.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_examples/MAF/plot_2d_3_Na2O1.5SiO2.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_MAF_plot_2d_3_Na2O1.5SiO2.py: 2D MAF data of 2Na2O.3SiO2 glass ================================ .. GENERATED FROM PYTHON SOURCE LINES 8-17 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 flipping (MAF) spectrum. In this example, we use the 2D MAF spectrum [#f1]_ of :math:`2\text{Na}_2\text{O}\cdot3\text{SiO}_2` glass. Before getting started ---------------------- Import all relevant packages. .. GENERATED FROM PYTHON SOURCE LINES 17-27 .. code-block:: default import csdmpy as cp import matplotlib.pyplot as plt import numpy as np from matplotlib import cm from mrinversion.kernel.nmr import ShieldingPALineshape from mrinversion.linear_model import SmoothLasso, TSVDCompression from mrinversion.utils import plot_3d, to_Haeberlen_grid .. GENERATED FROM PYTHON SOURCE LINES 29-30 Setup for the matplotlib figures. .. GENERATED FROM PYTHON SOURCE LINES 30-42 .. code-block:: default 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 MAF dataset in csdm format filename = "https://osu.box.com/shared/static/k405dsptwe1p43x8mfi1wc1geywrypzc.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 MAF dataset. The plot of the 2D MAF dataset is .. GENERATED FROM PYTHON SOURCE LINES 66-68 .. code-block:: default plot2D(data_object) .. image:: /auto_examples/MAF/images/sphx_glr_plot_2d_3_Na2O1.5SiO2_001.png :alt: plot 2d 3 Na2O1.5SiO2 :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 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[:, 220:280] plot2D(data_object_truncated) .. image:: /auto_examples/MAF/images/sphx_glr_plot_2d_3_Na2O1.5SiO2_002.png :alt: plot 2d 3 Na2O1.5SiO2 :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="500 Hz", label="x"), # the `x`-dimension. cp.LinearDimension(count=25, increment="500 Hz", label="y"), # the `y`-dimension. ] .. GENERATED FROM PYTHON SOURCE LINES 114-121 Generating the kernel ''''''''''''''''''''' For MAF datasets, the line-shape kernel corresponds to the pure nuclear shielding anisotropy line-shapes. Use the :class:`~mrinversion.kernel.nmr.ShieldingPALineshape` class to generate a shielding line-shape kernel. .. GENERATED FROM PYTHON SOURCE LINES 121-131 .. code-block:: default lineshape = ShieldingPALineshape( anisotropic_dimension=anisotropic_dimension, inverse_dimension=inverse_dimensions, channel="29Si", magnetic_flux_density="9.4 T", rotor_angle="90°", rotor_frequency="12 kHz", number_of_sidebands=4, ) .. GENERATED FROM PYTHON SOURCE LINES 132-147 Here, ``lineshape`` 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 MAF 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 MAF spectrum was acquired. The value of the `number_of_sidebands` argument is the number of sidebands calculated for each line-shape within the kernel. Unless, you have a lot of spinning sidebands in your MAF dataset, four sidebands should be enough. Once the ShieldingPALineshape instance is created, use the :meth:`~mrinversion.kernel.nmr.ShieldingPALineshape.kernel` method of the instance to generate the MAF line-shape kernel. .. GENERATED FROM PYTHON SOURCE LINES 147-150 .. code-block:: default K = lineshape.kernel(supersampling=1) print(K.shape) .. rst-class:: sphx-glr-script-out Out: .. code-block:: none (128, 625) .. GENERATED FROM PYTHON SOURCE LINES 151-154 The kernel ``K`` is a NumPy array of shape (128, 625), where the axes with 128 and 625 points are the anisotropic dimension and the features (x-y coordinates) corresponding to the :math:`25\times 25` `x`-`y` grid, respectively. .. GENERATED FROM PYTHON SOURCE LINES 156-161 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 161-167 .. 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 = 1.1851851851851851 truncation_index = 108 .. GENERATED FROM PYTHON SOURCE LINES 168-181 Solving the inverse problem --------------------------- Smooth LASSO cross-validation ''''''''''''''''''''''''''''' Solve the smooth-lasso problem. Ordinarily, one should use the statistical learning method to solve the inverse problem over a range of α and λ values and then determine the best nuclear shielding tensor parameter distribution for the given 2D MAF dataset. Considering the limited build time for the documentation, we skip this step and evaluate the distribution at pre-optimized α and λ values. The optimum values are :math:`\alpha = 2.2\times 10^{-8}` and :math:`\lambda = 1.27\times 10^{-6}`. The following commented code was used in determining the optimum α and λ values. .. GENERATED FROM PYTHON SOURCE LINES 183-214 .. code-block:: default # from mrinversion.linear_model import SmoothLassoCV # import numpy as np # # setup the pre-defined range of alpha and lambda values # lambdas = 10 ** (-4 - 3 * (np.arange(20) / 19)) # alphas = 10 ** (-4.5 - 5 * (np.arange(20) / 19)) # # 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=0.003, # The standard deviation of noise from the MAF data. # 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. # ) # # run fit using the compressed kernel and compressed data. # s_lasso.fit(compressed_K, compressed_s) # # the optimum hyper-parameters, alpha and lambda, from the cross-validation. # print(s_lasso.hyperparameters) # # {'alpha': 2.198392648862289e-08, 'lambda': 1.2742749857031348e-06} # # the solution # f_sol = s_lasso.f # # the cross-validation error curve # CV_metric = s_lasso.cross_validation_curve .. GENERATED FROM PYTHON SOURCE LINES 215-216 If you use the above ``SmoothLassoCV`` method, skip the following code-block. .. GENERATED FROM PYTHON SOURCE LINES 216-224 .. code-block:: default # Setup the smooth lasso class s_lasso = SmoothLasso( alpha=2.198e-8, lambda1=1.27e-6, inverse_dimension=inverse_dimensions ) # run the fit method on the compressed kernel and compressed data. s_lasso.fit(K=compressed_K, s=compressed_s) .. GENERATED FROM PYTHON SOURCE LINES 225-230 The optimum solution '''''''''''''''''''' The :attr:`~mrinversion.linear_model.SmoothLasso.f` attribute of the instance holds the solution, .. GENERATED FROM PYTHON SOURCE LINES 230-232 .. code-block:: default f_sol = s_lasso.f # f_sol is a CSDM object. .. GENERATED FROM PYTHON SOURCE LINES 233-240 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 240-246 .. 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/MAF/images/sphx_glr_plot_2d_3_Na2O1.5SiO2_003.png :alt: plot 2d 3 Na2O1.5SiO2 :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 247-248 The standard deviation of the residuals is .. GENERATED FROM PYTHON SOURCE LINES 248-250 .. code-block:: default residuals.std() .. rst-class:: sphx-glr-script-out Out: .. code-block:: none .. GENERATED FROM PYTHON SOURCE LINES 251-256 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 256-259 .. code-block:: default f_sol.save("2Na2O.3SiO2_inverse.csdf") # save the solution residuals.save("2Na2O.3SiO2_residue.csdf") # save the residuals .. GENERATED FROM PYTHON SOURCE LINES 260-271 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 MAF dataset. You may use any data visualization and interpretation tool of choice for further analysis. In the following sections, we provide minimal visualization to complete the case study. Visualizing the 3D solution ''''''''''''''''''''''''''' .. GENERATED FROM PYTHON SOURCE LINES 271-285 .. 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, elev=25, azim=-50, x_lim=[0, 150], y_lim=[0, 150], z_lim=[-60, -120]) plt.tight_layout() plt.show() .. image:: /auto_examples/MAF/images/sphx_glr_plot_2d_3_Na2O1.5SiO2_004.png :alt: plot 2d 3 Na2O1.5SiO2 :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 286-292 From the 3D plot, we observe three distinct regions corresponding to the :math:`\text{Q}^4`, :math:`\text{Q}^3`, and :math:`\text{Q}^2` sites, respectively. The :math:`\text{Q}^4` sites are resolved in the 3D distribution; however, we observe partial overlapping :math:`\text{Q}^3` and :math:`\text{Q}^2` sites. The following is a naive selection of the three regions. One may also apply sophisticated classification algorithms to better quantify the Q-species. .. GENERATED FROM PYTHON SOURCE LINES 292-302 .. code-block:: default Q4_region = f_sol[0:6, 0:6, 14:35] * 3 Q4_region.description = "Q4 region x 3" Q3_region = f_sol[0:8, 7:, 20:39] Q3_region.description = "Q3 region" Q2_region = f_sol[:10, 6:18, 36:52] Q2_region.description = "Q2 region" .. GENERATED FROM PYTHON SOURCE LINES 303-304 An approximate plot of the respective regions is shown below. .. GENERATED FROM PYTHON SOURCE LINES 304-362 .. 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, 150], # the x-limit y_lim=[0, 150], # the y-limit z_lim=[-60, -120], # 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 ) # plot for the Q3 region plot_3d( ax, Q3_region, x_lim=[0, 150], # the x-limit y_lim=[0, 150], # the y-limit z_lim=[-60, -120], # 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 ) # plot for the Q2 region plot_3d( ax, Q2_region, elev=25, # the elevation angle in the z plane azim=-50, # the azimuth angle in the x-y plane x_lim=[0, 150], # the x-limit y_lim=[0, 150], # the y-limit z_lim=[-60, -120], # 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.Oranges_r, # colormap box=False, # draw a box around the region ) ax.legend() plt.tight_layout() plt.show() .. image:: /auto_examples/MAF/images/sphx_glr_plot_2d_3_Na2O1.5SiO2_005.png :alt: plot 2d 3 Na2O1.5SiO2 :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 363-368 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 368-376 .. 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 377-379 The 3D plot ''''''''''' .. GENERATED FROM PYTHON SOURCE LINES 379-385 .. 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=[-60, -120], alpha=0.1) plt.tight_layout() plt.show() .. image:: /auto_examples/MAF/images/sphx_glr_plot_2d_3_Na2O1.5SiO2_006.png :alt: plot 2d 3 Na2O1.5SiO2 :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 386-394 References ---------- .. [#f1] Zhang, P., Dunlap, C., Florian, P., Grandinetti, P. J., Farnan, I., Stebbins , J. F. Silicon site distributions in an alkali silicate glass derived by two-dimensional 29Si nuclear magnetic resonance, J. Non. Cryst. Solids, **204**, (1996), 294–300. `doi:10.1016/S0022-3093(96)00601-1 `_. .. rst-class:: sphx-glr-timing **Total running time of the script:** ( 0 minutes 12.991 seconds) .. _sphx_glr_download_auto_examples_MAF_plot_2d_3_Na2O1.5SiO2.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/MAF/plot_2d_3_Na2O1.5SiO2.ipynb :alt: Launch binder :width: 150 px .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_2d_3_Na2O1.5SiO2.py ` .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_2d_3_Na2O1.5SiO2.ipynb ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_