{ "__type": "IngestedDoc", "__tag": 4010, "_content": { "Notes": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The Box-Cox log-likelihood function " }, { "__type": "InlineMath", "__tag": 4057, "value": "l" }, { "__type": "Text", "__tag": 4046, "value": " is defined here as" } ] }, { "__type": "Math", "__tag": 4058, "value": "l = (\\lambda - 1) \\sum_i^N \\log(x_i) -\n \\frac{N}{2} \\log\\left(\\sum_i^N (y_i - \\bar{y})^2 / N\\right)," }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "where " }, { "__type": "InlineMath", "__tag": 4057, "value": "N" }, { "__type": "Text", "__tag": 4046, "value": " is the number of data points " }, { "__type": "InlineCode", "__tag": 4051, "value": "data" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineMath", "__tag": 4057, "value": "y" }, { "__type": "Text", "__tag": 4046, "value": " is the Box-Cox transformed input data. This corresponds to the " }, { "__type": "Emphasis", "__tag": 4047, "children": [ { "__type": "Text", "__tag": 4046, "value": "profile log-likelihood" } ] }, { "__type": "Text", "__tag": 4046, "value": " of the original data " }, { "__type": "InlineMath", "__tag": 4057, "value": "x" }, { "__type": "Text", "__tag": 4046, "value": " with some constant terms dropped." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Strong", "__tag": 4048, "children": [ { "__type": "Text", "__tag": 4046, "value": "Array API Standard Support" } ] } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "CrossRef", "__tag": 4002, "value": "boxcox_llf", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "module", "path": "scipy.stats._morestats:boxcox_llf" }, "kind": "module" }, { "__type": "Text", "__tag": 4046, "value": " has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable " }, { "__type": "InlineCode", "__tag": 4051, "value": "SCIPY_ARRAY_API=1" }, { "__type": "Text", "__tag": 4046, "value": " and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported." } ] }, { "__type": "Code", "__tag": 4050, "value": "==================== ==================== ====================\nLibrary CPU GPU\n==================== ==================== ====================\nNumPy ✅ n/a \nCuPy n/a ✅ \nPyTorch ✅ ✅ \nJAX ✅ ✅ \nDask ✅ n/a \n==================== ==================== ====================", "execution_status": null }, { "__type": "Blockquote", "__tag": 4059, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "See " }, { "__type": "InlineRole", "__tag": 4003, "value": "dev-arrayapi", "domain": null, "role": "ref", "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " for more information." } ] } ] } ], "title": [], "level": 0, "target": null }, "Warns": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Raises": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Yields": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Methods": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Returns": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Parameters", "__tag": 4026, "children": [ { "__type": "DocParam", "__tag": 4016, "name": "llf", "annotation": "float or ndarray", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Box-Cox log-likelihood of " }, { "__type": "ParamRef", "__tag": 4071, "name": "data" }, { "__type": "Text", "__tag": 4046, "value": " given " }, { "__type": "ParamRef", "__tag": 4071, "name": "lmb" }, { "__type": "Text", "__tag": 4046, "value": ". A float for 1-D " }, { "__type": "ParamRef", "__tag": 4071, "name": "data" }, { "__type": "Text", "__tag": 4046, "value": ", an array otherwise." } ] } ] } ] } ], "title": [], "level": 0, "target": null }, "Summary": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The boxcox log-likelihood function." } ] } ], "title": [], "level": 0, "target": null }, "Receives": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Warnings": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Attributes": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Parameters": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Parameters", "__tag": 4026, "children": [ { "__type": "DocParam", "__tag": 4016, "name": "lmb", "annotation": "scalar", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Parameter for Box-Cox transformation. See " }, { "__type": "InlineRole", "__tag": 4003, "value": "boxcox", "domain": null, "role": null, "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " for details." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "data", "annotation": "array_like", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Data to calculate Box-Cox log-likelihood for. If " }, { "__type": "ParamRef", "__tag": 4071, "name": "data" }, { "__type": "Text", "__tag": 4046, "value": " is multi-dimensional, the log-likelihood is calculated along the first axis." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "axis", "annotation": "int, default: 0", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "If an int, the axis of the input along which to compute the statistic. The statistic of each axis-slice (e.g. row) of the input will appear in a corresponding element of the output. If " }, { "__type": "InlineCode", "__tag": 4051, "value": "None" }, { "__type": "Text", "__tag": 4046, "value": ", the input will be raveled before computing the statistic." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "nan_policy", "annotation": "{'propagate', 'omit', 'raise'", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Defines how to handle input NaNs." } ] }, { "__type": "BulletList", "__tag": 4053, "ordered": false, "start": 1, "children": [ { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "InlineCode", "__tag": 4051, "value": "propagate" }, { "__type": "Text", "__tag": 4046, "value": ": if a NaN is present in the axis slice (e.g. row) along which the statistic is computed, the corresponding entry of the output will be NaN." } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "InlineCode", "__tag": 4051, "value": "omit" }, { "__type": "Text", "__tag": 4046, "value": ": NaNs will be omitted when performing the calculation. If insufficient data remains in the axis slice along which the statistic is computed, the corresponding entry of the output will be NaN." } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "InlineCode", "__tag": 4051, "value": "raise" }, { "__type": "Text", "__tag": 4046, "value": ": if a NaN is present, a " }, { "__type": "InlineCode", "__tag": 4051, "value": "ValueError" }, { "__type": "Text", "__tag": 4046, "value": " will be raised." } ] } ] } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "keepdims", "annotation": "bool, default: False", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "If this is set to True, the axes which are reduced are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the input array." } ] } ] } ] } ], "title": [], "level": 0, "target": null }, "Extended Summary": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Other Parameters": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null } }, "_ordered_sections": [ "Summary", "Extended Summary", "Parameters", "Attributes", "Methods", "Returns", "Yields", "Receives", "Other Parameters", "Raises", "Warns", "Warnings", "Notes" ], "item_file": "/scipy/stats/_morestats.py", "item_line": 893, "item_type": "function", "aliases": [ "scipy.stats.boxcox_llf" ], "example_section_data": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Code", "__tag": 4050, "value": "import numpy as np\nfrom scipy import stats\nimport matplotlib.pyplot as plt\nfrom mpl_toolkits.axes_grid1.inset_locator import inset_axes\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nGenerate some random variates and calculate Box-Cox log-likelihood values\nfor them for a range of ``lmbda`` values:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "rng = np.random.default_rng()\nx = stats.loggamma.rvs(5, loc=10, size=1000, random_state=rng)\nlmbdas = np.linspace(-2, 10)\nllf = np.zeros(lmbdas.shape, dtype=float)\nfor ii, lmbda in enumerate(lmbdas):\n llf[ii] = stats.boxcox_llf(lmbda, x)\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nAlso find the optimal lmbda value with `boxcox`:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "x_most_normal, lmbda_optimal = stats.boxcox(x)\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nPlot the log-likelihood as function of lmbda. Add the optimal lmbda as a\nhorizontal line to check that that's really the optimum:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "fig = plt.figure()\nax = fig.add_subplot(111)\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "ax.plot(lmbdas, llf, 'b.-')\nax.axhline(stats.boxcox_llf(lmbda_optimal, x), color='r')\nax.set_xlabel('lmbda parameter')\nax.set_ylabel('Box-Cox log-likelihood')\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nNow add some probability plots to show that where the log-likelihood is\nmaximized the data transformed with `boxcox` looks closest to normal:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "locs = [3, 10, 4] # 'lower left', 'center', 'lower right'\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "for lmbda, loc in zip([-1, lmbda_optimal, 9], locs):\n xt = stats.boxcox(x, lmbda=lmbda)\n (osm, osr), (slope, intercept, r_sq) = stats.probplot(xt)\n ax_inset = inset_axes(ax, width=\"20%\", height=\"20%\", loc=loc)\n ax_inset.plot(osm, osr, 'c.', osm, slope*osm + intercept, 'k-')\n ax_inset.set_xticklabels([])\n ax_inset.set_yticklabels([])\n ax_inset.set_title(r'$\\lambda=%1.2f$' % lmbda)\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\n" }, { "__type": "Code", "__tag": 4050, "value": "plt.show()\n", "execution_status": "success" }, { "__type": "Figure", "__tag": 4024, "value": { "__type": "RefInfo", "__tag": 4000, "module": "scipy", "version": "1.17.1", "kind": "assets", "path": "fig-8dc6b125b5be3188.png" } } ], "title": [], "level": 0, "target": null }, "see_also": [ { "__type": "SeeAlsoItem", "__tag": 4028, "name": { "__type": "CrossRef", "__tag": 4002, "value": "boxcox", "reference": { "__type": "RefInfo", "__tag": 4000, "module": "current-module", "version": "current-version", "kind": "to-resolve", "path": "boxcox" }, "kind": "module" }, "descriptions": [], "type": "func" }, { "__type": "SeeAlsoItem", "__tag": 4028, "name": { "__type": "CrossRef", "__tag": 4002, "value": "boxcox_normmax", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "module", "path": "scipy.stats._morestats:boxcox_normmax" }, "kind": "module" }, "descriptions": [], "type": "func" }, { "__type": "SeeAlsoItem", "__tag": 4028, "name": { "__type": "CrossRef", "__tag": 4002, "value": "boxcox_normplot", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "module", "path": "scipy.stats._morestats:boxcox_normplot" }, "kind": "module" }, "descriptions": [], "type": "func" }, { "__type": "SeeAlsoItem", "__tag": 4028, "name": { "__type": "CrossRef", "__tag": 4002, "value": "probplot", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "module", "path": "scipy.stats._morestats:probplot" }, "kind": "module" }, "descriptions": [], "type": "func" } ], "signature": { "__type": "SignatureNode", "__tag": 4029, "kind": "function", "parameters": [ { "__type": "SigParam", "__tag": 4030, "name": "lmb", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "data", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "axis", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "KEYWORD_ONLY", "default": "0" }, { "__type": "SigParam", "__tag": 4030, "name": "keepdims", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "KEYWORD_ONLY", "default": "False" }, { "__type": "SigParam", "__tag": 4030, "name": "nan_policy", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "KEYWORD_ONLY", "default": "propagate" } ], "return_annotation": { "__type": "Empty", "__tag": 4031 }, "target_name": "boxcox_llf" }, "references": null, "qa": "scipy.stats._morestats:boxcox_llf", "arbitrary": [], "local_refs": [ "axis", "data", "keepdims", "llf", "lmb", "nan_policy" ] }