mle_parameters: MLE Parameters
In NORMA: Builds General Noise SVRs

Description Usage Arguments Details Value Author(s) References Examples

mle_parameters computes the optimal parameters via MLE of a given distribution.

zero_laplace_mle computes the optimal parameters via MLE assuming a zero-mean Laplace as noise distribution.

general_laplace_mle computes the optimal parameters via MLE assuming a general Laplace as noise distribution.

zero_gaussian_mle computes the optimal parameters via MLE assuming a zero-mean Gaussian as noise distribution.

general_gaussian_mle computes the optimal parameters via MLE assuming a general Gaussian as noise distribution.

beta_mle computes the optimal parameters via MLE assuming a Beta as noise distribution.

weibull_mle computes the optimal parameters via MLE assuming a Weibull as noise distribution.

moge_mle computes the optimal parameters via MLE assuming a MOGE as noise distribution.

mle_parameters(phi, dist = "nm", ...)

zero_laplace_mle(phi)

general_laplace_mle(phi)

zero_gaussian_mle(phi)

general_gaussian_mle(phi)

beta_mle(phi, m1 = mean(phi, na.rm = T), m2 = mean(phi^2, na.rm = T),
  alpha_0 = (m1 * (m1 - m2))/(m2 - m1^2), beta_0 = (alpha_0 * (1 - m1)/m1))

weibull_mle(phi, k_0 = 1)

moge_mle(phi, lambda_0 = 1, alpha_0 = 1, theta_0 = 1)

`phi`	a vector with residual values used to estimate the parameters.
`dist`	assumed distribution for the noise in the data. Possible values to take: l: Zero-mean Laplace distribution. lm: General Laplace distribution. n: Zero-mean Gaussian distribution. nm: General Gaussian distribution. b: Beta distribution. w: Weibull distribution. moge: MOGE distribution.
`...`	additional arguments to be passed to the low level functions (see below).
`m1`	first moment of the residuals. Used to compute `alpha_0`.
`m2`	second moment of the residuals. Used to compute `beta_0`.
`alpha_0`	initial value for Newton-Raphson method for the parameter α.
`beta_0`	initial value for Newton-Raphson method for the parameter β.
`k_0`	initial value for Newton-Raphson method for the parameter κ.
`lambda_0`	initial value for Newton-Raphson method for the parameter λ.
`theta_0`	initial value for Newton-Raphson method for the parameter θ. See also 'Details' and multiroot.

For the zero-μ Laplace distribution the optimal MLE parameters are

σ=mean(|φ_i|)

, where {φ_i} are the residuals passed as argument.

For the general Laplace distribution the optimal MLE parameters are

μ=median(φ_i)

σ=mean(|φ_i - μ|)

, where {φ_i} are the residuals passed as argument.

For the zero-μ Gaussian distribution the optimal MLE parameters are

σ^2=mean(φ_i^2)

, where {φ_i} are the residuals passed as argument.

For the general Gaussian distribution the optimal MLE parameters are

μ=mean(φ_i)

σ^2=mean((φ_i-μ)^2)

, where {φ_i} are the residuals passed as argument.

For the Beta distribution values of parameters α and β are estimated using Newton-Raphson method.

For the Weibull distribution value of parameter κ is estimated using Newton-Raphson method and then estimated value of λ is computed using the following closed form that depends on κ:

λ=mean(φ_i^kappa)^(1/κ)

For the MOGE distribution values of parameters λ, α and θ are estimated using Newton-Raphson method.

See also 'References'.

mle_parameters returns a list with the estimated parameters. Depending on the distribution these parameters will be one or more of the following ones:

sigma: scale parameter of the Laplace distribution.
mu: location or mean parameter of the Laplace or Gaussian distribution, respectively.
sigma_cuad: variance parameter of the Gaussian distribution.
alpha: shape1 parameter of the Beta distribution or second parameter of the MOGE distribution.
beta: shape2 parameter of the Beta distribution.
k: shape parameter of the Weibull distribution.
lambda: lambda scale parameter of the Weibull distribution or first parameter of the MOGE distribution.
theta: third parameter of the MOGE distribution.

Jesus Prada, jesus.prada@estudiante.uam.es

Link to the scientific paper

Prada, Jesus, and Jose Ramon Dorronsoro. "SVRs and Uncertainty Estimates in Wind Energy Prediction." Advances in Computational Intelligence. Springer International Publishing, 2015. 564-577,

with theoretical background for this package is provided below.

http://link.springer.com/chapter/10.1007/978-3-319-19222-2_47

# Estimate optimal parameters using default distribution ("nm").
mle_parameters(rnorm(100))

# Estimate optimal parameters using "lm" distribution.
mle_parameters(rnorm(100),dist="lm")

# Equivalent to mle_parameters(rnorm(100),dist="l")
zero_laplace_mle(rnorm(100))

# Equivalent to mle_parameters(rnorm(100),dist="lm")
general_laplace_mle(rnorm(100))

# Equivalent to mle_parameters(rnorm(100),dist="n")
zero_gaussian_mle(rnorm(100))

# Equivalent to mle_parameters(rnorm(100),dist="nm")
general_gaussian_mle(rnorm(100))

# Equivalent to mle_parameters(rnorm(100),dist="b")
beta_mle(rnorm(100))

# Equivalent to mle_parameters(rnorm(100),dist="w")
weibull_mle(rnorm(100))

# Equivalent to mle_parameters(rnorm(100),dist="moge")
moge_mle(rnorm(100))