Package 'SITSAR'

Description

Computes the usual accuracy measures of binary change maps: detection rate (DR, percentage of true changes that were detected), false alarm rate (FA, percentage of unchanged pixels flagged as changed), and the percentages of false positives (FP) and false negatives (FN) with respect to all pixels.

Usage

change_detection_metrics(truth, estimate)

Arguments

Value

A tibble with DR, FA, FP, FN (in percent) and the overall accuracy OA (in percent).

Examples

truth <- c(TRUE, TRUE, FALSE, FALSE, FALSE)
est <- c(TRUE, FALSE, TRUE, FALSE, FALSE)
change_detection_metrics(truth, est)

Description

Applies the entropy test of entropy_test() pixel by pixel, using the observations of a moving window in two co-registered images, and returns the map of p-values and the binary change map.

Usage

change_map(
  image1,
  image2,
  model = c("gamma", "g0i", "ggamma", "wishart", "g0p"),
  window = 5,
  L = NULL,
  channel = 1,
  level = 0.01
)

Arguments

Value

A list of class sitsar_change with the matrices statistic, p_value and change (logical), and the level.

Examples

set.seed(1)
a <- matrix(rgammasar(40 * 40, mu = 1, L = 4), 40)
b <- a
b[11:30, 11:30] <- rgammasar(400, mu = 4, L = 4)
cd <- change_map(a, b, model = "gamma", window = 5, L = 4)
cd
plot_map(cd$change, main = "Detected changes")

Description

Density and random generation for the compound truncated Poisson complex Wishart (CTPCW) and compound geometric complex Wishart (CGCW) laws (Ferreira, Nascimento and Frery, 2022). They describe the PolSAR return as a random sum $S = \sum_{i=1}^N Z_i$ of independent (unscaled) complex Wishart matrices $Z_i \sim \mathcal{W}^C_m(\Sigma, L)$ with $E(Z_i) = L\Sigma$, where the number of scatterers $N$ follows a zero-truncated Poisson law with mean parameter $\lambda$ (CTPCW) or a geometric law on $\{1, 2, \ldots\}$ with success probability $p$ (CGCW). Their densities are

$f(S) = \sum_{k=1}^{\infty} P(N = k) \frac{|S|^{kL - m}}{|\Sigma|^{kL}\Gamma_m(kL)} \exp\{-\mathrm{tr}(\Sigma^{-1}S)\},$

which are able to describe multimodal intensities. The series is evaluated in log-scale and truncated when the terms become negligible.

Usage

dctpcw(x, lambda, Sigma, L, log = FALSE, kmax = 500L)

rctpcw(n, lambda, Sigma, L, output = c("array", "vec"))

dcgcw(x, p, Sigma, L, log = FALSE, kmax = 500L)

rcgcw(n, p, Sigma, L, output = c("array", "vec"))

Arguments

Value

dctpcw()/dcgcw() return (log-)densities; rctpcw()/rcgcw() return the simulated sample, with the simulated numbers of components in the attribute "N".

References

Ferreira, J. A., Nascimento, A. D. C. and Frery, A. C. (2022). PolSAR models with multimodal intensities. Remote Sensing, 14(20), 5083. doi:10.3390/rs14205083

See Also

CompoundWishart for the R6 interface and EM estimation.

Examples

Sigma <- polsar_example_sigma("forest")
set.seed(1)
S <- rctpcw(3, lambda = 2, Sigma = Sigma, L = 4)
attr(S, "N")
dctpcw(S, lambda = 2, Sigma = Sigma, L = 4, log = TRUE)
S2 <- rcgcw(3, p = 0.6, Sigma = Sigma, L = 4)
dcgcw(S2, p = 0.6, Sigma = Sigma, L = 4, log = TRUE)

Description

R6 class for the compound truncated Poisson (CTPCW) and compound geometric (CGCW) complex Wishart laws of Ferreira, Nascimento and Frery (2022); see dctpcw(). Parameters are estimated by the expectation-maximisation (EM) algorithm, treating the number of scatterers $N$ as missing data:

• E-step: $\hat N_i = E(N \mid S_i)$;

• M-step: $\hat\Sigma = \bar S / (L \bar N)$ and $\hat\lambda/(1 - e^{-\hat\lambda}) = \bar N$ (CTPCW) or $\hat p = 1/\bar N$ (CGCW).

The Shannon entropy has no closed form and is approximated by Monte Carlo.

Value

An R6::R6Class generator. CompoundWishart$new() returns a new CompoundWishart object; its methods are described in the Methods section below.

Super class

SARModel -> CompoundWishart

Public fields

Methods

Public methods

• CompoundWishart$new()

• CompoundWishart$density()

• CompoundWishart$random()

• CompoundWishart$fit()

• CompoundWishart$entropy()

• CompoundWishart$entropy_variance()

• CompoundWishart$clone()

CompoundWishart$new()

Create a compound Wishart model.

Usage

CompoundWishart$new(
  counting = c("poisson", "geometric"),
  Sigma = diag(3) + (0+0i),
  L = 4,
  lambda = 1,
  p = 0.5
)

Arguments

CompoundWishart$density()

(Log-)density.

Usage

CompoundWishart$density(x, log = FALSE)

Arguments

CompoundWishart$random()

Simulate PolSAR observations.

Usage

CompoundWishart$random(n, output = "array")

Arguments

CompoundWishart$fit()

Fit by the EM algorithm.

Usage

CompoundWishart$fit(x, L, tol = 1e-06, max_iter = 500L)

Arguments

CompoundWishart$entropy()

Monte Carlo approximation of the Shannon entropy.

Usage

CompoundWishart$entropy(nsim = 2000L)

Arguments

CompoundWishart$entropy_variance()

Not available for compound models.

Usage

CompoundWishart$entropy_variance(n = self$n)

Arguments

CompoundWishart$clone()

The objects of this class are cloneable with this method.

Usage

CompoundWishart$clone(deep = FALSE)

Arguments

References

Ferreira, J. A., Nascimento, A. D. C. and Frery, A. C. (2022). PolSAR models with multimodal intensities. Remote Sensing, 14(20), 5083. doi:10.3390/rs14205083

Examples

set.seed(1)
S <- rctpcw(200, lambda = 1.5, Sigma = polsar_example_sigma("forest"), L = 4)
m <- CompoundWishart$new(counting = "poisson")$fit(S, L = 4)
m$params$lambda
m$iterations

Description

Density, distribution function, quantile function and random generation for the $\mathcal{G}_I^0(\alpha, \gamma, L)$ distribution, the multiplicative model for multilook SAR intensities with reciprocal gamma backscatter and gamma speckle (Frery et al., 1997).

Usage

dg0i(x, alpha, gamma, L, log = FALSE)

pg0i(q, alpha, gamma, L, lower.tail = TRUE, log.p = FALSE)

qg0i(p, alpha, gamma, L, lower.tail = TRUE, log.p = FALSE)

rg0i(n, alpha, gamma, L)

Arguments

Details

The density is

$f(z) = \frac{L^L \Gamma(L - \alpha)}{\gamma^{\alpha} \Gamma(-\alpha) \Gamma(L)} \frac{z^{L - 1}}{(\gamma + L z)^{L - \alpha}}, \quad z > 0,$

with roughness $\alpha < 0$, scale $\gamma > 0$ and number of looks $L > 0$. If $Z \sim \mathcal{G}_I^0(\alpha, \gamma, L)$ then $-\alpha Z / \gamma$ follows a Snedecor F distribution with $2L$ and $-2\alpha$ degrees of freedom, which is used to compute the distribution and quantile functions exactly.

Value

dg0i() gives the density, pg0i() the distribution function, qg0i() the quantile function and rg0i() generates random deviates.

References

Frery, A. C., Muller, H.-J., Yanasse, C. C. F. and Sant'Anna, S. J. S. (1997). A model for extremely heterogeneous clutter. IEEE Transactions on Geoscience and Remote Sensing, 35(3), 648–659.

Nascimento, A. D. C., Cintra, R. J. and Frery, A. C. (2010). Hypothesis testing in speckled data with stochastic distances. IEEE Transactions on Geoscience and Remote Sensing, 48(1), 373–385.

See Also

G0I for the R6 interface (estimation and entropy).

Examples

set.seed(1)
z <- rg0i(5, alpha = -1.5, gamma = 2, L = 8)
dg0i(z, alpha = -1.5, gamma = 2, L = 8)
pg0i(c(0.5, 1, 2), alpha = -1.5, gamma = 2, L = 8)
qg0i(c(0.1, 0.5, 0.9), alpha = -1.5, gamma = 2, L = 8)

Description

Density and random generation for the polarimetric $\mathcal{G}^0_m(\alpha, \Sigma, L)$ law (also known as $\mathcal{G}^0_P$), obtained from the multiplicative model $Z = X Y$ with $Y \sim \mathcal{W}_m(\Sigma, L)$ and reciprocal gamma texture $X \sim \Gamma^{-1}(-\alpha, -\alpha - 1)$ with unit mean (Freitas et al., 2005):

$f(Z) = \frac{L^{mL}\Gamma(mL - \alpha)|Z|^{L-m}} {|\Sigma|^L \Gamma_m(L) \Gamma(-\alpha)(-\alpha-1)^{\alpha}} \left[L\,\mathrm{tr}(\Sigma^{-1}Z) - \alpha - 1\right]^{\alpha - mL},$

with $\alpha < -1$. When $\alpha \to -\infty$ it converges to the scaled complex Wishart law.

Usage

dg0p(x, alpha, Sigma, L, log = FALSE)

rg0p(n, alpha, Sigma, L, output = c("array", "vec"))

Arguments

Value

dg0p() returns (log-)densities and rg0p() the simulated sample.

References

Freitas, C. C., Frery, A. C. and Correia, A. H. (2005). The polarimetric G distribution for SAR data analysis. Environmetrics, 16(1), 13–31.

Nascimento, A. D. C., Ferreira, J. A. and Frery, A. C. (2023). Unsupervised segmentation of PolSAR data with complex Wishart and G0m distributions and Shannon entropy. Statistics and Computing, 33, 129. doi:10.1007/s11222-023-10298-0

See Also

G0Polarimetric for the R6 interface.

Examples

Sigma <- polsar_example_sigma("urban")
set.seed(1)
Z <- rg0p(3, alpha = -3, Sigma = Sigma, L = 4)
dg0p(Z, alpha = -3, Sigma = Sigma, L = 4, log = TRUE)

Description

Density, distribution function, quantile function and random generation for the multilook intensity Gamma law parametrised by its mean $\mu$ and number of looks $L$:

$f(z) = \frac{L^L}{\mu^L \Gamma(L)} z^{L-1} \exp(-L z / \mu), \quad z > 0.$

This is the model for homogeneous areas (fully developed speckle).

Usage

dgammasar(x, mu, L, log = FALSE)

pgammasar(q, mu, L, lower.tail = TRUE, log.p = FALSE)

qgammasar(p, mu, L, lower.tail = TRUE, log.p = FALSE)

rgammasar(n, mu, L)

Arguments

Value

Density, distribution function, quantile function or random deviates.

See Also

GammaSAR for the R6 interface.

Examples

dgammasar(c(0.5, 1, 2), mu = 1, L = 4)
rgammasar(3, mu = 1, L = 4)

Description

Density, distribution function, quantile function and random generation for the generalized Gamma (Stacy) law with power $\alpha > 0$, shape $\phi > 0$ and inverse-scale $\mu > 0$:

$f(z) = \frac{\alpha \mu^{\alpha\phi}}{\Gamma(\phi)} z^{\alpha\phi - 1} \exp\{-(\mu z)^\alpha\}, \quad z > 0.$

For $\alpha = 1$ it reduces to the Gamma law with shape $\phi$ and rate $\mu$; in SAR applications $\phi$ plays the role of the number of looks.

Usage

dggamma(x, alpha, phi, mu, log = FALSE)

pggamma(q, alpha, phi, mu, lower.tail = TRUE, log.p = FALSE)

qggamma(p, alpha, phi, mu, lower.tail = TRUE, log.p = FALSE)

rggamma(n, alpha, phi, mu)

Arguments

Value

Density, distribution function, quantile function or random deviates.

References

Li, H.-C., Hong, W., Wu, Y.-R. and Fan, P.-Z. (2011). On the empirical- statistical modeling of SAR images with generalized gamma distribution. IEEE Journal of Selected Topics in Signal Processing, 5(3), 386–397.

See Also

GGammaSAR for the R6 interface.

Examples

dggamma(c(0.5, 1, 2), alpha = 1.5, phi = 2, mu = 1)
rggamma(3, alpha = 1.5, phi = 2, mu = 1)

Description

Computes stochastic distances between two $\mathcal{G}_I^0$ laws by numerical integration (Nascimento, Cintra and Frery, 2010).

Usage

distancesg0i(
  par_1,
  par_2,
  p = 1/2,
  type = c("KL", "Renyi", "Bhattacharyya", "Hellinger", "Arithmetic-Geometric",
    "Triangular", "Harmonic-Mean")
)

Arguments

Value

An object of class distancesg0i with the parameters, the type and the distance_value.

References

Nascimento, A. D. C., Cintra, R. J. and Frery, A. C. (2010). Hypothesis testing in speckled data with stochastic distances. IEEE Transactions on Geoscience and Remote Sensing, 48(1), 373–385.

Examples

distancesg0i(c(-1.5, 2, 4), c(-1.5, 2, 4))
res <- distancesg0i(c(-1.5, 2, 8), c(-3, 2, 4), type = "Hellinger")
res

Description

Fits the $\mathcal{G}_I^0$ law to each intensity channel of two rectangular regions of an image and computes all the stochastic distances of distancesg0i() between them.

Usage

distancesg0itable(image, marks1, marks2, p = 1/2, L = NULL)

Arguments

Value

An object of class distancesg0itable with the marks and the table of distances (distancestable, channels in rows).

Examples

res <- distancesg0itable(SanFrancisco150, c(10, 30, 10, 30),
                         c(10, 30, 120, 140), L = 4)
res

Description

divergence_classifier() defines the divergence-based classifier of Ferreira, Coelho and Nascimento (2021) as a 'parsnip' model specification, so it can be used with 'workflows', 'recipes', 'rsample', 'tune' and 'yardstick'. The engine selects the family of features: "gaussian" (multivariate normal intensity features, the default) or "wishart" (complex Wishart PolSAR matrix features). The predictors must be in the format of sar_features(); extra columns such as row/col are ignored by the engine (it is good practice to give them an "id" role in a recipe).

Usage

divergence_classifier(
  mode = "classification",
  divergence = NULL,
  order = NULL,
  kind = NULL,
  engine = "gaussian"
)

## S3 method for class 'divergence_classifier'
update(
  object,
  parameters = NULL,
  divergence = NULL,
  order = NULL,
  kind = NULL,
  fresh = FALSE,
  ...
)

Arguments

Value

A model specification object with classes divergence_classifier and model_spec.

See Also

DivergenceClassifier, sar_features()

Examples

spec <- divergence_classifier(divergence = "renyi", order = 0.5) |>
  parsnip::set_engine("wishart")
spec

feats <- sar_features(SARImage$new(SanFrancisco150)$crop(1:60, 1:60), window = 5)
feats$class <- factor(as.vector(SanFrancisco150_gt[1:60, 1:60]))
fit <- parsnip::fit(spec, class ~ ., data = feats)
head(predict(fit, feats))
head(predict(fit, feats, type = "prob"))

Description

Implements the family of divergence-based classifiers of Ferreira, Coelho and Nascimento (2021). Each pixel is represented by the model fitted to the observations of a moving window (see sar_features()) and each class by the model fitted to its training pixels. A pixel is assigned to the class that minimises the chosen divergence between the two fitted models:

$\hat\imath = \arg\min_{i \in \{1, \ldots, g\}} d(\hat\theta_h, \hat\theta_i).$

Two families of features are available:

"gaussian"

multivariate normal laws for the intensity vector, using the columns ⁠mean_*⁠ and ⁠cov_*⁠ of sar_features(). The class covariance combines the average local covariance and the covariance of the local means (law of total variance).

"wishart"

scaled complex Wishart laws for the full PolSAR covariance matrix, using the columns c11, c12_re, c12_im, ... of sar_features(). Since all laws share the number of looks, it only rescales the divergences and does not affect the classification.

The classifier is also available as the 'parsnip' model divergence_classifier(), for use in 'tidymodels' workflows.

Value

An R6::R6Class generator. DivergenceClassifier$new() returns a new DivergenceClassifier object; its methods are described in the Methods section below.

Public fields

Methods

Public methods

• DivergenceClassifier$new()

• DivergenceClassifier$fit()

• DivergenceClassifier$distances()

• DivergenceClassifier$predict()

• DivergenceClassifier$print()

• DivergenceClassifier$clone()

DivergenceClassifier$new()

Create a classifier.

Usage

DivergenceClassifier$new(
  family = c("gaussian", "wishart"),
  divergence = c("renyi", "kl", "jeffreys", "bhattacharyya"),
  order = 0.5,
  kind = 1
)

Arguments

DivergenceClassifier$fit()

Fit the class prototypes.

Usage

DivergenceClassifier$fit(x, y)

Arguments

DivergenceClassifier$distances()

Divergence between every row of x and every class.

Usage

DivergenceClassifier$distances(x)

Arguments

DivergenceClassifier$predict()

Predict classes, pseudo-probabilities or distances.

Usage

DivergenceClassifier$predict(x, type = c("class", "prob", "distance"))

Arguments

DivergenceClassifier$print()

Print the classifier.

Usage

DivergenceClassifier$print(...)

Arguments

DivergenceClassifier$clone()

The objects of this class are cloneable with this method.

Usage

DivergenceClassifier$clone(deep = FALSE)

Arguments

References

Ferreira, J. A., Coelho, H. and Nascimento, A. D. C. (2021). A family of divergence-based classifiers for polarimetric synthetic aperture radar (PolSAR) imagery vector and matrix features. International Journal of Remote Sensing, 42(4), 1201–1229. doi:10.1080/01431161.2020.1826060

Examples

img <- SARImage$new(SanFrancisco150)$crop(1:80, 1:80)
feats <- sar_features(img, window = 5)
feats$class <- factor(as.vector(SanFrancisco150_gt[1:80, 1:80]),
                      labels = c("ocean", "forest", "urban"))
set.seed(1)
train <- feats[sample(nrow(feats), 300), ]

clf <- DivergenceClassifier$new(family = "wishart", divergence = "renyi",
                                order = 0.5, kind = 3)
clf$fit(train, train$class)
pred <- clf$predict(feats)
mean(pred == feats$class)
head(clf$predict(feats, type = "distance"))

Description

Closed-form divergences between two multivariate normal laws $N_p(\mu_1, \Sigma_1)$ and $N_p(\mu_2, \Sigma_2)$, and between two scaled complex Wishart laws $\mathcal{W}_m(\Sigma_1, L)$ and $\mathcal{W}_m(\Sigma_2, L)$ with a common number of looks. They are the building blocks of the divergence-based classifiers of Ferreira, Coelho and Nascimento (2021).

Usage

divergence_gaussian(
  mu1,
  Sigma1,
  mu2,
  Sigma2,
  type = c("kl", "jeffreys", "bhattacharyya", "renyi"),
  order = 0.5,
  kind = 1
)

divergence_wishart(
  Sigma1,
  Sigma2,
  L,
  type = c("kl", "jeffreys", "bhattacharyya", "renyi"),
  order = 0.5,
  kind = 1
)

Arguments

Details

Available measures (type):

"kl"

Kullback-Leibler divergence $D_{KL}(\theta_1 \| \theta_2)$.

"jeffreys"

Jeffreys divergence $D_{KL}(\theta_1 \| \theta_2) + D_{KL}(\theta_2 \| \theta_1)$.

"bhattacharyya"

Bhattacharyya distance $-\log \int \sqrt{f_1 f_2}$.

"renyi"

Rényi divergence of order $\alpha$ (order), in one of four forms (kind): 1, $D^\alpha_R(\theta_1\|\theta_2)$; 2, $D^\alpha_R(\theta_2\|\theta_1)$; 3, the symmetrised mean $[D^\alpha_R(\theta_1\|\theta_2) + D^\alpha_R(\theta_2\|\theta_1)]/2$; and 4, $c \log\{[K(\theta_1\|\theta_2) + K(\theta_2\|\theta_1)]/2\}$, where $K(\theta_1\|\theta_2) = \int f_1^\alpha f_2^{1-\alpha}$.

For Gaussian laws the Rényi divergence uses the normalisation of Pardo (2006), $c = 1/[\alpha(\alpha - 1)]$, which gives

$D^\alpha_R = \frac{1}{2}\Delta^\top M^{-1}\Delta - \frac{1}{2\alpha(\alpha-1)}\log\frac{|M|}{|\Sigma_1|^{1-\alpha} |\Sigma_2|^{\alpha}},$

with $\Delta = \mu_1 - \mu_2$ and $M = \alpha\Sigma_2 + (1-\alpha)\Sigma_1$. For Wishart laws the normalisation of Frery, Nascimento and Cintra (2014), $c = 1/(\alpha - 1)$, is used:

$D^\alpha_R = \frac{L}{\alpha - 1}\log\frac{|(\alpha\Sigma_1^{-1} + (1-\alpha)\Sigma_2^{-1})^{-1}|}{|\Sigma_1|^\alpha|\Sigma_2|^{1-\alpha}}.$

Both converge to the Kullback-Leibler divergence when $\alpha \to 1$.

Value

A single non-negative number.

References

Ferreira, J. A., Coelho, H. and Nascimento, A. D. C. (2021). A family of divergence-based classifiers for polarimetric synthetic aperture radar (PolSAR) imagery vector and matrix features. International Journal of Remote Sensing, 42(4), 1201–1229. doi:10.1080/01431161.2020.1826060

Frery, A. C., Nascimento, A. D. C. and Cintra, R. J. (2014). Analytic expressions for stochastic distances between relaxed complex Wishart distributions. IEEE Transactions on Geoscience and Remote Sensing, 52(2), 1213–1226.

Pardo, L. (2006). Statistical Inference Based on Divergence Measures. Chapman & Hall/CRC.

Examples

S1 <- diag(3); S2 <- diag(c(2, 1, 0.5))
divergence_gaussian(c(0, 0, 0), S1, c(1, 0, 0), S2, type = "kl")
divergence_gaussian(c(0, 0, 0), S1, c(1, 0, 0), S2, type = "renyi",
                    order = 0.5, kind = 3)

W1 <- polsar_example_sigma("forest"); W2 <- polsar_example_sigma("urban")
divergence_wishart(W1, W2, L = 4, type = "jeffreys")
divergence_wishart(W1, W2, L = 4, type = "renyi", order = 0.9)

Description

Density and random generation for the scaled complex Wishart law $\mathcal{W}_m(\Sigma, L)$, the reference model for multilook PolSAR covariance (coherency) matrices in homogeneous areas:

$f(Z) = \frac{L^{mL} |Z|^{L-m}}{|\Sigma|^L \Gamma_m(L)} \exp\{-L\, \mathrm{tr}(\Sigma^{-1} Z)\},$

where $\Gamma_m(L) = \pi^{m(m-1)/2}\prod_{i=0}^{m-1}\Gamma(L-i)$ and $E(Z) = \Sigma$.

Usage

dwishart_sar(x, Sigma, L, log = FALSE)

rwishart_sar(n, Sigma, L, output = c("array", "vec"))

Arguments

Details

Random matrices are generated with the complex Bartlett decomposition, so non-integer numbers of looks $L > m - 1$ are allowed.

Value

dwishart_sar() returns a vector of (log-)densities; rwishart_sar() returns the simulated sample.

References

Goodman, N. R. (1963). Statistical analysis based on a certain multivariate complex Gaussian distribution. The Annals of Mathematical Statistics, 34(1), 152–177.

Lee, J.-S., Grunes, M. R. and Kwok, R. (1994). Classification of multi-look polarimetric SAR imagery based on complex Wishart distribution. International Journal of Remote Sensing, 15(11), 2299–2311.

See Also

WishartSAR for the R6 interface.

Examples

Sigma <- polsar_example_sigma("forest")
set.seed(1)
Z <- rwishart_sar(3, Sigma, L = 4)
dim(Z)
dwishart_sar(Z, Sigma, L = 4, log = TRUE)

Description

Closed-form Shannon entropies $H = E\{-\log f(Z)\}$ for the models used in SAR (intensity) and PolSAR (matrix) image analysis. All functions are vectorised in the scalar parameters.

Usage

entropy_gamma(mu, L)

entropy_g0i(alpha, gamma, L)

entropy_ggamma(alpha, phi, mu)

entropy_wishart(Sigma, L)

entropy_g0p(alpha, Sigma, L)

Arguments

Details

Gamma

$H = L - \log L + \log\mu + \log\Gamma(L) + (1 - L)\psi^{(0)}(L)$.

G0I

(Ferreira and Nascimento, 2020) $H = -\log\frac{L^L\Gamma(L-\alpha)}{\Gamma(-\alpha)\gamma^\alpha \Gamma(L)} + (1 - L)[\psi^{(0)}(L) - \log L + \log\gamma - \psi^{(0)}(-\alpha)] + (L - \alpha)[\psi^{(0)}(L - \alpha) + \log\gamma - \psi^{(0)}(-\alpha)]$.

Generalized Gamma

$H = -\log\alpha + \log\Gamma(\phi) + \phi - (\phi - 1/\alpha)\psi^{(0)}(\phi) - \log\mu$.

Scaled complex Wishart

(Nascimento, Ferreira and Frery, 2023) $H = m\log|\Sigma| - m^2\log L + \log\Gamma_m(L) + mL + (m - L)\psi^{(0)}_m(L)$, with $\psi^{(0)}_m(L) = \sum_{i=0}^{m-1}\psi^{(0)}(L - i)$.

G0 polarimetric

(Nascimento, Ferreira and Frery, 2023) obtained from $E\log|Z| = m[\log(-\alpha-1) - \psi^{(0)}(-\alpha)] + \log|\Sigma| - m\log L + \psi^{(0)}_m(L)$ and $E\log[L\,\mathrm{tr}(\Sigma^{-1}Z) - \alpha - 1] = \log(-\alpha-1) + \psi^{(0)}(mL - \alpha) - \psi^{(0)}(-\alpha)$.

Value

Numeric vector of entropies.

References

Ferreira, J. A. and Nascimento, A. D. C. (2020). Shannon entropy for the G0I model: a new segmentation approach. IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing, 13, 2547–2553. doi:10.1109/JSTARS.2020.2997666

Nascimento, A. D. C., Ferreira, J. A. and Frery, A. C. (2023). Unsupervised segmentation of PolSAR data with complex Wishart and G0m distributions and Shannon entropy. Statistics and Computing, 33, 129. doi:10.1007/s11222-023-10298-0

Examples

entropy_gamma(mu = 1, L = 4)
entropy_g0i(alpha = -3, gamma = 2, L = 4)
entropy_ggamma(alpha = 1, phi = 4, mu = 4) # equals the Gamma entropy
S <- polsar_example_sigma("forest")
entropy_wishart(S, L = 4)
entropy_g0p(alpha = -5, Sigma = S, L = 4)

Description

Estimates, at every pixel, the parameters of a SAR/PolSAR model with the observations inside a square moving window and returns the map of Shannon entropies (Ferreira and Nascimento, 2020; Nascimento, Ferreira and Frery, 2023). Optionally, the map of asymptotic variances of the entropy estimators is returned as well.

Usage

entropy_map(
  image,
  model = c("gamma", "g0i", "ggamma", "wishart", "g0p"),
  window = 7,
  L = NULL,
  channels = NULL,
  variance = FALSE
)

Arguments

Details

For the Gamma and Wishart models the computations are fully vectorised (local sufficient statistics are obtained from integral images), so the maps are computed in a fraction of a second. The G0I, generalized Gamma and G0 polarimetric models require a numerical maximisation per pixel and are therefore slower (use small images or crops to explore them).

Value

A matrix (one channel or matrix model) or an array ⁠nrow x ncol x channels⁠ of entropies. If variance = TRUE, a list with elements entropy, variance and n (window sizes).

References

Ferreira, J. A. and Nascimento, A. D. C. (2020). Shannon entropy for the G0I model: a new segmentation approach. IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing, 13, 2547–2553. doi:10.1109/JSTARS.2020.2997666

Nascimento, A. D. C., Ferreira, J. A. and Frery, A. C. (2023). Unsupervised segmentation of PolSAR data with complex Wishart and G0m distributions and Shannon entropy. Statistics and Computing, 33, 129. doi:10.1007/s11222-023-10298-0

Examples

img <- SARImage$new(SanFrancisco150)
H <- entropy_map(img, model = "wishart", window = 7, L = 4)
dim(H)
plot_map(H, main = "Wishart entropy")

Hg <- entropy_map(img, model = "gamma", window = 7, channels = "HV")

small <- img$crop(1:15, 1:15)
Hi <- entropy_map(small, model = "g0i", window = 5, L = 4, channels = 1)

Description

Tests whether two samples come from the same distribution by comparing their estimated Shannon entropies. Under the null hypothesis of equal entropies, and by the asymptotic normality of the maximum likelihood estimator of the entropy,

$S = \frac{(\hat H_1 - \hat H_2)^2}{\widehat{\mathrm{Var}}(\hat H_1) + \widehat{\mathrm{Var}}(\hat H_2)}$

follows asymptotically a $\chi^2_1$ distribution. This statistic is the basis of the entropy-based change detectors of the thesis of J. A. Ferreira (2021).

Usage

entropy_test(
  x,
  y,
  model = c("gamma", "g0i", "ggamma", "wishart", "g0p"),
  L = NULL
)

Arguments

Value

An object of class htest.

References

Ferreira, J. A. (2021). Entropias e somas compostas para matrizes estocásticas aplicadas a sistemas PolSAR. PhD thesis, Universidade Federal de Pernambuco.

Examples

set.seed(1)
x <- rg0i(200, alpha = -3, gamma = 2, L = 4)
y <- rg0i(200, alpha = -3, gamma = 2, L = 4)
z <- rg0i(200, alpha = -1.5, gamma = 1, L = 4)
entropy_test(x, y, model = "g0i", L = 4)
entropy_test(x, z, model = "g0i", L = 4)$p.value

Description

Unsupervised segmentation of SAR/PolSAR images from maps of estimated Shannon entropies (Ferreira and Nascimento, 2020; Nascimento, Ferreira and Frery, 2023). By the asymptotic normality of the entropy estimators, the entropy vector of a pixel of the $g$-th region is approximately Gaussian, so the image is modelled as a finite mixture of Gaussian laws (with diagonal covariance matrices) fitted by the expectation-maximisation algorithm (method = "em") or by its stochastic version (method = "sem"), in which the E-step is followed by a random allocation of every pixel to a class drawn from its posterior probabilities. Pixels are then labelled by the maximum a posteriori rule.

Value

An R6::R6Class generator. EntropySegmenter$new() returns a new EntropySegmenter object; its methods are described in the Methods section below.

Public fields

Methods

Public methods

• EntropySegmenter$new()

• EntropySegmenter$fit()

• EntropySegmenter$predict()

• EntropySegmenter$label_map()

• EntropySegmenter$print()

• EntropySegmenter$clone()

EntropySegmenter$new()

Create a segmenter.

Usage

EntropySegmenter$new(
  k = 2,
  method = c("em", "sem"),
  max_iter = 200L,
  tol = 1e-08
)

Arguments

EntropySegmenter$fit()

Fit the mixture.

Usage

EntropySegmenter$fit(x, init = NULL, is_map = TRUE)

Arguments

EntropySegmenter$predict()

Labels of new observations (maximum a posteriori).

Usage

EntropySegmenter$predict(x, is_map = TRUE)

Arguments

EntropySegmenter$label_map()

Labels as a matrix with the spatial dimensions of the fitted map.

Usage

EntropySegmenter$label_map()

EntropySegmenter$print()

Print a summary.

Usage

EntropySegmenter$print(...)

Arguments

EntropySegmenter$clone()

The objects of this class are cloneable with this method.

Usage

EntropySegmenter$clone(deep = FALSE)

Arguments

References

Ferreira, J. A. and Nascimento, A. D. C. (2020). Shannon entropy for the G0I model: a new segmentation approach. IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing, 13, 2547–2553. doi:10.1109/JSTARS.2020.2997666

Nascimento, A. D. C., Ferreira, J. A. and Frery, A. C. (2023). Unsupervised segmentation of PolSAR data with complex Wishart and G0m distributions and Shannon entropy. Statistics and Computing, 33, 129. doi:10.1007/s11222-023-10298-0

Examples

H <- entropy_map(SanFrancisco150, model = "wishart", window = 7, L = 4)
set.seed(1)
seg <- EntropySegmenter$new(k = 3)$fit(H)
seg
plot_map(seg$label_map(), main = "Entropy-based segmentation")
kappa_statistics(SanFrancisco150_gt, seg$label_map())

Description

R6 class for the $\mathcal{G}_I^0(\alpha, \gamma, L)$ law (see dg0i()) with maximum likelihood estimation (analytic score, quasi-Newton optimisation on a transformed scale with several starting points), closed-form Shannon entropy (Ferreira and Nascimento, 2020) and the asymptotic variance of its estimator.

Value

An R6::R6Class generator. G0I$new() returns a new G0I object; its methods are described in the Methods section below.

Super class

SARModel -> G0I

Methods

Public methods

• G0I$new()

• G0I$density()

• G0I$random()

• G0I$fit()

• G0I$entropy()

• G0I$clone()

G0I$new()

Create a G0I model.

Usage

G0I$new(alpha = -2, gamma = 1, L = 1)

Arguments

G0I$density()

(Log-)density.

Usage

G0I$density(x, log = FALSE)

Arguments

G0I$random()

Simulate intensities.

Usage

G0I$random(n)

Arguments

G0I$fit()

Fit by maximum likelihood.

Usage

G0I$fit(x, L = NULL, start = NULL)

Arguments

G0I$entropy()

Shannon entropy.

Usage

G0I$entropy()

G0I$clone()

The objects of this class are cloneable with this method.

Usage

G0I$clone(deep = FALSE)

Arguments

References

Ferreira, J. A. and Nascimento, A. D. C. (2020). Shannon entropy for the G0I model: a new segmentation approach. IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing, 13, 2547–2553. doi:10.1109/JSTARS.2020.2997666

Examples

set.seed(1)
z <- rg0i(1000, alpha = -3, gamma = 2, L = 4)
m <- G0I$new()$fit(z, L = 4)
m
m$entropy_variance()
G0I$new()$fit(z)$params # L estimated

Description

Maximum likelihood estimation of the $\mathcal{G}_I^0$ parameters. This is a convenience wrapper around G0I⁠$fit()⁠, kept for compatibility with previous versions of the package.

Usage

g0iestimation(Theta, x, L_fixed = FALSE)

Arguments

Value

Named vector c(alpha, gamma, L) of estimates.

Examples

set.seed(1)
y <- rg0i(1000, alpha = -1.5, gamma = 2, L = 8)
g0iestimation(c(-1, 1, 1), y)
g0iestimation(c(-1, 1, 8), y, L_fixed = TRUE)

Description

R6 class for the $\mathcal{G}^0_m(\alpha, \Sigma, L)$ law (see dg0p()). Following Nascimento, Ferreira and Frery (2023), $\Sigma$ is estimated by the sample mean (since $E(Z) = \Sigma$) and $(\alpha, L)$ by maximising the likelihood (analytic score). The variance of the entropy estimator combines the inverse observed information of $(\alpha, L)$ with a moment estimate of the variance of $m\log|\hat\Sigma|$.

Value

An R6::R6Class generator. G0Polarimetric$new() returns a new G0Polarimetric object; its methods are described in the Methods section below.

Super class

SARModel -> G0Polarimetric

Methods

Public methods

• G0Polarimetric$new()

• G0Polarimetric$density()

• G0Polarimetric$random()

• G0Polarimetric$fit()

• G0Polarimetric$entropy()

• G0Polarimetric$clone()

G0Polarimetric$new()

Create a G0 polarimetric model.

Usage

G0Polarimetric$new(alpha = -3, Sigma = diag(3) + (0+0i), L = 4)

Arguments

G0Polarimetric$density()

(Log-)density.

Usage

G0Polarimetric$density(x, log = FALSE)

Arguments

G0Polarimetric$random()

Simulate PolSAR observations.

Usage

G0Polarimetric$random(n, output = "array")

Arguments

G0Polarimetric$fit()

Fit by maximum likelihood.

Usage

G0Polarimetric$fit(x, L = NULL)

Arguments

G0Polarimetric$entropy()

Shannon entropy.

Usage

G0Polarimetric$entropy()

G0Polarimetric$clone()

The objects of this class are cloneable with this method.

Usage

G0Polarimetric$clone(deep = FALSE)

Arguments

Examples

set.seed(1)
Z <- rg0p(400, alpha = -4, Sigma = polsar_example_sigma("urban"), L = 4)
m <- G0Polarimetric$new()$fit(Z, L = 4)
m
m$entropy_variance()

Description

R6 class for the multilook Gamma law $\Gamma(\mu, L)$; see dgammasar(). The maximum likelihood estimator of $\mu$ is the sample mean and $L$ solves $\log L - \psi^{(0)}(L) = \log\bar z - \overline{\log z}$.

Value

An R6::R6Class generator. GammaSAR$new() returns a new GammaSAR object; its methods are described in the Methods section below.

Super class

SARModel -> GammaSAR

Methods

Public methods

• GammaSAR$new()

• GammaSAR$density()

• GammaSAR$random()

• GammaSAR$fit()

• GammaSAR$entropy()

• GammaSAR$entropy_variance()

• GammaSAR$clone()

GammaSAR$new()

Create a Gamma model.

Usage

GammaSAR$new(mu = 1, L = 1)

Arguments

GammaSAR$density()

(Log-)density.

Usage

GammaSAR$density(x, log = FALSE)

Arguments

GammaSAR$random()

Simulate intensities.

Usage

GammaSAR$random(n)

Arguments

GammaSAR$fit()

Fit by maximum likelihood.

Usage

GammaSAR$fit(x, L = NULL)

Arguments

GammaSAR$entropy()

Shannon entropy.

Usage

GammaSAR$entropy()

GammaSAR$entropy_variance()

Asymptotic variance of the entropy estimator (closed form).

Usage

GammaSAR$entropy_variance(n = self$n)

Arguments

GammaSAR$clone()

The objects of this class are cloneable with this method.

Usage

GammaSAR$clone(deep = FALSE)

Arguments

Examples

set.seed(1)
z <- rgammasar(500, mu = 2, L = 4)
m <- GammaSAR$new()$fit(z)
m
m$entropy()
m$entropy_variance()
generics::tidy(m)

Description

R6 class for the generalized Gamma law (see dggamma()). Estimation is carried out by profiling: for a given power $\alpha$, the maximum likelihood estimates of $\phi$ and $\mu$ are available in closed form (up to a one-dimensional root), so the profile likelihood is maximised over $\alpha$ only.

Value

An R6::R6Class generator. GGammaSAR$new() returns a new GGammaSAR object; its methods are described in the Methods section below.

Super class

SARModel -> GGammaSAR

Methods

Public methods

• GGammaSAR$new()

• GGammaSAR$density()

• GGammaSAR$random()

• GGammaSAR$fit()

• GGammaSAR$entropy()

• GGammaSAR$clone()

GGammaSAR$new()

Create a generalized Gamma model.

Usage

GGammaSAR$new(alpha = 1, phi = 1, mu = 1)

Arguments

GGammaSAR$density()

(Log-)density.

Usage

GGammaSAR$density(x, log = FALSE)

Arguments

GGammaSAR$random()

Simulate intensities.

Usage

GGammaSAR$random(n)

Arguments

GGammaSAR$fit()

Fit by (profile) maximum likelihood.

Usage

GGammaSAR$fit(x, phi = NULL)

Arguments

GGammaSAR$entropy()

Shannon entropy.

Usage

GGammaSAR$entropy()

GGammaSAR$clone()

The objects of this class are cloneable with this method.

Usage

GGammaSAR$clone(deep = FALSE)

Arguments

Examples

set.seed(1)
z <- rggamma(1000, alpha = 1.5, phi = 3, mu = 2)
m <- GGammaSAR$new()$fit(z)
m

Description

Computes Cohen's kappa coefficient of agreement, its asymptotic variance (Congalton and Green, 2009) and the overall accuracy from a confusion matrix or from two vectors (or matrices) of labels. Unlike yardstick::kap(), the variance of the estimator is also returned, which allows comparing classifiers with Z tests.

Usage

kappa_statistics(truth, estimate = NULL, match = FALSE)

Arguments

Details

Labels of unsupervised methods are arbitrary: use match = TRUE to relabel the estimated classes so that they best agree with the truth (an exhaustive search over permutations for up to 8 classes).

Value

A tibble with kappa, variance, std_error, overall_accuracy and the number of observations n.

References

Congalton, R. G. and Green, K. (2009). Assessing the Accuracy of Remotely Sensed Data: Principles and Practices. CRC Press.

Examples

truth <- c(1, 1, 2, 2, 3, 3, 3)
est <- c(1, 2, 2, 2, 3, 3, 1)
kappa_statistics(truth, est)
kappa_statistics(table(est, truth))

Description

Displays a matrix as an image with the first row on top: continuous maps (e.g. entropy maps) use a sequential palette and integer/factor maps (e.g. segmentation or classification results) use a qualitative one.

Usage

plot_map(x, palette = NULL, main = NULL, legend = FALSE)

Arguments

Value

Invisibly, x.

Examples

plot_map(SanFrancisco150_gt, main = "Ground truth", legend = TRUE)
plot_map(matrix(rnorm(100), 10), main = "Noise")

Description

Typical $3 \times 3$ covariance matrices (scaled to unit magnitude) for urban, forest and pasture areas, taken from the simulation studies of the authors' dissertation and thesis. They are handy to simulate PolSAR data.

Usage

polsar_example_sigma(type = c("urban", "forest", "pasture"))

Arguments

Value

A complex Hermitian positive definite $3 \times 3$ matrix.

Examples

polsar_example_sigma("forest")

Description

PolSAR observations are Hermitian positive definite matrices. SITSAR accepts them either as a complex array of dimension ⁠m x m x n⁠ or as an ⁠n x p⁠ complex matrix in which each row stores the $p = m(m+1)/2$ distinct entries in the order $C_{11}, \ldots, C_{mm}$ (the intensities) followed by the upper triangle $C_{12}, C_{13}, \ldots, C_{23}, \ldots$. For $m = 3$ this is the usual layer order ⁠HH, HV, VV, HH.HV*, HH.VV*, HV.VV*⁠ of the SanFrancisco150 image.

Usage

as_polsar_array(x)

as_polsar_vec(x)

Arguments

Value

as_polsar_array() returns a complex array ⁠m x m x n⁠; as_polsar_vec() returns an ⁠n x p⁠ complex matrix.

Examples

S <- diag(3) + 0i
S[1, 2] <- 0.2 + 0.1i
S[2, 1] <- Conj(S[1, 2])
v <- as_polsar_vec(S)
v
as_polsar_array(v)[, , 1]

Description

'dials' parameter objects for divergence_classifier(): renyi_order() for the order of the Rényi divergence and divergence_type() for the divergence measure. Both require the 'dials' package.

Usage

renyi_order(range = c(0.05, 0.95), trans = NULL)

divergence_type(values = c("renyi", "kl", "jeffreys", "bhattacharyya"))

renyi_kind(range = c(1L, 4L))

Arguments

Value

A 'dials' parameter object.

Examples

if (requireNamespace("dials", quietly = TRUE)) {
  renyi_order()
  divergence_type()
}

Description

A $150 \times 150$ crop of the fully polarimetric L-band AIRSAR image of the San Francisco Bay area (nominal 4 looks), used in the dissertation and thesis of J. A. Ferreira. The scene contains ocean, vegetation (forest/park) and urban areas; see SanFrancisco150_gt.

Usage

SanFrancisco150

Format

A complex array of dimension ⁠150 x 150 x 6⁠. For each pixel the six layers are the distinct entries of the $3 \times 3$ Hermitian covariance matrix: $|S_{HH}|^2$, $|S_{HV}|^2$, $|S_{VV}|^2$ (real intensities), $S_{HH}S_{HV}^*$, $S_{HH}S_{VV}^*$ and $S_{HV}S_{VV}^*$ (complex products).

Source

NASA/JPL AIRSAR mission.

Examples

dim(SanFrancisco150)
img <- SARImage$new(SanFrancisco150, looks = 4)
img$plot()

Description

Reference map of the SanFrancisco150 image with three classes.

Usage

SanFrancisco150_gt

Format

An integer matrix ⁠150 x 150⁠ with values 1 (ocean), 2 (vegetation) and 3 (urban area).

Examples

table(SanFrancisco150_gt)
plot_map(SanFrancisco150_gt)

Description

Computes moving-window statistics around every pixel, in the format used by DivergenceClassifier and the divergence_classifier() 'parsnip' model. Window statistics are computed with integral images, so the cost does not depend on the window size. Windows are truncated at the image borders.

Usage

sar_features(image, window = 3, type = c("all", "gaussian", "wishart"))

Arguments

Details

Two groups of columns are produced:

Gaussian features

⁠mean_<ch>⁠: local mean of each intensity channel; ⁠cov_<ch1>_<ch2>⁠: local (unbiased) covariance between the intensities. Together they define the multivariate normal fitted to the intensity vector in the window.

Wishart features (PolSAR only)

⁠c<i><i>⁠ and ⁠c<i><j>_re⁠/⁠c<i><j>_im⁠: entries of the local mean covariance matrix, i.e. the maximum likelihood estimate of $\Sigma$ for the scaled complex Wishart law in the window.

Value

A tibble with the columns row, col and the features.

Examples

img <- SARImage$new(SanFrancisco150)$crop(1:20, 1:20)
f <- sar_features(img, window = 3)
names(f)

Description

SARImage stores a SAR image as an array of dimension ⁠nrow x ncol x layers⁠. Two types are supported:

"polsar"

a complex array whose p = m(m+1)/2 layers are the distinct entries of the $m \times m$ covariance matrix of each pixel, in the order $C_{11}, \ldots, C_{mm}, C_{12}, C_{13}, \ldots$ (for $m = 3$: ⁠HH, HV, VV, HH.HV*, HH.VV*, HV.VV*⁠), as in SanFrancisco150;

"intensity"

a real (or real-valued complex) array whose layers are intensity channels.

The class offers cropping, Pauli colour composition, plotting, local (moving-window) features for classification, entropy maps and conversion to tibbles.

Value

An R6::R6Class generator. SARImage$new() returns a new SARImage object; its methods are described in the Methods section below.

Public fields

Active bindings

Methods

Public methods

• SARImage$new()

• SARImage$intensity()

• SARImage$intensities()

• SARImage$crop()

• SARImage$pixel()

• SARImage$window()

• SARImage$pauli()

• SARImage$plot()

• SARImage$features()

• SARImage$entropy_map()

• SARImage$as_tibble()

• SARImage$print()

• SARImage$clone()

SARImage$new()

Create a SAR image.

Usage

SARImage$new(data, type = NULL, looks = NULL, channels = NULL)

Arguments

SARImage$intensity()

Intensity of one channel.

Usage

SARImage$intensity(channel = 1)

Arguments

SARImage$intensities()

All intensity channels as a real array.

Usage

SARImage$intensities()

SARImage$crop()

Crop the image.

Usage

SARImage$crop(rows, cols)

Arguments

SARImage$pixel()

Observation at a pixel: the Hermitian covariance matrix (PolSAR) or the vector of intensities.

Usage

SARImage$pixel(row, col)

Arguments

SARImage$window()

Observations inside a square window centred at a pixel (truncated at the borders). Returns an ⁠n x layers⁠ matrix.

Usage

SARImage$window(row, col, window = 3)

Arguments

SARImage$pauli()

Pauli RGB composition scaled to [0, 1] (PolSAR images with three channels); for other images the first three intensities are used.

Usage

SARImage$pauli(stretch = 0.98, gamma = 0.5)

Arguments

SARImage$plot()

Plot the image (Pauli composition or one channel in dB).

Usage

SARImage$plot(type = c("pauli", "intensity"), channel = 1, main = NULL, ...)

Arguments

SARImage$features()

Local features for classification; see sar_features().

Usage

SARImage$features(window = 3, type = c("all", "gaussian", "wishart"))

Arguments

SARImage$entropy_map()

Entropy map; see entropy_map().

Usage

SARImage$entropy_map(...)

Arguments

SARImage$as_tibble()

Convert to a tibble with one row per pixel.

Usage

SARImage$as_tibble()

SARImage$print()

Print a summary.

Usage

SARImage$print(...)

Arguments

SARImage$clone()

The objects of this class are cloneable with this method.

Usage

SARImage$clone(deep = FALSE)

Arguments

Examples

img <- SARImage$new(SanFrancisco150, looks = 4)
img
small <- img$crop(1:40, 1:40)
dim(small$intensities())
small$pixel(1, 1)
head(small$features(window = 3))

img$plot()

Description

SARModel is the abstract parent of all model classes of SITSAR (GammaSAR, G0I, GGammaSAR, WishartSAR, G0Polarimetric and CompoundWishart). Every model can evaluate densities, simulate data, be fitted by maximum likelihood (⁠$fit()⁠) and report its Shannon entropy and the asymptotic variance of the entropy estimator.

Fitted models work with the generics verbs tidy(), glance(), as well as with stats::logLik() and stats::AIC().

Value

An R6::R6Class generator. SARModel$new() returns a new SARModel object; its methods are described in the Methods section below.

Public fields

Methods

Public methods

• SARModel$new()

• SARModel$density()

• SARModel$random()

• SARModel$fit()

• SARModel$log_likelihood()

• SARModel$entropy()

• SARModel$entropy_variance()

• SARModel$n_parameters()

• SARModel$tidy()

• SARModel$glance()

• SARModel$print()

• SARModel$clone()

SARModel$new()

Abstract constructor.

Usage

SARModel$new(...)

Arguments

SARModel$density()

(Log-)density of observations.

Usage

SARModel$density(x, log = FALSE)

Arguments

SARModel$random()

Generate random observations.

Usage

SARModel$random(n)

Arguments

SARModel$fit()

Fit the model by maximum likelihood.

Usage

SARModel$fit(x, ...)

Arguments

SARModel$log_likelihood()

Log-likelihood of a sample.

Usage

SARModel$log_likelihood(x)

Arguments

SARModel$entropy()

Shannon entropy at the current parameters.

Usage

SARModel$entropy()

SARModel$entropy_variance()

Asymptotic variance of the entropy estimator (delta method with the inverse observed Fisher information).

Usage

SARModel$entropy_variance(n = self$n)

Arguments

SARModel$n_parameters()

Number of free parameters.

Usage

SARModel$n_parameters()

SARModel$tidy()

Parameters as a tibble.

Usage

SARModel$tidy()

SARModel$glance()

One-row summary of the fit.

Usage

SARModel$glance()

SARModel$print()

Print the model.

Usage

SARModel$print(...)

Arguments

SARModel$clone()

The objects of this class are cloneable with this method.

Usage

SARModel$clone(deep = FALSE)

Arguments

Description

Tidy and glance methods for SITSAR models

Usage

## S3 method for class 'SARModel'
tidy(x, ...)

## S3 method for class 'SARModel'
glance(x, ...)

## S3 method for class 'SARModel'
logLik(object, ...)

Arguments