4. formel¶
Physical equations and useful formulas as quadrature of vector functions, viscosity, compressibility of water, scatteringLengthDensityCalc or sedimentationProfile. Use scipy.constants for physical constants.
Each topic is not enough for a single module, so this is a collection.
Return values are dataArrays were useful. To get only Y values use .Y
All scipy functions can be used. See http://docs.scipy.org/doc/scipy/reference/special.html.
Statistical functions http://docs.scipy.org/doc/scipy/reference/stats.html.
- Mass and scattering length of all elements in Elements are taken from :
Neutron scattering length: http://www.ncnr.nist.gov/resources/n-lengths/list.html
Units converted to amu for mass and nm for scattering length.
4.1. Functions¶
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Log like sequence between mini and maxi. |
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Normalized Gaussian function. |
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Normalized Lorentz function |
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Voigt function for peak analysis (normalized). |
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Lognormal distribution function. |
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Box function. |
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Mittag-Leffler function for real z and real a,b with 0<a, b<0. |
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Bose distribution for integer spin particles in non-condensed state (hw>0). |
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Schulz-Zimm distribution for polymeric particles/chains. |
Schulz-Zimm distribution as scipy.stats.rv_continuous object |
4.2. Quadrature¶
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Vectorized quadrature over one parameter with weights using the adaptive Simpson rule. |
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Vectorized definite integral using fixed-order Gaussian quadrature. |
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Vectorized fixed-order Gauss-Legendre quadrature in definite interval in 1,2,3 dimensions. |
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Vectorized definite integral using fixed-tolerance Gaussian quadrature. |
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Vectorized adaptive multidimensional integration (cubature) . |
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Vectorized spherical average - non-parallel |
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Parallel evaluation of spherical average of function. |
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Convolve A and B with proper tracking of the output X axis. |
4.3. Distribution of parameters¶
Experimental data might be influenced by multimodal parameters (like multiple sizes) or by one or several parameters distributed around a mean value.
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Vectorized average assuming a single parameter is distributed with width sig. |
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Vectorized average assuming multiple parameters are distributed in intervals. |
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Average function assuming one multimodal parameter like bimodal. |
4.4. Centrifugation¶
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Sedimentation coefficient of a sphere in a solvent. |
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Concentration profile of sedimenting particles in a centrifuge including bottom equilibrium distribution. |
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Faxen solution to the Lamm equation of sedimenting particles in centrifuge; no bottom part. |
4.5. NMR¶
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Rotational correlation time from T1/T2 or T1 and T2 from NMR proton relaxation measurement. |
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Calculates the T1/T2 from a given rotational correlation time tr or Drot for proton relaxation measurement. |
4.6. Material Data¶
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Scattering length density of composites and water with inorganic components for xrays and neutrons. |
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Density of water with inorganic substances (salts). |
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Viscosity of water with inorganic substances as used in biological buffers. |
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Dielectric constant of H2O and D2O buffer solutions. |
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Isothermal compressibility of H2O and D2O mixtures. |
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Overlap concentration |
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Calculates the molarity. |
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Viscosity of pure solvents. |
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Translational diffusion of a sphere. |
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Rotational diffusion of a sphere. |
4.7. other Stuff¶
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A least-recently-used cache decorator to cache expensive function evaluations. |
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Transformation cartesian coordinates [X,Y,Z] to spherical coordinates [r,phi,theta]. |
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Transformation spherical coordinates [r,phi,theta] to cartesian coordinates [x,y,z]. |
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Create a rotation matrix corresponding to rotation around vector v by a specified angle. |
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Fibonacci lattice points on a sphere with radius r (default r=1) |
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N quasi random points on sphere of radius r based on low-discrepancy sequence. |
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N quasi random points in cube of edge 1 based on low-discrepancy sequence. |
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Smooth data by convolution with window function or fft/ifft. |
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Creates image hash of an image to find duplicates or similar images in a fast way using the Hamming difference. |
4.8. Constants and Tables¶
Antisymmetric Levi-Civita symbol |
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Cross section of electron in nm |
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Elements Dictionary with: { symbol : (electron number; mass; neutron coherent scattering length, neutron incoherent scattering length, name) }; units amu for mass and nm for scattering length |
Physical equations and useful formulas as quadrature of vector functions, viscosity, compressibility of water, scatteringLengthDensityCalc or sedimentationProfile. Use scipy.constants for physical constants.
Each topic is not enough for a single module, so this is a collection.
Return values are dataArrays were useful. To get only Y values use .Y
All scipy functions can be used. See http://docs.scipy.org/doc/scipy/reference/special.html.
Statistical functions http://docs.scipy.org/doc/scipy/reference/stats.html.
- Mass and scattering length of all elements in Elements are taken from :
Neutron scattering length: http://www.ncnr.nist.gov/resources/n-lengths/list.html
Units converted to amu for mass and nm for scattering length.
-
jscatter.formel.
D0
(Rh, Temp=293.15, solvent='h2o', visc=None)¶ Translational diffusion of a sphere.
- Parameters
- Rhfloat
Hydrodynamic radius in nm.
- Tempfloat
Temperature in K.
- solventfloat
Solvent type as in viscosity; used if visc==None.
- viscfloat
Viscosity in Pas => H2O ~ 0.001 Pas =1 cPoise. If visc=None the solvent viscosity is calculated from function viscosity(solvent ,temp) with solvent eg ‘h2o’ (see viscosity).
- Returns
- float
Translational diffusion coefficient : float in nm^2/ns.
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jscatter.formel.
Drot
(Rh, Temp=293.15, solvent='h2o', visc=None)[source]¶ Rotational diffusion of a sphere.
- Parameters
- Rhfloat
Hydrodynamic radius in nm.
- Tempfloat
Temperature in K.
- solventfloat
Solvent type as in viscosity; used if visc==None.
- viscfloat
Viscosity in Pas => H2O ~ 0.001 Pa*s =1 cPoise. If visc=None the solvent viscosity is calculated from function viscosity(solvent ,temp) with solvent eg ‘h2o’.
- Returns
- float
Rotational diffusion coefficient in 1/ns.
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jscatter.formel.
DrotfromT12
(t12=None, Drot=None, F0=20000000.0, Tm=None, Ts=None, T1=None, T2=None)[source]¶ Rotational correlation time from T1/T2 or T1 and T2 from NMR proton relaxation measurement.
Allows to rescale by temperature and viscosity.
- Parameters
- t12float
T1/T2 from NMR with unit seconds
- Drotfloat
!=None means output Drot instead of rotational correlation time.
- F0float
Resonance frequency of NMR instrument. For Hydrogen F0=20 MHz => w0=F0*2*np.pi
- Tm: float
Temperature of measurement in K.
- Tsfloat
Temperature needed for Drot -> rescaled by visc(T)/T.
- T1float
NMR T1 result in s
- T2float
NMR T2 result in s to calc t12 directly remeber if the sequence has a factor of 2
- Returns
- float
Correlation time or Drot
Notes
See T1overT2
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jscatter.formel.
Dtrans
(Rh, Temp=293.15, solvent='h2o', visc=None)[source]¶ Translational diffusion of a sphere.
- Parameters
- Rhfloat
Hydrodynamic radius in nm.
- Tempfloat
Temperature in K.
- solventfloat
Solvent type as in viscosity; used if visc==None.
- viscfloat
Viscosity in Pas => H2O ~ 0.001 Pas =1 cPoise. If visc=None the solvent viscosity is calculated from function viscosity(solvent ,temp) with solvent eg ‘h2o’ (see viscosity).
- Returns
- float
Translational diffusion coefficient : float in nm^2/ns.
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jscatter.formel.
Ea
(z, a, b=1)[source]¶ Mittag-Leffler function for real z and real a,b with 0<a, b<0.
Evaluation of the Mittag-Leffler (ML) function with 1 or 2 parameters by means of the OPC algorithm [1]. The routine evaluates an approximation Et of the ML function E such that
- Parameters
- zreal array
Values
- afloat, real positive
Parameter alpha
- bfloat, real positive, default=1
Parameter beta
- Returns
- array
Notes
Mittag Leffler function defined as
The code uses code from K.Hinsen at https://github.com/khinsen/mittag-leffler which is a Python port of Matlab implementation of the generalized Mittag-Leffler function as described in [1].
References
- 1
R. Garrappa, Numerical evaluation of two and three parameter Mittag-Leffler functions, SIAM Journal of Numerical Analysis, 2015, 53(3), 1350-1369
Examples
import numpy as np import jscatter as js from scipy import special x=np.r_[-10:10:0.1] # tests np.all(js.formel.Ea(x,1,1)-np.exp(x)<1e-10) z = np.linspace(0., 2., 50) np.allclose(js.formel.Ea(np.sqrt(z), 0.5), np.exp(z)*special.erfc(-np.sqrt(z))) z = np.linspace(-2., 2., 50) np.allclose(js.formel.Ea(z**2, 2.), np.cosh(z))
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jscatter.formel.
Elements
= {'ac': (89, 227.0, 0, 0, 'Actinium'), 'ag': (47, 107.8682, 5.921999999999999e-06, 2.1483699284957805e-06, 'Silver'), 'al': (13, 26.9815386, 3.4489999999999995e-06, 2.5544769849751453e-07, 'Aluminium'), 'am': (95, 243.0, 8.3e-06, 1.5450968080927583e-06, 'Americium'), 'ar': (18, 39.948, 1.909e-06, 1.3380930871145783e-06, 'Argon'), 'as': (33, 74.9216, 6.58e-06, 6.90988298942671e-07, 'Arsenic'), 'at': (85, 210.0, 0, 0, 'Astatine'), 'au': (79, 196.966569, 7.63e-06, 1.8498192550829797e-06, 'Gold'), 'b': (5, 10.81, (5.299999999999999e-06-2.13e-07j), 3.6780660900548136e-06, 'Boron'), 'ba': (56, 137.327, 5.07e-06, 1.092548430592079e-06, 'Barium'), 'be': (4, 9.012182, 7.79e-06, 1.196826841204298e-07, 'Beryllium'), 'bi': (83, 208.9804, 8.531999999999999e-06, 2.585441472913205e-07, 'Bismuth'), 'br': (35, 79.904, 6.794999999999999e-06, 8.920620580763855e-07, 'Bromine'), 'c': (6, 12.011, 6.6459999999999996e-06, 8.920620580763856e-08, 'Carbon'), 'ca': (20, 40.078, 4.7e-06, 6.3078313050504e-07, 'Calcium'), 'cd': (48, 112.411, (4.87e-06-7e-07j), 5.247266445960019e-06, 'Cadmium'), 'ce': (58, 140.116, 4.839999999999999e-06, 8.920620580763856e-08, 'Cerium'), 'cl': (17, 35.45, 9.577e-06, 6.494309810853827e-06, 'Chlorine'), 'cm': (96, 247.0, 0, 0, 'Curium'), 'co': (27, 58.933195, 2.4900000000000003e-06, 6.180387232371033e-06, 'Cobalt'), 'cr': (24, 51.9961, 3.6349999999999995e-06, 3.8161076102369575e-06, 'Chromium'), 'cs': (55, 132.9054519, 5.42e-06, 1.2927207364566026e-06, 'Caesium'), 'cu': (29, 63.546, 7.718e-06, 2.0920709679710014e-06, 'Copper'), 'd': (1, 2.01410178, 6.671e-06, 4.038982751500589e-06, 'Deuterium'), 'dy': (66, 162.5, (1.6899999999999997e-05-2.7600000000000004e-07j), 2.0806283791440397e-05, 'Dysprosium'), 'er': (68, 167.259, 7.79e-06, 2.9586351363515986e-06, 'Erbium'), 'eu': (63, 151.964, (7.2199999999999995e-06-1.26e-06j), 4.460310290381928e-06, 'Europium'), 'f': (9, 18.9984032, 5.6539999999999996e-06, 7.978845608028653e-08, 'Fluorine'), 'fe': (26, 55.845, 9.45e-06, 1.784124116152771e-06, 'Iron'), 'fr': (87, 223.0, 0, 0, 'Francium'), 'ga': (31, 69.723, 7.2879999999999995e-06, 1.1283791670955127e-06, 'Gallium'), 'gd': (64, 157.25, (6.5e-06-1.382e-05j), 3.4664388359580353e-05, 'Gadolinium'), 'ge': (32, 72.63, 8.185e-06, 1.196826841204298e-06, 'Germanium'), 'h': (1, 1.008, -3.7406e-06, 2.527386721693619e-05, 'Hydrogen'), 'he': (2, 4.002602, 3.2599999999999997e-06, 0.0, 'Helium'), 'hf': (72, 178.49, 7.699999999999999e-06, 4.548641841467231e-06, 'Hafnium'), 'hg': (80, 200.592, 1.2692e-05, 7.247146419131151e-06, 'Mercury'), 'ho': (67, 164.93032, 8.01e-06, 1.6925687506432688e-06, 'Holmium'), 'i': (53, 126.90447, 5.28e-06, 1.5706373285785543e-06, 'Iodine'), 'in': (49, 114.818, (4.065e-06-5.39e-08j), 2.072964896828013e-06, 'Indium'), 'ir': (77, 192.217, 1.0599999999999998e-05, 0.0, 'Iridium'), 'k': (19, 39.0983, 3.6699999999999996e-06, 1.4658075357087598e-06, 'Potassium'), 'kr': (36, 83.798, 7.81e-06, 2.820947917738782e-07, 'Krypton'), 'la': (57, 138.90547, 8.239999999999999e-06, 2.9987087695693435e-06, 'Lanthanum'), 'li': (3, 6.94, -1.8999999999999998e-06, 2.705758189903005e-06, 'Lithium'), 'lu': (71, 174.9668, 7.21e-06, 2.360174359706574e-06, 'Lutetium'), 'mg': (12, 24.305, 5.374999999999999e-06, 7.978845608028654e-07, 'Magnesium'), 'mn': (25, 54.938045, -3.73e-06, 1.784124116152771e-06, 'Manganese'), 'mo': (42, 95.96, 6.714999999999999e-06, 5.641895835477563e-07, 'Molybdenum'), 'n': (7, 14.007, 9.359999999999998e-06, 1.9947114020071636e-06, 'Nitrogen'), 'na': (11, 22.98976928, 3.6299999999999995e-06, 3.5904805236128943e-06, 'Sodium'), 'nb': (41, 92.90638, 7.054e-06, 1.381976597885342e-07, 'Niobium'), 'nd': (60, 144.242, 7.69e-06, 8.556358677747904e-06, 'Neodymium'), 'ne': (10, 20.1797, 4.566e-06, 2.5231325220201604e-07, 'Neon'), 'ni': (28, 58.6934, 1.03e-05, 6.432750982580687e-06, 'Nickel'), 'np': (93, 237.0, 1.055e-05, 1.9947114020071636e-06, 'Neptunium'), 'o': (8, 15.999, 5.802999999999999e-06, 7.978845608028653e-08, 'Oxygen'), 'os': (76, 190.23, 1.07e-05, 1.5450968080927583e-06, 'Osmium'), 'p': (15, 30.973762, 5.13e-06, 1.9947114020071635e-07, 'Phosphorus'), 'pa': (91, 231.03588, 0, 0, 'Protactinium'), 'pb': (82, 207.2, 9.404999999999999e-06, 1.5450968080927585e-07, 'Lead'), 'pd': (46, 106.42, 5.91e-06, 8.602734945221278e-07, 'Palladium'), 'pm': (61, 145.0, 1.26e-05, 3.2163754912903434e-06, 'Promethium'), 'po': (84, 209.0, 0, 0, 'Polonium'), 'pr': (59, 140.90765, 4.58e-06, 3.454941494713355e-07, 'Praseodymium'), 'pt': (78, 195.084, 9.6e-06, 1.017107236282055e-06, 'Platinum'), 'pu': (94, 244.0, 0, 0, 'Plutonium'), 'ra': (88, 226.0, 0, 0, 'Radium'), 'rb': (37, 85.4678, 7.09e-06, 1.9947114020071636e-06, 'Rubidium'), 're': (75, 186.207, 9.199999999999998e-06, 2.6761861742291566e-06, 'Rhenium'), 'rh': (45, 102.9055, 5.88e-06, 1.5450968080927583e-06, 'Rhodium'), 'rn': (86, 222.0, 0, 0, 'Radon'), 'ru': (44, 101.07, 7.03e-06, 1.784124116152771e-06, 'Ruthenium'), 's': (16, 32.06, 2.847e-06, 2.360174359706574e-07, 'Sulfur'), 'sb': (51, 121.76, 5.57e-06, 2.360174359706574e-07, 'Antimony'), 'sc': (21, 44.955912, 1.229e-05, 5.98413420602149e-06, 'Scandium'), 'se': (34, 78.96, 7.97e-06, 1.5957691216057308e-06, 'Selenium'), 'si': (14, 28.085, 4.1491e-06, 1.7841241161527711e-07, 'Silicon'), 'sm': (62, 150.36, (8e-07-1.6499999999999999e-06j), 1.761681409986482e-05, 'Samarium'), 'sn': (50, 118.71, 6.225e-06, 4.1841419359420025e-07, 'Tin'), 'sr': (38, 87.62, 7.019999999999999e-06, 6.90988298942671e-07, 'Strontium'), 'ta': (73, 180.94788, 6.91e-06, 2.820947917738782e-07, 'Tantalum'), 'tb': (65, 158.92535, 7.38e-06, 1.7841241161527711e-07, 'Terbium'), 'tc': (43, 97.0, 6.799999999999999e-06, 1.9947114020071636e-06, 'Technetium'), 'te': (52, 127.6, 5.7999999999999995e-06, 8.462843753216344e-07, 'Tellurium'), 'th': (90, 232.03806, 1.031e-05, 0.0, 'Thorium'), 'ti': (22, 47.867, -3.438e-06, 4.77898884008814e-06, 'Titanium'), 'tl': (81, 204.38, 8.775999999999999e-06, 1.2927207364566026e-06, 'Thallium'), 'tm': (69, 168.93421, 7.07e-06, 8.920620580763855e-07, 'Thulium'), 'u': (92, 238.02891, 8.417e-06, 1.9947114020071635e-07, 'Uranium'), 'v': (23, 50.9415, -3.824e-07, 6.358093703724523e-06, 'Vanadium'), 'w': (74, 183.84, 4.86e-06, 3.601545204768291e-06, 'Tungsten'), 'xe': (54, 131.293, 4.9199999999999995e-06, 0.0, 'Xenon'), 'y': (39, 88.90585, 7.75e-06, 1.092548430592079e-06, 'Yttrium'), 'yb': (70, 173.054, 1.243e-05, 5.6418958354775635e-06, 'Ytterbium'), 'zn': (30, 65.38, 5.68e-06, 7.827812790964007e-07, 'Zinc'), 'zr': (40, 91.224, 7.16e-06, 3.989422804014327e-07, 'Zirconium')}¶ Elements Dictionary with: { symbol : (electron number; mass; neutron coherent scattering length, neutron incoherent scattering length, name) }; units amu for mass and nm for scattering length
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jscatter.formel.
Nscatlength
= {'ac': [0, 0], 'ag': [5.921999999999999e-06, 2.1483699284957805e-06], 'al': [3.4489999999999995e-06, 2.5544769849751453e-07], 'am': [8.3e-06, 1.5450968080927583e-06], 'ar': [1.909e-06, 1.3380930871145783e-06], 'as': [6.58e-06, 6.90988298942671e-07], 'at': [0, 0], 'au': [7.63e-06, 1.8498192550829797e-06], 'b': [(5.299999999999999e-06-2.13e-07j), 3.6780660900548136e-06], 'ba': [5.07e-06, 1.092548430592079e-06], 'be': [7.79e-06, 1.196826841204298e-07], 'bi': [8.531999999999999e-06, 2.585441472913205e-07], 'br': [6.794999999999999e-06, 8.920620580763855e-07], 'c': [6.6459999999999996e-06, 8.920620580763856e-08], 'ca': [4.7e-06, 6.3078313050504e-07], 'cd': [(4.87e-06-7e-07j), 5.247266445960019e-06], 'ce': [4.839999999999999e-06, 8.920620580763856e-08], 'cl': [9.577e-06, 6.494309810853827e-06], 'cm': [0, 0], 'co': [2.4900000000000003e-06, 6.180387232371033e-06], 'cr': [3.6349999999999995e-06, 3.8161076102369575e-06], 'cs': [5.42e-06, 1.2927207364566026e-06], 'cu': [7.718e-06, 2.0920709679710014e-06], 'd': [6.671e-06, 4.038982751500589e-06], 'dy': [(1.6899999999999997e-05-2.7600000000000004e-07j), 2.0806283791440397e-05], 'er': [7.79e-06, 2.9586351363515986e-06], 'eu': [(7.2199999999999995e-06-1.26e-06j), 4.460310290381928e-06], 'f': [5.6539999999999996e-06, 7.978845608028653e-08], 'fe': [9.45e-06, 1.784124116152771e-06], 'fr': [0, 0], 'ga': [7.2879999999999995e-06, 1.1283791670955127e-06], 'gd': [(6.5e-06-1.382e-05j), 3.4664388359580353e-05], 'ge': [8.185e-06, 1.196826841204298e-06], 'h': [-3.7406e-06, 2.527386721693619e-05], 'he': [3.2599999999999997e-06, 0.0], 'hf': [7.699999999999999e-06, 4.548641841467231e-06], 'hg': [1.2692e-05, 7.247146419131151e-06], 'ho': [8.01e-06, 1.6925687506432688e-06], 'i': [5.28e-06, 1.5706373285785543e-06], 'in': [(4.065e-06-5.39e-08j), 2.072964896828013e-06], 'ir': [1.0599999999999998e-05, 0.0], 'k': [3.6699999999999996e-06, 1.4658075357087598e-06], 'kr': [7.81e-06, 2.820947917738782e-07], 'la': [8.239999999999999e-06, 2.9987087695693435e-06], 'li': [-1.8999999999999998e-06, 2.705758189903005e-06], 'lu': [7.21e-06, 2.360174359706574e-06], 'mg': [5.374999999999999e-06, 7.978845608028654e-07], 'mn': [-3.73e-06, 1.784124116152771e-06], 'mo': [6.714999999999999e-06, 5.641895835477563e-07], 'n': [9.359999999999998e-06, 1.9947114020071636e-06], 'na': [3.6299999999999995e-06, 3.5904805236128943e-06], 'nb': [7.054e-06, 1.381976597885342e-07], 'nd': [7.69e-06, 8.556358677747904e-06], 'ne': [4.566e-06, 2.5231325220201604e-07], 'ni': [1.03e-05, 6.432750982580687e-06], 'np': [1.055e-05, 1.9947114020071636e-06], 'o': [5.802999999999999e-06, 7.978845608028653e-08], 'os': [1.07e-05, 1.5450968080927583e-06], 'p': [5.13e-06, 1.9947114020071635e-07], 'pa': [0, 0], 'pb': [9.404999999999999e-06, 1.5450968080927585e-07], 'pd': [5.91e-06, 8.602734945221278e-07], 'pm': [1.26e-05, 3.2163754912903434e-06], 'po': [0, 0], 'pr': [4.58e-06, 3.454941494713355e-07], 'pt': [9.6e-06, 1.017107236282055e-06], 'pu': [0, 0], 'ra': [0, 0], 'rb': [7.09e-06, 1.9947114020071636e-06], 're': [9.199999999999998e-06, 2.6761861742291566e-06], 'rh': [5.88e-06, 1.5450968080927583e-06], 'rn': [0, 0], 'ru': [7.03e-06, 1.784124116152771e-06], 's': [2.847e-06, 2.360174359706574e-07], 'sb': [5.57e-06, 2.360174359706574e-07], 'sc': [1.229e-05, 5.98413420602149e-06], 'se': [7.97e-06, 1.5957691216057308e-06], 'si': [4.1491e-06, 1.7841241161527711e-07], 'sm': [(8e-07-1.6499999999999999e-06j), 1.761681409986482e-05], 'sn': [6.225e-06, 4.1841419359420025e-07], 'sr': [7.019999999999999e-06, 6.90988298942671e-07], 't': [4.791999999999999e-06, 1.0555020614111882e-06], 'ta': [6.91e-06, 2.820947917738782e-07], 'tb': [7.38e-06, 1.7841241161527711e-07], 'tc': [6.799999999999999e-06, 1.9947114020071636e-06], 'te': [5.7999999999999995e-06, 8.462843753216344e-07], 'th': [1.031e-05, 0.0], 'ti': [-3.438e-06, 4.77898884008814e-06], 'tl': [8.775999999999999e-06, 1.2927207364566026e-06], 'tm': [7.07e-06, 8.920620580763855e-07], 'u': [8.417e-06, 1.9947114020071635e-07], 'v': [-3.824e-07, 6.358093703724523e-06], 'w': [4.86e-06, 3.601545204768291e-06], 'xe': [4.9199999999999995e-06, 0.0], 'y': [7.75e-06, 1.092548430592079e-06], 'yb': [1.243e-05, 5.6418958354775635e-06], 'zn': [5.68e-06, 7.827812790964007e-07], 'zr': [7.16e-06, 3.989422804014327e-07]}¶ Dictionary with coherent and incoherent neutron scattering length. units nm
-
jscatter.formel.
T1overT2
(tr=None, Drot=None, F0=20000000.0, T1=None, T2=None)[source]¶ Calculates the T1/T2 from a given rotational correlation time tr or Drot for proton relaxation measurement.
tr=1/(6*D_rot) with rotational diffusion D_rot and correlation time tr.
- Parameters
- trfloat
Rotational correlation time.
- Drotfloat
If given tr is calculated from Drot.
- F0float
NMR frequency e.g. F0=20e6 Hz=> w0=F0*2*np.pi is for Bruker Minispec with B0=0.47 Tesla
- T1float
NMR T1 result in s
- T2float
NMR T2 resilt in s to calc t12 directly
- Returns
- float
T1/T2
Notes
References
- 1
Intermolecular electrostatic interactions and Brownian tumbling in protein solutions. Krushelnitsky A Physical chemistry chemical physics 8, 2117-28 (2006)
- 2
The principle of nuclear magnetism A. Abragam Claredon Press, Oxford,1961
-
jscatter.formel.
boseDistribution
(w, temp)[source]¶ Bose distribution for integer spin particles in non-condensed state (hw>0).
- Parameters
- warray
Frequencies in units 1/ns
- tempfloat
Temperature in K
- Returns
- dataArray
-
jscatter.formel.
box
(x, edges=None, edgevalue=0, rtol=1e-05, atol=1e-08)[source]¶ Box function.
For equal edges and edge value > 0 the delta function is given.
- Parameters
- xarray
- edgeslist of float, float, default=[0]
Edges of the box. If only one number is given the box goes from [-edge:edge]
- edgevaluefloat, default=0
Value to use if x==edge for both edges.
- rtol,atolfloat
The relative/absolute tolerance parameter for the edge detection. See numpy.isclose.
- Returns
- dataArray
Notes
Edges may be smoothed by convolution with a Gaussian.:
import jscatter as js import numpy as np edge=2 x=np.r_[-4*edge:4*edge:200j] f=js.formel.box(x,edges=edge) res=js.formel.convolve(f,js.formel.gauss(x,0,0.2)) # p=js.mplot() p.Plot(f,li=1,le='box') p.Plot(res,li=2,le='smooth box') p.Legend() #p.savefig(js.examples.imagepath+'/box.jpg')
-
jscatter.formel.
bufferviscosity
(composition, T=293.15, show=False)[source]¶ Viscosity of water with inorganic substances as used in biological buffers.
Solvent with composition of H2O and D2O and additional components at temperature T. Ternary solutions allowed. Units are mol; 1l h2o = 55.50843 mol Based on data from ULTRASCAN3 [1] supplemented by the viscosity of H2O/D2O mixturees for conc=0.
- Parameters
- compositionlist of compositional strings
Compositional strings of chemical name as ‘float’+’name’ First float is content in Mol followed by component name as ‘h2o’ or ‘d2o’ light and heavy water were mixed with prepended fractions.
[‘1.5urea’,’0.1sodiumchloride’,’2h2o’,’1d2o’] for 1.5 M urea + 100 mM NaCl in a 2:1 mixture of h2o/d2o. By default ‘1h2o’ is assumed.
- Tfloat, default 293.15
Temperature in K
- showbool, default False
Show composition and validity range of components and result in mPas.
- Returns
- float
Viscosity in Pa*s
Notes
Viscosities of H2O/D2O mixtures mix by linear interpolation between concentrations (accuracy 0.2%) [2].
The change in viscosity due to components is added based on data from Ultrascan3 [1].
Multicomponent mixtures are composed of binary mixtures.
Glycerol%” is in unit “%weight/weight” for range=”0-32%, here the unit is changed to weight% insthead of M.
Propanol1, Propanol2 are 1-Propanol, 2-Propanol
References
- 1(1,2)
- 2
Viscosity of light and heavy water and their mixtures Kestin Imaishi Nott Nieuwoudt Sengers, Physica A: Statistical Mechanics and its Applications 134(1):38-58
- 3
Thermal Offset Viscosities of Liquid H2O, D2O, and T2O C. H. Cho, J. Urquidi, S. Singh, and G. Wilse Robinson J. Phys. Chem. B 1999, 103, 1991-1994
availible components:
h2o1 d2o1 aceticacid acetone ammoniumacetate ammoniumchloride ammoniumhydroxide ammoniumsulfate bariumchloride cadmiumchloride cadmiumsulfate calciumchloride cesiumchloride citricacid cobaltouschloride creatinine cupricsulfate edtadisodium ethanol ethyleneglycol ferricchloride formicacid fructose glucose glycerol glycerol% guanidinehydrochloride hydrochloricacid inulin lacticacid lactose lanthanumnitrate leadnitrate lithiumchloride magnesiumchloride magnesiumsulfate maltose manganoussulfate mannitol methanol nickelsulfate nitricacid oxalicacid phosphoricacid potassiumbicarbonate potassiumbiphthalate potassiumbromide potassiumcarbonate potassiumchloride potassiumchromate potassiumdichromate potassiumferricyanide potassiumferrocyanide potassiumhydroxide potassiumiodide potassiumnitrate potassiumoxalate potassiumpermanganate potassiumphosphate,dibasic potassiumphosphate,monobasic potassiumsulfate potassiumthiocyanate procainehydrochloride propanol1 propanol2 propyleneglycol silvernitrate sodiumacetate sodiumbicarbonate sodiumbromide sodiumcarbonate sodiumchloride sodiumcitrate sodiumdiatrizoate sodiumdichromate sodiumferrocyanide sodiumhydroxide sodiummolybdate sodiumphosphate,dibasic sodiumphosphate,monobasic sodiumphosphate,tribasic sodiumsulfate sodiumtartrate sodiumthiocyanate sodiumthiosulfate sodiumtungstate strontiumchloride sucrose sulfuricacid tartaricacid tetracainehydrochloride trichloroaceticacid trifluoroethanol tris(hydroxymethyl)aminomethane urea zincsulfate
-
jscatter.formel.
convolve
(A, B, mode='same', normA=False, normB=False)[source]¶ Convolve A and B with proper tracking of the output X axis.
Approximate the convolution integral as the discrete, linear convolution of two one-dimensional sequences. Missing values are linear interpolated to have matching steps. Values outside of X ranges are set to zero.
- Parameters
- A,BdataArray, ndarray
- To be convolved arrays (length N and M).
dataArray convolves Y with Y values
ndarray A[0,:] is X and A[1,:] is Y
- normA,normBbool, default False
Determines if A or B should be normalised that
.
- mode‘full’,’same’,’valid’, default ‘same’
- See example for the difference in range.
- ‘full’ Returns the convolution at each point of overlap,
with an output shape of (N+M-1,). At the end-points of the convolution, the signals do not overlap completely, and boundary effects may be seen.
- ‘same’ Returns output of length max(M, N).
Boundary effects are still visible.
‘valid’ Returns output of length M-N+1.
- Returns
- dataArray
with attributes from A
Notes
If A,B are only 1d array use np.convolve.
If attributes of B are needed later use .setattr(B,’B-‘) to prepend ‘B-‘ for B attributes.
References
- 1
Wikipedia, “Convolution”, http://en.wikipedia.org/wiki/Convolution.
Examples
Demonstrate the difference between modes
import jscatter as js;import numpy as np s1=3;s2=4;m1=50;m2=10 G1=js.formel.gauss(np.r_[0:100.1:0.1],mean=m1,sigma=s1) G2=js.formel.gauss(np.r_[-30:30.1:0.2],mean=m2,sigma=s2) p=js.grace() p.title('Convolution of Gaussians (width s mean m)') p.subtitle(r's1\S2\N+s2\S2\N=s_conv\S2\N ; m1+m2=mean_conv') p.plot(G1,le='mean 50 sigma 3') p.plot(G2,le='mean 10 sigma 4') ggf=js.formel.convolve(G1,G2,'full') p.plot(ggf,le='full') gg=js.formel.convolve(G1,G2,'same') p.plot(gg,le='same') gg=js.formel.convolve(G1,G2,'valid') p.plot(gg,le='valid') gg.fit(js.formel.gauss,{'mean':40,'sigma':1},{},{'x':'X'}) p.plot(gg.modelValues(),li=1,sy=0,le='fit m=$mean s=$sigma') p.legend(x=100,y=0.1) p.xaxis(max=150,label='x axis') p.yaxis(min=0,max=0.15,label='y axis') p.save(js.examples.imagepath+'/convolve.jpg')
-
jscatter.formel.
cstar
(Rg, Mw)[source]¶ Overlap concentration
for a polymer.
Equation 3 in [1] (Cotton) defines
as overlap concentration of space filling volumes corresponding to a cube or sphere with edge/radius equal to
while equ. 4 uses cubes with
(Graessley)
.
- Parameters
- Rgfloat in nm
radius of gyration
- Mwfloat
molecular weight
- Returns
- floatx3
Concentration limits [cube_rg, sphere_rg, cube_2rg] in units g/l.
References
- 1
Overlap concentration of macromolecules in solution Ying, Q. & Chu, B. Macromolecules 20, 362–366 (1987)
-
jscatter.formel.
debug
= False¶ Variable to allow printout for debugging as if debug:print(‘message’)
-
jscatter.formel.
dielectricConstant
(material='d2o', T=293.15, conc=0, delta=5.5)[source]¶ Dielectric constant of H2O and D2O buffer solutions.
Dielectric constant
of H2O and D2O (error +- 0.02) with added buffer salts.
- Parameters
- materialstring ‘d2o’ (default) or ‘h2o’
Material ‘d2o’ (default) or ‘h2o’
- Tfloat
Temperature in °C
- concfloat
Salt concentration in mol/l.
- deltafloat
Total excess polarisation dependent on the salt and presumably on the temperature!
- Returns
- float
Dielectric constant
Notes
Salt
delta(+-1)
deltalambda (not used here)
HCl
-10
0
LiCl
7
-3.5
NaCl
5.5
- -4
default
KCl
5
-4
RbCl
5
-4.5
NaF
6
-4
KF
6.5
-3.5
NaI
-7.5
-9.5
KI
-8
-9.5
MgCI,
-15
-6
BaCl2
-14
-8.5
LaCI.
-22
-13.5
NaOH
-10.5
-3
Na2SO.
-11
-9.5
References
- 1
Dielectric Constant of Water from 0 to 100 C. G . Malmberg and A. A. Maryott Journal of Research of the National Bureau of Standards, 56,1 ,369131-56–1 (1956) Research Paper 2641
- 2
Dielectric Constant of Deuterium Oxide C.G Malmberg, Journal of Research of National Bureau of Standards, Vol 60 No 6, (1958) 2874 http://nvlpubs.nist.gov/nistpubs/jres/60/jresv60n6p609_A1b.pdf
- 3
Dielectric Properties of Aqueous Ionic Solutions. Parts I and II Hasted et al. J Chem Phys 16 (1948) 1 http://link.aip.org/link/doi/10.1063/1.1746645
-
jscatter.formel.
eijk
= array([[[ 0., 0., 0.], [ 0., 0., 1.], [ 0., -1., 0.]], [[ 0., 0., -1.], [ 0., 0., 0.], [ 1., 0., 0.]], [[ 0., 1., 0.], [-1., 0., 0.], [ 0., 0., 0.]]])¶ Antisymmetric Levi-Civita symbol
-
jscatter.formel.
felectron
= 2.8179403267e-06¶ Cross section of electron in nm
-
jscatter.formel.
fibonacciLatticePointsOnSphere
(NN, r=1)[source]¶ FibonacciLatticePointsOnSphere; see
fibonacciLatticePointsOnSphere()
-
jscatter.formel.
gauss
(x, mean=1, sigma=1)[source]¶ Normalized Gaussian function.
- Parameters
- xfloat
Values
- meanfloat
Mean value
- sigmafloat
1/e width
- Returns
- dataArray
-
jscatter.formel.
loglist
(mini=0.1, maxi=5, number=100)[source]¶ Log like sequence between mini and maxi.
- Parameters
- mini,maxifloat, default 0.1, 5
Start and endpoint.
- numberint, default 100
Number of points in sequence.
- Returns
- ndarray
-
jscatter.formel.
lognorm
(x, mean=1, sigma=1)[source]¶ Lognormal distribution function.
- Parameters
- xarray
x values
- meanfloat
mean
- sigmafloat
sigma
- Returns
- dataArray
-
jscatter.formel.
lorentz
(x, mean=1, gamma=1)[source]¶ Normalized Lorentz function
- Parameters
- xarray
X values
- gammafloat
Half width half maximum
- meanfloat
Mean value
- Returns
- dataArray
-
jscatter.formel.
mPDA
(funktion, sigs, parnames, types='normal', N=30, ncpu=1, **kwargs)¶ Vectorized average assuming multiple parameters are distributed in intervals.
Function average over multiple distributed parameters with weights determined from probability distribution. The probabilities for the parameters are multiplied as weights and a weighted sum is calculated by Monte-Carlo integration.
- Parameters
- funktionfunction
Function to integrate with distribution weight.
- sigsfloat
List of widths for parameters, see Notes.
- parnamesstring
List of names of the parameters which show a distribution.
- typeslist of ‘normal’, ‘lognorm’, ‘gamma’, ‘lorentz’, ‘uniform’, ‘poisson’, ‘schulzZimm’, ‘duniform’, default ‘normal’
List of types of the distributions. If types list is shorter than parnames the last is repeated.
- kwargsparameters
Any additonal kword parameter to pass to function. The value of parnames that are distributed will be the mean value of the distribution.
- Nfloat , default=30
Number of points over distribution ranges. Distributions are integrated in probability intervals
.
- ncpuint, default=1, optional
Number of cpus in the pool for parallel excecution. Set this to 1 if the integrated function uses multiprocessing to avoid errors.
0 -> all cpus are used
int>0 min (ncpu, mp.cpu_count)
int<0 ncpu not to use
- Returns
- dataArray
- as returned from function with
.parname_mean = mean of parname
.parname_std = standard deviation of parname
Notes
Calculation of an average over D multiple distributed parameters by conventional integration requires
function evaluations which is quite time consuming. Monte-Carlo integration at N points with random combinations of parameters requires only N evaluations.
The given function of fixed parameters
and polydisperse parameters
with width
related to the indicated distribution (types) is integrated as
Each parameter
is distributed along values
with probability
describing the probability distribution with mean
and sigma
. Intervals for a parameter
are choosen to represent the distribution in the interval
The distributed values
are determined as pseudorandom numbers of N points with dimension len(i) for Monte-Carlo integration.
For a single polydisperse parameter use parDistributedAverage.
During fitting it has to be accounted for the information content of the experimental data. As in the example below it might be better to use a single width for all parameters to reduce the number of redundant parameters.
The used distributions are from scipy.stats. Choose the distribution according to the problem and check needed number of points N.
mean is the value in kwargs[parname]. mean and sig are used as:
- norm :
- mean , stdstats.norm(loc=mean,scale=sig)
- lognorm :
- mean and sig evaluate to mean and stdmu=math.log(mean**2/(sig+mean**2)**0.5)nu=(math.log(sig/mean**2+1))**0.5stats.lognorm(s=nu,scale=math.exp(mu))
- gamma :
- mean and sig evaluate to mean and stdstats.gamma(a=mean**2/sig**2,scale=sig**2/mean)
- lorentz = cauchy:
- mean and std are not defined. Use FWHM instead to describe width.sig=FWHMstats.cauchy(loc=mean,scale=sig))
- uniform :
- Continuous distribution.sig is widthstats.uniform(loc=mean-sig/2.,scale=sig))
- poisson:
stats.poisson(mu=mean,loc=sig)
- schulzZimm
- duniform:
- Uniform distribution integer values.sig>1stats.randint(low=mean-sig, high=mean+sig)
For more distribution look into this source code and use it appropriate with scipy.stats. Schulz-Zimm in formel is a new defined distribution not in scipy.stats.
Examples
The example of a cuboid with independent polydispersity on all edges. To use the function in fitting please encapsulate it in a model function hiding the list parameters.
import jscatter as js type=['norm','schulzZimm'] p=js.grace() q=js.loglist(0.1,5,500) sp=js.ff.cuboid(q=q,a=4,b=4.1,c=4.3) p.plot(sp,sy=[1,0.2],legend='single cube') p.yaxis(scale='l',label='I(Q)') p.xaxis(scale='n',label='Q / nm') p.title('Cuboid with independent polydispersity on all 3 edges') p.subtitle('Using Monte Carlo integration; 30 points are enough here!') sp1=js.formel.mPDA(js.ff.cuboid,sigs=[0.2,0.3,0.1],parnames=['a','b','c'],types=type,q=q,a=4,b=4.1,c=4.2,N=10) p.plot(sp1,li=[1,2,2],sy=0,legend='normal 10 points') sp2=js.formel.mPDA(js.ff.cuboid,sigs=[0.2,0.3,0.1],parnames=['a','b','c'],types=type,q=q,a=4,b=4.1,c=4.2,N=30) p.plot(sp2,li=[1,2,3],sy=0,legend='normal 30 points') sp3=js.formel.mPDA(js.ff.cuboid,sigs=[0.2,0.3,0.1],parnames=['a','b','c'],types=type,q=q,a=4,b=4.1,c=4.2,N=90) p.plot(sp3,li=[3,2,4],sy=0,legend='normal 100 points') p.legend(x=2,y=1000) # p.save(js.examples.imagepath+'/multiParDistributedAverage.jpg')
During fitting encapsulation might be done like this
def polyCube(a,b,c,sig,N): res = js.formel.mPDA(js.ff.cuboid,sigs=[sig,sig,sig],parnames=['a','b','c'],types='normal',q=q,a=a,b=b,c=c,N=N) return res
-
jscatter.formel.
memoize
(**memkwargs)[source]¶ A least-recently-used cache decorator to cache expensive function evaluations.
Memoize caches results and retrieves from cache if same parameters are used again. This can speedup computation in a model if a part is computed with same parameters several times. During fits it may be faster to calc result for a list and take from cache.
- Parameters
- functionfunction
Function to evaluate as e.g. f(Q,a,b,c,d)
- memkwargsdict
Keyword args with substitute values to cache for later interpolation. Empty for normal caching of a function. E.g. memkwargs={‘Q’:np.r_[0:10:0.1],’t’:np.r_[0:100:5]} caches with these values. The needed values can be interpolated from the returned result. See example below.
- maxsizeint, default 128
maximum size of the cache. Last is dropped.
- Returns
- function
- cached function with new methods
last(i) to retrieve the ith evaluation result in cache (last is i=-1).
clear() to clear the cached results.
hitsmisses counts hits and misses.
Notes
Only keyword arguments for the memoized function are supported!!!! Only one attribute and X are supported for fitting as .interpolate works only for two cached attributes.
Examples
The example uses a model that computes like I(q,n,..)=F(q)*B(t,n,..). F(q) is cheap to calculate B(t,n,..) not. In the following its better to calc the function for a list of q , put it to cache and take in the fit from there. B is only calculated once inside of the function.
Use it like this:
import jscatter as js import numpy as np # define some data TT=js.loglist(0.01,80,30) QQ=np.r_[0.1:1.5:0.15] # in the data we have 'q' and 'X' data=js.dynamic.finiteZimm(t=TT,q=QQ,NN=124,pmax=100,tintern=10,ll=0.38,Dcm=0.01,mu=0.5,viscosity=1.001,Temp=300) # makes a unique list of all X values -> interpolation is exact for X # one may also use a smaller list of values and only interpolate tt=list(set(data.X.flatten));tt.sort() # define memoized function which will always use the here defined q and t # use correct values from data for q -> interpolation is exact for q memfZ=js.formel.memoize(q=data.q,t=tt)(js.dynamic.finiteZimm) def fitfunc(Q,Ti,NN,tint,ll,D,mu,viscosity,Temp): # use the memoized function as usual (even if given t and q are used from above definition) res= memfZ(NN=NN,tintern=tint,ll=ll,Dcm=D,pmax=40,mu=mu,viscosity=viscosity,Temp=Temp) # interpolate to the here needed q and t (which is X) resint=res.interpolate(q=Q,X=Ti,deg=2)[0] return resint # do the fit data.setlimit(tint=[0.5,40],D=[0,1]) data.makeErrPlot(yscale='l') NN=20 data.fit(model=fitfunc, freepar={'tint':10,'D':0.1,}, fixpar={'NN':20,'ll':0.38/(NN/124.)**0.5,'mu':0.5,'viscosity':0.001,'Temp':300}, mapNames={'Ti':'X','Q':'q'},)
Second example
Use memoize as a decorator (@ in front) acting on the following function. This is a shortcut for the above and works in the same way
# define the function to memoize @js.formel.memoize(Q=np.r_[0:3:0.2],Time=np.r_[0:50:0.5,50:100:5]) def fZ(Q,Time,NN,tintern,ll,Dcm,mu,viscosity,Temp): # finiteZimm accepts t and q as array and returns a dataList with different Q and same X=t res=js.dynamic.finiteZimm(t=Time,q=Q,NN=NN,pmax=20,tintern=tintern, ll=ll,Dcm=Dcm,mu=mu,viscosity=viscosity,Temp=Temp) return res # define the fitfunc def fitfunc(Q,Ti,NN,tint,ll,D,mu,viscosity,Temp): #this is the cached result for the list of Q res= fZ(Time=Ti,Q=Q,NN=NN,tintern=tint,ll=ll,Dcm=D,mu=mu,viscosity=viscosity,Temp=Temp) # interpolate for the single Q value the cached result has again 'q' return res.interpolate(q=Q,X=Ti,deg=2)[0] # do the fit data.setlimit(tint=[0.5,40],D=[0,1]) data.makeErrPlot(yscale='l') data.fit(model=fitfunc, freepar={'tint':6,'D':0.1,}, fixpar={'NN':20,'ll':0.38/(20/124.)**0.5,'mu':0.5,'viscosity':0.001,'Temp':300}, mapNames={'Ti':'X','Q':'q'}) # the result depends on the interpolation;
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jscatter.formel.
molarity
(objekt, c, total=None)[source]¶ Calculates the molarity.
- Parameters
- objektobject,float
Objekt with method .mass() or molecular weight in Da.
- cfloat
Concentration in g/ml -> mass/Volume
- totalfloat, default None
Total volume in milliliter [ml] Concentration is calculated by c[g]/total[ml] if given.
- Returns
- float
molarity in mol/liter (= mol/1000cm^3)
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jscatter.formel.
multiParDistributedAverage
(funktion, sigs, parnames, types='normal', N=30, ncpu=1, **kwargs)[source]¶ Vectorized average assuming multiple parameters are distributed in intervals.
Function average over multiple distributed parameters with weights determined from probability distribution. The probabilities for the parameters are multiplied as weights and a weighted sum is calculated by Monte-Carlo integration.
- Parameters
- funktionfunction
Function to integrate with distribution weight.
- sigsfloat
List of widths for parameters, see Notes.
- parnamesstring
List of names of the parameters which show a distribution.
- typeslist of ‘normal’, ‘lognorm’, ‘gamma’, ‘lorentz’, ‘uniform’, ‘poisson’, ‘schulzZimm’, ‘duniform’, default ‘normal’
List of types of the distributions. If types list is shorter than parnames the last is repeated.
- kwargsparameters
Any additonal kword parameter to pass to function. The value of parnames that are distributed will be the mean value of the distribution.
- Nfloat , default=30
Number of points over distribution ranges. Distributions are integrated in probability intervals
.
- ncpuint, default=1, optional
Number of cpus in the pool for parallel excecution. Set this to 1 if the integrated function uses multiprocessing to avoid errors.
0 -> all cpus are used
int>0 min (ncpu, mp.cpu_count)
int<0 ncpu not to use
- Returns
- dataArray
- as returned from function with
.parname_mean = mean of parname
.parname_std = standard deviation of parname
Notes
Calculation of an average over D multiple distributed parameters by conventional integration requires
function evaluations which is quite time consuming. Monte-Carlo integration at N points with random combinations of parameters requires only N evaluations.
The given function of fixed parameters
and polydisperse parameters
with width
related to the indicated distribution (types) is integrated as
Each parameter
is distributed along values
with probability
describing the probability distribution with mean
and sigma
. Intervals for a parameter
are choosen to represent the distribution in the interval
The distributed values
are determined as pseudorandom numbers of N points with dimension len(i) for Monte-Carlo integration.
For a single polydisperse parameter use parDistributedAverage.
During fitting it has to be accounted for the information content of the experimental data. As in the example below it might be better to use a single width for all parameters to reduce the number of redundant parameters.
The used distributions are from scipy.stats. Choose the distribution according to the problem and check needed number of points N.
mean is the value in kwargs[parname]. mean and sig are used as:
- norm :
- mean , stdstats.norm(loc=mean,scale=sig)
- lognorm :
- mean and sig evaluate to mean and stdmu=math.log(mean**2/(sig+mean**2)**0.5)nu=(math.log(sig/mean**2+1))**0.5stats.lognorm(s=nu,scale=math.exp(mu))
- gamma :
- mean and sig evaluate to mean and stdstats.gamma(a=mean**2/sig**2,scale=sig**2/mean)
- lorentz = cauchy:
- mean and std are not defined. Use FWHM instead to describe width.sig=FWHMstats.cauchy(loc=mean,scale=sig))
- uniform :
- Continuous distribution.sig is widthstats.uniform(loc=mean-sig/2.,scale=sig))
- poisson:
stats.poisson(mu=mean,loc=sig)
- schulzZimm
- duniform:
- Uniform distribution integer values.sig>1stats.randint(low=mean-sig, high=mean+sig)
For more distribution look into this source code and use it appropriate with scipy.stats. Schulz-Zimm in formel is a new defined distribution not in scipy.stats.
Examples
The example of a cuboid with independent polydispersity on all edges. To use the function in fitting please encapsulate it in a model function hiding the list parameters.
import jscatter as js type=['norm','schulzZimm'] p=js.grace() q=js.loglist(0.1,5,500) sp=js.ff.cuboid(q=q,a=4,b=4.1,c=4.3) p.plot(sp,sy=[1,0.2],legend='single cube') p.yaxis(scale='l',label='I(Q)') p.xaxis(scale='n',label='Q / nm') p.title('Cuboid with independent polydispersity on all 3 edges') p.subtitle('Using Monte Carlo integration; 30 points are enough here!') sp1=js.formel.mPDA(js.ff.cuboid,sigs=[0.2,0.3,0.1],parnames=['a','b','c'],types=type,q=q,a=4,b=4.1,c=4.2,N=10) p.plot(sp1,li=[1,2,2],sy=0,legend='normal 10 points') sp2=js.formel.mPDA(js.ff.cuboid,sigs=[0.2,0.3,0.1],parnames=['a','b','c'],types=type,q=q,a=4,b=4.1,c=4.2,N=30) p.plot(sp2,li=[1,2,3],sy=0,legend='normal 30 points') sp3=js.formel.mPDA(js.ff.cuboid,sigs=[0.2,0.3,0.1],parnames=['a','b','c'],types=type,q=q,a=4,b=4.1,c=4.2,N=90) p.plot(sp3,li=[3,2,4],sy=0,legend='normal 100 points') p.legend(x=2,y=1000) # p.save(js.examples.imagepath+'/multiParDistributedAverage.jpg')
During fitting encapsulation might be done like this
def polyCube(a,b,c,sig,N): res = js.formel.mPDA(js.ff.cuboid,sigs=[sig,sig,sig],parnames=['a','b','c'],types='normal',q=q,a=a,b=b,c=c,N=N) return res
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jscatter.formel.
pAC
(func, lowlimit, uplimit, parnames, fdim=None, adaptive='p', abserr=1e-08, relerr=0.001, *args, **kwargs)¶ Vectorized adaptive multidimensional integration (cubature) .
We use the cubature module written by SG Johnson [2] for h-adaptive (recursively partitioning the integration domain into smaller subdomains) and p-adaptive (Clenshaw-Curtis quadrature, repeatedly doubling the degree of the quadrature rules). This function is a wrapper around the package cubature which can be used also directly.
- Parameters
- funcfunction
The function to integrate. The return value should be of shape [len(parnames), fdim]
- parnameslist of string
Parameter names of variables to integrate.
- lowlimitlist of float
Lower limits of the integration variables with same length as parnames.
- uplimit: list of float
Upper limits of the integration variables with same length as parnames.
- fdimint, None, optional
Second dimension size of the func return array. If None, the function is evaluated with the uplimit values to determine the size. For complex valued function it is twice the complex array length.
- adaptive‘h’, ‘p’, default=’p’
- Type of adaption algorithm.
- ‘h’ Multidimensional h-adaptive integration by subdividing the integration interval into smaller intervals
where the same rule is applied. The value and error in each interval is calculated from 7-point rule and difference to 5-point rule. For higher dimensions only the worst dimension is subdivided [3]. This algorithm is best suited for a moderate number of dimensions (say, < 7), and is superseded for high-dimensional integrals by other methods (e.g. Monte Carlo variants or sparse grids).
- ‘p’ Multidimensional p-adaptive integration by increasing the degree of the quadrature rule according to
Clenshaw-Curtis quadrature (in each iteration the number of points is doubled and the previous values are reused). Clenshaw-Curtis has similar error compared to Gaussian quadrature even if the used error estimate is worse. This algorithm is often superior to h-adaptive integration for smooth integrands in a few (≤ 3) dimensions,
- abserr, relerrfloat default = 1e-8, 1e-3
Absolute and relative error to stop. The integration will terminate when either the relative OR the absolute error tolerances are met. abserr=0, which means that it is ignored. The real error is much smaller than this stop criterion.
- maxEvalint, default 0, optional
Maximum number of function evaluations. 0 is infinite.
- normint, default=None, optional
- Norm to evaluate the error.
None: 0,1 automatically choosen for real or complex functions.
0: individual for each integrand (real valued functions)
1: paired error (L2 distance) for complex values as distance in complex plane.
Other values as mentioned in cubature documentation.
- args,kwargsoptional
Additional arguments and keyword arguments passed to func.
- Returns
- array
Notes
We use here the Python interface of S.G.P. Castro [1] to access C-module of S.G. Johnson [2] . Only the vectorized form is realized here.
Internal: For complex valued functions the complex has to be split in real and imaginary to pass to the integration and later the result has to be converted to complex again. This is done automatically dependent on the return value of the function. For the example the real valued function is about 9 times faster
References
- 1
- 2(1,2)
- 3
An adaptive algorithm for numeric integration over an N-dimensional rectangular region A. C. Genz and A. A. Malik, J. Comput. Appl. Math. 6 (4), 295–302 (1980).
- 4
Examples
Integration of the sphere to get the sphere formfactor. In the first example the symmetry is used to return real valued amplitude. In the second the complex amplitude is used. Both can be compared to the analytic formfactor. Errors are much smaller than the abserr/relerr stop criterion. The stop seems to be related to the minimal point at q=2.8 as critical point. h-adaptive is for dim=3 less accurate and slower than p-adaptive.
import jscatter as js import numpy as np R=5 q = np.r_[0.01:3:0.02] def sphere_real(r, theta, phi, b, q): res = b*np.cos(q[:,None]*r*np.cos(theta))*r**2*np.sin(theta)*2 return res.T pn = ['r','theta','phi'] fa_r,err = js.formel.pAC(sphere_real, [0,0,0], [R,np.pi/2,np.pi*2], pn, b=1, q=q) fa_rh,errh = js.formel.pAC(sphere_real, [0,0,0], [R,np.pi/2,np.pi*2], pn, b=1, q=q,adaptive='h') # As complex function def sphere_complex(r, theta, phi, b, q): fac = b * np.exp(1j * q[:, None] * r * np.cos(theta)) * r ** 2 * np.sin(theta) return fac.T fa_c, err = js.formel.pAC(sphere_complex, [0, 0, 0], [R, np.pi, np.pi * 2], pn, b=1, q=q) sp = js.ff.sphere(q, R) p = js.grace() p.multi(2,1,vgap=0) p[0].plot(q, fa_r ** 2, le='real integrand p-adaptive') p[0].plot(q, fa_rh ** 2, le='real integrand h-adaptive') p[0].plot(q, np.real(fa_c * np.conj(fa_c)),sy=[8,0.5,3], le='complex integrand') p[0].plot(q, sp.Y, li=1, sy=0, le='analytic') p[1].plot(q,np.abs(fa_r**2 -sp.Y), le='real integrand') p[1].plot(q,np.abs(fa_rh**2 -sp.Y), le='real integrand h-adaptive') p[1].plot(q,np.abs(np.real(fa_c * np.conj(fa_c)) -sp.Y),sy=[8,0.5,3]) p[0].yaxis(scale='l',label='F(q)',ticklabel=['power',0]) p[0].xaxis(ticklabel=0) p[0].legend(x=2,y=1e6) p[1].yaxis(scale='l',label=r'error', ticklabel=['power',0],min=1e-13,max=5e-6) p[1].xaxis(label=r'q / nm\S-1') p[1].text(r'error = abs(F(Q) - F(q)\sanalytic\N)',x=1.5,y=1e-9) p[0].title('Numerical quadrature sphere formfactor ') p[0].subtitle('stop criterion relerror=1e-3, real errors are smaller') #p.save(js.examples.imagepath+'/cubature.jpg')
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jscatter.formel.
pDA
(funktion, sig, parname, type='normal', nGauss=30, **kwargs)¶ Vectorized average assuming a single parameter is distributed with width sig.
Function average over a parameter with weights determined from probability distribution. Gaussian quadrature over given distribution or summation with weights is used.
- Parameters
- funktionfunction
Function to integrate with distribution weight. Function needs to return dataArray.
- sigfloat
width parameter of the distribution, see Notes
- parnamestring
Name of the parameter of funktion which shows a distribution
- type‘normal’,’lognorm’,’gamma’,’lorentz’,’uniform’,’poisson’,’schulzZimm’,’duniform’, default ‘normal’
Type of the distribution
- kwargsparameters
Any additonal kword parameter to pass to function. The value of parname will be the mean value of the distribution.
- nGaussfloat , default=30
Order of quadrature integration as number of intervals in Gauss–Legendre quadrature over distribution. Distribution is integrated in probability interval [0.001..0.999].
- ncpuint, optional
Number of cpus in the pool. Set this to 1 if the integrated function uses multiprocessing to avoid errors.
0 -> all cpus are used
int>0 min (ncpu, mp.cpu_count)
int<0 ncpu not to use
- Returns
- dataArray
- as returned from function with
.parname_mean = mean of parname
.parname_std = standard deviation of parname
Notes
The used distributions are from scipy.stats. Choose the distribution according to the problem.
mean is the value in kwargs[parname]. mean and sig are used as:
- norm :
- mean , stdstats.norm(loc=mean,scale=sig)
- lognorm :
- mean and sig evaluate to mean and stdmu=math.log(mean**2/(sig+mean**2)**0.5)nu=(math.log(sig/mean**2+1))**0.5stats.lognorm(s=nu,scale=math.exp(mu))
- gamma :
- mean and sig evaluate to mean and stdstats.gamma(a=mean**2/sig**2,scale=sig**2/mean)
- lorentz = cauchy:
- mean and std are not defined. Use FWHM instead to describe width.sig=FWHMstats.cauchy(loc=mean,scale=sig))
- uniform :
- Continuous distribution.sig is widthstats.uniform(loc=mean-sig/2.,scale=sig))
- schulzZimm
- poisson:
stats.poisson(mu=mean,loc=sig)
- duniform:
- Uniform distribution integer values.sig>1stats.randint(low=mean-sig, high=mean+sig)
For more distribution look into this source code and use it appropriate with scipy.stats.
Examples
import jscatter as js p=js.grace() q=js.loglist(0.1,5,500) sp=js.ff.sphere(q=q,radius=5) p.plot(sp,sy=[1,0.2],legend='single radius') p.yaxis(scale='l',label='I(Q)') p.xaxis(scale='n',label='Q / nm') sig=0.2 p.title('radius distribution with width %.g' %(sig)) sp2=js.formel.pDA(js.ff.sphere,sig,'radius',type='normal',q=q,radius=5,nGauss=100) p.plot(sp2,li=[1,2,2],sy=0,legend='normal 100 points Gauss ') sp4=js.formel.pDA(js.ff.sphere,sig,'radius',type='normal',q=q,radius=5,nGauss=30) p.plot(sp4,li=[1,2,3],sy=0,legend='normal 30 points Gauss ') sp5=js.formel.pDA(js.ff.sphere,sig,'radius',type='normal',q=q,radius=5,nGauss=5) p.plot(sp5,li=[1,2,5],sy=0,legend='normal 5 points Gauss ') sp3=js.formel.pDA(js.ff.sphere,sig,'radius',type='lognormal',q=q,radius=5) p.plot(sp3,li=[3,2,4],sy=0,legend='lognormal') sp6=js.formel.pDA(js.ff.sphere,sig,'radius',type='gamma',q=q,radius=5) p.plot(sp6,li=[2,2,6],sy=0,legend='gamma ') sp9=js.formel.pDA(js.ff.sphere,sig,'radius',type='schulzZimm',q=q,radius=5) p.plot(sp9,li=[3,2,2],sy=0,legend='SchulzZimm ') # an unrealistic example sp7=js.formel.pDA(js.ff.sphere,1,'radius',type='poisson',q=q,radius=5) p.plot(sp7,li=[1,2,6],sy=0,legend='poisson ') sp8=js.formel.pDA(js.ff.sphere,1,'radius',type='duniform',q=q,radius=5) p.plot(sp8,li=[1,2,6],sy=0,legend='duniform ') p.legend()
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jscatter.formel.
pQAG
(func, lowlimit, uplimit, parname, weights=None, tol=1e-08, rtol=1e-08, maxiter=50, miniter=8, **kwargs)¶ Vectorized definite integral using fixed-tolerance Gaussian quadrature.
Adaptive integration of func from a to b using Gaussian quadrature adaptivly increasing number of points by 8. If function returns ndarray each element is integrated and returned as ndarray of same dimension. If function returns dataArray, only the .Y values is integrated and returned as dataArray.
- Parameters
- funcfunction
A function or method to integrate.
- lowlimitfloat
Lower limit of integration.
- uplimitfloat
Upper limit of integration.
- parnamestring
name of the integration variable which should be a scalar.
- weightsndarray shape(2,N),default=None
Weights for integration along parname as a Gaussian with a<weights[0]<b and weights[1] contains weight values.
Missing values are linear interpolated (faster). If None equal weights are used.
- kwargsdict, optional
Extra keyword arguments to pass to function, if any.
- tol, rtolfloat, optional
Iteration stops when error between last two iterates is less than tol OR the relative change is less than rtol.
- maxiterint, default 50, optional
Maximum order of Gaussian quadrature.
- miniterint, default 8, optional
Minimum order of Gaussian quadrature.
- ncpuint, default=1, optional
Number of cpus in the pool. Set this to 1 if the integrated function uses multiprocessing to avoid errors.
0 -> all cpus are used
int>0 min (ncpu, mp.cpu_count)
int<0 ncpu not to use
- Returns
- valfloat
Gaussian quadrature approximation (within tolerance) to integral for all vector elements.
- errfloat
Difference between last two estimates of the integral.
Notes
Reimplementation of scipy.integrate.quadrature.quadrature to work with vector output of the integrand function.
References
Examples
t=np.r_[1:100] gg=js.formel.gauss(t,50,10) js.formel.parQuadratureAdaptiveGauss(js.formel.gauss,0,100,'x',mean=50,sigma=10)
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jscatter.formel.
pQFG
(func, lowlimit, uplimit, parname, n=5, weights=None, **kwargs)¶ Vectorized definite integral using fixed-order Gaussian quadrature.
Integrate func over parname from a to b using Gauss-Legendre quadrature [1] of order n for all .X. If function returns ndarray all is integrated and returned as ndarray. If function returns dataArray, only the .Y values are integrated and returned as dataArray.
- Parameters
- funccallable
A Python function or method returning a vector array of dimension 1. If func returns dataArray .Y is integrated.
- lowlimitfloat
Lower limit of integration.
- uplimitfloat
Upper limit of integration.
- parnamestring
Name of the integration variable which should be a scalar. After evaluation the corresponding attribute has the mean value with weights.
- weightsndarray shape(2,N),default=None
Weights for integration along parname with a<weights[0]<b and weights[1] contains weight values.
Missing values are linear interpolated (faster). If None equal weights are used.
- kwargsdict, optional
Extra keyword arguments to pass to function, if any.
- nint, optional
Order of quadrature integration. Default is 5.
- ncpuint,default=1, optional
Use parallel processing for the function with ncpu parallel processes in the pool. Set this to 1 if the function is already fast enough or if the integrated function uses multiprocessing.
0 -> all cpus are used
int>0 min (ncpu, mp.cpu_count)
int<0 ncpu not to use
- Returns
- array or dataArray
Notes
Reimplementation of scipy.integrate.quadrature.fixed_quad to work with vector output of the integrand function and weights.
References
Examples
t=np.r_[1:100] gg=js.formel.gauss(t,50,10) js.formel.parQuadratureFixedGauss(js.formel.gauss,0,100,'x',mean=50,sigma=10,n=50)
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jscatter.formel.
pQFGxD
(func, lowlimit, uplimit, parnames, n=5, weights0=None, weights1=None, weights2=None, **kwargs)¶ Vectorized fixed-order Gauss-Legendre quadrature in definite interval in 1,2,3 dimensions.
Integrate func over parnames between limits using Gauss-Legendre quadrature [1] of order n.
- Parameters
- funccallable
Function to integrate. The return value should be 2 dimensional with first dimension along integration variable and second along array to calculate. See examples.
- parnameslist of string, max len=3
Name of the integration variables which should be scalar in the function.
- lowlimitlist of float
Lower limits a of integration for parnames.
- uplimitlist of float
Upper limits b of integration for parnames.
- weights0,weights1,weights3ndarray shape(2,N), default=None
Weights for integration along parname with a<weightsi[0]<b and weightsi[1] contains weight values.
Missing values are linear interpolated (faster). If None equal weights are used.
- kwargsdict, optional
Extra keyword arguments to pass to function, if any.
- nint, optional
Order of quadrature integration for all parnames. Default is 5.
- Returns
- array or dataArray
Notes
To get a speedy integration the function should use numpy ufunctions which operate on numpy arrays with compiled code.
References
Examples
The following integrals in 1-3 dimensions over a normalised Gaussian give always 1 which achieved with reasonable accuracy with n=15.
The examples show different ways to return 2dim arrays with x,y,z in first dimension and vector q in second. x[:,None] adds a second dimension to array x.
import jscatter as js import numpy as np q=np.r_[0.1:5.1:0.1] def gauss(x,mean,sigma): return np.exp(-0.5 * (x - mean) ** 2 / sigma ** 2) / sigma / np.sqrt(2 * np.pi) # 1dimensional def gauss1(q,x,mean=0,sigma=1): g=np.exp(-0.5 * (x[:,None] - mean) ** 2 / sigma ** 2) / sigma / np.sqrt(2 * np.pi) return g*q js.formel.pQFGxD(gauss1,0,100,parnames='x',mean=50,sigma=10,q=q,n=15) # 2 dimensional def gauss2(q,x,y=0,mean=0,sigma=1): g=gauss(x[:,None],mean,sigma)*gauss(y[:,None],mean,sigma) return g*q js.formel.pQFGxD(gauss2,[0,0],[100,100],parnames=['x','y'],mean=50,sigma=10,q=q,n=15) # 3 dimensional def gauss3(q,x,y=0,z=0,mean=0,sigma=1): g=gauss(x,mean,sigma)*gauss(y,mean,sigma)*gauss(z,mean,sigma) return g[:,None]*q js.formel.pQFGxD(gauss3,[0,0,0],[100,100,100],parnames=['x','y','z'],mean=50,sigma=10,q=q,n=15)
Usage of weights allows weights for dimensions e.g. to realise a spherical average with weight
in the integral
. (Using the weight in the function is more accurate.) The weight needs to be normalised by unit sphere area
.
import jscatter as js import numpy as np q=np.r_[0,0.1:5.1:0.1] def cuboid(q, phi, theta, a, b, c): pi2 = np.pi * 2 fa = (np.sinc(q * a * np.sin(theta[:,None]) * np.cos(phi[:,None]) / pi2) * np.sinc(q * b * np.sin(theta[:,None]) * np.sin(phi[:,None]) / pi2) * np.sinc(q * c * np.cos(theta[:,None]) / pi2)) return fa**2*(a*b*c)**2 # generate weight for sin(theta) dtheta integration (better to integrate in cuboid function) # and normalise for unit sphere t = np.r_[0:np.pi:180j] wt = np.c_[t,np.sin(t)/np.pi/4].T Fq=js.formel.pQFGxD(cuboid,[0,0],[2*np.pi,np.pi],parnames=['phi','theta'],weights1=wt,q=q,n=15,a=1.9,b=2,c=2) # compare the result to the ff solution (which does the same with weights in the function). p=js.grace() p.plot(q,Fq) p.plot(js.ff.cuboid(q,1.9,2,2),li=1,sy=0) p.yaxis(scale='l')
-
jscatter.formel.
pQS
(funktion, lowlimit, uplimit, parname, weights=None, tol=1e-06, rtol=1e-06, dX=None, **kwargs)¶ Vectorized quadrature over one parameter with weights using the adaptive Simpson rule.
Integrate by adaptive Simpson integration for all .X values at once. Only .Y values are integrated and checked for tol criterion. Attributes and non .Y columns correspond to the weighted mean of parname.
- Parameters
- funktionfunction
Function returning dataArray or array
- lowlimit,uplimitfloat
Interval borders to integrate
- parnamestring
Parname to integrate
- weightsndarray shape(2,N),default=None
Weights for integration along parname as a Gaussian with a<weights[0]<b and weights[1] contains weight values.
Missing values are linear interpolated (faster). If None equal weights are used.
- tol,rtolfloat, default=1e-6
- Relative error or absolute error to stop integration. Stop if one is full filled.Tol is divided for each new interval that the sum of tol is kept..IntegralErr_funktEvaluations in dataArray contains error and number of points in interval.
- dXfloat, default=None
Minimal distance between integration points to determine a minimal step for integration variable.
- kwargs :
Additional parameters to pass to funktion. If parname is in kwargs it is overwritten.
- Returns
- dataArray or array
dataArrays have additional parameters as error and weights.
Notes
What is the meaning of tol in simpson method? If the error in an interval exceeds tol, the algorithm subdivides the interval in two equal parts with each
and applies the method to each subinterval in a recursive manner. The condition in interval i is
. The recursion stops in an interval if the improvement is smaller than tol. Thus tol is the upper estimate for the total error.
Here we use a absolute (tol) and relative (rtol) criterion:
with
as the next improved value As this is tested for all .X the worst case is better than tol, rtol.
The algorithm is efficient as it memoizes function evaluation at each interval border and reuses the result. This reduces computing time by about a factor 3-4.
Different distribution can be found in scipy.stats. But any distribution given explicitly can be used. E.g. triangular np.c_[[-1,0,1],[0,1,0]].T
References
Examples
Integrate Gaussian as test case
import jscatter as js import numpy as np import scipy # testcase: integrate over x of a function # area under normalized gaussian is 1 js.formel.parQuadratureSimpson(js.formel.gauss,-10,10,'x',mean=0,sigma=1)
Integrate a function over one parameter with a weighting function. If weight is 1 the result is a simple integration. Here the weight corresponds to a normal distribution and the result is a weighted average as implemented in parDistributedAverage using fixedGaussian quadrature.
# normal distribtion of parameter D with width ds t=np.r_[0:150:0.5] D=0.3 ds=0.1 diff=js.dynamic.simpleDiffusion(t=t,q=0.5,D=D) distrib =scipy.stats.norm(loc=D,scale=ds) x=np.r_[D-5*ds:D+5*ds:30j] pdf=np.c_[x,distrib.pdf(x)].T diff_g=js.formel.parQuadratureSimpson(js.dynamic.simpleDiffusion,-3*ds+D,3*ds+D,parname='D', weights=pdf,tol=0.01,q=0.5,t=t) # compare it p=js.grace() p.plot(diff,le='monodisperse') p.plot(diff_g,le='polydisperse') p.xaxis(scale='l') p.legend()
-
jscatter.formel.
parAdaptiveCubature
(func, lowlimit, uplimit, parnames, fdim=None, adaptive='p', abserr=1e-08, relerr=0.001, *args, **kwargs)[source]¶ Vectorized adaptive multidimensional integration (cubature) .
We use the cubature module written by SG Johnson [2] for h-adaptive (recursively partitioning the integration domain into smaller subdomains) and p-adaptive (Clenshaw-Curtis quadrature, repeatedly doubling the degree of the quadrature rules). This function is a wrapper around the package cubature which can be used also directly.
- Parameters
- funcfunction
The function to integrate. The return value should be of shape [len(parnames), fdim]
- parnameslist of string
Parameter names of variables to integrate.
- lowlimitlist of float
Lower limits of the integration variables with same length as parnames.
- uplimit: list of float
Upper limits of the integration variables with same length as parnames.
- fdimint, None, optional
Second dimension size of the func return array. If None, the function is evaluated with the uplimit values to determine the size. For complex valued function it is twice the complex array length.
- adaptive‘h’, ‘p’, default=’p’
- Type of adaption algorithm.
- ‘h’ Multidimensional h-adaptive integration by subdividing the integration interval into smaller intervals
where the same rule is applied. The value and error in each interval is calculated from 7-point rule and difference to 5-point rule. For higher dimensions only the worst dimension is subdivided [3]. This algorithm is best suited for a moderate number of dimensions (say, < 7), and is superseded for high-dimensional integrals by other methods (e.g. Monte Carlo variants or sparse grids).
- ‘p’ Multidimensional p-adaptive integration by increasing the degree of the quadrature rule according to
Clenshaw-Curtis quadrature (in each iteration the number of points is doubled and the previous values are reused). Clenshaw-Curtis has similar error compared to Gaussian quadrature even if the used error estimate is worse. This algorithm is often superior to h-adaptive integration for smooth integrands in a few (≤ 3) dimensions,
- abserr, relerrfloat default = 1e-8, 1e-3
Absolute and relative error to stop. The integration will terminate when either the relative OR the absolute error tolerances are met. abserr=0, which means that it is ignored. The real error is much smaller than this stop criterion.
- maxEvalint, default 0, optional
Maximum number of function evaluations. 0 is infinite.
- normint, default=None, optional
- Norm to evaluate the error.
None: 0,1 automatically choosen for real or complex functions.
0: individual for each integrand (real valued functions)
1: paired error (L2 distance) for complex values as distance in complex plane.
Other values as mentioned in cubature documentation.
- args,kwargsoptional
Additional arguments and keyword arguments passed to func.
- Returns
- array
Notes
We use here the Python interface of S.G.P. Castro [1] to access C-module of S.G. Johnson [2] . Only the vectorized form is realized here.
Internal: For complex valued functions the complex has to be split in real and imaginary to pass to the integration and later the result has to be converted to complex again. This is done automatically dependent on the return value of the function. For the example the real valued function is about 9 times faster
References
- 1
- 2(1,2)
- 3
An adaptive algorithm for numeric integration over an N-dimensional rectangular region A. C. Genz and A. A. Malik, J. Comput. Appl. Math. 6 (4), 295–302 (1980).
- 4
Examples
Integration of the sphere to get the sphere formfactor. In the first example the symmetry is used to return real valued amplitude. In the second the complex amplitude is used. Both can be compared to the analytic formfactor. Errors are much smaller than the abserr/relerr stop criterion. The stop seems to be related to the minimal point at q=2.8 as critical point. h-adaptive is for dim=3 less accurate and slower than p-adaptive.
import jscatter as js import numpy as np R=5 q = np.r_[0.01:3:0.02] def sphere_real(r, theta, phi, b, q): res = b*np.cos(q[:,None]*r*np.cos(theta))*r**2*np.sin(theta)*2 return res.T pn = ['r','theta','phi'] fa_r,err = js.formel.pAC(sphere_real, [0,0,0], [R,np.pi/2,np.pi*2], pn, b=1, q=q) fa_rh,errh = js.formel.pAC(sphere_real, [0,0,0], [R,np.pi/2,np.pi*2], pn, b=1, q=q,adaptive='h') # As complex function def sphere_complex(r, theta, phi, b, q): fac = b * np.exp(1j * q[:, None] * r * np.cos(theta)) * r ** 2 * np.sin(theta) return fac.T fa_c, err = js.formel.pAC(sphere_complex, [0, 0, 0], [R, np.pi, np.pi * 2], pn, b=1, q=q) sp = js.ff.sphere(q, R) p = js.grace() p.multi(2,1,vgap=0) p[0].plot(q, fa_r ** 2, le='real integrand p-adaptive') p[0].plot(q, fa_rh ** 2, le='real integrand h-adaptive') p[0].plot(q, np.real(fa_c * np.conj(fa_c)),sy=[8,0.5,3], le='complex integrand') p[0].plot(q, sp.Y, li=1, sy=0, le='analytic') p[1].plot(q,np.abs(fa_r**2 -sp.Y), le='real integrand') p[1].plot(q,np.abs(fa_rh**2 -sp.Y), le='real integrand h-adaptive') p[1].plot(q,np.abs(np.real(fa_c * np.conj(fa_c)) -sp.Y),sy=[8,0.5,3]) p[0].yaxis(scale='l',label='F(q)',ticklabel=['power',0]) p[0].xaxis(ticklabel=0) p[0].legend(x=2,y=1e6) p[1].yaxis(scale='l',label=r'error', ticklabel=['power',0],min=1e-13,max=5e-6) p[1].xaxis(label=r'q / nm\S-1') p[1].text(r'error = abs(F(Q) - F(q)\sanalytic\N)',x=1.5,y=1e-9) p[0].title('Numerical quadrature sphere formfactor ') p[0].subtitle('stop criterion relerror=1e-3, real errors are smaller') #p.save(js.examples.imagepath+'/cubature.jpg')
-
jscatter.formel.
parDistributedAverage
(funktion, sig, parname, type='normal', nGauss=30, **kwargs)[source]¶ Vectorized average assuming a single parameter is distributed with width sig.
Function average over a parameter with weights determined from probability distribution. Gaussian quadrature over given distribution or summation with weights is used.
- Parameters
- funktionfunction
Function to integrate with distribution weight. Function needs to return dataArray.
- sigfloat
width parameter of the distribution, see Notes
- parnamestring
Name of the parameter of funktion which shows a distribution
- type‘normal’,’lognorm’,’gamma’,’lorentz’,’uniform’,’poisson’,’schulzZimm’,’duniform’, default ‘normal’
Type of the distribution
- kwargsparameters
Any additonal kword parameter to pass to function. The value of parname will be the mean value of the distribution.
- nGaussfloat , default=30
Order of quadrature integration as number of intervals in Gauss–Legendre quadrature over distribution. Distribution is integrated in probability interval [0.001..0.999].
- ncpuint, optional
Number of cpus in the pool. Set this to 1 if the integrated function uses multiprocessing to avoid errors.
0 -> all cpus are used
int>0 min (ncpu, mp.cpu_count)
int<0 ncpu not to use
- Returns
- dataArray
- as returned from function with
.parname_mean = mean of parname
.parname_std = standard deviation of parname
Notes
The used distributions are from scipy.stats. Choose the distribution according to the problem.
mean is the value in kwargs[parname]. mean and sig are used as:
- norm :
- mean , stdstats.norm(loc=mean,scale=sig)
- lognorm :
- mean and sig evaluate to mean and stdmu=math.log(mean**2/(sig+mean**2)**0.5)nu=(math.log(sig/mean**2+1))**0.5stats.lognorm(s=nu,scale=math.exp(mu))
- gamma :
- mean and sig evaluate to mean and stdstats.gamma(a=mean**2/sig**2,scale=sig**2/mean)
- lorentz = cauchy:
- mean and std are not defined. Use FWHM instead to describe width.sig=FWHMstats.cauchy(loc=mean,scale=sig))
- uniform :
- Continuous distribution.sig is widthstats.uniform(loc=mean-sig/2.,scale=sig))
- schulzZimm
- poisson:
stats.poisson(mu=mean,loc=sig)
- duniform:
- Uniform distribution integer values.sig>1stats.randint(low=mean-sig, high=mean+sig)
For more distribution look into this source code and use it appropriate with scipy.stats.
Examples
import jscatter as js p=js.grace() q=js.loglist(0.1,5,500) sp=js.ff.sphere(q=q,radius=5) p.plot(sp,sy=[1,0.2],legend='single radius') p.yaxis(scale='l',label='I(Q)') p.xaxis(scale='n',label='Q / nm') sig=0.2 p.title('radius distribution with width %.g' %(sig)) sp2=js.formel.pDA(js.ff.sphere,sig,'radius',type='normal',q=q,radius=5,nGauss=100) p.plot(sp2,li=[1,2,2],sy=0,legend='normal 100 points Gauss ') sp4=js.formel.pDA(js.ff.sphere,sig,'radius',type='normal',q=q,radius=5,nGauss=30) p.plot(sp4,li=[1,2,3],sy=0,legend='normal 30 points Gauss ') sp5=js.formel.pDA(js.ff.sphere,sig,'radius',type='normal',q=q,radius=5,nGauss=5) p.plot(sp5,li=[1,2,5],sy=0,legend='normal 5 points Gauss ') sp3=js.formel.pDA(js.ff.sphere,sig,'radius',type='lognormal',q=q,radius=5) p.plot(sp3,li=[3,2,4],sy=0,legend='lognormal') sp6=js.formel.pDA(js.ff.sphere,sig,'radius',type='gamma',q=q,radius=5) p.plot(sp6,li=[2,2,6],sy=0,legend='gamma ') sp9=js.formel.pDA(js.ff.sphere,sig,'radius',type='schulzZimm',q=q,radius=5) p.plot(sp9,li=[3,2,2],sy=0,legend='SchulzZimm ') # an unrealistic example sp7=js.formel.pDA(js.ff.sphere,1,'radius',type='poisson',q=q,radius=5) p.plot(sp7,li=[1,2,6],sy=0,legend='poisson ') sp8=js.formel.pDA(js.ff.sphere,1,'radius',type='duniform',q=q,radius=5) p.plot(sp8,li=[1,2,6],sy=0,legend='duniform ') p.legend()
-
jscatter.formel.
parQuadratureAdaptiveGauss
(func, lowlimit, uplimit, parname, weights=None, tol=1e-08, rtol=1e-08, maxiter=50, miniter=8, **kwargs)[source]¶ Vectorized definite integral using fixed-tolerance Gaussian quadrature.
Adaptive integration of func from a to b using Gaussian quadrature adaptivly increasing number of points by 8. If function returns ndarray each element is integrated and returned as ndarray of same dimension. If function returns dataArray, only the .Y values is integrated and returned as dataArray.
- Parameters
- funcfunction
A function or method to integrate.
- lowlimitfloat
Lower limit of integration.
- uplimitfloat
Upper limit of integration.
- parnamestring
name of the integration variable which should be a scalar.
- weightsndarray shape(2,N),default=None
Weights for integration along parname as a Gaussian with a<weights[0]<b and weights[1] contains weight values.
Missing values are linear interpolated (faster). If None equal weights are used.
- kwargsdict, optional
Extra keyword arguments to pass to function, if any.
- tol, rtolfloat, optional
Iteration stops when error between last two iterates is less than tol OR the relative change is less than rtol.
- maxiterint, default 50, optional
Maximum order of Gaussian quadrature.
- miniterint, default 8, optional
Minimum order of Gaussian quadrature.
- ncpuint, default=1, optional
Number of cpus in the pool. Set this to 1 if the integrated function uses multiprocessing to avoid errors.
0 -> all cpus are used
int>0 min (ncpu, mp.cpu_count)
int<0 ncpu not to use
- Returns
- valfloat
Gaussian quadrature approximation (within tolerance) to integral for all vector elements.
- errfloat
Difference between last two estimates of the integral.
Notes
Reimplementation of scipy.integrate.quadrature.quadrature to work with vector output of the integrand function.
References
Examples
t=np.r_[1:100] gg=js.formel.gauss(t,50,10) js.formel.parQuadratureAdaptiveGauss(js.formel.gauss,0,100,'x',mean=50,sigma=10)
-
jscatter.formel.
parQuadratureFixedGauss
(func, lowlimit, uplimit, parname, n=5, weights=None, **kwargs)[source]¶ Vectorized definite integral using fixed-order Gaussian quadrature.
Integrate func over parname from a to b using Gauss-Legendre quadrature [1] of order n for all .X. If function returns ndarray all is integrated and returned as ndarray. If function returns dataArray, only the .Y values are integrated and returned as dataArray.
- Parameters
- funccallable
A Python function or method returning a vector array of dimension 1. If func returns dataArray .Y is integrated.
- lowlimitfloat
Lower limit of integration.
- uplimitfloat
Upper limit of integration.
- parnamestring
Name of the integration variable which should be a scalar. After evaluation the corresponding attribute has the mean value with weights.
- weightsndarray shape(2,N),default=None
Weights for integration along parname with a<weights[0]<b and weights[1] contains weight values.
Missing values are linear interpolated (faster). If None equal weights are used.
- kwargsdict, optional
Extra keyword arguments to pass to function, if any.
- nint, optional
Order of quadrature integration. Default is 5.
- ncpuint,default=1, optional
Use parallel processing for the function with ncpu parallel processes in the pool. Set this to 1 if the function is already fast enough or if the integrated function uses multiprocessing.
0 -> all cpus are used
int>0 min (ncpu, mp.cpu_count)
int<0 ncpu not to use
- Returns
- array or dataArray
Notes
Reimplementation of scipy.integrate.quadrature.fixed_quad to work with vector output of the integrand function and weights.
References
Examples
t=np.r_[1:100] gg=js.formel.gauss(t,50,10) js.formel.parQuadratureFixedGauss(js.formel.gauss,0,100,'x',mean=50,sigma=10,n=50)
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jscatter.formel.
parQuadratureFixedGaussxD
(func, lowlimit, uplimit, parnames, n=5, weights0=None, weights1=None, weights2=None, **kwargs)[source]¶ Vectorized fixed-order Gauss-Legendre quadrature in definite interval in 1,2,3 dimensions.
Integrate func over parnames between limits using Gauss-Legendre quadrature [1] of order n.
- Parameters
- funccallable
Function to integrate. The return value should be 2 dimensional with first dimension along integration variable and second along array to calculate. See examples.
- parnameslist of string, max len=3
Name of the integration variables which should be scalar in the function.
- lowlimitlist of float
Lower limits a of integration for parnames.
- uplimitlist of float
Upper limits b of integration for parnames.
- weights0,weights1,weights3ndarray shape(2,N), default=None
Weights for integration along parname with a<weightsi[0]<b and weightsi[1] contains weight values.
Missing values are linear interpolated (faster). If None equal weights are used.
- kwargsdict, optional
Extra keyword arguments to pass to function, if any.
- nint, optional
Order of quadrature integration for all parnames. Default is 5.
- Returns
- array or dataArray
Notes
To get a speedy integration the function should use numpy ufunctions which operate on numpy arrays with compiled code.
References
Examples
The following integrals in 1-3 dimensions over a normalised Gaussian give always 1 which achieved with reasonable accuracy with n=15.
The examples show different ways to return 2dim arrays with x,y,z in first dimension and vector q in second. x[:,None] adds a second dimension to array x.
import jscatter as js import numpy as np q=np.r_[0.1:5.1:0.1] def gauss(x,mean,sigma): return np.exp(-0.5 * (x - mean) ** 2 / sigma ** 2) / sigma / np.sqrt(2 * np.pi) # 1dimensional def gauss1(q,x,mean=0,sigma=1): g=np.exp(-0.5 * (x[:,None] - mean) ** 2 / sigma ** 2) / sigma / np.sqrt(2 * np.pi) return g*q js.formel.pQFGxD(gauss1,0,100,parnames='x',mean=50,sigma=10,q=q,n=15) # 2 dimensional def gauss2(q,x,y=0,mean=0,sigma=1): g=gauss(x[:,None],mean,sigma)*gauss(y[:,None],mean,sigma) return g*q js.formel.pQFGxD(gauss2,[0,0],[100,100],parnames=['x','y'],mean=50,sigma=10,q=q,n=15) # 3 dimensional def gauss3(q,x,y=0,z=0,mean=0,sigma=1): g=gauss(x,mean,sigma)*gauss(y,mean,sigma)*gauss(z,mean,sigma) return g[:,None]*q js.formel.pQFGxD(gauss3,[0,0,0],[100,100,100],parnames=['x','y','z'],mean=50,sigma=10,q=q,n=15)
Usage of weights allows weights for dimensions e.g. to realise a spherical average with weight
in the integral
. (Using the weight in the function is more accurate.) The weight needs to be normalised by unit sphere area
.
import jscatter as js import numpy as np q=np.r_[0,0.1:5.1:0.1] def cuboid(q, phi, theta, a, b, c): pi2 = np.pi * 2 fa = (np.sinc(q * a * np.sin(theta[:,None]) * np.cos(phi[:,None]) / pi2) * np.sinc(q * b * np.sin(theta[:,None]) * np.sin(phi[:,None]) / pi2) * np.sinc(q * c * np.cos(theta[:,None]) / pi2)) return fa**2*(a*b*c)**2 # generate weight for sin(theta) dtheta integration (better to integrate in cuboid function) # and normalise for unit sphere t = np.r_[0:np.pi:180j] wt = np.c_[t,np.sin(t)/np.pi/4].T Fq=js.formel.pQFGxD(cuboid,[0,0],[2*np.pi,np.pi],parnames=['phi','theta'],weights1=wt,q=q,n=15,a=1.9,b=2,c=2) # compare the result to the ff solution (which does the same with weights in the function). p=js.grace() p.plot(q,Fq) p.plot(js.ff.cuboid(q,1.9,2,2),li=1,sy=0) p.yaxis(scale='l')
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jscatter.formel.
parQuadratureSimpson
(funktion, lowlimit, uplimit, parname, weights=None, tol=1e-06, rtol=1e-06, dX=None, **kwargs)[source]¶ Vectorized quadrature over one parameter with weights using the adaptive Simpson rule.
Integrate by adaptive Simpson integration for all .X values at once. Only .Y values are integrated and checked for tol criterion. Attributes and non .Y columns correspond to the weighted mean of parname.
- Parameters
- funktionfunction
Function returning dataArray or array
- lowlimit,uplimitfloat
Interval borders to integrate
- parnamestring
Parname to integrate
- weightsndarray shape(2,N),default=None
Weights for integration along parname as a Gaussian with a<weights[0]<b and weights[1] contains weight values.
Missing values are linear interpolated (faster). If None equal weights are used.
- tol,rtolfloat, default=1e-6
- Relative error or absolute error to stop integration. Stop if one is full filled.Tol is divided for each new interval that the sum of tol is kept..IntegralErr_funktEvaluations in dataArray contains error and number of points in interval.
- dXfloat, default=None
Minimal distance between integration points to determine a minimal step for integration variable.
- kwargs :
Additional parameters to pass to funktion. If parname is in kwargs it is overwritten.
- Returns
- dataArray or array
dataArrays have additional parameters as error and weights.
Notes
What is the meaning of tol in simpson method? If the error in an interval exceeds tol, the algorithm subdivides the interval in two equal parts with each
and applies the method to each subinterval in a recursive manner. The condition in interval i is
. The recursion stops in an interval if the improvement is smaller than tol. Thus tol is the upper estimate for the total error.
Here we use a absolute (tol) and relative (rtol) criterion:
with
as the next improved value As this is tested for all .X the worst case is better than tol, rtol.
The algorithm is efficient as it memoizes function evaluation at each interval border and reuses the result. This reduces computing time by about a factor 3-4.
Different distribution can be found in scipy.stats. But any distribution given explicitly can be used. E.g. triangular np.c_[[-1,0,1],[0,1,0]].T
References
Examples
Integrate Gaussian as test case
import jscatter as js import numpy as np import scipy # testcase: integrate over x of a function # area under normalized gaussian is 1 js.formel.parQuadratureSimpson(js.formel.gauss,-10,10,'x',mean=0,sigma=1)
Integrate a function over one parameter with a weighting function. If weight is 1 the result is a simple integration. Here the weight corresponds to a normal distribution and the result is a weighted average as implemented in parDistributedAverage using fixedGaussian quadrature.
# normal distribtion of parameter D with width ds t=np.r_[0:150:0.5] D=0.3 ds=0.1 diff=js.dynamic.simpleDiffusion(t=t,q=0.5,D=D) distrib =scipy.stats.norm(loc=D,scale=ds) x=np.r_[D-5*ds:D+5*ds:30j] pdf=np.c_[x,distrib.pdf(x)].T diff_g=js.formel.parQuadratureSimpson(js.dynamic.simpleDiffusion,-3*ds+D,3*ds+D,parname='D', weights=pdf,tol=0.01,q=0.5,t=t) # compare it p=js.grace() p.plot(diff,le='monodisperse') p.plot(diff_g,le='polydisperse') p.xaxis(scale='l') p.legend()
-
jscatter.formel.
psphereAverage
(funktion, relError=300, *args, **kwargs)[source]¶ parallel sphereAverage; see
psphereAverage()
-
jscatter.formel.
randomPointsInCube
(NN, skip=0, dim=3)[source]¶ Random points in cube; see
randomPointsInCube()
-
jscatter.formel.
randomPointsOnSphere
(NN, r=1, skip=0)[source]¶ Random points on sphere; see
randomPointsOnSphere()
-
jscatter.formel.
rotationMatrix
(vector, angle)[source]¶ Create a rotation matrix corresponding to rotation around vector v by a specified angle.
See Notes for scipy rotation matrix.
- Parameters
- vectorarray
Rotation around a general vector
- anglefloat
Angle in rad
- Returns
- array
Rotation matrix
Notes
A convenient way to define more complex rotations is found in scipy.spatial.transform.Rotation . E.g. by Euler angles and returned as rotation matrix
from scipy.spatial.transform import Rotation as Rot R=Rot.from_euler('YZ',[90,10],1).as_dcm()
Examples
Examples show how to use a rotation matrix with vectors.
import jscatter as js import numpy as np from matplotlib import pyplot R=js.formel.rotationMatrix([0,0,1],np.deg2rad(-90)) v=[1,0,0] # rotated vector rv=np.dot(R,v) # # rotate Fibonacci Grid qfib=js.formel.fibonacciLatticePointsOnSphere(300,1) qfib=qfib[qfib[:,2]<np.pi/2,:] # select half sphere qfib[:,2]*=(30/90.) # shrink to cone of 30° qfx=js.formel.rphitheta2xyz(qfib) # transform to cartesian R=js.formel.rotationMatrix([0,1,0],np.deg2rad(-90)) # rotation matrix around y axis Rfx=np.einsum('ij,kj->ki',R,qfx) # do rotation fig = pyplot.figure() ax = fig.add_subplot(111, projection='3d') sc=ax.scatter(qfx[:,0], qfx[:,1], qfx[:,2], s=2, color='r') sc=ax.scatter(Rfx[:,0], Rfx[:,1], Rfx[:,2], s=2, color='b') ax.scatter(0,0,0, s=55, color='g',alpha=0.5) fig.axes[0].set_title('rotate red points to blue around origin (green)') pyplot.show(block=False) fig.savefig(js.examples.imagepath+'/rotationMatrix.jpg')
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jscatter.formel.
rphitheta2xyz
(RPT, transpose=False)[source]¶ Transformation spherical coordinates [r,phi,theta] to cartesian coordinates [x,y,z].
- Parameters
- RPTarray Nx3
- Coordinates with [r,phi,theta]
r : float length
phi : float azimuth -pi < phi < pi
theta : float polar angle 0 < theta < pi
- transposebool
Transpose RPT before transformation.
- Returns
- array Nx3
[x,y,z] coordinates
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jscatter.formel.
sQS
(funktion, lowlimit, uplimit, parname, weights=None, tol=1e-06, rtol=1e-06, **kwargs)¶ Integrate a scalar function over one of its parameters with weights using the adaptive Simpson rule.
Integrate by adaptive Simpson integration for scalar function.
- Parameters
- funktionfunction
function to integrate
- lowlimit,uplimitfloat
interval to integrate
- parnamestring
parname to integrate
- weightsndarray shape(2,N),default=None
Weights for integration along parname as a Gaussian with a<weights[0]<b and weights[1] contains weight values.
Missing values are linear interpolated (faster). If None equal weights are used.
- tol,rtolfloat, default=1e-6
- Relative error for intervals or absolute integral error to stop integration.
- kwargs :
additional parameters to pass to funktion if parname is in kwargs it is overwritten
- Returns
- float
Notes
What is the meaning of tol in simpson method? See parQuadratureSimpson.
References
Examples
distrib =scipy.stats.norm(loc=1,scale=0.2) x=np.linspace(0,1,1000) pdf=np.c_[x,distrib.pdf(x)].T # define function f1=lambda x,p1,p2,p3:js.dA(np.c_[x,x*p1+x*x*p2+p3].T) # calc the weighted integral result=js.formel.parQuadratureSimpson(f1,0,1,parname='p2',weights=pdf,tol=0.01,p1=1,p3=1e-2,x=x) # something simple should be 1 js.formel.simpleQuadratureSimpson(js.formel.gauss,-10,10,'x',mean=0,sigma=1)
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jscatter.formel.
scatteringFromSizeDistribution
(q, sizedistribution, size=None, func=None, weight=None, **kwargs)[source]¶ Average function assuming one multimodal parameter like bimodal.
Distributions might be mixtures of small and large particles bi or multimodal. For predefined distributions see formel.parDistributedAverage with examples. The weighted average over given sizedistribution is calculated.
- Parameters
- qarray of float;
Wavevectors to calculate scattering; unit = 1/unit(size distribution)
- sizedistributiondataArray or array
Explicit given distribution of sizes as [ [list size],[list probability]]
- sizestring
Name of the parameter describing the size (may be also something different than size).
- funclambda or function, default beaucage
Function that describes the form factor with first arguments (q,size,…) and should return dataArray with .Y as result as eg func=js.ff.sphere.
- kwargs :
Any additional keyword arguments passed to for func.
- weightfunction
Weight function dependent on size. E.g. weight = lambda R:rho**2 * (4/3*np.pi*R**3)**2 with V= 4pi/3 R**3 for normalized form factors to account for forward scattering of volume objects of dimension 3.
- Returns
- dataArray
Columns [q,I(q)]
Notes
We have to discriminate between formfactor normalized to 1 (e.g. beaucage) and form factors returning the absolute scattering (e.g. sphere) including the contrast. The later contains already
, the first not.
We need for normalized formfactors P(q)
with
as number density
as difference in average scattering length (contrast), V as volume of particle (~r³ ~ mass) and use
For a gaussian chain with
and monomer number N (nearly 2D object) we find
and the forward scattering as weight
Examples
The contribution of different simple sizes to Beaucage
import jscatter as js q=js.loglist(0.01,6,100) p=js.grace() # bimodal with equal concentration bimodal=[[12,70],[1,1]] Iq=js.formel.scatteringFromSizeDistribution(q=q,sizedistribution=bimodal, d=3,weight=lambda r:(r/12)**6,func=js.ff.beaucage) p.plot(Iq,legend='bimodal 1:1 weight ~r\S6 ') Iq=js.formel.scatteringFromSizeDistribution(q=q,sizedistribution=bimodal,d=3,func=js.ff.beaucage) p.plot(Iq,legend='bimodal 1:1 weight equal') # 2:1 concentration bimodal=[[12,70],[1,5]] Iq=js.formel.scatteringFromSizeDistribution(q=q,sizedistribution=bimodal,d=2.5,func=js.ff.beaucage) p.plot(Iq,legend='bimodal 1:5 d=2.5') p.yaxis(label='I(q)',scale='l') p.xaxis(scale='l',label='q / nm\S-1') p.title('Bimodal size distribution Beaucage particle') p.legend(x=0.2,y=10000) #p.save(js.examples.imagepath+'/scatteringFromSizeDistribution.jpg')
Three sphere sizes:
import jscatter as js q=js.loglist(0.001,6,1000) p=js.grace() # trimodal with equal concentration trimodal=[[10,50,500],[1,0.01,0.00001]] Iq=js.formel.scatteringFromSizeDistribution(q=q,sizedistribution=trimodal,size='radius',func=js.ff.sphere) p.plot(Iq,legend='with aggregates') p.yaxis(label='I(q)',scale='l',max=1e13,min=1) p.xaxis(scale='l',label='q / nm\S-1') p.text(r'minimum \nlargest',x=0.002,y=1e10) p.text(r'minimum \nmiddle',x=0.02,y=1e7) p.text(r'minimum \nsmallest',x=0.1,y=1e5) p.title('trimodal spheres') p.subtitle('first minima indicated') #p.save(js.examples.imagepath+'/scatteringFromSizeDistributiontrimodal.jpg')
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jscatter.formel.
scatteringLengthDensityCalc
(composition, density=None, T=293, units='mol', mode='all')[source]¶ Scattering length density of composites and water with inorganic components for xrays and neutrons.
- Parameters
- compositionlist of concentration + chemical formula
A string with chemical formula as letter + number and prepended concentration in mol or mmol. E.g. ‘0.1C3H8O3’ or ‘0.1C3H1D7O3’ for glycerol and deuterated glycerol (‘D’ for deuterium).
For single atoms append 1 to avoid confusion. Fractional numbers allowed, but think of meaning (Isotope mass fraction??)
For compositions use a list of strings preceded by mass fraction or concentration in mol of component. This will be normalized to total amount
- Examples:
[‘4.5H2O1’,’0.5D2O1’] mixture of 10% heavy and 90% light water.
[‘1.0h2o’,’0.1c3h8o3’] for 10% mass glycerol added to 100% h2o with units=’mass’
[‘55000H2O1’,’50Na3P1O4’,’137Na1Cl1’] for a 137mMol NaCl +50mMol phophate H2O buffer (1l is 55000 mM H2O)
[‘1Au1’] gold with density 19.302 g/ml
Remember to adjust density.
- densityfloat, default=None
Density in g/cm**3 = g/ml. If not given function waterdensity is tried to calculate the solution density with inorganic components. In this case ‘h2o1’ and/or ‘d2o1’ need to be in composition. Otherwise measure by weighting a volume from pipette (lower accuracy) or densiometry (higher accuracy).
- units‘mol’
Anything except ‘mol’ prepended unit is mass fraction (default). ‘mol’ prepended units is mol and mass fraction is calculated as mass=[mol]*mass_of_molecule e.g. 1l Water with 123mmol NaCl [‘55.5H2O1’,’0.123Na1Cl1’]
- mode :
‘xsld’ xray scattering length density in nm**-2
‘edensity’ electron density in e/nm**3
‘ncohsld’ coherent scattering length density in nm**-2
‘incsld’ incoherent scattering length density in nm**-2
- ‘all’ [xsld, edensity, ncohsld, incsld,
masses, masses full protonated, masses full deuterated, d2o/h2o fraction in composition]
- Tfloat, default=293
Temperature in °K
- Returns
- float, list
sld corresponding to mode
Notes
edensity=be*massdensity/weightpermol*sum_atoms(numberofatomi*chargeofatomi)
be = scattering length electron =µ0*e**2/4/pi/m=2.8179403267e-6 nm
masses, masses full protonated, masses full deuterated for each chemical in composition.
In mode ‘all’ the masses can be used to calc the deuterated density if same volume is assumed. e.g. fulldeuterated_density=protonated_density/massfullprotonated*massfulldeuterated
For density reference of H2O/D2O see waterdensity.
Examples
# 5% D2O in H2O with 10% mass NaCl js.formel.scatteringLengthDensityCalc(['9.5H2O1','0.5D2O1','1Na1Cl1'],units='mass') # protein NaPi buffer in D2O prevalue in mmol; 55000 mmol H2O is 1 liter. js.formel.scatteringLengthDensityCalc(['55000D2O1','50Na3P1O4','137Na1Cl1']) # silica js.formel.scatteringLengthDensityCalc('1Si1O2',density=2.65) # gold js.formel.scatteringLengthDensityCalc(['1Au1'],density=19.32) # PEG1000 js.formel.scatteringLengthDensityCalc(['1C44H88O23'],density=1.21)
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jscatter.formel.
schulzZimm
(r, mean, sigma)[source]¶ Schulz-Zimm distribution for polymeric particles/chains.
- Distribution describing a polymerisation like radical polymerization:
constant number of chains growth till termination,
concentration of active centers constant
start of chain growth not necessarily at the same time
- Parameters
- rarray
Distribution variable such as relative molecular mass or degree of polymerization.
- meanfloat
Mean
- sigmafloat
Width of the distribution. z = (mean/sigma)² -1 < 600
- Returns
- dataArrayColumns [x,p]
- .z ==> z+1 = k is degree of coupling = number of chain combined to dead chain in termination reaction
z = (mean/sigma)² -1
Notes
The Schulz-Zimm distribution
Normalized to
.
- Nth order average
number average
weight average
z average
References
- 1
Schulz, G. V. Z. Phys. Chem. 1939, 43, 25
- 2
Theory of dynamic light scattering from polydisperse systems S. R. Aragón and R. Pecora The Journal of Chemical Physics, 64, 2395 (1976)
Examples
import jscatter as js N=np.r_[1:200] p=js.grace(1.4,1) p.multi(1,2) m=50 for i,s in enumerate([10,20,40,50,75,100,150],1): SZ = js.formel.schulzZimm(N,mean=m,sigma=s) p[0].plot(SZ.X/m,SZ.Y,sy=0,li=[1,3,i],le=f'sigma/mean={s/m:.1f}') p[1].plot(SZ.X/m,SZ.Y*SZ.X,sy=0,li=[1,3,i],le=f'sigma/mean={s/m:.1f}') p[0].xaxis(label='N/mean') p[0].yaxis(label='h(N)') p[0].subtitle('number distribution') p[1].xaxis(label='N/mean') p[1].yaxis(label='N h(N)') p[1].subtitle('mass distribution') p[1].legend(x=2,y=1.5) p[0].title('Schulz-Zimm distribution') #p.save(js.examples.imagepath+'/schulzZimm.jpg')
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jscatter.formel.
schulzZimmDistribution
= <jscatter.formel._schulzZimm_gen object>¶ Schulz-Zimm distribution as scipy.stats.rv_continuous object
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jscatter.formel.
sedimentationCoefficient
(M, partialVol=None, density=None, visc=None)[source]¶ Sedimentation coefficient of a sphere in a solvent.
with
- Parameters
- Mfloat
Mass of the sphere or protein in units Da.
- partialVolfloat
Partial specific volume
of the particle in units ml/g = l/kg. Default is 0.73 ml/g for proteins.
- densityfloat
Density
of the solvent in units g/ml=kg/l. Default is H2O at 293.15K
- viscfloat
Solvent viscosity
in Pas. Default H2O at 293.15K
- Returns
- float
Sedimentation coefficient in units Svedberg (1S = 1e-13 sec )
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jscatter.formel.
sedimentationProfile
(t=1000.0, rm=48, rb=85, number=100, rlist=None, c0=0.01, S=None, Dt=None, omega=246, Rh=10, temp=293, densitydif=0.37, visc='h2o')[source]¶ Concentration profile of sedimenting particles in a centrifuge including bottom equilibrium distribution.
Approximate solution to the Lamm equation including the bottom equilibrium distribution which is not included in the Faxen solution. This calculates equ. 28 in [1]. Results in particle concentration profile between rm and rb for time t with a equal distribution at the start.
- Parameters
- tfloat or list
Time after centrifugation start in seconds. If list, a dataList for all times is returned.
- rmfloat
Axial position of meniscus in mm.
- rbfloat
Axial position of bottom in mm.
- numberint
Number of points between rm and rb to calculate.
- rlistlist of float
Explicit list of positions where to calculate eg to zoom bottom.
- c0float
Initial concentration in cell; just a scaling factor.
- Sfloat
Sedimentation coefficient in units Svedberg; 82 S is r=10 nm protein in H2O at T=20C.
- Dtfloat
Translational diffusion coefficient in m**2/s; 1.99e-11 is r=10 nm particle.
- omegafloat
Radial velocity rounds per second; 14760rpm = 246 rps = 1545 rad/s is 20800g in centrifuge fresco 21.
- Rhfloat
Hydrodynamic radius in nm ; if given the Dt and s are calculated from this.
- densitydiffloat
Density difference between solvent and particle in g/ml; Protein in ‘h2o’ => 1.37-1.= 0.37 g/cm**3.
- viscfloat, ‘h2o’, ‘d2o’
Viscosity of the solvent in Pas. (H2O ~ 0.001 Pa*s =1 cPoise) If ‘h2o’ or ‘d2o’ the corresponding viscosity at given temperature is used.
- tempfloat
temperature in K.
- Returns
- dataArray, dataList
Concentration profile Columns [position in [mm]; conc ; conc_meniscus_part; conc_bottom_part]
Notes
From [1]:”The deviations from the expected results are smaller than 1% for simulated curves and are valid for a great range of molecular masses from 0.4 to at least 7000 kDa. The presented approximate solution, an essential part of LAMM allows the estimation of s and D with an accuracy comparable to that achieved using numerical solutions, e.g the program SEDFIT of Schuck et al.”
Default values are for Heraeus Fresco 21 at 21000g.
References
- 1(1,2)
A new approximate whole boundary solution of the Lamm equation for the analysis of sedimentation velocity experiments J. Behlke, O. Ristau Biophysical Chemistry 95 (2002) 59–68
Examples
Cleaning from aggregates by sedimantation. Sedimentation of protein (R=2 nm) with aggregates of 100nm size.
import numpy as np import jscatter as js t1=np.r_[60:1.15e3:11j] # time in seconds # open plot() p=js.grace(1.5,1.5) p.multi(2,1) # calculate sedimentation profile for two different particles # data correspond to Fresco 21 with dual rotor # default is solvent='h2o',temp=293 g=21000. # g # RZB number omega=g*246/21000 D2nm=js.formel.sedimentationProfile(t=t1,Rh=2,densitydif=0.37, number=1000) D50nm=js.formel.sedimentationProfile(t=t1,Rh=50,densitydif=0.37, number=1000) # plot it p[0].plot(D2nm,li=[1,2,-1],sy=0,legend='t=$time s' ) p[1].plot(D50nm,li=[1,2,-1],sy=0,legend='t=$time s' ) # pretty up p[0].yaxis(min=0,max=0.05,label='concentration') p[1].yaxis(min=0,max=0.05,label='concentration') p[1].xaxis(label='position mm') p[0].xaxis(label='') p[0].text(x=70,y=0.04,string=r'R=2 nm \nno sedimantation') p[1].text(x=70,y=0.04,string=r'R=50 nm \nfast sedimentation') p[0].legend(x=42,y=0.05,charsize=0.5) p[1].legend(x=42,y=0.05,charsize=0.5) p[0].title('Concentration profile in first {0:} s'.format(np.max(t1))) p[0].subtitle('2nm and 50 nm particles at 21000g ') #p.save(js.examples.imagepath+'/sedimentation.jpg')
Sedimentation (up concentration) of unilamellar liposomes of DOPC. The density of DOPC is about 1.01 g/ccm in water with 1 g/ccm. Lipid volume fraction is 33% for 50nm radius (10% for 200 nm) for a bilayer thickness of 6.5 nm.
import numpy as np import jscatter as js t1=np.r_[100:6e3:11j] # time in seconds # open plot() p=js.grace(1.5,1.5) p.multi(2,1) # calculate sedimentation profile for two different particles # data correspond to Fresco 21 with dual rotor # default is solvent='h2o',temp=293 g=21000. # g # RZB number omega=g*246/21000 D100nm=js.formel.sedimentationProfile(t=t1,Rh=50,c0=0.05,omega=omega,rm=48,rb=85,densitydif=0.003) D400nm=js.formel.sedimentationProfile(t=t1,Rh=200,c0=0.05,omega=omega,rm=48,rb=85,densitydif=0.001) # plot it p[0].plot(D100nm,li=[1,2,-1],sy=0,legend='t=$time s' ) p[1].plot(D400nm,li=[1,2,-1],sy=0,legend='t=$time s' ) # pretty up p[0].yaxis(min=0,max=0.2,label='concentration') p[1].yaxis(min=0,max=0.2,label='concentration') p[1].xaxis(label='position mm') p[0].xaxis(label='') p[0].text(x=70,y=0.15,string='D=100 nm') p[1].text(x=70,y=0.15,string='D=400 nm') p[0].legend(x=42,y=0.2,charsize=0.5) p[1].legend(x=42,y=0.2,charsize=0.5) p[0].title('Concentration profile in first {0:} s'.format(np.max(t1))) p[0].subtitle('at 21000g ')
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jscatter.formel.
sedimentationProfileFaxen
(t=1000.0, rm=48, rb=85, number=100, rlist=None, c0=0.01, s=None, Dt=1.99e-11, w=246, Rh=10, visc='h2o', temp=293, densitydif=None)[source]¶ Faxen solution to the Lamm equation of sedimenting particles in centrifuge; no bottom part.
Bottom equillibrium distribution is not in Faxen solution included. Results in particle distribution along axis for time t.
- Parameters
- tfloat
Time after start in seconds. If list, results at these times is given as dataList.
- rmfloat
Axial position of meniscus in mm.
- rbfloat
Axial position of bottom in mm.
- rlistarray, optional
Explicit list of radial values to use between rm=max(rlist) and rb=min(rlist)
- numberinteger
Number of points between rm and rb to calculate.
- c0float
Initial concentration in cell; just a scaling factor.
- sfloat
Sedimentation coefficient in Svedberg; 77 S is r=10 nm particle in H2O.
- Dtfloat
Translational diffusion coefficient in m**2/s; 1.99e-11 is r=10 nm particle.
- wfloat
Radial velocity rounds per second; 246 rps=2545 rad/s is 20800g in centrifuge fresco 21.
- Rhfloat
Hydrodynamic radius in nm ; if given Dt and s are calculated from Rh.
- viscfloat, ‘h2o’,’d2o’
Viscosity in units Pas. If ‘h2o’ or ‘d2o’ the corresponding viscosity at given temperature is used.
- densitydiffloat
Density difference between solvent and particle in g/ml. Protein in ‘h2o’=> is used =>1.37-1.= 0.37 g/cm**3
- tempfloat
temperature in K.
- Returns
- dataArray, dataList
- Concentration distributiondataArray, dataList
.pelletfraction is the content in pellet as fraction already diffused out .rmeniscus
Notes
Default values are for Heraeus Fresco 21 at 21000g.
References
- 1
Über eine Differentialgleichung aus der physikalischen Chemie. Faxén, H. Ark. Mat. Astr. Fys. 21B:1-6 (1929)
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jscatter.formel.
simpleQuadratureSimpson
(funktion, lowlimit, uplimit, parname, weights=None, tol=1e-06, rtol=1e-06, **kwargs)[source]¶ Integrate a scalar function over one of its parameters with weights using the adaptive Simpson rule.
Integrate by adaptive Simpson integration for scalar function.
- Parameters
- funktionfunction
function to integrate
- lowlimit,uplimitfloat
interval to integrate
- parnamestring
parname to integrate
- weightsndarray shape(2,N),default=None
Weights for integration along parname as a Gaussian with a<weights[0]<b and weights[1] contains weight values.
Missing values are linear interpolated (faster). If None equal weights are used.
- tol,rtolfloat, default=1e-6
- Relative error for intervals or absolute integral error to stop integration.
- kwargs :
additional parameters to pass to funktion if parname is in kwargs it is overwritten
- Returns
- float
Notes
What is the meaning of tol in simpson method? See parQuadratureSimpson.
References
Examples
distrib =scipy.stats.norm(loc=1,scale=0.2) x=np.linspace(0,1,1000) pdf=np.c_[x,distrib.pdf(x)].T # define function f1=lambda x,p1,p2,p3:js.dA(np.c_[x,x*p1+x*x*p2+p3].T) # calc the weighted integral result=js.formel.parQuadratureSimpson(f1,0,1,parname='p2',weights=pdf,tol=0.01,p1=1,p3=1e-2,x=x) # something simple should be 1 js.formel.simpleQuadratureSimpson(js.formel.gauss,-10,10,'x',mean=0,sigma=1)
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jscatter.formel.
smooth
(data, windowlen=7, window='flat')[source]¶ Smooth data by convolution with window function or fft/ifft.
Smoothing based on position ignoring information on .X.
- Parameters
- dataarray, dataArray
Data to smooth. If is dataArray the .Y is smoothed and returned.
- windowlenint, default = 7
The length/size of the smoothing window; should be an odd integer. Smaller 3 returns unchanged data. For ‘fourier’ the high frequency cutoff is 2*size_data/windowlen.
- window‘hanning’, ‘hamming’, ‘bartlett’, ‘blackman’,’gaussian’,’fourier’ default =’flat’
- Type of window/smoothing.
‘flat’ will produce a moving average smoothing.
‘gaussian’ normalized Gaussian window with sigma=windowlen/7.
‘fourier’ cuts high frequencies above cutoff frequency between rfft and irfft.
- Returns
- array (only the smoothed array)
Notes
- ‘hanning’, ‘hamming’, ‘bartlett’, ‘blackman’,’gaussian’, ‘flat’ :
These methods convolve a scaled window function with the signal. The signal is prepared by introducing reflected copies of the signal (with the window size) at both ends so that transient parts are minimized in the beginning and end part of the output signal. Adapted from SciPy/Cookbook.
See numpy.hanning, numpy.hamming, numpy.bartlett, numpy.blackman, numpy.convolve, scipy.signal.gaussian
- fourier :
The real valued signal is mirrored at left side, Fourier transformed, the high frequencies are cut and the signal is back transformed. This is the simplest form as a hard cutoff frequency is used (ideal low pass filter) and may be improved using a specific window function in frequency domain.
rft = np.fft.rfft(np.r_[data[::-1],data]) rft[int(2*len(data)/windowlen):] = 0 smoothed = np.fft.irfft(rft)
Examples
Usage:
import jscatter as js import numpy as np t=np.r_[-5:5:0.01] data=np.sin(t)+np.random.randn(len(t))*0.1 y=js.formel.smooth(data) # 1d array # # smooth dataArray and replace .Y values. data2=js.dA(np.vstack([t,data])) data2.Y=js.formel.smooth(data2, windowlen=40, window='gaussian')
Comparison of some filters:
import jscatter as js import numpy as np t=np.r_[-5:5:0.01] data=js.dA(np.vstack([t,np.sin(t)+np.random.randn(len(t))*0.1])) p=js.grace() p.multi(4,2) windowlen=31 for i,window in enumerate(['flat','gaussian','hanning','fourier']): p[2*i].plot(data,sy=[1,0.1,6],le='original + noise') p[2*i].plot(t,js.formel.smooth(data,windowlen,window),sy=[2,0.1,4],le='filtered') p[2*i].plot(t,np.sin(t),li=[1,0.5,1],sy=0,le='noiseless') p[2*i+1].plot(data,sy=[1,0.1,6],le='original noise') p[2*i+1].plot(t,js.formel.smooth(data,windowlen,window),sy=[2,0.1,4],le=window) p[2*i+1].plot(t,np.sin(t),li=[1,2,1],sy=0,le='noiseless') p[2*i+1].text(window,x=-2.8,y=-1.2) p[2*i+1].xaxis(min=-3,max=-1,) p[2*i+1].yaxis(min=-1.5,max=-0.2,ticklabel=[None,None,None,'opposite']) p[2*i].yaxis(label='y') p[0].legend(x=10,y=4.5) p[6].xaxis(label='x') p[7].xaxis(label='x') p[0].title(f'Comparison of smoothing windows') p[0].subtitle(f'with windowlen {windowlen}') #p.save(js.examples.imagepath+'/smooth.jpg')
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jscatter.formel.
sphereAverage
(function, relError=300, passPoints=False, *args, **kwargs)[source]¶ Vectorized spherical average - non-parallel
A Fibonacci lattice or Monte Carlo integration with pseudo random grid is used.
- Parameters
- functionfunction
Function to evaluate returning a list of return values (all are integrated) function gets cartesian coordinate of point on unit sphere as first argument
- relErrorfloat, default 300
- Determines how points on sphere are selected for integration
>=1 Fibonacci Lattice with relError*2+1 points (min 15 points)
- 0<1 Pseudo random points on sphere (see randomPointsOnSphere).
Stops if relative improvement in mean is less than relError (uses steps of 20*ncpu new points). Final error is (stddev of N points) /sqrt(N) as for Monte Carlo methods. even if it is not a correct 1-sigma error in this case.
- passPointsbool
If the function accepts an Nx3 array of points these will be passed. The function return value should index the points result in axis 0. This might speedup dependent on the function.
- args,kwargs :
Forwarded to function.
- Returns
- array
Values from function and appended Monte Carlo error estimates. To separate use values, error = res.reshape(2,-1)
Examples
import jscatter as js import numpy as np def fp(singlepoint): return js.formel.xyz2rphitheta(singlepoint)[1:] js.formel.sphereAverage(fp,relError=500) js.formel.sphereAverage(fp,relError=0.01) def fps(allpoints): res = js.formel.xyz2rphitheta(allpoints)[:,1:] return res js.formel.sphereAverage(fps,r=1,passPoints=1)
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jscatter.formel.
viscosity
(mat='h2o', T=293.15)[source]¶ Viscosity of pure solvents. For buffer solvents use bufferviscosity.
- Parameters
- matstring ‘h2o’,’d2o’,’toluol’,’methylcyclohexan’, default h2o
Solvent
- Tfloat
Temperature T in Kelvin default 293K
- Returns
- float
- viscosity in Pa*s
water H2O ~ 0.001 Pa*s =1 cPoise # Poise=0.1 Pa*s
References
- 1
The Viscosity of Toluene in the Temperature Range 210 to 370 K M. J. Assael, N.K. Dalaouti, J.H., Dymond International Journal of Thermophysics, Vol. 21,291 No. 2, 2000 # accuracy +- 0.4 % laut paper Max error von Experiment data
- 2
Thermal Offset Viscosities of Liquid H2O, D2O, and T2O C. H. Cho, J. Urquidi, S. Singh, and G. Wilse Robinson J. Phys. Chem. B 1999, 103, 1991-1994
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jscatter.formel.
voigt
(x, center=0, fwhm=1, lg=1, asym=0, amplitude=1)[source]¶ Voigt function for peak analysis (normalized).
The Voigt function is a convolution of gaussian and lorenzian shape peaks for peak analysis. The Lorenzian shows a stronger contribution outside FWHM with a sharper peak. Asymmetry of the shape can be added by a sigmoidal change of the FWHM [2].
- Parameters
- xarray
Axis values.
- centerfloat
Center of the distribution.
- fwhmfloat
Full width half maximum of the Voigt function.
- lgfloat, default = 1
- Lorenzian/gaussian fraction of both FWHM, describes the contributions of gaussian and lorenzian shape.
lorenzian/gaussian >> 1 lorenzian,
lorenzian/gaussian ~ 1 central part gaussian, outside lorenzian wings
lorenzian/gaussian << 1. gaussian
- asymfloat, default=0
asymmetry factor in sigmoidal as
For a=0 the Voigt is symmetric with fwhm.
- amplitudefloat, default = 1
amplitude
- Returns
- dataArray
.center .sigma .gamma .fwhm .asymmetry .lorenzianOverGaussian
References
- 1
- 2
A simple asymmetric lineshape for fitting infrared absorption spectra Aaron L. Stancik, Eric B. Brauns Vibrational Spectroscopy 47 (2008) 66–69
- 3
Empirical fits to the Voigt line width: A brief review Olivero, J. J.; R. L. Longbothum Journal of Quantitative Spectroscopy and Radiative Transfer. 17, 233–236. doi:10.1016/0022-4073(77)90161-3
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jscatter.formel.
watercompressibility
(d2ofract=1, T=278, units='psnmg')[source]¶ Isothermal compressibility of H2O and D2O mixtures.
Compressibility in units ps^2*nm/(g/mol) or in 1/bar. Linear mixture according to d2ofract.
- Parameters
- d2ofractfloat, default 1
Fraction D2O
- Tfloat, default 278K
Temperature in K
- unitsstring ‘psnmg’
ps^2*nm/(g/mol) or 1/bar
- Returns
- float
Notes
To get kT*compressibility =compr*k_B/Nav*300/cm**3 in hwater 1.91e-24 cm**3 at 20°C
References
- 1
Isothermal compressibility of Deuterium Oxide at various Temperatures Millero FJ and Lepple FK Journal of chemical physics 54,946-949 (1971) http://dx.doi.org/10.1063/1.1675024
- 2
Precise representation of volume properties of water at one atmosphere G. S. Kell J. Chem. Eng. Data, 1967, 12 (1), pp 66–69 http://dx.doi.org/10.1021/je60032a018
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jscatter.formel.
waterdensity
(composition, T=293.15, units='mol', showvalidity=False)[source]¶ Density of water with inorganic substances (salts).
Solvent with composition of H2O and D2O and additional inorganic components at temperature T. Ternary solutions allowed. Units are mol; 1l h2o = 55.50843 mol
- Parameters
- compositionlist of compositional strings
- Compositional string of chemical formula as ‘float’+’chemical char’ + integerFirst float is content in mol (is later normalised to sum of contents)chemical letter + number of atoms in formula (single atoms append 1 ,fractional numbers allowed)e.g.‘h2o1’ or ‘d2o1’ light and heavy water with ‘d1’ for deuterium‘c3h8o3’ or ‘c3h1d7o3’ partial deuterated glycerol[‘55.55h2o1’,’2.5Na1Cl1’] for 2.5 mol NaCl added to 1l h2o (55.55 mol)[‘20H2O1’,’35.55D2O1’,’0.1Na1Cl1’] h2o/d2o mixture with 100mMol NaCl
- unitsdefault=’mol’
Anything except ‘mol’ unit is mass fraction ‘mol’ units is mol and mass fraction is calculated as mass=[mol]*mass_of_molecule e.g. 1l Water with 123mM NaCl [‘55.5H2O1’,’0.123Na1Cl1’]
- Tfloat, default=293.15
temperature in K
- showvaliditybool, default False
Show additionally validity range for temperature and concentration according to [4]. - Temperature range in °C - concentration in wt % or up to a saturated solution (satd) - error in 1/100 % see [4].
- Returns
- float
Density in g/ml
Notes
D2O maximum density 1.10596 at T=273.15+11.23 K [1] .
For mixtures of H2O/D2O molar volumes add with an accuracy of about 2e-4 cm**3/mol compared to 18 cm**3/mol molar volume [3].
Additional densities of binary aqueous solutions [4].
References
- 1
The dilatation of heavy water K. Stokland, E. Ronaess and L. Tronstad Trans. Faraday Soc., 1939,35, 312-318 DOI: 10.1039/TF9393500312
- 2
Effects of Isotopic Composition, Temperature, Pressure, and Dissolved Gases on the Density of Liquid Water George S. Kell JPCRD 6(4) pp. 1109-1131 (1977)
- 3
Excess volumes for H2O + D2O liquid mixtures Bottomley G Scott R Australian Journal of Chemistry 1976 vol: 29 (2) pp: 427
- 4(1,2,3)
Densities of binary aqueous solutions of 306 inorganic substances P. Novotny, O. Sohnel J. Chem. Eng. Data, 1988, 33 (1), pp 49–55 DOI: 10.1021/je00051a018
availible components:
h2o1 d2o1 TRIS c4h11n1o3 TABS c8h19n1o6s1 ag1cl104 ag1n1o3 al1cl3012 al1n3o9 al2s3o12 ba1br2 ba1cl2 ba1cl2o6 ba1i2 ba1n2o6 c4h11n1o3 c8h19n1o6s1 ca1br2 ca1cl2 ca1i2 ca1n2o6 cd1br2 cd1c12 cd1i2 cd1n2o6 cd1so4 co1n2o6 co1s1o4 cs1br1 cs1cl1 cs1f1 cs1i1 cs1n1o3 cs2s1o4 cu1cl2 cu1n2o6 cu1s1o4 dy1cl3 er1cl3 fe1cl3 fe1s1o4 gd1cl3 h1br1 h1cl1 h1cl104 h1i1 h1n1o3 h2s1o4 h3b103 h3p1o4 hg1c2n2 hg1cl2 k1ag1c2n2 k1al1s2o8 k1br1 k1br1o3 k1cl1 k1cl103 k1cl104 k1f1 k1h1c1o3 k1h1s1o4 k1h2p1o4 k1i1 k1i103 k1mn1o4 k1n1o2 k1n1o3 k1n3 k1o1h1 k1s1c1n1 k1tart1 k2c1o3 k2cr104 k2cr207 k2s1o4 k3co1c6n6 k3fe1c6n6 k4fe1c6n6 k4mo1c8n8 la1cl3 li1br1 li1cl1 li1cl104 li1i1 li1n1o3 li1o1h1 li2s1o4 mg1br2 mg1br206 mg1cl2 mg1cl2o8 mg1i2 mg1n2o6 mg1s1o4 mn1cl1 mn1s1o4 n1h3 n1h40x1 n1h4ac1 n1h4al1s2o8 n1h4cl1 n1h4cl104 n1h4fe1s2o8 n1h4h1s1o4 n1h4h2p1o4 n1h4i1 n1h4n1o3 n2h8ni1s2o8 n2h8s1o4 n2s203 na1ac1 na1br1 na1br1o3 na1cl1 na1cl103 na1cl104 na1f1 na1form1 na1h1s1o4 na1h2p1o4 na1i1 na1i103 na1k1tart1 na1mn104 na1n1o2 na1n1o3 na1n3 na1o1h1 na1ox1 na1s1c1n1 na1tart1 na2b407 na2c1o3 na2cr1o4 na2cr207 na2h1p1o4 na2mo104 na2s1 na2s103 na2s1o4 na2w104 na3p1o4 na4p207 na5p307 nd1cl3 ni1cl2 ni1n2o6 ni1s1o4 pb1n2o6 pr1cl3 rb1br1 rb1cl1 rb1f1 rb1i1 rb1n1o3 rb2so4 sm1cl3 sr1ac2 sr1br2 sr1br206 sr1cl2 sr1i2 sr1n2o6 ti1n1o3 ti2s1o4 u1o2n2o6 u1o2s104 yb1cl3 zn1br2 zn1cl2 zn1i2 zn1n2o6 zn1s1o1
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jscatter.formel.
xyz2rphitheta
(XYZ, transpose=False)[source]¶ Transformation cartesian coordinates [X,Y,Z] to spherical coordinates [r,phi,theta].
- Parameters
- XYZarray Nx3
Coordinates with [x,y,z] ( XYZ.shape[1]==3).
- transposebool
Transpose XYZ before transformation.
- Returns
- array Nx3
- Coordinates with [r,phi,theta]
phi : float azimuth -pi < phi < pi
theta : float polar angle 0 < theta < pi
r : float length
Examples
Single coordinates
js.formel.xyz2rphitheta([1,0,0])
Transform Fibonacci lattice on sphere to xyz coordinates
rpc=js.formel.randomPointsInCube(10) js.formel.xyz2rphitheta(rpc)
Tranformation 2D X,Y plane coordinates to r,phi coordinates (Z=0)
rp=js.formel.xyz2rphitheta([data.X,data.Z,abs(data.X*0)],transpose=True) )[:,:2]
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class
jscatter.formel.
imageHash
(image, type=None, hashsize=8, highfreq_factor=4)[source]¶ Bases:
object
Creates image hash of an image to find duplicates or similar images in a fast way using the Hamming difference.
Implements * average hashing (aHash) * perception hashing (pHash) * difference hashing (dHash)
- Parameters
- imagefilename, hexstr, PIL image
Image to calculate a hash. If a hexstr is given to restore a saved hash it must prepend ‘0x’ and the 0-padded length determines the hash size. type needs to be given additionally.
- type‘ahash’, ‘dhash’, ‘phash’
Hash type.
- hashsizeint , default 16
Hash size as hashsize x hashsize array.
- highfreq_factorint, default=4
For ‘phash’ increase initial image size to hashsize*highfreq_factor for cos-transform to catch high frequencies.
- Returns imageHash object
.bin, .hex, .int return respective representations
.similarity(other) returns relative Hamming distance.
imageHash subtractions returns Hamming distance.
equality checks hashtype and Hamming distance equal zero.
Notes
Images similarity cannot be done by bit comparison (e.g. md5sum) but using a simplified image representation converted to a unique bit representation representing a hash for the image.
Similar images should be different only in some bits measured by the Hamming distance (number of different bits).
- A typical procedure is
Reduce color by converting to grayscale.
Reduce size to e.g. 8x8 pixels by averaging over blocks.
Calc binary pixel hash pased on pixel values:
ahash - average hash: hash[i,j] = pixel > average(pixels)
dhash - difference hash: hash[i,j] = pixel[i,j+1] > pixel[i,j]
- phash - perceptual hash:
The low frequency part of the image cos-transform are most perceptual. The cos-transform of the image is used for an average hash. hash[i,j] = ahash(cos_tranform(pixels))
radial variance: See radon tranform in [1] (not implemented)
ahash and dhash are faster but phash dicriminates best.
Image similarity is decribed by the Hamming difference as number of different bits. A good measure is the relative Hamming difference (my similarity) as Hamming_diff/hash.size.
Similar images have similarity < 0.1 .
Random pixel difference results in similarity=0.5, an iverted image in similarity =1 (all bits different)
References
- 1
Rihamark: perceptual image hash benchmarking C. Zauner, M. Steinebach, E. Hermann Proc. SPIE 7880, Media Watermarking, Security, and Forensics III https://doi.org/10.1117/12.876617
Started based on photohash and imagehash
Copyright (c) 2013 Christopher J Pickett, MIT license, https://github.com/bunchesofdonald/photohash
Copyright (c) 2013-2016, Johannes Buchner, BSD 2-Clause “Simplified” License, https://github.com/JohannesBuchner/imagehash
Copyright (c) 2019, Ralf Biehl, BSD 2-Clause “Simplified” License, imagehash.py see https://gitlab.com/biehl/jscatter/tree/master/src/jscatter/libs
Examples
The calibration image migth be not the best choice as demo or a good one. rotate works not at the center of the beam but for the image center.
import jscatter as js from jscatter.formel import imageHash from PIL import Image image = Image.open(js.examples.datapath+'/calibration.tiff') type='dhash' original_hash = imageHash(image=image, type=type) rotate_image = image.rotate(-1) rotate_hash = imageHash(image=rotate_image,type=type) sim1 = original_hash.similarity(rotate_hash) rotate_image = image.rotate(-90) rotate_hash = imageHash(image=rotate_image, type=type) sim2 = original_hash.similarity(rotate_hash)
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property
bin
¶ Binary representation
-
property
hex
¶ Hexadecimal representation
-
property
int
¶ Integer representation
-
property
shape
¶ Shape of the hash.
-
similarity
(other)[source]¶ Relative Hamming difference.
Similar <0.1
Random pixels are close to 0.5
Inverted 1
-
property
size
¶ Size of the hash.