{ "__type": "IngestedDoc", "__tag": 4010, "_content": { "Notes": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The methods to estimate the optimal number of bins are well founded in literature, and are inspired by the choices R provides for histogram visualisation. Note that having the number of bins proportional to " }, { "__type": "InlineMath", "__tag": 4057, "value": "n^{1/3}" }, { "__type": "Text", "__tag": 4046, "value": " is asymptotically optimal, which is why it appears in most estimators. These are simply plug-in methods that give good starting points for number of bins. In the equations below, " }, { "__type": "InlineMath", "__tag": 4057, "value": "h" }, { "__type": "Text", "__tag": 4046, "value": " is the binwidth and " }, { "__type": "InlineMath", "__tag": 4057, "value": "n_h" }, { "__type": "Text", "__tag": 4046, "value": " is the number of bins. 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For small datasets the Sturges value will usually be chosen, while larger datasets will usually default to FD. Avoids the overly conservative behaviour of FD and Sturges for small and large datasets respectively. Switchover point is usually " }, { "__type": "InlineMath", "__tag": 4057, "value": "a.size \\approx 1000" }, { "__type": "Text", "__tag": 4046, "value": "." } ] } ] }, { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "'fd' (Freedman Diaconis Estimator)" } ] }, "dd": [ { "__type": "Math", "__tag": 4058, "value": "h = 2 \\frac{IQR}{n^{1/3}}" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The binwidth is proportional to the interquartile range (IQR) and inversely proportional to cube root of a.size. Can be too conservative for small datasets, but is quite good for large datasets. 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Values are very similar to the Freedman-Diaconis estimator in the absence of outliers." } ] } ] }, { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "'rice'" } ] }, "dd": [ { "__type": "Math", "__tag": 4058, "value": "n_h = 2n^{1/3}" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The number of bins is only proportional to cube root of " }, { "__type": "InlineCode", "__tag": 4051, "value": "a.size" }, { "__type": "Text", "__tag": 4046, "value": ". It tends to overestimate the number of bins and it does not take into account data variability." } ] } ] }, { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "'sturges'" } ] }, "dd": [ { "__type": "Math", "__tag": 4058, "value": "n_h = \\log _{2}(n) + 1" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The number of bins is the base 2 log of " }, { "__type": "InlineCode", "__tag": 4051, "value": "a.size" }, { "__type": "Text", "__tag": 4046, "value": ". This estimator assumes normality of data and is too conservative for larger, non-normal datasets. This is the default method in R's " }, { "__type": "InlineCode", "__tag": 4051, "value": "hist" }, { "__type": "Text", "__tag": 4046, "value": " method." } ] } ] }, { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "'doane'" } ] }, "dd": [ { "__type": "Math", "__tag": 4058, "value": "n_h = 1 + \\log_{2}(n) +\\log_{2}\\left(1 + \\frac{|g_1|}{\\sigma_{g_1}}\\right)" }, { "__type": "Blockquote", "__tag": 4059, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "g_1 = ean\\lft[\\lft(\\frac{x - mu}{sigma}right)^3right]" } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "sigma_{g_1} = srt{\\frac{6(n - 2)}{(n + 1)(n + 3)}}" } ] } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "An improved version of Sturges' formula that produces better estimates for non-normal datasets. This estimator attempts to account for the skew of the data." } ] } ] }, { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "'sqrt'" } ] }, "dd": [ { "__type": "Math", "__tag": 4058, "value": "n_h = \\sqrt n" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The simplest and fastest estimator. 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The histogram is computed over the flattened array." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "bins", "annotation": "int or sequence of scalars or str, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "If " }, { "__type": "ParamRef", "__tag": 4071, "name": "bins" }, { "__type": "Text", "__tag": 4046, "value": " is an int, it defines the number of equal-width bins in the given range (10, by default). 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