{ localUrl: '../page/logistic_function.html', arbitalUrl: 'https://arbital.com/p/logistic_function', rawJsonUrl: '../raw/558.json', likeableId: '3013', likeableType: 'page', myLikeValue: '0', likeCount: '2', dislikeCount: '0', likeScore: '2', individualLikes: [ 'EricBruylant', 'TravisRivera' ], pageId: 'logistic_function', edit: '1', editSummary: '', prevEdit: '0', currentEdit: '1', wasPublished: 'true', type: 'wiki', title: 'Logistic function', clickbait: 'A monotonic function from the real numbers to the open unit interval.', textLength: '1470', alias: 'logistic_function', externalUrl: '', sortChildrenBy: 'likes', hasVote: 'false', voteType: '', votesAnonymous: 'false', editCreatorId: 'JoeZeng', editCreatedAt: '2016-07-07 01:33:37', pageCreatorId: 'JoeZeng', pageCreatedAt: '2016-07-07 01:33:37', seeDomainId: '0', editDomainId: 'AlexeiAndreev', submitToDomainId: '0', isAutosave: 'false', isSnapshot: 'false', isLiveEdit: 'true', isMinorEdit: 'false', indirectTeacher: 'false', todoCount: '0', isEditorComment: 'false', isApprovedComment: 'true', isResolved: 'false', snapshotText: '', anchorContext: '', anchorText: '', anchorOffset: '0', mergedInto: '', isDeleted: 'false', viewCount: '46', text: '[summary: The logistic function is a [-sigmoid] function that maps the [4bc real numbers] to the unit interval $(0, 1)$ using the formula $\\displaystyle f(x) = \\frac{1}{1 + e^{-x}}$ or variants of this formula.]\n\nThe logistic function is a [-sigmoid] function that maps the [4bc real numbers] to the unit interval $(0, 1)$ using the formula $\\displaystyle f(x) = \\frac{1}{1 + e^{-x}}$.\n\nMore generally, there exists a [family_of_functions family] of logistic functions that can be written as $\\displaystyle f(x) = \\frac{M}{1 + \\alpha^{c(x_0 - x)}}$, where:\n\n* $M$ is the upper bound of the function (in which case the function maps to the interval $(0, M)$ instead). When $M = 1$, the logistic function is usually being used to calculate some [-1rf] or [-4w3] of a total.\n\n* $x_0$ is the inflection point of the curve, or the value of $x$ where the function's growth stops speeding up and starts slowing down.\n\n* $\\alpha$ is a variable controlling the steepness of the curve.\n\n* $c$ is a scaling factor for the distance.\n\n## Applications\n\n* The logistic function is used to model growth rates of populations in an ecosystem with a limited carrying capacity.\n\n* The inverse logistic function (with $\\alpha = 2$) is used to convert a probability to log-odds form for use in [1lz].\n\n* The logistic function (with $\\alpha = 10$ and $c = 1/400$) is used to calculate the expected probability of a player winning given a specific difference in rating in the Elo rating system.', metaText: '', isTextLoaded: 'true', isSubscribedToDiscussion: 'false', isSubscribedToUser: 'false', isSubscribedAsMaintainer: 'false', discussionSubscriberCount: '2', maintainerCount: '2', userSubscriberCount: '0', lastVisit: '', hasDraft: 'false', votes: [], voteSummary: 'null', muVoteSummary: '0', voteScaling: '0', currentUserVote: '-2', voteCount: '0', lockedVoteType: '', maxEditEver: '0', redLinkCount: '0', lockedBy: '', lockedUntil: '', nextPageId: '', prevPageId: '', usedAsMastery: 'false', proposalEditNum: '0', permissions: { edit: { has: 'false', reason: 'You don't have domain permission to edit this page' }, proposeEdit: { has: 'true', reason: '' }, delete: { has: 'false', reason: 'You don't have domain permission to delete this page' }, comment: { has: 'false', reason: 'You can't comment in this domain because you are not a member' }, proposeComment: { has: 'true', reason: '' } }, summaries: {}, creatorIds: [ 'JoeZeng' ], childIds: [], parentIds: [ 'math' ], commentIds: [], questionIds: [], tagIds: [ 'start_meta_tag', 'needs_parent_meta_tag' ], relatedIds: [], markIds: [], explanations: [], learnMore: [], requirements: [], subjects: [], lenses: [], lensParentId: '', pathPages: [], learnMoreTaughtMap: {}, learnMoreCoveredMap: {}, learnMoreRequiredMap: {}, editHistory: {}, domainSubmissions: {}, answers: [], answerCount: '0', commentCount: '0', newCommentCount: '0', linkedMarkCount: '0', changeLogs: [ { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '16108', pageId: 'logistic_function', userId: 'EricBruylant', edit: '0', type: 'newTag', createdAt: '2016-07-07 23:53:03', auxPageId: 'needs_parent_meta_tag', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '16107', pageId: 'logistic_function', userId: 'EricBruylant', edit: '0', type: 'newParent', createdAt: '2016-07-07 23:52:27', auxPageId: 'math', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '15884', pageId: 'logistic_function', userId: 'JoeZeng', edit: '0', type: 'newTag', createdAt: '2016-07-07 01:36:08', auxPageId: 'start_meta_tag', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '15883', pageId: 'logistic_function', userId: 'JoeZeng', edit: '1', type: 'newEdit', createdAt: '2016-07-07 01:33:37', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' } ], feedSubmissions: [], searchStrings: {}, hasChildren: 'false', hasParents: 'true', redAliases: {}, improvementTagIds: [], nonMetaTagIds: [], todos: [], slowDownMap: 'null', speedUpMap: 'null', arcPageIds: 'null', contentRequests: {} }