# Examples

## Control systems

This example shows how to simulate control systems (using ControlSystems.jl) with uncertain parameters. We calculate and display Bode diagrams, Nyquist diagrams and time-domain responses. We also illustrate how the package ControlSystemIdentification.jl interacts with MonteCarloMeasurements to facilitate the creation and analysis of uncertain systems.

We also perform some limited benchmarks.

The package RobustAndOptimalControl.jl contains lots of additional tools to work with linear systems with uncertainty represented as `Particles`

. See the documentation on Uncertainty modeling for several examples.

## Latin Hypercube Sampling

We show how to initialize particles with LHS and how to make sure the sample gets the desired moments. We also visualize the statistics of the sample.

## How MC uncertainty propagation works

We produce the first figure in this readme and explain in visual detail how different forms of uncertainty propagation propagates a probability distribution through a nonlinear function. Also see Comparison between linear uncertainty propagation and Monte-Carlo sampling for more visual examples.

## Robust probabilistic optimization

Here, we use MonteCarloMeasurements to perform robust optimization. With robust and probabilistic, we mean that we place some kind of bound on a quantile of an uncertain value, or otherwise make use of the probability distribution of some value that depend on the optimized parameters.

The application we consider is optimization of a PID controller. Normally, we are interested in controller performance and robustness against uncertainty. The robustness is often introduced by placing an upper bound on the, so called, sensitivity function. When the system to be controlled is parameterized by `Particles`

, we can penalize both variance in the performance measure as well as the 90:th quantile of the maximum of the sensitivity function. This example illustrates how easy it is to incorporate probabilistic constrains or cost functions in an optimization problem using `Particles`

.

## Autodiff and Robust optimization

Another example using MonteCarloMeasurements to perform robust optimization, this time with automatic differentiation. We use Optim.jl to solve a linear program with probabilistic constraints using 4 different methods, two gradient free, one first-order and one second-order method. We demonstrate calculation of gradients of uncertain functions with uncertain inputs using both Zygote.jl and ForwardDiff.jl.

## Unitful interaction

Particles with units can be created using the package Unitful.jl for uncertainty propagation with automatic unit checks. The interaction is only supported through the construct `Particles{Quantity}`

, whereas the reverse construct `Quantity{Particles}`

is likely to result in problems. Unitful particles are thus created like this

```
julia> using MonteCarloMeasurements, Unitful
julia> (1 ± 0.1)u"V"
1.0 V ± 0.09999231528930488 V Particles{Quantity{Float64, 𝐋^2 𝐌 𝐈^-1 𝐓^-3, Unitful.FreeUnits{(V,), 𝐋^2 𝐌 𝐈^-1 𝐓^-3, nothing}}, 2000}
julia> (1..2)u"m"
1.4999999999999998 m ± 0.28874729435961816 m Particles{Quantity{Float64, 𝐋, Unitful.FreeUnits{(m,), 𝐋, nothing}}, 2000}
```

## Monte-Carlo sampling properties

The variance introduced by Monte-Carlo sampling has some fortunate and some unfortunate properties. It decreases as 1/N, where N is the number of particles/samples. This unfortunately means that to get half the standard deviation in your estimate, you need to quadruple the number of particles. On the other hand, this variance does not depend on the dimension of the space, which is very fortunate.

In this package, we perform *systematic sampling* whenever possible. This approach exhibits lower variance than standard random sampling. Below, we investigate the variance of the mean estimator of a random sample from the normal distribution. The variance of the estimate of the mean is known to decrease as 1/N

```
default(l=(3,))
N = 1000
svec = round.(Int, exp10.(LinRange(1, 3, 50)))
vars = map(svec) do i
var(mean(randn(i)) for _ in 1:1000)
end
plot(svec, vars, yscale=:log10, xscale=:log10, lab="Random sampling", xlabel="\$N\$", ylabel="Variance")
plot!(svec, N->1/N, lab="\$1/N\$", l=(:dash,))
vars = map(svec) do i
var(mean(systematic_sample(i)) for _ in 1:1000)
end
plot!(svec, vars, lab="Systematic sampling")
plot!(svec, N->1/N^2, lab="\$1/N^2\$", l=(:dash,))
```

As we can see, the variance of the standard random sampling decreases as expected. We also see that the variance for the systematic sample is considerably lower, and also scales as (almost) 1/N².

A simplified implementation of the systematic sampler is given below

```
function systematic_sample(N, d=Normal(0,1))
e = rand()/N
y = e:1/N:1
map(x->quantile(d,x), y)
end
```

~~As we can see, a single random number is generated to seed the entire sample.~~ (This has been changed to `e=0.5/N`

to have a correct mean.) The samples are then drawn deterministically from the quantile function of the distribution.

## Variational inference

See blog post by @cscherrer for an example of variational inference using `Particles`

## Differential Equations

The tutorial for solving differential equations using `Measurement`

works for `Particles`

as well. A word of caution for actually using Measurements.jl in this example: while solving the pendulum on short time scales, linear uncertainty propagation works well, as evidenced by the below simulation of a pendulum with uncertain properties

```
function sim(±, tspan, plotfun=plot!; kwargs...)
g = 9.79 ± 0.02; # Gravitational constant
L = 1.00 ± 0.01; # Length of the pendulum
u₀ = [0 ± 0, π / 3 ± 0.02] # Initial speed and initial angle
#Define the problem
function simplependulum(du,u,p,t)
θ = u[1]
dθ = u[2]
du[1] = dθ
du[2] = -(g/L) * sin(θ)
end
prob = ODEProblem(simplependulum, u₀, tspan)
sol = solve(prob, Tsit5(), reltol = 1e-6)
plotfun(sol.t, getindex.(sol.u, 2); kwargs...)
end
tspan = (0.0, 5)
plot()
sim(Measurements.:±, tspan, label = "Linear", xlims=(tspan[2]-5,tspan[2]))
sim(MonteCarloMeasurements.:±, tspan, label = "MonteCarlo", xlims=(tspan[2]-5,tspan[2]))
```

The mean and errorbars for both Measurements and MonteCarloMeasurements line up perfectly when integrating over 5 seconds.

However, the uncertainty in the pendulum coefficients implies that the frequency of the pendulum oscillation is uncertain, when solving on longer time scales, this should result in the phase being completely unknown, something linear uncertainty propagation does not handle

```
tspan = (0.0, 200)
plot()
sim(Measurements.:±, tspan, label = "Linear", xlims=(tspan[2]-5,tspan[2]))
sim(MonteCarloMeasurements.:±, tspan, label = "MonteCarlo", xlims=(tspan[2]-5,tspan[2]))
```

We now integrated over 200 seconds and look at the last 5 seconds. This result maybe looks a bit confusing, the linear uncertainty propagation is very sure about the amplitude at certain points but not at others, whereas the Monte-Carlo approach is completely unsure. Furthermore, the linear approach thinks that the amplitude at some points is actually much higher than the starting amplitude, implying that energy somehow has been added to the system! The picture might become a bit more clear by plotting the individual trajectories of the particles

```
plot()
sim(Measurements.:±, tspan, label = "Linear", xlims=(tspan[2]-5,tspan[2]), l=(5,))
sim(MonteCarloMeasurements.:∓, tspan, mcplot!, label = "", xlims=(tspan[2]-5,tspan[2]), l=(:black,0.1))
```

It now becomes clear that each trajectory has a constant amplitude (although individual trajectories amplitudes vary slightly due to the uncertainty in the initial angle), but the phase is all mixed up due to the slightly different frequencies!

These problems grow with increasing uncertainty and increasing integration time. In fact, the uncertainty reported by Measurements.jl goes to infinity as the integration time does the same.

Of course, the added accuracy from using MonteCarloMeasurements does not come for free, as it costs some additional computation. We have the following timings for integrating the above system 100 seconds using three different uncertainty representations

```
Measurements.:± 14.596 ms (729431 allocations: 32.43 MiB) # Measurements.Measurement
MonteCarloMeasurements.:∓ 25.115 ms (25788 allocations: 24.68 MiB) # 100 StaticParticles
MonteCarloMeasurements.:± 345.730 ms (696212 allocations: 838.50 MiB) # 500 Particles
```

# MCMC inference using Soss.jl or Turing.jl

The probabilistic programming language Soss.jl has native support for converting the inference result to `Particles`

for further processing, see the Soss readme for further instruction.

Turing.jl is another probabilistic programming language, and an interface between Turing and MonteCarloMeasurements is provided by Turing2MonteCarloMeasurements.jl with instructions and examples in the readme.