This manual is for the GNU Octave GP Emulation package, version 0.0.1. The sources of this manual are derived from Oliver Heimlich’s Interval package manual.

Copyright © 2016–2017 Juan Pablo Carbajal

Permission is granted to copy, distribute and/or modify this document under a Creative Commons Attribution 4.0 International License. A copy of the license is included in Creative Commons License.

Permission is granted to copy, distribute and/or modify the source files used for this document under the terms of GNU General Public License, Version 3 or any later version published by the Free Software Foundation. A copy of the license is included in GNU General Public License.

How to install and use the GP Emulation package for GNU Octave | ||
---|---|---|

• Preface: | Background information before usage | |

• Getting Started: | Quick-start guide for the basics | |

• Quick Introduction to Gaussian Processes: | Fundamental concepts | |

• Introduction to Gaussian Processes based emulation: | Guided tour | |

• Advanced Examples: | Advanced Emulation | |

Appendix | ||

• Creative Commons License: | The license for this manual | |

• GNU General Public License: | The license for the software used in this manual |

Next: Getting Started, Previous: Top, Up: Top [Contents]

Welcome to the user manual of the *GP Emulation package* for GNU Octave. This chapter presents background information and may safely be skipped. First-time users who want to cut right to the chase should read Getting Started, which teaches basic concepts and first steps with the package.

Next: Quick Introduction to Gaussian Processes, Previous: Preface, Up: Top [Contents]

The following chapter take you by the hand and gives a quick overview on the GP emulation package’s basic capabilities. More detailed information can usually be found in the functions’ documentation.

Unless the package is provided by a third-party distributor, the interval package can be installed with the `pkg`

command from within Octave.
Assuming you have got a file named *gpemulation-<version>.tar.gz*, where *<version>* is a three digits version (here I use *1.0.0*), and that you have this file in the folder returned by Octave’s `pwd`

command; the installation proceeds as follows:

pkg install gpemulation-1.0.0.tar.gz -| For information about changes from previous versions -| of the gpemulation package, run 'news gpemulation'.

During this kind of installation parts of the gpemulation package are compiled for the target system, which requires development libraries for GNU Octave (version ≥ 3.8.0) to be installed. The official Octave release for Microsoft Windows already includes these dependencies. For other systems it might be necessary to install the package “liboctave-dev”, which are provided by most GNU distributions (names may vary).

In order to use the gpemulation package during an Octave session, it must have been loaded, i. e., added to the path. In the following parts of the manual it is assumed that the package has been loaded, which can be accomplished with the `pkg load gpemulation`

command.

That’s it. After only two commands the package is ready to be used within Octave.

Next: Introduction to Gaussian Processes based emulation, Previous: Getting Started, Up: Top [Contents]

In this chapter I introduce Gaussian Processes succinctly and conceptually. For a more formal and detailed introduction you could refer to Guassian Processes for Machine Learning.

A stochastic process can be thought of the generalization of a probability distribution of a finite-dimensional random variable to functions.
That is, the process draws a function from a distribution of functions.
If the function values at each point in the functions domain follow a Gaussian distribution, the process is a *Gaussian Process* (GP).
The figure below shows 500 draws from a GP.
The vertical lines mark points at which the histograms are calculated.
These histograms are well described with a Gaussian curve (shown in red) and this is true for any point in the domain.

If a finite-dimensional random variable follows a Gaussian distribution, this distribution is fully determined by the mean value and the variance.
In the case of GPs, the distribution of functions is fully determined by a *mean function* and a *covariance function*.

The mean function `m`(x), is any function defined in the domain (i.e. the *inputs*) of the GP.
The covariance function `k`(x,y), often called *kernel*, is a function defined for pairs of inputs and it must have certain properties:

- It must be symmetric, i.e. for any two inputs
`x`,`y`then`k`(`x`,`y`) =`k`(`y`,`x`). - It must be positive definite. Roughly speaking this means that when we build a matrix by evaluating
`k`at a finite set of inputs, the matrix`K`so obtained , i.e. the*covariance matrix*, has only positive eigenvalues. See Wikipedia for a formal definition.

A commonly used kernel function is the squared exponential

k(x,y) = exp [ - (x-y)² / (2 s²) ]

Let’s see how we can sample functions from the GP defined by this kernel.
We first need to generate a set of input values and build the covariance matrix.
Then we need to sample a Gaussian distribution with this covariance.
To do this last step we can use the function `mvnrnd`

provided by the http://octave.sourceforge.net/statistics/.

x = linspace (0, 1, 100).'; K = exp ( -(x - x.').^2 / 2 / 0.025 ); # Lets take 10 samples from this GP and plot them. pkg load statistics; y = mvnrnd (0, K, 10).'; plot (x, y, "linewidth", 2); axis tight xlabel ("x"); ylabel ("y");

The covariance function of a distribution of functions is calculated as the expectation value of squared deviations from the mean:

k(x,x’) = cov[y(x),y(x’) ] =[ (Ey(x) -m(x) ) ⋅ (y(x’) -m(x’) )ᵀ ]

The expectation is calculated over realization of whatever random parameter defines the distribution. For example, take a distribution of parabolas with random concavities which follow a Gaussian distribution with mean value ã and variance v

y(x) =a(x-x₀ )²a~N(ã,v)

The mean function is

m(x) =[Ey(x) ] = ∫ap(a) (x-x₀ )² da= ã (x-x₀ )²

and the covariance function is then

k(x,x’) = (x-x₀ )²(x’ -x₀ )² v

x = linspace (0, 1, 100).'; ## Inputs a = -1; ## Mean concavity v = 0.5^2; ## Variance of concavity K = (x-0.5).^2 .* (x.'-0.5).^2 * v; ## Covariance matrix m = a * (x.' - 0.5).^2; ## Mean vector # Lets take 10 samples from this GP and plot them. pkg load statistics; y = mvnrnd (m, K, 10).'; plot (x, y, "linewidth", 2) axis tight xlabel ("x"); ylabel ("y");

This is how you build GPs from known function distributions. When you do not know the analytic distribution of the functions you can try to estimate it from a large number of samples trajectories. We do this in the example below, where the covariance is obtained from samples of the solutions of a stochastic differential equation. The result is compared with the analytical covariance of this SDE implemented in the function covLTIo1d1. See Covariance Function of a First Order Ordinary Differential Equation with Additive Gaussian Noise

# Comparing covariance calculated from n samples of a # stochastic differential equation and the theoretical calculation. nT = 1e3; # Time samples for Ito integration t_ = linspace (0, 1, nT); # Time vector for Ito integral dt = t_(2) - t_(1); # Time step a = -3; eA = exp (a * dt); # Evolution matrix # Ito integral s = 1; n = 1e3; dW = sqrt (s*dt) * randn (nT, n); # Wiener differential y_ = zeros (nT,n); y_(1,:) = 1; for i = 1:nT-1 y_(i+1,:) = y_(i,:) * eA + dW(i+1,:); endfor # Experimental covariance i0 = nT/2; # index at t0 = 0.5 ym = mean (y_, 2); dy = y_ - ym; k = (dy * dy.'(:,i0)) / n; # Theoretical covariance t0 = 0.5; # centered in the interval t = linspace (t_(1),t_(end), 1e2).'; K = covLTIo1d1 (t, t0, a, s, 0); h = plot (t_, k,'-r;Experimental;', t, K,'-k;Exact;'); set (h,'linewidth', 2); xlabel ('Time'); ylabel ('Covariance') axis tight

A first order linear ordinary differential equation (LODE) with additive Gaussian noise is also known as a linear stochastic differential equation (LSDE)

ẏ(t) =Ay(t) +u(t) +ξ(t)y(0) =y₀ +ξ₀ξ(t) ~N(0,𝛴) ,ξ₀ ~N(0,𝛴₀)

where `u`(`t`) is a deterministic function called *actuation*.

If there were no noise terms and no actuation, this equation has a unique solution (a.k.a. *homogeneous solution*)

yₕ(t) = exp(At)y₀

The effect of the actuation can be calculated using the
Green’s functions of the LODE.
This give raise to the *particular solution* of the LODE

yₚ(t) = ∫G(t,r)u(r) drG(t,r) =H(t-r)exp(A(t-r))

Where `H`(`t`-`r`) is the Heaviside function.
Note that the particular solution is a linear transformation of the actuation and we can write it as `y`ₚ = `G``u`.
The general solution of the LODE can be written in the form

yₘ(t) =yₕ(t) +yₚ(t)

The noise terms in the LSDE induce a distribution of functions whose mean value coincides with the deterministic solution given above.

To calculate the covariance function of the distribution of solutions we need the concept of the Green’s function of the adjoint differenital operator, which is a big name for something simple.

Gᵀ(t,r) =H(r-t)exp(Aᵀ(r-t))

This will be needed when we calculate the covariance function (See covariance function's formula).

k(t,r) = cov[y(t),y(r) ] = exp(At)𝛴₀ exp(Aᵀr) +G𝛴Gᵀ

When the last term is expressed as an integral^{1} we get

k(t,r) = exp(At)𝛴₀ exp(Aᵀr) + ∫G(t,z)𝛴Gᵀ(r,z) dz

The function covLTIo1d1 calculates this covariance function for one-dimensional LSDE, i.e. `A` ∈ ℝ

nT = 100; t = linspace (0, 1, nT).'; ## Inputs a = -5; ## Decay rate s = 0.3^2; ## Variance of noise s0 = 0.25^2; ## Variance of initial conditions K = covLTIo1d1 (t,t,a,s,s0); ## Covariance matrix m = [1; -1].* exp (a*t.'); ## Mean functions subplot (2,2,1) colormap (cubehelix); imagesc (t,t,K); xlabel("r"); ylabel("t"); subplot (2,2,2) plot (t,K(:,50)); axis tight xlabel("t"); ylabel("K(t,0.5)"); subplot (2,2,3:4) # Lets take 10 samples from this GP and plot them. pkg load statistics; y = zeros (nT,10); y(:,1:5) = mvnrnd (m(1,:), K, 5).'; y(:,6:end) = mvnrnd (m(2,:), K, 5).'; plot (t, y, "linewidth", 2) axis tight xlabel ("t"); ylabel ("y");

The function covLTIo1diag calculates a covariance function for d-dimensional LSDE, i.e. `A` ∈ ℝᵈˣᵈ, when the system matrix can be diagonalized, i.e. `A` = `V``D``V`⁻¹ where `D` is a diagonal matrix.

nT = 100; t = linspace (0,1,nT).'; # Oscillator w = 2*pi*5; M = [0 1; -(w)^2 -0.2*2*w]; [V d] = eig (M); d = diag (d); Vi = inv (V); S = diag([0 w]); # Covariance of noise S0 = zeros(2); # Covariance of inital conditions K = covLTIo1diag ({t,{V,d,Vi}}, [], S, S0); K = covmat2cell (K, [length(t) length(d)]); subplot (2,4,[1 2 5 6]) colormap (cubehelix); imagesc (cell2mat (K)); set(gca,"visible","off"); hold on line(nT*[0 1; 2 1], nT*[1 0; 1 2]); ax = [3 7 4 8]; for i=1:4 subplot (2,4,ax(i)); plot(t,K{i}(:,nT/2),"linewidth", 2); [k,l] = ind2sub([2 2],i); title (sprintf("Component %d with %d",k,l)); endfor

This function can also be used to calculate the covariance between two d-dimensional LSDE coupled by noise.

nT = 100; t = linspace (0,1,nT).'; # Oscillator w = 2*pi*5; M = [0 1; -(w)^2 -0.1*2*w]; [V d] = eig (M); Vi = inv (V); d = diag (d); # First order system a = -3; # combined eigenspace V = blkdiag (1,V); Vi = blkdiag (1,Vi); d = [a;d]; # coupling S = ones(3); # Covariance of noise S0 = zeros(3); # Covariance of inital conditions K = covLTIo1diag ({t,{V,d,Vi}}, [], S, S0); K = covmat2cell (K, [length(t) length(d)]); subplot (3,6,[1:3 7:9 13:15]) colormap (cubehelix); imagesc (cell2mat (K)); set(gca,"visible","off"); hold on line(nT*[0 1; 3 1], nT*[1 0; 1 3]); ax = [4 10 16 5 11 17 6 12 18]; for i=1:9 subplot (3,6,ax(i)); [k,l] = ind2sub([3 3],i); y = K{i}(:,nT/2); plot(t,y,"linewidth", 2); set(gca,"visible","off"); title (sprintf("%d-%d",k,l)); endfor

Given a GP and a collection of input-output data (`X`,`Y`), i.e. *the design data*, we can calculate another GP that is consistent with this given data.
This is done following Bayes’ theorem for conditional distributions.
The conditioned GP’s mean function is used for prediction and the formula reads

y(x) =k(x,X)(k(X,X) + 𝛌)⁻¹(Y(X) -m(X))+m(x)

The covariance function evaluated at the design data inputs `k`(`X`,`X`) is the covariance matrix of the observations.
That is, GPs allow the use of linear algebra in Bayesian inference, considerably simplifying computations.
This is a important reason for the popularity of GPs.

The predictive mean, the function that we use to predict values, is a linear combination of columns of the covariance matrix `k`(`x`,`X`):

y(x) =k(x,X) 𝛂 +m(x) 𝛂 =(k(X,X) + 𝛌)⁻¹(Y(X) -m(X))

These columns were plotted in all the figures shown before. For example, the columns of the covariance of a first order system (See this figure) is a two sided decaying exponential centered at the data. For a second oder system the columns show oscillatory behavior (See this figure). It s useful when deciding what covariance to use to look at the data and decide what functions would be good for prediction.

The parameter 𝛌 called *regularization parameter*, it is free and we need to decide its value.
It represents the trust we have on the mean function of our GP.
If 𝛌 is big, the conditioned GP will pay little attention to the design data and mostly use the mean function for prediction.
If 𝛌 is small, the conditioned GP will stick to the design data.
The extreme case of 𝛌 = 0 forces the prediction to go exactly through the design data, this is called *interpolation*.
The value of the regularization parameter can be selected based on prediction errors (like cross-validation) or it can be decided on knowledge about the data.

To keep things simple, lets use our stochastic parabolas (See the derivation here) for prediction. First let say we got this data consisting of 10 points (X,Y)

Clearly whatever process generated this data, it cannot be described with a parabola.
Indeed the data was generated with the function `Y = 0.5*(X - 0.5).^2 - 5*(X - 0.5).^4`

We build the covariance matrix by evaluating our covariance function at these points.

The figure also shows the columns of the covariance matrix `k`(`x`,X),
with `x` ∈ [0,1].
This are the functions that will be used for prediction.
As you can see our covariance function has columns that are parabolas of different concavity (which could be guesses by the way we built this covariance matrix).
As we know the prediction will be a linear combination of these columns (See this equation).
What do we get if we linearly combine parabolas of different concavity?

z(x) = ∑ aₙ (x-x₀)² = A (x-x₀)²

Well just another parabola.
This means that our GP can only interpolate parabolas, in other words, the
covariance matrix essentially generates one type of function.
When this happens, we will always get covariance matrices with rank 1, because the columns are proportional to each other.
Covariances with finite ranks are called *degenerate covariance functions* (or *degenerate kernels*).
This also means that when the rank of the covariance is lower than the number of inputs, we will not be able to set 𝛌 = 0 when conditioning (See this equation).

Whenever you are given a covariance matrix, check its eigenvalue or singular value spectrum to know how powerful it is. In this case

s = svd (KXX).'; round (s * 1e2) * 1e-2 ⇒0.05 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

illustrating the degeneracy of our kernel.

We apply the conditioning formula (See here) using two different mean functions to show the effect they have on the solution. The first mean function we use is

mA(x) = -5 (x - 0.5)⁴

which will give us parabolic residuals and a good solution to the interpolation problem. The second mean functions to be used is

mB(x) = (x - 0.5)²

which is ok, but it does not help our covariance function very much, as the residuals will still be a mix of quartic and parabolic functions. We use this last function to show the effect of two extreme values of the regularization parameter. An solution without regularization 𝛌 = 1e-8 and a strongly regularized solution 𝛌 = 1.

The figure below shows the results.

As you can see the solution using `mA`

(labeled A) is able to interpolate (𝛌 = 1e-8)
because `mA`

takes care of the quartic term and leaves parabolic residuals for the covariance, which is able to find the right concavity.

The second mean, `mB`

, is not so useful.
The regularized solution (labeled B reg) just sticks to the mean function with little correction.
The solution without regularization (labeled B), finds the best parabola it can build.
Indeed, the theory shows that this is the same result as doing a least square error regression
using a parabola that goes through the origin.
This is shown in the plot with the legend "fit".

Next: Advanced Examples, Previous: Quick Introduction to Gaussian Processes, Up: Top [Contents]

At this point you have a conceptual grasping of what a GP is and how they are built.
In particular you know that there is a GP associated to a LSDE (a LODE with Gaussian noise actuation).
In this chapter you will learn about GP emulators and how to build one to emulate a deterministic nonlinear system, i.e. the *simulator*.

The objective of an emulator is to reduce the computational cost needed to obtain a trajectory of a nonlinear complicated simulator, without losing too much accuracy.

To build an emulator you need samples from the simulator and the parameters you used to generate them. You will also need to make some decisions about the structure of the emulator. We discuss this in the subsequent sections.

The emulator will end up being a GP associated to a LSDE, similar to the ones we saw before (See Covariance Function of a First Order Ordinary Differential Equation with Additive Gaussian Noise). The particular way in which we construct this LSDE is what makes the associated GP an emulator of a nonlinear simulator.

The input to the emulator will be the time stamps (as before) and a parameter vector. That is, the emulator will make predictions of the simulator output at a given time and parameter value.

Lets say you have a collection of `n`

sample simulations from your simulator, i.e. the nonlinear model to emulate.
Each simulation was evaluated at `nT`

time stamps and the signal are `d`

dimensional (that is you observe `d`

signals from the simulator).
We call this collection of sample trajectories the *design data* for the emulator, and we will
organize them in a multidimensional array of size `[nT n d]`

.

The parameters used in the simulator to generate the `n`

datasets are stored
in a matrix of size `[n p]`

. The *i*-th row of this matrix contains the
parameters used to generate the corresponding dataset.

In the following example we generate design data with `n = 80`

samples from a nonlinear ODE (the Van der Pol oscillator)

ẋ=yẏ= μ ( 1 -x² )y-x

That has one parameter (i.e. `p = 1`

) controlling the shape of the oscillations of the system.
Note that we remove the transient of the system and take only the data at the limit cycle.
Emulating transient behavior is difficult so we skip it for now (See Advanced Examples)

nT = 300; # Number of time samples n = 10; # Number of design datasets T = 30; # Time to wait before observation t = linspace (0, 18, nT).'; # Time samples mu = linspace (0.3, 4, n)'; # The oscillator parameters d = 2; # Dimension of the simulator signals vdp = @(y,t, mu) [y(2); ... # Simulator is mu * (1 - y(1)^2) * y(2) - y(1)]; # Van der Pol oscillator y0 = [0.5 0]; # Initialization conditions simulator = @(t,p) lsode (@(x,t)vdp(x,t,p), ... lsode(@(x,t)vdp(x,t,p), y0, [0 T])(end,:), ... # Initial condition t); y = zeros (nT, n, d); # Matrix to store the design datasets for i=1:n tmp = lsode (@(x,t)vdp(x,t,mu(i)), y0, [0 T]); tmp = tmp(end,:); # Use final position as initial condition y(:,i,:) = lsode (@(x,t)vdp(x,t,mu(i)), tmp, t); endfor p = 1; # Dimension of simulator parameter vector nlp = mu; # Simulator parameter matrix

The figure shows the phase space with some of the trajectories on the limit cycle of
the Van der Pol oscillator.
To produce the figure we used a lot more time stamps than in the actual design data (`nT = 40`

) and it only shows some of the `n = 80`

datasets.

For different parameter values, the shape of the cycle changes, the ellipse observed at low parameter values (system behaves more linearly) gets deformed as the parameter increases (nonlinear regime). The lateral plots show the two components of the data.

The way we calculate the covariance matrix is valid only for linear systems.
Therefore, to build the emulator we propose a linear model.
This linear model is built as an aggregation of independent linear submodels.
Each of these submodels is called *replica* (when all submodels have the same structure) or *proxy*.
We build a proxy for each dataset and as before we will organize the parameters of the proxy in a matrix of size `[n q]`

(Note that number of parameters can change, i.e. `q`

can be different from `p`

).

Since we know the parameters of the dataset (this is part of the dataset as explained before) we need to define a way to obtain the parameters of the proxy.
There is no fixed rule to do this, we can define and ad-hoc formula or we can fit our linear model to the *i*-th dataset.
It all depends on how much information about the simulator you can use.

When we use the emulator to generate predictions at a previously unseen parameter vector, we will need the parameter vector for the corresponding (unseen) proxy. This means that we will need a mapping from the simulator parameters to the proxy parameters. If you used an ad-hoc formula to do this mapping, then you will use this formula for this as well. If you have fitted the datasets you will need to build an interpolant, which can be as simple as a linear interpolation or as complex as a GP.

In the following example we fit the design data See previous example and create a simulator-emulator parameter mapping with a linear interpolation. The proxies are all of the form

y(t) = A cos ( w t ) + (B/w) sin ( w t )

Which is the solution of the harmonic oscillator equation (second order LODE) with frequency w, initial deformation A and initial speed B. It will be important later to know that these functions are the first component of the solutions of the dynamical system

ẏ=zż= -w²y

with `y`(0) = A and `ẏ`(0) = `z`(0) = B.

Since we use the same proxy structure for all datasets we could call them replicas. Once we have estimated the values of A,B,w for each data set generated with paramter μ, we generate a linear interpolation to map from any μ to the corresponding A,B,w.

######################### ## Proxy calibration ## ######################### proxy = @(t,p) p(2) * cos (p(1) * t) + p(3)/p(1) * sin (p(1) * t); q = 3; # Parameter vector size z = zeros (nT, n, 1); # Matrix to store the proxies Q = zeros (n, q); # Matrix of proxy parameters tmp2 = [1 0 0.5]; for i=1:n tmp2 = sqp (tmp2, @(x)sumsq(y(:,i,1)-proxy(t,x))); Q(i,:) = tmp2; z(:,i) = proxy (t, tmp2); endfor ######################### ## Parameter mapping ## ######################### pp = interp1 (mu, Q, "pp"); parmap = @(x) ppval(pp, x);

The top plot shows some design data and the fitted proxies. The bottom plot shows several trajectories that are not in the design data and the proxies we obtain with the parameter mapping.

The aggregated linear model built from the proxies can be used for the mean function of the emulator.

Note that so far the proxies are independent, while the datasets might not be. Therefore, we need to couple these models and we are going to use the covariance function to do this (See the equation).

In an emulator the independent proxies, are coupled through a noisy actuation term. If two proxies are highly coupled, their noisy actuation will be very correlated.

Assuming that our simulator is deterministic implies that for each parameter vector there is a unique dataset. It is natural then to propose that if two datasets were generated with similar parameter vectors, then their proxies must be actuated with correlated noise. This all means that the noise correlation matrix 𝚺 (See the equation) depends on the parameter vectors. The closer the parameter vectors the higher the entry in the correlation matrix.

`gpemulator`

ObjectAt this point all the ingredients are in place. Our emulator is a GP associated with a system of linear models (independent proxies) which are actuated with correlated noise (the coupling).

The package includes a simple object called `gpemulator`

, that simplifies the use of an emulator.
Currently it does not reduce the work required for each of the items described
in the previous section, but it helps putting them all together.

The interface of the constructor @gpemulator/gpemulator tries to follow the style of the GPML Toolbox. So instantiation of the object looks like

# Covariance function definition covfunc = {@covFunc_emu}; # Mean function definition meanfunc = {@meanFun_emu}; emu = gpemulator ('covFunc', covfunc,'meanFunc', meanfunc);

The last line is the actual construction of the emulator and we need to pass a covariance function and a mean function. The previous lines define the mean and the covariance functions, which are just containers of function handles, that we will discuss in detail in the examples below.

To train (or condition) the emulator on some given datasets `Y`

we call the method
@gpemulator/designdata

emu = designdata (emu, Y);

After this the emulator is able to make predictions at inputs `X`

using the method
@gpemulator/emulate

y = emulate (emu, X);

The full code for this example can be found in the script file emulator_VdP.m inside the examples folder of the package.

We continue with the example from the previous sections (See The Anatomy of an Emulator). We assume that we have the design data and the parameter mapping. The script to emulate will then have three parts. In the first part we load the design data and define some helper functions. In the second part we use the design data to condition our emulator. In the third part we emulate at unseen parameter values and time stamps.

##################### #### Design data #### ##################### fname = "vanderpol.dat"; ddata = load (fname); vdp = @(y,t, mu) [y(2); ... # Simulator is mu * (1 - y(1)^2) * y(2) - y(1)]; # Van der Pol oscillator y0 = [0.5 0]; simulator = @(t,p) cell2mat( arrayfun ( @(r)lsode (@(x,t)vdp(x,t,r), ... lsode(@(x,t)vdp(x,t,r), y0, [0 ddata.T])(end,:), ... # Initial condition t)(:,1), ... p(:).', 'unif', 0)); proxy = @(x) x.ic(:,1).' .* cos (x.lp.' .* x.time) + ... (x.ic(:,2)./x.lp).' .* sin (x.lp.' .* x.time); # Proxy ############################### #### Emulator conditioning #### ############################### pkg load gpml pkg load gpemulation covfunc = {@covFunc_emu, {@covMaterniso,1},{@covSEiso}}; meanfunc = {proxy}; mdp = log (min (tril (abs (ddata.nlp - ddata.nlp.'),-1,'pack'))); hyp.S = [mdp+log(2) 0]; hyp.S0 = [mdp+log(1) 0]; emu = gpemulator ('covFunc', {covfunc{:},hyp}, ... 'meanFunc', meanfunc, ... 'regParam', 0); #### Condition the emulator on design data par = ddata.parmap (ddata.nlp); x = struct ('time', ddata.t, ... 'nlp', ddata.nlp, ... 'lp', par(:,1), ... 'ic', par(:,2:3), ... 'idx', 1); emu = designdata (emu, x, ddata.y(:,:,1)); ################### #### Emulation #### ################### nT = 2 * length (ddata.t); t = linspace (ddata.t(1), ddata.t(end)+2, nT).'; N = 5; nls = [min(ddata.nlp) max(ddata.nlp)]; nlp = linspace (nls(1)-exp(mdp), nls(2)+exp(mdp), N).'; Y = simulator (t, nlp); par = ddata.parmap (nlp); x = struct ('time', t, ... 'nlp', nlp, ... 'lp', par(:,1), ... 'ic', par(:,2:3), ... 'idx', 1); yp = emulate (emu, x);

In the first part, we define the function `simulator`

that takes an array of parameters for the Van der Pol oscillator and returns trajectories on the limit cycle for each parameter
value.
Also we use a function `proxy`

that takes an array of parameters and
evaluates the proxy.
The `proxy`

function expects and input structure `x`

with the fields
`time`

, `ic`

and `lp`

.
The field `time`

contains the time stamps, `ic`

the initial conditions
and `lp`

the three parameters that define the proxy.

In the second part we define the emulator using `proxy`

as a mean function and
the function `covFunc_emu`

for the covariance.
This function is described in detail below.
At this point it is relevant to know that among its inputs there are two other
covariance functions that it will use to couple the proxies.
In this case we are using the Matern kernel for the parameter coupling and
the squared exponential for the initial conditions, both functions are from the
the GPML Toolbox.
The parameters for these functions are given in the `hyp`

structure, fields
`hyp.S`

and `hyp.S0`

, respectively.

At the end of this second part we condition the emulator on the design data by calling the method

x = struct ('time', t, ... 'nlp', nlp, ... 'lp', par(:,1), ... 'ic', par(:,2:3), ... 'idx', 1); emu = designdata (emu, x, ddata.y(:,:,1));

You might have noticed that the input structure `x`

has also the field
`idx`

.
This field contains the component of our proxy that we will use to emulate the
design data.
In this case we use only the first component of our emulator for the first
component of the design data.

Finally, in the third part of the script we emulate at unseen parameter values and
time stamps.
The field `x.time`

covers the observed time span plus an extra time window so we can see
how good our emulator extrapolates outside of the data region.
It also has the double number of time stamps, so we are *intrapolating*.
The field `x.nlp`

contains the unseen parameters at which we are emulating.
This field contains values between the design data values, so again we are intrapolating.
It also contains values outside the seen region of parameter values, i.e. extrapolation.

The extrapolation at low parameter values presents no difficulty for the emulator because the oscillator behaves more linearly there (see the top panel of the figure below). However, when we extrapolate at higher parameter values, the results get worse and you can see this in the bottom panel of the figure.

These are results using `n = 80`

trajectories of the oscillator and `nT = 40`

time samples.

In the example shown above the code for the actual covariance of our emulator was not shown. Here we show and describe the details of the implementation.

We want to use the function covLTIo1diag which we described before (See Covariance Function of a LSDE).
The inputs argument to this function are the timestamps and the
diagonalized representation of the emulator system matrix `A` = `V``D``V`⁻¹, that is we need to pass `V`, `V`⁻¹ and the diagonal of `D`.
Therefore we first write a function that builds these matrices from a given vector of parameters.
As we said before, our mean function

y(t) = A cos ( w t ) + (B/w) sin ( w t )

gives us the paramters for the system

ẏ=zż= -w²y

with `y`(0) = A and `ẏ`(0) = `z`(0) = B.

The columns of the matrix `V` are

Vₙᵀ = [ 1Dₙₙ]

where the diagonal of `D` is

Dₙₙ = (-1)ⁿiw

with `i` the imaginary unit.

The construction of these matrices is implemented in the function `param2matrix`

that you see in the following code.

function A = param2matrix (lp) # Egiven-values and -vector functions Df =@(l) l.*[-J; J]; # Eigenvalues Vf =@(l) [ones(1,2); Df(l).']; # Eigenvectors Vif =@(l) 0.5 * [ones(2,1) 1./Df(l)]; # Inverse of Eigenvectors # Eigenspace V = arrayfun (Vf, lp, "unif", 0); V(1) = sparse (V{1}); V = blkdiag (V{:}); # Inverse eigenspace Vi = arrayfun (Vif, lp, "unif", 0); Vi(1) = sparse (Vi{1}); Vi = blkdiag (Vi{:}); # Eigenvalues D = cell2mat (arrayfun (Df, lp, "unif", 0)); A = {V,D,Vi}; endfunction function k = covFunc_emu (covS, covS0, h, x, y=[]) n = length(x.nlp); X = {x.time, param2matrix(x.lp)}; idxx = []; if ~isempty (x.idx) idxx = cell2mat (arrayfun ( ... @(i)(i-1)*2 + x.idx, 1:n, ... "unif", 0)); endif if isempty (y) || strcmp (y, 'diag') S = feval (covS{:}, h.S, x.nlp); S0 = feval (covS0{:}, h.S0, x.ic); S = kron (S, ones(2)); S0 = kron (S0, ones(2)); k = covLTIo1diag (X, y, S ,S0, idxx); else ny = length(y.nlp); Y = {y.time, param2matrix(y.lp)}; idxy = []; if ~isempty (y.idx) idxy = cell2mat (arrayfun ( ... @(i)(i-1)*2 + y.idx, 1:ny, ... "unif", 0)); endif S = feval (covS{:}, h.S, x.nlp, y.nlp); S0 = feval (covS0{:}, h.S0, x.ic, y.ic); S = kron (S, ones(2)); S0 = kron (S0, ones(2)); k = covLTIo1diag (X, Y, S ,S0, idxx, idxy); endif endfunction

The second function implements the covariance function using covLTIo1diag.

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- Conveying Modified Source Versions.
You may convey a work based on the Program, or the modifications to produce it from the Program, in the form of source code under the terms of section 4, provided that you also meet all of these conditions:

- The work must carry prominent notices stating that you modified it, and giving a relevant date.
- The work must carry prominent notices stating that it is released under this License and any conditions added under section 7. This requirement modifies the requirement in section 4 to “keep intact all notices”.
- You must license the entire work, as a whole, under this License to anyone who comes into possession of a copy. This License will therefore apply, along with any applicable section 7 additional terms, to the whole of the work, and all its parts, regardless of how they are packaged. This License gives no permission to license the work in any other way, but it does not invalidate such permission if you have separately received it.
- If the work has interactive user interfaces, each must display Appropriate Legal Notices; however, if the Program has interactive interfaces that do not display Appropriate Legal Notices, your work need not make them do so.

A compilation of a covered work with other separate and independent works, which are not by their nature extensions of the covered work, and which are not combined with it such as to form a larger program, in or on a volume of a storage or distribution medium, is called an “aggregate” if the compilation and its resulting copyright are not used to limit the access or legal rights of the compilation’s users beyond what the individual works permit. Inclusion of a covered work in an aggregate does not cause this License to apply to the other parts of the aggregate.

- Conveying Non-Source Forms.
You may convey a covered work in object code form under the terms of sections 4 and 5, provided that you also convey the machine-readable Corresponding Source under the terms of this License, in one of these ways:

- Convey the object code in, or embodied in, a physical product (including a physical distribution medium), accompanied by the Corresponding Source fixed on a durable physical medium customarily used for software interchange.
- Convey the object code in, or embodied in, a physical product (including a physical distribution medium), accompanied by a written offer, valid for at least three years and valid for as long as you offer spare parts or customer support for that product model, to give anyone who possesses the object code either (1) a copy of the Corresponding Source for all the software in the product that is covered by this License, on a durable physical medium customarily used for software interchange, for a price no more than your reasonable cost of physically performing this conveying of source, or (2) access to copy the Corresponding Source from a network server at no charge.
- Convey individual copies of the object code with a copy of the written offer to provide the Corresponding Source. This alternative is allowed only occasionally and noncommercially, and only if you received the object code with such an offer, in accord with subsection 6b.
- Convey the object code by offering access from a designated place (gratis or for a charge), and offer equivalent access to the Corresponding Source in the same way through the same place at no further charge. You need not require recipients to copy the Corresponding Source along with the object code. If the place to copy the object code is a network server, the Corresponding Source may be on a different server (operated by you or a third party) that supports equivalent copying facilities, provided you maintain clear directions next to the object code saying where to find the Corresponding Source. Regardless of what server hosts the Corresponding Source, you remain obligated to ensure that it is available for as long as needed to satisfy these requirements.
- Convey the object code using peer-to-peer transmission, provided you inform other peers where the object code and Corresponding Source of the work are being offered to the general public at no charge under subsection 6d.

A separable portion of the object code, whose source code is excluded from the Corresponding Source as a System Library, need not be included in conveying the object code work.

A “User Product” is either (1) a “consumer product”, which means any tangible personal property which is normally used for personal, family, or household purposes, or (2) anything designed or sold for incorporation into a dwelling. In determining whether a product is a consumer product, doubtful cases shall be resolved in favor of coverage. For a particular product received by a particular user, “normally used” refers to a typical or common use of that class of product, regardless of the status of the particular user or of the way in which the particular user actually uses, or expects or is expected to use, the product. A product is a consumer product regardless of whether the product has substantial commercial, industrial or non-consumer uses, unless such uses represent the only significant mode of use of the product.

“Installation Information” for a User Product means any methods, procedures, authorization keys, or other information required to install and execute modified versions of a covered work in that User Product from a modified version of its Corresponding Source. The information must suffice to ensure that the continued functioning of the modified object code is in no case prevented or interfered with solely because modification has been made.

If you convey an object code work under this section in, or with, or specifically for use in, a User Product, and the conveying occurs as part of a transaction in which the right of possession and use of the User Product is transferred to the recipient in perpetuity or for a fixed term (regardless of how the transaction is characterized), the Corresponding Source conveyed under this section must be accompanied by the Installation Information. But this requirement does not apply if neither you nor any third party retains the ability to install modified object code on the User Product (for example, the work has been installed in ROM).

The requirement to provide Installation Information does not include a requirement to continue to provide support service, warranty, or updates for a work that has been modified or installed by the recipient, or for the User Product in which it has been modified or installed. Access to a network may be denied when the modification itself materially and adversely affects the operation of the network or violates the rules and protocols for communication across the network.

Corresponding Source conveyed, and Installation Information provided, in accord with this section must be in a format that is publicly documented (and with an implementation available to the public in source code form), and must require no special password or key for unpacking, reading or copying.

- Additional Terms.
“Additional permissions” are terms that supplement the terms of this License by making exceptions from one or more of its conditions. Additional permissions that are applicable to the entire Program shall be treated as though they were included in this License, to the extent that they are valid under applicable law. If additional permissions apply only to part of the Program, that part may be used separately under those permissions, but the entire Program remains governed by this License without regard to the additional permissions.

When you convey a copy of a covered work, you may at your option remove any additional permissions from that copy, or from any part of it. (Additional permissions may be written to require their own removal in certain cases when you modify the work.) You may place additional permissions on material, added by you to a covered work, for which you have or can give appropriate copyright permission.

Notwithstanding any other provision of this License, for material you add to a covered work, you may (if authorized by the copyright holders of that material) supplement the terms of this License with terms:

- Disclaiming warranty or limiting liability differently from the terms of sections 15 and 16 of this License; or
- Requiring preservation of specified reasonable legal notices or author attributions in that material or in the Appropriate Legal Notices displayed by works containing it; or
- Prohibiting misrepresentation of the origin of that material, or requiring that modified versions of such material be marked in reasonable ways as different from the original version; or
- Limiting the use for publicity purposes of names of licensors or authors of the material; or
- Declining to grant rights under trademark law for use of some trade names, trademarks, or service marks; or
- Requiring indemnification of licensors and authors of that material by anyone who conveys the material (or modified versions of it) with contractual assumptions of liability to the recipient, for any liability that these contractual assumptions directly impose on those licensors and authors.

All other non-permissive additional terms are considered “further restrictions” within the meaning of section 10. If the Program as you received it, or any part of it, contains a notice stating that it is governed by this License along with a term that is a further restriction, you may remove that term. If a license document contains a further restriction but permits relicensing or conveying under this License, you may add to a covered work material governed by the terms of that license document, provided that the further restriction does not survive such relicensing or conveying.

If you add terms to a covered work in accord with this section, you must place, in the relevant source files, a statement of the additional terms that apply to those files, or a notice indicating where to find the applicable terms.

Additional terms, permissive or non-permissive, may be stated in the form of a separately written license, or stated as exceptions; the above requirements apply either way.

- Termination.
You may not propagate or modify a covered work except as expressly provided under this License. Any attempt otherwise to propagate or modify it is void, and will automatically terminate your rights under this License (including any patent licenses granted under the third paragraph of section 11).

However, if you cease all violation of this License, then your license from a particular copyright holder is reinstated (a) provisionally, unless and until the copyright holder explicitly and finally terminates your license, and (b) permanently, if the copyright holder fails to notify you of the violation by some reasonable means prior to 60 days after the cessation.

Moreover, your license from a particular copyright holder is reinstated permanently if the copyright holder notifies you of the violation by some reasonable means, this is the first time you have received notice of violation of this License (for any work) from that copyright holder, and you cure the violation prior to 30 days after your receipt of the notice.

Termination of your rights under this section does not terminate the licenses of parties who have received copies or rights from you under this License. If your rights have been terminated and not permanently reinstated, you do not qualify to receive new licenses for the same material under section 10.

- Acceptance Not Required for Having Copies.
You are not required to accept this License in order to receive or run a copy of the Program. Ancillary propagation of a covered work occurring solely as a consequence of using peer-to-peer transmission to receive a copy likewise does not require acceptance. However, nothing other than this License grants you permission to propagate or modify any covered work. These actions infringe copyright if you do not accept this License. Therefore, by modifying or propagating a covered work, you indicate your acceptance of this License to do so.

- Automatic Licensing of Downstream Recipients.
Each time you convey a covered work, the recipient automatically receives a license from the original licensors, to run, modify and propagate that work, subject to this License. You are not responsible for enforcing compliance by third parties with this License.

An “entity transaction” is a transaction transferring control of an organization, or substantially all assets of one, or subdividing an organization, or merging organizations. If propagation of a covered work results from an entity transaction, each party to that transaction who receives a copy of the work also receives whatever licenses to the work the party’s predecessor in interest had or could give under the previous paragraph, plus a right to possession of the Corresponding Source of the work from the predecessor in interest, if the predecessor has it or can get it with reasonable efforts.

You may not impose any further restrictions on the exercise of the rights granted or affirmed under this License. For example, you may not impose a license fee, royalty, or other charge for exercise of rights granted under this License, and you may not initiate litigation (including a cross-claim or counterclaim in a lawsuit) alleging that any patent claim is infringed by making, using, selling, offering for sale, or importing the Program or any portion of it.

- Patents.
A “contributor” is a copyright holder who authorizes use under this License of the Program or a work on which the Program is based. The work thus licensed is called the contributor’s “contributor version”.

A contributor’s “essential patent claims” are all patent claims owned or controlled by the contributor, whether already acquired or hereafter acquired, that would be infringed by some manner, permitted by this License, of making, using, or selling its contributor version, but do not include claims that would be infringed only as a consequence of further modification of the contributor version. For purposes of this definition, “control” includes the right to grant patent sublicenses in a manner consistent with the requirements of this License.

Each contributor grants you a non-exclusive, worldwide, royalty-free patent license under the contributor’s essential patent claims, to make, use, sell, offer for sale, import and otherwise run, modify and propagate the contents of its contributor version.

In the following three paragraphs, a “patent license” is any express agreement or commitment, however denominated, not to enforce a patent (such as an express permission to practice a patent or covenant not to sue for patent infringement). To “grant” such a patent license to a party means to make such an agreement or commitment not to enforce a patent against the party.

If you convey a covered work, knowingly relying on a patent license, and the Corresponding Source of the work is not available for anyone to copy, free of charge and under the terms of this License, through a publicly available network server or other readily accessible means, then you must either (1) cause the Corresponding Source to be so available, or (2) arrange to deprive yourself of the benefit of the patent license for this particular work, or (3) arrange, in a manner consistent with the requirements of this License, to extend the patent license to downstream recipients. “Knowingly relying” means you have actual knowledge that, but for the patent license, your conveying the covered work in a country, or your recipient’s use of the covered work in a country, would infringe one or more identifiable patents in that country that you have reason to believe are valid.

If, pursuant to or in connection with a single transaction or arrangement, you convey, or propagate by procuring conveyance of, a covered work, and grant a patent license to some of the parties receiving the covered work authorizing them to use, propagate, modify or convey a specific copy of the covered work, then the patent license you grant is automatically extended to all recipients of the covered work and works based on it.

A patent license is “discriminatory” if it does not include within the scope of its coverage, prohibits the exercise of, or is conditioned on the non-exercise of one or more of the rights that are specifically granted under this License. You may not convey a covered work if you are a party to an arrangement with a third party that is in the business of distributing software, under which you make payment to the third party based on the extent of your activity of conveying the work, and under which the third party grants, to any of the parties who would receive the covered work from you, a discriminatory patent license (a) in connection with copies of the covered work conveyed by you (or copies made from those copies), or (b) primarily for and in connection with specific products or compilations that contain the covered work, unless you entered into that arrangement, or that patent license was granted, prior to 28 March 2007.

Nothing in this License shall be construed as excluding or limiting any implied license or other defenses to infringement that may otherwise be available to you under applicable patent law.

- No Surrender of Others’ Freedom.
If conditions are imposed on you (whether by court order, agreement or otherwise) that contradict the conditions of this License, they do not excuse you from the conditions of this License. If you cannot convey a covered work so as to satisfy simultaneously your obligations under this License and any other pertinent obligations, then as a consequence you may not convey it at all. For example, if you agree to terms that obligate you to collect a royalty for further conveying from those to whom you convey the Program, the only way you could satisfy both those terms and this License would be to refrain entirely from conveying the Program.

- Use with the GNU Affero General Public License.
Notwithstanding any other provision of this License, you have permission to link or combine any covered work with a work licensed under version 3 of the GNU Affero General Public License into a single combined work, and to convey the resulting work. The terms of this License will continue to apply to the part which is the covered work, but the special requirements of the GNU Affero General Public License, section 13, concerning interaction through a network will apply to the combination as such.

- Revised Versions of this License.
The Free Software Foundation may publish revised and/or new versions of the GNU General Public License from time to time. Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns.

Each version is given a distinguishing version number. If the Program specifies that a certain numbered version of the GNU General Public License “or any later version” applies to it, you have the option of following the terms and conditions either of that numbered version or of any later version published by the Free Software Foundation. If the Program does not specify a version number of the GNU General Public License, you may choose any version ever published by the Free Software Foundation.

If the Program specifies that a proxy can decide which future versions of the GNU General Public License can be used, that proxy’s public statement of acceptance of a version permanently authorizes you to choose that version for the Program.

Later license versions may give you additional or different permissions. However, no additional obligations are imposed on any author or copyright holder as a result of your choosing to follow a later version.

- Disclaimer of Warranty.
THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY APPLICABLE LAW. EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM “AS IS” WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE. THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM IS WITH YOU. SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF ALL NECESSARY SERVICING, REPAIR OR CORRECTION.

- Limitation of Liability.
IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MODIFIES AND/OR CONVEYS THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS), EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF SUCH DAMAGES.

- Interpretation of Sections 15 and 16.
If the disclaimer of warranty and limitation of liability provided above cannot be given local legal effect according to their terms, reviewing courts shall apply local law that most closely approximates an absolute waiver of all civil liability in connection with the Program, unless a warranty or assumption of liability accompanies a copy of the Program in return for a fee.

If you develop a new program, and you want it to be of the greatest possible use to the public, the best way to achieve this is to make it free software which everyone can redistribute and change under these terms.

To do so, attach the following notices to the program. It is safest to attach them to the start of each source file to most effectively state the exclusion of warranty; and each file should have at least the “copyright” line and a pointer to where the full notice is found.

one line to give the program's name and a brief idea of what it does.Copyright (C)yearname of authorThis program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program. If not, see http://www.gnu.org/licenses/.

Also add information on how to contact you by electronic and paper mail.

If the program does terminal interaction, make it output a short notice like this when it starts in an interactive mode:

programCopyright (C)yearname of authorThis program comes with ABSOLUTELY NO WARRANTY; for details type ‘show w’. This is free software, and you are welcome to redistribute it under certain conditions; type ‘show c’ for details.

The hypothetical commands ‘`show w`’ and ‘`show c`’ should show
the appropriate parts of the General Public License. Of course, your
program’s commands might be different; for a GUI interface, you would
use an “about box”.

You should also get your employer (if you work as a programmer) or school, if any, to sign a “copyright disclaimer” for the program, if necessary. For more information on this, and how to apply and follow the GNU GPL, see http://www.gnu.org/licenses/.

The GNU General Public License does not permit incorporating your program into proprietary programs. If your program is a subroutine library, you may consider it more useful to permit linking proprietary applications with the library. If this is what you want to do, use the GNU Lesser General Public License instead of this License. But first, please read http://www.gnu.org/philosophy/why-not-lgpl.html.