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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[ 257310, 6971]*) (*NotebookOutlinePosition[ 258026, 6995]*) (* CellTagsIndexPosition[ 257982, 6991]*) (*WindowFrame->Normal*) Notebook[{ Cell["\<\ The Mathematics of Modeling Part 2: Monte Carlo Methods\ \>", "Subtitle", TextAlignment->Center], Cell[TextData[{ "\tOur goal in computational science is to build a model that realistically \ reflects some particular behavior of an actual, physical system. \ Occasionally the factors that affect the behavior are so numerous and varied \ that it is easier to consider the possibilities as the results of random \ chance, favored by particular probabilities. Problems that involve a random \ variable are called ", StyleBox["stochastic", FontSlant->"Italic"], " problems, and modeling methods that take advantage of random variables \ are called ", StyleBox["Monte Carlo", FontSlant->"Italic"], " methods, in reference to the famous casino and resort town in Monaco. " }], "Text", FontSize->14], Cell[TextData[{ StyleBox["LCG Random Number Generation\n", "Section"], StyleBox["\tThe key element to a Monte Carlo method is the ability to \ generate a random number. There is actually quite a bit of philosophical \ discussion about what makes a number truly 'random' - computers are naturally \ ", FontSize->14], StyleBox["deterministic", FontSize->14, FontSlant->"Italic"], StyleBox[", i.e. what comes out is pre-determined by what goes in, making \ it quite a challenge for programmers to create 'pseudo-random' algorithms to \ generate sequences of numbers that sufficiently fit mathematical definitions \ of randomness. Let's take a look at one common method, called a Linear \ Congruential Generator (LCG).", FontSize->14] }], "Text"], Cell[TextData[{ StyleBox["\t", FontSize->14, FontSlant->"Italic"], StyleBox["An LCG algorithm requires 4 starting numbers. The first number \ is known as the ", FontSize->14], StyleBox["seed", FontSize->14, FontSlant->"Italic"], StyleBox[", because it acts as the starting point for a list of numbers, \ and only gets used to generate the first number. Computers will often choose \ this number from some other sort of 'random' source (typically a time from \ the computer's clock) to add even more variability.", FontSize->14] }], "Text"], Cell[BoxData[ \(\(seed\ = \ 17;\)\)], "Input"], Cell[TextData[{ StyleBox["\t", FontSize->14, FontSlant->"Italic"], StyleBox["The other three numbers - modulus, scale and shift - are used for \ each step.", FontSize->14] }], "Text"], Cell[BoxData[{ \(\(scale\ = \ 5;\)\), "\[IndentingNewLine]", \(\(shift\ = \ 2;\)\), "\[IndentingNewLine]", \(\(modulus\ = \ 13;\)\)}], "Input"], Cell[TextData[{ StyleBox["\t", FontSize->14, FontSlant->"Italic"], StyleBox["We multiply the seed by the scale...", FontSize->14] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(num\ = \ seed*scale\)], "Input"], Cell[BoxData[ \(85\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["\t", FontSize->14, FontSlant->"Italic"], StyleBox["Add the shift...", FontSize->14] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(num\ = \ num\ + \ shift\)], "Input"], Cell[BoxData[ \(87\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["\t", FontSize->14, FontSlant->"Italic"], StyleBox["And take the ", FontSize->14], StyleBox["remainder ", FontSize->14, FontSlant->"Italic"], StyleBox["after dividing by the modulus. This gives us a number that is \ greater than or equal to zero, but less that the modulus. ", FontSize->14], StyleBox["Mathematica", FontSize->14, FontSlant->"Italic"], StyleBox[" uses the function ", FontSize->14], StyleBox["Mod", "Input", FontSize->14], StyleBox[" for modulo (remainder) division.", FontSize->14] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(num\ = \ Mod[num, modulus]\)], "Input"], Cell[BoxData[ \(9\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["\t", FontSize->14, FontSlant->"Italic"], StyleBox["The new number then becomes the seed for the next random \ number...", FontSize->14] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(seed\ = \ num;\)\), "\[IndentingNewLine]", \(num\ = \ Mod[scale*seed + shift, modulus]\)}], "Input"], Cell[BoxData[ \(8\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["\t", FontSize->14, FontSlant->"Italic"], StyleBox["We can make a function that takes our four numbers as arguments \ (as well as a list length) and gives us a list of pseudorandom numbers \ between (and including) zero and (but not including) the modulus.", FontSize->14] }], "Text"], Cell[BoxData[ \(LCG[length_, seed_, scale_, shift_, modulus_]\ := \[IndentingNewLine]\((num\ = \ seed; \[IndentingNewLine]Table[\[IndentingNewLine]num\ = \ \ Mod[ scale*num + shift, modulus]\[IndentingNewLine]\ \ \ , {i, length}])\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(LCG[20, 17, 5, 2, 13]\)], "Input"], Cell[BoxData[ \({9, 8, 3, 4, 9, 8, 3, 4, 9, 8, 3, 4, 9, 8, 3, 4, 9, 8, 3, 4}\)], "Output"] }, Open ]], Cell[TextData[StyleBox["\tNot very random, is it! Let's try a different \ scale...", FontSize->14]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(LCG[20, 17, 7, 2, 13]\)], "Input"], Cell[BoxData[ \({4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4}\)], "Output"] }, Open ]], Cell[TextData[StyleBox["\tYikes! that's even worse! But if we keep the 'bad' \ scale and change the shift...", FontSize->14]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(LCG[20, 17, 7, 3, 13]\)], "Input"], Cell[BoxData[ \({5, 12, 9, 1, 10, 8, 7, 0, 3, 11, 2, 4, 5, 12, 9, 1, 10, 8, 7, 0}\)], "Output"] }, Open ]], Cell[TextData[StyleBox["\tWhich initially looks better, until you realize \ that the series again starts to repeat, after selecting every number from 0 \ to 12 (except for 6 !?!). A major problem with this kind of generator is \ that once a number is repeated in a series, the entire series begins to \ repeat! We can make the distance between 'repeats' longer with a bigger \ modulus...", FontSize->14]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(LCG[20, 17, 7, 3, 17]\)], "Input"], Cell[BoxData[ \({3, 7, 1, 10, 5, 4, 14, 16, 13, 9, 15, 6, 11, 12, 2, 0, 3, 7, 1, 10}\)], "Output"] }, Open ]], Cell[TextData[StyleBox["\tBut the patterns still appear, unless we decide not \ to use just integers... ", FontSize->14]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(LCG[20, 17.2, 7.3, 3.01, 17.4]\)], "Input"], Cell[BoxData[ \({6.770000000000003`, 0.23100000000002296`, 4.696300000000168`, 2.492990000001228`, 3.808827000008968`, 13.414437100065463`, 13.935390830477893`, 0.33835306248862196`, 5.479977356166939`, 8.213834700018658`, 10.770993310136205`, 12.038251163994303`, 3.889233497158422`, 14.001404529256483`, 0.8202530635723377`, 8.997847364078066`, 16.49428575776988`, 1.6182860317201389`, 14.823488031557012`, 6.821462630366199`}\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["\tNow ", FontSize->14], StyleBox["these", FontSize->14, FontWeight->"Bold"], StyleBox[" numbers seem much more random, at least at first glance. ", FontSize->14], StyleBox["So the question is, are the patterns 'gone' or are they just \ harder to see? One way to get a picture of what is happening is to plot each \ random number along with one of it's neighbors on an x-y graph! 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Cell[TextData[StyleBox["\tPerhaps you can see now why there is alot of study \ and discussion about how to generate random numbers - even the most complex \ algorithms may well have underlying patterns that belie randomness.", FontSize->14]], "Text"], Cell[TextData[{ StyleBox["\tMathematica", FontSize->14, FontSlant->"Italic"], StyleBox[" has it's own 'uniformly distributed pseudorandom' number \ generator, contained in the function ", FontSize->14], StyleBox["Random", "Input", FontSize->14], StyleBox[". To simulate rolling a pair of dice, we ask ", FontSize->14], StyleBox["Mathematica", FontSize->14, FontSlant->"Italic"], StyleBox[" to generate random integers from the range of 1 to 6. We use \ the ", FontSize->14], StyleBox[":=", "Input", FontSize->14], StyleBox[" notation to tell ", FontSize->14], StyleBox["Mathematica", FontSize->14, FontSlant->"Italic"], StyleBox[" to 'evaluate later', that way, each time we ask for a ", FontSize->14], StyleBox["DieRoll", "Input", FontSize->14], StyleBox[", ", FontSize->14], StyleBox["Mathematica", FontSize->14, FontSlant->"Italic"], StyleBox[" will evaluate the ", FontSize->14], StyleBox["Random", "Input", FontSize->14], StyleBox[" function and select a new random integer.", FontSize->14] }], "Text"], Cell[BoxData[ \(DieRoll\ := \ Random[Integer, {1, 6}]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \({DieRoll, DieRoll}\)], "Input"], Cell[BoxData[ \({6, 1}\)], "Output"] }, Open ]], Cell[TextData[StyleBox["\tIn the casino spirit, we can also simulate drawing \ one card from a deck of cards by randomly choosing one of the four suits and \ one of the thirteen ranks. The integer we choose represents the position of \ the rank or suit in the appropriate list.", FontSize->14]], "Text"], Cell[BoxData[{ \(\(Suits\ = \ {"\", "\", "\", \ "\"};\)\), "\[IndentingNewLine]", \(\(Ranks\ = \ {"\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\"};\)\), "\[IndentingNewLine]", \(Suitpick\ := \ Random[Integer, {1, 4}]\), "\[IndentingNewLine]", \(Rankpick\ := \ Random[Integer, {1, 13}]\)}], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Card\ = \ Ranks[\([Rankpick]\)]\ <> "\< of \>" <> Suits[\([Suitpick]\)]\)], "Input"], Cell[BoxData[ \("Seven of Diamonds"\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["Estimation of Pi\n", "Section"], "\tThe 'classic' problem for introducing Monte Carlo methods is a model for \ estimating the value of \[Pi]. 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", StyleBox["Given that we know ", FontSize->14], Cell[BoxData[ \(TraditionalForm\`p\_in\)]], StyleBox["and ", FontSize->14], Cell[BoxData[ \(TraditionalForm\`p\_total\)]], StyleBox[", we can make an estimate for \[Pi], where ", FontSize->14], Cell[BoxData[ \(TraditionalForm\`\[Pi]\_est = \(\(4\) \( p\_in\/p\_total\)\(\ \ \)\)\)]], StyleBox[". ", FontSize->14] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(PiGuess\ = \ N[\((4*InCirc)\)/AllPoints]\)], "Input"], Cell[BoxData[ \(3.16`\)], "Output"] }, Open ]], Cell[TextData[StyleBox["\tNot a bad estimate! Probability tends to work even \ better 'in the long run', so we would expect better results if we select more \ random points. This is much easier if we compile everything we've done into \ one guessing function that takes the number of points as an argument, and \ returns an estimate:", FontSize->14]], "Text"], Cell[BoxData[ \(PiGuessFn[NumPoints_]\ := \[IndentingNewLine]\((Radius\ = \ 1.0; \n InCirc\ = \ 0; \nOutCirc\ = \ 0; \n PointList\ = \ Table[\[IndentingNewLine]\((Ranx\ = \ Random[Real, {\(-Radius\), Radius}]; \[IndentingNewLine]Rany\ = \ Random[Real, {\(-Radius\), Radius}]; \[IndentingNewLine]Dist\ = \ \@\(Ranx\^2 + \ \ Rany\^2\); \[IndentingNewLine]If[ Dist < 1, \(InCirc++\), \(OutCirc++\)]; \ \[IndentingNewLine]{Ranx, Rany})\), {NumPoints}]; \n N[\((4*InCirc)\)/NumPoints])\)\)], "Input"], Cell[TextData[StyleBox["\tThe general trade-off in computational modeling is \ that getting better results takes more computer time. Calculating with one \ hundred thousand points might take a bit longer, but should give us an even \ better estimate!", FontSize->14]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(PiGuessFn[100000]\)], "Input"], Cell[BoxData[ \(3.14236`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(N[Pi]\)], "Input"], Cell[BoxData[ \(3.141592653589793`\)], "Output"] }, Open ]], Cell[TextData[StyleBox["\tThe trade-off between accuracy and expense is \ interesting to many computational scientists. Because we have a good \ 'correct' answer to the Pi estimation, we can determine how much error each \ of our simulations has. We can then demonstrate how a sequence of \ increasingly 'expensive' (in terms of computer time) simulations will follow \ a general trend of smaller errors.", FontSize->14]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(PiGuessList\ = \ Table[{2^i, Abs[Pi - PiGuessFn[2^i]]}, {i, 2, 18}]\)], "Input"], Cell[BoxData[ \({{4, 0.8584073464102069`}, {8, 0.3584073464102069`}, {16, 0.3584073464102069`}, {32, 0.2665926535897931`}, {64, 0.016592653589793116`}, {128, 0.10840734641020688`}, {256, 0.06346765358979312`}, {512, 0.14159265358979312`}, {1024, 0.014657346410206884`}, {2048, 0.020498903589793116`}, {4096, 0.025381716089793116`}, {8192, 0.007821408910206884`}, {16384, 0.001473752660206884`}, {32768, 0.000497190160206884`}, {65536, 0.004708615941456884`}, {131072, 0.010110227269581884`}, {262144, 0.001242311792918116`}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ListPlot[PiGuessList[\([All, 2]\)], PlotJoined \[Rule] True, \[IndentingNewLine]AxesLabel \[Rule] \ {"\", 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