Objective :
Related material:
Readings :
- Rasmussen, Carl Edward. "Gaussian processes in machine learning." In Advanced lectures on machine learning, pp. 63-71. Springer Berlin Heidelberg, 2004.
- see http://www.GaussianProcess.org/gpml
- Chapters: 2, 4, 5.1, & 5.4.2
- Slides provided
Software :
- R-cran (https://cran.r-project.org/)
- R packages
- DiceKrigin (https://cran.r-project.org/web/packages/DiceKriging/index.html)
- lhs (https://cran.r-project.org/web/packages/lhs/index.html)
- Roustant, Olivier, David Ginsbourger, and Yves Deville. "DiceKriging, DiceOptim: Two R packages for the analysis of computer experiments by kriging-based metamodeling and optimization." (2012).
- Plus dependences ...This is Example 3.1 of [Tsilifis, Panagiotis, Ilias Bilionis, Ioannis Katsounaros, and Nicholas Zabaras. "Variational Reformulation of Bayesian Inverse Problems." arXiv preprint arXiv:1410.5522 (2014)].
Consider the catalytic conversion of nitrate ($\mbox{NO}_3^-$) to nitrogen ($\mbox{N}_2$) and other by-products by electrochemical means. The mechanism that is followed is complex and not well understood. The experiment of \cite{katsounaros} confirmed the production of nitrogen ($\mbox{N}_2$), ammonia ($\mbox{NH}_3$), and nitrous oxide ($\mbox{N}_2\mbox{O}$) as final products of the reaction, as well as the intermediate production of nitrite ($\mbox{NO}_2^-$). The time is measured in minutes and the conentrations are measured in $\mbox{mmol}\cdot\mbox{L}^{-1}$. Let's load the data into this notebook using the Pandas Python module:
This inconsistency suggests the existence of an intermediate unobserved reaction product X. [Katsounaros, Ioannis, Maria Dortsiou, Christos Polatides, Simon Preston, Theodore Kypraios, and Georgios Kyriacou. "Reaction pathways in the electrochemical reduction of nitrate on tin." Electrochimica Acta 71 (2012): 270-276.] suggested that the following reaction path shown in the following figure.
The dynamical system associated with the reaction is: $$ \begin{array}{cc} \frac{d \left[\mbox{NO}_3^-\right]}{dt} &= -k_1\left[\mbox{NO}_3^-\right], \\ \frac{d\left[\mbox{NO}_2^-\right]}{dt} &= k_1\left[\mbox{NO}_3^-\right] - (k_2 + k_4 + k_5)[\mbox{NO}_2^-], \\ \frac{d \left[\mbox{X}\right]}{dt} &= k_2 \left[\mbox{NO}_2^-\right] - k_3 [X],\\ \frac{d \left[\mbox{N}_2\right]}{dt} &= k_3 \left[\mbox{X}\right], \\ \frac{d \left[\mbox{NH}_3\right]}{dt} &= k_4 \left[\mbox{NO}_2^-\right],\\ \frac{d \left[\mbox{N}_2O\right]}{dt} &= k_5 \left[\mbox{NO}_2^-\right], \end{array} $$ where $[\cdot]$ denotes the concentration of a quantity, and $k_i > 0$, $i=1,...5$ are the kinetic rate constants.
We will develop a generic computational model for the solution of dynamical systems and we will use it to study the catalysis problem. The code relies on the Fourth-order Runge-Kutta method and is a modified copy of http://www.math-cs.gordon.edu/courses/ma342/python/diffeq.py developed by Jonathan Senning. The code solves:
$$ \begin{array}{ccc} \dot{\mathbf{y}} &=& f(\mathbf{y}, t),\\ \mathbf{y}(0) &=& \mathbf{y}_0. \end{array} $$The input values are:
| Variable | Value |
|---|---|
| $\xi_1$ | $1.35\pm 0.05$ |
| $\xi_2$ | $1.65\pm 0.08$ |
| $\xi_3$ | $1.34\pm 0.11$ |
| $\xi_4$ | $-0.16\pm 0.16$ |
| $\xi_5$ | $-3.84\pm 0.20$ |
The output values of the simulator are the concentrations (in $\mbox{mmol}\cdot\mbox{L}^{-1}$) of $\mbox{NO}_3^-$, $\mbox{NO}_2^-$, X ( unobserved reaction product), $\mbox{N}_2$, $\mbox{NH}_3$, $\mbox{N}_2\mbox{O}$, and $\mbox{NO}_2^-$.
The R code in './catalytic.R' provides a simulator that returns only one output value (selected by the user), given the values of the 5 inputs.
Tsilifis, Panagiotis, Ilias Bilionis, Ioannis Katsounaros, and Nicholas Zabaras. "Variational Reformulation of Bayesian Inverse Problems." arXiv preprint arXiv:1410.5522 (2014)
In [1]:
# DOWNLOAD THE R PACKAGES REQUIRED
install.packages('DiceKriging', repos = "http://cran.us.r-project.org")
install.packages('lhs', repos = "http://cran.us.r-project.org")
# install.packages('tcltk', repos = "http://cran.us.r-project.org")
# install.packages('aplpack', repos = "http://cran.us.r-project.org")
In [2]:
# LOAD THE R PACKAGES REQUIRED
library('lhs')
library('DiceKriging')
# library('tcltk')
# library('aplpack')
In [3]:
# THIS IS THE SIMULATOR AND THE MIN AND MAX OF THE INPUTS
source("./catalytic.R") # function: output_1d <- simulator(input_5d, jout=4)
input_min <- c(1.30, 1.57, 1.23, -0.32, -4.04)
input_max <- c(1.40, 1.73, 1.45, 0.00, -3.64)
input_d <- length(input_min)
# par(mfrow=c(2,3))
# for (i in 1:6) {
# n_data <- 500 ;
# n_dim <- input_d
# X_data <- t(input_min + (input_max-input_min)*t(matrix(runif(n_data*n_dim),n_data, n_dim))) ;
# myfun <-function(xi){ return(simulator(xi,i))}
# Y_data <- apply(X_data, 1, myfun) ;
# hist(Y_data)
# }
myfun <- function(xx) {return(simulator(xx, jout=4)) }
# # PLOT THE REAL FUNCTION TO SEE HOW IT LOOKS LIKE
# par(mfrow = c(3,3))
# for (i in 1:input_d) {
# for ( j in 1:input_d )
# if(i>j) {
# n.grid <- 100 ;
# x1.grid <-seq(input_min[i],input_max[i],length.out=n.grid) ;
# x2.grid <-seq(input_min[j],input_max[j],length.out=n.grid) ;
# X.grid <- expand.grid( x1=x1.grid, x2=x2.grid )
# myfun2d<-function(xx){
# zz<-0.5*(input_min+input_max) ;
# zz[i]<-xx[1]; zz[j]<-xx[2];
# return(myfun(zz))
# }
# y.grid <- apply(X.grid,1,myfun2d)
# contour(x1.grid, x2.grid, matrix(y.grid, n.grid, n.grid), 10,
# main = "Real function",
# xlab = paste("x", as.character(i)),
# ylab = paste("x", as.character(j)),
# xlim = c(input_min[i],input_max[i]),
# ylim = c(input_min[j],input_max[j]))
# }
# }
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In [5]:
# PRINT THE PARAMETERS
show(myfun_km)
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