From c43a5082b8727c65d5c7b7742ecd8c7582ef5e2f Mon Sep 17 00:00:00 2001 From: "Documenter.jl" Date: Wed, 18 Oct 2023 08:49:19 +0000 Subject: [PATCH] build based on 9cdb958 --- stable | 2 +- v1 | 2 +- v1.7 | 2 +- v1.7.4/assets/documenter.js | 331 ++++++++++++++++++++++ v1.7.4/assets/search.js | 267 +++++++++++++++++ v1.7.4/assets/themes/documenter-dark.css | 7 + v1.7.4/assets/themes/documenter-light.css | 9 + v1.7.4/assets/themeswap.js | 66 +++++ v1.7.4/assets/warner.js | 49 ++++ v1.7.4/index.html | 97 +++++++ v1.7.4/search/index.html | 2 + v1.7.4/search_index.js | 3 + v1.7.4/siteinfo.js | 1 + versions.js | 2 +- 14 files changed, 836 insertions(+), 4 deletions(-) create mode 100644 v1.7.4/assets/documenter.js create mode 100644 v1.7.4/assets/search.js create mode 100644 v1.7.4/assets/themes/documenter-dark.css create mode 100644 v1.7.4/assets/themes/documenter-light.css create mode 100644 v1.7.4/assets/themeswap.js create mode 100644 v1.7.4/assets/warner.js create mode 100644 v1.7.4/index.html create mode 100644 v1.7.4/search/index.html create mode 100644 v1.7.4/search_index.js create mode 100644 v1.7.4/siteinfo.js diff --git a/stable b/stable index 0d687f1..1de48e4 120000 --- a/stable +++ b/stable @@ -1 +1 @@ -v1.7.3 \ No newline at end of file +v1.7.4 \ No newline at end of file diff --git a/v1 b/v1 index 0d687f1..1de48e4 120000 --- a/v1 +++ b/v1 @@ -1 +1 @@ -v1.7.3 \ No newline at end of file +v1.7.4 \ No newline at end of file diff --git a/v1.7 b/v1.7 index 0d687f1..1de48e4 120000 --- a/v1.7 +++ b/v1.7 @@ -1 +1 @@ -v1.7.3 \ No newline at end of file +v1.7.4 \ No newline at end of file diff --git a/v1.7.4/assets/documenter.js b/v1.7.4/assets/documenter.js new file mode 100644 index 0000000..6adfbbb --- /dev/null +++ b/v1.7.4/assets/documenter.js @@ -0,0 +1,331 @@ +// Generated by Documenter.jl +requirejs.config({ + paths: { + 'highlight-julia': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.5.1/languages/julia.min', + 'headroom': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/headroom.min', + 'jqueryui': 'https://cdnjs.cloudflare.com/ajax/libs/jqueryui/1.12.1/jquery-ui.min', + 'katex-auto-render': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.24/contrib/auto-render.min', + 'jquery': 'https://cdnjs.cloudflare.com/ajax/libs/jquery/3.6.0/jquery.min', + 'headroom-jquery': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/jQuery.headroom.min', + 'katex': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.24/katex.min', + 'highlight': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.5.1/highlight.min', + 'highlight-julia-repl': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.5.1/languages/julia-repl.min', + }, + shim: { + "highlight-julia": { + "deps": [ + "highlight" + ] + }, + "katex-auto-render": { + "deps": [ + "katex" + ] + }, + "headroom-jquery": { + "deps": [ + "jquery", + "headroom" + ] + }, + "highlight-julia-repl": { + "deps": [ + "highlight" + ] + } +} +}); +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'katex', 'katex-auto-render'], function($, katex, renderMathInElement) { +$(document).ready(function() { + renderMathInElement( + document.body, + { + "delimiters": [ + { + "left": "$", + "right": "$", + "display": false + }, + { + "left": "$$", + "right": "$$", + "display": true + }, + { + "left": "\\[", + "right": "\\]", + "display": true + } + ] +} + + ); +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'highlight', 'highlight-julia', 'highlight-julia-repl'], function($) { +$(document).ready(function() { + hljs.highlightAll(); +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require([], function() { +function addCopyButtonCallbacks() { + for (const el of document.getElementsByTagName("pre")) { + const button = document.createElement("button"); + button.classList.add("copy-button", "fas", "fa-copy"); + el.appendChild(button); + + const success = function () { + button.classList.add("success", "fa-check"); + button.classList.remove("fa-copy"); + }; + + const failure = function () { + button.classList.add("error", "fa-times"); + button.classList.remove("fa-copy"); + }; + + button.addEventListener("click", function () { + copyToClipboard(el.innerText).then(success, failure); + + setTimeout(function () { + button.classList.add("fa-copy"); + button.classList.remove("success", "fa-check", "fa-times"); + }, 5000); + }); + } +} + +function copyToClipboard(text) { + // clipboard API is only available in secure contexts + if (window.navigator && window.navigator.clipboard) { + return window.navigator.clipboard.writeText(text); + } else { + return new Promise(function (resolve, reject) { + try { + const el = document.createElement("textarea"); + el.textContent = text; + el.style.position = "fixed"; + el.style.opacity = 0; + document.body.appendChild(el); + el.select(); + document.execCommand("copy"); + + resolve(); + } catch (err) { + reject(err); + } finally { + document.body.removeChild(el); + } + }); + } +} + +if (document.readyState === "loading") { + document.addEventListener("DOMContentLoaded", addCopyButtonCallbacks); +} else { + addCopyButtonCallbacks(); +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'headroom', 'headroom-jquery'], function($, Headroom) { + +// Manages the top navigation bar (hides it when the user starts scrolling down on the +// mobile). +window.Headroom = Headroom; // work around buggy module loading? +$(document).ready(function() { + $('#documenter .docs-navbar').headroom({ + "tolerance": {"up": 10, "down": 10}, + }); +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Modal settings dialog +$(document).ready(function() { + var settings = $('#documenter-settings'); + $('#documenter-settings-button').click(function(){ + settings.toggleClass('is-active'); + }); + // Close the dialog if X is clicked + $('#documenter-settings button.delete').click(function(){ + settings.removeClass('is-active'); + }); + // Close dialog if ESC is pressed + $(document).keyup(function(e) { + if (e.keyCode == 27) settings.removeClass('is-active'); + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Manages the showing and hiding of the sidebar. +$(document).ready(function() { + var sidebar = $("#documenter > .docs-sidebar"); + var sidebar_button = $("#documenter-sidebar-button") + sidebar_button.click(function(ev) { + ev.preventDefault(); + sidebar.toggleClass('visible'); + if (sidebar.hasClass('visible')) { + // Makes sure that the current menu item is visible in the sidebar. + $("#documenter .docs-menu a.is-active").focus(); + } + }); + $("#documenter > .docs-main").bind('click', function(ev) { + if ($(ev.target).is(sidebar_button)) { + return; + } + if (sidebar.hasClass('visible')) { + sidebar.removeClass('visible'); + } + }); +}) + +// Resizes the package name / sitename in the sidebar if it is too wide. +// Inspired by: https://github.com/davatron5000/FitText.js +$(document).ready(function() { + e = $("#documenter .docs-autofit"); + function resize() { + var L = parseInt(e.css('max-width'), 10); + var L0 = e.width(); + if(L0 > L) { + var h0 = parseInt(e.css('font-size'), 10); + e.css('font-size', L * h0 / L0); + // TODO: make sure it survives resizes? + } + } + // call once and then register events + resize(); + $(window).resize(resize); + $(window).on('orientationchange', resize); +}); + +// Scroll the navigation bar to the currently selected menu item +$(document).ready(function() { + var sidebar = $("#documenter .docs-menu").get(0); + var active = $("#documenter .docs-menu .is-active").get(0); + if(typeof active !== 'undefined') { + sidebar.scrollTop = active.offsetTop - sidebar.offsetTop - 15; + } +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +function set_theme(theme) { + var active = null; + var disabled = []; + for (var i = 0; i < document.styleSheets.length; i++) { + var ss = document.styleSheets[i]; + var themename = ss.ownerNode.getAttribute("data-theme-name"); + if(themename === null) continue; // ignore non-theme stylesheets + // Find the active theme + if(themename === theme) active = ss; + else disabled.push(ss); + } + if(active !== null) { + active.disabled = false; + if(active.ownerNode.getAttribute("data-theme-primary") === null) { + document.getElementsByTagName('html')[0].className = "theme--" + theme; + } else { + document.getElementsByTagName('html')[0].className = ""; + } + disabled.forEach(function(ss){ + ss.disabled = true; + }); + } + + // Store the theme in localStorage + if(typeof(window.localStorage) !== "undefined") { + window.localStorage.setItem("documenter-theme", theme); + } else { + console.error("Browser does not support window.localStorage"); + } +} + +// Theme picker setup +$(document).ready(function() { + // onchange callback + $('#documenter-themepicker').change(function themepick_callback(ev){ + var themename = $('#documenter-themepicker option:selected').attr('value'); + set_theme(themename); + }); + + // Make sure that the themepicker displays the correct theme when the theme is retrieved + // from localStorage + if(typeof(window.localStorage) !== "undefined") { + var theme = window.localStorage.getItem("documenter-theme"); + if(theme !== null) { + $('#documenter-themepicker option').each(function(i,e) { + e.selected = (e.value === theme); + }) + } else { + $('#documenter-themepicker option').each(function(i,e) { + e.selected = $("html").hasClass(`theme--${e.value}`); + }) + } + } +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// update the version selector with info from the siteinfo.js and ../versions.js files +$(document).ready(function() { + // If the version selector is disabled with DOCUMENTER_VERSION_SELECTOR_DISABLED in the + // siteinfo.js file, we just return immediately and not display the version selector. + if (typeof DOCUMENTER_VERSION_SELECTOR_DISABLED === 'boolean' && DOCUMENTER_VERSION_SELECTOR_DISABLED) { + return; + } + + var version_selector = $("#documenter .docs-version-selector"); + var version_selector_select = $("#documenter .docs-version-selector select"); + + version_selector_select.change(function(x) { + target_href = version_selector_select.children("option:selected").get(0).value; + window.location.href = target_href; + }); + + // add the current version to the selector based on siteinfo.js, but only if the selector is empty + if (typeof DOCUMENTER_CURRENT_VERSION !== 'undefined' && $('#version-selector > option').length == 0) { + var option = $(""); + version_selector_select.append(option); + } + + if (typeof DOC_VERSIONS !== 'undefined') { + var existing_versions = version_selector_select.children("option"); + var existing_versions_texts = existing_versions.map(function(i,x){return x.text}); + DOC_VERSIONS.forEach(function(each) { + var version_url = documenterBaseURL + "/../" + each; + var existing_id = $.inArray(each, existing_versions_texts); + // if not already in the version selector, add it as a new option, + // otherwise update the old option with the URL and enable it + if (existing_id == -1) { + var option = $(""); + version_selector_select.append(option); + } else { + var option = existing_versions[existing_id]; + option.value = version_url; + option.disabled = false; + } + }); + } + + // only show the version selector if the selector has been populated + if (version_selector_select.children("option").length > 0) { + version_selector.toggleClass("visible"); + } +}) + +}) diff --git a/v1.7.4/assets/search.js b/v1.7.4/assets/search.js new file mode 100644 index 0000000..c133f74 --- /dev/null +++ b/v1.7.4/assets/search.js @@ -0,0 +1,267 @@ +// Generated by Documenter.jl +requirejs.config({ + paths: { + 'lunr': 'https://cdnjs.cloudflare.com/ajax/libs/lunr.js/2.3.9/lunr.min', + 'lodash': 'https://cdnjs.cloudflare.com/ajax/libs/lodash.js/4.17.21/lodash.min', + 'jquery': 'https://cdnjs.cloudflare.com/ajax/libs/jquery/3.6.0/jquery.min', + } +}); +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'lunr', 'lodash'], function($, lunr, _) { + +$(document).ready(function() { + // parseUri 1.2.2 + // (c) Steven Levithan + // MIT License + function parseUri (str) { + var o = parseUri.options, + m = o.parser[o.strictMode ? "strict" : "loose"].exec(str), + uri = {}, + i = 14; + + while (i--) uri[o.key[i]] = m[i] || ""; + + uri[o.q.name] = {}; + uri[o.key[12]].replace(o.q.parser, function ($0, $1, $2) { + if ($1) uri[o.q.name][$1] = $2; + }); + + return uri; + }; + parseUri.options = { + strictMode: false, + key: ["source","protocol","authority","userInfo","user","password","host","port","relative","path","directory","file","query","anchor"], + q: { + name: "queryKey", + parser: /(?:^|&)([^&=]*)=?([^&]*)/g + }, + parser: { + strict: /^(?:([^:\/?#]+):)?(?:\/\/((?:(([^:@]*)(?::([^:@]*))?)?@)?([^:\/?#]*)(?::(\d*))?))?((((?:[^?#\/]*\/)*)([^?#]*))(?:\?([^#]*))?(?:#(.*))?)/, + loose: /^(?:(?![^:@]+:[^:@\/]*@)([^:\/?#.]+):)?(?:\/\/)?((?:(([^:@]*)(?::([^:@]*))?)?@)?([^:\/?#]*)(?::(\d*))?)(((\/(?:[^?#](?![^?#\/]*\.[^?#\/.]+(?:[?#]|$)))*\/?)?([^?#\/]*))(?:\?([^#]*))?(?:#(.*))?)/ + } + }; + + $("#search-form").submit(function(e) { + e.preventDefault() + }) + + // list below is the lunr 2.1.3 list minus the intersect with names(Base) + // (all, any, get, in, is, only, which) and (do, else, for, let, where, while, with) + // ideally we'd just filter the original list but it's not available as a variable + lunr.stopWordFilter = lunr.generateStopWordFilter([ + 'a', + 'able', + 'about', + 'across', + 'after', + 'almost', + 'also', + 'am', + 'among', + 'an', + 'and', + 'are', + 'as', + 'at', + 'be', + 'because', + 'been', + 'but', + 'by', + 'can', + 'cannot', + 'could', + 'dear', + 'did', + 'does', + 'either', + 'ever', + 'every', + 'from', + 'got', + 'had', + 'has', + 'have', + 'he', + 'her', + 'hers', + 'him', + 'his', + 'how', + 'however', + 'i', + 'if', + 'into', + 'it', + 'its', + 'just', + 'least', + 'like', + 'likely', + 'may', + 'me', + 'might', + 'most', + 'must', + 'my', + 'neither', + 'no', + 'nor', + 'not', + 'of', + 'off', + 'often', + 'on', + 'or', + 'other', + 'our', + 'own', + 'rather', + 'said', + 'say', + 'says', + 'she', + 'should', + 'since', + 'so', + 'some', + 'than', + 'that', + 'the', + 'their', + 'them', + 'then', + 'there', + 'these', + 'they', + 'this', + 'tis', + 'to', + 'too', + 'twas', + 'us', + 'wants', + 'was', + 'we', + 'were', + 'what', + 'when', + 'who', + 'whom', + 'why', + 'will', + 'would', + 'yet', + 'you', + 'your' + ]) + + // add . as a separator, because otherwise "title": "Documenter.Anchors.add!" + // would not find anything if searching for "add!", only for the entire qualification + lunr.tokenizer.separator = /[\s\-\.]+/ + + // custom trimmer that doesn't strip @ and !, which are used in julia macro and function names + lunr.trimmer = function (token) { + return token.update(function (s) { + return s.replace(/^[^a-zA-Z0-9@!]+/, '').replace(/[^a-zA-Z0-9@!]+$/, '') + }) + } + + lunr.Pipeline.registerFunction(lunr.stopWordFilter, 'juliaStopWordFilter') + lunr.Pipeline.registerFunction(lunr.trimmer, 'juliaTrimmer') + + var index = lunr(function () { + this.ref('location') + this.field('title',{boost: 100}) + this.field('text') + documenterSearchIndex['docs'].forEach(function(e) { + this.add(e) + }, this) + }) + var store = {} + + documenterSearchIndex['docs'].forEach(function(e) { + store[e.location] = {title: e.title, category: e.category, page: e.page} + }) + + $(function(){ + searchresults = $('#documenter-search-results'); + searchinfo = $('#documenter-search-info'); + searchbox = $('#documenter-search-query'); + searchform = $('.docs-search'); + sidebar = $('.docs-sidebar'); + function update_search(querystring) { + tokens = lunr.tokenizer(querystring) + results = index.query(function (q) { + tokens.forEach(function (t) { + q.term(t.toString(), { + fields: ["title"], + boost: 100, + usePipeline: true, + editDistance: 0, + wildcard: lunr.Query.wildcard.NONE + }) + q.term(t.toString(), { + fields: ["title"], + boost: 10, + usePipeline: true, + editDistance: 2, + wildcard: lunr.Query.wildcard.NONE + }) + q.term(t.toString(), { + fields: ["text"], + boost: 1, + usePipeline: true, + editDistance: 0, + wildcard: lunr.Query.wildcard.NONE + }) + }) + }) + searchinfo.text("Number of results: " + results.length) + searchresults.empty() + results.forEach(function(result) { + data = store[result.ref] + link = $(''+data.title+'') + link.attr('href', documenterBaseURL+'/'+result.ref) + if (data.category != "page"){ + cat = $('('+data.category+', '+data.page+')') + } else { + cat = $('('+data.category+')') + } + li = $('
  • ').append(link).append(" ").append(cat) + searchresults.append(li) + }) + } + + function update_search_box() { + querystring = searchbox.val() + update_search(querystring) + } + + searchbox.keyup(_.debounce(update_search_box, 250)) + searchbox.change(update_search_box) + + // Disable enter-key form submission for the searchbox on the search page + // and just re-run search rather than refresh the whole page. + searchform.keypress( + function(event){ + if (event.which == '13') { + if (sidebar.hasClass('visible')) { + sidebar.removeClass('visible'); + } + update_search_box(); + event.preventDefault(); + } + } + ); + + search_query_uri = parseUri(window.location).queryKey["q"] + if(search_query_uri !== undefined) { + search_query = decodeURIComponent(search_query_uri.replace(/\+/g, '%20')) + searchbox.val(search_query) + } + update_search_box(); + }) +}) + +}) diff --git a/v1.7.4/assets/themes/documenter-dark.css b/v1.7.4/assets/themes/documenter-dark.css new file mode 100644 index 0000000..89fbb49 --- /dev/null +++ b/v1.7.4/assets/themes/documenter-dark.css @@ -0,0 +1,7 @@ +@keyframes spinAround{from{transform:rotate(0deg)}to{transform:rotate(359deg)}}html.theme--documenter-dark .tabs,html.theme--documenter-dark .pagination-previous,html.theme--documenter-dark .pagination-next,html.theme--documenter-dark .pagination-link,html.theme--documenter-dark .pagination-ellipsis,html.theme--documenter-dark .breadcrumb,html.theme--documenter-dark .file,html.theme--documenter-dark .button,.is-unselectable,html.theme--documenter-dark .modal-close,html.theme--documenter-dark .delete{-webkit-touch-callout:none;-webkit-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none}html.theme--documenter-dark .navbar-link:not(.is-arrowless)::after,html.theme--documenter-dark .select:not(.is-multiple):not(.is-loading)::after{border:3px solid rgba(0,0,0,0);border-radius:2px;border-right:0;border-top:0;content:" 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html.theme--documenter-dark .button.is-dark.is-outlined,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-outlined{background-color:transparent;border-color:#363636;box-shadow:none;color:#363636}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#f5f5f5;color:#f5f5f5}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined:hover,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined:focus,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--documenter-dark .content 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rgba(37,154,18,0.25)}html.theme--documenter-dark .button.is-success:active,html.theme--documenter-dark .button.is-success.is-active{background-color:#20830f;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-success[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-success{background-color:#259a12;border-color:transparent;box-shadow:none}html.theme--documenter-dark .button.is-success.is-inverted{background-color:#fff;color:#259a12}html.theme--documenter-dark .button.is-success.is-inverted:hover,html.theme--documenter-dark .button.is-success.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--documenter-dark .button.is-success.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-success.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#259a12}html.theme--documenter-dark .button.is-success.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-success.is-outlined{background-color:transparent;border-color:#259a12;color:#259a12}html.theme--documenter-dark .button.is-success.is-outlined:hover,html.theme--documenter-dark .button.is-success.is-outlined.is-hovered,html.theme--documenter-dark .button.is-success.is-outlined:focus,html.theme--documenter-dark .button.is-success.is-outlined.is-focused{background-color:#259a12;border-color:#259a12;color:#fff}html.theme--documenter-dark .button.is-success.is-outlined.is-loading::after{border-color:transparent transparent #259a12 #259a12 !important}html.theme--documenter-dark .button.is-success.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-success.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-success.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-success.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-success.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-success.is-outlined{background-color:transparent;border-color:#259a12;box-shadow:none;color:#259a12}html.theme--documenter-dark .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-success.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-success.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-focused{background-color:#fff;color:#259a12}html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #259a12 #259a12 !important}html.theme--documenter-dark .button.is-success.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-warning{background-color:#f4c72f;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-warning:hover,html.theme--documenter-dark .button.is-warning.is-hovered{background-color:#f3c423;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-warning:focus,html.theme--documenter-dark .button.is-warning.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-warning:focus:not(:active),html.theme--documenter-dark .button.is-warning.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(244,199,47,0.25)}html.theme--documenter-dark .button.is-warning:active,html.theme--documenter-dark .button.is-warning.is-active{background-color:#f3c017;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-warning[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-warning{background-color:#f4c72f;border-color:transparent;box-shadow:none}html.theme--documenter-dark .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);color:#f4c72f}html.theme--documenter-dark .button.is-warning.is-inverted:hover,html.theme--documenter-dark .button.is-warning.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-warning.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark 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.button.is-warning.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#f4c72f}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #f4c72f #f4c72f !important}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-danger{background-color:#cb3c33;border-color:transparent;color:#fff}html.theme--documenter-dark 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.button.is-danger.is-inverted:hover,html.theme--documenter-dark .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--documenter-dark .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#cb3c33}html.theme--documenter-dark .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-danger.is-outlined{background-color:transparent;border-color:#cb3c33;color:#cb3c33}html.theme--documenter-dark .button.is-danger.is-outlined:hover,html.theme--documenter-dark .button.is-danger.is-outlined.is-hovered,html.theme--documenter-dark .button.is-danger.is-outlined:focus,html.theme--documenter-dark .button.is-danger.is-outlined.is-focused{background-color:#cb3c33;border-color:#cb3c33;color:#fff}html.theme--documenter-dark .button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #cb3c33 #cb3c33 !important}html.theme--documenter-dark .button.is-danger.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-danger.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-outlined{background-color:transparent;border-color:#cb3c33;box-shadow:none;color:#cb3c33}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#cb3c33}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #cb3c33 #cb3c33 !important}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{border-radius:2px;font-size:.75rem}html.theme--documenter-dark .button.is-normal{font-size:1rem}html.theme--documenter-dark .button.is-medium{font-size:1.25rem}html.theme--documenter-dark .button.is-large{font-size:1.5rem}html.theme--documenter-dark .button[disabled],fieldset[disabled] html.theme--documenter-dark .button{background-color:#fff;border-color:#dbdbdb;box-shadow:none;opacity:.5}html.theme--documenter-dark .button.is-fullwidth{display:flex;width:100%}html.theme--documenter-dark .button.is-loading{color:transparent !important;pointer-events:none}html.theme--documenter-dark .button.is-loading::after{position:absolute;left:calc(50% - (1em / 2));top:calc(50% - (1em / 2));position:absolute !important}html.theme--documenter-dark .button.is-static{background-color:#f5f5f5;border-color:#dbdbdb;color:#7a7a7a;box-shadow:none;pointer-events:none}html.theme--documenter-dark 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.button:active:hover,html.theme--documenter-dark .buttons.has-addons .button.is-active:hover,html.theme--documenter-dark .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--documenter-dark .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--documenter-dark .buttons.is-centered{justify-content:center}html.theme--documenter-dark .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--documenter-dark .buttons.is-right{justify-content:flex-end}html.theme--documenter-dark .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--documenter-dark .container{flex-grow:1;margin:0 auto;position:relative;width:auto}@media screen and (min-width: 1056px){html.theme--documenter-dark .container{max-width:992px}html.theme--documenter-dark .container.is-fluid{margin-left:32px;margin-right:32px;max-width:none}}@media screen and (max-width: 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a.navbar-item,html.theme--documenter-dark .navbar-link{cursor:pointer}html.theme--documenter-dark a.navbar-item:focus,html.theme--documenter-dark a.navbar-item:focus-within,html.theme--documenter-dark a.navbar-item:hover,html.theme--documenter-dark a.navbar-item.is-active,html.theme--documenter-dark .navbar-link:focus,html.theme--documenter-dark .navbar-link:focus-within,html.theme--documenter-dark .navbar-link:hover,html.theme--documenter-dark .navbar-link.is-active{background-color:#fafafa;color:#2ad7e6}html.theme--documenter-dark .navbar-item{display:block;flex-grow:0;flex-shrink:0}html.theme--documenter-dark .navbar-item img{max-height:1.75rem}html.theme--documenter-dark .navbar-item.has-dropdown{padding:0}html.theme--documenter-dark .navbar-item.is-expanded{flex-grow:1;flex-shrink:1}html.theme--documenter-dark .navbar-item.is-tab{border-bottom:1px solid transparent;min-height:3.25rem;padding-bottom:calc(0.5rem - 1px)}html.theme--documenter-dark .navbar-item.is-tab:focus,html.theme--documenter-dark .navbar-item.is-tab:hover{background-color:rgba(0,0,0,0);border-bottom-color:#2ad7e6}html.theme--documenter-dark .navbar-item.is-tab.is-active{background-color:rgba(0,0,0,0);border-bottom-color:#2ad7e6;border-bottom-style:solid;border-bottom-width:3px;color:#2ad7e6;padding-bottom:calc(0.5rem - 3px)}html.theme--documenter-dark .navbar-content{flex-grow:1;flex-shrink:1}html.theme--documenter-dark .navbar-link:not(.is-arrowless){padding-right:2.5em}html.theme--documenter-dark .navbar-link:not(.is-arrowless)::after{border-color:#2ad7e6;margin-top:-0.375em;right:1.125em}html.theme--documenter-dark .navbar-dropdown{font-size:0.875rem;padding-bottom:0.5rem;padding-top:0.5rem}html.theme--documenter-dark .navbar-dropdown .navbar-item{padding-left:1.5rem;padding-right:1.5rem}html.theme--documenter-dark .navbar-divider{background-color:#f5f5f5;border:none;display:none;height:2px;margin:0.5rem 0}@media screen and (max-width: 1055px){html.theme--documenter-dark .navbar>.container{display:block}html.theme--documenter-dark .navbar-brand .navbar-item,html.theme--documenter-dark .navbar-tabs .navbar-item{align-items:center;display:flex}html.theme--documenter-dark .navbar-link::after{display:none}html.theme--documenter-dark .navbar-menu{background-color:#fff;box-shadow:0 8px 16px rgba(10,10,10,0.1);padding:0.5rem 0}html.theme--documenter-dark .navbar-menu.is-active{display:block}html.theme--documenter-dark .navbar.is-fixed-bottom-touch,html.theme--documenter-dark .navbar.is-fixed-top-touch{left:0;position:fixed;right:0;z-index:30}html.theme--documenter-dark .navbar.is-fixed-bottom-touch{bottom:0}html.theme--documenter-dark .navbar.is-fixed-bottom-touch.has-shadow{box-shadow:0 -2px 3px rgba(10,10,10,0.1)}html.theme--documenter-dark .navbar.is-fixed-top-touch{top:0}html.theme--documenter-dark .navbar.is-fixed-top .navbar-menu,html.theme--documenter-dark .navbar.is-fixed-top-touch 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.navbar-link{border-radius:4px}html.theme--documenter-dark .navbar.is-transparent a.navbar-item:focus,html.theme--documenter-dark .navbar.is-transparent a.navbar-item:hover,html.theme--documenter-dark .navbar.is-transparent a.navbar-item.is-active,html.theme--documenter-dark .navbar.is-transparent .navbar-link:focus,html.theme--documenter-dark .navbar.is-transparent .navbar-link:hover,html.theme--documenter-dark .navbar.is-transparent .navbar-link.is-active{background-color:transparent !important}html.theme--documenter-dark .navbar.is-transparent .navbar-item.has-dropdown.is-active .navbar-link,html.theme--documenter-dark .navbar.is-transparent .navbar-item.has-dropdown.is-hoverable:focus .navbar-link,html.theme--documenter-dark .navbar.is-transparent .navbar-item.has-dropdown.is-hoverable:focus-within .navbar-link,html.theme--documenter-dark .navbar.is-transparent .navbar-item.has-dropdown.is-hoverable:hover .navbar-link{background-color:transparent 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rgba(10,10,10,0.1);top:auto}html.theme--documenter-dark .navbar-item.is-active .navbar-dropdown,html.theme--documenter-dark .navbar-item.is-hoverable:focus .navbar-dropdown,html.theme--documenter-dark .navbar-item.is-hoverable:focus-within .navbar-dropdown,html.theme--documenter-dark .navbar-item.is-hoverable:hover .navbar-dropdown{display:block}.navbar.is-spaced html.theme--documenter-dark .navbar-item.is-active .navbar-dropdown,html.theme--documenter-dark .navbar-item.is-active .navbar-dropdown.is-boxed,.navbar.is-spaced html.theme--documenter-dark .navbar-item.is-hoverable:focus .navbar-dropdown,html.theme--documenter-dark .navbar-item.is-hoverable:focus .navbar-dropdown.is-boxed,.navbar.is-spaced html.theme--documenter-dark .navbar-item.is-hoverable:focus-within .navbar-dropdown,html.theme--documenter-dark .navbar-item.is-hoverable:focus-within .navbar-dropdown.is-boxed,.navbar.is-spaced html.theme--documenter-dark .navbar-item.is-hoverable:hover 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a.navbar-item:hover{background-color:#f5f5f5;color:#0a0a0a}html.theme--documenter-dark .navbar-dropdown a.navbar-item.is-active{background-color:#f5f5f5;color:#2ad7e6}.navbar.is-spaced html.theme--documenter-dark .navbar-dropdown,html.theme--documenter-dark .navbar-dropdown.is-boxed{border-radius:6px;border-top:none;box-shadow:0 8px 8px rgba(10,10,10,0.1), 0 0 0 1px rgba(10,10,10,0.1);display:block;opacity:0;pointer-events:none;top:calc(100% + (-4px));transform:translateY(-5px);transition-duration:86ms;transition-property:opacity, transform}html.theme--documenter-dark .navbar-dropdown.is-right{left:auto;right:0}html.theme--documenter-dark .navbar-divider{display:block}html.theme--documenter-dark .navbar>.container .navbar-brand,html.theme--documenter-dark .container>.navbar .navbar-brand{margin-left:-.75rem}html.theme--documenter-dark .navbar>.container .navbar-menu,html.theme--documenter-dark .container>.navbar .navbar-menu{margin-right:-.75rem}html.theme--documenter-dark .navbar.is-fixed-bottom-desktop,html.theme--documenter-dark .navbar.is-fixed-top-desktop{left:0;position:fixed;right:0;z-index:30}html.theme--documenter-dark .navbar.is-fixed-bottom-desktop{bottom:0}html.theme--documenter-dark .navbar.is-fixed-bottom-desktop.has-shadow{box-shadow:0 -2px 3px rgba(10,10,10,0.1)}html.theme--documenter-dark .navbar.is-fixed-top-desktop{top:0}html.theme--documenter-dark html.has-navbar-fixed-top-desktop,html.theme--documenter-dark body.has-navbar-fixed-top-desktop{padding-top:3.25rem}html.theme--documenter-dark html.has-navbar-fixed-bottom-desktop,html.theme--documenter-dark body.has-navbar-fixed-bottom-desktop{padding-bottom:3.25rem}html.theme--documenter-dark html.has-spaced-navbar-fixed-top,html.theme--documenter-dark body.has-spaced-navbar-fixed-top{padding-top:5.25rem}html.theme--documenter-dark html.has-spaced-navbar-fixed-bottom,html.theme--documenter-dark body.has-spaced-navbar-fixed-bottom{padding-bottom:5.25rem}html.theme--documenter-dark a.navbar-item.is-active,html.theme--documenter-dark .navbar-link.is-active{color:#0a0a0a}html.theme--documenter-dark a.navbar-item.is-active:not(:focus):not(:hover),html.theme--documenter-dark .navbar-link.is-active:not(:focus):not(:hover){background-color:rgba(0,0,0,0)}html.theme--documenter-dark .navbar-item.has-dropdown:focus .navbar-link,html.theme--documenter-dark .navbar-item.has-dropdown:hover .navbar-link,html.theme--documenter-dark .navbar-item.has-dropdown.is-active .navbar-link{background-color:#fafafa}}html.theme--documenter-dark .hero.is-fullheight-with-navbar{min-height:calc(100vh - 3.25rem)}html.theme--documenter-dark .pagination{font-size:1rem;margin:-.25rem}html.theme--documenter-dark .pagination.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.pagination{font-size:.75rem}html.theme--documenter-dark .pagination.is-medium{font-size:1.25rem}html.theme--documenter-dark .pagination.is-large{font-size:1.5rem}html.theme--documenter-dark .pagination.is-rounded .pagination-previous,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.pagination .pagination-previous,html.theme--documenter-dark .pagination.is-rounded .pagination-next,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.pagination .pagination-next{padding-left:1em;padding-right:1em;border-radius:290486px}html.theme--documenter-dark .pagination.is-rounded .pagination-link,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.pagination .pagination-link{border-radius:290486px}html.theme--documenter-dark .pagination,html.theme--documenter-dark .pagination-list{align-items:center;display:flex;justify-content:center;text-align:center}html.theme--documenter-dark .pagination-previous,html.theme--documenter-dark .pagination-next,html.theme--documenter-dark .pagination-link,html.theme--documenter-dark .pagination-ellipsis{font-size:1em;justify-content:center;margin:.25rem;padding-left:.5em;padding-right:.5em;text-align:center}html.theme--documenter-dark .pagination-previous,html.theme--documenter-dark .pagination-next,html.theme--documenter-dark .pagination-link{border-color:#dbdbdb;color:#363636;min-width:2.25em}html.theme--documenter-dark .pagination-previous:hover,html.theme--documenter-dark .pagination-next:hover,html.theme--documenter-dark .pagination-link:hover{border-color:#b5b5b5;color:#97ebf3}html.theme--documenter-dark .pagination-previous:focus,html.theme--documenter-dark .pagination-next:focus,html.theme--documenter-dark .pagination-link:focus{border-color:#3c5dcd}html.theme--documenter-dark .pagination-previous:active,html.theme--documenter-dark .pagination-next:active,html.theme--documenter-dark .pagination-link:active{box-shadow:inset 0 1px 2px rgba(10,10,10,0.2)}html.theme--documenter-dark .pagination-previous[disabled],html.theme--documenter-dark .pagination-next[disabled],html.theme--documenter-dark .pagination-link[disabled]{background-color:#dbdbdb;border-color:#dbdbdb;box-shadow:none;color:#7a7a7a;opacity:0.5}html.theme--documenter-dark .pagination-previous,html.theme--documenter-dark .pagination-next{padding-left:0.75em;padding-right:0.75em;white-space:nowrap}html.theme--documenter-dark .pagination-link.is-current{background-color:#2ad7e6;border-color:#2ad7e6;color:#fff}html.theme--documenter-dark .pagination-ellipsis{color:#b5b5b5;pointer-events:none}html.theme--documenter-dark .pagination-list{flex-wrap:wrap}@media screen and (max-width: 768px){html.theme--documenter-dark .pagination{flex-wrap:wrap}html.theme--documenter-dark .pagination-previous,html.theme--documenter-dark .pagination-next{flex-grow:1;flex-shrink:1}html.theme--documenter-dark .pagination-list li{flex-grow:1;flex-shrink:1}}@media screen and (min-width: 769px),print{html.theme--documenter-dark .pagination-list{flex-grow:1;flex-shrink:1;justify-content:flex-start;order:1}html.theme--documenter-dark .pagination-previous{order:2}html.theme--documenter-dark .pagination-next{order:3}html.theme--documenter-dark .pagination{justify-content:space-between}html.theme--documenter-dark .pagination.is-centered .pagination-previous{order:1}html.theme--documenter-dark .pagination.is-centered .pagination-list{justify-content:center;order:2}html.theme--documenter-dark .pagination.is-centered .pagination-next{order:3}html.theme--documenter-dark .pagination.is-right .pagination-previous{order:1}html.theme--documenter-dark .pagination.is-right .pagination-next{order:2}html.theme--documenter-dark .pagination.is-right .pagination-list{justify-content:flex-end;order:3}}html.theme--documenter-dark .panel{font-size:1rem}html.theme--documenter-dark .panel:not(:last-child){margin-bottom:1.5rem}html.theme--documenter-dark .panel-heading,html.theme--documenter-dark .panel-tabs,html.theme--documenter-dark .panel-block{border-bottom:1px solid #dbdbdb;border-left:1px solid #dbdbdb;border-right:1px solid #dbdbdb}html.theme--documenter-dark .panel-heading:first-child,html.theme--documenter-dark .panel-tabs:first-child,html.theme--documenter-dark .panel-block:first-child{border-top:1px solid #dbdbdb}html.theme--documenter-dark .panel-heading{background-color:#f5f5f5;border-radius:4px 4px 0 0;color:#cdc3e3;font-size:1.25em;font-weight:300;line-height:1.25;padding:0.5em 0.75em}html.theme--documenter-dark .panel-tabs{align-items:flex-end;display:flex;font-size:.875em;justify-content:center}html.theme--documenter-dark .panel-tabs a{border-bottom:1px solid #dbdbdb;margin-bottom:-1px;padding:0.5em}html.theme--documenter-dark .panel-tabs a.is-active{border-bottom-color:#4a4a4a;color:#363636}html.theme--documenter-dark .panel-list a{color:#fbfbfb}html.theme--documenter-dark .panel-list a:hover{color:#2ad7e6}html.theme--documenter-dark .panel-block{align-items:center;color:#cdc3e3;display:flex;justify-content:flex-start;padding:0.5em 0.75em}html.theme--documenter-dark .panel-block input[type="checkbox"]{margin-right:0.75em}html.theme--documenter-dark .panel-block>.control{flex-grow:1;flex-shrink:1;width:100%}html.theme--documenter-dark .panel-block.is-wrapped{flex-wrap:wrap}html.theme--documenter-dark .panel-block.is-active{border-left-color:#2ad7e6;color:#363636}html.theme--documenter-dark .panel-block.is-active .panel-icon{color:#2ad7e6}html.theme--documenter-dark a.panel-block,html.theme--documenter-dark label.panel-block{cursor:pointer}html.theme--documenter-dark a.panel-block:hover,html.theme--documenter-dark label.panel-block:hover{background-color:#f5f5f5}html.theme--documenter-dark .panel-icon{display:inline-block;font-size:14px;height:1em;line-height:1em;text-align:center;vertical-align:top;width:1em;color:#7a7a7a;margin-right:0.75em}html.theme--documenter-dark .panel-icon .fa{font-size:inherit;line-height:inherit}html.theme--documenter-dark .tabs{-webkit-overflow-scrolling:touch;align-items:stretch;display:flex;font-size:1rem;justify-content:space-between;overflow:hidden;overflow-x:auto;white-space:nowrap}html.theme--documenter-dark .tabs a{align-items:center;border-bottom-color:#dbdbdb;border-bottom-style:solid;border-bottom-width:1px;color:#fbfbfb;display:flex;justify-content:center;margin-bottom:-1px;padding:0.5em 1em;vertical-align:top}html.theme--documenter-dark .tabs a:hover{border-bottom-color:#cdc3e3;color:#cdc3e3}html.theme--documenter-dark .tabs li{display:block}html.theme--documenter-dark .tabs li.is-active a{border-bottom-color:#2ad7e6;color:#2ad7e6}html.theme--documenter-dark .tabs ul{align-items:center;border-bottom-color:#dbdbdb;border-bottom-style:solid;border-bottom-width:1px;display:flex;flex-grow:1;flex-shrink:0;justify-content:flex-start}html.theme--documenter-dark .tabs ul.is-left{padding-right:0.75em}html.theme--documenter-dark .tabs ul.is-center{flex:none;justify-content:center;padding-left:0.75em;padding-right:0.75em}html.theme--documenter-dark .tabs ul.is-right{justify-content:flex-end;padding-left:0.75em}html.theme--documenter-dark .tabs .icon:first-child{margin-right:0.5em}html.theme--documenter-dark .tabs .icon:last-child{margin-left:0.5em}html.theme--documenter-dark .tabs.is-centered ul{justify-content:center}html.theme--documenter-dark .tabs.is-right ul{justify-content:flex-end}html.theme--documenter-dark .tabs.is-boxed a{border:1px solid transparent;border-radius:4px 4px 0 0}html.theme--documenter-dark .tabs.is-boxed a:hover{background-color:#f5f5f5;border-bottom-color:#dbdbdb}html.theme--documenter-dark .tabs.is-boxed li.is-active a{background-color:#fff;border-color:#dbdbdb;border-bottom-color:rgba(0,0,0,0) !important}html.theme--documenter-dark .tabs.is-fullwidth li{flex-grow:1;flex-shrink:0}html.theme--documenter-dark .tabs.is-toggle a{border-color:#dbdbdb;border-style:solid;border-width:1px;margin-bottom:0;position:relative}html.theme--documenter-dark .tabs.is-toggle a:hover{background-color:#f5f5f5;border-color:#b5b5b5;z-index:2}html.theme--documenter-dark .tabs.is-toggle li+li{margin-left:-1px}html.theme--documenter-dark .tabs.is-toggle li:first-child a{border-radius:4px 0 0 4px}html.theme--documenter-dark .tabs.is-toggle li:last-child a{border-radius:0 4px 4px 0}html.theme--documenter-dark .tabs.is-toggle li.is-active a{background-color:#2ad7e6;border-color:#2ad7e6;color:#fff;z-index:1}html.theme--documenter-dark .tabs.is-toggle ul{border-bottom:none}html.theme--documenter-dark .tabs.is-toggle.is-toggle-rounded li:first-child a{border-bottom-left-radius:290486px;border-top-left-radius:290486px;padding-left:1.25em}html.theme--documenter-dark .tabs.is-toggle.is-toggle-rounded li:last-child a{border-bottom-right-radius:290486px;border-top-right-radius:290486px;padding-right:1.25em}html.theme--documenter-dark 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100644 index 0000000..c58e993 --- /dev/null +++ b/v1.7.4/assets/themeswap.js @@ -0,0 +1,66 @@ +// Small function to quickly swap out themes. Gets put into the tag.. +function set_theme_from_local_storage() { + // Intialize the theme to null, which means default + var theme = null; + // If the browser supports the localstorage and is not disabled then try to get the + // documenter theme + if(window.localStorage != null) { + // Get the user-picked theme from localStorage. May be `null`, which means the default + // theme. + theme = window.localStorage.getItem("documenter-theme"); + } + // Check if the browser supports user color preference + var darkPreference = false; + // Check if the users preference is for dark color scheme + if(window.matchMedia('(prefers-color-scheme: dark)').matches === true) { + darkPreference = true; + } + // Initialize a few variables for the loop: + // + // - active: will contain the index of the theme that should be active. Note that there + // is no guarantee that localStorage contains sane values. If `active` stays `null` + // we either could not find the theme or it is the default (primary) theme anyway. + // Either way, we then need to stick to the primary theme. + // + // - disabled: style sheets that should be disabled (i.e. all the theme style sheets + // that are not the currently active theme) + var active = null; var disabled = []; var darkTheme = null; + for (var i = 0; i < document.styleSheets.length; i++) { + var ss = document.styleSheets[i]; + // The tag of each style sheet is expected to have a data-theme-name attribute + // which must contain the name of the theme. The names in localStorage much match this. + var themename = ss.ownerNode.getAttribute("data-theme-name"); + // attribute not set => non-theme stylesheet => ignore + if(themename === null) continue; + // To distinguish the default (primary) theme, it needs to have the data-theme-primary + // attribute set. + var isprimary = (ss.ownerNode.getAttribute("data-theme-primary") !== null); + // Check if the theme is primary dark theme + var isDarkTheme = (ss.ownerNode.getAttribute("data-theme-primary-dark") !== null); + // If ss is for dark theme then set the value of darkTheme to the name of the theme + if(isDarkTheme) darkTheme = themename; + // If we find a matching theme (and it's not the default), we'll set active to non-null + if(themename === theme) active = i; + // Store the style sheets of inactive themes so that we could disable them + if(themename !== theme) disabled.push(ss); + } + if(active !== null) { + // If we did find an active theme, we'll (1) add the theme--$(theme) class to + document.getElementsByTagName('html')[0].className = "theme--" + theme; + // and (2) disable all the other theme stylesheets + disabled.forEach(function(ss){ + ss.disabled = true; + }); + } + else if(darkTheme !== null && darkPreference === true) { + // If we did find an active theme, we'll (1) add the theme--$(theme) class to + document.getElementsByTagName('html')[0].className = "theme--" + darkTheme; + // and (2) disable all the other theme stylesheets + disabled.forEach(function(ss){ + if (ss.ownerNode.getAttribute("data-theme-name") !== darkTheme) { + ss.disabled = true; + } + }); + } +} +set_theme_from_local_storage(); diff --git a/v1.7.4/assets/warner.js b/v1.7.4/assets/warner.js new file mode 100644 index 0000000..5531c88 --- /dev/null +++ b/v1.7.4/assets/warner.js @@ -0,0 +1,49 @@ +function maybeAddWarning () { + // DOCUMENTER_NEWEST is defined in versions.js, DOCUMENTER_CURRENT_VERSION and DOCUMENTER_STABLE + // in siteinfo.js. + // If either of these are undefined something went horribly wrong, so we abort. + if ( + window.DOCUMENTER_NEWEST === undefined || + window.DOCUMENTER_CURRENT_VERSION === undefined || + window.DOCUMENTER_STABLE === undefined + ) { + return + }; + + // Current version is not a version number, so we can't tell if it's the newest version. Abort. + if (!/v(\d+\.)*\d+/.test(window.DOCUMENTER_CURRENT_VERSION)) { + return + }; + + // Current version is newest version, so no need to add a warning. + if (window.DOCUMENTER_NEWEST === window.DOCUMENTER_CURRENT_VERSION) { + return + }; + + // Add a noindex meta tag (unless one exists) so that search engines don't index this version of the docs. + if (document.body.querySelector('meta[name="robots"]') === null) { + const meta = document.createElement('meta'); + meta.name = 'robots'; + meta.content = 'noindex'; + + document.getElementsByTagName('head')[0].appendChild(meta); + }; + + const div = document.createElement('div'); + div.classList.add('outdated-warning-overlay'); + const closer = document.createElement('button'); + closer.classList.add('outdated-warning-closer', 'delete'); + closer.addEventListener('click', function () { + document.body.removeChild(div); + }); + const href = window.documenterBaseURL + '/../' + window.DOCUMENTER_STABLE; + div.innerHTML = 'This documentation is not for the latest stable release, but for either the development version or an older release.
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    FractalDimensions.jl

    FractalDimensions — Module

    FractalDimensions.jl

    CI codecov Package Downloads

    A Julia package that estimates various definitions of fractal dimension from data. It can be used as a standalone package, or as part of DynamicalSystems.jl.

    To install it, run import Pkg; Pkg.add("FractalDimensions").

    All further information is provided in the documentation, which you can either find online or build locally by running the docs/make.jl file.

    Previously, this package was part of ChaosTools.jl.

    Publication

    FractalDimensions.jl is used in a review article comparing various estimators for fractal dimensions. The paper is likely a relevant read if you are interested in the package. And if you use the package, please cite the paper.

    @article{FractalDimensions.jl,
+  doi = {10.1063/5.0160394},
+  url = {https://doi.org/10.1063/5.0160394},
+  year = {2023},
+  month = oct,
+  publisher = {{AIP} Publishing},
+  volume = {33},
+  number = {10},
+  author = {George Datseris and Inga Kottlarz and Anton P. Braun and Ulrich Parlitz},
+  title = {Estimating fractal dimensions: A comparative review and open source implementations},
+  journal = {Chaos: An Interdisciplinary Journal of Nonlinear Science}
+}
    source

    Introduction

    Note

    This package is accompanying a review paper on estimating the fractal dimension: https://arxiv.org/abs/2109.05937. The paper is continuing the discussion of chapter 5 of Nonlinear Dynamics, Datseris & Parlitz, Springer 2022.

    There are numerous methods that one can use to calculate a so-called "dimension" of a dataset which in the context of dynamical systems is called the Fractal dimension. One way to do this is to estimate the scaling behaviour of some quantity as a size/scale increases. In the Fractal dimension example below, one finds the scaling of the correlation sum versus a ball radius. In this case, it approximately holds $ \log(C) \approx \Delta\log(\varepsilon) $ for radius $\varepsilon$. The scaling of many other quantities can be estimated as well, such as the generalized entropy, the Higuchi length, or others provided here.

    To actually find $\Delta$, one needs to find a linearly scaling region in the graph $\log(C)$ vs. $\log(\varepsilon)$ and estimate its slope. Hence, identifying a linear region is central to estimating a fractal dimension. That is why, the section Linear scaling regions is of central importance for this documentation.

    Fractal dimension example

    In this simplest example we will calculate the fractal dimension of the chaotic attractor of the Hénon map (for default parameters). For this example, we will generate the data on the spot:

    using DynamicalSystemsBase # for simulating dynamical systems
+using CairoMakie           # for plotting
+
+henon_rule(x, p, n) = SVector(1.0 - p[1]*x[1]^2 + x[2], p[2]*x[1])
+u0 = zeros(2)
+p0 = [1.4, 0.3]
+henon = DeterministicIteratedMap(henon_rule, u0, p0)
+
+X, t = trajectory(henon, 20_000; Ttr = 100)
+scatter(X[:, 1], X[:, 2]; color = ("black", 0.01), markersize = 4)

    instead of simulating the set X we could load it from disk, e.g., if there was a text file with two columns as x and y coordinates, we would load it as

    using DelimitedFiles
+file = "path/to/file.csv"
+M = readdlm(file)    # here `M` is a metrix with two columns
+X = StateSpaceSet(M) # important to convert to a state space set

    After we have X, we can start computing a fractal dimension and for this example we will use the correlationsum. Our goal is to compute the correlation sum of X for many different sizes/radii ε. This is as simple as

    using FractalDimensions
+ες = 2 .^ (-15:0.5:5) # semi-random guess
+Cs = correlationsum(X, ες; show_progress = false)
    41-element Vector{Float64}:
+ 1.8799060046997648e-6
+ 2.884855757212139e-6
+ 4.514774261286935e-6
+ 7.204639768011599e-6
+ 1.1174441277936103e-5
+ 1.7944102794860256e-5
+ 2.8333583320833955e-5
+ 4.5037748112594364e-5
+ 6.943652817359132e-5
+ 0.0001071696415179241
+ ⋮
+ 0.9486205889705515
+ 0.9999999999999999
+ 0.9999999999999999
+ 0.9999999999999999
+ 0.9999999999999999
+ 0.9999999999999999
+ 0.9999999999999999
+ 0.9999999999999999
+ 0.9999999999999999

    For a fractal set X dynamical systems theory says that there should be an exponential relationship between the correlation sum and the sizes:

    xs = log2.(ες)
+ys = log2.(Cs)
+scatterlines(xs, ys; axis = (ylabel = L"\log(C_2)", xlabel = L"\log (\epsilon)"))

    The slope of the linear scaling region of the above plot is the fractal dimension (based on the correlation sum).

    Given that we see the plot, we can estimate where the linear scaling region starts and ends. This is generally done using LargestLinearRegion in slopefit. But first, let's visualize what the method does, as it uses linear_regions.

    lrs, slopes = linear_regions(xs, ys, tol = 0.25)
+fig = Figure()
+ax = Axis(fig[1,1]; ylabel = L"\log(C_2)", xlabel = L"\log (\epsilon)")
+for r in lrs
+    scatterlines!(ax, xs[r], ys[r])
+end
+fig

    The LargestLinearRegion method finds, and computes the slope of, the largest region:

    Δ = slopefit(xs, ys, LargestLinearRegion())
    (1.2318376178087478, 1.2233720116518771, 1.2403032239656184)

    This result is an approximation of a fractal dimension.

    The whole above pipeline we went through is bundled in grassberger_proccacia_dim. Similar work is done by generalized_dim and many other functions.

    Be wary when using `xxxxx_dim`

    As stated clearly by the documentation strings, all pre-made dimension estimating functions (ending in _dim) perform a lot of automated steps, each having its own heuristic choices for function default values. They are more like convenient bundles with on-average good defaults, rather than precise functions. You should be careful when considering the validity of the returned number!

    Linear scaling regions

    FractalDimensions.slopefit — Function
    slopefit(x, y [, t::SLopeFit]; kw...) → s, s05, s95

    Fit a linear scaling region in the curve of the two AbstractVectors y versus x using t as the estimation method. Return the estimated slope, as well as the confidence intervals for it.

    The methods t that can be used for the estimation are:

    The keyword ignore_saturation = true ignores saturation that (sometimes) happens at the start and end of the curve y(x), where the curve flattens. The keyword sat_threshold = 0.01 decides what saturation is: while abs(y[i]-y[i+1]) < sat_threshold we are in a saturation regime. Said differently, slopes with value sat_threshold/dx with dx = x[i+1] - x[i] are neglected.

    The keyword ci = 0.95 specifies which quantile (and the 1 - quantile) the confidence interval values are returned at, and by defualt it is 95% (and hence also 5%).

    source
    FractalDimensions.LinearRegression — Type
    LinearRegression <: SlopeFit
+LinearRegression()

    Standard linear regression fit to all available data. Estimation of the confidence intervals is based om the standard error of the slope following a T-distribution, see:

    https://stattrek.com/regression/slope-confidence-interval

    source
    FractalDimensions.LargestLinearRegion — Type
    LargestLinearRegion <: SlopeFit
+LargestLinearRegion(; dxi::Int = 1, tol = 0.25)

    Identify regions where the curve y(x) is linear, by scanning the x-axis every dxi indices sequentially (e.g. at x[1] to x[5], x[5] to x[10], x[10] to x[15] and so on if dxi=5).

    If the slope (calculated via linear regression) of a region of width dxi is approximatelly equal to that of the previous region, within relative tolerance tol and absolute tolerance 0, then these two regions belong to the same linear region.

    The largest such region is then used to estimate the slope via standard linear regression of all points belonging to the largest linear region. "Largest" here means the region that covers the more extent along the x-axis.

    Use linear_regions if you wish to obtain the decomposition into linear regions.

    source
    FractalDimensions.linear_regions — Function
    linear_regions(x, y; dxi, tol) → lrs, tangents

    Apply the algorithm described by LargestLinearRegion, and return the indices of x that correspond to the linear regions, lrs, and the tangents at each region (obtained via a second linear regression at each accumulated region). lrs is hence a vector of UnitRanges.

    source
    FractalDimensions.AllSlopesDistribution — Type
    AllSlopesDistribution <: SlopeFit
+AllSlopesDistribution()

    Estimate a slope by computing the distribution of all possible slopes that can be estimated from the curve y(x), according to the method by (Deshmukh et al., 2021). The returned slope is the distribution mean and the confidence intervals are simply the corresponding quantiles of the distribution.

    Not implemented yet, the method is here as a placeholder.

    source
    FractalDimensions.estimate_boxsizes — Function
    estimate_boxsizes(X::AbstractStateSpaceSet; kwargs...) → εs

    Return k exponentially spaced values: εs = base .^ range(lower + w, upper + z; length = k), that are a good estimate for sizes ε that are used in calculating a Fractal Dimension. It is strongly recommended to standardize input dataset before using this function.

    Let d₋ be the minimum pair-wise distance in X, d₋ = dminimum_pairwise_distance(X). Let d₊ be the average total length of X, d₊ = mean(ma - mi) with mi, ma = minmaxima(X). Then lower = log(base, d₋) and upper = log(base, d₊). Because by default w=1, z=-1, the returned sizes are an order of mangitude larger than the minimum distance, and an order of magnitude smaller than the maximum distance.

    Keywords

    • w = 1, z = -1, k = 16 : as explained above.
    • base = MathConstants.e : the base used in the log function.
    • warning = true: Print some warnings for bad estimates.
    • autoexpand = true: If the final estimated range does not cover at least 2 orders of magnitude, it is automatically expanded by setting w -= we and z -= ze. You can set different default values to the keywords we = w, ze = z.
    source
    FractalDimensions.minimum_pairwise_distance — Function
    minimum_pairwise_distance(X::StateSpaceSet, kdtree = dimension(X) < 10, metric = Euclidean())

    Return min_d, min_pair: the minimum pairwise distance of all points in the dataset, and the corresponding point pair. The third argument is a switch of whether to use KDTrees or a brute force search.

    source

    Generalized (entropy) dimension

    Based on the definition of the Generalized entropy (genentropy), one can calculate an appropriate dimension, called generalized dimension:

    FractalDimensions.generalized_dim — Function
    generalized_dim(X::StateSpaceSet [, sizes]; q = 1, base = 2) -> Δ_q

    Return the q order generalized dimension of X, by calculating its histogram-based Rényi entropy for each ε ∈ sizes.

    The case of q = 0 is often called "capacity" or "box-counting" dimension, while q = 1 is the "information" dimension.

    Description

    The returned dimension is approximated by the (inverse) power law exponent of the scaling of the Renyi entropy $H_q$, versus the box size ε, where ε ∈ sizes:

    \[H_q \approx -\Delta_q\log_{b}(\varepsilon)\]

    $H_q$ is calculated using ComplexityMeasures: Renyi, ValueHistogram, entropy, i.e., by doing a histogram of the data with a given box size.

    Calling this function performs a lot of automated steps:

    1. A vector of box sizes is decided by calling sizes = estimate_boxsizes(dataset), if sizes is not given.
    2. For each element of sizes the appropriate entropy is calculated as
      H = [entropy(Renyi(; q, base), ValueHistogram(ε), data) for ε ∈ sizes]
      Let x = -log.(sizes).
    3. The curve H(x) is decomposed into linear regions, using slopefit(x, h)[1].
    4. The biggest linear region is chosen, and a fit for the slope of that region is performed using the function linear_region, which does a simple linear regression fit using linreg. This slope is the return value of generalized_dim.

    By doing these steps one by one yourself, you can adjust the keyword arguments given to each of these function calls, refining the accuracy of the result. The source code of this function is only 3 lines of code.

    This approach to estimating the fractal dimension has been used (to our knowledge) for the first time in (Russell et al., 1980).

    source
    FractalDimensions.molteno_dim — Function
    molteno_dim(X::AbstractStateSpaceSet; k0::Int = 10, q = 1.0, base = 2)

    Return an estimate of the generalized_dim of X using the algorithm by (Molteno, 1993). This function is a simple utilization of the probabilities estimated by molteno_boxing so see that function for more details. Here the entropy of the probabilities is computed at each size, and a line is fitted in the entropy vs log(size) graph, just like in generalized_dim.

    source
    FractalDimensions.molteno_boxing — Function
    molteno_boxing(X::AbstractStateSpaceSet; k0::Int = 10) → (probs, εs)

    Distribute X into boxes whose size is halved in each step, according to the algorithm by (Molteno, 1993). Division stops if the average number of points per filled box falls below the threshold k0.

    Return probs, a vector of Probabilities of finding points in boxes for different box sizes, and the corresponding box sizes εs. These outputs are used in molteno_dim.

    Description

    Project the data onto the whole interval of numbers that is covered by UInt64. The projected data is distributed into boxes whose size decreases by factor 2 in each step. For each box that contains more than one point 2^D new boxes are created where D is the dimension of the data.

    The process of dividing the data into new boxes stops when the number of points over the number of filled boxes falls below k0. The box sizes εs are calculated and returned together with the probs.

    This algorithm is faster than the traditional approach of using ValueHistogram(ε::Real), but it is only suited for low dimensional data since it divides each box into 2^D new boxes if D is the dimension. For large D this leads to low numbers of box divisions before the threshold is passed and the divison stops. This results to a low number of data points to fit the dimension to and thereby a poor estimate.

    source

    Correlation sum based dimension

    FractalDimensions.grassberger_proccacia_dim — Function
    grassberger_proccacia_dim(X::AbstractStateSpaceSet, εs = estimate_boxsizes(data); kwargs...)

    Use the method of Grassberger and Proccacia (Grassberger and Procaccia, 1983), and the correction by (Theiler, 1986), to estimate the correlation dimension Δ_C of X.

    This function does something extremely simple:

    cm = correlationsum(data, εs; kwargs...)
+Δ_C = slopefit(rs, ys)(log2.(sizes), log2.(cm))[1]

    i.e. it calculates correlationsum for various radii and then tries to find a linear region in the plot of the log of the correlation sum versus log(ε).

    See correlationsum for the available keywords. See also takens_best_estimate, boxassisted_correlation_dim.

    source
    FractalDimensions.correlationsum — Function
    correlationsum(X, ε::Real; w = 0, norm = Euclidean(), q = 2) → C_q(ε)

    Calculate the q-order correlation sum of X (StateSpaceSet or timeseries) for a given radius ε and norm. They keyword show_progress = true can be used to display a progress bar for large X.

    correlationsum(X, εs::AbstractVector; w, norm, q) → C_q(ε)

    If εs is a vector, C_q is calculated for each ε ∈ εs more efficiently. Multithreading is also enabled over the available threads (Threads.nthreads()). The function boxed_correlationsum is typically faster if the dimension of X is small and if maximum(εs) is smaller than the size of X.

    Keyword arguments

    • q = 2: order of the correlation sum
    • norm = Euclidean(): distance norm
    • w = 0: Theiler window
    • show_progress = true: display a progress bar

    Description

    The correlation sum is defined as follows for q=2:

    \[C_2(\epsilon) = \frac{2}{(N-w)(N-w-1)}\sum_{i=1}^{N}\sum_{j=1+w+i}^{N} +B(||X_i - X_j|| < \epsilon)\]

    for as follows for q≠2

    \[C_q(\epsilon) = \left[ \sum_{i=1}^{N} \alpha_i +\left[\sum_{j:|i-j| > w} B(||X_i - X_j|| < \epsilon)\right]^{q-1}\right]^{1/(q-1)}\]

    where

    \[\alpha_i = 1 / (N (\max(N-w, i) - \min(w + 1, i))^{(q-1)})\]

    with $N$ the length of X and $B$ gives 1 if its argument is true. w is the Theiler window.

    See the article of Grassberger for the general definition (Grassberger, 2007) and the book "Nonlinear Time Series Analysis" (Kantz and Schreiber, 2003), Ch. 6, for a discussion around choosing best values for w, and Ch. 11.3 for the explicit definition of the q-order correlationsum. Note that the formula in 11.3 is incorrect, but corrected here, indices are adapted to take advantage of all available points and also note that we immediatelly exponentiate $C_q$ to $1/(q-1)$, so that it scales exponentially as $C_q \propto \varepsilon ^\Delta_q$ versus the size $\varepsilon$.

    source

    Box-assisted version

    FractalDimensions.boxed_correlationsum — Function
    boxed_correlationsum(X::AbstractStateSpaceSet, εs, r0 = maximum(εs); kwargs...) → Cs

    Estimate the correlationsum for each size ε ∈ εs using an optimized algorithm that first distributes data into boxes of size r0, and then computes the correlation sum for each box and each neighboring box of each box. This method is much faster than correlationsum, provided that the box size r0 is significantly smaller than the attractor length. Good choices for r0 are estimate_r0_buenoorovio or estimate_r0_theiler.

    boxed_correlationsum(X::AbstractStateSpaceSet; kwargs...) → εs, Cs

    In this method the minimum inter-point distance and estimate_r0_buenoorovio of X are used to estimate suitable εs for the calculation, which are also returned.

    Keyword arguments

    • q = 2 : The order of the correlation sum.
    • P = 2 : The prism dimension.
    • w = 0 : The Theiler window.
    • show_progress = false : Whether to display a progress bar for the calculation.
    • norm = Euclidean() : Distance norm.

    Description

    C_q(ε) is calculated for every ε ∈ εs and each of the boxes to then be summed up afterwards. The method of splitting the data into boxes was implemented according to (Theiler, 1987). w is the Theiler window. P is the prism dimension. If P is unequal to the dimension of the data, only the first P dimensions are considered for the box distribution (this is called the prism-assisted version). By default P is 2, which is the version suggested by [Bueno2007]. Alternative for P is the prismdim_theiler. Note that only when P = dimension(X) the boxed version is guaranteed to be exact to the original correlationsum. For any other P, some point pairs that should have been included may be skipped due to having smaller distance in the remaining dimensions, but larger distance in the first P dimensions.

    source
    FractalDimensions.estimate_r0_buenoorovio — Function
    estimate_r0_buenoorovio(X::AbstractStateSpaceSet, P = 2) → r0, ε0

    Estimate a reasonable size for boxing X, proposed by Bueno-Orovio and Pérez-García (Bueno-Orovio and P{é}rez-Garc{ı́}a, 2007), before calculating the correlation dimension as presented by Theiler1983. Return the size r0 and the minimum interpoint distance ε0 in the data.

    If instead of boxes, prisms are chosen everything stays the same but P is the dimension of the prism. To do so the dimension ν is estimated by running the algorithm by Grassberger and Procaccia (Grassberger and Procaccia, 1983) with √N points where N is the number of total data points. An effective size ℓ of the attractor is calculated by boxing a small subset of size N/10 into boxes of sidelength r_ℓ and counting the number of filled boxes η_ℓ.

    \[\ell = r_\ell \eta_\ell ^{1/\nu}\]

    The optimal number of filled boxes η_opt is calculated by minimising the number of calculations.

    \[\eta_\textrm{opt} = N^{2/3}\cdot \frac{3^\nu - 1}{3^P - 1}^{1/2}.\]

    P is the dimension of the data or the number of edges on the prism that don't span the whole dataset.

    Then the optimal boxsize $r_0$ computes as

    \[r_0 = \ell / \eta_\textrm{opt}^{1/\nu}.\]

    source
    FractalDimensions.estimate_r0_theiler — Function
    estimate_r0_theiler(X::AbstractStateSpaceSet) → r0, ε0

    Estimate a reasonable size for boxing the data X before calculating the boxed_correlationsum proposed by (Theiler, 1987). Return the boxing size r0 and minimum inter-point distance in X, ε0.

    To do so the dimension is estimated by running the algorithm by Grassberger and Procaccia (Grassberger and Procaccia, 1983) with √N points where N is the number of total data points. Then the optimal boxsize $r_0$ computes as

    \[r_0 = R (2/N)^{1/\nu}\]

    where $R$ is the size of the chaotic attractor and $\nu$ is the estimated dimension.

    source

    Fixed mass correlation sum

    FractalDimensions.fixedmass_correlation_dim — Function
    fixedmass_correlation_dim(X [, max_j]; kwargs...)

    Use the fixed mass algorithm for computing the correlation sum, and use the result to compute the correlation dimension Δ_M of X.

    This function does something extremely simple:

    rs, ys = fixedmass_correlationsum(X, args...; kwargs...)
+slopefit(rs, ys)[1]
    source
    FractalDimensions.fixedmass_correlationsum — Function
    fixedmass_correlationsum(X [, max_j]; metric = Euclidean(), M = length(X)) → rs, ys

    A fixed mass algorithm for the calculation of the correlationsum, and subsequently a fractal dimension $\Delta$, with max_j the maximum number of neighbours that should be considered for the calculation.

    By default max_j = clamp(N*(N-1)/2, 5, 32) with N the data length.

    Keyword arguments

    • M defines the number of points considered for the averaging of distances, randomly subsampling them from X.
    • metric = Euclidean() is the distance metric.
    • start_j = 4 computes the equation below starting from j = start_j. Typically the first j values have not converged to the correct scaling of the fractal dimension.

    Description

    "Fixed mass" algorithms mean that instead of trying to find all neighboring points within a radius, one instead tries to find the max radius containing j points. A correlation sum is obtained with this constrain, and equivalently the mean radius containing k points. Based on this, one can calculate $\Delta$ approximating the information dimension. The implementation here is due to to (Grassberger, 1988), which defines

    \[Ψ(j) - \log N \sim \Delta \times \overline{\log \left( r_{(j)}\right)}\]

    where $\Psi(j) = \frac{\text{d} \log Γ(j)}{\text{d} j}$ is the digamma function, rs = $\overline{\log \left( r_{(j)}\right)}$ is the mean logarithm of a radius containing j neighboring points, and ys = $\Psi(j) - \log N$ ($N$ is the length of the data). The amount of neighbors found $j$ range from 2 to max_j. The numbers are also converted to base $2$ from base $e$.

    $\Delta$ can be computed by using linear_region(rs, ys).

    source

    Takens best estimate

    FractalDimensions.takens_best_estimate_dim — Function
    takens_best_estimate_dim(X, εmax, metric = Chebyshev(), εmin = 0)

    Use the "Takens' best estimate" [Takens1985][Theiler1988] method for estimating the correlation dimension.

    The original formula is

    \[\Delta_C \approx \frac{C(\epsilon_\text{max})}{\int_0^{\epsilon_\text{max}}(C(\epsilon) / \epsilon) \, d\epsilon}\]

    where $C$ is the correlationsum and $\epsilon_\text{max}$ is an upper cutoff. Here we use the later expression

    \[\Delta_C \approx - \frac{1}{\eta},\quad \eta = \frac{1}{(N-1)^*}\sum_{[i, j]^*}\log(||X_i - X_j|| / \epsilon_\text{max})\]

    where the sum happens for all $i, j$ so that $i < j$ and $||X_i - X_j|| < \epsilon_\text{max}$. In the above expression, the bias in the original paper has already been corrected, as suggested in [Borovkova1999].

    According to [Borovkova1999], introducing a lower cutoff εmin can make the algorithm more stable (no divergence), this option is given but defaults to zero.

    If X comes from a delay coordinates embedding of a timseries x, a recommended value for $\epsilon_\text{max}$ is std(x)/4.

    You may also use

    Δ_C, Δu_C, Δl_C = FractalDimensions.takens_best_estimate(args...)

    to obtain the upper and lower 95% confidence intervals. The intervals are estimated from the log-likelihood function by finding the values of Δ_C where the function has fallen by 2 from its maximum, see e.g. [Barlow] chapter 5.3.

    source

    Kaplan-Yorke dimension

    FractalDimensions.kaplanyorke_dim — Function
    kaplanyorke_dim(λs::AbstractVector)

    Calculate the Kaplan-Yorke dimension, a.k.a. Lyapunov dimension (Kaplan and Yorke, 1979) from the given Lyapunov exponents λs.

    Description

    The Kaplan-Yorke dimension is simply the point where cumsum(λs) becomes zero (interpolated):

    \[ D_{KY} = k + \frac{\sum_{i=1}^k \lambda_i}{|\lambda_{k+1}|},\quad k = \max_j \left[ \sum_{i=1}^j \lambda_i > 0 \right].\]

    If the sum of the exponents never becomes negative the function will return the length of the input vector.

    Useful in combination with lyapunovspectrum from ChaosTools.jl.

    source

    Higuchi dimension

    FractalDimensions.higuchi_dim — Function
    higuchi_dim(x::AbstractVector [, ks])

    Estimate the Higuchi dimension (Higuchi, 1988) of the graph of x.

    Description

    The Higuchi dimension is a number Δ ∈ [1, 2] that quantifies the roughness of the graph of the function x(t), assuming here that x is equi-sampled, like in the original paper.

    The method estimates how the length of the graph increases as a function of the indices difference (which, in this context, is equivalent with differences in t). Specifically, we calculate the average length versus k as

    \[L_m(k) = \frac{N-1}{\lfloor \frac{N-m}{k} floor k^2} +\sum_{i=1}^{\lfloor \frac{N-m}{k} \rfloor} |X_N(m+ik)-X_N(m+(i-1)k)| \\ + +L(k) = \frac{1}{k} \sum_{m=1}^k L_m(k)\]

    and then use linear_region in -log2.(k) vs log2.(L) as per usual when computing a fractal dimension.

    The algorithm chooses default ks to be exponentially spaced in base-2, up to at most 2^8. A user can provide their own ks as a second argument otherwise.

    Use FractalDimensions.higuchi_length(x, ks) to obtain $L(k)$ directly.

    source

    Extreme value value theory dimension

    FractalDimensions.extremevaltheory_dims_persistences — Function
    extremevaltheory_dims_persistences(x::AbstractStateSpaceSet, p::Real; kwargs...)

    Return the local dimensions Δloc and the persistences θloc for each point in the given set for quantile probability p, according to the estimation done via extreme value theory (Lucarini et al., 2016). The computation is parallelized to available threads (Threads.nthreads()).

    See also extremevaltheory_gpdfit_pvalues for obtaining confidence on the results.

    Keyword arguments

    • show_progress = true: displays a progress bar.
    • estimator = :mm: how to estimate the σ parameter of the Generalized Pareto Distribution. The local fractal dimension is 1/σ. The possible values are: :exp, :mm, as in estimate_gpd_parameters.
    • compute_persistence = true: whether to aso compute local persistences θloc (also called extremal index). If false, θloc are NaNs.
    • allocate_matrix = false: If true, the code calls a method that attempts to allocate an N×N matrix (N = length(X)) that stores the pairwise Euclidean distances. This method is faster due to optimizations of Distances.pairwise but will error if the computer does not have enough available memory for the matrix allocation.

    Description

    For each state space point $\mathbf{x}_i$ in X we compute $g_i = -\log(||\mathbf{x}_i - \mathbf{x}_j|| ) \; \forall j = 1, \ldots, N$ with $||\cdot||$ the Euclidean distance. Next, we choose an extreme quantile probability $p$ (e.g., 0.99) for the distribution of $g_i$. We compute $g_p$ as the $p$-th quantile of $g_i$. Then, we collect the exceedances of $g_i$, defined as $E_i = \{ g_i - g_p: g_i \ge g_p \}$, i.e., all values of $g_i$ larger or equal to $g_p$, also shifted by $g_p$. There are in total $n = N(1-q)$ values in $E_i$. According to extreme value theory, in the limit $N \to \infty$ the values $E_i$ follow a two-parameter Generalized Pareto Distribution (GPD) with parameters $\sigma,\xi$ (the third parameter $\mu$ of the GPD is zero due to the positive-definite construction of $E$). Within this extreme value theory approach, the local dimension $\Delta^{(E)}_i$ assigned to state space point $\textbf{x}_i$ is given by the inverse of the $\sigma$ parameter of the GPD fit to the data[Lucarini2012], $\Delta^{(E)}_i = 1/\sigma$. $\sigma$ is estimated according to the estimator keyword.

    A more precise description of this process is given in the review paper (Datseris et al., 2023).

    source
    FractalDimensions.estimate_gpd_parameters — Function
    estimate_gpd_parameters(X::AbstractVector{<:Real}, estimator::Symbol)

    Estimate and return the parameters σ, ξ of a Generalized Pareto Distribution fit to X, assuming that minimum(X) ≥ 0 and hence the parameter μ is 0 (if not, simply shift X by its minimum), according to the methods provided in Pons2023.

    The estimator can be:

    • :exp: Assume the distribution is exponential instead of GP and get σ from mean of X and set ξ = 0.
    • mm: Standing for "method of moments", estimants are given by

      \[\xi = (\bar{x}^2/s^2 - 1)/2, \quad \sigma = \bar{x}(\bar{x}^2/s^2 + 1)/2\]

      with $\bar{x}$ the sample mean and $s^2$ the sample variance. This estimator only exists if the true distribution ξ value is < 0.5.
    source
    FractalDimensions.extremevaltheory_gpdfit_pvalues — Function
    extremevaltheory_gpdfit_pvalues(X, p; kw...)

    Return various computed quantities that may quantify the significance of the results of extremevaltheory_dims_persistences(X, p; kw...), terms of quantifying how well a Generalized Pareto Distribution (GPD) describes exceedences in the input data.

    Keyword arguments

    • show_progress, estimator as in extremevaltheory_dims_persistences
    • TestType = ApproximateOneSampleKSTest: the test type to use. It can be ApproximateOneSampleKSTest, ExactOneSampleKSTest, CramerVonMises. We noticed that OneSampleADTest sometimes yielded nonsensical results: all p-values were equal and were very small ≈ 1e-6.
    • nbins = round(Int, length(X)*(1-p)/20): number of bins to use when computing the histogram of the exceedances for computing the NRMSE. The default value will use equally spaced bins that are equal to the length of the exceedances divided by 20.

    Description

    The function computes the exceedances $E_i$ for each point $x_i \in X$ as in extremevaltheory_dims_persistences. It returns 5 quantities, all being vectors of length length(X):

    • Es, all exceedences, as a vector of vectors.
    • sigmas, xis the fitted σ, ξ to the GPD fits for each exceedance
    • nrmses the normalized root mean square distance of the fitted GPD to the histogram of the exceedances
    • pvalues the pvalues of a statistical test of the appropriateness of the GPD fit

    The output nrmses quantifies the distance between the fitted GPD and the empirical histogram of the exceedances. It is computed as

    \[NRMSE = \sqrt{\frac{\sum{(P_j - G_j)^2}{\sum{(P_j - U)^2}}\]

    where $P_j$ the empirical (observed) probability at bin $j$, $G_j$ the fitted GPD probability at the midpoint of bin $j$, and $U$ same as $G_j$ but for the uniform distribution. The divisor of the equation normalizes the expression, so that the error of the empirical distribution is normalized to the error of the empirical distribution with fitting it with the uniform distribution. It is expected that NRMSE < 1. The smaller it is, the better the data are approximated by GPD versus uniform distribution.

    The output pvalues is a vector of p-values. pvalues[i] corresponds to the p-value of the hypothesis: "The exceedences around point X[i] are sampled from a GPD" versus the alternative hypothesis that they are not. To extract the p-values, we perform a one-sample hypothesis via HypothesisTests.jl to the fitted GPD. Very small p-values then indicate that the hypothesis should be rejected and the data are not well described by a GPD. This can be an indication that we do not have enough data, or that we choose too high of a quantile probability p, or that the data are not suitable in general. This p-value based method for significance has been used in [Faranda2017], but it is unclear precisely how it was used.

    For more details on how these quantities may quantify significance, see our review paper.

    source

    Theiler window

    The Theiler window is a concept that is useful when finding neighbors in a dataset that is coming from the sampling of a continuous dynamical system. Itt tries to eliminate spurious "correlations" (wrongly counted neighbors) due to a potentially dense sampling of the trajectory. Typically a good choice for w coincides with the choice an optimal delay time, see DelayEmbeddings.estimate_delay, for any of the timeseries of the dataset.

    For more details, see Chapter 5 of Nonlinear Dynamics, Datseris & Parlitz, Springer 2022.

    StateSpaceSet reference

    StateSpaceSets.StateSpaceSet — Type
    StateSpaceSet{D, T} <: AbstractStateSpaceSet{D,T}

    A dedicated interface for sets in a state space. It is an ordered container of equally-sized points of length D. Each point is represented by SVector{D, T}. The data are a standard Julia Vector{SVector}, and can be obtained with vec(ssset::StateSpaceSet). Typically the order of points in the set is the time direction, but it doesn't have to be.

    When indexed with 1 index, StateSpaceSet is like a vector of points. When indexed with 2 indices it behaves like a matrix that has each of the columns be the timeseries of each of the variables. When iterated over, it iterates over its contained points. See description of indexing below for more.

    StateSpaceSet also supports almost all sensible vector operations like append!, push!, hcat, eachrow, among others.

    Description of indexing

    In the following let i, j be integers, typeof(X) <: AbstractStateSpaceSet and v1, v2 be <: AbstractVector{Int} (v1, v2 could also be ranges, and for performance benefits make v2 an SVector{Int}).

    • X[i] == X[i, :] gives the ith point (returns an SVector)
    • X[v1] == X[v1, :], returns a StateSpaceSet with the points in those indices.
    • X[:, j] gives the jth variable timeseries (or collection), as Vector
    • X[v1, v2], X[:, v2] returns a StateSpaceSet with the appropriate entries (first indices being "time"/point index, while second being variables)
    • X[i, j] value of the jth variable, at the ith timepoint

    Use Matrix(ssset) or StateSpaceSet(matrix) to convert. It is assumed that each column of the matrix is one variable. If you have various timeseries vectors x, y, z, ... pass them like StateSpaceSet(x, y, z, ...). You can use columns(dataset) to obtain the reverse, i.e. all columns of the dataset in a tuple.

    StateSpaceSets.standardize — Function
    standardize(d::StateSpaceSet) → r

    Create a standardized version of the input set where each column is transformed to have mean 0 and standard deviation 1.

    standardize(x::AbstractVector{<:Real}) = (x - mean(x))/std(x)

    References

    diff --git a/v1.7.4/search/index.html b/v1.7.4/search/index.html new file mode 100644 index 0000000..2e5a31a --- /dev/null +++ b/v1.7.4/search/index.html @@ -0,0 +1,2 @@ + +Search · FractalDimensions.jl

    Loading search...

      diff --git a/v1.7.4/search_index.js b/v1.7.4/search_index.js new file mode 100644 index 0000000..b39c141 --- /dev/null +++ b/v1.7.4/search_index.js @@ -0,0 +1,3 @@ +var documenterSearchIndex = {"docs": +[{"location":"#FractalDimensions.jl","page":"FractalDimensions.jl","title":"FractalDimensions.jl","text":"","category":"section"},{"location":"","page":"FractalDimensions.jl","title":"FractalDimensions.jl","text":"FractalDimensions","category":"page"},{"location":"#FractalDimensions","page":"FractalDimensions.jl","title":"FractalDimensions","text":"FractalDimensions.jl\n\n(Image: ) (Image: ) (Image: ) (Image: CI) (Image: codecov) (Image: Package Downloads)\n\nA Julia package that estimates various definitions of fractal dimension from data. It can be used as a standalone package, or as part of DynamicalSystems.jl.\n\nTo install it, run import Pkg; Pkg.add(\"FractalDimensions\").\n\nAll further information is provided in the documentation, which you can either find online or build locally by running the docs/make.jl file.\n\nPreviously, this package was part of ChaosTools.jl.\n\nPublication\n\nFractalDimensions.jl is used in a review article comparing various estimators for fractal dimensions. The paper is likely a relevant read if you are interested in the package. And if you use the package, please cite the paper.\n\n@article{FractalDimensions.jl,\n doi = {10.1063/5.0160394},\n url = {https://doi.org/10.1063/5.0160394},\n year = {2023},\n month = oct,\n publisher = {{AIP} Publishing},\n volume = {33},\n number = {10},\n author = {George Datseris and Inga Kottlarz and Anton P. Braun and Ulrich Parlitz},\n title = {Estimating fractal dimensions: A comparative review and open source implementations},\n journal = {Chaos: An Interdisciplinary Journal of Nonlinear Science}\n}\n\n\n\n\n\n","category":"module"},{"location":"#Introduction","page":"FractalDimensions.jl","title":"Introduction","text":"","category":"section"},{"location":"","page":"FractalDimensions.jl","title":"FractalDimensions.jl","text":"note: Note\nThis package is accompanying a review paper on estimating the fractal dimension: https://arxiv.org/abs/2109.05937. The paper is continuing the discussion of chapter 5 of Nonlinear Dynamics, Datseris & Parlitz, Springer 2022.","category":"page"},{"location":"","page":"FractalDimensions.jl","title":"FractalDimensions.jl","text":"There are numerous methods that one can use to calculate a so-called \"dimension\" of a dataset which in the context of dynamical systems is called the Fractal dimension. One way to do this is to estimate the scaling behaviour of some quantity as a size/scale increases. In the Fractal dimension example below, one finds the scaling of the correlation sum versus a ball radius. In this case, it approximately holds $ \\log(C) \\approx \\Delta\\log(\\varepsilon) $ for radius varepsilon. The scaling of many other quantities can be estimated as well, such as the generalized entropy, the Higuchi length, or others provided here.","category":"page"},{"location":"","page":"FractalDimensions.jl","title":"FractalDimensions.jl","text":"To actually find Delta, one needs to find a linearly scaling region in the graph log(C) vs. log(varepsilon) and estimate its slope. Hence, identifying a linear region is central to estimating a fractal dimension. That is why, the section Linear scaling regions is of central importance for this documentation.","category":"page"},{"location":"#Fractal-dimension-example","page":"FractalDimensions.jl","title":"Fractal dimension example","text":"","category":"section"},{"location":"","page":"FractalDimensions.jl","title":"FractalDimensions.jl","text":"In this simplest example we will calculate the fractal dimension of the chaotic attractor of the Hénon map (for default parameters). For this example, we will generate the data on the spot:","category":"page"},{"location":"","page":"FractalDimensions.jl","title":"FractalDimensions.jl","text":"using DynamicalSystemsBase # for simulating dynamical systems\nusing CairoMakie # for plotting\n\nhenon_rule(x, p, n) = SVector(1.0 - p[1]*x[1]^2 + x[2], p[2]*x[1])\nu0 = zeros(2)\np0 = [1.4, 0.3]\nhenon = DeterministicIteratedMap(henon_rule, u0, p0)\n\nX, t = trajectory(henon, 20_000; Ttr = 100)\nscatter(X[:, 1], X[:, 2]; color = (\"black\", 0.01), markersize = 4)","category":"page"},{"location":"","page":"FractalDimensions.jl","title":"FractalDimensions.jl","text":"instead of simulating the set X we could load it from disk, e.g., if there was a text file with two columns as x and y coordinates, we would load it as","category":"page"},{"location":"","page":"FractalDimensions.jl","title":"FractalDimensions.jl","text":"using DelimitedFiles\nfile = \"path/to/file.csv\"\nM = readdlm(file) # here `M` is a metrix with two columns\nX = StateSpaceSet(M) # important to convert to a state space set","category":"page"},{"location":"","page":"FractalDimensions.jl","title":"FractalDimensions.jl","text":"After we have X, we can start computing a fractal dimension and for this example we will use the correlationsum. Our goal is to compute the correlation sum of X for many different sizes/radii ε. This is as simple as","category":"page"},{"location":"","page":"FractalDimensions.jl","title":"FractalDimensions.jl","text":"using FractalDimensions\nες = 2 .^ (-15:0.5:5) # semi-random guess\nCs = correlationsum(X, ες; show_progress = false)","category":"page"},{"location":"","page":"FractalDimensions.jl","title":"FractalDimensions.jl","text":"For a fractal set X dynamical systems theory says that there should be an exponential relationship between the correlation sum and the sizes:","category":"page"},{"location":"","page":"FractalDimensions.jl","title":"FractalDimensions.jl","text":"xs = log2.(ες)\nys = log2.(Cs)\nscatterlines(xs, ys; axis = (ylabel = L\"\\log(C_2)\", xlabel = L\"\\log (\\epsilon)\"))","category":"page"},{"location":"","page":"FractalDimensions.jl","title":"FractalDimensions.jl","text":"The slope of the linear scaling region of the above plot is the fractal dimension (based on the correlation sum).","category":"page"},{"location":"","page":"FractalDimensions.jl","title":"FractalDimensions.jl","text":"Given that we see the plot, we can estimate where the linear scaling region starts and ends. This is generally done using LargestLinearRegion in slopefit. But first, let's visualize what the method does, as it uses linear_regions.","category":"page"},{"location":"","page":"FractalDimensions.jl","title":"FractalDimensions.jl","text":"lrs, slopes = linear_regions(xs, ys, tol = 0.25)\nfig = Figure()\nax = Axis(fig[1,1]; ylabel = L\"\\log(C_2)\", xlabel = L\"\\log (\\epsilon)\")\nfor r in lrs\n scatterlines!(ax, xs[r], ys[r])\nend\nfig","category":"page"},{"location":"","page":"FractalDimensions.jl","title":"FractalDimensions.jl","text":"The LargestLinearRegion method finds, and computes the slope of, the largest region:","category":"page"},{"location":"","page":"FractalDimensions.jl","title":"FractalDimensions.jl","text":"Δ = slopefit(xs, ys, LargestLinearRegion())","category":"page"},{"location":"","page":"FractalDimensions.jl","title":"FractalDimensions.jl","text":"This result is an approximation of a fractal dimension.","category":"page"},{"location":"","page":"FractalDimensions.jl","title":"FractalDimensions.jl","text":"The whole above pipeline we went through is bundled in grassberger_proccacia_dim. Similar work is done by generalized_dim and many other functions.","category":"page"},{"location":"","page":"FractalDimensions.jl","title":"FractalDimensions.jl","text":"danger: Be wary when using `xxxxx_dim`\nAs stated clearly by the documentation strings, all pre-made dimension estimating functions (ending in _dim) perform a lot of automated steps, each having its own heuristic choices for function default values. They are more like convenient bundles with on-average good defaults, rather than precise functions. You should be careful when considering the validity of the returned number!","category":"page"},{"location":"#Linear-scaling-regions","page":"FractalDimensions.jl","title":"Linear scaling regions","text":"","category":"section"},{"location":"","page":"FractalDimensions.jl","title":"FractalDimensions.jl","text":"slopefit\nLinearRegression\nLargestLinearRegion\nlinear_regions\nAllSlopesDistribution\nestimate_boxsizes\nminimum_pairwise_distance","category":"page"},{"location":"#FractalDimensions.slopefit","page":"FractalDimensions.jl","title":"FractalDimensions.slopefit","text":"slopefit(x, y [, t::SLopeFit]; kw...) → s, s05, s95\n\nFit a linear scaling region in the curve of the two AbstractVectors y versus x using t as the estimation method. Return the estimated slope, as well as the confidence intervals for it.\n\nThe methods t that can be used for the estimation are:\n\nLinearRegression\nLargestLinearRegion (default)\nAllSlopesDistribution\n\nThe keyword ignore_saturation = true ignores saturation that (sometimes) happens at the start and end of the curve y(x), where the curve flattens. The keyword sat_threshold = 0.01 decides what saturation is: while abs(y[i]-y[i+1]) < sat_threshold we are in a saturation regime. Said differently, slopes with value sat_threshold/dx with dx = x[i+1] - x[i] are neglected.\n\nThe keyword ci = 0.95 specifies which quantile (and the 1 - quantile) the confidence interval values are returned at, and by defualt it is 95% (and hence also 5%).\n\n\n\n\n\n","category":"function"},{"location":"#FractalDimensions.LinearRegression","page":"FractalDimensions.jl","title":"FractalDimensions.LinearRegression","text":"LinearRegression <: SlopeFit\nLinearRegression()\n\nStandard linear regression fit to all available data. Estimation of the confidence intervals is based om the standard error of the slope following a T-distribution, see:\n\nhttps://stattrek.com/regression/slope-confidence-interval\n\n\n\n\n\n","category":"type"},{"location":"#FractalDimensions.LargestLinearRegion","page":"FractalDimensions.jl","title":"FractalDimensions.LargestLinearRegion","text":"LargestLinearRegion <: SlopeFit\nLargestLinearRegion(; dxi::Int = 1, tol = 0.25)\n\nIdentify regions where the curve y(x) is linear, by scanning the x-axis every dxi indices sequentially (e.g. at x[1] to x[5], x[5] to x[10], x[10] to x[15] and so on if dxi=5).\n\nIf the slope (calculated via linear regression) of a region of width dxi is approximatelly equal to that of the previous region, within relative tolerance tol and absolute tolerance 0, then these two regions belong to the same linear region.\n\nThe largest such region is then used to estimate the slope via standard linear regression of all points belonging to the largest linear region. \"Largest\" here means the region that covers the more extent along the x-axis.\n\nUse linear_regions if you wish to obtain the decomposition into linear regions.\n\n\n\n\n\n","category":"type"},{"location":"#FractalDimensions.linear_regions","page":"FractalDimensions.jl","title":"FractalDimensions.linear_regions","text":"linear_regions(x, y; dxi, tol) → lrs, tangents\n\nApply the algorithm described by LargestLinearRegion, and return the indices of x that correspond to the linear regions, lrs, and the tangents at each region (obtained via a second linear regression at each accumulated region). lrs is hence a vector of UnitRanges.\n\n\n\n\n\n","category":"function"},{"location":"#FractalDimensions.AllSlopesDistribution","page":"FractalDimensions.jl","title":"FractalDimensions.AllSlopesDistribution","text":"AllSlopesDistribution <: SlopeFit\nAllSlopesDistribution()\n\nEstimate a slope by computing the distribution of all possible slopes that can be estimated from the curve y(x), according to the method by (Deshmukh et al., 2021). The returned slope is the distribution mean and the confidence intervals are simply the corresponding quantiles of the distribution.\n\nNot implemented yet, the method is here as a placeholder.\n\n\n\n\n\n","category":"type"},{"location":"#FractalDimensions.estimate_boxsizes","page":"FractalDimensions.jl","title":"FractalDimensions.estimate_boxsizes","text":"estimate_boxsizes(X::AbstractStateSpaceSet; kwargs...) → εs\n\nReturn k exponentially spaced values: εs = base .^ range(lower + w, upper + z; length = k), that are a good estimate for sizes ε that are used in calculating a Fractal Dimension. It is strongly recommended to standardize input dataset before using this function.\n\nLet d₋ be the minimum pair-wise distance in X, d₋ = dminimum_pairwise_distance(X). Let d₊ be the average total length of X, d₊ = mean(ma - mi) with mi, ma = minmaxima(X). Then lower = log(base, d₋) and upper = log(base, d₊). Because by default w=1, z=-1, the returned sizes are an order of mangitude larger than the minimum distance, and an order of magnitude smaller than the maximum distance.\n\nKeywords\n\nw = 1, z = -1, k = 16 : as explained above.\nbase = MathConstants.e : the base used in the log function.\nwarning = true: Print some warnings for bad estimates.\nautoexpand = true: If the final estimated range does not cover at least 2 orders of magnitude, it is automatically expanded by setting w -= we and z -= ze. You can set different default values to the keywords we = w, ze = z.\n\n\n\n\n\n","category":"function"},{"location":"#FractalDimensions.minimum_pairwise_distance","page":"FractalDimensions.jl","title":"FractalDimensions.minimum_pairwise_distance","text":"minimum_pairwise_distance(X::StateSpaceSet, kdtree = dimension(X) < 10, metric = Euclidean())\n\nReturn min_d, min_pair: the minimum pairwise distance of all points in the dataset, and the corresponding point pair. The third argument is a switch of whether to use KDTrees or a brute force search.\n\n\n\n\n\n","category":"function"},{"location":"#Generalized-(entropy)-dimension","page":"FractalDimensions.jl","title":"Generalized (entropy) dimension","text":"","category":"section"},{"location":"","page":"FractalDimensions.jl","title":"FractalDimensions.jl","text":"Based on the definition of the Generalized entropy (genentropy), one can calculate an appropriate dimension, called generalized dimension:","category":"page"},{"location":"","page":"FractalDimensions.jl","title":"FractalDimensions.jl","text":"generalized_dim\nmolteno_dim\nmolteno_boxing","category":"page"},{"location":"#FractalDimensions.generalized_dim","page":"FractalDimensions.jl","title":"FractalDimensions.generalized_dim","text":"generalized_dim(X::StateSpaceSet [, sizes]; q = 1, base = 2) -> Δ_q\n\nReturn the q order generalized dimension of X, by calculating its histogram-based Rényi entropy for each ε ∈ sizes.\n\nThe case of q = 0 is often called \"capacity\" or \"box-counting\" dimension, while q = 1 is the \"information\" dimension.\n\nDescription\n\nThe returned dimension is approximated by the (inverse) power law exponent of the scaling of the Renyi entropy H_q, versus the box size ε, where ε ∈ sizes:\n\nH_q approx -Delta_qlog_b(varepsilon)\n\nH_q is calculated using ComplexityMeasures: Renyi, ValueHistogram, entropy, i.e., by doing a histogram of the data with a given box size.\n\nCalling this function performs a lot of automated steps:\n\nA vector of box sizes is decided by calling sizes = estimate_boxsizes(dataset), if sizes is not given.\nFor each element of sizes the appropriate entropy is calculated as\nH = [entropy(Renyi(; q, base), ValueHistogram(ε), data) for ε ∈ sizes]\nLet x = -log.(sizes).\nThe curve H(x) is decomposed into linear regions, using slopefit(x, h)[1].\nThe biggest linear region is chosen, and a fit for the slope of that region is performed using the function linear_region, which does a simple linear regression fit using linreg. This slope is the return value of generalized_dim.\n\nBy doing these steps one by one yourself, you can adjust the keyword arguments given to each of these function calls, refining the accuracy of the result. The source code of this function is only 3 lines of code.\n\nThis approach to estimating the fractal dimension has been used (to our knowledge) for the first time in (Russell et al., 1980).\n\n\n\n\n\n","category":"function"},{"location":"#FractalDimensions.molteno_dim","page":"FractalDimensions.jl","title":"FractalDimensions.molteno_dim","text":"molteno_dim(X::AbstractStateSpaceSet; k0::Int = 10, q = 1.0, base = 2)\n\nReturn an estimate of the generalized_dim of X using the algorithm by (Molteno, 1993). This function is a simple utilization of the probabilities estimated by molteno_boxing so see that function for more details. Here the entropy of the probabilities is computed at each size, and a line is fitted in the entropy vs log(size) graph, just like in generalized_dim.\n\n\n\n\n\n","category":"function"},{"location":"#FractalDimensions.molteno_boxing","page":"FractalDimensions.jl","title":"FractalDimensions.molteno_boxing","text":"molteno_boxing(X::AbstractStateSpaceSet; k0::Int = 10) → (probs, εs)\n\nDistribute X into boxes whose size is halved in each step, according to the algorithm by (Molteno, 1993). Division stops if the average number of points per filled box falls below the threshold k0.\n\nReturn probs, a vector of Probabilities of finding points in boxes for different box sizes, and the corresponding box sizes εs. These outputs are used in molteno_dim.\n\nDescription\n\nProject the data onto the whole interval of numbers that is covered by UInt64. The projected data is distributed into boxes whose size decreases by factor 2 in each step. For each box that contains more than one point 2^D new boxes are created where D is the dimension of the data.\n\nThe process of dividing the data into new boxes stops when the number of points over the number of filled boxes falls below k0. The box sizes εs are calculated and returned together with the probs.\n\nThis algorithm is faster than the traditional approach of using ValueHistogram(ε::Real), but it is only suited for low dimensional data since it divides each box into 2^D new boxes if D is the dimension. For large D this leads to low numbers of box divisions before the threshold is passed and the divison stops. This results to a low number of data points to fit the dimension to and thereby a poor estimate.\n\n\n\n\n\n","category":"function"},{"location":"#Correlation-sum-based-dimension","page":"FractalDimensions.jl","title":"Correlation sum based dimension","text":"","category":"section"},{"location":"","page":"FractalDimensions.jl","title":"FractalDimensions.jl","text":"grassberger_proccacia_dim\ncorrelationsum","category":"page"},{"location":"#FractalDimensions.grassberger_proccacia_dim","page":"FractalDimensions.jl","title":"FractalDimensions.grassberger_proccacia_dim","text":"grassberger_proccacia_dim(X::AbstractStateSpaceSet, εs = estimate_boxsizes(data); kwargs...)\n\nUse the method of Grassberger and Proccacia (Grassberger and Procaccia, 1983), and the correction by (Theiler, 1986), to estimate the correlation dimension Δ_C of X.\n\nThis function does something extremely simple:\n\ncm = correlationsum(data, εs; kwargs...)\nΔ_C = slopefit(rs, ys)(log2.(sizes), log2.(cm))[1]\n\ni.e. it calculates correlationsum for various radii and then tries to find a linear region in the plot of the log of the correlation sum versus log(ε).\n\nSee correlationsum for the available keywords. See also takens_best_estimate, boxassisted_correlation_dim.\n\n\n\n\n\n","category":"function"},{"location":"#FractalDimensions.correlationsum","page":"FractalDimensions.jl","title":"FractalDimensions.correlationsum","text":"correlationsum(X, ε::Real; w = 0, norm = Euclidean(), q = 2) → C_q(ε)\n\nCalculate the q-order correlation sum of X (StateSpaceSet or timeseries) for a given radius ε and norm. They keyword show_progress = true can be used to display a progress bar for large X.\n\ncorrelationsum(X, εs::AbstractVector; w, norm, q) → C_q(ε)\n\nIf εs is a vector, C_q is calculated for each ε ∈ εs more efficiently. Multithreading is also enabled over the available threads (Threads.nthreads()). The function boxed_correlationsum is typically faster if the dimension of X is small and if maximum(εs) is smaller than the size of X.\n\nKeyword arguments\n\nq = 2: order of the correlation sum\nnorm = Euclidean(): distance norm\nw = 0: Theiler window\nshow_progress = true: display a progress bar\n\nDescription\n\nThe correlation sum is defined as follows for q=2:\n\nC_2(epsilon) = frac2(N-w)(N-w-1)sum_i=1^Nsum_j=1+w+i^N\nB(X_i - X_j epsilon)\n\nfor as follows for q≠2\n\nC_q(epsilon) = left sum_i=1^N alpha_i\nleftsum_ji-j w B(X_i - X_j epsilon)right^q-1right^1(q-1)\n\nwhere\n\nalpha_i = 1 (N (max(N-w i) - min(w + 1 i))^(q-1))\n\nwith N the length of X and B gives 1 if its argument is true. w is the Theiler window.\n\nSee the article of Grassberger for the general definition (Grassberger, 2007) and the book \"Nonlinear Time Series Analysis\" (Kantz and Schreiber, 2003), Ch. 6, for a discussion around choosing best values for w, and Ch. 11.3 for the explicit definition of the q-order correlationsum. Note that the formula in 11.3 is incorrect, but corrected here, indices are adapted to take advantage of all available points and also note that we immediatelly exponentiate C_q to 1(q-1), so that it scales exponentially as C_q propto varepsilon ^Delta_q versus the size varepsilon.\n\n\n\n\n\n","category":"function"},{"location":"#Box-assisted-version","page":"FractalDimensions.jl","title":"Box-assisted version","text":"","category":"section"},{"location":"","page":"FractalDimensions.jl","title":"FractalDimensions.jl","text":"boxassisted_correlation_dim\nboxed_correlationsum\nprismdim_theiler\nestimate_r0_buenoorovio\nestimate_r0_theiler","category":"page"},{"location":"#FractalDimensions.boxassisted_correlation_dim","page":"FractalDimensions.jl","title":"FractalDimensions.boxassisted_correlation_dim","text":"boxassisted_correlation_dim(X::AbstractStateSpaceSet; kwargs...)\n\nUse the box-assisted optimizations of (Bueno-Orovio and P{é}rez-Garc{ı́}a, 2007) to estimate the correlation dimension Δ_C of X.\n\nThis function does something extremely simple:\n\nεs, Cs = boxed_correlationsum(X; kwargs...)\nslopefit(log2.(εs), log2.(Cs))[1]\n\nand hence see boxed_correlationsum for more information and available keywords.\n\n\n\n\n\n","category":"function"},{"location":"#FractalDimensions.boxed_correlationsum","page":"FractalDimensions.jl","title":"FractalDimensions.boxed_correlationsum","text":"boxed_correlationsum(X::AbstractStateSpaceSet, εs, r0 = maximum(εs); kwargs...) → Cs\n\nEstimate the correlationsum for each size ε ∈ εs using an optimized algorithm that first distributes data into boxes of size r0, and then computes the correlation sum for each box and each neighboring box of each box. This method is much faster than correlationsum, provided that the box size r0 is significantly smaller than the attractor length. Good choices for r0 are estimate_r0_buenoorovio or estimate_r0_theiler.\n\nboxed_correlationsum(X::AbstractStateSpaceSet; kwargs...) → εs, Cs\n\nIn this method the minimum inter-point distance and estimate_r0_buenoorovio of X are used to estimate suitable εs for the calculation, which are also returned.\n\nKeyword arguments\n\nq = 2 : The order of the correlation sum.\nP = 2 : The prism dimension.\nw = 0 : The Theiler window.\nshow_progress = false : Whether to display a progress bar for the calculation.\nnorm = Euclidean() : Distance norm.\n\nDescription\n\nC_q(ε) is calculated for every ε ∈ εs and each of the boxes to then be summed up afterwards. The method of splitting the data into boxes was implemented according to (Theiler, 1987). w is the Theiler window. P is the prism dimension. If P is unequal to the dimension of the data, only the first P dimensions are considered for the box distribution (this is called the prism-assisted version). By default P is 2, which is the version suggested by [Bueno2007]. Alternative for P is the prismdim_theiler. Note that only when P = dimension(X) the boxed version is guaranteed to be exact to the original correlationsum. For any other P, some point pairs that should have been included may be skipped due to having smaller distance in the remaining dimensions, but larger distance in the first P dimensions.\n\n\n\n\n\n","category":"function"},{"location":"#FractalDimensions.prismdim_theiler","page":"FractalDimensions.jl","title":"FractalDimensions.prismdim_theiler","text":"prismdim_theiler(X)\n\nAn algorithm to find the ideal choice of a prism dimension for boxed_correlationsum using Theiler's original suggestion.\n\n\n\n\n\n","category":"function"},{"location":"#FractalDimensions.estimate_r0_buenoorovio","page":"FractalDimensions.jl","title":"FractalDimensions.estimate_r0_buenoorovio","text":"estimate_r0_buenoorovio(X::AbstractStateSpaceSet, P = 2) → r0, ε0\n\nEstimate a reasonable size for boxing X, proposed by Bueno-Orovio and Pérez-García (Bueno-Orovio and P{é}rez-Garc{ı́}a, 2007), before calculating the correlation dimension as presented by Theiler1983. Return the size r0 and the minimum interpoint distance ε0 in the data.\n\nIf instead of boxes, prisms are chosen everything stays the same but P is the dimension of the prism. To do so the dimension ν is estimated by running the algorithm by Grassberger and Procaccia (Grassberger and Procaccia, 1983) with √N points where N is the number of total data points. An effective size ℓ of the attractor is calculated by boxing a small subset of size N/10 into boxes of sidelength r_ℓ and counting the number of filled boxes η_ℓ.\n\nell = r_ell eta_ell ^1nu\n\nThe optimal number of filled boxes η_opt is calculated by minimising the number of calculations.\n\neta_textrmopt = N^23cdot frac3^nu - 13^P - 1^12\n\nP is the dimension of the data or the number of edges on the prism that don't span the whole dataset.\n\nThen the optimal boxsize r_0 computes as\n\nr_0 = ell eta_textrmopt^1nu\n\n\n\n\n\n","category":"function"},{"location":"#FractalDimensions.estimate_r0_theiler","page":"FractalDimensions.jl","title":"FractalDimensions.estimate_r0_theiler","text":"estimate_r0_theiler(X::AbstractStateSpaceSet) → r0, ε0\n\nEstimate a reasonable size for boxing the data X before calculating the boxed_correlationsum proposed by (Theiler, 1987). Return the boxing size r0 and minimum inter-point distance in X, ε0.\n\nTo do so the dimension is estimated by running the algorithm by Grassberger and Procaccia (Grassberger and Procaccia, 1983) with √N points where N is the number of total data points. Then the optimal boxsize r_0 computes as\n\nr_0 = R (2N)^1nu\n\nwhere R is the size of the chaotic attractor and nu is the estimated dimension.\n\n\n\n\n\n","category":"function"},{"location":"#Fixed-mass-correlation-sum","page":"FractalDimensions.jl","title":"Fixed mass correlation sum","text":"","category":"section"},{"location":"","page":"FractalDimensions.jl","title":"FractalDimensions.jl","text":"fixedmass_correlation_dim\nfixedmass_correlationsum","category":"page"},{"location":"#FractalDimensions.fixedmass_correlation_dim","page":"FractalDimensions.jl","title":"FractalDimensions.fixedmass_correlation_dim","text":"fixedmass_correlation_dim(X [, max_j]; kwargs...)\n\nUse the fixed mass algorithm for computing the correlation sum, and use the result to compute the correlation dimension Δ_M of X.\n\nThis function does something extremely simple:\n\nrs, ys = fixedmass_correlationsum(X, args...; kwargs...)\nslopefit(rs, ys)[1]\n\n\n\n\n\n","category":"function"},{"location":"#FractalDimensions.fixedmass_correlationsum","page":"FractalDimensions.jl","title":"FractalDimensions.fixedmass_correlationsum","text":"fixedmass_correlationsum(X [, max_j]; metric = Euclidean(), M = length(X)) → rs, ys\n\nA fixed mass algorithm for the calculation of the correlationsum, and subsequently a fractal dimension Delta, with max_j the maximum number of neighbours that should be considered for the calculation.\n\nBy default max_j = clamp(N*(N-1)/2, 5, 32) with N the data length.\n\nKeyword arguments\n\nM defines the number of points considered for the averaging of distances, randomly subsampling them from X.\nmetric = Euclidean() is the distance metric.\nstart_j = 4 computes the equation below starting from j = start_j. Typically the first j values have not converged to the correct scaling of the fractal dimension.\n\nDescription\n\n\"Fixed mass\" algorithms mean that instead of trying to find all neighboring points within a radius, one instead tries to find the max radius containing j points. A correlation sum is obtained with this constrain, and equivalently the mean radius containing k points. Based on this, one can calculate Delta approximating the information dimension. The implementation here is due to to (Grassberger, 1988), which defines\n\nΨ(j) - log N sim Delta times overlinelog left( r_(j)right)\n\nwhere Psi(j) = fractextd log Γ(j)textd j is the digamma function, rs = overlinelog left( r_(j)right) is the mean logarithm of a radius containing j neighboring points, and ys = Psi(j) - log N (N is the length of the data). The amount of neighbors found j range from 2 to max_j. The numbers are also converted to base 2 from base e.\n\nDelta can be computed by using linear_region(rs, ys).\n\n\n\n\n\n","category":"function"},{"location":"#Takens-best-estimate","page":"FractalDimensions.jl","title":"Takens best estimate","text":"","category":"section"},{"location":"","page":"FractalDimensions.jl","title":"FractalDimensions.jl","text":"takens_best_estimate_dim","category":"page"},{"location":"#FractalDimensions.takens_best_estimate_dim","page":"FractalDimensions.jl","title":"FractalDimensions.takens_best_estimate_dim","text":"takens_best_estimate_dim(X, εmax, metric = Chebyshev(), εmin = 0)\n\nUse the \"Takens' best estimate\" [Takens1985][Theiler1988] method for estimating the correlation dimension.\n\nThe original formula is\n\nDelta_C approx fracC(epsilon_textmax)int_0^epsilon_textmax(C(epsilon) epsilon) depsilon\n\nwhere C is the correlationsum and epsilon_textmax is an upper cutoff. Here we use the later expression\n\nDelta_C approx - frac1etaquad eta = frac1(N-1)^*sum_i j^*log(X_i - X_j epsilon_textmax)\n\nwhere the sum happens for all i j so that i j and X_i - X_j epsilon_textmax. In the above expression, the bias in the original paper has already been corrected, as suggested in [Borovkova1999].\n\nAccording to [Borovkova1999], introducing a lower cutoff εmin can make the algorithm more stable (no divergence), this option is given but defaults to zero.\n\nIf X comes from a delay coordinates embedding of a timseries x, a recommended value for epsilon_textmax is std(x)/4.\n\nYou may also use\n\nΔ_C, Δu_C, Δl_C = FractalDimensions.takens_best_estimate(args...)\n\nto obtain the upper and lower 95% confidence intervals. The intervals are estimated from the log-likelihood function by finding the values of Δ_C where the function has fallen by 2 from its maximum, see e.g. [Barlow] chapter 5.3.\n\n[Takens1985]: Takens, On the numerical determination of the dimension of an attractor, in: B.H.W. Braaksma, B.L.J.F. Takens (Eds.), Dynamical Systems and Bifurcations, in: Lecture Notes in Mathematics, Springer, Berlin, 1985, pp. 99–106.\n\n[Theiler1988]: Theiler, Lacunarity in a best estimator of fractal dimension. Physics Letters A, 133(4–5)\n\n[Borovkova1999]: Borovkova et al., Consistency of the Takens estimator for the correlation dimension. The Annals of Applied Probability, 9, 05 1999.\n\n[Barlow]: Barlow, R., Statistics - A Guide to the Use of Statistical Methods in the Physical Sciences. Vol 29. John Wiley & Sons, 1993\n\n\n\n\n\n","category":"function"},{"location":"#Kaplan-Yorke-dimension","page":"FractalDimensions.jl","title":"Kaplan-Yorke dimension","text":"","category":"section"},{"location":"","page":"FractalDimensions.jl","title":"FractalDimensions.jl","text":"kaplanyorke_dim","category":"page"},{"location":"#FractalDimensions.kaplanyorke_dim","page":"FractalDimensions.jl","title":"FractalDimensions.kaplanyorke_dim","text":"kaplanyorke_dim(λs::AbstractVector)\n\nCalculate the Kaplan-Yorke dimension, a.k.a. Lyapunov dimension (Kaplan and Yorke, 1979) from the given Lyapunov exponents λs.\n\nDescription\n\nThe Kaplan-Yorke dimension is simply the point where cumsum(λs) becomes zero (interpolated):\n\n D_KY = k + fracsum_i=1^k lambda_ilambda_k+1quad k = max_j left sum_i=1^j lambda_i 0 right\n\nIf the sum of the exponents never becomes negative the function will return the length of the input vector.\n\nUseful in combination with lyapunovspectrum from ChaosTools.jl.\n\n\n\n\n\n","category":"function"},{"location":"#Higuchi-dimension","page":"FractalDimensions.jl","title":"Higuchi dimension","text":"","category":"section"},{"location":"","page":"FractalDimensions.jl","title":"FractalDimensions.jl","text":"higuchi_dim","category":"page"},{"location":"#FractalDimensions.higuchi_dim","page":"FractalDimensions.jl","title":"FractalDimensions.higuchi_dim","text":"higuchi_dim(x::AbstractVector [, ks])\n\nEstimate the Higuchi dimension (Higuchi, 1988) of the graph of x.\n\nDescription\n\nThe Higuchi dimension is a number Δ ∈ [1, 2] that quantifies the roughness of the graph of the function x(t), assuming here that x is equi-sampled, like in the original paper.\n\nThe method estimates how the length of the graph increases as a function of the indices difference (which, in this context, is equivalent with differences in t). Specifically, we calculate the average length versus k as\n\nL_m(k) = fracN-1lfloor fracN-mk \rfloor k^2\nsum_i=1^lfloor fracN-mk rfloor X_N(m+ik)-X_N(m+(i-1)k) \n\nL(k) = frac1k sum_m=1^k L_m(k)\n\nand then use linear_region in -log2.(k) vs log2.(L) as per usual when computing a fractal dimension.\n\nThe algorithm chooses default ks to be exponentially spaced in base-2, up to at most 2^8. A user can provide their own ks as a second argument otherwise.\n\nUse FractalDimensions.higuchi_length(x, ks) to obtain L(k) directly.\n\n\n\n\n\n","category":"function"},{"location":"#Extreme-value-value-theory-dimension","page":"FractalDimensions.jl","title":"Extreme value value theory dimension","text":"","category":"section"},{"location":"","page":"FractalDimensions.jl","title":"FractalDimensions.jl","text":"extremevaltheory_dim\nextremevaltheory_dims_persistences\nextremevaltheory_local_dim_persistence\nextremal_index_sueveges\nestimate_gpd_parameters\nextremevaltheory_gpdfit_pvalues","category":"page"},{"location":"#FractalDimensions.extremevaltheory_dim","page":"FractalDimensions.jl","title":"FractalDimensions.extremevaltheory_dim","text":"extremevaltheory_dim(X::StateSpaceSet, p::Real; kwargs...) → Δ\n\nConvenience syntax that returns the mean of the local dimensions of extremevaltheory_dims_persistences, which approximates a fractal dimension of X using extreme value theory and quantile probability p.\n\nSee also extremevaltheory_gpdfit_pvalues for obtaining confidence on the results.\n\n\n\n\n\n","category":"function"},{"location":"#FractalDimensions.extremevaltheory_dims_persistences","page":"FractalDimensions.jl","title":"FractalDimensions.extremevaltheory_dims_persistences","text":"extremevaltheory_dims_persistences(x::AbstractStateSpaceSet, p::Real; kwargs...)\n\nReturn the local dimensions Δloc and the persistences θloc for each point in the given set for quantile probability p, according to the estimation done via extreme value theory (Lucarini et al., 2016). The computation is parallelized to available threads (Threads.nthreads()).\n\nSee also extremevaltheory_gpdfit_pvalues for obtaining confidence on the results.\n\nKeyword arguments\n\nshow_progress = true: displays a progress bar.\nestimator = :mm: how to estimate the σ parameter of the Generalized Pareto Distribution. The local fractal dimension is 1/σ. The possible values are: :exp, :mm, as in estimate_gpd_parameters.\ncompute_persistence = true: whether to aso compute local persistences θloc (also called extremal index). If false, θloc are NaNs.\nallocate_matrix = false: If true, the code calls a method that attempts to allocate an N×N matrix (N = length(X)) that stores the pairwise Euclidean distances. This method is faster due to optimizations of Distances.pairwise but will error if the computer does not have enough available memory for the matrix allocation.\n\nDescription\n\nFor each state space point mathbfx_i in X we compute g_i = -log(mathbfx_i - mathbfx_j ) forall j = 1 ldots N with cdot the Euclidean distance. Next, we choose an extreme quantile probability p (e.g., 0.99) for the distribution of g_i. We compute g_p as the p-th quantile of g_i. Then, we collect the exceedances of g_i, defined as E_i = g_i - g_p g_i ge g_p , i.e., all values of g_i larger or equal to g_p, also shifted by g_p. There are in total n = N(1-q) values in E_i. According to extreme value theory, in the limit N to infty the values E_i follow a two-parameter Generalized Pareto Distribution (GPD) with parameters sigmaxi (the third parameter mu of the GPD is zero due to the positive-definite construction of E). Within this extreme value theory approach, the local dimension Delta^(E)_i assigned to state space point textbfx_i is given by the inverse of the sigma parameter of the GPD fit to the data[Lucarini2012], Delta^(E)_i = 1sigma. sigma is estimated according to the estimator keyword.\n\nA more precise description of this process is given in the review paper (Datseris et al., 2023).\n\n\n\n\n\n","category":"function"},{"location":"#FractalDimensions.extremevaltheory_local_dim_persistence","page":"FractalDimensions.jl","title":"FractalDimensions.extremevaltheory_local_dim_persistence","text":"extremevaltheory_local_dim_persistence(X::StateSpaceSet, ζ, p::Real; kw...)\n\nReturn the local values Δ, θ of the fractal dimension and persistence of X around a state space point ζ. p and kw are as in extremevaltheory_dims_persistences.\n\n\n\n\n\n","category":"function"},{"location":"#FractalDimensions.extremal_index_sueveges","page":"FractalDimensions.jl","title":"FractalDimensions.extremal_index_sueveges","text":"extremal_index_sueveges(y::AbstractVector, p)\n\nCompute the extremal index θ of y through the Süveges formula for quantile probability p, using the algorithm of (Süveges, 2007).\n\n\n\n\n\n","category":"function"},{"location":"#FractalDimensions.estimate_gpd_parameters","page":"FractalDimensions.jl","title":"FractalDimensions.estimate_gpd_parameters","text":"estimate_gpd_parameters(X::AbstractVector{<:Real}, estimator::Symbol)\n\nEstimate and return the parameters σ, ξ of a Generalized Pareto Distribution fit to X, assuming that minimum(X) ≥ 0 and hence the parameter μ is 0 (if not, simply shift X by its minimum), according to the methods provided in Pons2023.\n\nThe estimator can be:\n\n:exp: Assume the distribution is exponential instead of GP and get σ from mean of X and set ξ = 0.\nmm: Standing for \"method of moments\", estimants are given by\nxi = (barx^2s^2 - 1)2 quad sigma = barx(barx^2s^2 + 1)2\nwith barx the sample mean and s^2 the sample variance. This estimator only exists if the true distribution ξ value is < 0.5.\n\n\n\n\n\n","category":"function"},{"location":"#FractalDimensions.extremevaltheory_gpdfit_pvalues","page":"FractalDimensions.jl","title":"FractalDimensions.extremevaltheory_gpdfit_pvalues","text":"extremevaltheory_gpdfit_pvalues(X, p; kw...)\n\nReturn various computed quantities that may quantify the significance of the results of extremevaltheory_dims_persistences(X, p; kw...), terms of quantifying how well a Generalized Pareto Distribution (GPD) describes exceedences in the input data.\n\nKeyword arguments\n\nshow_progress, estimator as in extremevaltheory_dims_persistences\nTestType = ApproximateOneSampleKSTest: the test type to use. It can be ApproximateOneSampleKSTest, ExactOneSampleKSTest, CramerVonMises. We noticed that OneSampleADTest sometimes yielded nonsensical results: all p-values were equal and were very small ≈ 1e-6.\nnbins = round(Int, length(X)*(1-p)/20): number of bins to use when computing the histogram of the exceedances for computing the NRMSE. The default value will use equally spaced bins that are equal to the length of the exceedances divided by 20.\n\nDescription\n\nThe function computes the exceedances E_i for each point x_i in X as in extremevaltheory_dims_persistences. It returns 5 quantities, all being vectors of length length(X):\n\nEs, all exceedences, as a vector of vectors.\nsigmas, xis the fitted σ, ξ to the GPD fits for each exceedance\nnrmses the normalized root mean square distance of the fitted GPD to the histogram of the exceedances\npvalues the pvalues of a statistical test of the appropriateness of the GPD fit\n\nThe output nrmses quantifies the distance between the fitted GPD and the empirical histogram of the exceedances. It is computed as\n\nNRMSE = sqrtfracsum(P_j - G_j)^2sum(P_j - U)^2\n\nwhere P_j the empirical (observed) probability at bin j, G_j the fitted GPD probability at the midpoint of bin j, and U same as G_j but for the uniform distribution. The divisor of the equation normalizes the expression, so that the error of the empirical distribution is normalized to the error of the empirical distribution with fitting it with the uniform distribution. It is expected that NRMSE < 1. The smaller it is, the better the data are approximated by GPD versus uniform distribution.\n\nThe output pvalues is a vector of p-values. pvalues[i] corresponds to the p-value of the hypothesis: \"The exceedences around point X[i] are sampled from a GPD\" versus the alternative hypothesis that they are not. To extract the p-values, we perform a one-sample hypothesis via HypothesisTests.jl to the fitted GPD. Very small p-values then indicate that the hypothesis should be rejected and the data are not well described by a GPD. This can be an indication that we do not have enough data, or that we choose too high of a quantile probability p, or that the data are not suitable in general. This p-value based method for significance has been used in [Faranda2017], but it is unclear precisely how it was used.\n\nFor more details on how these quantities may quantify significance, see our review paper.\n\n[Faranda2017]: Faranda et al. (2017), Dynamical proxies of North Atlantic predictability and extremes, Scientific Reports, 7\n\n\n\n\n\n","category":"function"},{"location":"#Theiler-window","page":"FractalDimensions.jl","title":"Theiler window","text":"","category":"section"},{"location":"","page":"FractalDimensions.jl","title":"FractalDimensions.jl","text":"The Theiler window is a concept that is useful when finding neighbors in a dataset that is coming from the sampling of a continuous dynamical system. Itt tries to eliminate spurious \"correlations\" (wrongly counted neighbors) due to a potentially dense sampling of the trajectory. Typically a good choice for w coincides with the choice an optimal delay time, see DelayEmbeddings.estimate_delay, for any of the timeseries of the dataset.","category":"page"},{"location":"","page":"FractalDimensions.jl","title":"FractalDimensions.jl","text":"For more details, see Chapter 5 of Nonlinear Dynamics, Datseris & Parlitz, Springer 2022.","category":"page"},{"location":"#StateSpaceSet-reference","page":"FractalDimensions.jl","title":"StateSpaceSet reference","text":"","category":"section"},{"location":"","page":"FractalDimensions.jl","title":"FractalDimensions.jl","text":"StateSpaceSet\nstandardize","category":"page"},{"location":"#StateSpaceSets.StateSpaceSet","page":"FractalDimensions.jl","title":"StateSpaceSets.StateSpaceSet","text":"StateSpaceSet{D, T} <: AbstractStateSpaceSet{D,T}\n\nA dedicated interface for sets in a state space. It is an ordered container of equally-sized points of length D. Each point is represented by SVector{D, T}. The data are a standard Julia Vector{SVector}, and can be obtained with vec(ssset::StateSpaceSet). Typically the order of points in the set is the time direction, but it doesn't have to be.\n\nWhen indexed with 1 index, StateSpaceSet is like a vector of points. When indexed with 2 indices it behaves like a matrix that has each of the columns be the timeseries of each of the variables. When iterated over, it iterates over its contained points. See description of indexing below for more.\n\nStateSpaceSet also supports almost all sensible vector operations like append!, push!, hcat, eachrow, among others.\n\nDescription of indexing\n\nIn the following let i, j be integers, typeof(X) <: AbstractStateSpaceSet and v1, v2 be <: AbstractVector{Int} (v1, v2 could also be ranges, and for performance benefits make v2 an SVector{Int}).\n\nX[i] == X[i, :] gives the ith point (returns an SVector)\nX[v1] == X[v1, :], returns a StateSpaceSet with the points in those indices.\nX[:, j] gives the jth variable timeseries (or collection), as Vector\nX[v1, v2], X[:, v2] returns a StateSpaceSet with the appropriate entries (first indices being \"time\"/point index, while second being variables)\nX[i, j] value of the jth variable, at the ith timepoint\n\nUse Matrix(ssset) or StateSpaceSet(matrix) to convert. It is assumed that each column of the matrix is one variable. If you have various timeseries vectors x, y, z, ... pass them like StateSpaceSet(x, y, z, ...). You can use columns(dataset) to obtain the reverse, i.e. all columns of the dataset in a tuple.\n\n\n\n\n\n","category":"type"},{"location":"#StateSpaceSets.standardize","page":"FractalDimensions.jl","title":"StateSpaceSets.standardize","text":"standardize(d::StateSpaceSet) → r\n\nCreate a standardized version of the input set where each column is transformed to have mean 0 and standard deviation 1.\n\n\n\n\n\nstandardize(x::AbstractVector{<:Real}) = (x - mean(x))/std(x)\n\n\n\n\n\n","category":"function"},{"location":"#References","page":"FractalDimensions.jl","title":"References","text":"","category":"section"},{"location":"","page":"FractalDimensions.jl","title":"FractalDimensions.jl","text":"","category":"page"}] +} diff --git a/v1.7.4/siteinfo.js b/v1.7.4/siteinfo.js new file mode 100644 index 0000000..69b2874 --- /dev/null +++ b/v1.7.4/siteinfo.js @@ -0,0 +1 @@ +var DOCUMENTER_CURRENT_VERSION = "v1.7.4"; diff --git a/versions.js b/versions.js index 58da214..35f3870 100644 --- a/versions.js +++ b/versions.js @@ -10,5 +10,5 @@ var DOC_VERSIONS = [ "v1.0", "dev", ]; -var DOCUMENTER_NEWEST = "v1.7.3"; +var DOCUMENTER_NEWEST = "v1.7.4"; var DOCUMENTER_STABLE = "stable";