Generate and return the natural logarithm of the normalization factors of the components of the Probability Density Function (PDF) of the (Truncated) PiwiPoweto distribution for the input parameter vectors \((\alpha, x_\mathrm{lim})\).
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Generate and return the natural logarithm of the normalization factors of the components of the Probability Density Function (PDF) of the (Truncated) PiwiPoweto distribution for the input parameter vectors \((\alpha, x_\mathrm{lim})\).
See the documentation of pm_distPiwiPoweto for the definition of the normalization factors.
The primary use of this interface is to compute the normalization factors of the (Truncated) PiwiPoweto distribution for a fixed set of parameters and use it in subsequent repeated calculations of the (Truncated) PiwiPoweto PDF to improve the runtime performance by eliminating redundant calculations.
- Parameters
-
[in] | alpha | : The input vector of type real of kind any supported by the processor (e.g., RK, RK32, RK64, or RK128), of the same size n as the number of the power-law components of the distribution, containing the shape parameter(s) of the distribution (i.e., the exponents (plus one) of the power-law components of the distribution).
|
[in] | logLimX | : The input vector of the same type and kind as alpha , of size size(alpha) + 1 containing the natural logarithm of the scale parameters (i.e., the break points, or the limits) of the n power-law components of the distribution.
The scale parameter(s) must be in ascending order, such that logLimX(1) <= x <= logLimX(size(logLimX)) .
Setting logLimX(1) <= -log(huge(logLimX)) effectively implies a left-opened semi-infinite support for the distribution.
Setting logLimX(size(logLimX)) >= log(huge(logLimX)) effectively implies a right-opened semi-infinite support for the distribution.
|
[out] | cumSumArea | : The output vector of the same type, kind, and size as logLimX , each element of which corresponds to the cumulative area underneath the distribution from the minimum of the support exp(logLimX(1)) to the corresponding element of exp(logLimX) .
By definition, the conditions cumSumArea(1) == 0. and cumSumArea(size(cumSumArea)) == 1. and isAscending(cumSumArea) hold.
This output vector is a side-product of the computation of the normalization factors.
It is also required for random number generation from the (Truncated) PiwiPoweto distribution.
Precomputing and supplying this vector to the random number generator routines significantly improves the runtime performance.
(optional. If missing, it will be computed implicitly within the algorithm and discarded upon return.) |
- Returns
logPDFNF
: The output vector of the same type, kind, and size as the input argument alpha
, containing the natural logarithm of the normalization factors of the power-law components of the (Truncated) PiwiPoweto distribution.
Possible calling interfaces ⛓
!
Generate and return the natural logarithm of the normalization factors of the components of the Proba...
This module contains classes and procedures for computing various statistical quantities related to t...
- Warning
- The condition
size(alpha) > 0
must hold for the corresponding input arguments.
The condition size(logLimX) == size(alpha) + 1
must hold for the corresponding input arguments.
The condition size(cumSumArea) == size(alpha) + 1
must hold for the corresponding input arguments.
The condition size(logPDFNF) == size(alpha)
must hold for the corresponding input arguments.
The condition alpha(1) > 0 .or. logLimX(1) > 0
must hold for the corresponding input arguments.
The condition alpha(size(alpha)) < 0 .or. logLimX(size(logLimX)) < log(huge(logLimX))
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. The procedures under this generic interface are always impure
when the output argument cumSumArea
is present.
- See also
- getPiwiPowetoLogPDF
setPiwiPowetoLogPDF
Example usage ⛓
13 integer(IK) ,
parameter :: NP
= 4_IK
14 real ,
allocatable :: alpha(:), logPDFNF(:), logLimX(:), cumSumArea(:)
16 type(display_type) :: disp
20 logLimX
= [
getLinSpace(
log(
0.001),
log(
20.), NP),
log(
huge(
0.))]
21 allocate(cumSumArea,
mold = logLimX)
22 allocate(logPDFNF,
mold = alpha)
25 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
26 call disp%show(
"! Compute the natural logarithm of the normalization factor of the PiwiPoweto distribution.")
27 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
35 call disp%show(
"logPDFNF = getPiwiPowetoLogPDFNF(alpha, logLimX)")
46 call disp%show(
"logPDFNF = getPiwiPowetoLogPDFNF(alpha, logLimX, cumSumArea)")
55 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
56 call disp%show(
"! Compute the natural logarithm of the normalization factor of the Truncated PiwiPoweto distribution.")
57 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
65 call disp%show(
"logPDFNF = getPiwiPowetoLogPDFNF(alpha(1 : NP - 1), logLimX(1 : NP), cumSumArea(1 : NP)) ! `logLimX(NP)` is serves as the upper bound of the support of the distribution.")
78 integer :: fileUnit, i
80 logLimX
= getLinSpace(
log(
0.001),
log(
20.), count
= size(alpha,
1,
IK)
+ 1_IK)
82 deallocate(cumSumArea);
allocate(cumSumArea,
mold = logLimX)
84 open(newunit
= fileUnit, file
= "getPiwiPowetoLogPDFNF.RK.txt")
85 write(fileUnit,
"(2(g0,:,' '))") (
exp(logLimX(i)), cumSumArea(i), i
= 1,
size(cumSumArea))
Generate count evenly spaced points over the interval [x1, x2] if x1 < x2, or [x2,...
Generate count evenly-logarithmically-spaced points over the interval [base**logx1,...
Generate and return a scalar or a contiguous array of rank 1 of length s1 of randomly uniformly distr...
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 computing various statistical quantities related to t...
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 ⛓
8+0.318665028,
-1.13193512,
-3.77399993,
-3.00000000
10-6.90775537,
-3.60659266,
-0.305429935,
+2.99573231,
+88.7228394
13+0.808620453E-1,
-5.15086126,
-5.95782709,
-8.27652359
17+0.318665028,
-1.13193512,
-3.77399993,
-3.00000000
19-6.90775537,
-3.60659266,
-0.305429935,
+2.99573231,
+88.7228394
22+0.808620453E-1,
-5.15086126,
-5.95782709,
-8.27652359
24+0.00000000,
+0.701559663,
+0.997830212,
+0.999999702,
+0.999999702
33+0.318665028,
-1.13193512,
-3.77399993
35-6.90775537,
-3.60659266,
-0.305429935,
+2.99573231
38+0.808625221E-1,
-5.15086126,
-5.95782709
40+0.00000000,
+0.701560020,
+0.997830629,
+1.00000012,
+0.999999702
Postprocessing of the example output ⛓
3import matplotlib.pyplot
as plt
15xlab = {
"CK" :
r"Scale Parameters: $x_{min}$ ( real/imaginary )"
16 ,
"IK" :
r"Scale Parameters: $x_{min}$ ( integer-valued )"
17 ,
"RK" :
r"Scale Parameters: $x_{min}$ ( 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(
"Cumulative Area Under PDF Components", 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_distPiwiPoweto
- Todo:
- 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 179 of file pm_distPiwiPoweto.F90.