>> print len(p)
3

Class functionals.oddpoly

class casacore.functionals.oddpoly(order, params=None, dtype=0)

Create an odd polynomial of specified degree.

Parameters:
  • order – the order of the polynomial
  • params – the values of the parameters as a list.
  • dtype – the optional data type. Default is float, but will be auto-detected from params. Can be set to ‘complex’.
f(x)

Calculate the value of the functional for the specified arguments (taking any specified mask into account).

Parameters:x – the value(s) to evaluate at

Example:

a = gaussian1d()
a.f(0.0)
# equivalent
a(0.0)
fdf(x)
Calculate the value of the functional for the specified arguments,
and the derivatives with respect to the parameters (taking any specified mask into account).
Parameters:x –

the value(s) to evaluate at

Example:

a = gaussian1d()
a.fdf(0.0)
# equivalent
a(0.0, derivative=True)
ndim()

Return the dimensionality of the functional

Retval:int
npar()

Return the numper of parameters of the functional :retval: int

Equivalent:

>>> p = poly(2)
>>> print p.npar()
3
>>> print len(p)
3

Class functionals.evenpoly

class casacore.functionals.evenpoly(order, params=None, dtype=0)

Create an even polynomial of specified degree.

Parameters:
  • order – the order of the polynomial
  • params – the values of the parameters as a list.
  • dtype – the optional data type. Default is float, but will be auto-detected from params. Can be set to ‘complex’.
f(x)

Calculate the value of the functional for the specified arguments (taking any specified mask into account).

Parameters:x – the value(s) to evaluate at

Example:

a = gaussian1d()
a.f(0.0)
# equivalent
a(0.0)
fdf(x)
Calculate the value of the functional for the specified arguments,
and the derivatives with respect to the parameters (taking any specified mask into account).
Parameters:x –

the value(s) to evaluate at

Example:

a = gaussian1d()
a.fdf(0.0)
# equivalent
a(0.0, derivative=True)
ndim()

Return the dimensionality of the functional

Retval:int
npar()

Return the numper of parameters of the functional :retval: int

Equivalent:

>>> p = poly(2)
>>> print p.npar()
3
>>> print len(p)
3

Class functionals.chebyshev

class casacore.functionals.chebyshev(order, params=None, xmin=-1.0, xmax=1.0, ooimode='constant', dtype=0)
f(x)

Calculate the value of the functional for the specified arguments (taking any specified mask into account).

Parameters:x – the value(s) to evaluate at

Example:

a = gaussian1d()
a.f(0.0)
# equivalent
a(0.0)
fdf(x)
Calculate the value of the functional for the specified arguments,
and the derivatives with respect to the parameters (taking any specified mask into account).
Parameters:x –

the value(s) to evaluate at

Example:

a = gaussian1d()
a.fdf(0.0)
# equivalent
a(0.0, derivative=True)
ndim()

Return the dimensionality of the functional

Retval:int
npar()

Return the numper of parameters of the functional :retval: int

Equivalent:

>>> p = poly(2)
>>> print p.npar()
3
>>> print len(p)
3

Class functionals.compound

class casacore.functionals.compound(dtype=0)
f(x)

Calculate the value of the functional for the specified arguments (taking any specified mask into account).

Parameters:x – the value(s) to evaluate at

Example:

a = gaussian1d()
a.f(0.0)
# equivalent
a(0.0)
fdf(x)
Calculate the value of the functional for the specified arguments,
and the derivatives with respect to the parameters (taking any specified mask into account).
Parameters:x –

the value(s) to evaluate at

Example:

a = gaussian1d()
a.fdf(0.0)
# equivalent
a(0.0, derivative=True)
ndim()

Return the dimensionality of the functional

Retval:int
npar()

Return the numper of parameters of the functional :retval: int

Equivalent:

>>> p = poly(2)
>>> print p.npar()
3
>>> print len(p)
3

Class functionals.combi

class casacore.functionals.combi(dtype=0)
f(x)

Calculate the value of the functional for the specified arguments (taking any specified mask into account).

Parameters:x – the value(s) to evaluate at

Example:

a = gaussian1d()
a.f(0.0)
# equivalent
a(0.0)
fdf(x)
Calculate the value of the functional for the specified arguments,
and the derivatives with respect to the parameters (taking any specified mask into account).
Parameters:x –

the value(s) to evaluate at

Example:

a = gaussian1d()
a.fdf(0.0)
# equivalent
a(0.0, derivative=True)
ndim()

Return the dimensionality of the functional

Retval:int
npar()

Return the numper of parameters of the functional :retval: int

Equivalent:

>>> p = poly(2)
>>> print p.npar()
3
>>> print len(p)
3

Class functionals.compiled

class casacore.functionals.compiled(code='', params=None, dtype=0)

Create a function based on the programable string. The string should be a single expression, which can use the standard operators and functions and parentheses, having a single value as a result. The parameters of the function can be addressed with the p variable. This variable can be indexed in two ways. The first way is using the standard a lgebraic way, where the parameters are: p (or p0), p1, p2, ... . The second way is by indexing, where the parameters are addressed as: p[0], p[1], ... . The arguments are accessed in the same way, but using the variable name x. The compilation determines the number of dimensions and parameters of the produced function.

Operators are the standard operators (including comparisons, which produce a zero or one result; and conditional expression).

In addition to the standard expected functions, there is an atan with either one or two arguments (although atan2 exists as well), and pi and ee with no or one argument. The functional created behaves as all other functionals, and hence can be used in combinations.

Examples:

>>> from casacore.functionals import compiled
>>> import math
>>> a = compiled('sin(pi(0.5) ) +pi');  # an example
>>> print a(0)
array([ 4.1415926535897931])
>>> b = compiled('p*exp(-(x/p[2])^2)')
>>> print b.get_parameters()
[0.0, 0.0]
>>> b.set_parameters([10, 1]) # change to height 10 and  halfwidth 1
>>> print b([-1,-0.5,0,.5,1])
array([ 3.6787944117144233,
 7.788007830714049,
 10.0,
 7.788007830714049,
 3.6787944117144233])
# the next one is sync(x), catering for x=0
# using the fact that comparisons deliver values. Note
# the extensive calculation to make sure no divison by 0
>>> synca = compiled('( (x==0) * 1)+( (x!=0) * sin(x+(x==0)*1)/(x+(x==0)*1) )')
>>> print synca([-1,0,1])
[0.841471, 10., 0.841471]
>>> print math.sin(1)/1
0.841471
# using conditional expressions:
print compiled('x==0 ? 1 : sin(x)/x')([-1,0,1])
[0.841471, 1.0, 0.841471]
f(x)

Calculate the value of the functional for the specified arguments (taking any specified mask into account).

Parameters:x – the value(s) to evaluate at

Example:

a = gaussian1d()
a.f(0.0)
# equivalent
a(0.0)
fdf(x)
Calculate the value of the functional for the specified arguments,
and the derivatives with respect to the parameters (taking any specified mask into account).
Parameters:x –

the value(s) to evaluate at

Example:

a = gaussian1d()
a.fdf(0.0)
# equivalent
a(0.0, derivative=True)
ndim()

Return the dimensionality of the functional

Retval:int
npar()

Return the numper of parameters of the functional :retval: int

Equivalent:

>>> p = poly(2)
>>> print p.npar()
3
>>> print len(p)
3