knitr::opts_chunk$set( collapse = TRUE, #dev="png", comment = "#>" ) library(targeted)
Mathematical and statistical software often relies on sequential computations. ordinary differential equations where
numerical approximations are based on looping over the evolving time. When using
high-level languages such as R
calculations can be very slow
unless the algorithms can be vectorized
There are various excellent (O)DE solvers (R: deSolve)
Here I will illustrate the above techniques using the targeted
R
-package based on
the target C++ library.
The ODE is specified using the specify_ode
function
args(targeted::specify_ode)
The differential equations are here specified as a string containing the C++
code defining the
differential equation via the code
argument.
The variable names are defined through the pname
argument which defaults to
dy : Vector with derivatives, i.e. the rhs of the ODE, (y'(t)) (the result).
x : Vector, with the first element being the time, and the following elements additional exogenous input variables, (x(t) = {t, x_{1}(t), \ldots, x_{k}(t)})
y : Dependent variable, (y(t) = {y_{1}(t),\ldots,y_{l}(t)})
p : Parameter vector [y'(t) = f_{p}(x(t), y(t))]
All variables are treated as armadillo vectors/matrices, arma::mat
.
As an example, we can specify the simple differential equation [y'(t) = y(t)-1]
dy <- targeted::specify_ode("dy = y - 1;")
This compiles the function and stores the pointer in the variable dy
.
To solve the ODE we must then use the function solve_ode
args(targeted::solve_ode)
The first argument is the external pointer, the second argument input
is the
input matrix ((x(t)) above), and the init
argument is the vector of initial
boundary conditions (y(0)). The argument par
is the vector of parameters
defining the ODE ((p)).
In this example the input variable does not depend on any exogenous variables so we only need to supply the time points, and the defined ODE does not depend on any parameters. To approximate the solution with initial condition (y(0)=0), we therefore run the following code
t <- seq(0, 10, length.out=1e4) y <- targeted::solve_ode(dy, t, init=0) plot(t, y, type='l', lwd=3)
As a more interesting example consider the Lorenz Equations [\frac{dx_{t}}{dt} = \sigma(y_{t}-x_{t})] [\frac{dy_{t}}{dt} = x_{t}(\rho-z_{t})-y_{t}] [\frac{dz_{t}}{dt} = x_{t}y_{t}-\beta z_{t}]
we may define them as
library(targeted) ode <- 'dy(0) = p(0)*(y(1)-y(0)); dy(1) = y(0)*(p(1)-y(2)); dy(2) = y(0)*y(1)-p(2)*y(2);' f <- specify_ode(ode)
With the choice of parameters given by (\sigma=10, \rho=28, \beta=8/3) and initial conditions ((x_0,y_0,z_0)=(1,1,1)), we can calculate the solution
tt <- seq(0, 100, length.out=2e4) y <- solve_ode(f, input=tt, init=c(1, 1, 1), par=c(10, 28, 8/3)) head(y)
colnames(y) <- c("x","y","z") scatterplot3d::scatterplot3d(y, cex.symbols=0.1, type='b', color=viridisLite::viridis(nrow(y)))
To illustrate the use of exogenous inputs, consider the following simulated data
n <- 1e4 tt <- seq(0, 10, length.out=n) # Time xx <- rep(0, n); xx[(n/3):(2*n/3)] <- 1 # Exogenous input, x(t) input <- cbind(tt, xx)
and the following ODE
[y'(t) = \beta_{0} + \beta_{1}y(t) + \beta_{2}y(t)x(t) + \beta_{3}x(t)\cdot t]
mod <- 'double t = x(0); dy = p(0) + p(1)*y + p(2)*x(1)*y + p(3)*x(1)*t;' dy <- specify_ode(mod)
With (y(0)=100) and (\beta_0=0, \beta_{1}=0.4, \beta_{2}=-0.5, \beta_{3}=-5) we obtain the following solution
yy <- solve_ode(dy, input=input, init=100, c(0, .4, -.5, -5)) plot(tt, yy, type='l', lwd=3, xlab='time', ylab='y')
sessionInfo()
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