{ "__type": "IngestedDoc", "__tag": 4010, "_content": {}, "_ordered_sections": [], "item_file": null, "item_line": null, "item_type": null, "aliases": [], "example_section_data": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "see_also": [], "signature": null, "references": null, "qa": "tutorial:stats:multiscale_graphcorr", "arbitrary": [ { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "With " }, { "__type": "CrossRef", "__tag": 4002, "value": "scipy.stats.multiscale_graphcorr", "reference": { "__type": "RefInfo", "__tag": 4000, "module": "scipy", "version": "*", "kind": "api", "path": "scipy.stats._mgc:multiscale_graphcorr" }, "kind": "module" }, { "__type": "Text", "__tag": 4046, "value": ", we can test for independence on high dimensional and nonlinear data. Before we start, let's import some useful packages:" } ] }, { "__type": "Blockquote", "__tag": 4059, "children": [ { "__type": "Code", "__tag": 4050, "value": ">>> import numpy as np\n>>> import matplotlib.pyplot as plt; plt.style.use('classic')\n>>> from scipy.stats import multiscale_graphcorr", "execution_status": null } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Let's use a custom plotting function to plot the data relationship:" } ] }, { "__type": "Blockquote", "__tag": 4059, "children": [ { "__type": "Code", "__tag": 4050, "value": ">>> def mgc_plot(x, y, sim_name, mgc_dict=None, only_viz=False,\n... only_mgc=False):\n... \"\"\"Plot sim and MGC-plot\"\"\"\n... if not only_mgc:\n... # simulation\n... plt.figure(figsize=(8, 8))\n... ax = plt.gca()\n... ax.set_title(sim_name + \" Simulation\", fontsize=20)\n... ax.scatter(x, y)\n... ax.set_xlabel('X', fontsize=15)\n... ax.set_ylabel('Y', fontsize=15)\n... ax.axis('equal')\n... ax.tick_params(axis=\"x\", labelsize=15)\n... ax.tick_params(axis=\"y\", labelsize=15)\n... plt.show()\n... if not only_viz:\n... # local correlation map\n... plt.figure(figsize=(8,8))\n... ax = plt.gca()\n... mgc_map = mgc_dict[\"mgc_map\"]\n... # draw heatmap\n... ax.set_title(\"Local Correlation Map\", fontsize=20)\n... im = ax.imshow(mgc_map, cmap='YlGnBu')\n... # colorbar\n... cbar = ax.figure.colorbar(im, ax=ax)\n... cbar.ax.set_ylabel(\"\", rotation=-90, va=\"bottom\")\n... ax.invert_yaxis()\n... # Turn spines off and create white grid.\n... for edge, spine in ax.spines.items():\n... spine.set_visible(False)\n... # optimal scale\n... opt_scale = mgc_dict[\"opt_scale\"]\n... ax.scatter(opt_scale[0], opt_scale[1],\n... marker='X', s=200, color='red')\n... # other formatting\n... ax.tick_params(bottom=\"off\", left=\"off\")\n... ax.set_xlabel('#Neighbors for X', fontsize=15)\n... ax.set_ylabel('#Neighbors for Y', fontsize=15)\n... ax.tick_params(axis=\"x\", labelsize=15)\n... ax.tick_params(axis=\"y\", labelsize=15)\n... ax.set_xlim(0, 100)\n... ax.set_ylim(0, 100)\n... plt.show()", "execution_status": null } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Let's look at some linear data first:" } ] }, { "__type": "Blockquote", "__tag": 4059, "children": [ { "__type": "Code", "__tag": 4050, "value": ">>> rng = np.random.default_rng()\n>>> x = np.linspace(-1, 1, num=100)\n>>> y = x + 0.3 * rng.random(x.size)", "execution_status": null } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The simulation relationship can be plotted below:" } ] }, { "__type": "Blockquote", "__tag": 4059, "children": [ { "__type": "Code", "__tag": 4050, "value": ">>> mgc_plot(x, y, \"Linear\", only_viz=True)", "execution_status": null } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Now, we can see the test statistic, p-value, and MGC map visualized below. The optimal scale is shown on the map as a red \"x\":" } ] }, { "__type": "Blockquote", "__tag": 4059, "children": [ { "__type": "Code", "__tag": 4050, "value": ">>> stat, pvalue, mgc_dict = multiscale_graphcorr(x, y)\n>>> print(\"MGC test statistic: \", round(stat, 1))\nMGC test statistic: 1.0\n>>> print(\"P-value: \", round(pvalue, 1))\nP-value: 0.0\n>>> mgc_plot(x, y, \"Linear\", mgc_dict, only_mgc=True)", "execution_status": null } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "It is clear from here, that MGC is able to determine a relationship between the input data matrices because the p-value is very low and the MGC test statistic is relatively high. The MGC-map indicates a " }, { "__type": "Strong", "__tag": 4048, "children": [ { "__type": "Text", "__tag": 4046, "value": "strongly linear relationship" } ] }, { "__type": "Text", "__tag": 4046, "value": ". Intuitively, this is because having more neighbors will help in identifying a linear relationship between " }, { "__type": "InlineMath", "__tag": 4057, "value": "x" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineMath", "__tag": 4057, "value": "y" }, { "__type": "Text", "__tag": 4046, "value": ". The optimal scale in this case is " }, { "__type": "Strong", "__tag": 4048, "children": [ { "__type": "Text", "__tag": 4046, "value": "equivalent to the global scale" } ] }, { "__type": "Text", "__tag": 4046, "value": ", marked by a red spot on the map." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The same can be done for nonlinear data sets. The following " }, { "__type": "InlineMath", "__tag": 4057, "value": "x" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineMath", "__tag": 4057, "value": "y" }, { "__type": "Text", "__tag": 4046, "value": " arrays are derived from a nonlinear simulation:" } ] }, { "__type": "Blockquote", "__tag": 4059, "children": [ { "__type": "Code", "__tag": 4050, "value": ">>> unif = np.array(rng.uniform(0, 5, size=100))\n>>> x = unif * np.cos(np.pi * unif)\n>>> y = unif * np.sin(np.pi * unif) + 0.4 * rng.random(x.size)", "execution_status": null } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The simulation relationship can be plotted below:" } ] }, { "__type": "Blockquote", "__tag": 4059, "children": [ { "__type": "Code", "__tag": 4050, "value": ">>> mgc_plot(x, y, \"Spiral\", only_viz=True)", "execution_status": null } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Now, we can see the test statistic, p-value, and MGC map visualized below. The optimal scale is shown on the map as a red \"x\":" } ] }, { "__type": "Blockquote", "__tag": 4059, "children": [ { "__type": "Code", "__tag": 4050, "value": ">>> stat, pvalue, mgc_dict = multiscale_graphcorr(x, y)\n>>> print(\"MGC test statistic: \", round(stat, 1))\nMGC test statistic: 0.2 # random\n>>> print(\"P-value: \", round(pvalue, 1))\nP-value: 0.0\n>>> mgc_plot(x, y, \"Spiral\", mgc_dict, only_mgc=True)", "execution_status": null } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "It is clear from here, that MGC is able to determine a relationship again because the p-value is very low and the MGC test statistic is relatively high. The MGC-map indicates a " }, { "__type": "Strong", "__tag": 4048, "children": [ { "__type": "Text", "__tag": 4046, "value": "strongly nonlinear relationship" } ] }, { "__type": "Text", "__tag": 4046, "value": ". The optimal scale in this case is " }, { "__type": "Strong", "__tag": 4048, "children": [ { "__type": "Text", "__tag": 4046, "value": "equivalent to the local scale" } ] }, { "__type": "Text", "__tag": 4046, "value": ", marked by a red spot on the map." } ] } ], "title": [ { "__type": "Text", "__tag": 4046, "value": "Multiscale Graph Correlation (MGC)" } ], "level": 0, "target": null } ], "local_refs": [] }