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ExponentialModel[]

指数関数を表す．

ExponentialModel[vars]

明示的な変数指定 vars を使う．

ExponentialModel[pars,vars]

提供された係数 pars を使う．

詳細
Details and Options Details and Options
例題  
例  
スコープ  
変数  
パラメータ  
評価  
情報  
フィット  
アプリケーション  
基本的なアプリケーション  
人口増加  
関連項目
関連するガイド
履歴
引用書式

ExponentialModel

ExponentialModel[]

指数関数を表す．

ExponentialModel[vars]

明示的な変数指定 vars を使う．

ExponentialModel[pars,vars]

提供された係数 pars を使う．

詳細

  • ExponentialModelは，記号あるいは数値による評価とフィットに相応しい形式の，指定された変数による指数関数を表す．
  • 指数モデルは，成長，減衰，緩和，平衡への接近などのように，変化が現在の状態に比例する過程を記述する．
  • 一変量ExponentialModel[]は TemplateBox[{1}, CTraditional]+c_2 e^(c_3 x)としてパラメータ化される．
  • 多変量指数関数は TemplateBox[{1}, CTraditional]+TemplateBox[{2}, CTraditional] product_(i=3)^nexp(TemplateBox[{i}, CTraditional] x_i)としてパラメータ化される．
    • 変数
  • 指定されない場合，変数は自動的にx[i]を用いて列挙される．
  • 有効な変数指定 vars には次のものがある．
  • n変数の数
    symb単一の変数の記号的表現
    {symb1,…}記号変数のリスト
    • パラメータ
  • 指定されていない場合，パラメータは自動的にC[i]を用いて列挙される．
  • 形式{par1,…}による有効なパラメータ pars の指定には以下がある．
  • val固定されたパラメータ値 val
    par記号的なパラメータ名 par
    parval固定値 val に設定された記号名 par
    {par,val0}初期値 val0を持つ記号的なパラメータ名 par
    • 特性
  • モデルの特性はInformation[PowerModel[…],prop]を用いて抽出できる．
  • 有効な基本特性には以下がある．
  • "BaseType"モデルの基本型
    "Name"モデル名
    "ShortName"ラベルとして使用する短い識別子
    "InputType"サポートされる入力型
    "OutputType"サポートされる出力型
  • 有効なデータ関連の特性には以下がある．
  • "ColumnNames"入力特徴の名前
    "ColumnVariableMap"列名とモデル変数との対応付け
    "InputSize"入力の次元数
    "OutputSize"出力の次元数
    "Trainable"モデルが完全に指定されており学習可能であるかどうか
    "Trained"モデルが数値的に評価可能であるかどうか
    "VariableColumnMap"モデル変数と列名との対応付け
    "Variables"モデル変数の名称
  • 最良モデルに関連する特性には以下がある．
  • "Expression"モデル式
    "Function"純関数としてのモデル
    "SymbolicExpression"記号パラメータを含むモデル式
    "TabularFunction"表形式の行に対する動作に適した純関数
  • パラメータに関連する特性には以下がある．
  • "ParameterAssociation"パラメータ名と値の連想
    "ParameterCount"パラメータの数
    "ParameterInitialValues"フィットのための初期値
    "ParameterNames"パラメータ名
    "ParameterRules"パラメータ名と値からなる規則のリスト
    "Parameters"存在する場合はパラメータ値，それ以外の場合は名前
    "ParameterValues"パラメータ値
    "Constraints"パラメータの制約条件

例題

すべて開く すべて閉じる

例  (3)

一般的な指数関数モデルを指定する：

Wolfram Language code: ExponentialModel[]

記号パラメータに明示的な入力サイズとカスタム名を使う：

Wolfram Language code: ExponentialModel[{a, b, c}, 1]

定数パラメータを使用してさまざまな成長因子をプロットする：

Wolfram Language code: Plot[{ExponentialModel[{0, 1, 1}, 1][{x}], ExponentialModel[{1, 1, -1}, 1][{x}], ExponentialModel[{-0.5, 1, .5}, 1][{x}]}, {x, -1, 1}]

スコープ  (22)

変数  (6)

一般的なモデルを指定する：

Wolfram Language code: ExponentialModel[]

変数の数は引数から推測される：

Wolfram Language code: ExponentialModel[][{x}] ExponentialModel[][{x, y, z}]

ModelFitは，変数の数がデータ点の次元よりも1少ないと仮定する：

Wolfram Language code: ModelFit[{{0, 0, 0}, {1, 9, 55}, {1, 1, 7}, {1, 4, 25}, {8, 9, 118}, {8, 10, 124}, {8, 5, 94}}, ExponentialModel[]]

変数の数を指定する：

Wolfram Language code: ExponentialModel[2]

変数にカスタムの記号表現を与える：

Wolfram Language code: ExponentialModel[{var1, var2}]

これらの名前はモデルが評価される際に上書きされる：

Wolfram Language code: ExponentialModel[{var1, var2}][{a, b}]

多変量指数関数モデルを可視化する：

Wolfram Language code: Plot3D[Evaluate@ExponentialModel[{0, 1, -0.5, .4}, 2][{x, y}], {x, -1, 1}, {y, -1, 1}, ColorFunction -> "Rainbow"]

パラメータ  (4)

パラメータ名は自動的に割り当てられる：

Wolfram Language code: ExponentialModel[2]

カスタムのパラメータ名を指定する：

Wolfram Language code: model = ExponentialModel[ {c, A, k}, 1]

パラメータを特定の値に設定する：

Wolfram Language code: model = ExponentialModel[{0, 2, k}, 1]

パラメータの名前と値の両方を指定する：

Wolfram Language code: model = ExponentialModel[{c, a -> 2, k}, 1]

順序は，C[n]の自動列挙と一致する：

Wolfram Language code: ExponentialModel[][x]

評価  (5)

一変数モデルを記号的に評価する：

Wolfram Language code: ExponentialModel[1][x]

二変数モデルを記号的に評価する：

Wolfram Language code: ExponentialModel[2][{x, y}]

変数の数は，指定されていない場合は自動的に推測される：

Wolfram Language code: ExponentialModel[][x]
Wolfram Language code: ExponentialModel[][{x, y}]

複数の記号変数でモデルを評価する：

Wolfram Language code: ExponentialModel[2][{{x, y}, {a, b}}]

点のリストについてモデルを評価する：

Wolfram Language code: ExponentialModel[1][{{1}, {2}, {3}}]

情報  (5)

モデルについての一般的な情報を見る：

Wolfram Language code: Information[ExponentialModel[]]

情報の中には変数またはパラメータが完全に指定されていないと取得できないものがある：

Wolfram Language code: Information[ExponentialModel[{x, y}]]

単一の特性を抽出する：

Wolfram Language code: Information[ExponentialModel[{a, b, c}, x], "Variables"]

複数の特性を取り出す：

Wolfram Language code: Information[ExponentialModel[{a, b, c}, x], {"Variables", "Parameters"}]

デフォルトのモデル値についての情報を取得する：

Wolfram Language code: Information[ExponentialModel[1], {"Variables", "Parameters"}]

フィット  (2)

指数関数モデルをフィットする：

Wolfram Language code: ModelFit[{...}, ExponentialModel[]]

モデルをフィットするときは，パラメータに初期値を使う：

Wolfram Language code: ModelFit[{...}, ExponentialModel[{0, 1 / 3, Subscript[k, "x"], Subscript[k, "y"]}, {x, y}]]

アプリケーション  (9)

基本的なアプリケーション  (8)

抵抗を介して充電されるとき，コンデンサの両端の電圧は供給電圧に指数関数的に近付く：

Wolfram Language code: model = ModelFit[data -> "Voltage", ExponentialModel[]]
Wolfram Language code: model = ModelFit[Tabular[Association["RawSchema" -> Association["ColumnProperties" -> Association["Time" -> Association["ElementType" -> "Real64"], "Voltage" -> Association["ElementType" -> "Real64"]], "KeyColumns" -> None, "Backend" -> "WolframKernel"], "Options" -> {}, "BackendData" -> Association["ColumnData" -> DataStructure["ColumnTable", {{TabularColumn[Association["Data" -> {{5.838988328758781, 6.738191996308977, 6.19608830911937, 5.80789503296751, 6.250604171731046, 5.179844714242305, 8.721810088941064, 1.7205702996647743, 8.577864810423153, 0.7222558265533063, 4.338023011638031, 6.7412711375946515, 1.9644724536929292, 5.046552471939199, 0.9969698240844305, 4.337188925837601, 7.020937526546442, 8.335043162393749, 4.763353714547913, 0.5137760333079378, 5.231694534023026, 8.7522634979224, 6.782498605460079, 4.11807952344139, 1.2977900867029324, 2.3997274593090556, 0.15838896234651623, 3.757328348884501, 0.5384642946347684, 1.1995761528111237, 1.4004601296516905, 3.224695090678109, 4.0767559979435575, 4.517724446184081, 6.8587291812564555, 2.8433778568332735, 8.34314308472577, 1.8803062830582484, 5.133561559861124, 7.346765956160082, 3.108323252777292, 6.569156322953152, 5.285372338196844, 1.1778497669069798, 1.559562748897858, 3.6811765405561103, 0.6636163376087456, 9.374473666847713, 2.4112841981970567, 2.8381854361673264, 8.058768704225601, 7.697273935110871, 2.047669915806134, 2.9506849572499427, 5.081294282256998, 7.069862276305976, 4.772907835970401, 6.676727517173527, 6.46957405748386, 8.766026240140445, 0.6607648124870602, 0.583556799047793, 4.975208740425168, 8.661731830268737, 1.6964602183192867, 2.6692931146107646, 6.4338767780326815, 4.868275843700294, 5.559284856045961, 6.068822400277101, 4.77948482866273, 5.639648265365826, 0.10472243170756235, 5.948881449956518, 2.9549286836210653, 8.383000763138773, 6.2399090997234445, 4.660007510168757, 0.9996706653056076, 5.994272170067719, 9.246351546390024, 1.8966964723831725, 0.475840391126654, 8.54632535772652, 4.560884513302382, 5.387913714615083, 6.302598648702816, 9.958759058383801, 1.7813742791318932, 7.316117239291017, 2.813291724138469, 9.809682227983348, 7.708060380700297, 9.028271919774538, 9.47123012552268, 7.095565769183219, 7.36683153755493, 2.467582202007166, 6.864612386346513, 5.0949618589117}, {}, None}, "ElementType" -> "Real64"]], TabularColumn[Association["Data" -> {{4.6966608878120155, 4.824312283883162, 4.78298109642428, 4.756908568314992, 4.795744206846373, 4.613573845705569, 5.036994622775709, 2.916493844373077, 4.941313497302164, 1.5658324440895717, 4.468553323719556, 4.787026785119092, 3.1330852522191046, 4.628386572976896, 2.01823141048421, 4.38772190469146, 4.794182509880719, 4.94913772577836, 4.521654973899749, 1.13220805192281, 4.562686218087409, 5.0063007968155055, 4.84214812366535, 4.314622952991398, 2.377299024846298, 3.5366648635880518, 0.3316489032278883, 4.220801929940632, 1.1986612361121582, 2.257225037459695, 2.56891695064928, 4.018350303759731, 4.387623581517519, 4.426518511935061, 4.863743038403332, 3.8817893471837897, 4.92080578245386, 3.0626310414609854, 4.617690614645442, 4.895675350806012, 4.003850885196173, 4.813508131823607, 4.6545568660916175, 2.2160045105389683, 2.7871854863828465, 4.2544136152602325, 1.4259331730637714, 4.8777533949039, 3.448987468376241, 3.811922409119405, 4.908711028902273, 4.945361180959812, 3.188539248063974, 3.9448004431009607, 4.5850582217453875, 4.867136949072108, 4.545185876088478, 4.875382578568834, 4.877041702088289, 4.944054324845519, 1.3976201021089794, 1.1471376885932159, 4.571544097879094, 4.94616066465977, 2.8335401861601826, 3.678572317281462, 4.804891531810396, 4.652652580553596, 4.747792640610623, 4.763850799884042, 4.5503630336586625, 4.676774078885084, 0.2779261670640416, 4.725688518479375, 3.9127058431000403, 4.912771073391018, 4.706956433628692, 4.368940658754038, 1.9889028914188887, 4.757798616779952, 4.848641841535179, 3.011532716133896, 1.0434839585621511, 4.94165839642236, 4.470539925798977, 4.595770638864039, 4.719122945110612, 4.927458297680816, 2.954987997076547, 4.875258039312025, 3.832285369585581, 4.90716312590821, 4.886081978517763, 5.014719857724921, 4.995918853828975, 4.8432863841043785, 4.978770965631156, 3.565109756764496, 4.814056522876913, 4.662826256104416}, {}, None}, "ElementType" -> "Real64"]]}}]]]] -> "Voltage", ExponentialModel[]]

記号的な結果と比較する：

Wolfram Language code: DSolveValue[{v'[t] == (V - v[t]) / (R C), v[0] == 0}, v[t], t]//Simplify

電圧ステップの後，インダクタを流れる電流は指数関数的に増加する：

Wolfram Language code: ModelFit[Tabular[Association["RawSchema" -> Association["ColumnProperties" -> Association["Time" -> Association["ElementType" -> "Real64"], "Current" -> Association["ElementType" -> "Real64"]], "KeyColumns" -> None, "Backend" -> "WolframKernel"], "Options" -> {}, "BackendData" -> Association["ColumnData" -> DataStructure["ColumnTable", {{TabularColumn[Association["Data" -> {{2.212919562876545, 1.1561178651027557, 9.661722821647654, 6.826158647065498, 8.467111066389252, 11.145589431454844, 4.0164328953783865, 3.1487457319666925, 5.967426177157303, 6.283330889447044, 6.559183506313567, 10.450766717688143, 5.388716469992724, 1.6407687545928509, 3.826826905357912, 4.020358188801509, 4.548356995042589, 1.0849943278070437, 9.627205118992809, 3.8495069668218598, 8.566874025173, 6.291687181071698, 7.185408538789895, 7.980988880384784, 6.582084043257244, 1.2450427735681382, 8.974161517773332, 2.6353694334312188, 4.4538041987787125, 7.035649949951655, 9.386210508571185, 8.398856899989845, 10.003345233853596, 5.340290471089001, 7.244720411501349, 7.426451343115991, 6.383335934136449, 11.766832240863653, 8.85995169534161, 10.613525352079126, 4.872899526511935, 0.1652475223955232, 11.821063139895735, 1.4432347542820132, 11.061328130710752, 1.3820234120557364, 0.22633214192125362, 7.557216260097409, 2.315817437202721, 9.067431498252416, 5.050004162603553, 11.645684857880571, 0.1718686672636922, 11.85539333888126, 10.190933888844167, 0.26517384139682587, 10.394316766180971, 9.739297683514028, 4.168599857982254, 2.739237528563864, 10.957131101468, 9.47443117614488, 3.9965251699120907, 2.626624077265528, 7.438837801707518, 9.580839897248143, 2.678984329416374, 4.154218756176009, 2.0797307235947065, 11.591582737904774, 3.836199061728779, 10.294469622826554, 5.146326790249952, 2.6572834034560007, 1.5110411278935683, 7.8585557765629055, 10.816602932120025, 11.514701763813106, 4.749624619990653, 2.8522849238016352, 11.560984001295422, 10.068374960966082, 1.3355721532621718, 5.044639825120428, 6.726474733164955, 10.148747709592602, 5.611651234603483, 7.0168978218759595, 6.414187894270681, 1.4376410136000457, 3.6664733145346844, 10.053652863953754, 5.590928658805095, 9.390482036277543, 4.115232724634687, 4.151777955416745, 9.911744741917094, 11.86938814669364, 4.582117504682287, 10.393489202778664}, {}, None}, "ElementType" -> "Real64"]], TabularColumn[Association["Data" -> {{1.0665008742997244, 0.6386618092410143, 1.9436105544589466, 1.833499561145964, 1.848070231829321, 1.9364418233380745, 1.4946079343849783, 1.3589508051999768, 1.67038444434536, 1.7143473483507539, 1.8070907891709187, 1.9338260576306177, 1.6022528986934232, 0.8088201687479399, 1.4471941571513358, 1.4218883137270468, 1.5512780370534607, 0.608798991749496, 1.9315167727110125, 1.4486833540267487, 1.8149399878825763, 1.8243044836066598, 1.7713553729895581, 1.8450521781067724, 1.8161542338630203, 0.6611099247728978, 1.9236034588876358, 1.1773420081560788, 1.5852709514564753, 1.761745503973351, 1.9713197207537436, 1.8934220515923166, 1.9730931068304023, 1.6685648946993237, 1.861434900646372, 1.8576027502955512, 1.7356082879859034, 1.952183949113016, 1.8999864335540495, 1.9796378836122033, 1.6459722068534972, 0.09852052635097334, 1.9614000003218306, 0.7329141354311155, 1.9575330118158254, 0.7278702063583689, 0.16088488001294152, 1.8779113354784576, 1.042939443275763, 1.9354476467896247, 1.643003214921751, 1.9795728729175788, 0.12164932475879965, 1.9798762378593187, 1.8907893793343777, 0.1365595785936381, 2.0204593147517516, 1.887184249892198, 1.5107753726319177, 1.2172315334271315, 2.0309469895745673, 1.8943944994563064, 1.4765579324396538, 1.1915375772639434, 1.8650320160310292, 1.8823758285754013, 1.1738779508292967, 1.4732091065264568, 0.9768855265028001, 1.9367302180113446, 1.430449451894662, 1.8803286404996058, 1.6176480308898658, 1.2060017007782775, 0.7748884938391409, 1.889666631624872, 1.9681112294397822, 1.9756262311259676, 1.583234277859401, 1.2249936977408697, 1.9838519874468457, 1.946052038831788, 0.695508691588814, 1.6162550464295729, 1.812447513633879, 1.932406718536364, 1.7382107669962683, 1.807284572204545, 1.7343703250707732, 0.7269834339065654, 1.4111610222679576, 2.0188002854874196, 1.6652293961390294, 1.9521817654946807, 1.4915165887846673, 1.509758908636861, 1.9908931952073223, 1.9572957214845867, 1.593302883046879, 1.9105412685144856}, {}, None}, "ElementType" -> "Real64"]]}}]]]] -> "Current", ExponentialModel[]]

物体の温度が周囲温度へ緩和する現象（ニュートンの冷却の法則）:

Wolfram Language code: ModelFit[Tabular[Association["RawSchema" -> Association["ColumnProperties" -> Association["Time" -> Association["ElementType" -> "Real64"], "Temperature" -> Association["ElementType" -> "Real64"]], "KeyColumns" -> None, "Backend" -> "WolframKernel"], "Options" -> {}, "BackendData" -> Association["ColumnData" -> DataStructure["ColumnTable", {{TabularColumn[Association["Data" -> {{1.285645967204907, 1.8712787246468885, 11.204913120079635, 0.9327675678511369, 15.425846238297659, 2.9432446478631284, 10.011174830279373, 14.280294208759141, 8.089061193735901, 3.4244126868497844, 10.42342957773948, 7.973059594729719, 0.5372116237468916, 6.445844934336577, 6.66719502988666, 14.240768031750843, 10.844702228069945, 3.0667952718343727, 7.193403198379748, 0.29756995361034555, 15.608090583002806, 9.291183862792476, 7.62477376237911, 14.692944707864182, 12.552080071979965, 1.5346289046881942, 8.585111582053713, 13.054109009600602, 10.71008665953136, 6.332958566980185, 9.912939739931236, 4.576546987021143, 3.653080870032202, 0.9270434000841128, 7.379653234880006, 2.5405786255933798, 14.425413094084231, 0.9317115372247606, 13.229840010706376, 12.182056264647791, 9.119997251194285, 8.677477533055857, 4.004294665303213, 14.098585500854991, 10.031476523129466, 12.052388613047434, 12.909548111079893, 14.494802283521, 8.149656047986014, 10.176429009076315, 5.995558871479105, 15.16337259593097, 15.059708147230058, 1.9234806507360602, 7.938169895939332, 4.604105582828073, 7.991512971771002, 3.592113657325992, 9.588668547317948, 12.02664466024801, 4.62761387609515, 7.987874307160666, 12.788743684802355, 6.690228322992255, 12.623802250712178, 6.819978472585561, 13.905358646884721, 10.90934225823441, 10.924063936535454, 0.8364614241072523, 15.876312926368634, 0.7521077420230888, 7.66127288304337, 7.552282274727435, 12.269911623115693, 3.899624081900644, 1.1660160923403495, 7.134024357742096, 15.175041619145833, 1.8418095621713917, 3.920421090264451, 10.719842385440703, 8.119348627747243, 9.546113706664233, 1.652998428395179, 13.962108809843933, 10.566290312888786, 0.08949753833467611, 10.16410005817507, 12.144472733474089, 4.3816421577122036, 5.24049016407055, 5.508739443567649, 6.986317602544833, 11.543622944961342, 8.270970639343538, 10.193783246820974, 7.985809127726881, 6.025896132194127, 3.95526363138357}, {}, None}, "ElementType" -> "Real64"]], TabularColumn[Association["Data" -> {{78.05020850105669, 70.8225882348392, 25.528868677817165, 83.27735387285071, 22.31953168527481, 59.14099455875653, 26.111202054083034, 22.382257784187438, 30.766308316372616, 54.165901851471915, 25.30921485543199, 31.648429614857086, 90.1538121117141, 36.299059739996984, 35.33655386537939, 22.135130644890644, 25.128348426862527, 57.70143330964701, 33.04098711634308, 93.21245225123337, 22.10534186887484, 27.191641685009948, 31.62956397818126, 22.95084548553296, 23.143025802376787, 75.03463349383095, 28.903097053526764, 22.10925110407935, 26.34973245548435, 38.821573983622, 25.740434554811287, 45.25062689348684, 52.46101264313618, 83.84579441335809, 33.49672737240911, 63.04975987493146, 22.33191979466187, 83.50125812808702, 23.586326731912866, 23.058956643367477, 27.79430505314852, 29.275782990720433, 49.489013764290945, 21.85520284114316, 26.800578757085336, 23.45653321835535, 23.93809230445902, 21.17388990903688, 31.56552027568414, 27.053648655404945, 38.40592932298102, 21.06603242779303, 21.99953514955764, 68.30385780859135, 31.37893367252733, 46.077474393949956, 29.783207984440097, 53.99810537652909, 27.106616138754134, 23.914213289468872, 44.49794583443664, 30.940142551782383, 24.325792094125546, 35.301020236621405, 22.702910755791315, 34.38021657314126, 22.976990522797234, 25.54876041677832, 24.45165264585479, 85.13497202963997, 21.102283124359335, 86.74034923903888, 31.703919833212385, 31.98256081466666, 24.018515791028754, 50.59571322635378, 79.46145275061856, 33.17147195610933, 22.364562553116425, 70.184473329525, 50.37470527789668, 26.006777958486055, 30.208350984038194, 27.491179054506325, 73.13020074022178, 21.944446544272402, 24.661303328092785, 98.6251899480942, 26.406881471436087, 24.062356153732413, 46.57404944608977, 42.025084404469574, 39.89434722507042, 32.60072888558425, 23.776124586255563, 30.168343795784224, 26.20609727030415, 30.85340262489411, 37.47373198632847, 49.36871094213025}, {}, None}, "ElementType" -> "Real64"]]}}]]]] -> "Temperature", ExponentialModel[]]

これは を解く：

Wolfram Language code: DSolveValue[{T'[t] == -(T[t] - Tenv) k, T[0] == T0}, T[t], t]//FullSimplify

一定の崩壊率を持つ放射性崩壊：

Wolfram Language code: ModelFit[Tabular[Association["RawSchema" -> Association["ColumnProperties" -> Association["Time" -> Association["ElementType" -> "Real64"], "Count" -> Association["ElementType" -> "Integer64"]], "KeyColumns" -> None, "Backend" -> "WolframKernel"], "Options" -> {}, "BackendData" -> Association["ColumnData" -> DataStructure["ColumnTable", {{TabularColumn[Association["Data" -> {{5.337700872089618, 10.98576811973911, 4.805041489332591, 1.8462731232197358, 5.314719883206279, 6.966733179599361, 0.9079259418314463, 4.550936233145684, 11.426694017193348, 6.708552586686331, 2.9597261863671545, 11.721311664412262, 5.458467631474851, 5.524725163560618, 11.286658688051173, 1.4340256719994962, 0.41644631569252866, 10.53714004334317, 8.37894164825412, 0.6944931314932488, 11.02177954943916, 11.914272411327754, 9.326687329175632, 0.8928551930058459, 4.647344748884753, 6.331444529701435, 7.320152824583512, 8.850486225114814, 9.45161594252961, 9.880598614130706, 9.374963369294719, 10.759224132438202, 7.832053677051045, 9.11984558250973, 6.446139970335793, 10.66008018455055, 1.7333039526394254, 6.703337153811383, 0.3945522750920123, 6.458424891243147, 6.997662843593305, 1.9977328636566192, 8.1187000749271, 9.237596115422544, 2.88870206966787, 3.498272876518028, 6.830524415633669, 10.252589705165105, 5.671786064321341, 11.406467227303846, 8.90071377141832, 11.667815108509963, 9.213884564762875, 1.4033388098120527, 9.668321942846665, 2.6449063847494543, 6.82212436876689, 4.997421343696847, 9.40149120995925, 7.617539156741947, 9.43787340901295, 4.8841158068762835, 2.827927027754461, 3.6276194321595536, 11.65130485956078, 11.233227684163374, 11.58443395995014, 10.564189184918401, 2.448868100820172, 11.936733190271905, 9.114357627148422, 1.729146365951836, 0.5097798708455858, 11.111897684270065, 6.222613438425448, 1.2216887548752506, 6.2893532033164945, 2.404995400986097, 0.6707515572483, 1.2866166925373452, 7.580194862656924, 11.57368819585116, 10.72966374687443, 7.605923897504848, 6.803855583242103, 8.772417720721972, 6.808543381706475, 7.103089487653111, 4.31028324973492, 6.544553430243834, 6.853991078805163, 0.5024427107244032, 9.515324659604346, 9.366693286505601, 8.768383655504078, 0.6685711219533257, 3.4362812454297345, 5.832347470115885, 1.0891060445212588, 3.207405066360627}, {}, None}, "ElementType" -> "Real64"]], TabularColumn[Association["Data" -> {{17, 0, 20, 55, 17, 9, 75, 21, 5, 9, 36, 3, 17, 15, 0, 62, 88, 2, 6, 80, 2, 1, 7, 75, 23, 16, 9, 5, 5, 0, 3, 4, 7, 3, 13, 4, 56, 10, 88, 13, 11, 50, 6, 4, 39, 31, 13, 2, 17, 3, 5, 2, 3, 64, 4, 43, 8, 19, 4, 11, 3, 21, 35, 30, 3, -1, 4, 1, 45, 1, 8, 56, 88, 2, 11, 67, 12, 47, 83, 66, 8, 5, 3, 8, 11, 5, 10, 10, 22, 9, 12, 84, 3, 6, 5, 82, 31, 13, 68, 32}, {}, None}, "ElementType" -> "Integer64"]]}}]]]] -> "Count", ExponentialModel[]]

この崩壊は を解く：

Wolfram Language code: DSolveValue[{n'[t] == -n[t]λ, n[0] == n0}, n[t], t]//FullSimplify

吸収媒質を通過する際に減衰する光強度：

Wolfram Language code: ModelFit[Tabular[Association["RawSchema" -> Association["ColumnProperties" -> Association["Distance" -> Association["ElementType" -> "Real64"], "Intensity" -> Association["ElementType" -> "Real64"]], "KeyColumns" -> None, "Backend" -> "WolframKernel"], "Options" -> {}, "BackendData" -> Association["ColumnData" -> DataStructure["ColumnTable", {{TabularColumn[Association["Data" -> {{1.452072527878494, 1.236002618200014, 1.6049896474976348, 0.2572655864506954, 0.6466459381956104, 1.4426923888274725, 0.6166120576468872, 0.9919963114765205, 2.074133448624451, 2.695164128026949, 2.9597688892220857, 1.2548696788941047, 2.2383239500128678, 3.259021505533342, 3.9894902480975185, 3.662721253875856, 2.826836314522512, 1.2527551187076273, 1.4411293120920687, 1.9588879650737328, 2.3396740178237394, 0.44298854498977924, 1.919523894214377, 2.510061970200727, 1.140562029973414, 1.5716767766502286, 3.0823675388887626, 1.3640653689539706, 2.1976710715437706, 1.4412374379880522, 1.8383538695684587, 1.9625931764074167, 3.02826164379191, 1.3113769013762733, 0.1938044031019226, 3.3043083735619296, 1.7337260256493705, 2.6975957834457747, 3.567353509804118, 2.8958514353552323, 0.8085759118739597, 2.6805644163983224, 3.820484470623575, 1.6484646059651906, 3.2375019586939704, 1.100767532067179, 2.0772846723721496, 0.9168188164788091, 2.722323714490166, 1.6872661212769557, 3.516632900963879, 2.663294978386836, 3.7916830729619626, 3.276443183593198, 2.8687796629622957, 0.665086712003621, 0.387298278816548, 2.999777220698493, 3.539622127579517, 3.7765269444122778, 2.1246834673085466, 1.1312021198362228, 0.5811036771660483, 3.57246708057058, 2.345014146876916, 0.915910279464466, 2.1659524109402355, 1.9560422171544865, 2.836763011197033, 2.5425482021339825, 0.49511030510964193, 0.03267677772872091, 1.1142722643459662, 2.343913579729233, 0.2822150990189911, 1.6746142776572732, 0.02130691518614558, 3.422739173996037, 1.1530032721384798, 0.22845067299486388, 2.924537877540156, 2.629183020060207, 3.664283518040702, 1.0536331317075796, 0.4182010958204616, 1.7792388177153677, 0.057637213108190366, 2.8407248398150013, 1.857964674791762, 2.338551195559705, 1.9372626707946665, 1.919905159244359, 0.6581844990845598, 1.7126880909361768, 0.9121804677244478, 3.9812372507975624, 1.9928466043212314, 0.9414606299346318, 1.9267632545704476, 3.553700399001892}, {}, None}, "ElementType" -> "Real64"]], TabularColumn[Association["Data" -> {{1.604897235399745, 2.0775386552481336, 1.3356210678299627, 7.143050218569366, 4.382782015967245, 1.4856336750436312, 4.495651731978334, 2.9269347018062097, 0.6396363511905463, 0.2891798498897677, 0.3817686572206104, 1.9845867760579046, 0.6476176914670387, 0.055674596734077986, -0.022860120546744606, 0.0566077785024775, 0.32575684406482236, 2.0290907757286947, 1.6764192769271722, 0.8837658468112412, 0.5654922792374334, 5.73556809497845, 1.0141484977072548, 0.5501188014186955, 2.540580244101438, 1.3198444227114672, 0.24674544381953623, 1.7476471480519251, 0.5449689311228909, 1.6495572005394414, 1.1080553256240215, 0.8373927070214104, 0.17490254256186255, 1.8351422652163643, 7.705576343665908, 0.20999704582285228, 1.0181410382457172, 0.47929377996102135, 0.0634271685493516, 0.18587860919923502, 3.525124095336061, 0.3833746107490176, 0.0899048137352011, 1.390930744561271, 0.26776003625046263, 2.5376563427692025, 0.7085217355400356, 3.130869670995621, 0.38286540693305826, 1.3293094388864, 0.08250893592593433, 0.39557369984105584, -0.013416603903613308, 0.20224366671873298, 0.19902420440566662, 4.300361948009942, 6.059234572729483, 0.23923376688126652, 0.18562122009025311, 0.05066517392580174, 0.7456860064152544, 2.507995789152173, 4.863714970678542, 0.09582001731152298, 0.5443489769514683, 3.1983092956128196, 0.6850239789302458, 0.960264823852436, 0.3113149038906888, 0.5413770705096295, 5.319419427957398, 9.574118440829361, 2.523334569105736, 0.44348326571165675, 6.947404859649886, 1.374554684588511, 9.697485475502367, 0.18056519115643674, 2.2585428193043775, 7.515345648454695, 0.10854566388189424, 0.4956926498734075, 0.16293670868058785, 2.7699734367649462, 5.856767702689511, 1.0309442150569106, 9.27136679277107, 0.2867262714797627, 0.9308659403898156, 0.5162509725295958, 0.8870448910204113, 0.8342012000293125, 4.3173058894531575, 1.0948764716024175, 3.0592769106867332, 0.12728315589910646, 0.8156723950681163, 3.1637694775619063, 0.7531656153817168, 0.040717721908193144}, {}, None}, "ElementType" -> "Real64"]]}}]]]] -> "Intensity", ExponentialModel[]]

これはBeer–Lambertの法則であり，を解く：

Wolfram Language code: DSolveValue[{i'[x] == -k i[x], i[0] == i0}, i[x], x]//FullSimplify

出生率が人口規模に比例する場合の人口増加：

Wolfram Language code: ModelFit[Tabular[Association["RawSchema" -> Association["ColumnProperties" -> Association["Time" -> Association["ElementType" -> "Real64"], "Population" -> Association["ElementType" -> TypeSpecifier["Quantity"]["Real64", "People"]]], "KeyColumns" -> None, "Backend" -> "WolframKernel"], "Options" -> {}, "BackendData" -> Association["ColumnData" -> DataStructure["ColumnTable", {{TabularColumn[Association["Data" -> {{1., 1.0707070707070707, 1.1414141414141414, 1.2121212121212122, 1.2828282828282829, 1.3535353535353536, 1.4242424242424243, 1.494949494949495, 1.5656565656565657, 1.6363636363636362, 1.7070707070707072, 1.7777777777777777, 1.8484848484848486, 1.9191919191919193, 1.98989898989899, 2.0606060606060606, 2.1313131313131315, 2.2020202020202024, 2.2727272727272725, 2.3434343434343434, 2.4141414141414144, 2.484848484848485, 2.555555555555556, 2.6262626262626263, 2.6969696969696972, 2.7676767676767677, 2.8383838383838387, 2.909090909090909, 2.97979797979798, 3.0505050505050506, 3.121212121212121, 3.191919191919192, 3.262626262626263, 3.3333333333333335, 3.4040404040404044, 3.474747474747475, 3.5454545454545454, 3.6161616161616164, 3.686868686868687, 3.757575757575758, 3.8282828282828287, 3.8989898989898997, 3.9696969696969697, 4.040404040404041, 4.111111111111112, 4.181818181818182, 4.252525252525253, 4.3232323232323235, 4.3939393939393945, 4.4646464646464645, 4.5353535353535355, 4.606060606060606, 4.676767676767677, 4.747474747474747, 4.818181818181818, 4.888888888888889, 4.95959595959596, 5.03030303030303, 5.101010101010101, 5.171717171717172, 5.242424242424242, 5.313131313131314, 5.383838383838384, 5.454545454545455, 5.525252525252526, 5.595959595959596, 5.666666666666667, 5.737373737373738, 5.808080808080809, 5.878787878787879, 5.94949494949495, 6.020202020202021, 6.090909090909091, 6.161616161616163, 6.232323232323233, 6.303030303030304, 6.373737373737374, 6.444444444444445, 6.515151515151516, 6.5858585858585865, 6.6565656565656575, 6.7272727272727275, 6.7979797979797985, 6.868686868686869, 6.9393939393939394, 7.01010101010101, 7.080808080808081, 7.151515151515152, 7.222222222222222, 7.292929292929294, 7.363636363636364, 7.434343434343434, 7.505050505050506, 7.575757575757576, 7.646464646464647, 7.717171717171718, 7.787878787878788, 7.858585858585859, 7.92929292929293, 8.}, {}, None}, "ElementType" -> "Real64"]], TabularColumn[ Association["Data" -> {100, {{{70.55680312304183, 68.43429018551174, 67.98732941144834, 68.17841236873511, 68.3698558756711, 70.68727838964895, 72.9884034537681, 69.50936910497005, 74.04199279765051, 80.02291182305879, 76.91707523019356, 82.68139487918276, 72.71249966265012, 82.06983299916436, 82.00175313779532, 79.06736300952436, 91.20374767487829, 91.92072518473292, 90.8940188483398, 90.73022909581155, 95.85206974037209, 99.32234050444407, 93.93029213658433, 96.40436282268392, 96.54164496641631, 99.06344690441688, 102.56994928384833, 105.10631351991887, 108.14795425739567, 106.93723957537912, 107.42273704159568, 113.0377942250788, 115.75496900462949, 109.39373978200138, 114.93254911012723, 116.25602605862963, 124.49850108688159, 121.46260126519319, 123.63573934247873, 121.19020468660436, 135.81316560569795, 134.6612789363785, 132.46706021925127, 138.8210957758649, 135.81764308831848, 138.22335946629926, 141.8360647328008, 148.56113222201924, 146.7501805273003, 155.27982016136247, 156.38954846457213, 154.31148934185944, 160.13708297107908, 164.01255547926567, 165.29406231736908, 167.88833012172563, 169.36662442082832, 177.01230580833453, 181.76037817280226, 187.01310878237487, 181.6440740185259, 189.79047247656393, 192.69304332863456, 193.94043510855846, 201.8205416374228, 207.0595447377911, 206.79430131119616, 209.821765011897, 216.03558605388085, 217.8628606309387, 220.4955459603509, 226.57455664329007, 226.42045988001965, 238.25271527902865, 241.94175878128738, 242.5449188810773, 242.5526454499557, 252.55076405051867, 258.93336516137794, 260.07748514711767, 265.59090633338405, 274.3020535365378, 272.6444276115051, 281.9026617823347, 278.2618114228327, 285.73354726097597, 296.8215376455581, 301.18178862264284, 299.63288199147127, 304.96292101840135, 315.5582993777776, 319.53362401268083, 326.5468654596027, 333.9922414727986, 335.90311656931965, 343.3905818039281, 350.2084688607777, 357.4993226120938, 363.35184744775376, 368.83910531219584}, {}, None}}, None}, "ElementType" -> TypeSpecifier["Quantity"][ "Real64", "People"]]]}}]]]] -> "Population", ExponentialModel[]]

初期の指数増殖段階における細菌集団：

Wolfram Language code: ModelFit[Tabular[Association["RawSchema" -> Association["ColumnProperties" -> Association["Time" -> Association["ElementType" -> "Real64"], "Bacteria" -> Association["ElementType" -> "Integer64"]], "KeyColumns" -> None, "Backend" -> "WolframKernel"], "Options" -> {}, "BackendData" -> Association["ColumnData" -> DataStructure["ColumnTable", {{TabularColumn[Association["Data" -> {{0.018327604515882424, 0.0516260679109893, 0.11774919627914182, 0.1880822583845465, 0.24205543787556605, 0.3124572637436951, 0.3819473823033358, 0.4169329022292048, 0.49994295585481074, 0.5334248241483606, 0.6190146508950924, 0.6219225221641801, 0.6926402999850353, 0.7713055420125987, 0.8403536972386229, 0.8967002399742096, 0.9628144311228186, 1.0293333932510849, 1.095002997813873, 1.1430741170465453, 1.2207002531852589, 1.2817760695823721, 1.3265232227905974, 1.3786622474460304, 1.4631361819206263, 1.526134332045102, 1.5715703670113716, 1.6417958657616547, 1.6876592817405272, 1.7536213545221695, 1.8229231424239272, 1.875326221272049, 1.9284570305134368, 1.997191132277343, 2.0455448706193806, 2.125091766107617, 2.174872286783137, 2.228979048560695, 2.2939925180278053, 2.353635512493176, 2.412343136124892, 2.4860852106033042, 2.544265662788614, 2.611766674040443, 2.674343895693982, 2.730990062788041, 2.7861231606293337, 2.847446808490808, 2.908062025066122, 2.945653248509133, 3.037607668053089, 3.101012884794662, 3.1457095743402874, 3.2045344320724873, 3.2855512862453797, 3.334525912522707, 3.3838741268528696, 3.447300111893052, 3.522422641762201, 3.5856239838462955, 3.6421824449851403, 3.684930107989741, 3.7329522613176627, 3.8259671897831167, 3.8671274077353206, 3.937812684190679, 3.9971029034056826, 4.059287582161241, 4.135304944040231, 4.186373677799128, 4.242661631443246, 4.298473455095522, 4.3745775981127, 4.423097479865286, 4.48801000073051, 4.551085108444547, 4.606074403727251, 4.664752997595886, 4.743294948700719, 4.793142916658574, 4.840419021818262, 4.909207462972745, 4.973515816564539, 5.0209678617744276, 5.085379455240456, 5.15122553411187, 5.202179717608479, 5.280046217897273, 5.315491087032563, 5.402105846547357, 5.46566252133749, 5.505912611361974, 5.594724623953701, 5.642392941048236, 5.695051492485362, 5.74655265817626, 5.822352701636927, 5.870607321034391, 5.950572414450936, 5.9953249612295165}, {}, None}, "ElementType" -> "Real64"]], TabularColumn[Association["Data" -> {{212, 196, 229, 215, 235, 226, 228, 227, 225, 258, 222, 242, 258, 253, 240, 256, 280, 290, 287, 284, 305, 308, 315, 316, 336, 330, 349, 342, 363, 362, 347, 375, 382, 385, 386, 405, 403, 420, 431, 441, 445, 457, 452, 487, 480, 483, 522, 507, 528, 533, 548, 557, 579, 575, 609, 617, 620, 634, 661, 657, 678, 690, 695, 726, 720, 736, 758, 778, 782, 804, 802, 843, 858, 867, 882, 912, 935, 931, 964, 970, 994, 1025, 1037, 1070, 1098, 1100, 1139, 1162, 1198, 1200, 1240, 1253, 1274, 1301, 1337, 1368, 1410, 1417, 1445, 1488}, {}, None}, "ElementType" -> "Integer64"]]}}]]]] -> "Bacteria", ExponentialModel[]]

安定した一次線形系のステップ応答：

Wolfram Language code: ModelFit[Tabular[Association["RawSchema" -> Association["ColumnProperties" -> Association["Time" -> Association["ElementType" -> "Real64"], "Response" -> Association["ElementType" -> "Real64"]], "KeyColumns" -> None, "Backend" -> "WolframKernel"], "Options" -> {}, "BackendData" -> Association["ColumnData" -> DataStructure["ColumnTable", {{TabularColumn[Association["Data" -> {{0., 0.08080808080808081, 0.16161616161616163, 0.24242424242424243, 0.32323232323232326, 0.4040404040404041, 0.48484848484848486, 0.5656565656565657, 0.6464646464646465, 0.7272727272727273, 0.8080808080808082, 0.888888888888889, 0.9696969696969697, 1.0505050505050506, 1.1313131313131315, 1.2121212121212122, 1.292929292929293, 1.373737373737374, 1.4545454545454546, 1.5353535353535355, 1.6161616161616164, 1.696969696969697, 1.777777777777778, 1.8585858585858588, 1.9393939393939394, 2.0202020202020203, 2.101010101010101, 2.181818181818182, 2.262626262626263, 2.3434343434343434, 2.4242424242424243, 2.505050505050505, 2.585858585858586, 2.666666666666667, 2.747474747474748, 2.8282828282828287, 2.909090909090909, 2.98989898989899, 3.070707070707071, 3.151515151515152, 3.2323232323232327, 3.3131313131313136, 3.393939393939394, 3.474747474747475, 3.555555555555556, 3.6363636363636367, 3.7171717171717176, 3.7979797979797985, 3.878787878787879, 3.95959595959596, 4.040404040404041, 4.121212121212122, 4.202020202020202, 4.282828282828283, 4.363636363636364, 4.444444444444445, 4.525252525252526, 4.606060606060606, 4.686868686868687, 4.767676767676768, 4.848484848484849, 4.92929292929293, 5.01010101010101, 5.090909090909092, 5.171717171717172, 5.252525252525253, 5.333333333333334, 5.414141414141414, 5.494949494949496, 5.575757575757576, 5.6565656565656575, 5.737373737373738, 5.818181818181818, 5.8989898989899, 5.97979797979798, 6.060606060606061, 6.141414141414142, 6.222222222222222, 6.303030303030304, 6.383838383838384, 6.464646464646465, 6.545454545454546, 6.626262626262627, 6.707070707070708, 6.787878787878788, 6.868686868686869, 6.94949494949495, 7.030303030303031, 7.111111111111112, 7.191919191919193, 7.272727272727273, 7.353535353535354, 7.434343434343435, 7.515151515151516, 7.595959595959597, 7.676767676767677, 7.757575757575758, 7.838383838383839, 7.91919191919192, 8.}, {}, None}, "ElementType" -> "Real64"]], TabularColumn[ Association["Data" -> {{0.001116659825296017, 0.2499247320838528, 0.3451326198581678, 0.4697963757216964, 0.6810619292185928, 0.7858916533093112, 0.9659318464890925, 1.0070705839877523, 1.1212210767959405, 1.140603158321751, 1.1854188938250885, 1.2430494799773009, 1.4131900894589284, 1.4106484599701368, 1.5088779798041965, 1.5001390132085632, 1.536679608789021, 1.6170460463506244, 1.6693062078709515, 1.7238567894805783, 1.744513457457441, 1.8171303247592494, 1.7485229131901938, 1.8392514999150502, 1.7769989018906434, 1.830246804683987, 1.853488933770925, 1.879642374462409, 1.9052456674125837, 1.913152973156993, 1.8529912974122988, 1.932049167626204, 1.8786913927497908, 1.938649650860597, 1.9282314177574253, 1.9471932000906016, 1.9255673074718593, 1.9363149766902135, 1.9808168704178895, 1.945596783958051, 1.9861600835638875, 1.9776754866636828, 2.023141858347108, 1.96745599376252, 2.0127720511795215, 1.9476158965117283, 1.9943228621683409, 2.064932754358817, 1.9195051035577526, 1.9249957824402375, 2.0537662040717715, 2.0189347358673593, 1.9513877051259043, 1.9831136779932548, 2.0279323222679033, 2.032847963096754, 2.0291940282826877, 1.9601146538652734, 1.9969975532639268, 2.0117848343405247, 2.0317102475905644, 2.0743806025723566, 1.998893341021143, 1.9822357218205557, 1.975539857717858, 1.9299977922560831, 2.0017557700562363, 1.9663126094321934, 2.000502959781868, 2.052068168897171, 2.014429544595991, 1.9707391377307109, 2.0619622583739856, 2.024656782297172, 1.982170253492918, 1.9336211601487863, 1.956719041161304, 2.030961619983265, 2.035651019472237, 1.9339923272537525, 2.0814775358114197, 1.964655635457677, 2.0613056509490417, 1.9199697577721258, 2.0196489542148712, 1.9613007168011862, 2.0102284018437775, 2.0129079178448523, 2.061614025712872, 2.0323271013584274, 1.9733887883398653, 2.040644348387311, 1.9558326300962676, 1.9745035982083827, 2.009743908852638, 2.033910145294105, 2.0124700494852235, 1.9822970192592306, 2.0269961299210943, 1.9779949194867361}, {}, None}, "ElementType" -> "Real64"]]}}]]]] -> "Response", ExponentialModel[]]

UnitStep入力に対する系の応答と比較する：

Wolfram Language code: InputOutputResponse[TransferFunctionModel[{{{1}}, s + λ}, s], UnitStep[t], t]//First//Simplify

人口増加  (1)

2000年以前の世界の人口増加に関する統計を取得する：

Wolfram Language code: population = CountryData["World", {{"Population"}, {1950, 2000}}]

指数関数モデルを成長率にフィットする：

Wolfram Language code: model = ModelFit[population, ExponentialModel[]]

予測の列を追加する：

Wolfram Language code: predictions = TransformColumns[population, {"Prediction" -> (model[#Timestamp]&)}]

フィットをデータと比較する：

Wolfram Language code: ListPlot[{population -> {"Timestamp", "Value"}, predictions -> {"Timestamp", "Prediction"}}, ...]

2020年の人口を予測する：

Wolfram Language code: model[DateObject[{2020}]]

これは，現行の値よりも大きい：

Wolfram Language code: CountryData["World", {"Population", 2020}]

2001年からの人口統計を取得する：

Wolfram Language code: millenniumPopulation = CountryData["World", {{"Population"}, {2001, 2020}}]

予測される継続的な成長率と現実を比較する：

Wolfram Language code: ListPlot[{millenniumPopulation -> {"Timestamp", "Value"}, TransformColumns[millenniumPopulation, {"Prediction" -> (model[#"Timestamp"]&)}] -> {"Timestamp", "Prediction"}}, ...]

指数関数的である傾向がどこで停止したかを調査するために，1980年から2020年までの限定されたデータ範囲に対してフィットを行い，その損失を測定する：

Wolfram Language code: allPopulation = Join[population, millenniumPopulation];

損失は約31年前に劇的に増加し始めているが，これは人類の人口増加が1995年頃に指数関数的ではなくなったことを示している．

Wolfram Language code: Table[{endDate, ModelFit[allPopulation[[ ;; endDate]], ExponentialModel[], "Report"]["Loss"]}, {endDate, Range[-40, -1]}]//ListLogPlot
Wolfram Research (2026), ExponentialModel, Wolfram言語関数， https://reference.wolfram.com/language/ref/ExponentialModel.html.

テキスト

Wolfram Research (2026), ExponentialModel, Wolfram言語関数， https://reference.wolfram.com/language/ref/ExponentialModel.html.

CMS

Wolfram Language. 2026. "ExponentialModel." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ExponentialModel.html.

APA

Wolfram Language. (2026). ExponentialModel. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ExponentialModel.html

BibTeX

@misc{reference.wolfram_2026_exponentialmodel, author="Wolfram Research", title="{ExponentialModel}", year="2026", howpublished="\url{https://reference.wolfram.com/language/ref/ExponentialModel.html}", note=[Accessed: 07-September-2026]}

BibLaTeX

@online{reference.wolfram_2026_exponentialmodel, organization={Wolfram Research}, title={ExponentialModel}, year={2026}, url={https://reference.wolfram.com/language/ref/ExponentialModel.html}, note=[Accessed: 07-September-2026]}