Package 'gaussratiovegind'

Ratio of two Gaussian distributions

Description

Density of the ratio of two Gaussian distributions.

Usage

dnormratio(z, bet, rho, delta)

Arguments

z	length $p$ numeric vector.
bet, rho, delta	numeric values. The parameters $(\beta, \rho, \delta)$ of the distribution, see Details.

Details

Let two independant random variables $X \sim N(\mu_x, \sigma_x)$ and $Y \sim N(\mu_y, \sigma_y)$.

If we denote $\beta = \frac{\mu_x}{\mu_y}$, $\rho = \frac{\sigma_y}{\sigma_x}$ and $\delta_y = \frac{\sigma_y}{\mu_y}$, the probability distribution function of the ratio $Z = \frac{X}{Y}$ is given by:

$\displaystyle{ f_Z(z; \beta, \rho, \delta_y) = \frac{\rho}{\pi (1 + \rho^2 z^2)} \left[ \exp{\left(-\frac{\rho^2 \beta^2 + 1}{2\delta_y^2}\right)} + \sqrt{\frac{\pi}{2}} \ q \ \text{erf}\left(\frac{q}{\sqrt{2}}\right) \exp\left(-\frac{\rho^2 (z-\beta)^2}{2 \delta_y^2 (1 + \rho^2 z^2)}\right) \right] }$

with $\displaystyle{ q = \frac{1 + \beta \rho^2 z}{\delta_y \sqrt{1 + \rho^2 z^2}} }$ and $\displaystyle{ \text{erf}\left(\frac{q}{\sqrt{2}}\right) = \frac{2}{\sqrt{\pi}} \int_0^{\frac{q}{\sqrt{2}}}{\exp{(-t^2)}\ dt} }$

Value

Numeric: the value of density.

Author(s)

Pierre Santagostini, Angélina El Ghaziri, Nizar Bouhlel

References

El Ghaziri, A., Bouhlel, N., Sapoukhina, N., Rousseau, D., On the importance of non-Gaussianity in chlorophyll fluorescence imaging. Remote Sensing 15(2), 528 (2023). doi:10.3390/rs15020528

Díaz-Francés, E., Rubio, F.J., On the existence of a normal approximation to the distribution of the ratio of two independent normal random variables. Stat Papers 54, 309–323 (2013). doi:10.1007/s00362-012-0429-2

See Also

rnormratio(): sample simulation.

estparnormratio(): parameter estimation.

Examples

# First example
beta1 <- 0.15
rho1 <- 5.75
delta1 <- 0.22
dnormratio(0, bet = beta1, rho = rho1, delta = delta1)
dnormratio(0.5, bet = beta1, rho = rho1, delta = delta1)
curve(dnormratio(x, bet = beta1, rho = rho1, delta = delta1), from = -0.1, to = 0.7)

# Second example
beta2 <- 2
rho2 <- 2
delta2 <- 2
dnormratio(0, bet = beta2, rho = rho2, delta = delta2)
dnormratio(0.5, bet = beta2, rho = rho2, delta = delta2)
curve(dnormratio(x, bet = beta2, rho = rho2, delta = delta2), from = -0.1, to = 0.7)

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Estimation of the Parameters of a Normal Ratio Distribution

Description

Estimation of the parameters of a ratio $\displaystyle{Z = \frac{X}{Y}}$, $X$ and $Y$ being two independant random variables distributed according to Gaussian distributions, using the EM (estimation-maximization) algorithm.

Usage

estparnormratio(z, eps = 1e-6)

Arguments

z	numeric matrix or data frame.
eps	numeric. Precision for the estimation of the parameters.

Details

Let a random variable: $\displaystyle{Z = \frac{X}{Y}}$,

$X$ and $Y$ being normally distributed: $X \sim N(\mu_x, \sigma_x)$ and $Y \sim N(\mu_y, \sigma_y)$.

The density probability of $Z$ is:

$\displaystyle{ f_Z(z; \beta, \rho, \delta_y) = \frac{\rho}{\pi (1 + \rho^2 z^2)} \ \exp{\left(-\frac{\rho^2 \beta^2 + 1}{2\delta_y^2}\right)} \ {}_1 F_1\left( 1, \frac{1}{2}; \frac{1}{2 \delta_y^2} \frac{(1 + \beta \rho^2 z)^2}{1 + \rho^2 z^2} \right) }$

with: $\displaystyle{\beta = \frac{\mu_x}{\mu}_y}$, $\displaystyle{\rho = \frac{\sigma_y}{\sigma_x}}$, $\displaystyle{\delta_y = \frac{\sigma_y}{\mu_y}}$.

and $_1 F_1\left(a, b; x\right)$ is the confluent hypergeometric function (Kummer's function):

$\displaystyle{ _1 F_1\left(a, b; x\right) = \sum_{n = 0}^{+\infty}{ \frac{ (a)_n }{ (b)_n } \frac{x^n}{n!} } }$

The parameters $\beta$, $\rho$, $\delta_y$ of the $Z$ distribution are estimated with the EM algorithm, as presented in El Ghaziri et al. The computation uses the kummerM function.

This uses an iterative algorithm.

The precision for the estimation of the parameters is given by the eps parameter.

Value

A list of 3 elements beta, rho, delta: the estimated parameters of the $Z$ distribution $\hat{\beta}$, $\hat{\rho}$, $\hat{\delta}_y$, with two attributes attr(, "epsilon") (precision of the result) and attr(, "k") (number of iterations).

Author(s)

Pierre Santagostini, Angélina El Ghaziri, Nizar Bouhlel

References

El Ghaziri, A., Bouhlel, N., Sapoukhina, N., Rousseau, D., On the importance of non-Gaussianity in chlorophyll fluorescence imaging. Remote Sensing 15(2), 528 (2023). doi:10.3390/rs15020528

See Also

dnormratio(): probability density of a normal ratio.

rnormratio(): sample simulation.

Examples

# First example
beta1 <- 0.15
rho1 <- 5.75
delta1 <- 0.22

set.seed(1234)
z1 <- rnormratio(800, bet = beta1, rho = rho1, delta = delta1)

estparnormratio(z1)

# Second example
beta2 <- 0.24
rho2 <- 4.21
delta2 <- 0.25

set.seed(1234)
z2 <- rnormratio(800, bet = beta2, rho = rho2, delta = delta2)

estparnormratio(z2)

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Confluent $D$-Hypergeometric Function

Description

Computes the Kummer's function, or confluent hypergeometric function.

Usage

kummerM(a, b, z, eps = 1e-06)

Arguments

a	numeric.
b	numeric
z	numeric vector.
eps	numeric. Precision for the sum (default 1e-06).

Details

The Kummer's confluent hypergeometric function is given by:

$\displaystyle{_1 F_1\left(a, b; z\right) = \sum_{n = 0}^{+\infty}{ \frac{ (a)_n }{ (b)_n } \frac{z^n}{n!} }}$

where $(z)_p$ is the Pochhammer symbol (see pochhammer).

The eps argument gives the required precision for its computation. It is the attr(, "epsilon") attribute of the returned value.

Value

A numeric value: the value of the Kummer's function, with two attributes attr(, "epsilon") (precision of the result) and attr(, "k") (number of iterations).

Author(s)

Pierre Santagostini, Angélina El Ghaziri, Nizar Bouhlel

References

El Ghaziri, A., Bouhlel, N., Sapoukhina, N., Rousseau, D., On the importance of non-Gaussianity in chlorophyll fluorescence imaging. Remote Sensing 15(2), 528 (2023). doi:10.3390/rs15020528

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Logarithm of the Pochhammer Symbol

Description

Computes the logarithm of the Pochhammer symbol.

Usage

lnpochhammer(x, n)

Arguments

x	numeric.
n	positive integer.

Details

The Pochhammer symbol is given by:

$\displaystyle{ (x)_n = \frac{\Gamma(x+n)}{\Gamma(x)} = x (x+1) ... (x+n-1) }$

So, if $n > 0$:

$\displaystyle{ log\left((x)_n\right) = log(x) + log(x+1) + ... + log(x+n-1) }$

If $n = 0$, $\displaystyle{ log\left((x)_n\right) = log(1) = 0}$

Value

Numeric value. The logarithm of the Pochhammer symbol.

Author(s)

Pierre Santagostini, Nizar Bouhlel

See Also

pochhammer, kummerM

Examples

lnpochhammer(2, 0)
lnpochhammer(2, 1)
lnpochhammer(2, 3)

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Pochhammer Symbol

Description

Computes the Pochhammer symbol.

Usage

pochhammer(x, n)

Arguments

x	numeric.
n	positive integer.

Details

The Pochhammer symbol is given by:

$\displaystyle{ (x)_n = \frac{\Gamma(x+n)}{\Gamma(x)} = x (x+1) ... (x+n-1) }$

Value

Numeric value. The value of the Pochhammer symbol.

Author(s)

Pierre Santagostini, Nizar Bouhlel

See Also

lnpochhammer, kummerM

Examples

pochhammer(2, 0)
pochhammer(2, 1)
pochhammer(2, 3)

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Ratio of two Gaussian distributions

Description

Simulate data from a ratio of two Gaussian distributions.

Usage

rnormratio(n, bet, rho, delta)

Arguments

n	integer. Number of observations. If length(n) > 1, the length is taken to be the nmber required.
bet, rho, delta	numeric values. The parameters $(\beta, \rho, \delta)$ of the distribution, see Details.

Details

Let two random variables $X \sim N(\mu_x, \sigma_x)$ and $Y \sim N(\mu_y, \sigma_y)$

with probability densities $f_X$ and $f_Y$.

The parameters of the distribution of the ratio $Z = \frac{X}{Y}$ are: $\displaystyle{\beta = \frac{\mu_x}{\mu_y}}$, $\displaystyle{\rho = \frac{\sigma_y}{\sigma_x}}$, $\displaystyle{\delta_y = \frac{\sigma_y}{\mu_y}}$.

$\mu_x$, $\sigma_x$, $\mu_y$ and $\sigma_y$ are computed from $\beta$, $\rho$ and $\delta_y$ (by fixing arbitrarily $\mu_x = 1$) and two random samples $\left( x_1, \dots, x_n \right)$ and $\left( y_1, \dots, y_n \right)$ are simulated.

Then $\displaystyle{\left( \frac{x_1}{y_1}, \dots, \frac{x_n}{y_n} \right)}$ is returned.

Value

A numeric vector: the produced sample.

Author(s)

Pierre Santagostini, Angélina El Ghaziri, Nizar Bouhlel

References

El Ghaziri, A., Bouhlel, N., Sapoukhina, N., Rousseau, D., On the importance of non-Gaussianity in chlorophyll fluorescence imaging. Remote Sensing 15(2), 528 (2023). doi:10.3390/rs15020528

See Also

dnormratio(): probability density of a normal ratio.

estparnormratio(): parameter estimation.

Examples

# First example
beta1 <- 0.15
rho1 <- 5.75
delta1 <- 0.22
rnormratio(20, bet = beta1, rho = rho1, delta = delta1)

# Second example
beta2 <- 0.24
rho2 <- 4.21
delta2 <- 0.25
rnormratio(20, bet = beta2, rho = rho2, delta = delta2)

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