# colorblind friendly palette palette(c("#000000", "#E69F00", "#56B4E9", "#009E73", "#F0E442", "#0072B2", "#D55E00", "#CC79A7")) library('knitr') knitr::opts_chunk$set(fig.width=7,fig.height=4.7,fig.show='hold')
The nonlinearTseries package provides functionality for nonlinear time series analysis. This package permits the computation of the most-used nonlinear statistics/algorithms including generalized correlation dimension, information dimension, largest Lyapunov exponent, sample entropy and Recurrence Quantification Analysis (RQA), among others. Basic routines for surrogate data testing are also included. This vignette provides a brief overview of the most important routines contained in nonlinearTseries.
To explore the routines included in nonlinearTseries, we will study the famous Lorenz system. nonlinearTseries offers different routines for simulating the best well-known nonlinear systems:
suppressMessages(library('nonlinearTseries')) library('plot3D') # by default, the simulation creates a RGL plot of the system's phase space lor = lorenz(do.plot = F) # let's plot the phase space of the simulated lorenz system scatter3D(lor$x, lor$y, lor$z, main = "Lorenz's system phase space", col = 1, type="o",cex = 0.3)
It must be noted that the
lorenz function returns the simulated components
of the system in a
list. Future versions of the package will allow to obtain
the same simulations as
A complete list of the available functions for nonlinear systems simulation
can be found in the
lorenz help page (
Usually, what we observe in a physical experiment is a single time series and not the complete phase space. For example, let's assume that we have only measured the $x$ component of the Lorenz system. Fortunately, we can still infer the properties of the phase space by constructing a set of vectors whose components are time delayed versions of the $x$ signal $[x(t), x(t+\tau), ..., x(t + m\tau)]$ (This theoretical result is referred to as the Takens' embedding theorem).
The nonlinearTseries package provides functions for estimating proper values of the embedding dimension $m$ and the delay-parameter $\tau$. First, the delay-parameter can be estimated by using the autocorrelation function or the average mutual information of the signal.
# suppose that we have only measured the x-component of the Lorenz system lor.x = lor$x old.par = par(mfrow = c(1, 2)) # tau-delay estimation based on the autocorrelation function tau.acf = timeLag(lor.x, technique = "acf", lag.max = 100, do.plot = T) # tau-delay estimation based on the mutual information function tau.ami = timeLag(lor.x, technique = "ami", lag.max = 100, do.plot = T) par(old.par)
Both techniques select a time-lag based on the behavior of the autocorrelation or the average mutual information function. Since the autocorrelation function is a linear statistic we usually obtain more appropriate values with the mutual information technique. Thus, for the remainder of this section, we will use the value obtained with this technique.
Once the time-lag parameter has been estimated, a proper embedding dimension can
be computed by using the well-known Cao's algorithm (see the documentation
estimateEmbeddingDim function for references):
emb.dim = estimateEmbeddingDim(lor.x, time.lag = tau.ami, max.embedding.dim = 15)
When applied to the Lorenz system, the Cao's algorithm suggests the use of
an embedding dimension of 4. The final phase space reconstruction can be obtained
tak = buildTakens(lor.x,embedding.dim = emb.dim, time.lag = tau.ami) scatter3D(tak[,1], tak[,2], tak[,3], main = "Lorenz's system reconstructed phase space", col = 1, type="o",cex = 0.3)
Note that the reconstructed and the original phase space, although different, share similar topological features.
In practical applications, some of the best well-known nonlinear statistics (such as the Lyapunov exponent, the generalized correlation dimensions or the sample entropies) share a similar estimation process. This process could be summarized as follows:
plotfunction can be used with all the objects involved in the computation of these statistics.
In the following sections we illustrate this procedure computing the correlation dimension, the sample entropy and the Lyapunov exponents of the Lorenz system.
The correlation dimension is a technique that measures the fractal dimension of
the phase space of a dynamical system. To verify
that the estimation of the correlation dimension does not depend on the
embedding dimension, we compute the correlation sums (
for several embedding dimensions. Once we have checked for the existence of the
linear regions in different embedding dimensions, we obtain an estimation of
the correlation dimension with the
estimate function. This function allows to
specify the range in which the linear behavior appears (
parameter) as well as the embedding dimensions
to be used for the estimation of the correlation dimension (
parameter). The final estimation of the correlation dimension is an average of
the slopes obtained for each embedding dimension.
cd = corrDim(lor.x, min.embedding.dim = emb.dim, max.embedding.dim = emb.dim + 5, time.lag = tau.ami, min.radius = 0.001, max.radius = 50, n.points.radius = 40, do.plot=FALSE) plot(cd) cd.est = estimate(cd, regression.range=c(0.75,3), use.embeddings = 5:7) cat("expected: 2.05 --- estimate: ",cd.est,"\n")
The generalized correlation dimensions can also be computed with the
function (by modifying the
q parameter). To estimate the information dimension,
nonlinearTseries provides the
infDim function (see
?infDim for more
The sample entropy is a technique for measuring the unpredictability of a time
series. It is possible to use the correlation sums for obtaining an estimation
of the sample entropy of a time series. In this case, the computations should
yield a function with a clear plateau. The value of this plateau is an estimation
of the sample entropy. The next chunk of code illustrates the procedure for
estimating the sample entropy from a previously computed
se = sampleEntropy(cd, do.plot = F) se.est = estimate(se, do.plot = F, regression.range = c(8,15)) cat("Sample entropy estimate: ", mean(se.est), "\n")
One of the more important characteristics of a chaotic system is its sensitivity
to initial conditions. As a consequence of this sensitivity, close trajectories
diverge exponentially fast. The maximum Lyapunov exponent measures the
average rate of divergence of close trajectories in the system. The
function can be used for computing this divergence through time. To define what is
a close trajectory we make use of the
parameter. After the computation of the divergence rates we can get an
estimate of the maximum Lyapunov exponent by performing a linear regression
estimate function), just as we did with the correlation dimension.
# get the sampling period of the lorenz simulation # computing the differences of time (all differences should be equal) sampling.period = diff(lor$time) ml = maxLyapunov(lor.x, sampling.period=0.01, min.embedding.dim = emb.dim, max.embedding.dim = emb.dim + 3, time.lag = tau.ami, radius=1, max.time.steps=1000, do.plot=FALSE) plot(ml,type="l", xlim = c(0,8)) ml.est = estimate(ml, regression.range = c(0,3), do.plot = T,type="l") cat("expected: 0.906 --- estimate: ", ml.est,"\n")
Although we have postponed its discussion until the end of this vignette, the first step before studying a system using nonlinear analysis techniques should be checking that the data shows indeed some degree of nonlinearity.
The preferred method for nonlinearity-test in literature is surrogate data testing. In surrogate data testing, a statistic $\mu$ quantifying some nonlinear feature of the data is computed and compared with the resulting values for an ensemble of comparable linear processes.
nonlinearTseries includes basic functionality for surrogate data testing. The next example performs surrogate data testing by measuring the time asymmetry of the data and the surrogates (since linear stochastic processes are symmetric under time reversal, a deviation from the distribution of the surrogates would be a strong sign of nonlinearity). From the resulting figure, it is clear that our time series shows some degree of nonlinearity.
st = surrogateTest(lor.x,significance = 0.05,one.sided = F, FUN = timeAsymmetry, do.plot=F) plot(st)
In this quickstart vignette we have only covered some of the main functions included in nonlinearTseries. Other interesting functions included in this package are enumerated below. The main reason for not including them in this quickstart guide is that these functions are quite simple to use.
rqa: performs Recurrence Quantification Analysis.
dfa: performs Detrended Fluctuation Analysis.
nonLinearNoiseReduction: denoises a given time series using phase space techniques.
poincareMap: computes a Poincaré map of the trajectories in phase space.
spaceTimePlot: shows the space time separation plot: broadly-used method of detecting non-stationarity and temporal correlations in time series.
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