.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_examples/sideband/plot_2D_KMg0.5O4SiO2.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note Click :ref:`here ` to download the full example code or to run this example in your browser via Binder .. rst-class:: sphx-glr-example-title .. _sphx_glr_auto_examples_sideband_plot_2D_KMg0.5O4SiO2.py: 2D MAT data of KMg0.5O 4.SiO2 glass =================================== .. GENERATED FROM PYTHON SOURCE LINES 8-14 The following example illustrates an application of the statistical learning method applied in determining the distribution of the nuclear shielding tensor parameters from a 2D magic-angle turning (MAT) spectrum. In this example, we use the 2D MAT spectrum [#f1]_ of :math:`\text{KMg}_{0.5}\text{O}\cdot4\text{SiO}_2` glass. Setup for the matplotlib figure. .. GENERATED FROM PYTHON SOURCE LINES 14-26 .. code-block:: default import csdmpy as cp import matplotlib.pyplot as plt import numpy as np from matplotlib import cm from csdmpy import statistics as stats from mrinversion.kernel.nmr import ShieldingPALineshape from mrinversion.kernel.utils import x_y_to_zeta_eta from mrinversion.linear_model import SmoothLassoCV, TSVDCompression from mrinversion.utils import plot_3d, to_Haeberlen_grid .. GENERATED FROM PYTHON SOURCE LINES 28-29 Setup for the matplotlib figures. .. GENERATED FROM PYTHON SOURCE LINES 29-42 .. code-block:: default # function for plotting 2D dataset def plot2D(csdm_object, **kwargs): plt.figure(figsize=(4.5, 3.5)) ax = plt.subplot(projection="csdm") ax.imshow(csdm_object, cmap="gist_ncar_r", aspect="auto", **kwargs) ax.invert_xaxis() ax.invert_yaxis() plt.tight_layout() plt.show() .. GENERATED FROM PYTHON SOURCE LINES 43-50 Dataset setup ------------- Import the dataset '''''''''''''''''' Load the dataset. Here, we import the dataset as the CSDM data-object. .. GENERATED FROM PYTHON SOURCE LINES 50-61 .. code-block:: default # The 2D MAT dataset in csdm format filename = "https://zenodo.org/record/3964531/files/KMg0_5-4SiO2-MAT.csdf" data_object = cp.load(filename) # For inversion, we only interest ourselves with the real part of the complex dataset. data_object = data_object.real # We will also convert the coordinates of both dimensions from Hz to ppm. _ = [item.to("ppm", "nmr_frequency_ratio") for item in data_object.dimensions] .. GENERATED FROM PYTHON SOURCE LINES 62-66 Here, the variable ``data_object`` is a `CSDM `_ object that holds the real part of the 2D MAT dataset. The plot of the MAT dataset is .. GENERATED FROM PYTHON SOURCE LINES 66-68 .. code-block:: default plot2D(data_object) .. image:: /auto_examples/sideband/images/sphx_glr_plot_2D_KMg0.5O4SiO2_001.png :alt: plot 2D KMg0.5O4SiO2 :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 69-89 There are two dimensions in this dataset. The dimension at index 0 is the pure anisotropic spinning sideband dimension, while the dimension at index 1 is the isotropic chemical shift dimension. Prepping the data for inversion ''''''''''''''''''''''''''''''' **Step-1: Data Alignment** When using the csdm objects with the ``mrinversion`` package, the dimension at index 0 must be the dimension undergoing the linear inversion. In this example, we plan to invert the pure anisotropic shielding line-shape. In the ``data_object``, the anisotropic dimension is already at index 0 and, therefore, no further action is required. **Step-2: Optimization** Also notice, the signal from the 2D MAF dataset occupies a small fraction of the two-dimensional frequency grid. For optimum performance, truncate the dataset to the relevant region before proceeding. Use the appropriate array indexing/slicing to select the signal region. .. GENERATED FROM PYTHON SOURCE LINES 89-92 .. code-block:: default data_object_truncated = data_object[:, 75:105] plot2D(data_object_truncated) .. image:: /auto_examples/sideband/images/sphx_glr_plot_2D_KMg0.5O4SiO2_002.png :alt: plot 2D KMg0.5O4SiO2 :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 93-103 Linear Inversion setup ---------------------- Dimension setup ''''''''''''''' **Anisotropic-dimension:** The dimension of the dataset that holds the pure anisotropic frequency contributions. In ``mrinversion``, this must always be the dimension at index 0 of the data object. .. GENERATED FROM PYTHON SOURCE LINES 103-105 .. code-block:: default anisotropic_dimension = data_object_truncated.dimensions[0] .. GENERATED FROM PYTHON SOURCE LINES 106-108 **x-y dimensions:** The two inverse dimensions corresponding to the `x` and `y`-axis of the `x`-`y` grid. .. GENERATED FROM PYTHON SOURCE LINES 108-113 .. code-block:: default inverse_dimensions = [ cp.LinearDimension(count=25, increment="370 Hz", label="x"), # the `x`-dimension. cp.LinearDimension(count=25, increment="370 Hz", label="y"), # the `y`-dimension. ] .. GENERATED FROM PYTHON SOURCE LINES 114-121 Generating the kernel ''''''''''''''''''''' For MAF/PASS datasets, the kernel corresponds to the pure nuclear shielding anisotropy sideband spectra. Use the :class:`~mrinversion.kernel.nmr.ShieldingPALineshape` class to generate a shielding spinning sidebands kernel. .. GENERATED FROM PYTHON SOURCE LINES 121-131 .. code-block:: default sidebands = ShieldingPALineshape( anisotropic_dimension=anisotropic_dimension, inverse_dimension=inverse_dimensions, channel="29Si", magnetic_flux_density="9.4 T", rotor_angle="54.735°", rotor_frequency="790 Hz", number_of_sidebands=anisotropic_dimension.count, ) .. GENERATED FROM PYTHON SOURCE LINES 132-161 Here, ``sidebands`` is an instance of the :class:`~mrinversion.kernel.nmr.ShieldingPALineshape` class. The required arguments of this class are the `anisotropic_dimension`, `inverse_dimension`, and `channel`. We have already defined the first two arguments in the previous sub-section. The value of the `channel` argument is the nucleus observed in the MAT/PASS experiment. In this example, this value is '29Si'. The remaining arguments, such as the `magnetic_flux_density`, `rotor_angle`, and `rotor_frequency`, are set to match the conditions under which the 2D MAT/PASS spectrum was acquired, which in this case corresponds to acquisition at the magic-angle and spinning at a rotor frequency of 790 Hz in a 9.4 T magnetic flux density. The value of the `rotor_frequency` argument is the effective anisotropic modulation frequency. In a MAT measurement, the anisotropic modulation frequency is the same as the physical rotor frequency. For other measurements like the extended chemical shift modulation sequences (XCS) [#f3]_, or its variants, the effective anisotropic modulation frequency is lower than the physical rotor frequency and should be set appropriately. The argument `number_of_sidebands` is the maximum number of computed sidebands in the kernel. For most two-dimensional isotropic v.s pure anisotropic spinning-sideband correlation measurements, the sampling along the sideband dimension is the rotor or the effective anisotropic modulation frequency. Therefore, the value of the `number_of_sidebands` argument is usually the number of points along the sideband dimension. In this example, this value is 32. Once the ShieldingPALineshape instance is created, use the kernel() method to generate the spinning sideband lineshape kernel. .. GENERATED FROM PYTHON SOURCE LINES 161-164 .. code-block:: default K = sidebands.kernel(supersampling=2) print(K.shape) .. rst-class:: sphx-glr-script-out Out: .. code-block:: none (32, 625) .. GENERATED FROM PYTHON SOURCE LINES 165-168 The kernel ``K`` is a NumPy array of shape (32, 625), where the axes with 32 and 625 points are the spinning sidebands dimension and the features (x-y coordinates) corresponding to the :math:`25\times 25` `x`-`y` grid, respectively. .. GENERATED FROM PYTHON SOURCE LINES 170-175 Data Compression '''''''''''''''' Data compression is optional but recommended. It may reduce the size of the inverse problem and, thus, further computation time. .. GENERATED FROM PYTHON SOURCE LINES 175-181 .. code-block:: default new_system = TSVDCompression(K, data_object_truncated) compressed_K = new_system.compressed_K compressed_s = new_system.compressed_s print(f"truncation_index = {new_system.truncation_index}") .. rst-class:: sphx-glr-script-out Out: .. code-block:: none compression factor = 1.032258064516129 truncation_index = 31 .. GENERATED FROM PYTHON SOURCE LINES 182-194 Solving the inverse problem --------------------------- Smooth LASSO cross-validation ''''''''''''''''''''''''''''' Solve the smooth-lasso problem. Use the statistical learning ``SmoothLassoCV`` method to solve the inverse problem over a range of α and λ values and determine the best nuclear shielding tensor parameter distribution for the given 2D MAT dataset. Considering the limited build time for the documentation, we'll perform the cross-validation over a smaller :math:`5 \times 5` `x`-`y` grid. You may increase the grid resolution for your problem if desired. .. GENERATED FROM PYTHON SOURCE LINES 194-213 .. code-block:: default # setup the pre-defined range of alpha and lambda values lambdas = 10 ** (-5.4 - 1 * (np.arange(5) / 4)) alphas = 10 ** (-4.5 - 1.5 * (np.arange(5) / 4)) # setup the smooth lasso cross-validation class s_lasso = SmoothLassoCV( alphas=alphas, # A numpy array of alpha values. lambdas=lambdas, # A numpy array of lambda values. sigma=0.00070, # The standard deviation of noise from the MAT dataset. folds=10, # The number of folds in n-folds cross-validation. inverse_dimension=inverse_dimensions, # previously defined inverse dimensions. verbose=1, # If non-zero, prints the progress as the computation proceeds. max_iterations=20000, # The maximum number of allowed interations. ) # run fit using the compressed kernel and compressed data. s_lasso.fit(compressed_K, compressed_s) .. rst-class:: sphx-glr-script-out Out: .. code-block:: none [Parallel(n_jobs=-1)]: Using backend ThreadingBackend with 2 concurrent workers. [Parallel(n_jobs=-1)]: Done 5 out of 5 | elapsed: 7.4s finished [Parallel(n_jobs=-1)]: Using backend ThreadingBackend with 2 concurrent workers. [Parallel(n_jobs=-1)]: Done 5 out of 5 | elapsed: 9.2s finished [Parallel(n_jobs=-1)]: Using backend ThreadingBackend with 2 concurrent workers. [Parallel(n_jobs=-1)]: Done 5 out of 5 | elapsed: 11.3s finished [Parallel(n_jobs=-1)]: Using backend ThreadingBackend with 2 concurrent workers. [Parallel(n_jobs=-1)]: Done 5 out of 5 | elapsed: 16.5s finished [Parallel(n_jobs=-1)]: Using backend ThreadingBackend with 2 concurrent workers. [Parallel(n_jobs=-1)]: Done 5 out of 5 | elapsed: 23.6s finished .. GENERATED FROM PYTHON SOURCE LINES 214-220 The optimum hyper-parameters '''''''''''''''''''''''''''' Use the :attr:`~mrinversion.linear_model.SmoothLassoCV.hyperparameters` attribute of the instance for the optimum hyper-parameters, :math:`\alpha` and :math:`\lambda`, determined from the cross-validation. .. GENERATED FROM PYTHON SOURCE LINES 220-222 .. code-block:: default print(s_lasso.hyperparameters) .. rst-class:: sphx-glr-script-out Out: .. code-block:: none {'alpha': 5.623413251903491e-06, 'lambda': 2.2387211385683376e-06} .. GENERATED FROM PYTHON SOURCE LINES 223-229 The cross-validation surface '''''''''''''''''''''''''''' Optionally, you may want to visualize the cross-validation error curve/surface. Use the :attr:`~mrinversion.linear_model.SmoothLassoCV.cross_validation_curve` attribute of the instance, as follows .. GENERATED FROM PYTHON SOURCE LINES 229-244 .. code-block:: default CV_metric = s_lasso.cross_validation_curve # `CV_metric` is a CSDM object. # plot of the cross validation surface plt.figure(figsize=(5, 3.5)) ax = plt.subplot(projection="csdm") ax.contour(np.log10(CV_metric), levels=25) ax.scatter( -np.log10(s_lasso.hyperparameters["alpha"]), -np.log10(s_lasso.hyperparameters["lambda"]), marker="x", color="k", ) plt.tight_layout(pad=0.5) plt.show() .. image:: /auto_examples/sideband/images/sphx_glr_plot_2D_KMg0.5O4SiO2_003.png :alt: plot 2D KMg0.5O4SiO2 :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 245-250 The optimum solution '''''''''''''''''''' The :attr:`~mrinversion.linear_model.SmoothLassoCV.f` attribute of the instance holds the solution corresponding to the optimum hyper-parameters, .. GENERATED FROM PYTHON SOURCE LINES 250-252 .. code-block:: default f_sol = s_lasso.f # f_sol is a CSDM object. .. GENERATED FROM PYTHON SOURCE LINES 253-260 where ``f_sol`` is the optimum solution. The fit residuals ''''''''''''''''' To calculate the residuals between the data and predicted data(fit), use the :meth:`~mrinversion.linear_model.SmoothLasso.residuals` method, as follows, .. GENERATED FROM PYTHON SOURCE LINES 260-266 .. code-block:: default residuals = s_lasso.residuals(K=K, s=data_object_truncated) # residuals is a CSDM object. # The plot of the residuals. plot2D(residuals, vmax=data_object_truncated.max(), vmin=data_object_truncated.min()) .. image:: /auto_examples/sideband/images/sphx_glr_plot_2D_KMg0.5O4SiO2_004.png :alt: plot 2D KMg0.5O4SiO2 :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 267-268 The standard deviation of the residuals is .. GENERATED FROM PYTHON SOURCE LINES 268-270 .. code-block:: default residuals.std() .. rst-class:: sphx-glr-script-out Out: .. code-block:: none .. GENERATED FROM PYTHON SOURCE LINES 271-276 Saving the solution ''''''''''''''''''' To serialize the solution to a file, use the `save()` method of the CSDM object, for example, .. GENERATED FROM PYTHON SOURCE LINES 276-279 .. code-block:: default f_sol.save("KMg_mixed_silicate_inverse.csdf") # save the solution residuals.save("KMg_mixed_silicate_residue.csdf") # save the residuals .. GENERATED FROM PYTHON SOURCE LINES 280-291 Data Visualization ------------------ At this point, we have solved the inverse problem and obtained an optimum distribution of the nuclear shielding tensor parameters from the 2D MAT dataset. You may use any data visualization and interpretation tool of choice for further analysis. In the following sections, we provide minimal visualization and analysis to complete the case study. Visualizing the 3D solution ''''''''''''''''''''''''''' .. GENERATED FROM PYTHON SOURCE LINES 291-305 .. code-block:: default # Normalize the solution f_sol /= f_sol.max() # Convert the coordinates of the solution, `f_sol`, from Hz to ppm. [item.to("ppm", "nmr_frequency_ratio") for item in f_sol.dimensions] # The 3D plot of the solution plt.figure(figsize=(5, 4.4)) ax = plt.subplot(projection="3d") plot_3d(ax, f_sol, x_lim=[0, 120], y_lim=[0, 120], z_lim=[-50, -150]) plt.tight_layout() plt.show() .. image:: /auto_examples/sideband/images/sphx_glr_plot_2D_KMg0.5O4SiO2_005.png :alt: plot 2D KMg0.5O4SiO2 :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 306-309 From the 3D plot, we observe two distinct regions: one for the :math:`\text{Q}^4` sites and another for the :math:`\text{Q}^3` sites. Select the respective regions by using the appropriate array indexing, .. GENERATED FROM PYTHON SOURCE LINES 309-316 .. code-block:: default Q4_region = f_sol[0:8, 0:8, 5:25] Q4_region.description = "Q4 region" Q3_region = f_sol[0:8, 7:24, 7:25] Q3_region.description = "Q3 region" .. GENERATED FROM PYTHON SOURCE LINES 317-318 The plot of the respective regions is shown below. .. GENERATED FROM PYTHON SOURCE LINES 318-364 .. code-block:: default # Calculate the normalization factor for the 2D contours and 1D projections from the # original solution, `f_sol`. Use this normalization factor to scale the intensities # from the sub-regions. max_2d = [ f_sol.sum(axis=0).max().value, f_sol.sum(axis=1).max().value, f_sol.sum(axis=2).max().value, ] max_1d = [ f_sol.sum(axis=(1, 2)).max().value, f_sol.sum(axis=(0, 2)).max().value, f_sol.sum(axis=(0, 1)).max().value, ] plt.figure(figsize=(5, 4.4)) ax = plt.subplot(projection="3d") # plot for the Q4 region plot_3d( ax, Q4_region, x_lim=[0, 120], # the x-limit y_lim=[0, 120], # the y-limit z_lim=[-50, -150], # the z-limit max_2d=max_2d, # normalization factors for the 2D contours projections max_1d=max_1d, # normalization factors for the 1D projections cmap=cm.Reds_r, # colormap box=True, # draw a box around the region ) # plot for the Q3 region plot_3d( ax, Q3_region, x_lim=[0, 120], # the x-limit y_lim=[0, 120], # the y-limit z_lim=[-50, -150], # the z-limit max_2d=max_2d, # normalization factors for the 2D contours projections max_1d=max_1d, # normalization factors for the 1D projections cmap=cm.Blues_r, # colormap box=True, # draw a box around the region ) ax.legend() plt.tight_layout() plt.show() .. image:: /auto_examples/sideband/images/sphx_glr_plot_2D_KMg0.5O4SiO2_006.png :alt: plot 2D KMg0.5O4SiO2 :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 365-371 Visualizing the isotropic projections. '''''''''''''''''''''''''''''''''''''' Because the :math:`\text{Q}^4` and :math:`\text{Q}^3` regions are fully resolved after the inversion, evaluating the contributions from these regions is trivial. For examples, the distribution of the isotropic chemical shifts for these regions are .. GENERATED FROM PYTHON SOURCE LINES 371-405 .. code-block:: default # Isotropic chemical shift projection of the 2D MAT dataset. data_iso = data_object_truncated.sum(axis=0) data_iso /= data_iso.max() # normalize the projection # Isotropic chemical shift projection of the tensor distribution dataset. f_sol_iso = f_sol.sum(axis=(0, 1)) # Isotropic chemical shift projection of the tensor distribution for the Q4 region. Q4_region_iso = Q4_region.sum(axis=(0, 1)) # Isotropic chemical shift projection of the tensor distribution for the Q3 region. Q3_region_iso = Q3_region.sum(axis=(0, 1)) # Normalize the three projections. f_sol_iso_max = f_sol_iso.max() f_sol_iso /= f_sol_iso_max Q4_region_iso /= f_sol_iso_max Q3_region_iso /= f_sol_iso_max # The plot of the different projections. plt.figure(figsize=(5.5, 3.5)) ax = plt.subplot(projection="csdm") ax.plot(f_sol_iso, "--k", label="tensor") ax.plot(Q4_region_iso, "r", label="Q4") ax.plot(Q3_region_iso, "b", label="Q3") ax.plot(data_iso, "-k", label="MAF") ax.plot(data_iso - f_sol_iso - 0.1, "gray", label="residuals") ax.set_title("Isotropic projection") ax.invert_xaxis() plt.legend() plt.tight_layout() plt.show() .. image:: /auto_examples/sideband/images/sphx_glr_plot_2D_KMg0.5O4SiO2_007.png :alt: Isotropic projection :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 406-417 Notice the shape of the isotropic chemical shift distribution for the :math:`\text{Q}^4` sites is skewed, which is expected. Analysis -------- For the analysis, we use the `statistics `_ module of the csdmpy package. Following is the moment analysis of the 3D volumes for both the :math:`\text{Q}^4` and :math:`\text{Q}^3` regions up to the second moment. .. GENERATED FROM PYTHON SOURCE LINES 417-436 .. code-block:: default int_Q4 = stats.integral(Q4_region) # volume of the Q4 distribution mean_Q4 = stats.mean(Q4_region) # mean of the Q4 distribution std_Q4 = stats.std(Q4_region) # standard deviation of the Q4 distribution int_Q3 = stats.integral(Q3_region) # volume of the Q3 distribution mean_Q3 = stats.mean(Q3_region) # mean of the Q3 distribution std_Q3 = stats.std(Q3_region) # standard deviation of the Q3 distribution print("Q4 statistics") print(f"\tpopulation = {100 * int_Q4 / (int_Q4 + int_Q3)}%") print("\tmean\n\t\tx:\t{0}\n\t\ty:\t{1}\n\t\tiso:\t{2}".format(*mean_Q4)) print("\tstandard deviation\n\t\tx:\t{0}\n\t\ty:\t{1}\n\t\tiso:\t{2}".format(*std_Q4)) print("Q3 statistics") print(f"\tpopulation = {100 * int_Q3 / (int_Q4 + int_Q3)}%") print("\tmean\n\t\tx:\t{0}\n\t\ty:\t{1}\n\t\tiso:\t{2}".format(*mean_Q3)) print("\tstandard deviation\n\t\tx:\t{0}\n\t\ty:\t{1}\n\t\tiso:\t{2}".format(*std_Q3)) .. rst-class:: sphx-glr-script-out Out: .. code-block:: none Q4 statistics population = 55.526682010553586% mean x: 8.668624339369668 ppm y: 8.895729603986638 ppm iso: -107.32386708833955 ppm standard deviation x: 4.73326212666404 ppm y: 4.929006419658588 ppm iso: 5.427012270570254 ppm Q3 statistics population = 44.47331798944642% mean x: 10.133763235406391 ppm y: 62.26200849066072 ppm iso: -97.01639426409565 ppm standard deviation x: 4.540084886749352 ppm y: 10.65716485254756 ppm iso: 4.741866191977004 ppm .. GENERATED FROM PYTHON SOURCE LINES 437-441 The statistics shown above are according to the respective dimensions, that is, the `x`, `y`, and the isotropic chemical shifts. To convert the `x` and `y` statistics to commonly used :math:`\zeta` and :math:`\eta` statistics, use the :func:`~mrinversion.kernel.utils.x_y_to_zeta_eta` function. .. GENERATED FROM PYTHON SOURCE LINES 441-461 .. code-block:: default mean_ζη_Q3 = x_y_to_zeta_eta(*mean_Q3[0:2]) # error propagation for calculating the standard deviation std_ζ = (std_Q3[0] * mean_Q3[0]) ** 2 + (std_Q3[1] * mean_Q3[1]) ** 2 std_ζ /= mean_Q3[0] ** 2 + mean_Q3[1] ** 2 std_ζ = np.sqrt(std_ζ) std_η = (std_Q3[1] * mean_Q3[0]) ** 2 + (std_Q3[0] * mean_Q3[1]) ** 2 std_η /= (mean_Q3[0] ** 2 + mean_Q3[1] ** 2) ** 2 std_η = (4 / np.pi) * np.sqrt(std_η) print("Q3 statistics") print(f"\tpopulation = {100 * int_Q3 / (int_Q4 + int_Q3)}%") print("\tmean\n\t\tζ:\t{0}\n\t\tη:\t{1}\n\t\tiso:\t{2}".format(*mean_ζη_Q3, mean_Q3[2])) print( "\tstandard deviation\n\t\tζ:\t{0}\n\t\tη:\t{1}\n\t\tiso:\t{2}".format( std_ζ, std_η, std_Q3[2] ) ) .. rst-class:: sphx-glr-script-out Out: .. code-block:: none Q3 statistics population = 44.47331798944642% mean ζ: 63.0813035582048 ppm η: 0.20543107026990554 iso: -97.01639426409565 ppm standard deviation ζ: 10.544005747153385 ppm η: 0.09682370563685412 iso: 4.741866191977004 ppm .. GENERATED FROM PYTHON SOURCE LINES 462-467 Convert the 3D tensor distribution in Haeberlen parameters ---------------------------------------------------------- You may re-bin the 3D tensor parameter distribution from a :math:`\rho(\delta_\text{iso}, x, y)` distribution to :math:`\rho(\delta_\text{iso}, \zeta_\sigma, \eta_\sigma)` distribution as follows. .. GENERATED FROM PYTHON SOURCE LINES 467-475 .. code-block:: default # Create the zeta and eta dimensions,, as shown below. zeta = cp.as_dimension(np.arange(40) * 4 - 40, unit="ppm", label="zeta") eta = cp.as_dimension(np.arange(16) / 15, label="eta") # Use the `to_Haeberlen_grid` function to convert the tensor parameter distribution. fsol_Hae = to_Haeberlen_grid(f_sol, zeta, eta) .. GENERATED FROM PYTHON SOURCE LINES 476-478 The 3D plot ''''''''''' .. GENERATED FROM PYTHON SOURCE LINES 478-484 .. code-block:: default plt.figure(figsize=(5, 4.4)) ax = plt.subplot(projection="3d") plot_3d(ax, fsol_Hae, x_lim=[0, 1], y_lim=[-40, 120], z_lim=[-50, -150], alpha=0.4) plt.tight_layout() plt.show() .. image:: /auto_examples/sideband/images/sphx_glr_plot_2D_KMg0.5O4SiO2_008.png :alt: plot 2D KMg0.5O4SiO2 :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 485-496 References ---------- .. [#f1] Walder, B. J., Dey, K. K., Kaseman, D. C., Baltisberger, J. H., Grandinetti, P. J. Sideband separation experiments in NMR with phase incremented echo train acquisition, J. Chem. Phys. 138, 4803142, (2013). `doi:10.1063/1.4803142. `_ .. [#f3] Gullion, T., Extended chemical-shift modulation, J. Mag. Res., **85**, 3, (1989). `10.1016/0022-2364(89)90253-9 `_ .. rst-class:: sphx-glr-timing **Total running time of the script:** ( 1 minutes 20.095 seconds) .. _sphx_glr_download_auto_examples_sideband_plot_2D_KMg0.5O4SiO2.py: .. only :: html .. container:: sphx-glr-footer :class: sphx-glr-footer-example .. container:: binder-badge .. image:: images/binder_badge_logo.svg :target: https://mybinder.org/v2/gh/DeepanshS/mrinversion/master?urlpath=lab/tree/docs/_build/html/../../notebooks/auto_examples/sideband/plot_2D_KMg0.5O4SiO2.ipynb :alt: Launch binder :width: 150 px .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_2D_KMg0.5O4SiO2.py ` .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_2D_KMg0.5O4SiO2.ipynb ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_