Intermediate Problem Answers

MyRange and MyLinSpace Problem

Part 1

In [4]:
struct MyRange
    start
    step
    stop
end

function _MyRange(a::MyRange,i::Int)
    tmp = a.start + a.step*(i-1)
    if tmp > a.stop
        error("Index is out of bounds!")
    else
        return tmp
    end
end
Out[4]:
_MyRange (generic function with 1 method)
In [5]:
a = MyRange(1,2,20)
_MyRange(a,5) == (1:2:20)[5]
Out[5]:
true
In [6]:
Base.getindex(a::MyRange,i::Int) = _MyRange(a,i)
a[5]
Out[6]:
9

Part 2

In [7]:
struct MyLinSpace
    start
    stop
    n
end

function Base.getindex(a::MyLinSpace,i::Int)
    dx = (a.stop-a.start)/a.n
    a.start + dx*(i-1)
end
In [8]:
l = MyLinSpace(1,2,50)
l[6]
Out[8]:
1.1
In [10]:
range(1,stop=2,length=50)[6]
Out[10]:
1.1020408163265305

Part 3

In [11]:
(a::MyRange)(x) = a.start + a.step*(x-1)
a = MyRange(1,2,20)
a(1.1)
Out[11]:
1.2000000000000002

Part 4

In [12]:
using Unitful
a = MyRange(1u"kg",2u"kg",20u"kg")
a[5]
Out[12]:
9 kg

Operator Problem

In [19]:
struct StrangMatrix end
A = StrangMatrix()

using LinearAlgebra
function LinearAlgebra.mul!(C,A::StrangMatrix,B::AbstractVector)
    for i in 2:length(B)-1
        C[i] = B[i-1] - 2B[i] + B[i+1]
    end
    C[1] = -2B[1] + B[2]
    C[end] = B[end-1] - 2B[end]
    C
end

Base.:*(A::StrangMatrix,B::AbstractVector) = (C = similar(B); mul!(C,A,B))
In [20]:
A*ones(10)
Out[20]:
10-element Array{Float64,1}:
 -1.0
  0.0
  0.0
  0.0
  0.0
  0.0
  0.0
  0.0
  0.0
 -1.0
In [21]:
struct SizedStrangMatrix
    size
end

Base.eltype(A::SizedStrangMatrix) = Float64
Base.size(A::SizedStrangMatrix) = A.size
Base.size(A::SizedStrangMatrix,i::Int) = A.size[i]
In [22]:
b = sin.(0:0.1:2π)
Out[22]:
63-element Array{Float64,1}:
  0.0                
  0.09983341664682815
  0.19866933079506122
  0.2955202066613396 
  0.3894183423086505 
  0.479425538604203  
  0.5646424733950355 
  0.6442176872376911 
  0.7173560908995228 
  0.7833269096274834 
  0.8414709848078965 
  0.8912073600614354 
  0.9320390859672264 
  ⋮                  
 -0.9258146823277321 
 -0.8834546557201531 
 -0.8322674422239008 
 -0.7727644875559871 
 -0.7055403255703919 
 -0.6312666378723208 
 -0.5506855425976376 
 -0.4646021794137566 
 -0.373876664830236  
 -0.27941549819892586
 -0.18216250427209502
 -0.0830894028174964
In [24]:
A = SizedStrangMatrix((length(b),length(b)))

function LinearAlgebra.mul!(C,A::SizedStrangMatrix,B)
    for i in 2:length(B)-1
        C[i] = B[i-1] - 2B[i] + B[i+1]
    end
    C[1] = -2B[1] + B[2]
    C[end] = B[end-1] - 2B[end]
    C
end
Base.:*(A::SizedStrangMatrix,B::AbstractVector) = (C = similar(B); mul!(C,A,B))
In [29]:
using IterativeSolvers
x = gmres(A,b,tol=1e-14)
Out[29]:
63-element Array{Float64,1}:
   -9.755738784527166
  -19.511766690859723
  -29.168509953717926
  -38.62733151652308 
  -47.79172939964289 
  -56.567920000176755
  -64.86624757504093 
  -72.60180867498484 
  -79.69549984767762 
  -86.07440664159829 
  -91.67293002909796 
  -96.43316292771152 
 -100.30568365156476 
    ⋮                
   92.73105600644514 
   88.6304815198612  
   83.64512976504756 
   77.82628337498842 
   71.23343662444809 
   63.93401241944272 
   56.00167853409517 
   47.51925322977693 
   38.572331686261165
   29.25162537589525 
   19.65105899819377 
    9.867868177542748
In [30]:
A*x - b
Out[30]:
63-element Array{Float64,1}:
 -0.00028912180539109045
 -0.0005487731724762024 
 -0.0007476307420101191 
 -0.0010965269760021545 
 -0.0012110597227052433 
 -0.0015625129345115951 
 -0.0018759984747689362 
 -0.00234775998656922   
 -0.0025717121273999277 
 -0.002943503206488285  
 -0.0031804959217944795 
 -0.00349518530111137   
 -0.00378335609624314   
  ⋮                     
 -0.001396616777037063  
 -0.0013226125095587848 
 -0.001227193021585582  
 -0.0012358729252088807 
 -0.0010371288946419144 
 -0.0016430424698667956 
  0.0005941236269499939 
  0.00010594021623389072
  9.189798008585326e-5  
 -0.0004445691366434801 
 -0.000461938677442153  
 -0.001587954074231443

Regression Problem

In [33]:
#### Prepare Data

X = rand(1000, 3)               # feature matrix
a0 = rand(3)                    # ground truths
y = X * a0 + 0.1 * randn(1000);  # generate response

X2 = hcat(X,ones(1000))
println(X2\y)

using MultivariateStats
println(llsq(X,y))

using DataFrames, GLM
data = DataFrame(X1=X[:,1], X2=X[:,2], X3=X[:,3],Y=y)
OLS = lm(@formula(Y ~ X1 + X2 + X3), data)


X = rand(100);
y = 2X  + 0.1 * randn(100);

using Plots
b = X\y
println(b)
gr()
scatter(X,y)
Plots.abline!(b[1],0.0, lw=3) # Slope,Intercept

[0.400269, 0.834389, 0.0869948, -0.00661511]

┌ Info: Precompiling MultivariateStats [6f286f6a-111f-5878-ab1e-185364afe411]
└ @ Base loading.jl:1186
ERROR: LoadError: UndefVarError: LinAlg not defined
Stacktrace:
 [1] include at ./boot.jl:317 [inlined]
 [2] include_relative(::Module, ::String) at ./loading.jl:1038
 [3] include(::Module, ::String) at ./sysimg.jl:29
 [4] top-level scope at none:2
 [5] eval at ./boot.jl:319 [inlined]
 [6] eval(::Expr) at ./client.jl:389
 [7] top-level scope at ./none:3
in expression starting at /home/chrisrackauckas/.julia/packages/MultivariateStats/wGpiN/src/MultivariateStats.jl:8

Failed to precompile MultivariateStats [6f286f6a-111f-5878-ab1e-185364afe411] to /home/chrisrackauckas/.julia/compiled/v1.0/MultivariateStats/l7I74.ji.

Stacktrace:
 [1] error(::String) at ./error.jl:33
 [2] macro expansion at ./logging.jl:313 [inlined]
 [3] compilecache(::Base.PkgId, ::String) at ./loading.jl:1184
 [4] macro expansion at ./logging.jl:311 [inlined]
 [5] _require(::Base.PkgId) at ./loading.jl:941
 [6] require(::Base.PkgId) at ./loading.jl:852
 [7] macro expansion at ./logging.jl:311 [inlined]
 [8] require(::Module, ::Symbol) at ./loading.jl:834
 [9] top-level scope at In[33]:10

Type Hierarchy Problem

In [34]:
abstract type AbstractPerson end
abstract type AbstractStudent <: AbstractPerson end

struct Person <: AbstractPerson
    name
end

struct Student <: AbstractStudent
    name
    grade
end

struct GraduateStudent <: AbstractStudent
    name
    grade
end

person_info(p::AbstractPerson) = println(p.name)
person_info(s::AbstractStudent) = (println(s.name); println(s.grade))
Out[34]:
person_info (generic function with 2 methods)
In [35]:
person_info(Person("Bob"))

Bob
In [36]:
person_info(Student("Bob",2))

Bob
2
In [37]:
person_info(GraduateStudent("Bob",2))

Bob
2

Distribution Dispatch Problem

This is from Josh Day's talk: https://www.youtube.com/watch?v=EwcTNzpQ6Sc

Solution is from: https://github.com/joshday/Talks/blob/master/SLG2016_IntroToJulia/Slides.ipynb

In [39]:
using Distributions
function myquantile(d::UnivariateDistribution, q::Number)
    θ = mean(d)
    tol = Inf
    while tol > 1e-5
        θold = θ
        θ = θ - (cdf(d, θ) - q) / pdf(d, θ)
        tol = abs(θold - θ)
    end
    θ
end

for dist in [Gamma(5, 1), Normal(0, 1), Beta(2, 4)]
    @show myquantile(dist, .75)
    @show quantile(dist, .75)
    println()
end

myquantile(dist, 0.75) = 6.274430698436519
quantile(dist, 0.75) = 6.2744306984446885

myquantile(dist, 0.75) = 0.6744897501960708
quantile(dist, 0.75) = 0.6744897501960818

myquantile(dist, 0.75) = 0.45418056477357555
quantile(dist, 0.75) = 0.4541805647736157