Julia has both a large number of useful, well written libraries and many incomplete poorly maintained proofs of concept.
A major advantage of Julia libraries is that, because Julia itself is sufficiently fast, there is less need to mix in low level languages like C and Fortran.
As a result, most Julia libraries are written exclusively in Julia.
Not only does this make the libraries more portable, it makes them much easier to dive into, read, learn from and modify.
In this lecture we introduce a few of the Julia libraries that we’ve found particularly useful for quantitative work in economics.
using InstantiateFromURL # optionally add arguments to force installation: instantiate = true, precompile = true github_project("QuantEcon/quantecon-notebooks-julia", version = "0.8.0")
using LinearAlgebra, Statistics using QuantEcon, QuadGK, FastGaussQuadrature, Distributions, Expectations using Interpolations, Plots, LaTeXStrings, ProgressMeter
Many applications require directly calculating a numerical derivative and calculating expectations.
using QuadGK @show value, tol = quadgk(cos, -2π, 2π);
(value, tol) = quadgk(cos, -2π, 2π) = (-1.5474478810961125e-14, 5.7846097329025695e-24)
This is an adaptive Gauss-Kronrod integration technique that’s relatively accurate for smooth functions.
However, its adaptive implementation makes it slow and not well suited to inner loops.
using FastGaussQuadrature x, w = gausslegendre( 100_000 ); # i.e. find 100,000 nodes # integrates f(x) = x^2 from -1 to 1 f(x) = x^2 @show w ⋅ f.(x); # calculate integral
w ⋅ f.(x) = 0.6666666666666667
The only problem with the
FastGaussQuadrature package is that you will need to deal with affine transformations to the non-default domains yourself.
QuantEcon.jl has routines for Gaussian quadrature that translate the domains.
using QuantEcon x, w = qnwlege(65, -2π, 2π); @show w ⋅ cos.(x); # i.e. on [-2π, 2π] domain
w ⋅ cos.(x) = -3.0064051806277455e-15
If the calculations of the numerical integral is simply for calculating mathematical expectations of a particular distribution, then Expectations.jl provides a convenient interface.
Under the hood, it is finding the appropriate Gaussian quadrature scheme for the distribution using
using Distributions, Expectations dist = Normal() E = expectation(dist) f(x) = x @show E(f) #i.e. identity # Or using as a linear operator f(x) = x^2 x = nodes(E) w = weights(E) E * f.(x) == f.(x) ⋅ w
E(f) = -6.991310601309959e-18
In economics we often wish to interpolate discrete data (i.e., build continuous functions that join discrete sequences of points).
The package we usually turn to for this purpose is Interpolations.jl.
There are a variety of options, but we will only demonstrate the convenient notations.
Univariate with a Regular Grid¶
Let’s start with the univariate case.
We begin by creating some data points, using a sine function
using Interpolations using Plots gr(fmt=:png); x = -7:7 # x points, coase grid y = sin.(x) # corresponding y points xf = -7:0.1:7 # fine grid plot(xf, sin.(xf), label = "sin function") scatter!(x, y, label = "sampled data", markersize = 4)
li = LinearInterpolation(x, y) li_spline = CubicSplineInterpolation(x, y) @show li(0.3) # evaluate at a single point scatter(x, y, label = "sampled data", markersize = 4) plot!(xf, li.(xf), label = "linear") plot!(xf, li_spline.(xf), label = "spline")
li(0.3) = 0.25244129544236954
Univariate with Irregular Grid¶
In the above, the
LinearInterpolation function uses a specialized function
for regular grids since
x is a
For an arbitrary, irregular grid
x = log.(range(1, exp(4), length = 10)) .+ 1 # uneven grid y = log.(x) # corresponding y points interp = LinearInterpolation(x, y) xf = log.(range(1, exp(4), length = 100)) .+ 1 # finer grid plot(xf, interp.(xf), label = "linear") scatter!(x, y, label = "sampled data", markersize = 4, size = (800, 400))
f(x,y) = log(x+y) xs = 1:0.2:5 ys = 2:0.1:5 A = [f(x,y) for x in xs, y in ys] # linear interpolation interp_linear = LinearInterpolation((xs, ys), A) @show interp_linear(3, 2) # exactly log(3 + 2) @show interp_linear(3.1, 2.1) # approximately log(3.1 + 2.1) # cubic spline interpolation interp_cubic = CubicSplineInterpolation((xs, ys), A) @show interp_cubic(3, 2) # exactly log(3 + 2) @show interp_cubic(3.1, 2.1) # approximately log(3.1 + 2.1);
interp_linear(3, 2) = 1.6094379124341003 interp_linear(3.1, 2.1) = 1.6484736801441782 interp_cubic(3, 2) = 1.6094379124341 interp_cubic(3.1, 2.1) = 1.6486586594237707
The standard library contains many useful routines for linear algebra, in
addition to standard functions such as
Routines are available for
- Cholesky factorization
- LU decomposition
- Singular value decomposition,
- Schur factorization, etc.
See here for further details.
using LaTeXStrings L"an equation: $1 + \alpha^2$"
using ProgressMeter @showprogress 1 "Computing..." for i in 1:50 sleep(0.1) # some computation.... end
Computing...100%|███████████████████████████████████████| Time: 0:00:05