(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 415841, 10601]*) (*NotebookOutlinePosition[ 416930, 10634]*) (* CellTagsIndexPosition[ 416886, 10630]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Curve fitting How-to", "Title"], Cell["by W. Garrett Mitchener", "Author"], Cell[TextData[{ "This worksheet goes over traditional linear and non-linear least squares \ curve fitting and different ways to do it in ", StyleBox["Mathematica. ", FontSlant->"Italic"], "It also goes over maximum likelihood curve fitting. Along the way, it \ shows different functions for finding maxima and minima of expressions." }], "Abstract"], Cell[CellGroupData[{ Cell["Least squares and linear regression", "Section"], Cell[TextData[{ "Let's say you have some data ", Cell[BoxData[ \(TraditionalForm\`{\((x\_1, y\_1)\), \ \((x\_2, y\_2)\) ... }\)]], " and you want to fit a curve to it so that you can say ", Cell[BoxData[ \(TraditionalForm\`y = f(x) + noise\)]], ". 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396}, {617.5, 440.562}} -> {-9.26072, 5.06281, \ 0.0366761, 0.0144585}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output", CellLabel->"Out[12]="] }, Open ]], Cell["\<\ But don't get carried away. If you give it too many degrees of \ freedom, it will start to fit the noise, as in this example:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(powerTable = Table[x^n, {n, 0, 25}]\)], "Input", CellLabel->"In[13]:="], Cell[BoxData[ \({1, x, x\^2, x\^3, x\^4, x\^5, x\^6, x\^7, x\^8, x\^9, x\^10, x\^11, x\^12, x\^13, x\^14, x\^15, x\^16, x\^17, x\^18, x\^19, x\^20, x\^21, x\^22, x\^23, x\^24, x\^25}\)], "Output", CellLabel->"Out[13]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(poly25fit = Fit[data2, powerTable, x]\)], "Input", CellLabel->"In[14]:="], Cell[BoxData[ RowBox[{\(General::"spell"\), \(\(:\)\(\ \)\), "\<\"Possible spelling \ error: new symbol name \\\"\\!\\(poly25fit\\)\\\" is similar to existing \ symbols \\!\\({poly2fit, poly5fit}\\). \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\ \\\", ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \ ButtonData:>\\\"General::spell\\\"]\\)\"\>"}]], "Message", CellLabel->"From In[14]:="], Cell[BoxData[ 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In fact, I get vastly \ different plots every time I run this worksheet, which indicates that the \ random noise added to the data is having a huge impact on the cosine fit, \ which isn't the case for the fifth degree polynomial. That indicates that \ these cosines are not a good way to fit this data. (Think about it: Why?) \ The first step to getting a good fit is to know what functions to \ include.\ \>", "Text"], Cell[CellGroupData[{ Cell[TextData[{ "Details on how ", ButtonBox["Fit", ButtonStyle->"RefGuideLink"], " works." }], "Subsection"], Cell[TextData[{ "The way ", ButtonBox["Fit", ButtonStyle->"RefGuideLink"], " works is called least squares, because it minimizes this:" }], "Text"], Cell[BoxData[ \(\[Sum]\+\(j = 1\)\%n\((f[x\_j] - y\_j)\)\^2\)], "DisplayFormula"], Cell[TextData[{ "In ", StyleBox["Mathematica", FontSlant->"Italic"], " notation" }], "Text"], Cell[BoxData[ \(LeastSquaresError[data_, f_] := Sum[\((f[data[\([j, 1]\)]] - data[\([j, 2]\)])\)^2, {j, 1, Length[data]}]\)], "Input", CellLabel->"In[18]:="], Cell["Let's return to our linear data.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(data\)], "Input", CellLabel->"In[19]:="], Cell[BoxData[ \({{\(-5\), \(-6.049648716247662`\)}, {\(-4.75`\), \ \(-5.587671709504661`\)}, {\(-4.5`\), \(-5.441053368522673`\)}, {\(-4.25`\), \ \(-5.259667271826949`\)}, {\(-4.`\), \(-5.477262129595641`\)}, {\(-3.75`\), \ \(-4.880523765045758`\)}, {\(-3.5`\), \(-3.2615292728763645`\)}, {\(-3.25`\), \ \(-3.0619991990959736`\)}, {\(-3.`\), \(-3.8282219729494527`\)}, {\(-2.75`\), \ \(-3.444864658045677`\)}, {\(-2.5`\), \(-1.3997853184593039`\)}, {\(-2.25`\), \ \(-1.1593555906061943`\)}, {\(-2.`\), \(-1.8271496953932282`\)}, {\(-1.75`\), \ \(-0.07178606360499229`\)}, {\(-1.5`\), 0.4688225020863652`}, {\(-1.25`\), 1.1939763012540714`}, {\(-1.`\), 1.5512952149122463`}, {\(-0.75`\), 1.5256772694875604`}, {\(-0.5`\), 1.6135232818913574`}, {\(-0.25`\), 2.513948383588697`}, {0.`, 2.092715931822976`}, {0.25`, 2.834611687625716`}, {0.5`, 3.0571802694861376`}, {0.75`, 4.077948832834285`}, {1.`, 4.142364648070639`}, {1.25`, 4.922283397130377`}, {1.5`, 5.49823363800881`}, {1.75`, 6.837616104661234`}, {2.`, 7.61962677766628`}, {2.25`, 8.302807162176135`}, {2.5`, 7.759762910885175`}, {2.75`, 9.399615303757209`}, {3.`, 9.447848750615734`}, {3.25`, 10.247671820221811`}, {3.5`, 10.159548229344479`}, {3.75`, 10.058970894363402`}, {4.`, 11.274998446008961`}, {4.25`, 10.819457883826804`}, {4.5`, 12.690725727258114`}, {4.75`, 12.36499459310933`}, {5.`, 13.723703231096716`}}\)], "Output", CellLabel->"Out[19]="] }, Open ]], Cell["Here's how to define a general line function:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(lineFunction[x_] = a\ x\ + \ b\)], "Input", CellLabel->"In[20]:="], Cell[BoxData[ \(b + a\ x\)], "Output", CellLabel->"Out[20]="] }, Open ]], Cell[TextData[{ "To fit this line to the data, we need to determine ", Cell[BoxData[ \(TraditionalForm\`a\)]], " and ", Cell[BoxData[ \(TraditionalForm\`b\)]], " such that the error is minimized." }], "Text"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " has several functions for finding the minimum of an expression. They all \ work a little differently. The ", ButtonBox["Minimize", ButtonStyle->"RefGuideLink"], " function works algebraically:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(minSolution = Minimize[LeastSquaresError[data, lineFunction], {a, b}]\)], "Input", CellLabel->"In[21]:="], Cell[BoxData[ \({15.3329159852948`, {a \[Rule] 1.9935041266480915`, b \[Rule] 3.059741718571124`}}\)], "Output", CellLabel->"Out[21]="] }, Open ]], Cell[TextData[{ "The result is a list ", Cell[BoxData[ \(TraditionalForm\`{min, \ args}\)]], " where min is the minimum value it found, and args is a rule table. \ Here's how to use a rule table. First, just so we're clear on what's going \ on, we'll unpack the list returned by ", ButtonBox["Minimize", ButtonStyle->"RefGuideLink"], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \({min, args} = minSolution\)], "Input", CellLabel->"In[22]:="], Cell[BoxData[ RowBox[{\(General::"spell1"\), \(\(:\)\(\ \)\), "\<\"Possible spelling \ error: new symbol name \\\"\\!\\(min\\)\\\" is similar to existing symbol \ \\\"\\!\\(Min\\)\\\". \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", \ ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \ ButtonData:>\\\"General::spell1\\\"]\\)\"\>"}]], "Message", CellLabel->"From In[22]:="], Cell[BoxData[ \({15.3329159852948`, {a \[Rule] 1.9935041266480915`, b \[Rule] 3.059741718571124`}}\)], "Output", CellLabel->"Out[22]="] }, Open ]], Cell[TextData[{ "That defined min to be the minimum value, and ", Cell[BoxData[ \(TraditionalForm\`args\)]], " to be the rule table giving the values of ", Cell[BoxData[ \(TraditionalForm\`a\)]], " and ", Cell[BoxData[ \(TraditionalForm\`b\)]], ":" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(args\)], "Input", CellLabel->"In[23]:="], Cell[BoxData[ \({a \[Rule] 1.9935041266480915`, b \[Rule] 3.059741718571124`}\)], "Output", CellLabel->"Out[23]="] }, Open ]], Cell["This was the function we were trying to fit:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(lineFunction[x]\)], "Input", CellLabel->"In[24]:="], Cell[BoxData[ \(b + a\ x\)], "Output", CellLabel->"Out[24]="] }, Open ]], Cell[TextData[{ "And we can connect the general function to the specific values of ", Cell[BoxData[ \(TraditionalForm\`a\)]], " and ", Cell[BoxData[ \(TraditionalForm\`b\)]], " by using the ", ButtonBox["/.", ButtonStyle->"RefGuideLink"], "\noperator. (You can also use the ", ButtonBox["ReplaceAll", ButtonStyle->"RefGuideLink"], " function. The ", ButtonBox["/.", ButtonStyle->"RefGuideLink"], " is short-hand for ", ButtonBox["ReplaceAll", ButtonStyle->"RefGuideLink"], ".)" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(lineFunction[x] /. args\)], "Input", CellLabel->"In[25]:="], Cell[BoxData[ \(\(\(3.059741718571124`\)\(\[InvisibleSpace]\)\) + 1.9935041266480915`\ x\)], "Output", CellLabel->"Out[25]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ReplaceAll[lineFunction[x], args]\)], "Input", CellLabel->"In[26]:="], Cell[BoxData[ \(\(\(3.059741718571124`\)\(\[InvisibleSpace]\)\) + 1.9935041266480915`\ x\)], "Output", CellLabel->"Out[26]="] }, Open ]], Cell["If you want to define a function representing the fit:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(fittedFunction[x_] = lineFunction[x] /. args\)], "Input", CellLabel->"In[27]:="], Cell[BoxData[ \(\(\(3.059741718571124`\)\(\[InvisibleSpace]\)\) + 1.9935041266480915`\ x\)], "Output", CellLabel->"Out[27]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(fittedFunction[3.5]\)], "Input", CellLabel->"In[28]:="], Cell[BoxData[ \(10.037006161839445`\)], "Output", CellLabel->"Out[28]="] }, Open ]], Cell[TextData[{ "The ", ButtonBox["Minimize", ButtonStyle->"RefGuideLink"], " function works algebraically, which means it sometimes doesn't do quite \ what you'd like. It generally works well on polynomial problems, but if you \ give it a nasty enough trancendental problem, you're out of luck." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Minimize[Exp[\(-x^2\)] + Exp[\((x - 2)\)^2], x]\)], "Input", CellLabel->"In[29]:="], Cell[BoxData[ \(Minimize[\[ExponentialE]\^\(\((\(-2\) + x)\)\^2\) + \ \[ExponentialE]\^\(-x\^2\), x]\)], "Output", CellLabel->"Out[29]="] }, Open ]], Cell[TextData[{ "So, sometimes you get better results working just numerically. For that, \ try ", ButtonBox["NMinimize", ButtonStyle->"RefGuideLink"], " . " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(NMinimize[\[ExponentialE]\^\(\((\(-2\) + x)\)\^2\) + \ \[ExponentialE]\^\(-x\^2\), x]\)], "Input", CellLabel->"In[30]:="], Cell[BoxData[ \({1.017122695506149`, {x \[Rule] 2.032606797778184`}}\)], "Output", CellLabel->"Out[30]="] }, Open ]], Cell["Here's our linear least squares fit again:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(NMinimize[LeastSquaresError[data, lineFunction], {a, b}]\)], "Input", CellLabel->"In[31]:="], Cell[BoxData[ \({15.3329159852948`, {a \[Rule] 1.9935041266480915`, b \[Rule] 3.059741718571124`}}\)], "Output", CellLabel->"Out[31]="] }, Open ]], Cell[TextData[{ "Another numerical function is ", ButtonBox["FindMinimum", ButtonStyle->"RefGuideLink"], ", which uses a different numerical algorithm. You have to specify a \ starting point for each unknown variable." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(FindMinimum[ LeastSquaresError[data, lineFunction], {{a, 1}, {b, 1}}]\)], "Input", CellLabel->"In[32]:="], Cell[BoxData[ \({15.332915985294802`, {a \[Rule] 1.9935041266480915`, b \[Rule] 3.059741718571124`}}\)], "Output", CellLabel->"Out[32]="] }, Open ]], Cell[TextData[{ "The ", ButtonBox["Fit", ButtonStyle->"RefGuideLink"], " function does exactly this same minimization conceptually, but it only \ works if the fit function looks like" }], "Text"], Cell[BoxData[ \(f[x] = \(\(a\_1\) f\_1[x] + \(a\_2\) f\_2[x] + ... \) + \(a\_n\) f\_n[x]\)], "DisplayFormula"], Cell[TextData[{ "and the ", Cell[BoxData[ \(TraditionalForm\`a\_1, \ a\_2, \(\(...\) \(a\_n\)\)\)]], " are the only unknowns. This is the traditional method of curve fitting \ (predating modern computers that can do more powerful techniques almost as \ fast) because if ", Cell[BoxData[ \(TraditionalForm\`f\)]], " has this form, you can take a short cut from linear algebra and do the \ computation very quickly. Otherwise, the minimization can be computationally \ intensive and may get stuck at a local minimum instead of finding the global \ minimum." }], "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Non-linear least squares", "Section"], Cell[CellGroupData[{ Cell["The linearization method", "Subsection"], Cell["\<\ To do non-linear curve fitting with least squares, there are a \ couple of alternatives. One is to linearize the data first, then proceed \ using Fit.\ \>", "Text"], Cell[TextData[{ "As before, let's make up some noisy data to play with. It's basically ", Cell[BoxData[ \(TraditionalForm\`4 x\^2.2\)]], " with noise added to the coefficient and the exponent." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(nonlinearData = Table[{x, \((4 + Random[Real, {\(-1\), 1}])\) x^\((2.2 + Random[Real, {\(-0.05\), 0.05}])\)}, \[IndentingNewLine]{x, 1, 10, 0.25}]\)], "Input", CellLabel->"In[33]:="], Cell[BoxData[ \({{1, 4.928235486636071`}, {1.25`, 6.3518130924858545`}, {1.5`, 9.885798328811687`}, {1.75`, 14.8501660828667`}, {2.`, 18.99448232981404`}, {2.25`, 28.501758013561375`}, {2.5`, 25.487387438751966`}, {2.75`, 41.988748638684`}, {3.`, 35.67930449151278`}, {3.25`, 51.38691901426361`}, {3.5`, 67.63654926607977`}, {3.75`, 69.29238926789924`}, {4.`, 91.36921298244707`}, {4.25`, 73.45045522683911`}, {4.5`, 95.85709975704381`}, {4.75`, 110.00597105740584`}, {5.`, 164.82225840663043`}, {5.25`, 154.04955036784767`}, {5.5`, 169.3334457125186`}, {5.75`, 222.83067441608668`}, {6.`, 207.93903221106282`}, {6.25`, 201.61916680077948`}, 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4.07506}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output", CellLabel->"Out[34]="] }, Open ]], Cell[TextData[{ "We'd like to fit this to a power function and find ", Cell[BoxData[ \(TraditionalForm\`a\)]], " and ", Cell[BoxData[ \(TraditionalForm\`b\)]], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(powerFunction[x_] = a\ x^b\)], "Input", CellLabel->"In[35]:="], Cell[BoxData[ \(a\ x\^b\)], "Output", CellLabel->"Out[35]="] }, Open ]], Cell["\<\ But we can't use Fit, because the unknowns aren't in the right \ place. So, we linearize the data first. Assuming the power function is \ correct for our data, we can take the logarithm, and get something in the \ right form for Fit:\ \>", "Text"], Cell[BoxData[ \(Log[ y] = \(Log[a\ x\^b] = \(Log[a] + Log[x\^b] = Log[a] + b\ Log[x]\)\)\)], "DisplayFormula"], Cell[TextData[{ "So even though our ", Cell[BoxData[ \(TraditionalForm\`\((x, y)\)\)]], " data is not in the right form for Fit, it turns out that ", Cell[BoxData[ \(TraditionalForm\`\((Log[x], Log[y])\)\)]], " is in the right form. Here's an incantation to linearize the data. The \ trick is that ", ButtonBox["/.", ButtonStyle->"RefGuideLink"], " can apply rules that involve patterns (see the ", StyleBox["Mathematica", FontSlant->"Italic"], " book ", ButtonBox["2.5", ButtonStyle->"MainBookLink"], " ), so this next command looks at ", Cell[BoxData[ \(TraditionalForm\`nonlinearData\)]], " and replaces anything that looks like ", Cell[BoxData[ \(TraditionalForm\`{x, y}\)]], " with ", Cell[BoxData[ \(TraditionalForm\`{Log[x], Log[y]}\)]], ". As in function definitions, the _ on the x and y indicates that these \ are pattern variables. Without the _, ", StyleBox["Mathematica", FontSlant->"Italic"], " will think you mean to replace only stuff with the symbols x and y. And \ you don't use the _ on the right hand side of the rule or in a function \ definition, only on the left." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(linearizedData = nonlinearData /. {x_, y_} \[Rule] {Log[x], Log[y]}\)], "Input", CellLabel->"In[36]:="], Cell[BoxData[ \({{0, 1.5949810105270454`}, {0.22314355131420976`, 1.8487402985254273`}, {0.4054651081081644`, 2.2910992150015637`}, {0.5596157879354227`, 2.6980110492175955`}, {0.6931471805599453`, 2.9441485332971244`}, {0.8109302162163288`, 3.349965770058547`}, {0.9162907318741551`, 3.2381837195593635`}, {1.0116009116784799`, 3.737401692839409`}, {1.0986122886681098`, 3.5745708145709436`}, {1.1786549963416462`, 3.9393836461861333`}, {1.252762968495368`, 4.214148506556328`}, {1.3217558399823195`, 4.238335077190836`}, {1.3862943611198906`, 4.51490858345091`}, {1.4469189829363254`, 4.296611100379293`}, {1.5040773967762742`, 4.562858538287812`}, {1.55814461804655`, 4.700534646659237`}, {1.6094379124341003`, 5.104867671502195`}, 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0007ool00`00ooooo`3oooloool00`00ooooo`1ool00`00ooooo`0Mool0023oo`03003ooooo0?oo oeSoo`T000koo`03003ooooo01coo`008?oo00<00?ooool0ooooF?oo2@003ooo00<00?ooool06ooo 000Pool2003ooomHool:000@ool00`00ooooo`0Jool0023oo`03003ooooo0?oooeSoo`T0017oo`03 003ooooo01Woo`008?oo00<00?ooool0ooooF?oo2@004_oo00<00?ooool06?oo000Pool00`00oooo o`3ooomHool8000Dool00`00ooooo`0Gool0023oo`03003ooooo0?oooeWoo`H001Koo`03003ooooo 00Goo`03003ooooo00koo`008?oo00<00?ooool0ooooG?oo00<00?ooool05_oo00<00?ooool01_oo 00<00?ooool03Ooo000Pool00`00ooooo`3ooomfool00`00ooooo`06ool00`00ooooo`0ool0023oo`03003ooooo0?ooohkoo`008?oo00<00?ooool0oooo S_oo000Pool00`00ooooo`3ooon>ool0023oo`800?ooohooo`008?oo00<00?ooool0ooooHooo1P00 9Ooo000Pool00`00ooooo`3ooomRool8000Tool0023oo`03003ooooo0?ooof;oo`T002?oo`008?oo 00<00?ooool0ooooH_oo2@008ooo000Pool00`00ooooo`3ooomQool:000Sool0023oo`03003ooooo 0?ooof;oo`T002?oo`008?oo00<00?ooool0ooooH_oo2@008ooo000Pool2003ooomSool8000Tool0 023oo`03003ooooo0?ooof?oo`H002Goo`008?oo00<00?ooool0ooooI_oo00<00?ooool09Ooo000P ool00`00ooooo`3ooon>ool00001\ \>"], ImageRangeCache->{{{109, 396}, {594.875, 417.938}} -> {-5.03253, 1651.79, \ 0.0385924, 4.07506}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output", CellLabel->"Out[47]="] }, Open ]], Cell["\<\ And you should be able to see that the function found by the \ linearization method isn't quite the same as the one found by the direct \ method:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \({nonlinearFit[x], directFit[x]}\)], "Input", CellLabel->"In[48]:="], Cell[BoxData[ \({4.230295881844876`\ x\^2.165182932635264`, 6.674363562787636`\ x\^1.9462447170378596`}\)], "Output", CellLabel->"Out[48]="] }, Open ]], Cell["\<\ Both functions are actually the optimum fit to the data, but under \ different notions of distance. And notice that neither one is exact compared \ to what we started with. For example, here's what I got on one run of this \ worksheet:\ \>", "Text"], Cell[BoxData[ \({3.5498\ x\^2.27148, 3.0792\ x\^2.34363}\)], "DisplayFormula", CellLabel->"Out[55]="], Cell[TextData[{ "Since the data set is constructed with random noise, the results will be a \ little different each time you run it. But neither of these gives back ", Cell[BoxData[ \(TraditionalForm\`4\ x\^2.2\)]], "exactly. (Think about it: Should they?) And which curve is \"better\"? \ " }], "Text"], Cell["\<\ This contention between the results illustrates the fundamental \ conceptual problems in curve fitting: (1) Guess the appropriate form of the \ function. (2) Determine an appropriate notion of \"distance\" between the \ curve and the data to minimize.\ \>", "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Maximum likelihood", "Section"], Cell[TextData[{ "An alternative notion of distance that is appropriate for modeling \ changing probabilities is likelihood. This notion assumes that the data is \ of the form ", Cell[BoxData[ \(TraditionalForm\`y\_j = f(x\_j, \[Omega]\_j; a, b, ... )\)]], " where ", Cell[BoxData[ \(TraditionalForm\`x\)]], " is an independent variable, such as distance or time, ", Cell[BoxData[ \(TraditionalForm\`\[Omega]\_j\)]], " are independent random numbers, and ", Cell[BoxData[ \(TraditionalForm\`a, b, ... \)]], " are the parameters that we want to find. Then, the likelihood of the \ data is the probability of getting exactly the ", Cell[BoxData[ \(TraditionalForm\`y\_j\)]], "'s. The curve fit procedure is to determine values of the parameters such \ that the likelihood of the data is maximal." }], "Text"], Cell[TextData[{ "As an example, let's suppose we have a biological experiment that succeeds \ or fails, for example, getting bacteria to accept a fragment of DNA. Let's \ suppose that the probability of success depends on temperature ", Cell[BoxData[ \(TraditionalForm\`x\)]], ". Furthermore, let's suppose that we have some knowledge of the \ biochemistry involved that tells us that the probability of success is \ actually an exponential function: ", Cell[BoxData[ \(TraditionalForm\`p[x] = e\^\(\(\ \)\(\(-\ a\)\ \((x - b)\)\)\)\)]], " where ", Cell[BoxData[ \(TraditionalForm\`a\)]], " and ", Cell[BoxData[ \(TraditionalForm\`b\)]], " are unknown. We are given results from experiments run at different \ temperatures, and they are simply given as success or failure, and our job is \ to find ", Cell[BoxData[ \(TraditionalForm\`\(\(a\)\(\ \)\)\)]], "and ", Cell[BoxData[ \(TraditionalForm\`b\)]], "." }], "Text"], Cell[TextData[{ "First, let's invent some data, using ", Cell[BoxData[ \(TraditionalForm\`a = 0.025\)]], " and ", Cell[BoxData[ \(TraditionalForm\`b = \(-5\)\)]], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(pSuccess[x_, a_, b_] = Exp[\(-a\) \((x - b)\)]\)], "Input", CellLabel->"In[49]:="], Cell[BoxData[ \(\[ExponentialE]\^\(\(-a\)\ \((\(-b\) + x)\)\)\)], "Output", CellLabel->"Out[49]="] }, Open ]], Cell[TextData[{ ButtonBox["Random", ButtonStyle->"RefGuideLink"], " returns a random number between 0 and 1, so the test ", StyleBox["Random[] < p", "Input"], " returns True with probability ", Cell[BoxData[ \(TraditionalForm\`p\)]], " and False with probability ", Cell[BoxData[ \(TraditionalForm\`1 - p\)]], ", so we can use it to simulate our experiment. We'll also assume that \ this experiment is fairly expensive and time consuming, so we can't run \ zillions of experiments." }], "Text"], Cell[BoxData[ \(runExperiment[x_, a_, b_] := \((Random[] < pSuccess[x, a, b])\)\)], "Input", CellLabel->"In[50]:="], Cell[CellGroupData[{ Cell[BoxData[ \(bioExampleData = Table[{x, runExperiment[x, 0.025, \(-5\)]}, \[IndentingNewLine]{x, 0, 80, 2}]\)], "Input", CellLabel->"In[51]:="], Cell[BoxData[ \({{0, True}, {2, True}, {4, True}, {6, True}, {8, True}, {10, True}, {12, False}, {14, False}, {16, True}, {18, True}, {20, True}, {22, True}, {24, True}, {26, True}, {28, False}, {30, False}, {32, True}, {34, False}, {36, True}, {38, True}, {40, True}, {42, False}, {44, True}, {46, False}, {48, False}, {50, False}, {52, False}, {54, False}, {56, False}, {58, True}, {60, False}, {62, True}, {64, False}, {66, False}, {68, False}, {70, True}, {72, False}, {74, False}, {76, False}, {78, False}, {80, False}}\)], "Output", CellLabel->"Out[51]="] }, Open ]], Cell["\<\ Here's a way to plot that data. 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ool60004ool60004ool60003ool60004ool60004ool6000Gool60004ool60003ool60004ool60004 ool60004ool6000Gool6000=ool60004ool60004ool6000=ool6000nool6000>ool6000Qool6000h ool001?oo`800003ool00?oo00Soo`03003ooooo00Ooo`03003ooooo00Ooo`03003ooooo00Koo`03 003ooooo00Ooo`03003ooooo00Ooo`03003ooooo01[oo`03003ooooo00Ooo`03003ooooo00Koo`03 003ooooo00Ooo`03003ooooo00Ooo`03003ooooo00Ooo`03003ooooo01[oo`03003ooooo013oo`03 003ooooo00Ooo`03003ooooo00Ooo`03003ooooo013oo`03003ooooo047oo`03003ooooo017oo`03 003ooooo02Coo`03003ooooo03Soo`005Ooo0P002Ooo00<00?ooool0ooooS_oo000Pool00`00oooo o`3ooon>ool00001\ \>"], ImageRangeCache->{{{109, 396}, {293.938, 117}} -> {-40.3135, 0.653605, \ 0.308942, 0.00624851}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output", CellLabel->"Out[52]="] }, Open ]], Cell[TextData[{ "The data we have isn't points ", Cell[BoxData[ \(TraditionalForm\`\((x, p[x, a, b])\)\)]], " which is what we'd need to do traditional curve fitting. In other words, \ we want to fit a curve but we don't have points from the curve plus noise \ like we did in earlier examples; we have something else entirely. \ Intuitively, it seems impossible for least squares to give anything useful \ for this type of data, so instead, we do maximum likelihood. The experiments \ are independent, so the probability of getting all this data is the product \ of the probabilities of getting each point. But the probability of getting \ each point depends on ", Cell[BoxData[ \(TraditionalForm\`a\)]], " and ", Cell[BoxData[ \(TraditionalForm\`b\)]], "." }], "Text"], Cell[BoxData[ \(likelihood[data_, a_, b_] := Product[\[IndentingNewLine]If[ data[\([j, 2]\)], \[IndentingNewLine]pSuccess[data[\([j, 1]\)], a, b], \[IndentingNewLine]1 - pSuccess[data[\([j, 1]\)], a, b]], \[IndentingNewLine]{j, 1, Length[data]}]\)], "Input", CellLabel->"In[53]:="], Cell["So for example:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(likelihood[bioExampleData, 0.001, \(-4\)]\)], "Input", CellLabel->"In[54]:="], Cell[BoxData[ \(3.212001024963687`*^-28\)], "Output", CellLabel->"Out[54]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(likelihood[bioExampleData, 0.02, 2]\)], "Input", CellLabel->"In[55]:="], Cell[BoxData[ \(3.5700018536038285`*^-10\)], "Output", CellLabel->"Out[55]="] }, Open ]], Cell["\<\ These are tiny numbers, and that causes trouble with the numerical \ maximization process, so instead of maximizing the likelihood, we maximize \ the log likelihood (Think about it: Why can we do this?)\ \>", "Text"], Cell[TextData[{ "We could do this, but then ", StyleBox["Mathematica", FontSlant->"Italic"], " will compute that tiny likelihood, then take the log." }], "Text"], Cell[BoxData[ \(logLikelihood1[data_, a_, b_] := Log[likelihood[data, a, b]]\)], "Input",\ CellLabel->"In[56]:="], Cell["\<\ A better way to compute the same thing is to expand the log of the \ product into a sum of logs. I'll do this computation using different \ notation:\ \>", "Text"], Cell[BoxData[ \(logLikelihood[data_, a_, b_] := Total[\[IndentingNewLine]data /. {x_, success_} \[Rule] If[success, \[IndentingNewLine]Log[ pSuccess[x, a, b]], \[IndentingNewLine]Log[ 1 - pSuccess[x, a, b]]]]\)], "Input", CellLabel->"In[57]:="], Cell["\<\ Just to check that we did this right, these should be the same \ number:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Log[likelihood[bioExampleData, 0.001, \(-4\)]]\)], "Input", CellLabel->"In[58]:="], Cell[BoxData[ \(\(-63.30548848864431`\)\)], "Output", CellLabel->"Out[58]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(logLikelihood[bioExampleData, 0.001, \(-4\)]\)], "Input", CellLabel->"In[59]:="], Cell[BoxData[ \(\(-63.30548848864429`\)\)], "Output", CellLabel->"Out[59]="] }, Open ]], Cell["Here's the first try at running the fit:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Maximize[{logLikelihood[bioExampleData, a, b], a > 0}, {a, b}]\)], "Input", CellLabel->"In[60]:="], Cell[BoxData[ \(Maximize[{Log[\[ExponentialE]\^\(\(-a\)\ \((2 - b)\)\)] + Log[\[ExponentialE]\^\(\(-a\)\ \((4 - b)\)\)] + Log[\[ExponentialE]\^\(\(-a\)\ \((6 - b)\)\)] + Log[\[ExponentialE]\^\(\(-a\)\ \((8 - b)\)\)] + Log[\[ExponentialE]\^\(\(-a\)\ \((10 - b)\)\)] + Log[\[ExponentialE]\^\(\(-a\)\ \((16 - b)\)\)] + Log[\[ExponentialE]\^\(\(-a\)\ \((18 - b)\)\)] + Log[\[ExponentialE]\^\(\(-a\)\ \((20 - b)\)\)] + Log[\[ExponentialE]\^\(\(-a\)\ \((22 - b)\)\)] + Log[\[ExponentialE]\^\(\(-a\)\ \((24 - b)\)\)] + Log[\[ExponentialE]\^\(\(-a\)\ \((26 - b)\)\)] + Log[\[ExponentialE]\^\(\(-a\)\ \((32 - b)\)\)] + Log[\[ExponentialE]\^\(\(-a\)\ \((36 - b)\)\)] + Log[\[ExponentialE]\^\(\(-a\)\ \((38 - b)\)\)] + Log[\[ExponentialE]\^\(\(-a\)\ \((40 - b)\)\)] + Log[\[ExponentialE]\^\(\(-a\)\ \((44 - b)\)\)] + Log[\[ExponentialE]\^\(\(-a\)\ \((58 - b)\)\)] + Log[\[ExponentialE]\^\(\(-a\)\ \((62 - b)\)\)] + Log[\[ExponentialE]\^\(\(-a\)\ \((70 - b)\)\)] + Log[\[ExponentialE]\^\(a\ b\)] + Log[1 - \[ExponentialE]\^\(\(-a\)\ \((12 - b)\)\)] + Log[1 - \[ExponentialE]\^\(\(-a\)\ \((14 - b)\)\)] + Log[1 - \[ExponentialE]\^\(\(-a\)\ \((28 - b)\)\)] + Log[1 - \[ExponentialE]\^\(\(-a\)\ \((30 - b)\)\)] + Log[1 - \[ExponentialE]\^\(\(-a\)\ \((34 - b)\)\)] + Log[1 - \[ExponentialE]\^\(\(-a\)\ \((42 - b)\)\)] + Log[1 - \[ExponentialE]\^\(\(-a\)\ \((46 - b)\)\)] + Log[1 - \[ExponentialE]\^\(\(-a\)\ \((48 - b)\)\)] + Log[1 - \[ExponentialE]\^\(\(-a\)\ \((50 - b)\)\)] + Log[1 - \[ExponentialE]\^\(\(-a\)\ \((52 - b)\)\)] + Log[1 - \[ExponentialE]\^\(\(-a\)\ \((54 - b)\)\)] + Log[1 - \[ExponentialE]\^\(\(-a\)\ \((56 - b)\)\)] + Log[1 - \[ExponentialE]\^\(\(-a\)\ \((60 - b)\)\)] + Log[1 - \[ExponentialE]\^\(\(-a\)\ \((64 - b)\)\)] + Log[1 - \[ExponentialE]\^\(\(-a\)\ \((66 - b)\)\)] + Log[1 - \[ExponentialE]\^\(\(-a\)\ \((68 - b)\)\)] + Log[1 - \[ExponentialE]\^\(\(-a\)\ \((72 - b)\)\)] + Log[1 - \[ExponentialE]\^\(\(-a\)\ \((74 - b)\)\)] + Log[1 - \[ExponentialE]\^\(\(-a\)\ \((76 - b)\)\)] + Log[1 - \[ExponentialE]\^\(\(-a\)\ \((78 - b)\)\)] + Log[1 - \[ExponentialE]\^\(\(-a\)\ \((80 - b)\)\)], a > 0}, {a, b}]\)], "Output", CellLabel->"Out[60]="] }, Open ]], Cell["\<\ Which means it couldn't solve the problem algebraically. So, let's \ try the numerical methods:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(NMaximize[logLikelihood[bioExampleData, a, b], {a, b}]\)], "Input", CellLabel->"In[61]:="], Cell[BoxData[ RowBox[{\(NMaximize::"nrnum"\), \(\(:\)\(\ \)\), "\<\"The function value \ \\!\\(\\(\\(-1524.7554251052638`\\)\\) - \\(\\(\\(\\(\[LeftSkeleton] 18 \ \[RightSkeleton]\\)\\)\\\\ \[ImaginaryI]\\)\\)\\) is not a real number at \\!\ \\({a, b}\\) = \\!\\({\\(\\(-0.9362927260317491`\\)\\), \\(\\(\[LeftSkeleton] \ 20 \[RightSkeleton]\\)\\)}\\). \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", \ ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \ ButtonData:>\\\"General::nrnum\\\"]\\)\"\>"}]], "Message", CellLabel->"From In[61]:="], Cell[BoxData[ \(NMaximize[ Log[\[ExponentialE]\^\(\(-a\)\ \((2 - b)\)\)] + Log[\[ExponentialE]\^\(\(-a\)\ \((4 - b)\)\)] + Log[\[ExponentialE]\^\(\(-a\)\ \((6 - b)\)\)] + Log[\[ExponentialE]\^\(\(-a\)\ \((8 - b)\)\)] + Log[\[ExponentialE]\^\(\(-a\)\ \((10 - b)\)\)] + Log[\[ExponentialE]\^\(\(-a\)\ \((16 - b)\)\)] + Log[\[ExponentialE]\^\(\(-a\)\ \((18 - b)\)\)] + Log[\[ExponentialE]\^\(\(-a\)\ \((20 - b)\)\)] + Log[\[ExponentialE]\^\(\(-a\)\ \((22 - b)\)\)] + Log[\[ExponentialE]\^\(\(-a\)\ \((24 - b)\)\)] + Log[\[ExponentialE]\^\(\(-a\)\ \((26 - b)\)\)] + Log[\[ExponentialE]\^\(\(-a\)\ \((32 - b)\)\)] + Log[\[ExponentialE]\^\(\(-a\)\ \((36 - b)\)\)] + Log[\[ExponentialE]\^\(\(-a\)\ \((38 - b)\)\)] + Log[\[ExponentialE]\^\(\(-a\)\ \((40 - b)\)\)] + Log[\[ExponentialE]\^\(\(-a\)\ \((44 - b)\)\)] + Log[\[ExponentialE]\^\(\(-a\)\ \((58 - b)\)\)] + Log[\[ExponentialE]\^\(\(-a\)\ \((62 - b)\)\)] + Log[\[ExponentialE]\^\(\(-a\)\ \((70 - b)\)\)] + Log[\[ExponentialE]\^\(a\ b\)] + Log[1 - \[ExponentialE]\^\(\(-a\)\ \((12 - b)\)\)] + Log[1 - \[ExponentialE]\^\(\(-a\)\ \((14 - b)\)\)] + Log[1 - \[ExponentialE]\^\(\(-a\)\ \((28 - b)\)\)] + Log[1 - \[ExponentialE]\^\(\(-a\)\ \((30 - b)\)\)] + Log[1 - \[ExponentialE]\^\(\(-a\)\ \((34 - b)\)\)] + Log[1 - \[ExponentialE]\^\(\(-a\)\ \((42 - b)\)\)] + Log[1 - \[ExponentialE]\^\(\(-a\)\ \((46 - b)\)\)] + Log[1 - \[ExponentialE]\^\(\(-a\)\ \((48 - b)\)\)] + Log[1 - \[ExponentialE]\^\(\(-a\)\ \((50 - b)\)\)] + Log[1 - \[ExponentialE]\^\(\(-a\)\ \((52 - b)\)\)] + Log[1 - \[ExponentialE]\^\(\(-a\)\ \((54 - b)\)\)] + Log[1 - \[ExponentialE]\^\(\(-a\)\ \((56 - b)\)\)] + Log[1 - \[ExponentialE]\^\(\(-a\)\ \((60 - b)\)\)] + Log[1 - \[ExponentialE]\^\(\(-a\)\ \((64 - b)\)\)] + Log[1 - \[ExponentialE]\^\(\(-a\)\ \((66 - b)\)\)] + Log[1 - \[ExponentialE]\^\(\(-a\)\ \((68 - b)\)\)] + Log[1 - \[ExponentialE]\^\(\(-a\)\ \((72 - b)\)\)] + Log[1 - \[ExponentialE]\^\(\(-a\)\ \((74 - b)\)\)] + Log[1 - \[ExponentialE]\^\(\(-a\)\ \((76 - b)\)\)] + Log[1 - \[ExponentialE]\^\(\(-a\)\ \((78 - b)\)\)] + Log[1 - \[ExponentialE]\^\(\(-a\)\ \((80 - b)\)\)], {a, b}]\)], "Output", CellLabel->"Out[61]="] }, Open ]], Cell[TextData[{ "You probably got an error message, either that an overflow occurred, or \ that it got a complex number somewhere along the way. That means that ", ButtonBox["NMaximize", ButtonStyle->"RefGuideLink"], " is trying values of ", Cell[BoxData[ \(TraditionalForm\`a\)]], " and ", Cell[BoxData[ \(TraditionalForm\`b\)]], " that yield logs of negative numbers, or perhaps log of 0. Let's use the \ constraint feature of ", ButtonBox["NMaximize", ButtonStyle->"RefGuideLink"], " to give it some additional hints: We ask it to maximize the log \ likelihood, but subject to some reasonable constraints. We know the \ exponential should decrease as ", Cell[BoxData[ \(TraditionalForm\`x\)]], " increases, so we add in ", Cell[BoxData[ \(TraditionalForm\`a > 0\)]], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(NMaximize[{logLikelihood[bioExampleData, a, b], a > 0}, {a, b}]\)], "Input", CellLabel->"In[62]:="], Cell[BoxData[ \({\(-21.274769392052303`\), {a \[Rule] 0.027743455209693187`, b \[Rule] 7.111012046168768`}}\)], "Output", CellLabel->"Out[62]="] }, Open ]], Cell[TextData[{ "Sometimes that works, sometimes it doesn't, depending on what random \ numbers appeared in our simulated experiment. Let's try ", ButtonBox["FindMaximum", ButtonStyle->"RefGuideLink"], " since it takes a starting point." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(FindMaximum[ logLikelihood[bioExampleData, a, b], {{a, 0}, {b, \(-1\)}}]\)], "Input",\ CellLabel->"In[63]:="], Cell[BoxData[ RowBox[{\(FindMaximum::"nnum"\), \(\(:\)\(\ \)\), "\<\"The function value \ \\!\\(Indeterminate\\) is not a number at \\!\\({a, b}\\) = \\!\\({0.`, \ \\(\\(-1.`\\)\\)}\\). \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", \ ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \ ButtonData:>\\\"General::nnum\\\"]\\)\"\>"}]], "Message", CellLabel->"From In[63]:="], Cell[BoxData[ \(FindMaximum[ logLikelihood[bioExampleData, a, b], {{a, 0}, {b, \(-1\)}}]\)], "Output",\ CellLabel->"Out[63]="] }, Open ]], Cell[TextData[{ "That probably didn't work because we get something like ", Cell[BoxData[ \(TraditionalForm\`Log[0]\)]], " somewhere if we start at ", Cell[BoxData[ \(TraditionalForm\`a = 0. \)]] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(FindMaximum[ logLikelihood[bioExampleData, a, b], {{a, 0.01}, {b, \(-1\)}}]\)], "Input", CellLabel->"In[64]:="], Cell[BoxData[ RowBox[{\(FindMaximum::"lstol"\), \(\(:\)\(\ \)\), "\<\"The line search \ decreased the step size to within tolerance specified by AccuracyGoal and \ PrecisionGoal but was unable to find a sufficient increase in the function. \ You may need more than \\!\\(MachinePrecision\\) digits of working precision \ to meet these tolerances. \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", \ ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \ ButtonData:>\\\"FindMaximum::lstol\\\"]\\)\"\>"}]], "Message", CellLabel->"From In[64]:="], Cell[BoxData[ \({\(-21.274769392052303`\), {a \[Rule] 0.02774345522736005`, b \[Rule] 7.11101205972265`}}\)], "Output", CellLabel->"Out[64]="] }, Open ]], Cell["\<\ This is pretty good, at least the time I ran it. Sometimes you \ have to poke around with different starting points to get reasonable results. \ There are also zillions of options, and you can spend lots of time playing \ with them, tweaking the numerical method, but unless you're desperate or know \ what all the tweaks mean, doing that is often a waste of time.\ \>", "Text"], Cell[TextData[{ "The result isn't perfect, but it has a definite interpretation: These \ values for ", Cell[BoxData[ \(TraditionalForm\`a\)]], " and ", Cell[BoxData[ \(TraditionalForm\`b\)]], " are the ones such that the probability of getting our observations is \ maximal. And they're reasonably close to the exact answer, which is great \ given how little information our data acutally contains." }], "Text"] }, Open ]] }, Open ]] }, FrontEndVersion->"5.0 for X", ScreenRectangle->{{0, 1280}, {0, 1024}}, ScreenStyleEnvironment->"Working", PrintingStyleEnvironment->"Printout", WindowSize->{843, 600}, WindowMargins->{{53, Automatic}, {Automatic, 32}}, PrintingPageRange->{Automatic, Automatic}, PrintingOptions->{"PaperSize"->{612, 792}, "PaperOrientation"->"Portrait", "PostScriptOutputFile":>FrontEnd`FileName[{$RootDirectory, "home", "postdoc", \ "wgm", "Teaching", "Mathematica how-to"}, "CurveFittingHowTo.nb.ps", \ CharacterEncoding -> "iso8859-1"], "Magnification"->1}, Magnification->1.5, StyleDefinitions -> "ArticleClassic.nb" ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. 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