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ExponentialModel []

represents an exponential function.

ExponentialModel [vars]

uses explicit variable specification vars.

ExponentialModel [pars,vars]

uses the provided coefficients pars.

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Variables  
Parameters  
Evaluation  
Information  
Fitting  
Applications  
Basic Applications  
Population Growth  
See Also
Related Guides
History
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ExponentialModel []

represents an exponential function.

ExponentialModel [vars]

uses explicit variable specification vars.

ExponentialModel [pars,vars]

uses the provided coefficients pars.

Details

  • ExponentialModel represents an exponential in the given variables in a format suitable for symbolic or numerical evaluation and fitting.
  • Exponential models describe processes where change is proportional to the current state, such as growth, decay, relaxation and approach to equilibrium.
  • Single-variate ExponentialModel is parametrized as TemplateBox[{1}, CTraditional]+c_2 e^(c_3 x).
  • Multivariate exponentials are parametrized as TemplateBox[{1}, CTraditional]+TemplateBox[{2}, CTraditional] product_(i=3)^nexp(TemplateBox[{i}, CTraditional] x_i).
    • Variables
  • When not specified, variables will automatically be enumerated using x [i].
  • Valid variable specifications vars include:
  • n the number of variables
    symb a symbolic representation of a single variable
    {symb1,…} a list of symbolic variables
    • Parameters
  • When not specified, parameters will automatically be enumerated using C [i].
  • Valid parameter pars specifications in the form {par1,…} include:
  • val a fixed parameter value val
    par a symbolic parameter name par
    parval a symbolic name par set to a fixed value val
    {par,val0} a symbolic parameter name par with the initial value val0
    • Properties
  • Model properties can be extracted using Information [PowerModel[…],prop].
  • Valid basic properties include:
  • "BaseType" model base type
    "Name" model name
    "ShortName" short identifier to use as label
    "InputType" supported input types
    "OutputType" supported output types
  • Valid data-related properties include:
  • "ColumnNames" names of the input features
    "ColumnVariableMap" map between column names and model variables
    "InputSize" dimensionality of the input
    "OutputSize" dimensionality of the output
    "Trainable" whether the model is fully specified and can be trained
    "Trained" whether the model can be evaluated numerically
    "VariableColumnMap" map between model variables and column names
    "Variables" name of the model variables
  • Best model-related properties include:
  • "Expression" model expression
    "Function" model as a pure function
    "SymbolicExpression" model expression with symbolic parameters
    "TabularFunction" pure function suitable to work on a tabular row
  • Parameter-related properties include:
  • "ParameterAssociation" association of parameter names and values
    "ParameterCount" the number of parameters
    "ParameterInitialValues" initial values for the fit
    "ParameterNames" parameter names
    "ParameterRules" list of rules with parameter names and values
    "Parameters" parameter values if present; names otherwise
    "ParameterValues" parameter values
    "Constraints" parameter constraints

Examples

open all close all

Basic Examples  (3)

Specify a generic exponential model:

Wolfram Language code: ExponentialModel[]

Use an explicit input size and custom names for the symbolic parameters:

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

Plot over a variety of growth factors using constant parameters:

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}]

Scope  (22)

Variables  (6)

Specify a generic model:

Wolfram Language code: ExponentialModel[]

The number of variables is inferred from the arguments:

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

ModelFit will assume the number of variables is one less than the dimensionality of data points:

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[]]

Specify the number of variables:

Wolfram Language code: ExponentialModel[2]

Give the variables a custom symbolic representation:

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

These names are overwritten when the model is evaluated:

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

Visualize a multivariate exponential model:

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

Parameters  (4)

Parameter names are assigned automatically:

Wolfram Language code: ExponentialModel[2]

Specify custom parameter names:

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

Set a parameter to a specific value:

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

Specify both parameter names and values:

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

The ordering matches the automatic enumeration of the C [n]:

Wolfram Language code: ExponentialModel[][x]

Evaluation  (5)

Symbolically evaluate a single-variable model:

Wolfram Language code: ExponentialModel[1][x]

Symbolically evaluate a two-variable model:

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

The number of variables is automatically inferred if not specified:

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

Evaluate the model on multiple symbolic variables:

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

Evaluate the model on a list of points:

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

Information  (5)

View general information about a model:

Wolfram Language code: Information[ExponentialModel[]]

Some information is only available when variables or parameters are fully specified:

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

Extract a single property:

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

Retrieve multiple properties:

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

Get information about the default model values:

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

Fitting  (2)

Fit an exponential model:

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

Use a set value for the parameter when fitting the model:

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

Applications  (9)

Basic Applications  (8)

Voltage across a capacitor approaches the supply voltage exponentially when charging through a resistor:

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[]]

Compare with the symbolic result:

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

Current through an inductor increases exponentially after a voltage step:

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[]]

Object temperature relaxing to ambient temperature (Newton's law of cooling):

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[]]

This solves :

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

Radioactive decay with constant decay rate:

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[]]

The decay solves :

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

Light intensity attenuated through an absorbing medium:

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[]]

The is the Beer–Lambert law, which solves :

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

Population growth when birth rate is proportional to population size:

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[]]

Bacterial population during an early exponential growth phase:

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[]]

Step response of a stable first-order linear system:

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[]]

Compare with the response of a system to a UnitStep input:

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

Population Growth  (1)

Retrieve statistics on worldwide population growth before the millennium:

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

Fit an exponential model to the growth rate:

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

Add a column of predictions:

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

Compare the fit to the data:

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

Predict the population in the year 2020:

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

This is greater than the current population:

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

Get the statistics of the population since the year 2001:

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

Compare projected continuing growth rate vs the reality:

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

To investigate where the trend stopped being exponential, fit over a restricted amount of data from 1980 to 2020 and measure the loss:

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

The loss starts increasing dramatically about 31 years ago, indicating that human population growth stopped being exponential in about 1995.

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

Text

Wolfram Research (2026), ExponentialModel, Wolfram Language function, 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: 15-August-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: 15-August-2026]}
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