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1.2. ANINTRODUCTORYEXAMPLE
21
git clone https://github.com/QuantEcon/QuantEcon.jl
Thisisjustgit cloneinfrontoftheURLfortherepository
Evenbetter,signuptoGitHub—it’sfree
Lookinto‘forking’GitHubrepositories
(Loosely speaking, , forkingmeansmaking g your r own copyof f aGitHub repository, stored on
GitHub)
TryforkingtheQuantEconrepositoryforthecourse
Nowtrycloningittosomelocaldirectory,makingedits,addingandcommittingthem,andpush-
ingthembackuptoyourforkedGitHubrepo
Forreadingontheseandothertopics,try
• TheofficialGitdocumentation
• ReadingthroughthedocsonGitHub
1.2 AnIntroductoryExample
Contents
• AnIntroductoryExample
– Overview
– Example:PlottingaWhiteNoiseProcess
– Exercises
– Solutions
Overview
We’renowreadytostartlearningtheJulialanguageitself
Ourapproachisaimedatthosewhoalreadyhaveatleastsomeknowledgeofprogramming—
perhapsexperiencewithPython,MATLAB,R,Corsimilar
Inparticular, weassumeyou havesomefamiliaritywithfundamentalprogrammingconcepts
suchas
• variables
• loops
• conditionals(if/else)
IfyouhavenosuchprogrammingexperiencewehumblysuggestyoutryPythonfirst
Pythonisagreatfirstlanguageand,moreimportantly,therearemany,manyintroductorytreat-
ments
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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1.2. ANINTRODUCTORYEXAMPLE
22
InfactourtreatmentofPythonismuchslowerthanourtreatmentofJulia,especiallyatthestart
OnceyouarecomfortablewithPythonyou’llfindtheleaptoJuliaiseasy
Approach InthislecturewewillwriteandthenpickapartsmallJuliaprograms
Atthisstagetheobjectiveistointroduceyoutobasicsyntaxanddatastructures
Deeperconcepts—howthingswork—willbecoveredinlaterlectures
Sincewearelookingforsimplicitytheexamplesarealittlecontrived
OtherReferences ThedefinitivereferenceisJulia’sowndocumentation
Themanualisthoughtfullywrittenbutalsoquitedense(andsomewhatevangelical)
Thepresentationinthisandourremaininglecturesismoreofatutorialstylebasedaroundexam-
ples
Example: Plotting g aWhiteNoiseProcess
Tobegin,let’ssupposethatwewanttosimulateandplotthewhitenoiseprocesse
0
,e
1
,...,e
T
,
whereeachdrawe
t
isindependentstandardnormal
Inotherwords,wewanttogeneratefiguresthatlooksomethinglikethis:
ThisisstraightforwardusingthePyPlotlibraryweinstalledearlier
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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1.2. ANINTRODUCTORYEXAMPLE
23
using PyPlot
ts_length 100
epsilon_values randn(ts_length)
plot(epsilon_values, "b-")
YoushouldbeabletorunthatcodeeitherinIJuliaorinthestandardREPL(thebasicinterpreter)
Inbrief,
• using PyPlotmakesthefunctionalityinPyPlotavailableforuse
– Inparticular,itpullsthenamesexportedbythePyPlotmoduleintotheglobalscope
– Oneoftheseisplot(),whichinturncallstheplotfunctionfromMatplotlib
• randn()isaJuliafunctionfromthestandardlibraryforgeneratingstandardnormals
ImportingFunctions Theeffectofthestatementusing PyPlotistomakeallthenamesexported
bythePyPlotmoduleavailableintheglobalscope
Ifyouprefertobemoreselectiveyoucanreplaceusing PyPlotwithimport PyPlot: : plot
Nowonlytheplotfunctionisaccessible
Sinceourprogramusesonlytheplotfunctionfromthismodule,eitherwouldhaveworkedinthe
previousexample
Arrays Thefunctioncallepsilon_values = randn(ts_length)createsoneofthemostfunda-
mentalJuliadatatypes:anarray
julia> typeof(epsilon_values)
Array{Float64,1}
julia> epsilon_values
100-element Array{Float64,1}:
-0.908823
-0.759142
-1.42078
0.792799
0.577181
1.74219
-0.912529
1.06259
0.5766
-0.0172788
-0.591671
-1.02792
...
-1.29412
-1.12475
0.437858
-0.709243
-1.96053
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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1.2. ANINTRODUCTORYEXAMPLE
24
1.31092
1.19819
1.54028
-0.246204
-1.23305
-1.16484
Theinformationfromtypeof()tellsusthatepsilon_valuesisanarrayof64bitfloatingpoint
values,ofdimension1
Juliaarraysarequiteflexible—theycanstoreheterogeneousdataforexample
julia> [10"foo", false]
3-element Array{Any,1}:
10
"foo"
false
NoticenowthatthedatatypeisrecordedasAny,sincethearraycontainsmixeddata
Thefirstelementofxisaninteger
julia> typeof(x[1])
Int64
Thesecondisastring
julia> typeof(x[2])
ASCIIString (constructor with 2 methods)
Thethirdisthebooleanvaluefalse
julia> typeof(x[3])
Bool
Noticefromtheabovethat
• arrayindicesstartat1(unlikePython,wherearraysarezero-based)
• arrayelementsarereferencedusingsquarebrackets(unlikeMATLABandFortran)
Juliacontainsmanyfunctionsforactingonarrays—we’llreviewthemlater
Fornowhere’sseveralexamples,appliedtothesamelistx = [10, "foo", false]
julia> length(x)
3
julia> pop!(x)
false
julia> x
2-element Array{Any,1}:
10
"foo"
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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1.2. ANINTRODUCTORYEXAMPLE
25
julia> push!(x, "bar")
3-element Array{Any,1}:
10
"foo"
"bar"
julia> x
3-element Array{Any,1}:
10
"foo"
"bar"
Thefirstexamplejustreturnsthelengthofthelist
Thesecond,pop!(),popsthelastelementoffthelistandreturnsit
Indoingsoitchangesthelist(bydroppingthelastelement)
Becauseofthiswecallpop!amutatingmethod
It’sconventionalinJuliathatmutatingmethodsendin!toremindtheuserthatthefunctionhas
othereffectsbeyondjustreturningavalue
Thefunctionpush!()issimilar,expectthatitappendsitssecondargumenttothearray
ForLoops Althoughthere’snoneedintermsofwhatwewantedtoachievewithourprogram,
forthesakeoflearningsyntaxlet’srewriteourprogramtouseaforloop
using PyPlot
ts_length 100
epsilon_values Array(Float64, ts_length)
for in 1:ts_length
epsilon_values[i] randn()
end
plot(epsilon_values, "b-")
Herewefirstdeclaredepsilon_valuestobeanemptyarrayforstoring64bitfloatingpointnum-
bers
Theforloopthenpopulatesthisarraybysuccessivecallstorandn()
• Calledwithoutanargument,randn()returnsasinglefloat
LikeallcodeblocksinJulia, theendoftheforloopcodeblock(whichisjustonelinehere)is
indicatedbythekeywordend
Thewordinfromtheforloopcanbereplacedbysymbol=
Theexpression1:ts_lengthcreatesaniteratorthatisloopedover—inthiscasetheintegersfrom
1tots_length
Iteratorsarememoryefficientbecausetheelementsaregeneratedontheflyratherthanstoredin
memory
InJuliayoucanalsoloopdirectlyoverarraysthemselves,likeso
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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1.2. ANINTRODUCTORYEXAMPLE
26
words ["foo""bar"]
for word in words
println("Hello $word")
end
Theoutputis
Hello foo
Hello bar
WhileLoops Thesyntaxforthewhileloopcontainsnosurprises
using PyPlot
ts_length 100
epsilon_values Array(Float64, ts_length)
1
while <= ts_length
epsilon_values[i] randn()
1
end
plot(epsilon_values, "b-")
Thenextexampledoesthesamethingwithaconditionandthebreakstatement
using PyPlot
ts_length 100
epsilon_values Array(Float64, ts_length)
1
while true
epsilon_values[i] randn()
1
if ts_length
break
end
end
plot(epsilon_values, "b-")
User-DefinedFunctions Forthesakeoftheexercise,let’snowgobacktotheforloopbutre-
structureourprogramsothatgenerationofrandomvariablestakesplacewithinauser-defined
function
using PyPlot
function generate_data(n)
epsilon_values Array(Float64, n)
for 1:n
epsilon_values[i] randn()
end
return epsilon_values
end
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
1.2. ANINTRODUCTORYEXAMPLE
27
ts_length 100
data generate_data(ts_length)
plot(data, "b-")
Here
• functionisaJuliakeywordthatindicatesthestartofafunctiondefinition
• generate_dataisanarbitrarynameforthefunction
• returnisakeywordindicatingthereturnvalue
ASlightlyMoreUsefulFunction Ofcoursethefunctiongenerate_dataiscompletelycontrived
Wecouldjustwritethefollowingandbedone
ts_length 100
data randn(ts_length)
plot(data, "b-")
Let’smakeaslightlymoreusefulfunction
Thisfunctionwillbepassedachoiceofprobabilitydistributionandrespondbyplottingahis-
togramofobservations
Indoingsowe’llmakeuseoftheDistributionspackage
julia> Pkg.add("Distributions")
Here’sthecode
using PyPlot
using Distributions
function plot_histogram(distribution, n)
epsilon_values rand(distribution, , n) # n draws from distribution
plt[:hist](epsilon_values)
end
lp Laplace()
plot_histogram(lp, 500)
Theresultingfigurelookslikethis
Let’shaveacasualdiscussionofhowallthisworkswhileleavingtechnicaldetailsforlaterinthe
lectures
First,lp = Laplace()createsaninstanceofadatatypedefinedintheDistributionsmodulethat
representstheLaplacedistribution
Thenamelpisboundtothisobject
Whenwemakethefunctioncallplot_histogram(lp, 500)thecodeinthebodyofthefunction
plot_histogramisrunwith
• thenamedistributionboundtothesameobjectaslp
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
1.2. ANINTRODUCTORYEXAMPLE
28
• thenamenboundtotheinteger500
AMystery Nowconsiderthefunctioncallrand(distribution, n)
Thislookslikesomethingofamystery
Thefunctionrand()isdefinedinthebaselibrarysuchthatrand(n)returnsnuniformrandom
variableson[0,1)
julia> rand(3)
3-element Array{Float64,1}:
0.856817
0.981502
0.510947
Ontheotherhand,distributionpointstoadatatyperepresentingtheLaplacedistributionthat
hasbeendefinedinathirdpartypackage
Sohowcanitbethatrand()isabletotakethiskindofobjectasanargumentandreturntheoutput
thatwewant?
Theanswerinanutshellismultipledispatch
ThisreferstotheideathatfunctionsinJuliacanhavedifferentbehaviordependingonthepartic-
ularargumentsthatthey’repassed
HenceinJuliawecantakeanexistingfunctionandgiveitanewbehaviorbydefininghowitacts
onanewtypeofobject
Theinterpreterknowswhichfunctiondefinitiontoapplyinagivensettingbylookingatthetypes
oftheobjectsthefunctioniscalledon
InJuliathesealternativeversionsofafunctionarecalledmethods
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
1.2. ANINTRODUCTORYEXAMPLE
29
Exercises
Exercise1 Recallthatn!isreadas“nfactorial”anddefinedasn!=n(1)21
InJuliayoucancomputethisvaluewithfactorial(n)
Writeyourownversionofthisfunction,calledfactorial2,usingaforloop
Exercise2 ThebinomialrandomvariableYBin(n,p)represents
• numberofsuccessesinnbinarytrials
• eachtrialsucceedswithprobabilityp
Usingonlyrand()fromthesetofJulia’sbuiltinrandomnumbergenerators(nottheDistributions
package),writeafunctionbinomial_rvsuchthatbinomial_rv(n, p)generatesonedrawofY
Hint: IfUisuniformon(0,1)and2(0,1),thentheexpressionU U < pevaluatestotruewith
probabilityp
Exercise3 ComputeanapproximationtopusingMonteCarlo
Forrandomnumbergenerationuseonlyrand()
Yourhintsareasfollows:
• IfUisabivariateuniformrandomvariableontheunitsquare(0,1)
2
,thentheprobability
thatUliesinasubsetBof(0,1)
2
isequaltotheareaofB
• IfU
1
,...,U
n
areiidcopiesofU,then,asngetslarge,thefractionthatfallsinBconvergesto
theprobabilityoflandinginB
• Foracircle,area=pi*radius^2
Exercise4 Writeaprogramthatprintsonerealizationofthefollowingrandomdevice:
• Flipanunbiasedcoin10times
• If3consecutiveheadsoccuroneormoretimeswithinthissequence,payonedollar
• Ifnot,paynothing
Onceagainuseonlyrand()asyourrandomnumbergenerator
Exercise5 Simulateandplotthecorrelatedtimeseries
x
t+1
=ax
t
+e
t+1
where x
0
=0 and t=0,...,T
Thesequenceofshocksfe
t
gisassumedtobeiidandstandardnormal
SetT=200anda=0.9
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
1.2. ANINTRODUCTORYEXAMPLE
30
Exercise6 Todothenextexercise,youwillneedtoknowhowtoproduceaplotlegend
Thefollowingexampleshouldbesufficienttoconveytheidea
using PyPlot
randn(100)
plot(x, "b-", label="white noise")
legend()
Runningitproducesafigurelikeso
Now,plotthreesimulatedtimeseries,oneforeachofthecasesa=0,a=0.8anda=0.98
Inparticular,youshouldproduce(modulorandomness)afigurethatlooksasfollows
(Thefigureillustrateshowtimeserieswiththesameone-step-aheadconditionalvolatilities, as
thesethreeprocesseshave,canhaveverydifferentunconditionalvolatilities.)
Hints:
• Ifyoucalltheplot()function n multipletimesbeforecallingshow(), allofthelinesyou
producewillenduponthesamefigure
• Ifyouomittheargument"b-"totheplotfunction,PyPlotwillautomaticallyselectdifferent
colorsforeachline
Solutions
Solutionnotebook
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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