Generate and return the unnormalized density function (UDF) of the Band spectral model/distribution.
See the documentation of pm_distBand for more information on the Band distribution.
The unnormalized unit-less density function of the Band model can be written as,
\begin{equation}
\large
f_{\ms{BAND}}(E | \alpha, \beta, \ebreak) =
\begin{cases}
E^\alpha \exp\left(-\frac{E}{\efold}\right) &,~ \ms{if} & 0 < \ms{lb} \leq E < \ebreak \\
\zeta E^\beta &,~ \ms{if} & \ebreak \leq E < \ms{ub} < +\infty \\
\end{cases}
\end{equation}
where,
-
\(E\) is the (presumably unitless) energy at which the distribution must be computed,
-
\(\ebreak\) is the (presumably unitless or of the same unit as \(E\)) spectral break energy,
-
\(\efold = \frac{\alpha - \beta}{\ebreak}\) is the e-folding energy, a measure of the scale of the distribution below and above which the distribution approaches power-law behavior with exponents \(\alpha\) and \(\beta\) respectively,
-
the factor \(\eta(\alpha, \beta, \ebreak)\) is a normalization constant that properly normalizes the PDF,
-
the factor \(\zeta = \ebreak^{\alpha - \beta} \exp\left(\beta - \alpha\right)\) is the coefficient of continuity of the distribution that makes the distribution continuously differentiable.
-
the constants \((\ms{lb}, \ms{ub})\) represent the lower and upper bounds of the PDF respectively.
- Parameters
-
[in] | energy | : The input scalar or array of the same shape as other array like arguments of type real of kind any supported by the processor (e.g., RK, RK32, RK64, or RK128), containing the energy at which the UDF must be computed.
|
[in] | alpha | : The input scalar or array of the same shape as other array-like arguments of the same type and kind as energy , containing the first shape parameter of the distribution.
|
[in] | beta | : The input scalar or array of the same shape as other array-like arguments of the same type and kind as energy , containing the second shape parameter of the distribution.
|
[in] | ebreak | : The input scalar or array of the same shape as other array-like arguments of the same type and kind as energy , containing the spectral break energy values.
|
[in] | zeta | : The input scalar or array of the same shape as other array-like arguments of the same type and kind as energy , containing the containing the coefficient of continuity of the distribution.
(optional. default = getBandZeta(alpha, beta, ebreak). Its presence can expedite the computations.)
This means that the output udf is computed from the lower tail of the distribution.) |
[in] | invEfold | : The input scalar or array of the same shape as other array-like arguments of the same type and kind as energy , containing the inverse of the e-folding energy of the Band model: \(\frac{1}{\efold} = \frac{\alpha - \beta}{\ebreak}\).
(optional. default = (alpha - beta) / ebreak . Its presence can expedite the computations.)
|
- Returns
udf
: The output scalar or array of the same shape as any input array-like argument, of the same type and kind as the input argument energy
, containing the distribution UDF.
-
Possible calling interfaces ⛓
udf
= getBandUDF(energy, alpha, beta, ebreak, zeta
= zeta, invEfold
= invEfold)
Generate and return the unnormalized density function (UDF) of the Band spectral model/distribution.
This module contains procedures and generic interfaces for computing the Band photon distribution wid...
- Warning
- The condition
0 < energy
must hold for the corresponding input arguments.
The condition alpha /= -2
must hold for the corresponding input arguments.
The condition 0 < ebreak
must hold for the corresponding input arguments.
The condition 0 < invEfold
must hold for the corresponding input arguments.
The condition beta < alpha
must hold for the corresponding input arguments.
The condition ebreak = (alpha - beta) * invEfold
must hold for the corresponding input arguments.
The condition zeta = getZeta(alpha, beta, ebreak)
must hold for the corresponding input arguments.
These conditions are verified only if the library is built with the preprocessor macro CHECK_ENABLED=1
.
-
The
pure
procedure(s) documented herein become impure
when the ParaMonte library is compiled with preprocessor macro CHECK_ENABLED=1
.
By default, these procedures are pure
in release
build and impure
in debug
and testing
builds.
- Note
- The normalization (and the physical units) of the input
energy
is irrelevant as long as the input values ebreak
and zeta
are computed in the same physical dimensions and with the same normalizations.
- See also
- getBandUDF
setBandUCDF
getBandZeta
getBandEpeak
getBandEbreak
Example usage ⛓
12 integer(IK),
parameter :: NP
= 999
13 real :: udf(NP), point(NP), alpha, beta, ebreak
15 type(display_type) :: disp
23 call disp%show(
"alpha = -.5; beta = -1.5; ebreak = 100.")
24 alpha
= -.
5; beta
= -1.5; ebreak
= 100.
25 call disp%show(
"udf(1) = getBandUDF(point(1), alpha, beta, ebreak)")
26 udf(
1)
= getBandUDF(point(
1), alpha, beta, ebreak)
34 call disp%show(
"alpha = -.5; beta = -1.5; ebreak = 100.")
35 alpha
= -.
5; beta
= -1.5; ebreak
= 100.
36 call disp%show(
"udf(1) = getBandUDF(point(1), alpha, beta, ebreak, zeta = getBandZeta(alpha, beta, ebreak), invEfold = (alpha - beta) / ebreak)")
37 udf(
1)
= getBandUDF(point(
1), alpha, beta, ebreak, zeta
= getBandZeta(alpha, beta, ebreak), invEfold
= (alpha
- beta)
/ ebreak)
47 integer(IK) :: fileUnit, i
48 open(newunit
= fileUnit, file
= "getBandUDF.RK.txt")
50 write(fileUnit,
"(*(g0,:,' '))") point(i),
getBandUDF(point(i), [.
5,
1.5,
-.
5,
+1.5],
-[.
5,
1.0,
2.,
3.], [.
5,
1.0,
2.0,
5.])
Generate count evenly spaced points over the interval [x1, x2] if x1 < x2, or [x2,...
Return the linSpace output argument with size(linSpace) elements of evenly-spaced values over the int...
Generate and return the coefficient of continuity of the Band spectral model/distribution from the Ba...
This is a generic method of the derived type display_type with pass attribute.
This is a generic method of the derived type display_type with pass attribute.
This module contains procedures and generic interfaces for generating arrays with linear or logarithm...
This module contains classes and procedures for input/output (IO) or generic display operations on st...
type(display_type) disp
This is a scalar module variable an object of type display_type for general display.
This module defines the relevant Fortran kind type-parameters frequently used in the ParaMonte librar...
integer, parameter IK
The default integer kind in the ParaMonte library: int32 in Fortran, c_int32_t in C-Fortran Interoper...
integer, parameter SK
The default character kind in the ParaMonte library: kind("a") in Fortran, c_char in C-Fortran Intero...
Generate and return an object of type display_type.
Example Unix compile command via Intel ifort
compiler ⛓
3ifort -fpp -standard-semantics -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe
Example Windows Batch compile command via Intel ifort
compiler ⛓
2set PATH=..\..\..\lib;%PATH%
3ifort /fpp /standard-semantics /O3 /I:..\..\..\include main.F90 ..\..\..\lib\libparamonte*.lib /exe:main.exe
Example Unix / MinGW compile command via GNU gfortran
compiler ⛓
3gfortran -cpp -ffree-line-length-none -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe
Example output ⛓
4alpha
= -.
5; beta
= -1.5; ebreak
= 100.
5udf(
1)
= getBandUDF(point(
1), alpha, beta, ebreak)
12alpha
= -.
5; beta
= -1.5; ebreak
= 100.
13udf(
1)
= getBandUDF(point(
1), alpha, beta, ebreak, zeta
= getBandZeta(alpha, beta, ebreak), invEfold
= (alpha
- beta)
/ ebreak)
Postprocessing of the example output ⛓
3import matplotlib.pyplot
as plt
16xlab = {
"CK" :
"X ( real/imaginary components )"
17 ,
"IK" :
"X ( integer-valued )"
18 ,
"RK" :
"X ( real-valued )"
20label = [
r"$\alpha, \beta = +0.5, -0.5, x_b = 0.5$"
21 ,
r"$\alpha, \beta = +1.5, -1.0, x_b = 1.0$"
22 ,
r"$\alpha, \beta = -0.5, -2.0, x_b = 2.0$"
23 ,
r"$\alpha, \beta = +1.5, -3.0, x_b = 5.0$"
26for kind
in [
"IK",
"CK",
"RK"]:
28 pattern =
"*." + kind +
".txt"
29 fileList = glob.glob(pattern)
30 if len(fileList) == 1:
32 df = pd.read_csv(fileList[0], delimiter =
" ")
34 fig = plt.figure(figsize = 1.25 * np.array([6.4, 4.8]), dpi = 200)
38 plt.plot( df.values[:, 0]
41 , linewidth = linewidth
44 plt.plot( df.values[:, 1]
47 , linewidth = linewidth
51 plt.plot( df.values[:, 0]
54 , linewidth = linewidth
61 plt.xticks(fontsize = fontsize - 2)
62 plt.yticks(fontsize = fontsize - 2)
65 ax.set_xlabel(xlab[kind], fontsize = 17)
66 ax.set_ylabel(
"Unnormalized Density Function (UDF)", fontsize = 17)
68 plt.grid(visible =
True, which =
"both", axis =
"both", color =
"0.85", linestyle =
"-")
69 ax.tick_params(axis =
"y", which =
"minor")
70 ax.tick_params(axis =
"x", which =
"minor")
73 plt.savefig(fileList[0].replace(
".txt",
".png"))
75 elif len(fileList) > 1:
77 sys.exit(
"Ambiguous file list exists.")
Visualization of the example output ⛓
- Test:
- test_pm_distBand
Final Remarks ⛓
If you believe this algorithm or its documentation can be improved, we appreciate your contribution and help to edit this page's documentation and source file on GitHub.
For details on the naming abbreviations, see this page.
For details on the naming conventions, see this page.
This software is distributed under the MIT license with additional terms outlined below.
-
If you use any parts or concepts from this library to any extent, please acknowledge the usage by citing the relevant publications of the ParaMonte library.
-
If you regenerate any parts/ideas from this library in a programming environment other than those currently supported by this ParaMonte library (i.e., other than C, C++, Fortran, MATLAB, Python, R), please also ask the end users to cite this original ParaMonte library.
This software is available to the public under a highly permissive license.
Help us justify its continued development and maintenance by acknowledging its benefit to society, distributing it, and contributing to it.
- Copyright
- Computational Data Science Lab
- Author:
- Amir Shahmoradi, Oct 16, 2009, 11:14 AM, Michigan
Definition at line 656 of file pm_distBand.F90.