{ "__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:continuous_kstwo", "arbitrary": [ { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "This is the distribution of the maximum absolute differences between an empirical distribution function, computed from " }, { "__type": "InlineMath", "__tag": 4057, "value": "n" }, { "__type": "Text", "__tag": 4046, "value": " samples or observations, and a comparison (or target) cumulative distribution function, which is assumed to be continuous. (The \"two\" in the name is because this is the two-sided difference. " }, { "__type": "InlineCode", "__tag": 4051, "value": "ksone" }, { "__type": "Text", "__tag": 4046, "value": " is the distribution of the positive differences, " }, { "__type": "InlineMath", "__tag": 4057, "value": "D_n^+" }, { "__type": "Text", "__tag": 4046, "value": ", hence it concerns one-sided differences. " }, { "__type": "InlineCode", "__tag": 4051, "value": "kstwobign" }, { "__type": "Text", "__tag": 4046, "value": " is the limiting distribution of the " }, { "__type": "Emphasis", "__tag": 4047, "children": [ { "__type": "Text", "__tag": 4046, "value": "normalized" } ] }, { "__type": "Text", "__tag": 4046, "value": " maximum absolute differences " }, { "__type": "InlineMath", "__tag": 4057, "value": "\\sqrt{n} D_n" }, { "__type": "Text", "__tag": 4046, "value": ".)" } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Writing " }, { "__type": "InlineMath", "__tag": 4057, "value": "D_n = \\sup_t \\left|F_{empirical,n}(t)-F_{target}(t)\\right|" }, { "__type": "Text", "__tag": 4046, "value": ", " }, { "__type": "InlineCode", "__tag": 4051, "value": "kstwo" }, { "__type": "Text", "__tag": 4046, "value": " is the distribution of the " }, { "__type": "InlineMath", "__tag": 4057, "value": "D_n" }, { "__type": "Text", "__tag": 4046, "value": " values." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "InlineCode", "__tag": 4051, "value": "kstwo" }, { "__type": "Text", "__tag": 4046, "value": " can also be used with the differences between two empirical distribution functions, for sets of observations with " }, { "__type": "InlineMath", "__tag": 4057, "value": "m" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineMath", "__tag": 4057, "value": "n" }, { "__type": "Text", "__tag": 4046, "value": " samples respectively. Writing " }, { "__type": "InlineMath", "__tag": 4057, "value": "D_{m,n} = \\sup_t \\left|F_{1,m}(t)-F_{2,n}(t)\\right|" }, { "__type": "Text", "__tag": 4046, "value": ", where " }, { "__type": "InlineMath", "__tag": 4057, "value": "F_{1,m}" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineMath", "__tag": 4057, "value": "F_{2,n}" }, { "__type": "Text", "__tag": 4046, "value": " are the two empirical distribution functions, then " }, { "__type": "InlineMath", "__tag": 4057, "value": "Pr(D_{m,n} \\le x) \\approx Pr(D_N \\le x)" }, { "__type": "Text", "__tag": 4046, "value": " under appropriate conditions, where " }, { "__type": "InlineMath", "__tag": 4057, "value": "N = \\sqrt{\\left(\\frac{mn}{m+n}\\right)}" }, { "__type": "Text", "__tag": 4046, "value": "." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "There is one shape parameter " }, { "__type": "InlineMath", "__tag": 4057, "value": "n" }, { "__type": "Text", "__tag": 4046, "value": ", a positive integer, and the support is " }, { "__type": "InlineMath", "__tag": 4057, "value": "x\\in\\left[0,1\\right]" }, { "__type": "Text", "__tag": 4046, "value": "." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The implementation follows Simard & L'Ecuyer, which combines exact algorithms of Durbin and Pomeranz with asymptotic estimates of Li-Chien, Pelz and Good to compute the CDF with 5-15 accurate digits." } ] } ], "title": [ { "__type": "Text", "__tag": 4046, "value": "KStwo Distribution" } ], "level": 0, "target": "continuous-kstwo" }, { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Code", "__tag": 4050, "value": ">>> import numpy as np\n>>> from scipy.stats import kstwo", "execution_status": null }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Show the probability of a gap at least as big as 0, 0.5 and 1.0 for a sample of size 5" } ] }, { "__type": "Code", "__tag": 4050, "value": ">>> kstwo.sf([0, 0.5, 1.0], 5)\narray([1. , 0.112, 0. ])", "execution_status": null }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Compare a sample of size 5 drawn from a source N(0.5, 1) distribution against a target N(0, 1) CDF." } ] }, { "__type": "Code", "__tag": 4050, "value": ">>> from scipy.stats import norm\n>>> n = 5\n>>> gendist = norm(0.5, 1) # Normal distribution, mean 0.5, stddev 1\n>>> x = np.sort(gendist.rvs(size=n, random_state=np.random.default_rng()))\n>>> x\narray([-1.59113056, -0.66335147, 0.54791569, 0.78009321, 1.27641365]) # may vary\n>>> target = norm(0, 1)\n>>> cdfs = target.cdf(x)\n>>> cdfs\narray([0.0557901 , 0.25355274, 0.7081251 , 0.78233199, 0.89909533]) # may vary\n>>> # Construct the Empirical CDF and the K-S statistics (Dn+, Dn-, Dn)\n>>> ecdfs = np.arange(n+1, dtype=float)/n\n>>> cols = np.column_stack([x, ecdfs[1:], cdfs, cdfs - ecdfs[:n], ecdfs[1:] - cdfs])\n>>> np.set_printoptions(precision=3)\n>>> cols\narray([[-1.591, 0.2 , 0.056, 0.056, 0.144], # may vary\n [-0.663, 0.4 , 0.254, 0.054, 0.146],\n [ 0.548, 0.6 , 0.708, 0.308, -0.108],\n [ 0.78 , 0.8 , 0.782, 0.182, 0.018],\n [ 1.276, 1. , 0.899, 0.099, 0.101]])\n>>> gaps = cols[:, -2:]\n>>> Dnpm = np.max(gaps, axis=0)\n>>> Dn = np.max(Dnpm)\n>>> iminus, iplus = np.argmax(gaps, axis=0)\n>>> print('Dn- = %f (at x=%.2f)' % (Dnpm[0], x[iminus]))\nDn- = 0.246201 (at x=-0.14)\n>>> print('Dn+ = %f (at x=%.2f)' % (Dnpm[1], x[iplus]))\nDn+ = 0.224726 (at x=0.19)\n>>> print('Dn = %f' % (Dn))\nDn = 0.246201", "execution_status": null }, { "__type": "Code", "__tag": 4050, "value": ">>> probs = kstwo.sf(Dn, n)\n>>> print(chr(10).join(['For a sample of size %d drawn from a N(0, 1) distribution:' % n,\n... ' Kolmogorov-Smirnov 2-sided n=%d: Prob(Dn >= %f) = %.4f' % (n, Dn, probs)]))\nFor a sample of size 5 drawn from a N(0, 1) distribution:\n Kolmogorov-Smirnov 2-sided n=5: Prob(Dn >= 0.246201) = 0.8562", "execution_status": null }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Plot the Empirical CDF against the target N(0, 1) CDF" } ] }, { "__type": "Code", "__tag": 4050, "value": ">>> import matplotlib.pyplot as plt\n>>> plt.step(np.concatenate([[-3], x]), ecdfs, where='post', label='Empirical CDF')\n>>> x3 = np.linspace(-3, 3, 100)\n>>> plt.plot(x3, target.cdf(x3), label='CDF for N(0, 1)')\n>>> plt.ylim([0, 1]); plt.grid(True); plt.legend();\n>>> plt.vlines([x[iminus]], ecdfs[iminus], cdfs[iminus], color='r', linestyle='solid', lw=4)\n>>> plt.vlines([x[iplus]], cdfs[iplus], ecdfs[iplus+1], color='m', linestyle='solid', lw=4)\n>>> plt.annotate('Dn-', xy=(x[iminus], (ecdfs[iminus]+ cdfs[iminus])/2),\n... xytext=(x[iminus]+1, (ecdfs[iminus]+ cdfs[iminus])/2 - 0.02),\n... arrowprops=dict(facecolor='white', edgecolor='r', shrink=0.05), size=15, color='r');\n>>> plt.annotate('Dn+', xy=(x[iplus], (ecdfs[iplus+1]+ cdfs[iplus])/2),\n... xytext=(x[iplus]-2, (ecdfs[iplus+1]+ cdfs[iplus])/2 - 0.02),\n... arrowprops=dict(facecolor='white', edgecolor='m', shrink=0.05), size=15, color='m');\n>>> plt.show()", "execution_status": null } ], "title": [ { "__type": "Text", "__tag": 4046, "value": "Examples" } ], "level": 1, "target": null }, { "__type": "Section", "__tag": 4015, "children": [ { "__type": "BulletList", "__tag": 4053, "ordered": false, "start": 1, "children": [ { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "\"Kolmogorov-Smirnov test\", Wikipedia https://en.wikipedia.org/wiki/Kolmogorov-Smirnov_test" } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Durbin J. \"The Probability that the Sample Distribution Function Lies Between Two Parallel Straight Lines.\" " }, { "__type": "Emphasis", "__tag": 4047, "children": [ { "__type": "Text", "__tag": 4046, "value": "Ann. Math. Statist" } ] }, { "__type": "Text", "__tag": 4046, "value": "., 39 (1968) 39, 398-411." } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Pomeranz J. \"Exact Cumulative Distribution of the Kolmogorov-Smirnov Statistic for Small Samples (Algorithm 487).\" " }, { "__type": "Emphasis", "__tag": 4047, "children": [ { "__type": "Text", "__tag": 4046, "value": "Communications of the ACM" } ] }, { "__type": "Text", "__tag": 4046, "value": ", 17(12), (1974) 703-704." } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Li-Chien, C. \"On the exact distribution of the statistics of A. N. Kolmogorov and their asymptotic expansion.\" " }, { "__type": "Emphasis", "__tag": 4047, "children": [ { "__type": "Text", "__tag": 4046, "value": "Acta Matematica Sinica" } ] }, { "__type": "Text", "__tag": 4046, "value": ", 6, (1956) 55-81." } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Pelz W, Good IJ. \"Approximating the Lower Tail-areas of the Kolmogorov-Smirnov One-sample Statistic.\" " }, { "__type": "Emphasis", "__tag": 4047, "children": [ { "__type": "Text", "__tag": 4046, "value": "Journal of the Royal Statistical Society" } ] }, { "__type": "Text", "__tag": 4046, "value": ", Series B, (1976) 38(2), 152-156." } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Simard, R., L'Ecuyer, P. \"Computing the Two-Sided Kolmogorov-Smirnov Distribution\", " }, { "__type": "Emphasis", "__tag": 4047, "children": [ { "__type": "Text", "__tag": 4046, "value": "Journal of Statistical Software" } ] }, { "__type": "Text", "__tag": 4046, "value": ", Vol 39, (2011) 11." } ] } ] } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Implementation: " }, { "__type": "InlineRole", "__tag": 4003, "value": "scipy.stats.kstwo", "domain": null, "role": null, "inventory": null } ] } ], "title": [ { "__type": "Text", "__tag": 4046, "value": "References" } ], "level": 1, "target": null } ], "local_refs": [] }