Generate and return the natural logarithm of the normalization factor of the PDF of the (Truncated) Power distribution for an input parameter set \((\alpha, x_\mathrm{min}, x_\mathrm{max})\).
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Generate and return the natural logarithm of the normalization factor of the PDF of the (Truncated) Power distribution for an input parameter set \((\alpha, x_\mathrm{min}, x_\mathrm{max})\).
The normalization factor \(\eta(\alpha, x_\mathrm{min}, x_\mathrm{max})\) of the PDF of the Power distribution is,
\begin{equation}
\large
\eta(\alpha, x_\mathrm{min}, x_\mathrm{max}) = \frac{\alpha}{x_\mathrm{max}^\alpha - x_\mathrm{min}^\alpha} ~,~ 0 < \alpha < +\infty
\end{equation}
When \(x_\mathrm{min} \rightarrow 0\), the Truncated Power distribution simplifies to the Power Distribution.
The equation for \(\eta(\cdot)\) for the Power distribution simplifies to,
\begin{equation}
\large
\lim_{x_\mathrm{min} \rightarrow 0} \eta(\alpha, x_\mathrm{min}, x_\mathrm{max}) = \alpha x_\mathrm{max}^{-\alpha} ~,~ 0 < \alpha < +\infty.
\end{equation}
See the documentation of pm_distPower for more information on the (Truncated) Power distribution.
The primary use of this interface is to compute the normalization factor of the PDF of the (Truncated) Power distribution for a fixed set of parameters and use it in subsequent repeated calculations of the properties of the (Truncated) Power distribution to improve the runtime performance by eliminating redundant calculations.
- Parameters
-
[in] | alpha | : 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 shape parameter of the distribution.
|
[in] | logMinX | : The input scalar or array of the same shape as other array-like arguments, of the same type and kind as alpha , containing the natural logarithm of the first scale parameter of the distribution, representing the minimum of the support of the distribution.
(optional, default = \(-\infty\)) |
[in] | logMaxX | : The input scalar or array of the same shape as other array-like arguments, of the same type and kind as alpha , containing the natural logarithm of the second scale parameter of the distribution, representing the maximum of the support of the distribution.
|
- Returns
logPDFNF
: 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 alpha
, containing the natural logarithm of the normalization factor of the PDF of (Truncated) Power distribution.
Possible calling interfaces ⛓
!
Generate and return the natural logarithm of the normalization factor of the PDF of the (Truncated) P...
This module contains classes and procedures for computing various statistical quantities related to t...
- Warning
- The condition
0 < alpha
must hold for the corresponding input arguments.
The condition logMinX < logMaxX
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.
- See also
- getPowerLogPDF
setPowerLogPDF
Example usage ⛓
13 type(display_type) :: disp
17 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
18 call disp%show(
"! Compute the natural logarithm of the normalization factor of the (Truncated) Power distribution PDF.")
19 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
23 call disp%show(
"logPDFNF(1) = getPowerLogPDFNF(alpha = 1., logMaxX = 1.)")
30 call disp%show(
"logPDFNF(1:3) = getPowerLogPDFNF(alpha = 2., logMaxX = log([1., 2., 3.]))")
37 call disp%show(
"logPDFNF(1:3) = getPowerLogPDFNF(alpha = [+2., +3., +4.], logMaxX = log([1., 2., 3.]))")
38 logPDFNF(
1:
3)
= getPowerLogPDFNF(alpha
= [
+2.,
+3.,
+4.], logMaxX
= log([
1.,
2.,
3.]))
44 call disp%show(
"logPDFNF(1:3) = getPowerLogPDFNF(alpha = [+2., +3., +4.], logMinX = log([1., 2., 3.]), logMaxX = log(20.))")
45 logPDFNF(
1:
3)
= getPowerLogPDFNF(alpha
= [
+2.,
+3.,
+4.], logMinX
= log([
1.,
2.,
3.]), logMaxX
= log(
20.))
55 integer :: fileUnit, i
56 real,
allocatable :: alpha(:), logPDFNF(:)
59 open(newunit
= fileUnit, file
= "getPowerLogPDFNF.RK.txt")
60 write(fileUnit,
"(2(g0,:,' '))") (alpha(i),
exp(logPDFNF(i)), i
= 1,
size(logPDFNF))
Generate count evenly-logarithmically-spaced points over the interval [base**logx1,...
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 ⛓
14+0.693147182,
-0.693147182,
-1.50407743
17logPDFNF(
1:
3)
= getPowerLogPDFNF(alpha
= [
+2.,
+3.,
+4.], logMaxX
= log([
1.,
2.,
3.]))
19+0.693147182,
-0.980829239,
-3.00815487
22logPDFNF(
1:
3)
= getPowerLogPDFNF(alpha
= [
+2.,
+3.,
+4.], logMinX
= log([
1.,
2.,
3.]), logMaxX
= log(
20.))
24-5.29581451,
-7.88758421,
-10.5961285
Postprocessing of the example output ⛓
3import matplotlib.pyplot
as plt
15xlab = {
"CK" :
r"$\alpha$ ( real/imaginary )"
16 ,
"IK" :
r"$\alpha$ ( integer-valued )"
17 ,
"RK" :
r"$\alpha$ ( real-valued )"
20for kind
in [
"IK",
"CK",
"RK"]:
22 pattern =
"*." + kind +
".txt"
23 fileList = glob.glob(pattern)
24 if len(fileList) == 1:
26 df = pd.read_csv(fileList[0], delimiter =
" ")
28 fig = plt.figure(figsize = 1.25 * np.array([6.4, 4.8]), dpi = 200)
32 plt.plot( df.values[:, 0]
37 plt.plot( df.values[:, 1]
43 plt.plot( df.values[:, 0]
49 plt.xticks(fontsize = fontsize - 2)
50 plt.yticks(fontsize = fontsize - 2)
51 ax.set_xlabel(xlab[kind], fontsize = fontsize)
52 ax.set_ylabel(
"Normalization Factor of the PDF", fontsize = fontsize)
56 plt.grid(visible =
True, which =
"both", axis =
"both", color =
"0.85", linestyle =
"-")
57 ax.tick_params(axis =
"y", which =
"minor")
58 ax.tick_params(axis =
"x", which =
"minor")
61 plt.savefig(fileList[0].replace(
".txt",
".png"))
63 elif len(fileList) > 1:
65 sys.exit(
"Ambiguous file list exists.")
Visualization of the example output ⛓
- Test:
- test_pm_distPower
- Todo:
- Very Low Priority: This generic interface can be extended to
complex
arguments.
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 269 of file pm_distPower.F90.