| Type: | Package |
| Title: | Sparse Identification of Nonlinear Dynamics |
| Version: | 0.2.4 |
| Date: | 2024-04-30 |
| Author: | Rick Dale and Harish S. Bhat |
| Maintainer: | Rick Dale <racdale@gmail.com> |
| Description: | This implements the Brunton et al (2016; PNAS <doi:10.1073/pnas.1517384113>) sparse identification algorithm for finding ordinary differential equations for a measured system from raw data (SINDy). The package includes a set of additional tools for working with raw data, with an emphasis on cognitive science applications (Dale and Bhat, 2018 <doi:10.1016/j.cogsys.2018.06.020>). See https://github.com/racdale/sindyr for examples and updates. |
| Depends: | R (≥ 3.4), arrangements, matrixStats, igraph, graphics, grDevices |
| Imports: | pracma |
| License: | GPL-2 | GPL-3 [expanded from: GPL (≥ 2)] |
| Collate: | 'windowed_sindy.R' 'sindy.R' 'features.R' 'finite_differences.R' 'finite_difference.R' 'lorenzattractor.R' |
| NeedsCompilation: | no |
| Packaged: | 2024-05-01 05:36:17 UTC; rd |
| Repository: | CRAN |
| Date/Publication: | 2024-05-01 06:00:02 UTC |
Sparse Identification of Nonlinear Dynamics
Description
This implements the Brunton et al (2016; PNAS, doi: 10.1073/pnas.1517384113) sparse identification algorithm for finding ordinary differential equations for a measured system from raw data (SINDy). The package includes a set of additional tools for working with raw data, with an emphasis on cognitive science applications (Dale and Bhat, 2018, doi: 10.1016/j.cogsys.2018.06.020). See <https://github.com/racdale/sindyr> for examples and updates.
Details
| Package: | sindyr |
| Type: | Package |
| Version: | 0.2.1 |
| Date: | 2018-09-10 |
| License: | GPL >= 2 |
sindy: Main function to infer coefficient matrix for set of ODEs.
windowed_sindy: Sliding window function to obtain SINDy results across segments of a time series.
features: Function for generation feature space from measured variables.
finite_differences: Numerical differentiation over multiple columns.
finite_difference: Numerical differential of a vector.
Author(s)
Rick Dale and Harish S. Bhat
References
Dale, R. and Bhat, H. S. (2018). Equations of mind: data science for inferring nonlinear dynamics of socio-cognitive systems. Cognitive Systems Research, 52, 275-290.
Brunton, S. L., Proctor, J. L., and Kutz, J. N. (2016). Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proceedings of the National Academy of Sciences, 113(15), 3932-3937.
For further examples and links to other materials see: https://github.com/racdale/sindyr
Examples
# example to reconstruct of
# the Lorenz system
library(sindyr)
set.seed(666)
dt = .001
numsteps = 10000; dt = dt; sigma = 10; r = 28; b = 2.6;
xs = data.frame(lorenzattractor(numsteps, dt, sigma, r, b))
colnames(xs) = list('x','y','z')
xs = xs[2000:nrow(xs),] # cut out initialization
Theta = features(xs,3) # grid of features
par(mfrow=c(7,3),oma = c(2,0,0,0) + 0.1,mar = c(1,1,1,1) + 0.1)
for (i in 2:ncol(Theta)) {
plot(Theta[,i],xlab='t',main=gsub(':','',colnames(Theta)[i]),type='l',xaxt='n',yaxt='n')
}
sindy.obj = sindy(xs=xs,dt=dt,lambda=.5) # let's reconstruct
sindy.obj$B # Lorenz equations
Build a matrix of features for SINDy
Description
Takes a raw matrix of data and converts into polynomial features
Arguments
x |
Raw data to be converted into features |
polyorder |
Order of polynomials (including k-th self products) |
intercept |
Include column of 1s in features to represent intercept (default = TRUE) |
Details
Expands raw data into a set of polynomial features.
Value
Returns a new matrix of data with features from raw data
Author(s)
Rick Dale and Harish S. Bhat
Estimate derivative of variable with finite differences
Description
Estimates first-order derivatives of a vector
Arguments
x |
Raw data to be differentiated |
S |
Sample rate of data to return derivatives using raw time |
Details
Uses simplest version of finite-difference method (window size 2) to numerically estimate derivative of a time series.
Value
Returns first-order numerical derivatives estimated from data.
Author(s)
Rick Dale and Harish S. Bhat
Estimate derivatives of multiple variables with finite differences
Description
Estimates first-order derivatives of column vectors of a matrix
Arguments
xs |
Raw data to be differentiated (matrix) |
S |
Sample rate of data to return derivatives using raw time |
Details
Uses simplest version of finite-difference method (window size 2) to numerically estimate derivative of multiple columnar time series.
Value
Returns first-order numerical derivatives estimated from data.
Author(s)
Rick Dale and Harish S. Bhat
Simulate the Lorenz Attractor
Description
An implementation of the Lorenz dynamical system, which describes the motion of a possible particle, which will neither converge to a steady state, nor diverge to infinity; but rather stay in a bounded but 'chaotically' defined region, i.e., an attractor.
Usage
lorenzattractor(numsteps, dt, sigma, r, b)
Arguments
numsteps |
The number of simulated points |
dt |
System parameter |
sigma |
System parameter |
r |
System parameter |
b |
System parameter |
Value
It returns a matrix with the 3 dimensions of the Lorenz
Author(s)
Moreno I. Coco (moreno.cocoi@gmail.com)
References
Lorenz, Edward Norton (1963). Deterministic nonperiodic flow. Journal of the Atmospheric Sciences 20(2) 130-141.
Examples
## initialize the parameters
numsteps = 2 ^ 11; dt = .01; sigma = 10; r = 28; b = 8/3;
res = lorenzattractor(numsteps, dt, sigma, r, b)
Run main SINDy function
Description
Estimates coefficients for set of ordinary differential equations governing system variables.
Arguments
xs |
Matrix of raw data |
dx |
Matrix of main system variable dervatives; if NULL, it estimates with finite differences from xs |
dt |
Sample interval, if data continuously sampled; default = 1 |
Theta |
Matrix of features; if not supplied, assumes polynomial features of order 3 |
lambda |
Threshold to use for iterated least squares sparsification (Brunton et al.) |
B.expected |
The function will compute a goodness of fit if supplied with an expected coefficient matrix B; default = NULL |
verbose |
Verbose mode outputs Theta and dx values in their entirety; default = FALSE |
fit.its |
Number of iterations to conduct the least-square threshold sparsification; default = 10 |
plot.eq.graph |
When set to TRUE, prints an igraph plot of variables as a graph structure; default = FALSE |
Details
Uses the "left-division" approach of Brunton et al. (2016), and implements least-squares sparsification, and outputs coefficients after iterations stabilize.
Value
Returns a matrix B of coefficients specifying the relationship between dx and Theta
Author(s)
Rick Dale and Harish S. Bhat
References
Dale, R. and Bhat, H. S. (in press). Equations of mind: data science for inferring nonlinear dynamics of socio-cognitive systems. Cognitive Systems Research.
Brunton, S. L., Proctor, J. L., and Kutz, J. N. (2016). Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proceedings of the National Academy of Sciences, 113(15), 3932-3937.
Run SINDy over time windows
Description
Run SINDy on raw data with a sliding window approach
Arguments
xs |
Matrix of raw data |
dx |
Matrix of main system variable dervatives; if NULL, it estimates with finite differences from xs |
dt |
Sample interval, if data continuously sampled; default = 1 |
Theta |
Matrix of features; if not supplied, assumes polynomial features of order 3 |
lambda |
Threshold to use for iterated least squares sparsification (Brunton et al.) |
fit.its |
Number of iterations to conduct the least-square threshold sparsification; default = 10 |
B.expected |
The function will compute a goodness of fit if supplied with an expected coefficient matrix B; default = NULL |
window.size |
Size of window to segment raw data as separate time series; defaults to deciles |
window.shift |
Step sizes across windows, permitting overlap; defaults to deciles |
Details
A convenience function for extracting a list of coefficients on segments of a time series. This facilitates using SINDy output as source of descriptive measures of dynamics.
Value
It returns a list of coefficients Bs containing B coefficients at each window
Author(s)
Rick Dale and Harish S. Bhat
References
Dale, R. and Bhat, H. S. (in press). Equations of mind: data science for inferring nonlinear dynamics of socio-cognitive systems. Cognitive Systems Research.