## BasisFunctionExpansions.jl

Basis Function Expansions for Julia
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23 Stars
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February 2017

# BasisFunctionExpansions

A Julia toolbox for approximation of functions using basis-function expansions (BFEs).

BFEs are useful when one wants to estimate an arbitrary/unknown/complicated functional relationship between (in the simple case) two variables, `y` and `v`. In simple linear regression, we might consider a functional relationship `y = ϕ(v) = αv + β`, with parameters `α` and `β`. However, if the function `ϕ` has an arbitrary nonlinar form, it might be hard to come up with suitable basis functions to use for linear regression. This package provides a set of convenient methods to estimate `ϕ(v)` as a linear combination of basis functions, such as radial basis functions, for situations where `v` has a single or multiple dimensions.

Currently supported basis functions are

• Radial Basis Functions `UniformRBFE, MultiRBFE, MultiUniformRBFE, MultiDiagonalRBFE`

# Usage

We demonstrate typical usage with some examples. Further usage examples in the context of reinforcement learning are provided at baggepinnen.github.io

The idea is to create an object representing an expansion. This object contains information regarding the domain of the expansion, which type of basis functions used and how many. These objects are, once created, callable with a scheduling vector/matrix. A call like this returns a vector/matrix of basis function activations.

To reconstruct a signal, a linear combination of basis functions must be estimated. To facilitate this, a second type of object is available: `BasisFunctionApproximation`. Once created, `BasisFunctionApproximation`s are callable with a scheduling signal and return an estimate of the output. The parameter estimation is performed behind the scenes using standard linear regression (least-squares). An optional regularization parameter can be supplied if needed, see `?BasisFunctionApproximation` for help.

Plotting functionality requires `Plots.jl`

## Single dimension

We start by simulating a signal `y` and a scheduling signal `v`. The task is to estimate a function `y = ϕ(v)`, where `ϕ` is a basis-function expansion.

```using DSP # For filt
N = 1000
v = range(0, stop=10, length=N) # Scheduling signal
y = randn(N) # Signal to be approximated
y = filt(ones(500)/500,[1],y)```

Next, we setup the basis-function expansion object `rbf` and use it to create a reconstruction object `bfa`

```using BasisFunctionExpansions, Plots
Nv  = 10 # Number of basis functions
rbf = UniformRBFE(v,Nv, normalize=true) # Approximate using radial basis functions with constant width
bfa = BasisFunctionApproximation(y,v,rbf,1) # Create approximation object
ŷ   = bfa(v) # Reconstruct signal using approximation object
scatter(v,y, lab="Signal")
scatter!(v,ŷ, lab="Reconstruction")```

For comparison, we can also plot the regular linear regression `y = α₀ + α₁x + α₂x²... αₙxⁿ` for varying orders of `n`.

```A = v.^(0:3)'
ŷ_linreg = [A[:,1:i]*(A[:,1:i]\y) for i=2:4]
plot!(v,hcat(ŷ_linreg...), lab=reshape(["Linear regression order \$i" for i=1:3],1,:))```

As we can see from the figure, the linear combination of basis functions forming the reconstruction has learnt the overall structure of the signal `y`. To capture more detail, one can try to increase the number of basis functions. The final choice of this number is a tradeoff between reconstruction bias and variance, where a high number of basis functions can model the signal in great detail, but may increase the variance if data is sparse.

## Multiple dimensions

We now demonstrate the same thing but with `v ∈ ℜ²`. To create a nice plot, we let `v` form a spiral with increasing radius.

```using BasisFunctionExpansions, DSP
N = 1000
x = range(0, stop=4pi, length=N)
v = [cos.(x) sin.(x)].*x # Scheduling signal
y = randn(N) # Signal to be approximated
y = filt(ones(500)./500,[1],y)```

Now we're creating a two-dimensional basis-function expansion using ten functions in each dimension (for a total of 10*10=100 parameters).

```Nv  = [10,10] # Number of basis functions along each dimension
rbf = MultiUniformRBFE(v,Nv, normalize=true) # Approximate using radial basis functions with constant width (Not isotropic, but all functions have the same diagonal covariance matrix)
bfa = BasisFunctionApproximation(y,v,rbf,0.0001) # Create approximation object
ŷ   = bfa(v) # Reconstruct signal using approximation object
scatter3d(v[:,1],v[:,2],y, lab="Signal")
scatter3d!(v[:,1],v[:,2],ŷ, lab="Reconstruction")```

To visualize also the basis functions, we can simply call `plot!(rbf)` (the exclamation mark adds to the current plot instead of creating a new one). Below is an example when a 5x5 BFE is visualized using `plotly` as backend.

### Nonuniform covariance

We can let all centers have different (diagonal) covariance matrices using the type `MultiDiagonalRBFE`. In this case, good center locations and covariances are estimated using K-means clustering. With this strategy, we can usually get away with much fewer basis functions compared to a uniform grid. A drawback is that we must know in advance which area of the scheduling signal space is of interest.

```Nc   = 8
rbf  = MultiDiagonalRBFE(v,Nc, normalize=true)
bfa  = BasisFunctionApproximation(y,v,rbf,0.0001)
yhat = bfa(v)
scatter3d(v[:,1],v[:,2],y, lab="Signal")
scatter3d!(v[:,1],v[:,2],yhat, lab="Reconstruction")```

### Full covariance

For the type `MultiRBFE` The covariance matrix and center locations are esimated using K-means.

```Nc   = 8                            # Number of centers/BFs
rbf  = MultiRBFE(v,Nc, normalize=true)
bfa  = BasisFunctionApproximation(y,v,rbf,0.0001)
yhat = bfa(v)
scatter3d(v[:,1],v[:,2],y, lab="Signal")
scatter3d!(v[:,1],v[:,2],yhat, lab="Reconstruction")```

# Dynamics modeling

## LPV ARX modeling

We can use basis-function expansions for identification of elementary, non-linear dynamics models. Consider the following dynamical system, with a non-linearity on the input `A(z)y = B(z)√(|u|)` We can simulate this system using the code

```A = [1,2*0.7*1,1] # A(z) coeffs
B = [10,5]        # B(z) coeffs
u = randn(100)    # Simulate 100 time steps with Gaussian input
y = filt(B,A,sqrt.(abs.(u)))```

We can now try to fit a regular ARX model to this input-output data

```yr,A  = getARXregressor(y,u,2,2) # We assume that we know the system order 3,2
x     = A\yr                     # Fit using standard least-squares
e_arx = √(mean((yr - A*x).^2))   # Calculate RMS error (11.2)
plot([yr A*x], lab=["Signal" "ARX prediction"])```

Due to the non-linearity at the input of the system, the linear model fails to fit the data well. Our next attempt is a non-linear model based on BFEs. We select the simplest form of multi-dimensional BFE, `MultiUniformRBFE` and further select to cover the state-space with a single basis functions along each dimension corresponding to `y`, which falls back onto a linear function, and 4 basis functions along each dimension corresponding to `u` for a total of 1^2*4^2=16 parameters (4 basis functions is the smallest number that can somewhat accurately fit `√(|u|)`). The number of parameters in this case is large compared to the number of data points, we will need some regularization to fit this model properly. The regularization choice is made when forming the `BasisFunctionApproximation` and the strength is determined by the last argument `1e-3` in this case.

```bfe   = MultiUniformRBFE(A,[1,1,4,4], normalize=true)
bfa   = BasisFunctionApproximation(yr,A,bfe, 1e-3)
e_bfe = √(mean((yr - bfa(A)).^2)) # (3.11)```

The non-linear model fits the data much better!

We also note that if we knew in advance that the system is linear with a non-linearity on the input, we could do this in a slightly more efficient way by incorporating lagged values of `y` directly in the regressor, instead of expanding the lagged values of `y` in a BFE. If we knew the exact non-linearity, we could simply transform our measured signal `u` and use it as input. With the LPV model, however, we can estimate the shape of the non-linearity.

## LPV State-space modeling

We can also estimate a state-space model with varying coefficient matrices, i.e. a model on the form `x(t+1) = A(v)x(t) + B(v)u(t)`

This is accomplished using the built in convenience functions

```nc    = 10                                        # Number of centers
model = LPVSS(x, u, nc; normalize=true, λ = 1e-3) # Estimate a model
xh    = model(x,u)                                # Form prediction```

See `?LPVSS` for more details and a runnable example that produces a plot.

Functionality in this package is used in the packages

And in the papers

BasisFunctionExpansions plays nice with ReverseDiff.jl and ForwardDiff.jl

```julia> using ReverseDiff
julia> a = randn(1,2)
julia> ReverseDiff.gradient(bfa,a) # bfa here comes from the Multi-dim example
1×2 Array{Float64,2}:
1.29364  -0.536586

julia> h = 0.0001 # Finite difference for validation
0.0001

julia> [(bfa(a+[h 0]) - bfa(a))/h (bfa(a+[0 h]) - bfa(a))/h]
1×2 Array{Float64,2}:
1.29363  -0.536488```

Note: for `ForwardDiff.jl` to work, you have to use `ForwardDiff.jacobian` instead of `ForwardDiff.gradient`.

See `?ReverseDiff.gradient` for tips regarding high performance gradient calculation through preallocation of GradientConfig and prerecording of `bfa`.

# Another example

```N   = 200
v   = range(0, stop=10, length=N)
y   = 0.1.*(v.-2).*(v.-7) .+ 0.2randn(N)
rbf = UniformRBFE(v, 5, normalize = true)
bfa = BasisFunctionApproximation(y,v,rbf)

scatter(v,y,lab="Signal",c=:orange, subplot=1, xlabel="\\$v\\$", size=(600,300))
plot!(rbf)
plot!(v,bfa(v),lab="Reconstruction",c=:blue,linewidth=2)```

# Selecting the number of basis functions

A simple way of choosing the number of basis functions is to plot an L-curve (parameter vs. error). A suitable number is where the kink in the curve occurs, for this example at around 6 basis functions.

```N    = 200
v    = range(0, stop=10, length=N)
y    = 0.1.*(v.-2).*(v.-7) .+ 0.2randn(N)
nvec = 2:100
lcurve = map(nvec) do n
rbf = UniformRBFE(v, n, normalize = true)
bfa = BasisFunctionApproximation(y,v,rbf)
std(y-bfa(v))
end

plot(nvec, lcurve, yscale=:log10, ylabel="RMS Error", xlabel="Number of basis functions")```

# Citing

This package was developed for the thesis Bagge Carlson, F., "Machine Learning and System Identification for Estimation in Physical Systems" (PhD Thesis 2018).

```@thesis{bagge2018,
title        = {Machine Learning and System Identification for Estimation in Physical Systems},
author       = {Bagge Carlson, Fredrik},
keyword      = {Machine Learning,System Identification,Robotics,Spectral estimation,Calibration,State estimation},
month        = {12},
type         = {PhD Thesis},
number       = {TFRT-1122},
institution  = {Dept. Automatic Control, Lund University, Sweden},
year         = {2018},
url          = {https://lup.lub.lu.se/search/publication/ffb8dc85-ce12-4f75-8f2b-0881e492f6c0},
}```

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