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                "<style type=\"text/css\">",
                "/*!",
                " * Nomad Beaker Notebook Template",
                " *",
                " * @copyright  Copyright 2017 Fritz Haber Institute of the Max Planck Society,",
                " *             Benjamin Regler - Apache 2.0 License",
                " * @license    http://www.apache.org/licenses/LICENSE-2.0",
                " * @author     Benjamin Regler",
                " * @version    1.0.0",
                " *",
                " * Licensed under the Apache License, Version 2.0 (the \"License\");",
                " * you may not use this file except in compliance with the License.",
                " * You may obtain a copy of the License at",
                " * ",
                " *     http://www.apache.org/licenses/LICENSE-2.0",
                " *",
                " * Unless required by applicable law or agreed to in writing, software",
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                " * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.",
                " * See the License for the specific language governing permissions and",
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                "",
                "<div id=\"teaser\" style='background-color: rgba(149,170,79, 1.0); background-position:  right center; background-size: 200px; background-repeat: no-repeat; ",
                "    padding-top: 20px;",
                "    padding-right: 10px;",
                "    padding-bottom: 50px;",
                "    padding-left: 80px;' > ",
                "",
                "  <div class=\"nomad--header\">",
                "   <div style=\"text-align:center\">",
                "    <h2> <img id=\"nomad\" src=\"https://nomad-coe.eu/uploads/nomad/images/NOMAD_Logo2.png\" height=\"100\" alt=\"NOMAD Logo\">  NOMAD Analytics Toolkit  <img id=\"nomad\" src=\"https://www.nomad-coe.eu/uploads/nomad/backgrounds/head_big-data_analytics_2.png\" height=\"80\" alt=\"NOMAD Logo\"> </h2>",
                "  </div>",
                "    <h3>Hands-on Workshop Density-Functional Theory and Beyond:<br> Compressed sensing for identifying descriptors ",
                "    </h3>",
                "    <p class=\"nomad--description\">",
                "      created by:",
                " Emre Ahmetcik<sup>1</sup>, ",
                " Angelo Ziletti<sup> 1</sup>,",
                " Runhai Ouyang<sup>1</sup>,",
                " Luca Ghiringhelli<sup>1</sup>,",
                " and Matthias Scheffler<sup>1</sup> <br><br>",
                "   ",
                "      <sup>1</sup> Fritz Haber Institute of the Max Planck Society, Faradayweg 4-6, D-14195 Berlin, Germany <br>",
                "      <span class=\"nomad--last-updated\" data-version=\"v1.0.0\">[Last updated: July 23, 2017]</span>",
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                "",
                "<div style='text-align: right;'>",
                "<a href=\"https://analytics-toolkit.nomad-coe.eu/home/\" class=\"btn btn-primary\" style=\"font-size:larger;\">Back to Analytics Home</a> ",
                "<a href=\"https://www.nomad-coe.eu/\" class=\"btn btn-primary\" style=\"font-size:larger;\">Back to NOMAD CoE Home</a> ",
                "</div>  ",
                "",
                "",
                "",
                "<br><br><br>",
                "This tutorial shows how to find descriptive parameters (short formulas) to predict materials properties using compressed sensing tools. As an example we adress the prediction of the crystal structure stability of 82 octet binary compounds. We provide scripts which access the relevant data from the NOMAD Archive and determine descriptors for both regression models (predicting the rocksalt (RS) vs zincblende (ZB) structure energy difference) and classification (a compound is predicted to be most stable in either RS, ZB, CsCl, NiAs or CrB structure) .",
                "",
                "The idea of using compressed sensing tools: Starting from simple physical quantities (\"building blocks\", here properties of the constituent free atoms such as orbital radii), millions (or billions) of candidate formulas are generated by applying arithmetic operations combining building blocks, for example forming sums and products of them. These candidate formulas constitute the so-called \"feature space\". Then a feature selection method is used to select only a few of these formulas that explain the data. In this tutorial we use the methods LASSO+$\\ell_0$ as introduced in ",
                "<div style=\"padding: 1ex; margin-top: 1ex; margin-bottom: 1ex; border-style: dotted; border-width: 1pt; border-color: blue; border-radius: 3px;\">",
                "L. M. Ghiringhelli, J. Vybiral, S. V. Levchenko, C. Draxl, M. Scheffler: <span style=\"font-style: italic;\">Big Data of Materials Science: Critical Role of the Descriptor</span>,  Phys. Rev. Lett. 114, 105503 (2015) <a href=\"http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.114.105503\" target=\"_blank\">[PDF]</a></div>",
                " and Sure Independence Screening Sparse Operator  (SISSO) as proposed in",
                "<div style=\"padding: 1ex; margin-top: 1ex; margin-bottom: 1ex; border-style: dotted; border-width: 1pt; border-color: blue; border-radius: 3px;\">",
Emre Ahmetcik's avatar
Emre Ahmetcik committed
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                "R. Ouyang, S. Curtarolo, E. Ahmetcik, M. Scheffler, L. M. Ghiringhelli: <span style=\"font-style: italic;\">SISSO: a compressed-sensing method for identifying the best low-dimensional descriptor in an immensity of offered candidates</span>, Phys Rev. Materials, in print (2018) <a href=\"https://arxiv.org/abs/1710.03319\" target=\"_blank\">https://arxiv.org/abs/1710.03319</a> .",
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            "id": "sectionFrtJgy",
            "type": "section",
            "title": "Introduction to the compressed sensing methods",
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                "The feature space is generated by creating a list of analytical expressions (the derived features), obtained by combining the primary features and arithmetic operations. We put all $m$ derived features into a descriptor matrix $\\mathbf{D} \\in \\mathbb{R}^{82 \\times m}$ where each column stands for a derived feature and each row for a compound. An $\\ell_0$-regularization ",
                "",
                "$\\text{argmin}_{\\mathbf{c} \\in \\mathbb{R}^{m}} \\{\\|\\mathbf{P} - \\mathbf{D}\\mathbf{c}\\|^2_2 +\\lambda \\|\\mathbf{c}\\|_0\\}$",
                "",
                "determines those few feature columns which approximate a property vector $\\mathbf{P} \\in \\mathbb{R}^{82}$ (i.e RS vs. ZB energy differences) best. The subscript 0 stays for the $\\ell_0$-quasinorm, that counts the number of non-zero elements of $\\mathbf{c}$ and $\\lambda > 0$ is called the regularization parameter. Performing the $\\ell_0$-regularization becomes fast computational infeasable and often approximations (i.e. LASSO, LASSO+L0, SIS+L0) are needed since in practice the $\\ell_0$-regularization needs to be solved combinatorial: All singletons, pairs, triplets, ... $n$-tuples (up to the selected maximum dimension of the descriptor) are listed and for each set a least-square regression is performed. The $n$-tuple that gives the lowest mean square error for the least-square regression fit is selected as the resulting $n$-dimensional descriptor."
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            "id": "sectionS8rXSG",
            "type": "section",
            "title": "The LASSO+$\\ell_0$ method",
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                " LASSO+$\\ell_0$  combines the Least Absolute Shrinkage and Selection Operator (LASSO)  and the $\\ell_0$-regularization. In the first step the LASSO minimization",
                "",
                "$\\text{argmin}_{\\mathbf{c} \\in \\mathbb{R}^{m}} \\{\\|\\mathbf{P} - \\mathbf{D}\\mathbf{c}\\|^2_2 +\\lambda \\|\\mathbf{c}\\|_1\\}$",
                "",
                "is performed repetitively by decreasing $\\lambda$ in small steps starting from the largest value that gives one non-zero element in $\\mathbf{c}$,  until a desired number of features (i.e. 30) that have non-zero coefficient in $\\mathbf{c}$ are collected/saved.  Using these collected features  the $\\ell_0$-regularization is performed subsequently.",
                ""
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            "id": "sectionbYEx1l",
            "type": "section",
            "title": "The SISSO for regression",
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                "SISSO works iteratively. In the first iteration, a number $k$ of features is collected that have the largest correlation (scalar product) with $\\mathbf{P}$. The feature with the largest correlation is simply the 1D descriptor. Next, a residual is constructed as the error made at the first iteration. A new set of $k$ features is now selected as those having the largest correlation with the residual. The 2D descriptor is the pair of features that yield the smallest fitting error upon least-square regression, among all possible pairs contained in the union of the sets selected in this and the first iteration. In each next iteration a new residual is constructed as the error made in the previous iteration, then a new set of $k$ features is extracted as those that have largest correlation with each new residual. The $n$D descriptor is the $n$-tuple of features that yield the smallest fitting error upon least square regression, among all possible $n$-tuples contained in the union of the sets obtained in each new iteration and all the previous iterations. If $k=1$ the method collapses to the so-called orthogonal matching pursuit."
            ],
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            "id": "sectionH1x7Pk",
            "type": "section",
            "title": "The SISSO for classification",
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            "body": [
                "For classification (categorial target),  the SISSO is applied with a different loss function. Given the convex hull of each target class, now,  the number of data points located in overlapping convex hull regions is minimized. "
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            "id": "sectionB3Bqbw",
            "type": "section",
            "title": "Get the data from the NOMAD Archive",
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            "body": [
                "<summary>",
                "<li>See that the data from the repository can be accessed easily with simple python scripts.</li>",
                "<li>Consider the specific case of 82 octet binary compounds in the RS and ZB structure. Process the data to build the target property $\\mathbf{P} \\in \\mathbb{R}^{82}$ of RS vs ZB energy differences.</li>",
                "<li>Construct the descriptor matrix $\\mathbf{D} \\in \\mathbb{R}^{82 \\times m}$ using primary features from the NOMAD atomic data collection.</li>",
                "</summary>"
            ],
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                "In the following cell we declare the paths to the JSON files of the DFT calculations. The list provides 2 x 82 paths of single point calculations at (close to) the equilibrium of the RS and ZB crystal symmetry."
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                    "'/parsed/prod-022/FhiAimsParser2.0.0-2-gf9335c4/RWApItBGtGUDsfMVlHKqrjUQ4rShT/PAzsSjYMU1-CdGulNpG_KzgFlfRrK.json',",
                    "'/parsed/prod-022/FhiAimsParser2.0.0-2-gf9335c4/RWApItBGtGUDsfMVlHKqrjUQ4rShT/PVlh5JoGn6jHWlE96SKt6eRMYTIVK.json',",
                    "'/parsed/prod-022/FhiAimsParser2.0.0-2-gf9335c4/RWApItBGtGUDsfMVlHKqrjUQ4rShT/PHfNgOoPEHjzs9iOh900vIUv-GVJl.json',",
                    "'/parsed/prod-022/FhiAimsParser2.0.0-2-gf9335c4/RWApItBGtGUDsfMVlHKqrjUQ4rShT/PVTUPQyCTvrAWV0DEN_xnPgrBPmM2.json',",
                    "'/parsed/prod-022/FhiAimsParser2.0.0-2-gf9335c4/RWApItBGtGUDsfMVlHKqrjUQ4rShT/Pws96oc5f7jIltD9Vvqc3svzL4mcW.json',",
                    "'/parsed/prod-022/FhiAimsParser2.0.0-2-gf9335c4/RWApItBGtGUDsfMVlHKqrjUQ4rShT/PLJBz0uY-AywnUhGMCXMounM-_Af3.json',",
                    "'/parsed/prod-022/FhiAimsParser2.0.0-2-gf9335c4/RWApItBGtGUDsfMVlHKqrjUQ4rShT/Pkole11VWAOiu91qHeq6lOzIM2Y1Y.json',",
                    "'/parsed/prod-022/FhiAimsParser2.0.0-2-gf9335c4/RWApItBGtGUDsfMVlHKqrjUQ4rShT/P2R4Ds9DFm8USF_AgHtQnWK1TkQiR.json',",
                    "'/parsed/prod-022/FhiAimsParser2.0.0-2-gf9335c4/RWApItBGtGUDsfMVlHKqrjUQ4rShT/PxrF4NRKjX9jsmVIocs7uQuLwD_cS.json',",
                    "'/parsed/prod-022/FhiAimsParser2.0.0-2-gf9335c4/RWApItBGtGUDsfMVlHKqrjUQ4rShT/PMxYGoRCMDXQWrNytWJHc-vUgRKTT.json',",
                    "'/parsed/prod-022/FhiAimsParser2.0.0-2-gf9335c4/RWApItBGtGUDsfMVlHKqrjUQ4rShT/PK3-3e-av7nkv5AOEwjZyyjkI9Hgy.json',",
                    "'/parsed/prod-022/FhiAimsParser2.0.0-2-gf9335c4/RWApItBGtGUDsfMVlHKqrjUQ4rShT/PC8N-y0PPPHeAwhkYGyYYI9H1UUHy.json',",
                    "'/parsed/prod-022/FhiAimsParser2.0.0-2-gf9335c4/RWApItBGtGUDsfMVlHKqrjUQ4rShT/PqWPF7Pn3u9LPGyrxipPfrpfm31zz.json',",
                    "'/parsed/prod-022/FhiAimsParser2.0.0-2-gf9335c4/RWApItBGtGUDsfMVlHKqrjUQ4rShT/PM_ADyGOaL4e2biSXvxQWrEDM78Z3.json',",
                    "'/parsed/prod-022/FhiAimsParser2.0.0-2-gf9335c4/RWApItBGtGUDsfMVlHKqrjUQ4rShT/PfGXdJkORwLQ-aX-d9bla7obqtnkt.json',",
                    "'/parsed/prod-022/FhiAimsParser2.0.0-2-gf9335c4/RWApItBGtGUDsfMVlHKqrjUQ4rShT/PmILc9BsSYjJ9OKH4MkPr0D4LGYGC.json',",
                    "'/parsed/prod-022/FhiAimsParser2.0.0-2-gf9335c4/RWApItBGtGUDsfMVlHKqrjUQ4rShT/Py-J0ezaQ_Fdsh_196hT-XgsYNQAs.json',",
                    "'/parsed/prod-022/FhiAimsParser2.0.0-2-gf9335c4/RWApItBGtGUDsfMVlHKqrjUQ4rShT/PUJUPZHk2jrE1KVUS7H13mKBH4oVR.json',",
                    "'/parsed/prod-022/FhiAimsParser2.0.0-2-gf9335c4/RWApItBGtGUDsfMVlHKqrjUQ4rShT/POIYfYCEIron9yzowfHWhVea-VEFW.json',",
                    "'/parsed/prod-022/FhiAimsParser2.0.0-2-gf9335c4/RWApItBGtGUDsfMVlHKqrjUQ4rShT/P_1mfRE8eDZ7zCLQwGT_3n8YC34dE.json'",
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                    "]"
                ]
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            },
            "output": {
                "state": {},
                "selectedType": "Hidden",
                "pluginName": "IPython",
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                "shellId": "3DF25244EDA34E38A482AD12D464F1C9"
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            },
            "evaluatorReader": true,
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            "lineCount": 166,
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            "tags": "json_list"
        },
        {
            "id": "markdownjLPoKc",
            "type": "markdown",
            "body": [
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                "Now, import and use the class 'NOMADStructure'  to get the chemical formula, the total energy and the space group from each json file. You can type 'help(NOMADStructure)' to find out, what else can be extracted from a structure."
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            ],
            "evaluatorReader": false
        },
        {
            "id": "code7GfA9p",
            "type": "code",
            "evaluator": "IPython",
            "input": {
                "body": [
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                    "import h5py",
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                    "from nomad_sim.nomad_structures import NOMADStructure",
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                    "",
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                    "nomad_structure_list = [NOMADStructure(in_file=json_path, file_format='NOMAD', take_first='False') for json_path in json_list]",
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                    "",
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                    "# Consider first element of nomad_structure_list as an example",
                    "first_structure = nomad_structure_list[0]",
                    "print first_structure.chemical_symbols[0,0]",
                    "print first_structure.chemical_formula[0,0]",
                    "print first_structure.spacegroup_analyzer[0,0].get_space_group_number() ",
                    "print first_structure.energy_total__eV[0,0] # energy per unit cell",
                    "print first_structure.energy_total__eV[0,0]/ len(first_structure.atoms[0,0]) # energy per atom"
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                ]
            },
            "output": {
                "state": {},
                "selectedType": "Results",
                "pluginName": "IPython",
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                "shellId": "3DF25244EDA34E38A482AD12D464F1C9",
                "height": 237
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            },
            "evaluatorReader": true,
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            "lineCount": 12,
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            "tags": "nomad_structure_list"
        },
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        {
            "id": "markdownI6KjT2",
            "type": "markdown",
            "body": [
                "Find the function 'get_energies'  in the following cell which returns a data frame of  2 x 82 rows containing the chemical formula, the total energy per atom in eV, the space group and the json path."
            ],
            "evaluatorReader": false
        },
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        {
            "id": "codeE3BU1L",
            "type": "code",
            "evaluator": "IPython",
            "input": {
                "body": [
                    "import pandas as pd",
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                    "",
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                    "def get_energies(nomad_structure_list):",
                    "    chemical_formula_list = [nomad_structure.chemical_formula[0,0] for nomad_structure in nomad_structure_list]",
                    "    energy_list = [nomad_structure.energy_total__eV[0,0] / len(nomad_structure.atoms[0,0]) for nomad_structure in nomad_structure_list ]",
                    "    space_group_list = [nomad_structure.spacegroup_analyzer[0,0].get_space_group_number() for nomad_structure in nomad_structure_list]",
                    "    ",
                    "    data = zip(chemical_formula_list, energy_list, space_group_list, json_list)",
                    "    df_out = pd.DataFrame(data, columns=['chemical_formula', 'energy', 'space_group', 'json_path'])",
                    "    return df_out"
                ]
            },
            "output": {
                "state": {},
                "selectedType": "Hidden",
                "pluginName": "IPython",
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                "shellId": "3DF25244EDA34E38A482AD12D464F1C9",
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                "height": 5705
            },
            "evaluatorReader": true,
            "lineCount": 10,
            "tags": "get_energies"
        },
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        {
            "id": "markdownbEHiV5",
            "type": "markdown",
            "body": [
                "Call 'get_energies' and print the data frame sorted by the chemical formula."
            ],
            "evaluatorReader": false
        },
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        {
            "id": "codeheXP3D",
            "type": "code",
            "evaluator": "IPython",
            "input": {
                "body": [
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                    "# set pandas option to display whole data frame",
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                    "pd.set_option('display.max_rows', 200)",
                    "pd.set_option('display.expand_frame_repr', False)",
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                    "",
                    "df_energies = get_energies(nomad_structure_list)",
                    "print df_energies.sort_values('chemical_formula')"
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                ]
            },
            "output": {
                "state": {},
                "selectedType": "Results",
                "pluginName": "IPython",
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                "shellId": "3DF25244EDA34E38A482AD12D464F1C9",
                "height": 3009
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            },
            "evaluatorReader": true,
            "lineCount": 6
        },
        {
            "id": "markdownnT2K2b",
            "type": "markdown",
            "body": [
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                "In the following cell, a function to construct the property vector $\\mathbf{P}$ of RS vs ZB energy differences is defined using the just introduced function 'get_energies'. The structures are separated by their space groups: 225 and 221 for RS and 216 and 227 for ZB (the second space group of each structure takes account of the elemental solids). Furthermore the json path of the minimum structure (RS or ZB) is returned."
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            ],
            "evaluatorReader": false
        },
        {
            "id": "code2z75Py",
            "type": "code",
            "evaluator": "IPython",
            "input": {
                "body": [
                    "def get_energy_diffs(nomad_structure_list):    ",
                    "    df = get_energies(nomad_structure_list)",
                    "    ",
                    "    spacegroup_tuples = [(225, 221), (216, 227)] # [(RS), (ZB)]",
                    "    selected_space_groups, spacegroups_to_be_replace = zip(*spacegroup_tuples)",
                    "    ",
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                    "    # replace all 221 by 225 and all 227 by 216 to get one RS and one ZB column",
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                    "    df['space_group'] = df['space_group'].replace( spacegroups_to_be_replace , selected_space_groups )   ",
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                    "",
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                    "    # df with json_path of minimum energy per chemical_formula",
                    "    df_path_of_min = df.sort_values(by='energy').groupby(['chemical_formula'], as_index=True).first()['json_path']",
                    "    ",
                    "    # transform space_group values to column names",
                    "    df = df.pivot_table('energy', 'chemical_formula', 'space_group')",
                    "    df.columns = [216, 225]",
                    "    ",
                    "    df['energy_diff'] = df[225] - df[216]",
                    "    df['json_path'] = df_path_of_min   ",
                    "    return df[['energy_diff', 'json_path']]"
                ]
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            },
            "output": {
                "state": {},
                "selectedType": "Hidden",
                "pluginName": "IPython",
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                "shellId": "3DF25244EDA34E38A482AD12D464F1C9",
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                "height": 1435
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            },
            "evaluatorReader": true,
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            "lineCount": 19,
            "tags": "get_energy_diffs"
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        },
        {
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            "id": "markdownOBfktj",
            "type": "markdown",
            "body": [
                "Now let's consider a function get_descriptors which creates the descriptor matrix $\\mathbf{D}$ . This will use a materials-to-descritpors dictionary (cell below) and a function combine_features (defined two cells below) to create new derived descriptors with arithmetic operations. The functions are rather technical, so you can ignore the definitions in the next three cells. But you need to run them!"
            ],
            "evaluatorReader": false
        },
        {
            "id": "codebV3mYa",
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            "type": "code",
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            "evaluator": "IPython",
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            "input": {
                "body": [
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                    "features_dict = {u'EA(B)': {'SeZn': -2.750999927520752, 'InSb': -1.8466999530792236, 'SZn': -2.844899892807007, 'BN': -1.8674999475479126, 'OSr': -3.0058999061584473, 'BrRb': -3.739300012588501, 'BaTe': -2.6659998893737793, 'BeSe': -2.750999927520752, 'MgS': -2.844899892807007, 'ClRb': -3.9707999229431152, 'BrNa': -3.739300012588501, 'BP': -1.9199999570846558, 'MgSe': -2.750999927520752, 'FK': -4.273499965667725, 'BrLi': -3.739300012588501, 'BSb': -1.8466999530792236, 'AsB': -1.8392000198364258, 'GeSn': -0.9490000009536743, 'GeSi': -0.9929999709129333, 'CaTe': -2.6659998893737793, 'ClK': -3.9707999229431152, 'CsI': -3.5134999752044678, 'MgO': -3.0058999061584473, 'BrCs': -3.739300012588501, 'CsF': -4.273499965667725, 'BrCu': -3.739300012588501, 'ILi': -3.5134999752044678, 'FLi': -4.273499965667725, 'CuF': -4.273499965667725, 'INa': -3.5134999752044678, 'Ge2': -0.9490000009536743, 'FNa': -4.273499965667725, 'C2': -0.8723999857902527, 'AgBr': -3.739300012588501, 'AsGa': -1.8392000198364258, 'CuI': -3.5134999752044678, 'AlN': -1.8674999475479126, 'Si2': -0.9929999709129333, 'SiSn': -0.9929999709129333, 'ClLi': -3.9707999229431152, 'ClNa': -3.9707999229431152, 'AsIn': -1.8392000198364258, 'OZn': -3.0058999061584473, 'CGe': -0.8723999857902527, 'CdO': -3.0058999061584473, 'InP': -1.9199999570846558, 'SSr': -2.844899892807007, 'InN': -1.8674999475479126, 'BaSe': -2.750999927520752, 'BrK': -3.739300012588501, 'BeTe': -2.6659998893737793, 'CdS': -2.844899892807007, 'CdTe': -2.6659998893737793, 'TeZn': -2.6659998893737793, 'GaP': -1.9199999570846558, 'CdSe': -2.750999927520752, 'MgTe': -2.6659998893737793, 'AlP': -1.9199999570846558, 'BeO': -3.0058999061584473, 'CaSe': -2.750999927520752, 'FRb': -4.273499965667725, 'SeSr': -2.750999927520752, 'CSi': -0.8723999857902527, 'AgCl': -3.9707999229431152, 'AgI': -3.5134999752044678, 'GaN': -1.8674999475479126, 'CaS': -2.844899892807007, 'AgF': -4.273499965667725, 'GaSb': -1.8466999530792236, 'IK': -3.5134999752044678, 'IRb': -3.5134999752044678, 'BaS': -2.844899892807007, 'CaO': -3.0058999061584473, 'AlAs': -1.8392000198364258, 'Sn2': -1.039199948310852, 'ClCu': -3.9707999229431152, 'CSn': -0.8723999857902527, 'BaO': -3.0058999061584473, 'ClCs': -3.9707999229431152, 'AlSb': -1.8466999530792236, 'SrTe': -2.6659998893737793, 'BeS': -2.844899892807007}, u'r_p(A)': {'SeZn': 1.5499999523162842, 'InSb': 1.5, 'SZn': 1.5499999523162842, 'BN': 0.8299999833106995, 'OSr': 2.549999952316284, 'BrRb': 3.200000047683716, 'BaTe': 2.630000114440918, 'BeSe': 1.2100000381469727, 'MgS': 1.899999976158142, 'ClRb': 3.200000047683716, 'BrNa': 2.5999999046325684, 'BP': 0.8299999833106995, 'MgSe': 1.899999976158142, 'FK': 2.440000057220459, 'BrLi': 2.0, 'BSb': 0.8299999833106995, 'AsB': 0.8299999833106995, 'GeSn': 1.340000033378601, 'GeSi': 1.159999966621399, 'CaTe': 2.319999933242798, 'ClK': 2.440000057220459, 'CsI': 3.1600000858306885, 'MgO': 1.899999976158142, 'BrCs': 3.1600000858306885, 'CsF': 3.1600000858306885, 'BrCu': 1.6799999475479126, 'ILi': 2.0, 'FLi': 2.0, 'CuF': 1.6799999475479126, 'INa': 2.5999999046325684, 'Ge2': 1.159999966621399, 'FNa': 2.5999999046325684, 'C2': 0.6299999952316284, 'AgBr': 1.8799999952316284, 'AsGa': 1.3300000429153442, 'CuI': 1.6799999475479126, 'AlN': 1.3899999856948853, 'Si2': 1.1299999952316284, 'SiSn': 1.340000033378601, 'ClLi': 2.0, 'ClNa': 2.5999999046325684, 'AsIn': 1.5, 'OZn': 1.5499999523162842, 'CGe': 1.159999966621399, 'CdO': 1.7400000095367432, 'InP': 1.5, 'SSr': 2.549999952316284, 'InN': 1.5, 'BaSe': 2.630000114440918, 'BrK': 2.440000057220459, 'BeTe': 1.2100000381469727, 'CdS': 1.7400000095367432, 'CdTe': 1.7400000095367432, 'TeZn': 1.5499999523162842, 'GaP': 1.3300000429153442, 'CdSe': 1.7400000095367432, 'MgTe': 1.899999976158142, 'AlP': 1.3899999856948853, 'BeO': 1.2100000381469727, 'CaSe': 2.319999933242798, 'FRb': 3.200000047683716, 'SeSr': 2.549999952316284, 'CSi': 1.1299999952316284, 'AgCl': 1.8799999952316284, 'AgI': 1.8799999952316284, 'GaN': 1.3300000429153442, 'CaS': 2.319999933242798, 'AgF': 1.8799999952316284, 'GaSb': 1.3300000429153442, 'IK': 2.440000057220459, 'IRb': 3.200000047683716, 'BaS': 2.630000114440918, 'CaO': 2.319999933242798, 'AlAs': 1.3899999856948853, 'Sn2': 1.340000033378601, 'ClCu': 1.6799999475479126, 'CSn': 1.340000033378601, 'BaO': 2.630000114440918, 'ClCs': 3.1600000858306885, 'AlSb': 1.3899999856948853, 'SrTe': 2.549999952316284, 'BeS': 1.2100000381469727}, u'IP(A)': {'SeZn': -10.135499954223633, 'InSb': -5.537399768829346, 'SZn': -10.135499954223633, 'BN': -8.1899995803833, 'OSr': -6.031599998474121, 'BrRb': -4.288899898529053, 'BaTe': -5.515699863433838, 'BeSe': -9.459400177001953, 'MgS': -8.037099838256836, 'ClRb': -4.288899898529053, 'BrNa': -5.223100185394287, 'BP': -8.1899995803833, 'MgSe': -8.037099838256836, 'FK': -4.433199882507324, 'BrLi': -5.329100131988525, 'BSb': -8.1899995803833, 'AsB': -8.1899995803833, 'GeSn': -7.042799949645996, 'GeSi': -7.566999912261963, 'CaTe': -6.427999973297119, 'ClK': -4.433199882507324, 'CsI': -4.006199836730957, 'MgO': -8.037099838256836, 'BrCs': -4.006199836730957, 'CsF': -4.006199836730957, 'BrCu': -8.388799667358398, 'ILi': -5.329100131988525, 'FLi': -5.329100131988525, 'CuF': -8.388799667358398, 'INa': -5.223100185394287, 'Ge2': -7.566999912261963, 'FNa': -5.223100185394287, 'C2': -10.851699829101562, 'AgBr': -8.058099746704102, 'AsGa': -5.81820011138916, 'CuI': -8.388799667358398, 'AlN': -5.7804999351501465, 'Si2': -7.757699966430664, 'SiSn': -7.042799949645996, 'ClLi': -5.329100131988525, 'ClNa': -5.223100185394287, 'AsIn': -5.537399768829346, 'OZn': -10.135499954223633, 'CGe': -7.566999912261963, 'CdO': -9.581399917602539, 'InP': -5.537399768829346, 'SSr': -6.031599998474121, 'InN': -5.537399768829346, 'BaSe': -5.515699863433838, 'BrK': -4.433199882507324, 'BeTe': -9.459400177001953, 'CdS': -9.581399917602539, 'CdTe': -9.581399917602539, 'TeZn': -10.135499954223633, 'GaP': -5.81820011138916, 'CdSe': -9.581399917602539, 'MgTe': -8.037099838256836, 'AlP': -5.7804999351501465, 'BeO': -9.459400177001953, 'CaSe': -6.427999973297119, 'FRb': -4.288899898529053, 'SeSr': -6.031599998474121, 'CSi': -7.757699966430664, 'AgCl': -8.058099746704102, 'AgI': -8.058099746704102, 'GaN': -5.81820011138916, 'CaS': -6.427999973297119, 'AgF': -8.058099746704102, 'GaSb': -5.81820011138916, 'IK': -4.433199882507324, 'IRb': -4.288899898529053, 'BaS': -5.515699863433838, 'CaO': -6.427999973297119, 'AlAs': -5.7804999351501465, 'Sn2': -7.042799949645996, 'ClCu': -8.388799667358398, 'CSn': -7.042799949645996, 'BaO': -5.515699863433838, 'ClCs': -4.006199836730957, 'AlSb': -5.7804999351501465, 'SrTe': -6.031599998474121, 'BeS': -9.459400177001953}, u'r_p(B)': {'SeZn': 0.949999988079071, 'InSb': 1.2300000190734863, 'SZn': 0.8500000238418579, 'BN': 0.5099999904632568, 'OSr': 0.4300000071525574, 'BrRb': 0.8799999952316284, 'BaTe': 1.1399999856948853, 'BeSe': 0.949999988079071, 'MgS': 0.8500000238418579, 'ClRb': 0.7599999904632568, 'BrNa': 0.8799999952316284, 'BP': 0.9700000286102295, 'MgSe': 0.949999988079071, 'FK': 0.3700000047683716, 'BrLi': 0.8799999952316284, 'BSb': 1.2300000190734863, 'AsB': 1.0399999618530273, 'GeSn': 1.159999966621399, 'GeSi': 1.1299999952316284, 'CaTe': 1.1399999856948853, 'ClK': 0.7599999904632568, 'CsI': 1.0700000524520874, 'MgO': 0.4300000071525574, 'BrCs': 0.8799999952316284, 'CsF': 0.3700000047683716, 'BrCu': 0.8799999952316284, 'ILi': 1.0700000524520874, 'FLi': 0.3700000047683716, 'CuF': 0.3700000047683716, 'INa': 1.0700000524520874, 'Ge2': 1.159999966621399, 'FNa': 0.3700000047683716, 'C2': 0.6299999952316284, 'AgBr': 0.8799999952316284, 'AsGa': 1.0399999618530273, 'CuI': 1.0700000524520874, 'AlN': 0.5099999904632568, 'Si2': 1.1299999952316284, 'SiSn': 1.1299999952316284, 'ClLi': 0.7599999904632568, 'ClNa': 0.7599999904632568, 'AsIn': 1.0399999618530273, 'OZn': 0.4300000071525574, 'CGe': 0.6299999952316284, 'CdO': 0.4300000071525574, 'InP': 0.9700000286102295, 'SSr': 0.8500000238418579, 'InN': 0.5099999904632568, 'BaSe': 0.949999988079071, 'BrK': 0.8799999952316284, 'BeTe': 1.1399999856948853, 'CdS': 0.8500000238418579, 'CdTe': 1.1399999856948853, 'TeZn': 1.1399999856948853, 'GaP': 0.9700000286102295, 'CdSe': 0.949999988079071, 'MgTe': 1.1399999856948853, 'AlP': 0.9700000286102295, 'BeO': 0.4300000071525574, 'CaSe': 0.949999988079071, 'FRb': 0.3700000047683716, 'SeSr': 0.949999988079071, 'CSi': 0.6299999952316284, 'AgCl': 0.7599999904632568, 'AgI': 1.0700000524520874, 'GaN': 0.5099999904632568, 'CaS': 0.8500000238418579, 'AgF': 0.3700000047683716, 'GaSb': 1.2300000190734863, 'IK': 1.0700000524520874, 'IRb': 1.0700000524520874, 'BaS': 0.8500000238418579, 'CaO': 0.4300000071525574, 'AlAs': 1.0399999618530273, 'Sn2': 1.340000033378601, 'ClCu': 0.7599999904632568, 'CSn': 0.6299999952316284, 'BaO': 0.4300000071525574, 'ClCs': 0.7599999904632568, 'AlSb': 1.2300000190734863, 'SrTe': 1.1399999856948853, 'BeS': 0.8500000238418579}, u'E_LUMO(B)': {'SeZn': 1.315999984741211, 'InSb': 0.10499999672174454, 'SZn': 0.6420000195503235, 'BN': 3.056999921798706, 'OSr': 2.5409998893737793, 'BrRb': 0.7080000042915344, 'BaTe': 0.0989999994635582, 'BeSe': 1.315999984741211, 'MgS': 0.6420000195503235, 'ClRb': 0.5740000009536743, 'BrNa': 0.7080000042915344, 'BP': 0.18299999833106995, 'MgSe': 1.315999984741211, 'FK': 1.2510000467300415, 'BrLi': 0.7080000042915344, 'BSb': 0.10499999672174454, 'AsB': 0.06400000303983688, 'GeSn': 2.174999952316284, 'GeSi': 0.4399999976158142, 'CaTe': 0.0989999994635582, 'ClK': 0.5740000009536743, 'CsI': 0.21299999952316284, 'MgO': 2.5409998893737793, 'BrCs': 0.7080000042915344, 'CsF': 1.2510000467300415, 'BrCu': 0.7080000042915344, 'ILi': 0.21299999952316284, 'FLi': 1.2510000467300415, 'CuF': 1.2510000467300415, 'INa': 0.21299999952316284, 'Ge2': 2.174999952316284, 'FNa': 1.2510000467300415, 'C2': 1.9919999837875366, 'AgBr': 0.7080000042915344, 'AsGa': 0.06400000303983688, 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0.6420000195503235, 'CaO': 2.5409998893737793, 'AlAs': 0.06400000303983688, 'Sn2': 0.00800000037997961, 'ClCu': 0.5740000009536743, 'CSn': 1.9919999837875366, 'BaO': 2.5409998893737793, 'ClCs': 0.5740000009536743, 'AlSb': 0.10499999672174454, 'SrTe': 0.0989999994635582, 'BeS': 0.6420000195503235}, u'IP(B)': {'SeZn': -10.946000099182129, 'InSb': -8.468299865722656, 'SZn': -11.795100212097168, 'BN': -13.585200309753418, 'OSr': -16.43320083618164, 'BrRb': -12.6496000289917, 'BaTe': -9.866700172424316, 'BeSe': -10.946000099182129, 'MgS': -11.795100212097168, 'ClRb': -13.901800155639648, 'BrNa': -12.6496000289917, 'BP': -9.75059986114502, 'MgSe': -10.946000099182129, 'FK': -19.404300689697266, 'BrLi': -12.6496000289917, 'BSb': -8.468299865722656, 'AsB': -9.261899948120117, 'GeSn': -7.566999912261963, 'GeSi': -7.757699966430664, 'CaTe': -9.866700172424316, 'ClK': -13.901800155639648, 'CsI': -11.257100105285645, 'MgO': -16.43320083618164, 'BrCs': -12.6496000289917, 'CsF': -19.404300689697266, 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1.649999976158142, 'BSb': 0.8100000023841858, 'AsB': 0.8100000023841858, 'GeSn': 1.059999942779541, 'GeSi': 0.9200000166893005, 'CaTe': 1.7599999904632568, 'ClK': 2.130000114440918, 'CsI': 2.4600000381469727, 'MgO': 1.3300000429153442, 'BrCs': 2.4600000381469727, 'CsF': 2.4600000381469727, 'BrCu': 1.2000000476837158, 'ILi': 1.649999976158142, 'FLi': 1.649999976158142, 'CuF': 1.2000000476837158, 'INa': 1.7100000381469727, 'Ge2': 0.9200000166893005, 'FNa': 1.7100000381469727, 'C2': 0.6399999856948853, 'AgBr': 1.3200000524520874, 'AsGa': 0.9900000095367432, 'CuI': 1.2000000476837158, 'AlN': 1.090000033378601, 'Si2': 0.9399999976158142, 'SiSn': 1.059999942779541, 'ClLi': 1.649999976158142, 'ClNa': 1.7100000381469727, 'AsIn': 1.1299999952316284, 'OZn': 1.100000023841858, 'CGe': 0.9200000166893005, 'CdO': 1.2300000190734863, 'InP': 1.1299999952316284, 'SSr': 1.909999966621399, 'InN': 1.1299999952316284, 'BaSe': 2.1500000953674316, 'BrK': 2.130000114440918, 'BeTe': 1.0800000429153442, 'CdS': 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2.0399999618530273, 'BaTe': 2.359999895095825, 'BeSe': 1.2000000476837158, 'MgS': 1.7100000381469727, 'ClRb': 2.0399999618530273, 'BrNa': 1.4900000095367432, 'BP': 0.8100000023841858, 'MgSe': 1.7100000381469727, 'FK': 1.909999966621399, 'BrLi': 1.350000023841858, 'BSb': 0.8100000023841858, 'AsB': 0.8100000023841858, 'GeSn': 1.3700000047683716, 'GeSi': 1.159999966621399, 'CaTe': 1.9900000095367432, 'ClK': 1.909999966621399, 'CsI': 2.25, 'MgO': 1.7100000381469727, 'BrCs': 2.25, 'CsF': 2.25, 'BrCu': 1.0700000524520874, 'ILi': 1.350000023841858, 'FLi': 1.350000023841858, 'CuF': 1.0700000524520874, 'INa': 1.4900000095367432, 'Ge2': 1.159999966621399, 'FNa': 1.4900000095367432, 'C2': 0.6299999952316284, 'AgBr': 1.2400000095367432, 'AsGa': 1.2300000190734863, 'CuI': 1.0700000524520874, 'AlN': 1.2699999809265137, 'Si2': 1.090000033378601, 'SiSn': 1.3700000047683716, 'ClLi': 1.350000023841858, 'ClNa': 1.4900000095367432, 'AsIn': 1.4800000190734863, 'OZn': 1.409999966621399, 'CGe': 1.159999966621399, 'CdO': 1.5499999523162842, 'InP': 1.4800000190734863, 'SSr': 2.2200000286102295, 'InN': 1.4800000190734863, 'BaSe': 2.359999895095825, 'BrK': 1.909999966621399, 'BeTe': 1.2000000476837158, 'CdS': 1.5499999523162842, 'CdTe': 1.5499999523162842, 'TeZn': 1.409999966621399, 'GaP': 1.2300000190734863, 'CdSe': 1.5499999523162842, 'MgTe': 1.7100000381469727, 'AlP': 1.2699999809265137, 'BeO': 1.2000000476837158, 'CaSe': 1.9900000095367432, 'FRb': 2.0399999618530273, 'SeSr': 2.2200000286102295, 'CSi': 1.090000033378601, 'AgCl': 1.2400000095367432, 'AgI': 1.2400000095367432, 'GaN': 1.2300000190734863, 'CaS': 1.9900000095367432, 'AgF': 1.2400000095367432, 'GaSb': 1.2300000190734863, 'IK': 1.909999966621399, 'IRb': 2.0399999618530273, 'BaS': 2.359999895095825, 'CaO': 1.9900000095367432, 'AlAs': 1.2699999809265137, 'Sn2': 1.3700000047683716, 'ClCu': 1.0700000524520874, 'CSn': 1.3700000047683716, 'BaO': 2.359999895095825, 'ClCs': 2.25, 'AlSb': 1.2699999809265137, 'SrTe': 2.2200000286102295, 'BeS': 1.2000000476837158}, u'period(B)': {'SeZn': 4.0, 'InSb': 5.0, 'SZn': 3.0, 'BN': 2.0, 'OSr': 2.0, 'BrRb': 4.0, 'BaTe': 5.0, 'BeSe': 4.0, 'MgS': 3.0, 'ClRb': 3.0, 'BrNa': 4.0, 'BP': 3.0, 'MgSe': 4.0, 'FK': 2.0, 'BrLi': 4.0, 'BSb': 5.0, 'AsB': 4.0, 'GeSn': 4.0, 'GeSi': 3.0, 'CaTe': 5.0, 'ClK': 3.0, 'CsI': 5.0, 'MgO': 2.0, 'BrCs': 4.0, 'CsF': 2.0, 'BrCu': 4.0, 'ILi': 5.0, 'FLi': 2.0, 'CuF': 2.0, 'INa': 5.0, 'Ge2': 4.0, 'FNa': 2.0, 'C2': 2.0, 'AgBr': 4.0, 'AsGa': 4.0, 'CuI': 5.0, 'AlN': 2.0, 'Si2': 3.0, 'SiSn': 3.0, 'ClLi': 3.0, 'ClNa': 3.0, 'AsIn': 4.0, 'OZn': 2.0, 'CGe': 2.0, 'CdO': 2.0, 'InP': 3.0, 'SSr': 3.0, 'InN': 2.0, 'BaSe': 4.0, 'BrK': 4.0, 'BeTe': 5.0, 'CdS': 3.0, 'CdTe': 5.0, 'TeZn': 5.0, 'GaP': 3.0, 'CdSe': 4.0, 'MgTe': 5.0, 'AlP': 3.0, 'BeO': 2.0, 'CaSe': 4.0, 'FRb': 2.0, 'SeSr': 4.0, 'CSi': 2.0, 'AgCl': 3.0, 'AgI': 5.0, 'GaN': 2.0, 'CaS': 3.0, 'AgF': 2.0, 'GaSb': 5.0, 'IK': 5.0, 'IRb': 5.0, 'BaS': 3.0, 'CaO': 2.0, 'AlAs': 4.0, 'Sn2': 5.0, 'ClCu': 3.0, 'CSn': 2.0, 'BaO': 2.0, 'ClCs': 3.0, 'AlSb': 5.0, 'SrTe': 5.0, 'BeS': 3.0}, u'E_HOMO(A)': {'SeZn': -6.2170000076293945, 'InSb': -2.697000026702881, 'SZn': -6.2170000076293945, 'BN': -3.7149999141693115, 'OSr': -3.6410000324249268, 'BrRb': -2.359999895095825, 'BaTe': -3.3459999561309814, 'BeSe': -5.599999904632568, 'MgS': -4.7820000648498535, 'ClRb': -2.359999895095825, 'BrNa': -2.819000005722046, 'BP': -3.7149999141693115, 'MgSe': -4.7820000648498535, 'FK': -2.4260001182556152, 'BrLi': -2.874000072479248, 'BSb': -3.7149999141693115, 'AsB': -3.7149999141693115, 'GeSn': -3.865999937057495, 'GeSi': -4.046000003814697, 'CaTe': -3.864000082015991, 'ClK': -2.4260001182556152, 'CsI': -2.2200000286102295, 'MgO': -4.7820000648498535, 'BrCs': -2.2200000286102295, 'CsF': -2.2200000286102295, 'BrCu': -4.855999946594238, 'ILi': -2.874000072479248, 'FLi': -2.874000072479248, 'CuF': -4.855999946594238, 'INa': -2.819000005722046, 'Ge2': -4.046000003814697, 'FNa': -2.819000005722046, 'C2': 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-2.7320001125335693, 'IK': -2.4260001182556152, 'IRb': -2.359999895095825, 'BaS': -3.3459999561309814, 'CaO': -3.864000082015991, 'AlAs': -2.7839999198913574, 'Sn2': -3.865999937057495, 'ClCu': -4.855999946594238, 'CSn': -3.865999937057495, 'BaO': -3.3459999561309814, 'ClCs': -2.2200000286102295, 'AlSb': -2.7839999198913574, 'SrTe': -3.6410000324249268, 'BeS': -5.599999904632568}, u'period(A)': {'SeZn': 4.0, 'InSb': 5.0, 'SZn': 4.0, 'BN': 2.0, 'OSr': 5.0, 'BrRb': 5.0, 'BaTe': 6.0, 'BeSe': 2.0, 'MgS': 3.0, 'ClRb': 5.0, 'BrNa': 3.0, 'BP': 2.0, 'MgSe': 3.0, 'FK': 4.0, 'BrLi': 2.0, 'BSb': 2.0, 'AsB': 2.0, 'GeSn': 5.0, 'GeSi': 4.0, 'CaTe': 4.0, 'ClK': 4.0, 'CsI': 6.0, 'MgO': 3.0, 'BrCs': 6.0, 'CsF': 6.0, 'BrCu': 4.0, 'ILi': 2.0, 'FLi': 2.0, 'CuF': 4.0, 'INa': 3.0, 'Ge2': 4.0, 'FNa': 3.0, 'C2': 2.0, 'AgBr': 5.0, 'AsGa': 4.0, 'CuI': 4.0, 'AlN': 3.0, 'Si2': 3.0, 'SiSn': 5.0, 'ClLi': 2.0, 'ClNa': 3.0, 'AsIn': 5.0, 'OZn': 4.0, 'CGe': 4.0, 'CdO': 5.0, 'InP': 5.0, 'SSr': 5.0, 'InN': 5.0, 'BaSe': 6.0, 'BrK': 4.0, 'BeTe': 2.0, 'CdS': 5.0, 'CdTe': 5.0, 'TeZn': 4.0, 'GaP': 4.0, 'CdSe': 5.0, 'MgTe': 3.0, 'AlP': 3.0, 'BeO': 2.0, 'CaSe': 4.0, 'FRb': 5.0, 'SeSr': 5.0, 'CSi': 3.0, 'AgCl': 5.0, 'AgI': 5.0, 'GaN': 4.0, 'CaS': 4.0, 'AgF': 5.0, 'GaSb': 4.0, 'IK': 4.0, 'IRb': 5.0, 'BaS': 6.0, 'CaO': 4.0, 'AlAs': 3.0, 'Sn2': 5.0, 'ClCu': 4.0, 'CSn': 5.0, 'BaO': 6.0, 'ClCs': 6.0, 'AlSb': 3.0, 'SrTe': 5.0, 'BeS': 2.0}, u'd(B)': {'SeZn': 1.0800000429153442, 'InSb': 1.2300000190734863, 'SZn': 0.949999988079071, 'BN': 0.550000011920929, 'OSr': 0.6000000238418579, 'BrRb': 1.1399999856948853, 'BaTe': 1.2699999809265137, 'BeSe': 1.0800000429153442, 'MgS': 0.949999988079071, 'ClRb': 0.9900000095367432, 'BrNa': 1.1399999856948853, 'BP': 0.9399999976158142, 'MgSe': 1.0800000429153442, 'FK': 0.6899999976158142, 'BrLi': 1.1399999856948853, 'BSb': 1.2300000190734863, 'AsB': 1.0399999618530273, 'GeSn': 1.159999966621399, 'GeSi': 1.090000033378601, 'CaTe': 1.2699999809265137, 'ClK': 0.9900000095367432, 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'CsF': 55.0, 'BrCu': 29.0, 'ILi': 3.0, 'FLi': 3.0, 'CuF': 29.0, 'INa': 11.0, 'Ge2': 32.0, 'FNa': 11.0, 'C2': 6.0, 'AgBr': 47.0, 'AsGa': 31.0, 'CuI': 29.0, 'AlN': 13.0, 'Si2': 14.0, 'SiSn': 50.0, 'ClLi': 3.0, 'ClNa': 11.0, 'AsIn': 49.0, 'OZn': 30.0, 'CGe': 32.0, 'CdO': 48.0, 'InP': 49.0, 'SSr': 38.0, 'InN': 49.0, 'BaSe': 56.0, 'BrK': 19.0, 'BeTe': 4.0, 'CdS': 48.0, 'CdTe': 48.0, 'TeZn': 30.0, 'GaP': 31.0, 'CdSe': 48.0, 'MgTe': 12.0, 'AlP': 13.0, 'BeO': 4.0, 'CaSe': 20.0, 'FRb': 37.0, 'SeSr': 38.0, 'CSi': 14.0, 'AgCl': 47.0, 'AgI': 47.0, 'GaN': 31.0, 'CaS': 20.0, 'AgF': 47.0, 'GaSb': 31.0, 'IK': 19.0, 'IRb': 37.0, 'BaS': 56.0, 'CaO': 20.0, 'AlAs': 13.0, 'Sn2': 50.0, 'ClCu': 29.0, 'CSn': 50.0, 'BaO': 56.0, 'ClCs': 55.0, 'AlSb': 13.0, 'SrTe': 38.0, 'BeS': 4.0}, u'r_d(A)': {'SeZn': 2.25, 'InSb': 3.109999895095825, 'SZn': 2.25, 'BN': 1.9500000476837158, 'OSr': 1.2000000476837158, 'BrRb': 1.9600000381469727, 'BaTe': 1.350000023841858, 'BeSe': 2.880000114440918, 'MgS': 3.1700000762939453, 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1.2000000476837158, 'InN': 3.109999895095825, 'BaSe': 1.350000023841858, 'BrK': 1.7899999618530273, 'BeTe': 2.880000114440918, 'CdS': 2.5999999046325684, 'CdTe': 2.5999999046325684, 'TeZn': 2.25, 'GaP': 2.1600000858306885, 'CdSe': 2.5999999046325684, 'MgTe': 3.1700000762939453, 'AlP': 1.940000057220459, 'BeO': 2.880000114440918, 'CaSe': 0.6800000071525574, 'FRb': 1.9600000381469727, 'SeSr': 1.2000000476837158, 'CSi': 1.8899999856948853, 'AgCl': 2.9700000286102295, 'AgI': 2.9700000286102295, 'GaN': 2.1600000858306885, 'CaS': 0.6800000071525574, 'AgF': 2.9700000286102295, 'GaSb': 2.1600000858306885, 'IK': 1.7899999618530273, 'IRb': 1.9600000381469727, 'BaS': 1.350000023841858, 'CaO': 0.6800000071525574, 'AlAs': 1.940000057220459, 'Sn2': 2.0299999713897705, 'ClCu': 2.5799999237060547, 'CSn': 2.0299999713897705, 'BaO': 1.350000023841858, 'ClCs': 1.9700000286102295, 'AlSb': 1.940000057220459, 'SrTe': 1.2000000476837158, 'BeS': 2.880000114440918}, u'r_s(B)': {'SeZn': 0.800000011920929, 'InSb': 1.0, 'SZn': 0.7400000095367432, 'BN': 0.5400000214576721, 'OSr': 0.46000000834465027, 'BrRb': 0.75, 'BaTe': 0.9399999976158142, 'BeSe': 0.800000011920929, 'MgS': 0.7400000095367432, 'ClRb': 0.6800000071525574, 'BrNa': 0.75, 'BP': 0.8299999833106995, 'MgSe': 0.800000011920929, 'FK': 0.4099999964237213, 'BrLi': 0.75, 'BSb': 1.0, 'AsB': 0.8500000238418579, 'GeSn': 0.9200000166893005, 'GeSi': 0.9399999976158142, 'CaTe': 0.9399999976158142, 'ClK': 0.6800000071525574, 'CsI': 0.8999999761581421, 'MgO': 0.46000000834465027, 'BrCs': 0.75, 'CsF': 0.4099999964237213, 'BrCu': 0.75, 'ILi': 0.8999999761581421, 'FLi': 0.4099999964237213, 'CuF': 0.4099999964237213, 'INa': 0.8999999761581421, 'Ge2': 0.9200000166893005, 'FNa': 0.4099999964237213, 'C2': 0.6399999856948853, 'AgBr': 0.75, 'AsGa': 0.8500000238418579, 'CuI': 0.8999999761581421, 'AlN': 0.5400000214576721, 'Si2': 0.9399999976158142, 'SiSn': 0.9399999976158142, 'ClLi': 0.6800000071525574, 'ClNa': 0.6800000071525574, 'AsIn': 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0.6800000071525574, 'AlSb': 1.0, 'SrTe': 0.9399999976158142, 'BeS': 0.7400000095367432}, u'r_d(B)': {'SeZn': 2.180000066757202, 'InSb': 2.059999942779541, 'SZn': 2.369999885559082, 'BN': 1.5399999618530273, 'OSr': 2.2200000286102295, 'BrRb': 1.8700000047683716, 'BaTe': 1.8300000429153442, 'BeSe': 2.180000066757202, 'MgS': 2.369999885559082, 'ClRb': 1.6699999570846558, 'BrNa': 1.8700000047683716, 'BP': 1.7699999809265137, 'MgSe': 2.180000066757202, 'FK': 1.4299999475479126, 'BrLi': 1.8700000047683716, 'BSb': 2.059999942779541, 'AsB': 2.0199999809265137, 'GeSn': 2.369999885559082, 'GeSi': 1.8899999856948853, 'CaTe': 1.8300000429153442, 'ClK': 1.6699999570846558, 'CsI': 1.7200000286102295, 'MgO': 2.2200000286102295, 'BrCs': 1.8700000047683716, 'CsF': 1.4299999475479126, 'BrCu': 1.8700000047683716, 'ILi': 1.7200000286102295, 'FLi': 1.4299999475479126, 'CuF': 1.4299999475479126, 'INa': 1.7200000286102295, 'Ge2': 2.369999885559082, 'FNa': 1.4299999475479126, 'C2': 1.6299999952316284, 'AgBr': 1.8700000047683716, 'AsGa': 2.0199999809265137, 'CuI': 1.7200000286102295, 'AlN': 1.5399999618530273, 'Si2': 1.8899999856948853, 'SiSn': 1.8899999856948853, 'ClLi': 1.6699999570846558, 'ClNa': 1.6699999570846558, 'AsIn': 2.0199999809265137, 'OZn': 2.2200000286102295, 'CGe': 1.6299999952316284, 'CdO': 2.2200000286102295, 'InP': 1.7699999809265137, 'SSr': 2.369999885559082, 'InN': 1.5399999618530273, 'BaSe': 2.180000066757202, 'BrK': 1.8700000047683716, 'BeTe': 1.8300000429153442, 'CdS': 2.369999885559082, 'CdTe': 1.8300000429153442, 'TeZn': 1.8300000429153442, 'GaP': 1.7699999809265137, 'CdSe': 2.180000066757202, 'MgTe': 1.8300000429153442, 'AlP': 1.7699999809265137, 'BeO': 2.2200000286102295, 'CaSe': 2.180000066757202, 'FRb': 1.4299999475479126, 'SeSr': 2.180000066757202, 'CSi': 1.6299999952316284, 'AgCl': 1.6699999570846558, 'AgI': 1.7200000286102295, 'GaN': 1.5399999618530273, 'CaS': 2.369999885559082, 'AgF': 1.4299999475479126, 'GaSb': 2.059999942779541, 'IK': 1.7200000286102295, 'IRb': 1.7200000286102295, 'BaS': 2.369999885559082, 'CaO': 2.2200000286102295, 'AlAs': 2.0199999809265137, 'Sn2': 2.0299999713897705, 'ClCu': 1.6699999570846558, 'CSn': 1.6299999952316284, 'BaO': 2.2200000286102295, 'ClCs': 1.6699999570846558, 'AlSb': 2.059999942779541, 'SrTe': 1.8300000429153442, 'BeS': 2.369999885559082}, u'EA(A)': {'SeZn': 1.0807000398635864, 'InSb': -0.2563000023365021, 'SZn': 1.0807000398635864, 'BN': -0.10740000009536743, 'OSr': 0.34310001134872437, 'BrRb': -0.590399980545044, 'BaTe': 0.27799999713897705, 'BeSe': 0.6305000185966492, 'MgS': 0.6924999952316284, 'ClRb': -0.590399980545044, 'BrNa': -0.7156999707221985, 'BP': -0.10740000009536743, 'MgSe': 0.6924999952316284, 'FK': -0.6212999820709229, 'BrLi': -0.6980999708175659, 'BSb': -0.10740000009536743, 'AsB': -0.10740000009536743, 'GeSn': -1.039199948310852, 'GeSi': -0.9490000009536743, 'CaTe': 0.30390000343322754, 'ClK': -0.6212999820709229, 'CsI': -0.569599986076355, 'MgO': 0.6924999952316284, 'BrCs': -0.569599986076355, 'CsF': -0.569599986076355, 'BrCu': -1.6384999752044678, 'ILi': -0.6980999708175659, 'FLi': -0.6980999708175659, 'CuF': -1.6384999752044678, 'INa': -0.7156999707221985, 'Ge2': -0.9490000009536743, 'FNa': -0.7156999707221985, 'C2': -0.8723999857902527, 'AgBr': -1.666599988937378, 'AsGa': -0.10809999704360962, 'CuI': -1.6384999752044678, 'AlN': -0.3125, 'Si2': -0.9929999709129333, 'SiSn': -1.039199948310852, 'ClLi': -0.6980999708175659, 'ClNa': -0.7156999707221985, 'AsIn': -0.2563000023365021, 'OZn': 1.0807000398635864, 'CGe': -0.9490000009536743, 'CdO': 0.838699996471405, 'InP': -0.2563000023365021, 'SSr': 0.34310001134872437, 'InN': -0.2563000023365021, 'BaSe': 0.27799999713897705, 'BrK': -0.6212999820709229, 'BeTe': 0.6305000185966492, 'CdS': 0.838699996471405, 'CdTe': 0.838699996471405, 'TeZn': 1.0807000398635864, 'GaP': -0.10809999704360962, 'CdSe': 0.838699996471405, 'MgTe': 0.6924999952316284, 'AlP': -0.3125, 'BeO': 0.6305000185966492, 'CaSe': 0.30390000343322754, 'FRb': -0.590399980545044, 'SeSr': 0.34310001134872437, 'CSi': -0.9929999709129333, 'AgCl': -1.666599988937378, 'AgI': -1.666599988937378, 'GaN': -0.10809999704360962, 'CaS': 0.30390000343322754, 'AgF': -1.666599988937378, 'GaSb': -0.10809999704360962, 'IK': -0.6212999820709229, 'IRb': -0.590399980545044, 'BaS': 0.27799999713897705, 'CaO': 0.30390000343322754, 'AlAs': -0.3125, 'Sn2': -1.039199948310852, 'ClCu': -1.6384999752044678, 'CSn': -1.039199948310852, 'BaO': 0.27799999713897705, 'ClCs': -0.569599986076355, 'AlSb': -0.3125, 'SrTe': 0.34310001134872437, 'BeS': 0.6305000185966492}, u'E_LUMO(A)': {'SeZn': -1.194000005722046, 'InSb': 0.36800000071525574, 'SZn': -1.194000005722046, 'BN': 2.247999906539917, 'OSr': -1.378999948501587, 'BrRb': -0.7049999833106995, 'BaTe': -2.128999948501587, 'BeSe': -2.0980000495910645, 'MgS': -1.3580000400543213, 'ClRb': -0.7049999833106995, 'BrNa': -0.7179999947547913, 'BP': 2.247999906539917, 'MgSe': -1.3580000400543213, 'FK': -0.6970000267028809, 'BrLi': -0.9779999852180481, 'BSb': 2.247999906539917, 'AsB': 2.247999906539917, 'GeSn': 0.00800000037997961, 'GeSi': 2.174999952316284, 'CaTe': -2.132999897003174, 'ClK': -0.6970000267028809, 'CsI': -0.5479999780654907, 'MgO': -1.3580000400543213, 'BrCs': -0.5479999780654907, 'CsF': -0.5479999780654907, 'BrCu': -0.640999972820282, 'ILi': -0.9779999852180481, 'FLi': -0.9779999852180481, 'CuF': -0.640999972820282, 'INa': -0.7179999947547913, 'Ge2': 2.174999952316284, 'FNa': -0.7179999947547913, 'C2': 1.9919999837875366, 'AgBr': -0.4790000021457672, 'AsGa': 0.12999999523162842, 'CuI': -0.640999972820282, 'AlN': 0.6949999928474426, 'Si2': 0.4399999976158142, 'SiSn': 0.00800000037997961, 'ClLi': -0.9779999852180481, 'ClNa': -0.7179999947547913, 'AsIn': 0.36800000071525574, 'OZn': -1.194000005722046, 'CGe': 2.174999952316284, 'CdO': -1.309000015258789, 'InP': 0.36800000071525574, 'SSr': -1.378999948501587, 'InN': 0.36800000071525574, 'BaSe': -2.128999948501587, 'BrK': -0.6970000267028809, 'BeTe': -2.0980000495910645, 'CdS': -1.309000015258789, 'CdTe': -1.309000015258789, 'TeZn': -1.194000005722046, 'GaP': 0.12999999523162842, 'CdSe': -1.309000015258789, 'MgTe': -1.3580000400543213, 'AlP': 0.6949999928474426, 'BeO': -2.0980000495910645, 'CaSe': -2.132999897003174, 'FRb': -0.7049999833106995, 'SeSr': -1.378999948501587, 'CSi': 0.4399999976158142, 'AgCl': -0.4790000021457672, 'AgI': -0.4790000021457672, 'GaN': 0.12999999523162842, 'CaS': -2.132999897003174, 'AgF': -0.4790000021457672, 'GaSb': 0.12999999523162842, 'IK': -0.6970000267028809, 'IRb': -0.7049999833106995, 'BaS': -2.128999948501587, 'CaO': -2.132999897003174, 'AlAs': 0.6949999928474426, 'Sn2': 0.00800000037997961, 'ClCu': -0.640999972820282, 'CSn': 0.00800000037997961, 'BaO': -2.128999948501587, 'ClCs': -0.5479999780654907, 'AlSb': 0.6949999928474426, 'SrTe': -1.378999948501587, 'BeS': -2.0980000495910645}, u'Z(B)': {'SeZn': 34.0, 'InSb': 51.0, 'SZn': 16.0, 'BN': 7.0, 'OSr': 8.0, 'BrRb': 35.0, 'BaTe': 52.0, 'BeSe': 34.0, 'MgS': 16.0, 'ClRb': 17.0, 'BrNa': 35.0, 'BP': 15.0, 'MgSe': 34.0, 'FK': 9.0, 'BrLi': 35.0, 'BSb': 51.0, 'AsB': 33.0, 'GeSn': 32.0, 'GeSi': 14.0, 'CaTe': 52.0, 'ClK': 17.0, 'CsI': 53.0, 'MgO': 8.0, 'BrCs': 35.0, 'CsF': 9.0, 'BrCu': 35.0, 'ILi': 53.0, 'FLi': 9.0, 'CuF': 9.0, 'INa': 53.0, 'Ge2': 32.0, 'FNa': 9.0, 'C2': 6.0, 'AgBr': 35.0, 'AsGa': 33.0, 'CuI': 53.0, 'AlN': 7.0, 'Si2': 14.0, 'SiSn': 14.0, 'ClLi': 17.0, 'ClNa': 17.0, 'AsIn': 33.0, 'OZn': 8.0, 'CGe': 6.0, 'CdO': 8.0, 'InP': 15.0, 'SSr': 16.0, 'InN': 7.0, 'BaSe': 34.0, 'BrK': 35.0, 'BeTe': 52.0, 'CdS': 16.0, 'CdTe': 52.0, 'TeZn': 52.0, 'GaP': 15.0, 'CdSe': 34.0, 'MgTe': 52.0, 'AlP': 15.0, 'BeO': 8.0, 'CaSe': 34.0, 'FRb': 9.0, 'SeSr': 34.0, 'CSi': 6.0, 'AgCl': 17.0, 'AgI': 53.0, 'GaN': 7.0, 'CaS': 16.0, 'AgF': 9.0, 'GaSb': 51.0, 'IK': 53.0, 'IRb': 53.0, 'BaS': 16.0, 'CaO': 8.0, 'AlAs': 33.0, 'Sn2': 50.0, 'ClCu': 17.0, 'CSn': 6.0, 'BaO': 8.0, 'ClCs': 17.0, 'AlSb': 51.0, 'SrTe': 52.0, 'BeS': 16.0}, u'E_HOMO(B)': {'SeZn': -6.6539998054504395, 'InSb': -4.991000175476074, 'SZn': -7.105999946594238, 'BN': -7.238999843597412, 'OSr': -9.196999549865723, 'BrRb': -8.00100040435791, 'BaTe': -6.109000205993652, 'BeSe': -6.6539998054504395, 'MgS': -7.105999946594238, 'ClRb': -8.699999809265137, 'BrNa': -8.00100040435791, 'BP': -5.5960001945495605, 'MgSe': -6.6539998054504395, 'FK': -11.293999671936035, 'BrLi': -8.00100040435791, 'BSb': -4.991000175476074, 'AsB': -5.341000080108643, 'GeSn': -4.046000003814697, 'GeSi': -4.163000106811523, 'CaTe': -6.109000205993652, 'ClK': -8.699999809265137, 'CsI': -7.236000061035156, 'MgO': -9.196999549865723, 'BrCs': -8.00100040435791, 'CsF': -11.293999671936035, 'BrCu': -8.00100040435791, 'ILi': -7.236000061035156, 'FLi': -11.293999671936035, 'CuF': -11.293999671936035, 'INa': -7.236000061035156, 'Ge2': -4.046000003814697, 'FNa': -11.293999671936035, 'C2': -5.415999889373779, 'AgBr': -8.00100040435791, 'AsGa': -5.341000080108643, 'CuI': -7.236000061035156, 'AlN': -7.238999843597412, 'Si2': -4.163000106811523, 'SiSn': -4.163000106811523, 'ClLi': -8.699999809265137, 'ClNa': -8.699999809265137, 'AsIn': -5.341000080108643, 'OZn': -9.196999549865723, 'CGe': -5.415999889373779, 'CdO': -9.196999549865723, 'InP': -5.5960001945495605, 'SSr': -7.105999946594238, 'InN': -7.238999843597412, 'BaSe': -6.6539998054504395, 'BrK': -8.00100040435791, 'BeTe': -6.109000205993652, 'CdS': -7.105999946594238, 'CdTe': -6.109000205993652, 'TeZn': -6.109000205993652, 'GaP': -5.5960001945495605, 'CdSe': -6.6539998054504395, 'MgTe': -6.109000205993652, 'AlP': -5.5960001945495605, 'BeO': -9.196999549865723, 'CaSe': -6.6539998054504395, 'FRb': -11.293999671936035, 'SeSr': -6.6539998054504395, 'CSi': -5.415999889373779, 'AgCl': -8.699999809265137, 'AgI': -7.236000061035156, 'GaN': -7.238999843597412, 'CaS': -7.105999946594238, 'AgF': -11.293999671936035, 'GaSb': -4.991000175476074, 'IK': -7.236000061035156, 'IRb': -7.236000061035156, 'BaS': -7.105999946594238, 'CaO': -9.196999549865723, 'AlAs': -5.341000080108643, 'Sn2': -3.865999937057495, 'ClCu': -8.699999809265137, 'CSn': -5.415999889373779, 'BaO': -9.196999549865723, 'ClCs': -8.699999809265137, 'AlSb': -4.991000175476074, 'SrTe': -6.109000205993652, 'BeS': -7.105999946594238}}"
                ],
                "hidden": true
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            },
            "output": {
                "state": {},
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                "selectedType": "Hidden",
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                "pluginName": "IPython",
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                "shellId": "3DF25244EDA34E38A482AD12D464F1C9"
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            },
            "evaluatorReader": true,
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            "lineCount": 1,
            "tags": "get_descriptors"
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        },
        {
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            "id": "codeNlZOhl",
            "type": "code",
            "evaluator": "IPython",
            "input": {
                "body": [
                    "import itertools",
                    "from math import exp, sqrt",
                    "import math",
                    "",
                    "def _my_power_2(row):",
                    "    return pow(row[0], 2)         ",
                    "",
                    "def _my_power_3(row):",
                    "    return pow(row[0], 3)    ",
                    "",
                    "def _my_power_m1(row):",
                    "    return pow(row[0],-1)",
                    "",
                    "def _my_power_m2(row):",
                    "    return pow(row[0],-2)",
                    "",
                    "def _my_power_m3(row):",
                    "    return pow(row[0],-3)",
                    "",
                    "def _my_abs_sqrt(row):",
                    "    return math.sqrtabs(abs(row[0]))",
                    "    ",
                    "def _my_exp(row):",
                    "    return exp(row[0])",
                    "",
                    "def _my_exp_power_2(row):",
                    "    return exp(pow(row[0], 2))",
                    "",
                    "def _my_exp_power_3(row):",
                    "    return exp(pow(row[0], 3))",
                    "",
                    "def _my_sum(row):",
                    "    return row[0] + row[1]",
                    "    ",
                    "def _my_abs_sum(row):",
                    "    return abs(row[0] + row[1])",
                    "",
                    "def _my_abs_diff(row):",
                    "    return abs(row[0] - row[1])   ",
                    "",
                    "def _my_diff(row):",
                    "    return row[0] - row[1] ",
                    "",
                    "def _my_div(row):",
                    "    return row[0]/row[1]",
                    "    ",
                    "def _my_sum_power_2(row):",
                    "    return pow((row[0] + row[1]), 2)",
                    "",
                    "def _my_sum_power_3(row):",
                    "    return pow((row[0] + row[1]), 3)",
                    "    ",
                    "def _my_sum_exp(row):",
                    "    return exp(row[0] + row[1])",
                    "",
                    "def _my_sum_exp_power_2(row):",
                    "    return exp(pow(row[0] + row[1], 2))",
                    "",
                    "def _my_sum_exp_power_3(row):",
                    "    return exp(pow(row[0] + row[1], 3))",
                    "  ",
                    "def combine_features(df=None, allowed_operations=None):",
                    "    \"\"\"Generate combination of features given a dataframe and a list of allowed operations.",
                    "    ",
                    "    For the exponentials, we introduce a characteristic energy/length",
                    "    converting the ",
                    "    ..todo:: Fix under/overflow errors, and introduce handling of exceptions.",
                    "",
                    "    \"\"\"",
                    "        ",
                    "    if allowed_operations:",
                    "        print('Selected operations:\\n {0}'.format(allowed_operations)) ",
                    "    else:",
                    "        print('No allowed operations selected.') ",
                    "        ",
                    "    columns_ = df.columns.tolist()    ",
                    "    ",
                    "    dict_features = {",
                    "        'period':'a0', ",
                    "        'Z': 'a0', ",
                    "        'group': 'a0', ",
                    "",
                    "        'IP': 'a1', ",
                    "        'EA': 'a1', ",
                    "",
                    "        'E_HOMO': 'a2', ",
                    "        'E_LUMO': 'a2', ",
                    "",
                    "",
                    "        'r_s': 'a3',",
                    "        'r_p': 'a3',",
                    "        'r_d': 'a3',",
                    "        'd': 'a3', ",
                    "   ",
                    "        }",
                    "        ",
                    "",
                    "    df_a0 = df[[col for col in columns_ if dict_features.get(col.split('(', 1)[0])=='a0']].astype('float32')    ",
                    "    df_a1 = df[[col for col in columns_ if dict_features.get(col.split('(', 1)[0])=='a1']].astype('float32')    ",
                    "    df_a2 = df[[col for col in columns_ if dict_features.get(col.split('(', 1)[0])=='a2']].astype('float32')    ",
                    "    df_a3 = df[[col for col in columns_ if dict_features.get(col.split('(', 1)[0])=='a3']].astype('float32')   ",
                    "",
                    "    ",
                    "    col_a0 = df_a0.columns.tolist()",
                    "    col_a1 = df_a1.columns.tolist()",
                    "    col_a2 = df_a2.columns.tolist()",
                    "    col_a3 = df_a3.columns.tolist()",
                    "",
                    "    #  this list will at the end all the dataframes created",
                    "    df_list = []",
                    "",
                    "    df_b0_list = []    ",
                    "    df_b1_list = []",
                    "    df_b2_list = []",
                    "    df_b3_list = []",
                    "    df_c3_list = []",
                    "    df_d3_list = []",
                    "    df_e3_list = []",
                    "    df_f1_list = []",
                    "    df_f2_list = []",
                    "    df_f3_list = []",
                    "    df_x1_list = []",
                    "    df_x2_list = []",
                    "    df_x_list = []",
                    "",
                    "",
                    "    # create b0: absolute differences and sums of a0   ",
                    "    # this is not in the PRL. ",
                    "    for subset in itertools.combinations(col_a0, 2):",
                    "        if '+' in allowed_operations:",
                    "            cols = ['('+subset[0]+'+'+subset[1]+')']        ",
                    "            data = df_a0[list(subset)].apply(_my_sum, axis=1)            ",
                    "            df_b0_list.append(pd.DataFrame(data, columns=cols))         ",
                    "            ",
                    "        if '-' in allowed_operations:",
                    "            cols = ['('+subset[0]+'-'+subset[1]+')']        ",
                    "            data = df_a0[list(subset)].apply(_my_diff, axis=1)            ",
                    "            df_b0_list.append(pd.DataFrame(data, columns=cols))   ",
                    "            ",
                    "            cols = ['('+subset[1]+'-'+subset[0]+')']        ",
                    "            data = df_a0[list(subset)].apply(_my_diff, axis=1)            ",
                    "            df_b0_list.append(pd.DataFrame(data, columns=cols))  ",
                    "        ",
                    "        if '|+|' in allowed_operations:",
                    "            cols = ['|'+subset[0]+'+'+subset[1]+'|']        ",
                    "            data = df_a0[list(subset)].apply(_my_abs_sum, axis=1)            ",
                    "            df_b0_list.append(pd.DataFrame(data, columns=cols))     ",
                    "        ",
                    "        if '|-|' in allowed_operations:",
                    "            cols = ['|'+subset[0]+'-'+subset[1]+'|']        ",
                    "            data = df_a0[list(subset)].apply(_my_abs_diff, axis=1)            ",
                    "            df_b0_list.append(pd.DataFrame(data, columns=cols))  ",
                    "            ",
                    "        if '/' in allowed_operations:",
                    "            cols = [subset[0]+'/'+subset[1]]        ",
                    "            data = df_a0[list(subset)].apply(_my_div, axis=1)            ",
                    "            df_b0_list.append(pd.DataFrame(data, columns=cols))  ",
                    "",
                    "            cols = [subset[1]+'/'+subset[0]]        ",
                    "            data = df_a0[list(subset)].apply(_my_div, axis=1)            ",
                    "            df_b0_list.append(pd.DataFrame(data, columns=cols))  ",
                    "",
                    "    ",
                    "    # we kept itertools.combinations to make the code more uniform with the binary operations",
                    "    for subset in itertools.combinations(col_a0, 1):",
                    "        if '^2' in allowed_operations:",
                    "            cols = [subset[0]+'^2']        ",
                    "            data = df_a0[list(subset)].apply(_my_power_2, axis=1)            ",
                    "            df_b0_list.append(pd.DataFrame(data, columns=cols))    ",
                    "            ",
                    "        if '^3' in allowed_operations:",
                    "            cols = [subset[0]+'^3']   ",
                    "            data = df_a0[list(subset)].apply(_my_power_3, axis=1)            ",
                    "            df_b0_list.append(pd.DataFrame(data, columns=cols)) ",
                    "",
                    "        if 'exp' in allowed_operations:",
                    "            cols = ['exp('+subset[0]+')']       ",
                    "            data = df_a0[list(subset)].apply(_my_exp, axis=1)            ",
                    "            df_b0_list.append(pd.DataFrame(data, columns=cols))        ",
                    "        ",
                    "        ",
                    "    # create b1: absolute differences and sums of a1    ",
                    "    for subset in itertools.combinations(col_a1, 2):",
                    "        if '+' in allowed_operations:",
                    "            cols = ['('+subset[0]+'+'+subset[1]+')']        ",
                    "            data = df_a1[list(subset)].apply(_my_sum, axis=1)            ",
                    "            df_b1_list.append(pd.DataFrame(data, columns=cols))         ",
                    "            ",
                    "        if '-' in allowed_operations:",
                    "            cols = ['('+subset[0]+'-'+subset[1]+')']        ",
                    "            data = df_a1[list(subset)].apply(_my_diff, axis=1)            ",
                    "            df_b1_list.append(pd.DataFrame(data, columns=cols))   ",
                    "",
                    "        if '|+|' in allowed_operations:",
                    "            cols = ['|'+subset[0]+'+'+subset[1]+'|']        ",
                    "            data = df_a1[list(subset)].apply(_my_abs_sum, axis=1)            ",
                    "            df_b1_list.append(pd.DataFrame(data, columns=cols))     ",
                    "            ",
                    "        if '|-|' in allowed_operations:",
                    "            cols = ['|'+subset[0]+'-'+subset[1]+'|']        ",
                    "            data = df_a1[list(subset)].apply(_my_abs_diff, axis=1)            ",
                    "            df_b1_list.append(pd.DataFrame(data, columns=cols))  ",
                    "",
                    "    # create b2: absolute differences and sums of a2    ",
                    "    for subset in itertools.combinations(col_a2, 2):",
                    "        if '+' in allowed_operations:",
                    "            cols = ['('+subset[0]+'+'+subset[1]+')']        ",
                    "            data = df_a2[list(subset)].apply(_my_sum, axis=1)            ",
                    "            df_b2_list.append(pd.DataFrame(data, columns=cols))         ",
                    "            ",
                    "        if '-' in allowed_operations:",
                    "            cols = ['('+subset[0]+'-'+subset[1]+')']        ",
                    "            data = df_a2[list(subset)].apply(_my_diff, axis=1)            ",
                    "            df_b2_list.append(pd.DataFrame(data, columns=cols))   ",
                    "",
                    "        if '|+|' in allowed_operations:",
                    "            cols = ['|'+subset[0]+'+'+subset[1]+'|']        ",
                    "            data = df_a2[list(subset)].apply(_my_abs_sum, axis=1)            ",
                    "            df_b2_list.append(pd.DataFrame(data, columns=cols))         ",
                    "            ",
                    "        if '|-|' in allowed_operations:",
                    "            cols = ['|'+subset[0]+'-'+subset[1]+'|']        ",
                    "            data = df_a2[list(subset)].apply(_my_abs_diff, axis=1)            ",
                    "            df_b2_list.append(pd.DataFrame(data, columns=cols))   ",
                    " ",
                    "    # create b3: absolute differences and sums of a3    ",
                    "    for subset in itertools.combinations(col_a3, 2):",
                    "        if '+' in allowed_operations:",
                    "            cols = ['('+subset[0]+'+'+subset[1]+')']        ",
                    "            data = df_a3[list(subset)].apply(_my_sum, axis=1)            ",
                    "            df_b3_list.append(pd.DataFrame(data, columns=cols))         ",
                    "            ",
                    "        if '-' in allowed_operations:",
                    "            cols = ['('+subset[0]+'-'+subset[1]+')']        ",
                    "            data = df_a3[list(subset)].apply(_my_diff, axis=1)            ",
                    "            df_b3_list.append(pd.DataFrame(data, columns=cols))              ",
                    "",
                    "        if '|+|' in allowed_operations:",
                    "            cols = ['|'+subset[0]+'+'+subset[1]+'|']        ",
                    "            data = df_a3[list(subset)].apply(_my_abs_sum, axis=1)            ",
                    "            df_b3_list.append(pd.DataFrame(data, columns=cols))  ",
                    "            ",
                    "        if '|-|' in allowed_operations:",
                    "            cols = ['|'+subset[0]+'-'+subset[1]+'|']        ",
                    "            data = df_a3[list(subset)].apply(_my_abs_diff, axis=1)            ",
                    "            df_b3_list.append(pd.DataFrame(data, columns=cols))              ",
                    "",
                    "    # create c3: two steps:",
                    "    # 1) squares of a3 - unary operations ",
                    "    # we kept itertools.combinations to make the code more uniform with the binary operations",
                    "    for subset in itertools.combinations(col_a3, 1):",
                    "        if '^2' in allowed_operations:",
                    "            cols = [subset[0]+'^2']        ",
                    "            data = df_a3[list(subset)].apply(_my_power_2, axis=1)            ",
                    "            df_c3_list.append(pd.DataFrame(data, columns=cols))    ",
                    "        if '^3' in allowed_operations:",
                    "            cols = [subset[0]+'^3']   ",
                    "            data = df_a3[list(subset)].apply(_my_power_3, axis=1)            ",
                    "            df_c3_list.append(pd.DataFrame(data, columns=cols)) ",
                    "",
                    "            ",
                    "    # 2) squares of b3 (only sums) --> sum squared of a3",
                    "    for subset in itertools.combinations(col_a3, 2):",
                    "        if '^2' in allowed_operations:",
                    "            cols = ['('+subset[0]+'+'+subset[1]+')^2']   ",
                    "            data = df_a3[list(subset)].apply(_my_sum_power_2, axis=1)            ",
                    "            df_c3_list.append(pd.DataFrame(data, columns=cols))        ",
                    "            ",
                    "        if '^3' in allowed_operations:",
                    "            cols = ['('+subset[0]+'+'+subset[1]+')^3']        ",
                    "            data = df_a3[list(subset)].apply(_my_sum_power_3, axis=1)            ",
                    "            df_c3_list.append(pd.DataFrame(data, columns=cols))",
                    "",
                    "    # create d3: two steps:",
                    "    # 1) exponentials of a3 - unary operations ",
                    "    # we kept itertools.combinations to make the code more uniform with the binary operations",
                    "    for subset in itertools.combinations(col_a3, 1):",
                    "        if 'exp' in allowed_operations:",
                    "            cols = ['exp('+subset[0]+')']      ",
                    "            df_subset = df_a3[list(subset)]",
                    "            data = df_subset.apply(_my_exp, axis=1)            ",
                    "            df_d3_list.append(pd.DataFrame(data, columns=cols))    ",
                    "            ",
                    "    # 2) exponentials of b3 (only sums) --> exponential of sum of a3",
                    "    for subset in itertools.combinations(col_a3, 2):",
                    "        if 'exp' in allowed_operations:",
                    "            cols = ['exp('+subset[0]+'+'+subset[1]+')']    ",
                    "            df_subset = df_a3[list(subset)]",
                    "            data = df_subset.apply(_my_sum_exp, axis=1)               ",
                    "            df_d3_list.append(pd.DataFrame(data, columns=cols))        ",
                    "",
                    "    # create e3: two steps:",
                    "    # 1) exponentials of squared a3 - unary operations ",
                    "    # we kept itertools.combinations to make the code more uniform with the binary operations",
                    "    for subset in itertools.combinations(col_a3, 1):",
                    "        operations={'exp', '^2'}",
                    "        if operations <= set(allowed_operations):",
                    "            cols = ['exp('+subset[0]+'^2)']",
                    "            df_subset = df_a3[list(subset)]",
                    "            data = df_subset.apply(_my_exp_power_2, axis=1)            ",
                    "            df_e3_list.append(pd.DataFrame(data, columns=cols))    ",
                    "            ",
                    "        operations={'exp', '^3'}",
                    "        if operations <= set(allowed_operations):",
                    "            try:",
                    "                cols = ['exp('+subset[0]+'^3)']",
                    "                df_subset = df_a3[list(subset)]",
                    "                data = df_subset.apply(_my_exp_power_3, axis=1)            ",
                    "                df_e3_list.append(pd.DataFrame(data, columns=cols)) ",
                    "            except OverflowError as e:",
                    "                print('Dropping feature combination that caused under/overflow.\\n')",
                    "",
                    "            ",
                    "    # 2) exponentials of b3 (only sums) --> exponential of sum of a3",
                    "    for subset in itertools.combinations(col_a3, 2):",
                    "        operations={'exp', '^2'}",
                    "        if operations <= set(allowed_operations):",
                    "            cols = ['exp(('+subset[0]+'+'+subset[1]+')^2)']",
                    "            df_subset = df_a3[list(subset)]",
                    "            data = df_subset.apply(_my_sum_exp_power_2, axis=1)            ",
                    "            df_e3_list.append(pd.DataFrame(data, columns=cols))        ",
                    "",
                    "        operations={'exp', '^3'}",
                    "        if operations <= set(allowed_operations):",
                    "            try:",
                    "                cols = ['exp(('+subset[0]+'+'+subset[1]+')^3)']",
                    "                df_subset = df_a3[list(subset)]",
                    "                data = df_subset.apply(_my_sum_exp_power_3, axis=1)            ",
                    "                df_e3_list.append(pd.DataFrame(data, columns=cols))   ",
                    "            except OverflowError as e:",
                    "                print('Dropping feature combination that caused under/overflow.\\n')",
                    "",
                    "    # make dataframes from lists, check if they are not empty",
                    "    # we make there here because they are going to be used to further",
                    "    # combine the features",
                    "    if not df_a0.empty: ",
                    "        df_list.append(df_a0)",
                    "        ",
                    "    if not df_a1.empty: ",
                    "        df_x1_list.append(df_a1)",
                    "        df_list.append(df_a1)",
                    "",
                    "    if not df_a2.empty: ",
                    "        df_x1_list.append(df_a2)",
                    "        df_list.append(df_a2)",
                    "        ",
                    "    if not df_a3.empty: ",
                    "        df_x1_list.append(df_a3)",
                    "        df_list.append(df_a3)",
                    "",
                    "",
                    "",
                    "    if df_b0_list: ",
                    "        df_b0 = pd.concat(df_b0_list, axis=1)",
                    "        col_b0 = df_b0.columns.tolist()",
                    "        df_b0.to_csv('./df_b0.csv', index=True)",
                    "        df_list.append(df_b0)",
                    "        ",
                    "    if df_b1_list: ",
                    "        df_b1 = pd.concat(df_b1_list, axis=1)",
                    "        col_b1 = df_b1.columns.tolist()",
                    "        df_x1_list.append(df_b1)",
                    "        df_list.append(df_b1)",
                    "",
                    "    if df_b2_list: ",
                    "        df_b2 = pd.concat(df_b2_list, axis=1)",
                    "        col_b2 = df_b2.columns.tolist()",
                    "        df_x1_list.append(df_b2)",
                    "        df_list.append(df_b2)",
                    "        ",
                    "    if df_b3_list: ",
                    "        df_b3 = pd.concat(df_b3_list, axis=1)",
                    "        col_b3 = df_b3.columns.tolist()        ",
                    "        df_x1_list.append(df_b3)",
                    "        df_list.append(df_b3)",
                    "    ",
                    "    if df_c3_list:",
                    "        df_c3 = pd.concat(df_c3_list, axis=1)",
                    "        col_c3 = df_c3.columns.tolist()",
                    "        df_x2_list.append(df_c3)",
                    "        df_list.append(df_c3)",
                    "",
                    "    if df_d3_list:",
                    "        df_d3 = pd.concat(df_d3_list, axis=1)",
                    "        col_d3 = df_d3.columns.tolist()",
                    "        df_x2_list.append(df_d3)",
                    "        df_list.append(df_d3)",
                    "",
                    "    if df_e3_list:",
                    "        df_e3 = pd.concat(df_e3_list, axis=1)",
                    "        col_e3 = df_e3.columns.tolist()",
                    "        df_x2_list.append(df_e3)",
                    "        df_list.append(df_e3)",
                    "",
                    "    if df_x1_list:",
                    "        df_x1 = pd.concat(df_x1_list, axis=1)",
                    "        col_x1 = df_x1.columns.tolist()",
                    "                ",
                    "    if df_x2_list:",
                    "        df_x2 = pd.concat(df_x2_list, axis=1)",
                    "        col_x2 = df_x2.columns.tolist()",
                    "",
                    "    if df_x1_list and df_x2_list:",
                    "        for el_x1 in col_x1:",
                    "            for el_x2 in col_x2:",
                    "                if '/' in allowed_operations:",
                    "                    cols = [el_x1+'/'+el_x2] ",
                    "                    #now the operation is between two dataframes",
                    "                    data = df_x1[el_x1].divide(df_x2[el_x2])     ",
                    "                    df_x_list.append(pd.DataFrame(data, columns=cols))   ",
                    "     ",
                    "",
                    "    if df_f1_list:",
                    "        df_f1 = pd.concat(df_f1_list, axis=1)",
                    "        col_f1 = df_f1.columns.tolist()",
                    "        df_list.append(df_f1)",
                    "",
                    "                ",
                    "    if df_x_list:",
                    "        df_x = pd.concat(df_x_list, axis=1)",
                    "        col_x = df_x.columns.tolist()",
                    "        df_list.append(df_x)",
                    "",
                    "",
                    "",
                    "",
                    "    if df_list:",
                    "        df_combined_features = pd.concat(df_list, axis=1)",
                    "    else:",
                    "        print('No features selected. Please select at least two primary features.')",
                    "        ",
                    "",
                    "    ",
                    "    print('Number of total features generated: {0}'.format(df_combined_features.shape[1]))",
                    "    ",
                    "    return df_combined_features",
                    ""
                ],
                "hidden": true
            },
            "output": {
                "state": {},
                "selectedType": "Hidden",
                "pluginName": "IPython",
                "shellId": "3DF25244EDA34E38A482AD12D464F1C9"
            },
            "evaluatorReader": true,
            "lineCount": 437,
            "tags": "get_descriptors"
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        },
        {
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            "id": "codeWJJ3l4",
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            "type": "code",
            "evaluator": "IPython",
            "input": {
                "body": [
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                    "from nomad_sim.utils_binaries import get_chemical_formula_binaries",
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                    "",
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                    "def get_descriptors(nomad_structure_list=[] ,selected_feature_list=[], allowed_operations=[], **kwargs):",
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                    "        ",
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                    "    # keep only one nomad_structure per chemical_formula since in our case the descriptors of same chemical_formula ",
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                    "    # but different structures are the same     ",
                    "    list_of_chemical_formulas = [get_chemical_formula_binaries(nomad_structure.atoms[0,0]) for nomad_structure in nomad_structure_list]",
                    "    # remove chemical_formula duplicates  ",
                    "    list_of_chemical_formulas = list(set(list_of_chemical_formulas))",
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                    "    ",
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                    "    # add both '(A)', '(B)' to each feature",
                    "    selected_featureAB_list = [f+A_or_B for f in selected_feature_list for A_or_B in ['(A)', '(B)']]",
                    "",
                    "    # make descriptors DataFrame out of descriptors_dict",
                    "    df_features = pd.DataFrame(features_dict)",
                    "    ",
                    "    # reduce df_descriptors to chemical formulas of passed nomad_structure_list and selected features",
                    "    df_features = df_features[selected_featureAB_list].loc[list_of_chemical_formulas]",
                    "    ",
                    "    # name the index (column of the compounds) 'chemical formula'",
                    "    df_features.index.name = 'chemical_formula'",
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                    "",
                    "    # convert numerical columns in float",
                    "    for col in df_features.columns.tolist():",
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                    "        df_features[col] = df_features[col].astype(float)",
                    "    ",
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                    "    # arethmetic combinations    ",
                    "    df_combined = combine_features(df=df_features, allowed_operations=allowed_operations)",
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                    "    ",
                    "    return df_combined"
                ]
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            },
            "output": {
                "state": {},
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                "selectedType": "BeakerDisplay",
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                "pluginName": "IPython",
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                "shellId": "3DF25244EDA34E38A482AD12D464F1C9",
                "height": 81
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            },
            "evaluatorReader": true,
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            "lineCount": 30,
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            "tags": "get_descriptors"
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        },
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        {
            "id": "markdownS5NEVZ",
            "type": "markdown",
            "body": [
                "Use 'get_descriptors' to obtain the descriptor matrix $\\mathbf{D}$ of the radii at the maximum value of  the s, p, and d valence radial probability density:  $r_s(A), r_s(B), r_p(A), r_p(B), r_d(A), r_d(B)$. If the list of allowed (arithmetic) operations is empty only the six primary features are returned. If strings of arithmetic operations are included (i.e. \"$+$\", \"$-$\", \"$/$\", \"$\\exp$\") also derived features will be added to the matrix."
            ],
            "evaluatorReader": false
        },
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        {
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            "id": "codeWjXSr9",
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            "type": "code",
            "evaluator": "IPython",
            "input": {
                "body": [
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                    "kwargs = {",
                    "          'nomad_structure_list': nomad_structure_list,",
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                    "          'selected_feature_list': ['r_s', 'r_p', 'r_d'],",
                    "          'allowed_operations': [],                ",
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                    "         }",
                    "",
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                    "#kwargs['selected_feature_list'] = ['IP', 'EA', 'E_HOMO', 'E_LUMO', 'r_s', 'r_p', 'r_d', 'Z', 'period', 'd']",
                    "#kwargs['allowed_operations'] = ['+', '-', '|-|', '*', '/' '^2', '^3',  'exp']",
                    "",
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                    "df_desc = get_descriptors(**kwargs)",
                    "print df_desc"
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                ]
            },
            "output": {
                "state": {},
                "selectedType": "Results",
                "pluginName": "IPython",
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                "shellId": "3DF25244EDA34E38A482AD12D464F1C9",
                "height": 1587
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            },
            "evaluatorReader": true,
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            "lineCount": 11
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        },
        {
            "id": "markdownjES4yc",
            "type": "markdown",
            "body": [
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                "Let's summarize the untill now declared functions to a new function 'get_data' which returns the numpy arrays  $\\mathbf{P}$ and $\\mathbf{D}$, the list of feature name strings, the list of compound strings ( the order matches to the ones of the rows of both $\\mathbf{P}$ and $\\mathbf{D}$) and the list of json paths of the minimum energy structures (also in the right order). We will need this function in the next chapters."
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            ],
            "evaluatorReader": false
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        },
        {
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            "id": "code57Huu9",
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            "type": "code",
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            "evaluator": "IPython",
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            "input": {
                "body": [
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                    "def get_data(selected_feature_list, allowed_operations):",
                    "    kwargs = {",
                    "          'nomad_structure_list': nomad_structure_list,",
                    "          'selected_feature_list': selected_feature_list,",
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                    "          'allowed_operations': allowed_operations,               ",
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                    "         }    ",
                    "    D_df = get_descriptors(**kwargs)",
                    "    P_df = get_energy_diffs(nomad_structure_list)",
                    "    ",
                    "    feature_list = D_df.columns.tolist()",
                    "    compounds_list = D_df.index.tolist()",
                    "    P_df = P_df.reindex(compounds_list)",
                    "    ",
                    "    json_paths = P_df['json_path'].tolist()",
                    "    P_df =  P_df['energy_diff']",
                    "    ",
                    "    return P_df.values, D_df.values, feature_list, compounds_list, json_paths"
                ]
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            },
            "output": {
                "state": {},
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                "selectedType": "BeakerDisplay",
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                "pluginName": "IPython",
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                "shellId": "3DF25244EDA34E38A482AD12D464F1C9",
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                "height": 89
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            },
            "evaluatorReader": true,
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            "lineCount": 17,
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            "tags": "get_data"
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        },
        {
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            "id": "code6YvnAC",
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            "type": "code",
            "evaluator": "IPython",
            "input": {
                "body": [
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                    "# check that get_data works",
                    "selected_feature_list = ['r_s', 'r_p', 'r_d']",
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                    "allowed_operations = []",
                    "P, D, feature_list, compounds_list, json_paths = get_data(selected_feature_list, allowed_operations)",
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                    "print P.shape, D.shape, len(feature_list), len(json_paths)"
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                ]
            },
            "output": {
                "state": {},
                "selectedType": "Results",
                "pluginName": "IPython",
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                "shellId": "3DF25244EDA34E38A482AD12D464F1C9",
                "height": 93
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            },
            "evaluatorReader": true,
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            "lineCount": 5
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        },
        {
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            "id": "sectionCs89Kj",
            "type": "section",
            "title": "Determining low-dimensional descriptors with the $\\ell_0$ method",
            "level": 1,
            "evaluatorReader": false,
            "collapsed": true
        },
        {
            "id": "markdownUK5ZfK",
            "type": "markdown",
            "body": [
                "<summary>",
                "<li> Perform an $\\ell_0$-regularization to identify the best low dimensional descriptors using the primary features.</li>",
                "<li> Show that non-linear functions of the primary features improve the models significantly. </li>",
                "<li> See that the $\\ell_0$-regularization can rapidly become computational infeasible.</li>",
                "</summary>"
            ],
            "evaluatorReader": false
        },
        {
            "id": "markdownJaNdzs",
            "type": "markdown",
            "body": [
                "For the case you have skipped a chapter, at the beginning of each chapter a JavaScript cell is provided which runs the relevant cells from chapters before with functions we will need in this chapter."
            ],
            "evaluatorReader": false
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        },
        {
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            "id": "coderkkthv",
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            "type": "code",
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            "evaluator": "JavaScript",
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            "input": {
                "body": [
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                    "// 'Import' relevant functions from chapter before",
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                    "var functions_list = [\"json_list\", \"nomad_structure_list\", \"get_energies\", \"get_energy_diffs\", \"get_descriptors\", \"get_data\"];",
                    "var n_functions = functions_list.length",
                    "var i;",
                    "for (i = 0; i < n_functions; i++) {",
                    "    beaker.evaluate(functions_list[i]);",
                    "    beaker.print('import '+functions_list[i]);",
                    "}"
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                ]
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            },
            "output": {
                "state": {},
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                "selectedType": "Results",
                "pluginName": "JavaScript",
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                "height": 0
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            },
            "evaluatorReader": true,
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            "lineCount": 8
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        },
        {
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            "id": "markdownxZfCEL",
            "type": "markdown",
            "body": [
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                "Our target is to find the best low dimensional descriptor for a linear model. The L0 regularization",
                "",
                "$\\text{argmin}_{\\mathbf{c} \\in \\mathbb{R}^{m}} \\{\\|\\mathbf{P} - \\mathbf{D}\\mathbf{c}\\|^2_2 +\\lambda \\|\\mathbf{c}\\|_0\\}$",
                "",
                "provides exactly what we want. It is defined in the follwing and solved combinatorial:"
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            ],
            "evaluatorReader": false
        },
        {
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            "id": "codeVTzLtQ",
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            "type": "code",
            "evaluator": "IPython",
            "input": {
                "body": [
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                    "from itertools import combinations",
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                    "",
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                    "def L0(P, D, dimension):",
                    "    n_rows, n_columns = D.shape",
                    "    D = np.column_stack((D,np.ones(n_rows)))",
                    "    SE_min = np.inner(P,P)",
                    "    coef_min, permu_min = None, None",
                    "    for permu in combinations(range(n_columns),dimension):",
                    "        D_ls = D[:,permu+(-1,)]",
                    "        coef, SE, __1, __2 = np.linalg.lstsq(D_ls,P)",
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                    "        try:",
                    "            if SE[0] < SE_min: ",
                    "                SE_min = SE[0]",
                    "                coef_min, permu_min = coef, permu",
                    "        except:",
                    "            pass",
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                    "    RMSE = np.sqrt(SE_min/n_rows)",
                    "    return RMSE, coef_min, permu_min"
                ]
            },
            "output": {
                "state": {},
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                "selectedType": "BeakerDisplay",
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                "pluginName": "IPython",
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                "shellId": "3DF25244EDA34E38A482AD12D464F1C9"
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            },
            "evaluatorReader": true,
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            "lineCount": 18,
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            "tags": "L0"
        },
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        {
            "id": "markdownkIHqDn",
            "type": "markdown",
            "body": [
                "Perform the $\\ell_0$-regularization for different dimensions (numbers of non-zero coefficients in the model) and see the root mean square errors (RMSE) and the selected features."
            ],
            "evaluatorReader": false
        },
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        {
            "id": "code5rtN3R",
            "type": "code",
            "evaluator": "IPython",
            "input": {
                "body": [
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                    "selected_feature_list = ['r_s', 'r_p', 'r_d']",
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                    "allowed_operations = []",
                    "P, D, feature_list, compounds_list, json_paths = get_data(selected_feature_list, allowed_operations)",
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                    "",
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                    "print \"    RMSE           Best desriptor\"",
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                    "for dim in range(1,7):",
                    "    RMSE, coefficients, selected_indices = L0(P,D,dim)",
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                    "    print '%sD:' %dim, RMSE, [feature_list[i] for i in selected_indices]"
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                ]
            },
            "output": {
                "state": {},
                "selectedType": "Results",
                "pluginName": "IPython",
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                "shellId": "3DF25244EDA34E38A482AD12D464F1C9",
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                "height": 0
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            },
            "evaluatorReader": true,
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            "lineCount": 8
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        },
        {
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            "id": "markdown7wwBua",
            "type": "markdown",
            "body": [
                "Improvements can be obtained by increasing the feature space using more complex features (the derived features). Run the following script and plot the results afterwards. How does the accuracy of the models change? How does the feature space size and the dimension of the descriptors depend on the needed time to solve the $\\ell_0$-problem?"
            ],
            "evaluatorReader": false
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        },
        {
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            "id": "codeLJovcN",
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            "type": "code",
            "evaluator": "IPython",
            "input": {
                "body": [
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                    "from time import time",
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                    "selected_feature_list = ['r_s', 'r_p', 'r_d']",
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                    "op_lists = [[], ['+','|-|'], ['+','|-|','exp'], ['+','|-|','exp', '^2'] ]",
                    "X  = []",
                    "Errors, Time = np.empty([3,len(op_lists)]), np.empty([3,len(op_lists)])",
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                    "",
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                    "for n_op, allowed_operations in enumerate(op_lists):",
                    "    P, D, feature_list, compounds_list, json_paths = get_data(selected_feature_list, allowed_operations)",
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                    "    number_of_features = len(feature_list)",
                    "    X.append(number_of_features)",
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                    "    for dim in range(1,4):",
                    "        t1= time()",
                    "        RMSE, coefficients, selected_indices = L0(P,D,dim)",
                    "        t2 = time()-t1             ",
                    "        ",
                    "        Time [dim-1][n_op] = t2",
                    "        Errors[dim-1][n_op] = RMSE ",
                    "        ",
                    "        print \"features: %s; %sD  RMSE: %s  best features: %s\" %(len(feature_list), dim, RMSE, [feature_list[i] for i in selected_indices])"
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                ]
            },
            "output": {
                "state": {},
                "selectedType": "Results",
                "pluginName": "IPython",
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                "shellId": "3DF25244EDA34E38A482AD12D464F1C9",
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                "height": 0
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            },
            "evaluatorReader": true,
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            "lineCount": 19
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        },
        {
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            "id": "codeZtBOQi",
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            "type": "code",
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            "evaluator": "IPython",
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            "input": {
                "body": [
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                    "#plot",
                    "f, (ax1, ax2) = plt.subplots(1,2, sharex=True, figsize=(12,8))",
                    "ax1.set_xlabel('Number of features')",
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                    "ax2.set_xlabel('Number of features'",
                    ")",
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                    "ax1.set_ylabel('RMSE [eV]')",
                    "ax2.set_ylabel('Time [s]')",
                    "#ax2.set_yscale('log')",
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                    "",
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                    "for dim in range(1,4):",
                    "    ax1.plot(X, Errors[dim-1], 's-', label='%sD' %dim)",
                    "    ax2.plot(X, Time[dim-1], 's-', label='%sD' %dim)",
                    "ax2.legend(loc='best')"
                ]
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            },
            "output": {
                "state": {},
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                "selectedType": "OutputContainer",
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                "pluginName": "IPython",
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                "shellId": "3DF25244EDA34E38A482AD12D464F1C9",
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                "height": 0
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            },
            "evaluatorReader": true,
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            "lineCount": 13
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        },
        {
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            "id": "markdownJAAraB",
            "type": "markdown",
            "body": [
                "Assume now that we would like to include thousands or millions of (more) complex features to obtain more accurate models..."
            ],
            "evaluatorReader": false
        },
        {
            "id": "sectionX7Z2F0",
            "type": "section",
            "title": "Approximations to the $\\ell_0$ method",
            "level": 1,
            "evaluatorReader": false,
            "collapsed": true
        },
        {
            "id": "markdowneXLmPW",
            "type": "markdown",
            "body": [
                "<summary>",
                "<li >Perform a LASSO minimization and the LASSO+$\\ell_0$ method.</li>",
                "<li >Compare the solutions with the ones from the $\\ell_0$ method.</li>",
                "</summary>"
            ],
            "evaluatorReader": false
        },
        {
            "id": "codeSGrV0g",
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            "type": "code",
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            "evaluator": "JavaScript",
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            "input": {
                "body": [
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                    "// 'Import' relevant functions from chapters before",
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                    "var functions_list = [\"json_list\", \"nomad_structure_list\", \"get_energies\", \"get_energy_diffs\", \"get_descriptors\", \"get_data\", \"L0\"];",
                    "var n_functions = functions_list.length",
                    "var i;",
                    "for (i = 0; i < n_functions; i++) {",
                    "    beaker.evaluate(functions_list[i]);",
                    "    beaker.print('import '+functions_list[i]);",
                    "}"
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                ]
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            },
            "output": {
                "state": {},
                "selectedType": "Results",
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                "pluginName": "JavaScript",
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                "height": 0
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            },
            "evaluatorReader": true,
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            "lineCount": 8
        },
        {
            "id": "section96VfCa",
            "type": "section",
            "title": "The LASSO",
            "level": 2,
            "evaluatorReader": false,
            "collapsed": false
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        },
        {
            "id": "markdownHjYp4E",
            "type": "markdown",
            "body": [
                "",
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                "One state-of-the art approximation to the $\\ell_0$-method is the LASSO: ",
                "",
                "$\\text{argmin}_{\\mathbf{c} \\in \\mathbb{R}^{m}} \\{\\|\\mathbf{P} - \\mathbf{D}\\mathbf{c}\\|^2_2 +\\lambda \\|\\mathbf{c}\\|_1\\}$.",
                "",
                "Before performing the LASSO regression we standardize the data to have mean 0 and variance 1, since otherwise the $\\ell_2$-norm of a column would affect bias its contribution to the model. <br>",
                "Note that we can use the LASSO also only for feature selection. We can use then a least-square model with the selected features afterwards instead of the LASSO model directly."
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            ],
            "evaluatorReader": false
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        },
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        {
            "id": "code3B4gWJ",
            "type": "code",
            "evaluator": "IPython",
            "input": {
                "body": [
                    "from sklearn.linear_model import Lasso",
                    "import scipy.stats as ss",
                    "",
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                    "def lasso_fit(lam, P, D, feature_list):",
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                    "    #LASSO",
                    "    D_standardized = ss.zscore(D)",
                    "    lasso =  Lasso(alpha=lam)",
                    "    lasso.fit(D_standardized, P)",
                    "    coef =  lasso.coef_",
                    "    ",
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                    "    # get strings of selected features",
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                    "    selected_indices = coef.nonzero()[0]",
                    "    selected_features = [feature_list[i] for i in selected_indices]",
                    "    ",
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                    "    # get RMSE of LASSO model",
                    "    P_predict = lasso.predict(D_standardized)",
                    "    RMSE_LASSO = np.linalg.norm(P-P_predict) / np.sqrt(82.)",
                    "       ",
                    "    #get RMSE for least-square fit",
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                    "    D_new = D[:, selected_indices]",
                    "    D_new = np.column_stack((D_new, np.ones(82)))",
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                    "    RMSE_LS = np.sqrt(np.linalg.lstsq(D_new,P)[1][0]/82.)",
                    "        ",
                    "    return RMSE_LASSO, RMSE_LS, coef, selected_features"
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                ]
            },
            "output": {
                "state": {},
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                "selectedType": "BeakerDisplay",
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                "pluginName": "IPython",
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                "shellId": "3DF25244EDA34E38A482AD12D464F1C9",
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                "height": 78
            },
            "evaluatorReader": true,
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            "lineCount": 24
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        },
        {
            "id": "markdown5p8ptN",
            "type": "markdown",
            "body": [
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                "$\\lambda$ regulates the sparsity of the coefficient vector of the model. Get the data and try different $\\lambda$ by shifting the lever along the range. How good does LASSO (directly or with a least square fit afterwards) approximate the L0-method (when the same feature space is used for both)?"
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            ],
            "evaluatorReader": false
        },
        {
            "id": "codeQQDvNE",
            "type": "code",
            "evaluator": "IPython",
            "input": {
                "body": [
                    "#import Data",
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                    "selected_feature_list = ['r_s', 'r_p', 'r_d']",
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                    "allowed_operations = ['+','|-|','exp', '^2']",
                    "P, D, feature_list, compounds_list, json_paths = get_data(selected_feature_list, allowed_operations)"
                ]
            },
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            "output": {
                "state": {},
                "selectedType": "Results",
                "pluginName": "IPython",
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                "shellId": "3DF25244EDA34E38A482AD12D464F1C9",
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                "height": 0
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            },
            "evaluatorReader": true,
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            "lineCount": 4
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        },
        {
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            "id": "codeQMp7hF",
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            "type": "code",
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            "evaluator": "HTML",
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            "input": {
                "body": [
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                    "<input id=\"valBox\" type=\"range\" min=\"0.02\" max=\"0.36\" step=\"0.01\" ",
                    "   oninput=\"showVal(this.value)\" >",
                    "<output for=value id=\"output\">lambda: </output>",
                    "<script>",
                    "function showVal(newVal){",
                    "  beaker.lam = newVal;",
                    "  beaker.evaluate(\"lambda_cell\")",
                    "  document.querySelector('#output').value = \"lambda: \"+newVal;;",
                    "}",
                    "</script>"
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                ],
                "hidden": true
            },
            "output": {
                "state": {},