# An Intro to Expectations via DiffEqUncertainty.jl

## System Model

First, lets consider the following linear model.

$u' = p u$

f(u,p,t) = p.*u

f (generic function with 1 method)


We then wish to solve this model on the timespan t=0.0 to t=10.0, with an intial condition u0=10.0 and parameter p=-0.3. We can then setup the differential equations, solve, and plot as follows

using DifferentialEquations, Plots
u0 = [10.0]
p = [-0.3]
tspan = (0.0,10.0)
prob = ODEProblem(f,u0,tspan,p)
sol = solve(prob)
plot(sol)
ylims!(0.0,10.0)


However, what if we wish to consider a random initial condition? Assume u0 is distributed uniformly from -10.0 to 10.0, i.e.,

using Distributions
u0_dist = [Uniform(-10.0,10.0)]

1-element Array{Distributions.Uniform{Float64},1}:
Distributions.Uniform{Float64}(a=-10.0, b=10.0)


We can then run a Monte Carlo simulation of 100,000 trajectories by solving an EnsembleProblem.

prob_func(prob,i,repeat) = remake(prob, u0 = rand.(u0_dist))
ensemble_prob = EnsembleProblem(prob,prob_func=prob_func)


EnsembleSolution Solution of length 100000 with uType:
DiffEqBase.ODESolution{Float64,2,Array{Array{Float64,1},1},Nothing,Nothing,
Array{Float64,1},Array{Array{Array{Float64,1},1},1},DiffEqBase.ODEProblem{A
rray{Float64,1},Tuple{Float64,Float64},false,Array{Float64,1},DiffEqBase.OD
EFunction{false,typeof(Main.##WeaveSandBox#253.f),LinearAlgebra.UniformScal
ing{Bool},Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,N
othing,Nothing,Nothing,Nothing},Base.Iterators.Pairs{Union{},Union{},Tuple{
},NamedTuple{(),Tuple{}}},DiffEqBase.StandardODEProblem},OrdinaryDiffEq.Tsi
t5,OrdinaryDiffEq.InterpolationData{DiffEqBase.ODEFunction{false,typeof(Mai
n.##WeaveSandBox#253.f),LinearAlgebra.UniformScaling{Bool},Nothing,Nothing,
Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Not
hing},Array{Array{Float64,1},1},Array{Float64,1},Array{Array{Array{Float64,
1},1},1},OrdinaryDiffEq.Tsit5ConstantCache{Float64,Float64}},DiffEqBase.DES
tats}


Plotting the first 250 trajectories produces

plot(ensemblesol, vars = (0,1), lw=1,alpha=0.1, label=nothing, idxs = 1:250)