Scaling#
MTZ IO#
ess.nmx has MTZ IO helper functions. They can be used as providers in a workflow of scaling routine.
They are wrapping MTZ IO functions of gemmi.
[1]:
%matplotlib inline
[2]:
import gemmi
from ess.nmx.mtz_io import (
read_mtz_file,
mtz_to_pandas,
MTZFilePath,
get_unique_space_group,
MtzDataFrame,
merge_mtz_dataframes,
)
from ess.nmx.data import get_small_random_mtz_samples
small_mtz_sample = get_small_random_mtz_samples()[0]
mtz = read_mtz_file(MTZFilePath(small_mtz_sample))
df = mtz_to_pandas(mtz)
df.head()
Downloading file 'mtz_random_samples.tar.gz' from 'https://public.esss.dk/groups/scipp/ess/nmx/mtz_random_samples.tar.gz' to '/home/runner/.cache/essnmx/0'.
Untarring contents of '/home/runner/.cache/essnmx/0/mtz_random_samples.tar.gz' to '/home/runner/.cache/essnmx/0/mtz_random_samples.tar.gz.untar'
[2]:
| I | SIGI | LAMBDA | H | K | L | |
|---|---|---|---|---|---|---|
| 0 | 121.346222 | 16.221380 | 3.199910 | 68.0 | 66.0 | 60.0 |
| 1 | 120.356422 | 14.619821 | 3.199888 | 5.0 | -18.0 | -43.0 |
| 2 | 118.749969 | 17.432007 | 3.199803 | -24.0 | -15.0 | -52.0 |
| 3 | 118.462883 | 12.729899 | 3.199750 | 86.0 | -78.0 | 91.0 |
| 4 | 116.012108 | 14.932206 | 3.199700 | 42.0 | 3.0 | 18.0 |
Build Pipeline#
Scaling routine includes: - Reducing individual MTZ dataset - Merging MTZ dataset - Reducing merged MTZ dataset
These operations are done on pandas dataframe as recommended in gemmi. And multiple MTZ files are expected, so we need to use sciline.ParamTable.
[3]:
import pandas as pd
import sciline as sl
import scipp as sc
from ess.nmx.mtz_io import providers as mtz_io_providers, default_parameters as mtz_io_params
from ess.nmx.mtz_io import SpaceGroupDesc
from ess.nmx.scaling import providers as scaling_providers, default_parameters as scaling_params
from ess.nmx.scaling import (
WavelengthBins,
FilteredEstimatedScaledIntensities,
ReferenceWavelength,
ScaledIntensityLeftTailThreshold,
ScaledIntensityRightTailThreshold,
)
pl = sl.Pipeline(
providers=mtz_io_providers + scaling_providers,
params={
SpaceGroupDesc: "C 1 2 1",
ReferenceWavelength: sc.scalar(
3, unit=sc.units.angstrom
), # Remove it if you want to use the middle of the bin
ScaledIntensityLeftTailThreshold: sc.scalar(
0.1, # Increase it to remove more outliers
),
ScaledIntensityRightTailThreshold: sc.scalar(
4.0, # Decrease it to remove more outliers
),
**mtz_io_params,
**scaling_params,
WavelengthBins: 250,
},
)
pl
[3]:
| Name | Value | Source |
|---|---|---|
| <Generic>[SpaceGroup, NoneType] |
get_space_group_from_mtzess.nmx.mtz_io.get_space_group_from_mtz | |
| EstimatedScaleFactor |
estimate_scale_factor_per_hkl_asu_from_referenceess.nmx.scaling.estimate_scale_factor_per_hkl_asu_from_reference | |
| EstimatedScaledIntensities |
average_roughly_scaled_intensitiesess.nmx.scaling.average_roughly_scaled_intensities | |
| FilteredEstimatedScaledIntensities |
cut_tailsess.nmx.scaling.cut_tails | |
| FittingResult |
fit_wavelength_scale_factor_polynomialess.nmx.scaling.fit_wavelength_scale_factor_polynomial | |
| IntensityColumnName | I | |
| MTZFilePath | ||
| Mtz |
read_mtz_fileess.nmx.mtz_io.read_mtz_file | |
| MtzDataFrame |
process_single_mtz_to_dataframeess.nmx.mtz_io.process_single_mtz_to_dataframe | |
| NMXMtzDataArray |
nmx_mtz_dataframe_to_scipp_dataarrayess.nmx.mtz_io.nmx_mtz_dataframe_to_scipp_dataarray | |
| NMXMtzDataFrame |
process_mtz_dataframeess.nmx.mtz_io.process_mtz_dataframe | |
| ReciprocalAsu |
get_reciprocal_asuess.nmx.mtz_io.get_reciprocal_asu | |
| ReferenceIntensities |
get_reference_intensitiesess.nmx.scaling.get_reference_intensities | |
| ReferenceWavelength |
<scipp.Variable> () ...<scipp.Variable> () int64 [Å] 3 |
|
| ScaledIntensityLeftTailThreshold |
<scipp.Variable> () f...<scipp.Variable> () float64 [dimensionless] 0.1 |
|
| ScaledIntensityRightTailThreshold |
<scipp.Variable> () f...<scipp.Variable> () float64 [dimensionless] 2 |
|
| SelectedReferenceWavelength |
get_reference_wavelengthess.nmx.scaling.get_reference_wavelength | |
| SpaceGroup | ||
| SpaceGroupDesc | C 1 2 1 | |
| StdDevColumnName | SIGI | |
| WavelengthBinned |
get_wavelength_binnedess.nmx.scaling.get_wavelength_binned | |
| WavelengthBins | 250 | |
| WavelengthColumnName | LAMBDA | |
| WavelengthFittingPolynomialDegree | 7 | |
| WavelengthScaleFactors |
calculate_wavelength_scale_factoress.nmx.scaling.calculate_wavelength_scale_factor |
[4]:
file_paths = pd.DataFrame({MTZFilePath: get_small_random_mtz_samples()}).rename_axis(
"mtzfile"
)
mapped = pl.map(file_paths)
pl[gemmi.SpaceGroup] = mapped[gemmi.SpaceGroup | None].reduce(
index='mtzfile', func=get_unique_space_group
)
pl[MtzDataFrame] = mapped[MtzDataFrame].reduce(
index='mtzfile', func=merge_mtz_dataframes
)
Build Workflow#
[5]:
from ess.nmx.scaling import WavelengthScaleFactors
scaling_nmx_workflow = pl.get(WavelengthScaleFactors)
scaling_nmx_workflow.visualize(graph_attr={"rankdir": "LR"})
[5]:
Compute Desired Type#
[6]:
from ess.nmx.scaling import (
SelectedReferenceWavelength,
FittingResult,
WavelengthScaleFactors,
)
results = scaling_nmx_workflow.compute(
(
FilteredEstimatedScaledIntensities,
SelectedReferenceWavelength,
FittingResult,
WavelengthScaleFactors,
)
)
results[WavelengthScaleFactors]
/home/runner/work/essnmx/essnmx/.tox/docs/lib/python3.10/site-packages/scipp/curve_fit.py:243: OptimizeWarning: Covariance of the parameters could not be estimated
popt, pcov = opt.curve_fit(
[6]:
scipp.DataArray (3.90 KB)
- wavelength: 247
- wavelength(wavelength)float32Å2.8012676, 2.8044639, ..., 3.1944056, 3.196004
Values:
array([2.8012676, 2.8044639, 2.8060622, 2.80766 , 2.8092585, 2.8108563, 2.8124547, 2.8140526, 2.815651 , 2.8172488, 2.8188472, 2.820445 , 2.8220434, 2.8236413, 2.8252397, 2.8268375, 2.828436 , 2.8300338, 2.8316321, 2.83323 , 2.8348284, 2.8364263, 2.8380246, 2.8396225, 2.8412209, 2.8428187, 2.844417 , 2.846015 , 2.8476133, 2.8492112, 2.8508096, 2.8524075, 2.8540058, 2.8556037, 2.857202 , 2.8588 , 2.8603983, 2.8619962, 2.8635945, 2.8651924, 2.8667908, 2.8683887, 2.869987 , 2.871585 , 2.8731833, 2.8747811, 2.8763795, 2.8779774, 2.8795757, 2.8811736, 2.882772 , 2.8843699, 2.8859682, 2.887566 , 2.8891644, 2.8907623, 2.8923607, 2.8939586, 2.895557 , 2.8971548, 2.8987532, 2.9003513, 2.9019494, 2.9035475, 2.9051456, 2.906744 , 2.908342 , 2.9099402, 2.9115381, 2.9131365, 2.9147344, 2.9163327, 2.9179306, 2.919529 , 2.9211268, 2.9227252, 2.924323 , 2.9259214, 2.9275193, 2.9291177, 2.9307156, 2.932314 , 2.9339118, 2.9355102, 2.937108 , 2.9387064, 2.9403043, 2.9419026, 2.9435005, 2.9450989, 2.9466968, 2.948295 , 2.949893 , 2.9514914, 2.9530892, 2.9546876, 2.9562855, 2.9578838, 2.9594817, 2.96108 , 2.962678 , 2.9642763, 2.9658742, 2.9674726, 2.9690704, 2.9706688, 2.9722667, 2.973865 , 2.975463 , 2.9770613, 2.9786592, 2.9802575, 2.9818554, 2.9834538, 2.9850516, 2.98665 , 2.9882479, 2.9898462, 2.991444 , 2.9930425, 2.9946404, 2.9962387, 2.9978366, 2.999435 , 3.0010328, 3.0026312, 3.004229 , 3.0058274, 3.0074253, 3.0090237, 3.0106215, 3.01222 , 3.0138178, 3.0154161, 3.017014 , 3.0186124, 3.0202103, 3.0218086, 3.0234065, 3.0250049, 3.0266027, 3.028201 , 3.029799 , 3.0313973, 3.0329952, 3.0345936, 3.0361915, 3.0377898, 3.0393877, 3.040986 , 3.042584 , 3.0441823, 3.0457802, 3.0473785, 3.0489764, 3.0505748, 3.0521727, 3.053771 , 3.055369 , 3.0569673, 3.0585651, 3.0601635, 3.0617614, 3.0633597, 3.0649576, 3.066556 , 3.0681539, 3.0697522, 3.07135 , 3.0729485, 3.0745463, 3.0761447, 3.0777426, 3.079341 , 3.0809388, 3.0825372, 3.084135 , 3.0857334, 3.0873313, 3.0889297, 3.0905275, 3.092126 , 3.0937238, 3.0953221, 3.0969203, 3.0985184, 3.1001165, 3.1017146, 3.1033127, 3.1049109, 3.106509 , 3.108107 , 3.1097052, 3.1113033, 3.1129014, 3.1144996, 3.1160977, 3.1176958, 3.119294 , 3.120892 , 3.1224904, 3.1240883, 3.1256866, 3.1272845, 3.128883 , 3.1304808, 3.1320791, 3.133677 , 3.1352754, 3.1368732, 3.1384716, 3.1400695, 3.1416678, 3.1432657, 3.144864 , 3.146462 , 3.1480603, 3.1496582, 3.1512566, 3.1528544, 3.1544528, 3.1560507, 3.157649 , 3.159247 , 3.1608453, 3.1624432, 3.1640415, 3.1656394, 3.1672378, 3.1688356, 3.170434 , 3.1720319, 3.1736302, 3.175228 , 3.1768265, 3.1784244, 3.1800227, 3.1816206, 3.183219 , 3.1848168, 3.1864152, 3.188013 , 3.1896114, 3.1912093, 3.1928077, 3.1944056, 3.196004 ], dtype=float32)
- (wavelength)float64𝟙0.944, 0.945, ..., 1.064, 1.065
Values:
array([0.94430435, 0.94513162, 0.94554609, 0.94596096, 0.94637649, 0.94679241, 0.94720899, 0.94762595, 0.94804359, 0.94846161, 0.94888029, 0.94929938, 0.94971912, 0.95013926, 0.95056006, 0.95098128, 0.95140315, 0.95182542, 0.95224837, 0.95267172, 0.95309574, 0.95352017, 0.95394526, 0.95437077, 0.95479695, 0.95522355, 0.95565081, 0.95607849, 0.95650685, 0.95693563, 0.95736508, 0.95779495, 0.9582255 , 0.95865646, 0.95908811, 0.95952019, 0.95995293, 0.96038612, 0.96081998, 0.96125428, 0.96168926, 0.96212466, 0.96256076, 0.96299727, 0.96343451, 0.96387214, 0.96431049, 0.96474927, 0.96518874, 0.96562865, 0.96606925, 0.96651029, 0.96695203, 0.96739421, 0.96783709, 0.96828043, 0.96872445, 0.96916892, 0.96961409, 0.97005971, 0.97050604, 0.97095289, 0.97140031, 0.97184829, 0.97229687, 0.97274611, 0.97319579, 0.97364619, 0.97409704, 0.9745486 , 0.97500061, 0.97545336, 0.97590657, 0.97636048, 0.97681486, 0.97726996, 0.97772553, 0.97818182, 0.97863859, 0.97909606, 0.97955402, 0.98001269, 0.98047183, 0.98093171, 0.98139207, 0.98185313, 0.98231469, 0.98277697, 0.98323974, 0.98370325, 0.98416722, 0.98463193, 0.98509713, 0.98556306, 0.98602949, 0.98649665, 0.9869643 , 0.98743269, 0.98790155, 0.98837117, 0.98884131, 0.98931215, 0.9897835 , 0.99025561, 0.99072819, 0.99120155, 0.99167538, 0.99215 , 0.99262507, 0.99310094, 0.99357729, 0.9940544 , 0.99453201, 0.99501038, 0.99548926, 0.99596891, 0.99644905, 0.99692996, 0.99741138, 0.99789357, 0.99837626, 0.99885974, 0.99934371, 0.99982847, 1.00031373, 1.00079978, 1.00128634, 1.00177368, 1.00226152, 1.00275015, 1.0032393 , 1.00372925, 1.00421969, 1.00471095, 1.00520271, 1.00569526, 1.00618834, 1.00668222, 1.00717659, 1.0076718 , 1.00816751, 1.00866403, 1.00916106, 1.00965892, 1.01015729, 1.01065647, 1.01115616, 1.01165667, 1.01215773, 1.01265959, 1.01316197, 1.01366519, 1.01416894, 1.01467349, 1.01517857, 1.01568449, 1.01619095, 1.01669822, 1.01720603, 1.01771468, 1.01822384, 1.01873385, 1.01924441, 1.01975578, 1.02026772, 1.02078049, 1.02129379, 1.02180795, 1.02232263, 1.02283817, 1.02335425, 1.02387118, 1.02438865, 1.02490699, 1.02542586, 1.02594559, 1.02646587, 1.02698701, 1.0275087 , 1.02803124, 1.02855434, 1.02907829, 1.02960283, 1.03012822, 1.03065423, 1.03118096, 1.03170841, 1.03223657, 1.03276544, 1.03329506, 1.03382538, 1.03435641, 1.03488816, 1.03542065, 1.03595386, 1.03648779, 1.03702246, 1.03755783, 1.03809396, 1.0386308 , 1.03916846, 1.03970668, 1.04024579, 1.0407855 , 1.04132611, 1.04186726, 1.04240932, 1.04295199, 1.04349552, 1.04403965, 1.04458469, 1.0451303 , 1.04567684, 1.04622394, 1.04677199, 1.04732058, 1.04787012, 1.04842022, 1.04897125, 1.04952285, 1.05007539, 1.05062854, 1.05118258, 1.05173724, 1.05229282, 1.05284898, 1.05340609, 1.05396379, 1.05452241, 1.05508164, 1.05564181, 1.05620259, 1.05676429, 1.05732658, 1.05788985, 1.05845371, 1.05901849, 1.0595839 , 1.06015027, 1.06071722, 1.06128515, 1.06185368, 1.06242317, 1.06299327, 1.06356433, 1.06413599, 1.06470863])
Plots#
Here are plotting examples of the fitting/estimation results.
Estimated Scaled Intensities.#
[7]:
import scipy.stats as stats
import matplotlib.pyplot as plt
fig, (density_ax, prob_ax) = plt.subplots(1, 2, figsize=(10, 5))
densities = sc.values(results[FilteredEstimatedScaledIntensities].data).values
sc.values(results[FilteredEstimatedScaledIntensities].data).hist(intensity=50).plot(
title="Filtered Estimated Scaled Intensities\nDensity Plot",
grid=True,
linewidth=3,
ax=density_ax,
)
stats.probplot(densities, dist="norm", plot=prob_ax)
prob_ax.set_title("Filtered Estimated Scaled Intensities\nProbability Plot")
[7]:
Text(0.5, 1.0, 'Filtered Estimated Scaled Intensities\nProbability Plot')
Curve Fitting#
[8]:
import plopp as pp
import numpy as np
from ess.nmx.scaling import FittingResult
chebyshev_func = np.polynomial.chebyshev.Chebyshev(np.array([1, -1, 1]))
scale_function = np.vectorize(
chebyshev_func / chebyshev_func(results[SelectedReferenceWavelength].value)
)
pp.plot(
{
"Original Data": results[FilteredEstimatedScaledIntensities],
"Fit Result": results[FittingResult].fit_output,
},
grid=True,
title="Fit Result [Intensities vs Wavelength]",
marker={"Chebyshev": None, "Fit Result": None},
linestyle={"Chebyshev": "solid", "Fit Result": "solid"},
)
[8]:
[9]:
reference_wavelength = sc.DataArray(
data=sc.concat(
[
results[WavelengthScaleFactors].data.min(),
results[WavelengthScaleFactors].data.max(),
],
"wavelength",
),
coords={
"wavelength": sc.broadcast(
results[SelectedReferenceWavelength], dims=["wavelength"], shape=[2]
)
},
)
wavelength_scale_factor_plot = pp.plot(
{
"scale_factor": results[WavelengthScaleFactors],
"reference_wavelength": reference_wavelength,
},
title="Wavelength Scale Factors",
grid=True,
marker={"reference_wavelength": None},
linestyle={"reference_wavelength": "solid"},
)
wavelength_scale_factor_plot.ax.set_xlim(2.8, 3.2)
reference_wavelength = results[SelectedReferenceWavelength].value
wavelength_scale_factor_plot.ax.text(
3.0,
0.25,
f"{reference_wavelength=:} [{results[SelectedReferenceWavelength].unit}]",
fontsize=8,
color="black",
)
wavelength_scale_factor_plot
[9]:
Change Provider#
Here is an example of how to insert different filter function.
In this example, we will swap a provider that filters EstimatedScaledIntensities and provide FilteredEstimatedScaledIntensities.
After updating the providers, you can go back to Compute Desired Type and start over.
[10]:
from typing import NewType
import scipp as sc
from ess.nmx.scaling import (
EstimatedScaledIntensities,
FilteredEstimatedScaledIntensities,
)
# Define the new types for the filtering function
NRoot = NewType("NRoot", int)
"""The n-th root to be taken for the standard deviation."""
NRootStdDevCut = NewType("NRootStdDevCut", float)
"""The number of standard deviations to be cut from the n-th root data."""
def _calculate_sample_standard_deviation(var: sc.Variable) -> sc.Variable:
"""Calculate the sample variation of the data.
This helper function is a temporary solution before
we release new scipp version with the statistics helper.
"""
import numpy as np
return sc.scalar(np.nanstd(var.values))
# Define the filtering function with right argument types and return type
def cut_estimated_scaled_intensities_by_n_root_std_dev(
scaled_intensities: EstimatedScaledIntensities,
n_root: NRoot,
n_root_std_dev_cut: NRootStdDevCut,
) -> FilteredEstimatedScaledIntensities:
"""Filter the mtz data array by the quad root of the sample standard deviation.
Parameters
----------
scaled_intensities:
The scaled intensities to be filtered.
n_root:
The n-th root to be taken for the standard deviation.
Higher n-th root means cutting is more effective on the right tail.
More explanation can be found in the notes.
n_root_std_dev_cut:
The number of standard deviations to be cut from the n-th root data.
Returns
-------
:
The filtered scaled intensities.
"""
# Check the range of the n-th root
if n_root < 1:
raise ValueError("The n-th root should be equal to or greater than 1.")
copied = scaled_intensities.copy(deep=False)
nth_root = copied.data ** (1 / n_root)
# Calculate the mean
nth_root_mean = nth_root.nanmean()
# Calculate the sample standard deviation
nth_root_std_dev = _calculate_sample_standard_deviation(nth_root)
# Calculate the cut value
half_window = n_root_std_dev_cut * nth_root_std_dev
keep_range = (nth_root_mean - half_window, nth_root_mean + half_window)
# Filter the data
return FilteredEstimatedScaledIntensities(
copied[(nth_root > keep_range[0]) & (nth_root < keep_range[1])]
)
pl.insert(cut_estimated_scaled_intensities_by_n_root_std_dev)
pl[NRoot] = 4
pl[NRootStdDevCut] = 1.0
pl.compute(FilteredEstimatedScaledIntensities)
[10]:
scipp.DataArray (3.11 KB)
- wavelength: 180
- wavelength(wavelength)float32Å2.857202, 2.8588, ..., 3.1432657, 3.144864
Values:
array([2.857202 , 2.8588 , 2.8619962, 2.8635945, 2.8651924, 2.8667908, 2.8683887, 2.869987 , 2.871585 , 2.8731833, 2.8747811, 2.8763795, 2.8779774, 2.8795757, 2.8811736, 2.882772 , 2.8843699, 2.8859682, 2.887566 , 2.8891644, 2.8907623, 2.8923607, 2.8939586, 2.895557 , 2.8971548, 2.8987532, 2.9003513, 2.9019494, 2.9035475, 2.9051456, 2.906744 , 2.908342 , 2.9099402, 2.9115381, 2.9131365, 2.9147344, 2.9163327, 2.9179306, 2.919529 , 2.9211268, 2.9227252, 2.924323 , 2.9259214, 2.9275193, 2.9291177, 2.9307156, 2.932314 , 2.9339118, 2.9355102, 2.937108 , 2.9387064, 2.9403043, 2.9419026, 2.9435005, 2.9450989, 2.9466968, 2.948295 , 2.949893 , 2.9514914, 2.9530892, 2.9546876, 2.9562855, 2.9578838, 2.9594817, 2.96108 , 2.962678 , 2.9642763, 2.9658742, 2.9674726, 2.9690704, 2.9706688, 2.9722667, 2.973865 , 2.975463 , 2.9770613, 2.9786592, 2.9802575, 2.9818554, 2.9834538, 2.9850516, 2.98665 , 2.9882479, 2.9898462, 2.991444 , 2.9930425, 2.9946404, 2.9962387, 2.9978366, 2.999435 , 3.0010328, 3.0026312, 3.004229 , 3.0058274, 3.0074253, 3.0090237, 3.0106215, 3.01222 , 3.0138178, 3.0154161, 3.017014 , 3.0186124, 3.0202103, 3.0218086, 3.0234065, 3.0250049, 3.0266027, 3.028201 , 3.029799 , 3.0313973, 3.0329952, 3.0345936, 3.0361915, 3.0377898, 3.0393877, 3.040986 , 3.042584 , 3.0441823, 3.0457802, 3.0473785, 3.0489764, 3.0505748, 3.0521727, 3.053771 , 3.055369 , 3.0569673, 3.0585651, 3.0601635, 3.0617614, 3.0633597, 3.0649576, 3.066556 , 3.0681539, 3.0697522, 3.07135 , 3.0729485, 3.0745463, 3.0761447, 3.0777426, 3.079341 , 3.0809388, 3.0825372, 3.084135 , 3.0857334, 3.0873313, 3.0889297, 3.0905275, 3.092126 , 3.0937238, 3.0953221, 3.0969203, 3.0985184, 3.1001165, 3.1017146, 3.1033127, 3.1049109, 3.106509 , 3.108107 , 3.1097052, 3.1113033, 3.1129014, 3.1144996, 3.1160977, 3.1176958, 3.119294 , 3.120892 , 3.1224904, 3.1240883, 3.1256866, 3.1272845, 3.128883 , 3.1304808, 3.1320791, 3.133677 , 3.1352754, 3.1368732, 3.1384716, 3.1400695, 3.1416678, 3.1432657, 3.144864 ], dtype=float32)
- (wavelength)float32𝟙0.6070114, 0.6354325, ..., 1.4262968, 1.4329288σ = 0.007581526, 0.009966516, ..., 0.015530329, 0.015853887
Values:
array([0.6070114 , 0.6354325 , 0.61014795, 0.6068784 , 0.6406334 , 0.64477855, 0.63060856, 0.63542855, 0.6324167 , 0.6801704 , 0.64791816, 0.652414 , 0.6717658 , 0.6621639 , 0.6798379 , 0.7657368 , 0.6897617 , 0.6949786 , 0.68923384, 0.6942186 , 0.71312326, 0.71762073, 0.719248 , 0.71350837, 0.7397376 , 0.7398708 , 0.7309188 , 0.7652437 , 0.7870944 , 0.7594039 , 0.7531101 , 0.75556797, 0.76490945, 0.8025271 , 0.7721254 , 0.78140897, 0.7865576 , 0.80898607, 0.7910712 , 0.80424136, 0.81415737, 0.8031877 , 0.8225756 , 0.82421577, 0.8191481 , 0.83883417, 0.8289338 , 0.83124685, 0.83585805, 0.8385015 , 0.84103566, 0.8469526 , 0.8568808 , 0.8694568 , 0.85906804, 0.9218925 , 0.8691482 , 0.910457 , 0.88111806, 0.9104874 , 0.9135022 , 0.92319083, 0.90916634, 0.9476996 , 0.9178994 , 0.9017219 , 0.91642 , 1.0024371 , 0.9270845 , 0.9234043 , 0.9451939 , 0.9580411 , 0.95741826, 0.94907516, 0.9626036 , 0.9487712 , 0.9515659 , 0.9979782 , 0.9831484 , 0.9640661 , 0.9879607 , 0.9713198 , 0.9780923 , 1.0047737 , 0.98358345, 0.9882377 , 0.99075764, 1.018909 , 1.0000254 , 1.0028013 , 1.0172088 , 1.024566 , 1.0286235 , 1.0433916 , 1.1104821 , 1.0264289 , 1.0312595 , 1.0576755 , 1.0381942 , 1.0659459 , 1.0475025 , 1.0555083 , 1.0599737 , 1.0647993 , 1.1044763 , 1.0716054 , 1.085836 , 1.1003727 , 1.0791866 , 1.096032 , 1.0906415 , 1.1422418 , 1.1413152 , 1.1131383 , 1.1150991 , 1.1492105 , 1.1475223 , 1.1539438 , 1.1373259 , 1.12481 , 1.1319934 , 1.1355053 , 1.1449871 , 1.1513128 , 1.1601958 , 1.1492685 , 1.2059618 , 1.1931733 , 1.2093873 , 1.172439 , 1.1882433 , 1.2029705 , 1.2417048 , 1.1958982 , 1.192574 , 1.2114031 , 1.2002187 , 1.2199563 , 1.2094076 , 1.256482 , 1.2189393 , 1.2234412 , 1.2631079 , 1.2533275 , 1.246985 , 1.2448152 , 1.2749248 , 1.2598064 , 1.2584211 , 1.26102 , 1.2679971 , 1.2830582 , 1.2782192 , 1.2803183 , 1.2984796 , 1.2955241 , 1.308334 , 1.3730578 , 1.307733 , 1.3506969 , 1.316869 , 1.3708428 , 1.3499488 , 1.3652765 , 1.3402617 , 1.3605239 , 1.3775644 , 1.3633896 , 1.385249 , 1.3687915 , 1.381712 , 1.3882626 , 1.3878992 , 1.4007676 , 1.4101408 , 1.4342211 , 1.4354299 , 1.4360559 , 1.4262968 , 1.4329288 ], dtype=float32)
Variances (σ²):
array([5.74795376e-05, 9.93314388e-05, 3.98175434e-05, 5.14143139e-05, 6.59856814e-05, 6.35042234e-05, 5.12313964e-05, 5.03618030e-05, 5.48519383e-05, 6.64681502e-05, 5.44289178e-05, 5.28388628e-05, 6.18483537e-05, 6.49031790e-05, 5.17519657e-05, 1.58387702e-04, 5.97466023e-05, 6.94945847e-05, 6.17728219e-05, 6.55613185e-05, 6.64384133e-05, 6.10811840e-05, 6.79638761e-05, 6.27376867e-05, 6.04419656e-05, 5.94244702e-05, 6.86752101e-05, 6.63347600e-05, 1.16998461e-04, 7.06466308e-05, 7.18489682e-05, 6.07973707e-05, 6.64159816e-05, 9.21256287e-05, 7.51157058e-05, 7.64880751e-05, 7.49804167e-05, 8.41489455e-05, 8.22133006e-05, 8.44401584e-05, 8.60130967e-05, 8.42609152e-05, 9.90535991e-05, 8.61488079e-05, 8.50271390e-05, 8.76785271e-05, 8.34146922e-05, 7.02328107e-05, 1.00364952e-04, 9.05520501e-05, 9.42670522e-05, 9.19481608e-05, 9.79183460e-05, 9.65009240e-05, 8.78656865e-05, 1.20414938e-04, 9.97904062e-05, 1.23560938e-04, 9.40254249e-05, 1.04686864e-04, 1.14489587e-04, 1.22188212e-04, 1.13109927e-04, 1.28414715e-04, 1.25353123e-04, 1.21911195e-04, 9.33227857e-05, 1.74027431e-04, 1.04913488e-04, 1.25347389e-04, 1.10791552e-04, 1.26701852e-04, 1.09050285e-04, 1.24742073e-04, 1.30781860e-04, 1.05761384e-04, 1.11854650e-04, 1.41779499e-04, 1.31608744e-04, 1.30743283e-04, 1.20826429e-04, 1.26319152e-04, 1.34927235e-04, 1.38158532e-04, 1.27356092e-04, 1.19864315e-04, 1.11187808e-04, 1.48789157e-04, 1.01213933e-04, 1.20698089e-04, 1.43554687e-04, 1.38697913e-04, 1.25700622e-04, 1.37931595e-04, 1.61406861e-04, 1.37677387e-04, 1.58749288e-04, 1.38160191e-04, 1.55284506e-04, 1.42545759e-04, 1.27560212e-04, 1.42550489e-04, 1.49925720e-04, 1.52065113e-04, 1.55441434e-04, 1.23287304e-04, 1.55261441e-04, 1.44539532e-04, 1.24982442e-04, 1.57909060e-04, 1.49287778e-04, 2.06205310e-04, 1.90313207e-04, 1.77068432e-04, 1.71384701e-04, 1.79194976e-04, 1.81565265e-04, 1.82234216e-04, 1.63582881e-04, 1.53892339e-04, 1.61651551e-04, 1.55816175e-04, 1.77663547e-04, 1.43336118e-04, 1.79886978e-04, 1.44532270e-04, 2.15440523e-04, 1.78006536e-04, 1.68227794e-04, 1.43395737e-04, 1.68639366e-04, 2.08071913e-04, 2.88074225e-04, 1.69707899e-04, 2.09134174e-04, 1.68616767e-04, 1.68781975e-04, 1.89460785e-04, 1.70585758e-04, 2.13972060e-04, 1.83197044e-04, 1.77821799e-04, 2.12173894e-04, 1.94450797e-04, 2.00403243e-04, 2.15643304e-04, 1.80441697e-04, 2.11129067e-04, 2.12913204e-04, 2.02940195e-04, 1.92602383e-04, 1.97268106e-04, 2.11366874e-04, 2.23142299e-04, 2.23695009e-04, 1.99477669e-04, 2.05906850e-04, 2.75295926e-04, 2.21289927e-04, 2.34401392e-04, 2.12831786e-04, 2.33427883e-04, 2.35852815e-04, 2.32453123e-04, 2.25265467e-04, 2.60025292e-04, 2.28782679e-04, 2.31453407e-04, 2.34481500e-04, 2.85590824e-04, 3.04085901e-04, 2.48227967e-04, 2.44201190e-04, 2.40701804e-04, 2.76635052e-04, 2.38082663e-04, 2.61420006e-04, 2.59264576e-04, 2.41191126e-04, 2.51345773e-04], dtype=float32)