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Each univariate marginal distribution is stratified, placing exactly one point in " }, { "__type": "InlineMath", "__tag": 4057, "value": "[j/n, (j+1)/n)" }, { "__type": "Text", "__tag": 4046, "value": " for " }, { "__type": "InlineMath", "__tag": 4057, "value": "j=0,1,...,n-1" }, { "__type": "Text", "__tag": 4046, "value": ". They are still applicable when " }, { "__type": "InlineMath", "__tag": 4057, "value": "n << d" }, { "__type": "Text", "__tag": 4046, "value": "." } ] } ], "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/_qmc.py", "item_line": 1286, "item_type": "ABCMeta", "aliases": [ "scipy.stats._qmc.LatinHypercube" ], "example_section_data": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Text", "__tag": 4046, "value": "Generate samples from a Latin hypercube generator.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "from scipy.stats import qmc\nsampler = qmc.LatinHypercube(d=2)\nsample = sampler.random(n=5)\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "sample\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nCompute the quality of the sample using the discrepancy criterion.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "qmc.discrepancy(sample)\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nSamples can be scaled to bounds.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "l_bounds = [0, 2]\nu_bounds = [10, 5]\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "qmc.scale(sample, l_bounds, u_bounds)\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nBelow are other examples showing alternative ways to construct LHS with\neven better coverage of the space.\n\nUsing a base LHS as a baseline.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "sampler = qmc.LatinHypercube(d=2)\nsample = sampler.random(n=5)\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "qmc.discrepancy(sample)\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nUse the `optimization` keyword argument to produce a LHS with\nlower discrepancy at higher computational cost.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "sampler = qmc.LatinHypercube(d=2, optimization=\"random-cd\")\nsample = sampler.random(n=5)\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "qmc.discrepancy(sample)\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nUse the `strength` keyword argument to produce an orthogonal array based\nLHS of strength 2. In this case, the number of sample points must be the\nsquare of a prime number.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "sampler = qmc.LatinHypercube(d=2, strength=2)\nsample = sampler.random(n=9)\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "qmc.discrepancy(sample)\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nOptions could be combined to produce an optimized centered\northogonal array based LHS. After optimization, the result would not\nbe guaranteed to be of strength 2.\n\n**Real-world example**\n\nIn [9]_, a Latin Hypercube sampling (LHS) strategy was used to sample a\nparameter space to study the importance of each parameter of an epidemic\nmodel. Such analysis is also called a sensitivity analysis.\n\nSince the dimensionality of the problem is high (6), it is computationally\nexpensive to cover the space. When numerical experiments are costly, QMC\nenables analysis that may not be possible if using a grid.\n\nThe six parameters of the model represented the probability of illness,\nthe probability of withdrawal, and four contact probabilities. The\nauthors assumed uniform distributions for all parameters and generated\n50 samples.\n\nUsing `scipy.stats.qmc.LatinHypercube` to replicate the protocol,\nthe first step is to create a sample in the unit hypercube:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "from scipy.stats import qmc\nsampler = qmc.LatinHypercube(d=6)\nsample = sampler.random(n=50)\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nThen the sample can be scaled to the appropriate bounds:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "l_bounds = [0.000125, 0.01, 0.0025, 0.05, 0.47, 0.7]\nu_bounds = [0.000375, 0.03, 0.0075, 0.15, 0.87, 0.9]\nsample_scaled = qmc.scale(sample, l_bounds, u_bounds)\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nSuch a sample was used to run the model 50 times, and a polynomial\nresponse surface was constructed. This allowed the authors to study the\nrelative importance of each parameter across the range of possibilities\nof every other parameter.\n\nIn this computer experiment, they showed a 14-fold reduction in the\nnumber of samples required to maintain an error below 2% on their\nresponse surface when compared to a grid sampling." } ], "title": [], "level": 0, "target": null }, "see_also": [ { "__type": "SeeAlsoItem", "__tag": 4028, "name": { "__type": "CrossRef", "__tag": 4002, "value": "quasi-monte-carlo", "reference": { "__type": "RefInfo", "__tag": 4000, "module": "current-module", "version": "current-version", "kind": "to-resolve", "path": "quasi-monte-carlo" }, "kind": "module" }, "descriptions": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "ref" } ] } ], "type": null } ], "signature": { "__type": "SignatureNode", "__tag": 4029, "kind": "function", "parameters": [ { "__type": "SigParam", "__tag": 4030, "name": "d", "annotation": "int | numpy.integer", "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "scramble", "annotation": "bool", "kind": "KEYWORD_ONLY", "default": "True" }, { "__type": "SigParam", "__tag": 4030, "name": "strength", "annotation": "int", "kind": "KEYWORD_ONLY", "default": "1" }, { "__type": "SigParam", "__tag": 4030, "name": "optimization", "annotation": "Literal['random-cd', 'lloyd'] | None", "kind": "KEYWORD_ONLY", "default": "None" }, { "__type": "SigParam", "__tag": 4030, "name": "rng", "annotation": "int | numpy.integer | numpy.random._generator.Generator | numpy.random.mtrand.RandomState | None", "kind": "KEYWORD_ONLY", "default": "None" }, { "__type": "SigParam", "__tag": 4030, "name": "seed", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "KEYWORD_ONLY", "default": "None" } ], "return_annotation": "None", "target_name": "LatinHypercube" }, "references": [ ".. [1] Mckay et al., \"A Comparison of Three Methods for Selecting Values", " of Input Variables in the Analysis of Output from a Computer Code.\"", " Technometrics, 1979.", ".. [2] M. Stein, \"Large sample properties of simulations using Latin", " hypercube sampling.\" Technometrics 29, no. 2: 143-151, 1987.", ".. [3] A. B. Owen, \"Monte Carlo variance of scrambled net quadrature.\"", " SIAM Journal on Numerical Analysis 34, no. 5: 1884-1910, 1997", ".. [4] Loh, W.-L. \"On Latin hypercube sampling.\" The annals of statistics", " 24, no. 5: 2058-2080, 1996.", ".. [5] Fang et al. \"Design and modeling for computer experiments\".", " Computer Science and Data Analysis Series, 2006.", ".. [6] Damblin et al., \"Numerical studies of space filling designs:", " optimization of Latin Hypercube Samples and subprojection properties.\"", " Journal of Simulation, 2013.", ".. [7] A. B. Owen , \"Orthogonal arrays for computer experiments,", " integration and visualization.\" Statistica Sinica, 1992.", ".. [8] B. Tang, \"Orthogonal Array-Based Latin Hypercubes.\"", " Journal of the American Statistical Association, 1993.", ".. [9] Seaholm, Susan K. et al. (1988). Latin hypercube sampling and the", " sensitivity analysis of a Monte Carlo epidemic model. Int J Biomed", " Comput, 23(1-2), 97-112. :doi:`10.1016/0020-7101(88)90067-0`" ], "qa": "scipy.stats._qmc:LatinHypercube", "arbitrary": [], "local_refs": [ "d", "optimization", "rng", "scramble", "strength" ] }