### Install nonlinearTseries package

Source: https://github.com/constantino-garcia/nonlineartseries/blob/master/README.md

Use these commands to install the package from CRAN or the development version from GitHub.

```r
# You can install the he latest released version from CRAN:
install.packages("nonlinearTseries")

# Or the the development version from GitHub:
devtools::install_github("constantino-garcia/nonlinearTseries")
```

--------------------------------

### Construct Takens Vectors

Source: https://context7.com/constantino-garcia/nonlineartseries/llms.txt

Builds delay embedding vectors for phase space reconstruction and visualizes the result.

```r
# Build Takens vectors for phase space reconstruction
takens <- buildTakens(ts.data,
                      embedding.dim = emb.dim,
                      time.lag = tau.ami)

# Takens vectors stored as matrix (one vector per row)
dim(takens)

# Plot reconstructed phase space (first 3 dimensions)
plot(takens[,1], takens[,2], type = "l",
     main = "Reconstructed Phase Space",
     xlab = "x(t)", ylab = "x(t + tau)")
```

--------------------------------

### Phase Space Reconstruction

Source: https://context7.com/constantino-garcia/nonlineartseries/llms.txt

Tools for estimating parameters required for Takens' embedding theorem.

```APIDOC
## timeLag(ts.data, technique, selection.method, lag.max, do.plot)

### Description
Estimates an appropriate time lag for Takens' vectors using autocorrelation or average mutual information.

### Parameters
- **ts.data** (vector) - Required - The input time series data.
- **technique** (string) - Optional - 'acf' or 'ami'.
- **selection.method** (string) - Optional - Method for selecting the lag (e.g., 'first.minimum').
- **lag.max** (integer) - Optional - Maximum lag to search.

## estimateEmbeddingDim(ts.data, ...)

### Description
Determines the minimum embedding dimension using Cao's algorithm based on false nearest neighbors.
```

--------------------------------

### Perform Recurrence Quantification Analysis (RQA)

Source: https://context7.com/constantino-garcia/nonlineartseries/llms.txt

Quantifies recurrence patterns in phase space using the rqa function. Requires a time series and embedding parameters.

```r
# Generate Rossler system data
r <- rossler(time = seq(0, 10, by = 0.01), do.plot = FALSE)

# Perform RQA
rqa.result <- rqa(time.series = r$x,
                  embedding.dim = 2,
                  time.lag = 1,
                  radius = 1.2,
                  lmin = 2,
                  vmin = 2,
                  distanceToBorder = 2,
                  save.RM = TRUE,
                  do.plot = FALSE)

# Access RQA measures
cat("Recurrence Rate (REC):", rqa.result$REC, "\n")
cat("Determinism (DET):", rqa.result$DET, "\n")
cat("Laminarity (LAM):", rqa.result$LAM, "\n")
cat("Longest diagonal (Lmax):", rqa.result$Lmax, "\n")
cat("Mean diagonal length (Lmean):", rqa.result$Lmean, "\n")
cat("Entropy (ENTR):", rqa.result$ENTR, "\n")
cat("Trapping time (Vmean):", rqa.result$Vmean, "\n")
cat("Trend:", rqa.result$TREND, "\n")

# Plot recurrence plot (computationally expensive)
plot(rqa.result)
```

--------------------------------

### Generate Surrogate Data via Phase Randomization

Source: https://context7.com/constantino-garcia/nonlineartseries/llms.txt

Creates surrogate time series using FFT-based phase randomization to preserve the power spectrum of the original data.

```r
# Generate surrogate data sets
original <- arima.sim(list(order = c(1, 0, 1), ar = 0.6, ma = 0.5),
                      n = 500)
surrogates <- FFTsurrogate(original, n.samples = 20)

# Surrogates have same autocorrelation but randomized phases
dim(surrogates)  # 20 x 500 matrix

# Compare original and surrogate
par(mfrow = c(2, 1))
plot(original, type = "l", main = "Original")
plot(surrogates[1,], type = "l", main = "Surrogate 1")
par(mfrow = c(1, 1))
```

--------------------------------

### Generate Lorenz System Time Series

Source: https://context7.com/constantino-garcia/nonlineartseries/llms.txt

Generate a 3D time series using Lorenz differential equations with default chaotic parameters.

```r
library(nonlinearTseries)

# Generate Lorenz system with default chaotic parameters
lor <- lorenz(sigma = 10, rho = 28, beta = 8/3,
              start = c(-13, -14, 47),
              time = seq(0, 50, by = 0.01),
              do.plot = FALSE)

# Access time series components
head(lor$x)  # x-component
head(lor$y)  # y-component
head(lor$z)  # z-component
head(lor$time)  # time vector

# Plot the x-component time series
plot(lor$time, lor$x, type = "l",
     xlab = "Time", ylab = "x(t)",
     main = "Lorenz System x-component")
```

--------------------------------

### Compute Sample Entropy

Source: https://context7.com/constantino-garcia/nonlineartseries/llms.txt

Calculates Kolmogorov-Sinai entropy to measure the complexity of a time series.

```r
# First compute correlation sums
h <- henon(n.sample = 15000, n.transient = 100, do.plot = FALSE)
cd <- corrDim(time.series = h$x,
              min.embedding.dim = 2,
              max.embedding.dim = 9,
              time.lag = 1,
              min.radius = 0.05,
              max.radius = 1,
              n.points.radius = 100,
              theiler.window = 20,
              do.plot = FALSE)

# Compute sample entropy from correlation sums
se <- sampleEntropy(cd, do.plot = TRUE)

# Estimate from plateau region
se.est <- estimate(se,
                   regression.range = c(0.4, 0.6),
                   use.embeddings = 6:9,
                   do.plot = TRUE)
cat("Sample entropy estimates:", se.est, "\n")
cat("Mean sample entropy:", mean(se.est), "\n")
```

--------------------------------

### Test for Nonlinearity using Surrogate Data

Source: https://context7.com/constantino-garcia/nonlineartseries/llms.txt

Tests the null hypothesis of a linear stochastic process by comparing original data statistics against surrogate data.

```r
# Generate nonlinear data
lor <- lorenz(do.plot = FALSE)

# Perform surrogate data test using time asymmetry statistic
sdt <- surrogateTest(time.series = lor$x,
                     significance = 0.05,
                     one.sided = FALSE,
                     K = 5,
                     FUN = timeAsymmetry,
                     verbose = TRUE,
                     do.plot = FALSE)

# Plot comparison of original vs surrogate statistics
plot(sdt)

# Access results
cat("Original data statistic:", sdt$data.statistic, "\n")
cat("Surrogate statistics range:",
    range(sdt$surrogates.statistics), "\n")
# If original statistic falls outside surrogate range,
# reject null hypothesis -> data is nonlinear
```

--------------------------------

### Generate Rössler System Time Series

Source: https://context7.com/constantino-garcia/nonlineartseries/llms.txt

Generate a 3D continuous time series using the Rössler system.

```r
# Generate Rossler system
r <- rossler(a = 0.2, b = 0.2, w = 5.7,
             start = c(-2, -10, 0.2),
             time = seq(0, 100, by = 0.1),
             do.plot = FALSE)

# Plot phase space projection
plot(r$x, r$y, type = "l",
     main = "Rössler System Phase Space (x-y projection)",
     xlab = "x(t)", ylab = "y(t)")
```

--------------------------------

### Compute Average Mutual Information (AMI)

Source: https://context7.com/constantino-garcia/nonlineartseries/llms.txt

Calculates the AMI function to assist in determining the optimal time lag for phase space reconstruction.

```r
# Generate sample data
sx <- sinaiMap(a = 0.3, n.sample = 5000,
               start = c(0.23489, 0.8923),
               do.plot = FALSE)$x

# Compute mutual information
mutinf <- mutualInformation(sx,
                            lag.max = 100,
                            n.partitions = 20,
                            units = "Bits",
                            do.plot = TRUE)

# Access AMI values
head(mutinf$mutual.information)
head(mutinf$time.lag)

# Find first minimum for optimal time lag
first.min <- which.min(mutinf$mutual.information[2:50]) + 1
cat("Suggested time lag:", first.min, "\n")
```

--------------------------------

### Estimate Embedding Dimension

Source: https://context7.com/constantino-garcia/nonlineartseries/llms.txt

Uses Cao's algorithm to determine the appropriate embedding dimension for phase space reconstruction.

```r
emb.dim <- estimateEmbeddingDim(ts.data,
                                 time.lag = tau.ami,
                                 max.embedding.dim = 15,
                                 threshold = 0.95,
                                 do.plot = TRUE)
cat("Estimated embedding dimension:", emb.dim, "\n")
```

--------------------------------

### Generate Hénon Map Time Series

Source: https://context7.com/constantino-garcia/nonlineartseries/llms.txt

Generate a 2D discrete time series using the Hénon map.

```r
# Generate Henon map time series
h <- henon(a = 1.4, b = 0.3,
           start = c(0.63954883, 0.04772637),
           n.sample = 5000,
           n.transient = 500,
           do.plot = FALSE)

# Access the x and y coordinates
plot(h$x, h$y, pch = ".",
     main = "Hénon Attractor",
     xlab = "x[n]", ylab = "y[n]")

# The x-component can be used for time series analysis
ts.plot(h$x[1:500], main = "Hénon x-coordinate")
```

--------------------------------

### Compute Information Dimension

Source: https://context7.com/constantino-garcia/nonlineartseries/llms.txt

Estimates the information dimension (D1) using the fixed-mass algorithm on Henon map data.

```r
# Generate Henon map data for information dimension
h <- henon(n.sample = 3000, n.transient = 100, do.plot = FALSE)

# Compute information dimension
id <- infDim(time.series = h$x,
             min.embedding.dim = 2,
             max.embedding.dim = 5,
             time.lag = 1,
             min.fixed.mass = 0.04,
             max.fixed.mass = 0.2,
             number.fixed.mass.points = 100,
             radius = 0.001,
             theiler.window = 10,
             do.plot = FALSE)

# Plot and estimate
plot(id, type = "l")
id.est <- estimate(id,
                   regression.range = c(0.05, 0.15),
                   use.embeddings = 3:5,
                   do.plot = TRUE)
cat("Information dimension estimate:", id.est, "\n")
```

--------------------------------

### Perform Detrended Fluctuation Analysis (DFA)

Source: https://context7.com/constantino-garcia/nonlineartseries/llms.txt

Estimates the scaling exponent alpha to detect long-range correlations in time series data.

```r
# Generate fractional Gaussian noise-like data
set.seed(42)
white.noise <- rnorm(5000)

# Perform DFA
dfa.result <- dfa(time.series = white.noise,
                  window.size.range = c(10, 1000),
                  npoints = 20,
                  do.plot = FALSE)

# Plot fluctuation function
plot(dfa.result)

# Estimate scaling exponent
alpha <- estimate(dfa.result,
                  regression.range = c(10, 1000),
                  do.plot = TRUE)
cat("DFA scaling exponent (alpha):", alpha, "\n")
# Expected for white noise: approximately 0.5
# alpha < 0.5: anti-correlated
# alpha = 0.5: uncorrelated (white noise)
# alpha > 0.5: long-range correlated
# alpha = 1.0: 1/f noise
# alpha > 1.0: non-stationary
```

--------------------------------

### Chaotic System Generators

Source: https://context7.com/constantino-garcia/nonlineartseries/llms.txt

Functions to generate time series data from various chaotic dynamical systems.

```APIDOC
## lorenz(sigma, rho, beta, start, time, do.plot)

### Description
Generates a 3-dimensional time series using the Lorenz differential equations.

### Parameters
- **sigma** (numeric) - Optional - Lorenz parameter sigma.
- **rho** (numeric) - Optional - Lorenz parameter rho.
- **beta** (numeric) - Optional - Lorenz parameter beta.
- **start** (vector) - Optional - Initial conditions c(x, y, z).
- **time** (vector) - Optional - Time sequence.
- **do.plot** (boolean) - Optional - Whether to plot the result.

## henon(a, b, start, n.sample, n.transient, do.plot)

### Description
Generates a 2-dimensional discrete time series using the Hénon map.

### Parameters
- **a** (numeric) - Optional - Hénon parameter a.
- **b** (numeric) - Optional - Hénon parameter b.
- **start** (vector) - Optional - Initial conditions c(x, y).
- **n.sample** (integer) - Optional - Number of samples to generate.
- **n.transient** (integer) - Optional - Number of transient points to discard.

## logisticMap(r, start, n.sample, n.transient, do.plot)

### Description
Generates a 1-dimensional time series using the logistic map.

### Parameters
- **r** (numeric) - Optional - Logistic map parameter r.
- **start** (numeric) - Optional - Initial value.
- **n.sample** (integer) - Optional - Number of samples.
```

--------------------------------

### Generate Logistic Map Time Series

Source: https://context7.com/constantino-garcia/nonlineartseries/llms.txt

Generate a 1D time series using the logistic map.

```r
# Generate chaotic logistic map (r=4)
lm <- logisticMap(r = 4,
                  start = 0.1,
                  n.sample = 1000,
                  n.transient = 100,
                  do.plot = FALSE)

# Plot the time series
plot(lm, type = "l",
     main = "Logistic Map (r=4)",
     xlab = "n", ylab = "x[n]")
```

--------------------------------

### Compute Maximum Lyapunov Exponent

Source: https://context7.com/constantino-garcia/nonlineartseries/llms.txt

Measures the rate of divergence of nearby trajectories to identify chaotic behavior.

```r
# Compute divergence of nearby trajectories
ml <- maxLyapunov(time.series = ts.data,
                  sampling.period = 0.01,
                  min.embedding.dim = 4,
                  max.embedding.dim = 7,
                  time.lag = tau.ami,
                  radius = 1,
                  theiler.window = 50,
                  min.neighs = 5,
                  min.ref.points = 500,
                  max.time.steps = 1000,
                  do.plot = FALSE)

# Plot S(t) function showing exponential divergence
plot(ml, type = "l", xlim = c(0, 8))

# Estimate Lyapunov exponent from linear region
ml.est <- estimate(ml,
                   regression.range = c(0, 3),
                   use.embeddings = 5:7,
                   do.plot = TRUE)
cat("Maximum Lyapunov exponent:", ml.est, "\n")

# Positive value indicates chaos
if (ml.est > 0) {
  cat("System exhibits chaotic behavior\n")
}
```

--------------------------------

### Estimate Time Lag Parameter

Source: https://context7.com/constantino-garcia/nonlineartseries/llms.txt

Estimate the time lag for Takens' vectors using autocorrelation or average mutual information.

```r
# Generate sample data
lor <- lorenz(do.plot = FALSE)
ts.data <- lor$x

# Estimate time lag using autocorrelation function
tau.acf <- timeLag(ts.data,
                   technique = "acf",
                   selection.method = "first.e.decay",
                   lag.max = 100,
                   do.plot = TRUE)
cat("Time lag (ACF):", tau.acf, "\n")

# Estimate time lag using average mutual information (preferred)
tau.ami <- timeLag(ts.data,
                   technique = "ami",
                   selection.method = "first.minimum",
                   lag.max = 100,
                   do.plot = TRUE)
cat("Time lag (AMI):", tau.ami, "\n")
```

--------------------------------

### Compute Correlation Dimension

Source: https://context7.com/constantino-garcia/nonlineartseries/llms.txt

Estimates the fractal dimension using the Grassberger-Procaccia algorithm.

```r
# Compute correlation sums for multiple embedding dimensions
cd <- corrDim(time.series = ts.data,
              min.embedding.dim = 4,
              max.embedding.dim = 9,
              time.lag = tau.ami,
              min.radius = 0.001,
              max.radius = 50,
              n.points.radius = 40,
              theiler.window = 100,
              do.plot = FALSE)

# Plot correlation sums with local scaling exponents
plot(cd)

# Estimate correlation dimension from linear scaling region
cd.est <- estimate(cd,
                   regression.range = c(0.75, 3),
                   use.embeddings = 5:7,
                   do.plot = TRUE)
cat("Correlation dimension estimate:", cd.est, "\n")
```

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