Package ‘AdequacyModel’
February 19, 2015
Type Package
Title Adequacy of probabilistic models and generation of pseudo-random numbers.
Version 1.0.8 Date 2013-09-12
Maintainer Pedro Rafael Diniz Marinho <pedro.rafael.marinho@gmail.com> Description This package provides some useful functions for calculating quality
measures for
adjustment, including statistics Kolmogorov-Smirnov, Cramer-von Mises and Anderson-Darling. These statistics are often used to compare models not equipped. Also provided are other measures of fitness, such as AIC, CAIC, BIC and HQIC. The package also provides functions for generating pseudo-random number of random variables that follow the following probability distribu-tions: Kumaraswamy Birnbaum-Saunders, Kumaraswamy Pareto, Exponentiated Weibull and Kumaraswamy Weibull.
URL http://www.r-project.org
License GPL (>= 2)
Author Pedro Rafael Diniz Marinho [aut, cre], Marcelo Bourguignon [aut, ctb], Cicero Rafael Bar-ros Dias [aut, ctb]
NeedsCompilation no Repository CRAN Date/Publication 2013-12-20 17:01:24
R topics documented:
AdequacyModel-package . . . 2 carbone . . . 2 descriptive . . . 3 goodness.fit . . . 4 kwbs . . . 7 kwp . . . 9 rew . . . 10 rkwweibull . . . 11 TTT . . . 12 12 carbone
Index 14
AdequacyModel-package Adequacy of probabilistic models and generation of pseudo-random numbers.
Description
This package provides some useful functions for calculating quality measures for adjustment, in-cluding statistics Kolmogorov-Smirnov, Cramer-von Mises and Anderson-Darling. These statistics are often used to compare models not equipped. Also provided are other measures of fitness, such as AIC, CAIC, BIC and HQIC. The package also provides functions for generating pseudo-random number of random variables that follow the following probability distributions: Kumaraswamy Birnbaum-Saunders, Kumaraswamy Pareto, Exponentiated Weibull and Kumaraswamy Weibull. Details Package: AdequacyModel Type: Package Version: 1.0 Date: 2012-11-15 License: GPL-2 Depends: R (>= 2.10.0) References
Chen, G., Balakrishnan, N. (1995). A general purpose approximate goodness-of-fit test. Journal of Quality Technology 27, 154-161.
carbone Breaking stress of carbon fibres
Description
The first real data set corresponds to an uncensored data set from Nichols and Padgett (2006) on breaking stress of carbon fibres (in Gba).
Usage
descriptive 3 Format
The format is: num [1:100] 3.7 2.74 2.73 2.5 3.6 3.11 3.27 2.87 1.47 3.11 ...
References
Nichols, M.D, Padgett, W.J. (2006). A Bootstrap control chart for Weibull percentiles. Quality and Reliability Engineering International 22, 141-151.
Examples
data(carbone) hist(carbone)
descriptive descriptive - Calculation of descriptive statistics
Description
The function descriptive calculates the main descriptive statistics of a vector of data.
Usage
descriptive(x)
Arguments
x Data vector.
Author(s)
Pedro Rafael Diniz Marinho <pedro.rafael.marinho@gmail.com> Marcelo Bourguignon <m.p.bourguignon@gmail.com>
References
Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole.
Examples
data(carbone) descriptive(carbone)
4 goodness.fit
goodness.fit Adequacy of models
Description
This function provides some useful statistics to assess the quality of fit of probabilistic models, including the statistics Cramer-von Mises and Anderson-Darling. These statistics are often used to compare models not fitted. You can also calculate other goodness of fit such as AIC, CAIC, BIC, HQIC and Kolmogorov-Smirnov test.
Usage
goodness.fit(pdf, cdf, starts, data, method="L-BFGS-B", domain=c(0,Inf), mle=NULL)
Arguments
pdf Probability density function; cdf Cumulative distribution function;
starts Initial parameters to maximize the likelihood function; data Data vector;
method Method used for minimization of the function -log(likelihood). The meth-ods supported are: L-BFGS-B (default), BFGS, Nelder-Mead, SANN, CG. Can also be transmitted only the first letter of the methodology, ie, L, B, N, S or C respec-tively;
domain Domain of probability density function. By default the domain of probability density function is the open interval 0 to infinity.This option must be an vector with two values;
mle Vector with the estimation maximum likelihood. This option should be used if you already have knowledge of the maximum likelihood estimates. The default is NULL, ie, the function will try to obtain the estimates of maximum likelihoods. Details
The function goodness.fit returns statistics KS (Kolmogorov-Smirnov), A (Anderson-Darling), W (Cramer-von Misses). Are also calculated other measures of goodness of fit. These functions are: AIC (Akaike Information Criterion), CAIC (Consistent Akaikes Information Criterion), BIC (Bayesian Information Criterion) and HQIC (Hannan-Quinn information criterion).
The Kolmogorov-Smirnov test may return NA with a certain frequency. The return NA informs that the statistical KS is not reliable for the data set used. More details about this issue can be obtained fromks.test.
By default, the function calculates the maximum likelihood estimates. The errors of the estimates are also calculated. In cases that the function can not obtain the maximum likelihood estimates, the change of the values initial, in some cases, resolve the problem. You can also enter with the maximum likelihood estimation if there is already prior knowledge.
goodness.fit 5 Value
W Statistic Cramer-von Misses; A Statistic Anderson Darling; KS Kolmogorov Smirnov test; mle Maximum likelihood estimates; AIC Akaike Information Criterion;
CAIC Consistent Akaikes Information Criterion; BIC Bayesian Information Criterion;
HQIC Hannan-Quinn information criterion;
Erro Standard errors of the maximum likelihood estimates; Value Minimum value of the function -log(likelihood);
Convergence 0 indicates successful completion and 1 indicates that the iteration limit maxit had been reached. More details atoptim.
Note
It is not necessary to define the likelihood function or log-likelihood. You only need to define the probability density function and distribution function.
Author(s)
Pedro Rafael Diniz Marinho <pedro.rafael.marinho@gmail.com> References
Chen, G., Balakrishnan, N. (1995). A general purpose approximate goodness-of-fit test. Journal of Quality Technology, 27, 154-161.
Nocedal, J. and Wright, S. J. (1999) Numerical Optimization. Springer.
Sakamoto, Y., Ishiguro, M., and Kitagawa G. (1986). Akaike Information Criterion Statistics. D. Reidel Publishing Company.
Hannan, E. J. and Quinn, B. G. (1979). The Determination of the Order of an Autoregression. Journal of the Royal Statistical Society, Series B, 41, 190-195.
See Also
For details about the optimization methodologies may view the functionsoptim,ks.test,nlminb. Examples
# Example 1: data(carbone)
# Exponentiated Weibull - Probability density function. pdf_expweibull <- function(par,x){
6 goodness.fit
beta = par[1] c = par[2] a = par[3]
a * beta * c * exp(-(beta*x)^c) * (beta*x)^(c-1) * (1 - exp(-(beta*x)^c))^(a-1) }
# Exponentiated Weibull - Cumulative distribution function. cdf_expweibull <- function(par,x){ beta = par[1] c = par[2] a = par[3] (1 - exp(-(beta*x)^c))^a } goodness.fit(pdf=pdf_expweibull, cdf=cdf_expweibull, starts = c(1,1,1), data = carbone,
method="L-BFGS-B", domain=c(0,Inf),mle=NULL) # Example 2: data = c(0.08, 2.09, 3.48, 4.87, 6.94, 8.66, 13.11, 23.63, 0.20, 2.23, 3.52, 4.98, 6.97, 9.02, 13.29, 0.40, 2.26, 3.57, 5.06, 7.09, 9.22, 13.80, 25.74, 0.50, 2.46, 3.64, 5.09, 7.26, 9.47, 14.24, 25.82, 0.51, 2.54, 3.70, 5.17, 7.28, 9.74, 14.76, 26.31, 0.81, 2.62, 3.82, 5.32, 7.32, 10.06, 14.77, 32.15, 2.64, 3.88, 5.32, 7.39, 10.34, 14.83, 34.26, 0.90, 2.69, 4.18, 5.34, 7.59, 10.66, 15.96, 36.66, 1.05, 2.69, 4.23, 5.41, 7.62, 10.75, 16.62, 43.01, 1.19, 2.75, 4.26, 5.41, 7.63, 17.12, 46.12, 1.26, 2.83, 4.33, 5.49, 7.66, 11.25, 17.14, 79.05, 1.35, 2.87, 5.62, 7.87, 11.64, 17.36, 1.40, 3.02, 4.34, 5.71, 7.93, 11.79, 18.10, 1.46, 4.40, 5.85, 8.26, 11.98, 19.13, 1.76, 3.25, 4.50, 6.25, 8.37, 12.02, 2.02, 3.31, 4.51, 6.54, 8.53, 12.03, 20.28, 2.02, 3.36, 6.76, 12.07, 21.73, 2.07, 3.36, 6.93, 8.65, 12.63, 22.69) # Kumaraswamy Weibull Poisson - Probability density function. pdf_kwweibullpoisson <- function(par,x){ a = par[1] b = par[2] c = par[3] lambda = par[4] beta = par[5] (a*b*c*lambda*(beta^c)*(x^(c-1))*((1-exp(-(x*beta)^c))^(a-1)) * ((1-(1-exp(-(beta*x)^c))^a)^(b-1)) * exp(-lambda*(1-(1-(1-exp(-(beta*x)^c))^a)^b) - (beta*x)^c))/(1-exp(-lambda)) }
# Kumaraswamy Weibull Poisson - Cumulative distribution function. cdf_kwweibullpoisson <- function(par,x){
kwbs 7 b = par[2] c = par[3] lambda = par[4] beta = par[5] (1 - exp(lambda*(-(1-(1-(1-exp(-(x*beta)^c))^a)^b))))/(1-exp(-lambda)) } goodness.fit(pdf=pdf_kwweibullpoisson, cdf=cdf_kwweibullpoisson, starts = c(0.120,1.010,1.000,1.000,0.100), data = data, method="L-BFGS-B", domain=c(0,Inf),mle=NULL)
data = rweibull(250,2,3)
goodness.fit(pdf=pdf_kwweibullpoisson, cdf=cdf_kwweibullpoisson, starts = c(0.120,1.010,1.000,1.000,0.100), data = data, method="L-BFGS-B", domain=c(0,Inf),mle=NULL)
# Example 3:
# Kumaraswamy Beta - Probability density function. pdf_kwbeta <- function(par,x){ beta = par[1] a = par[2] alpha = par[3] b = par[4] (a*b*x^(alpha-1)*(1-x)^(beta-1)*(pbeta(x,alpha,beta))^(a-1)* (1-pbeta(x,alpha,beta)^a)^(b-1))/beta(alpha,beta) }
# Kumaraswamy Beta - Cumulative distribution function. cdf_kwbeta <- function(par,x){ beta = par[1] a = par[2] alpha = par[3] b = par[4] 1 - (1 - pbeta(x,alpha,beta)^a)^b } data = rbeta(1000,2,2.2)
goodness.fit(pdf = pdf_kwbeta, cdf=cdf_kwbeta, starts=c(0.5,1.02,2,1), data=data, method="L-BFGS-B", domain=c(0,1), mle=NULL)
kwbs Pseudo-Random Numbers - Kumaraswamy Birnbaum-Saunders Dis-tribution
Description
8 kwbs Usage
rkwbs(n, alpha, beta, a, b) Arguments
n Amount of generated numbers;
alpha Shape parameter of the Kumaraswamy Birnbaum-Saunders distribution; beta Scale parameter Kumaraswamy Birnbaum-Saunders distribution; a Shape parameter Kumaraswamy Birnbaum-Saunders distribution; b Shape parameter Kumaraswamy Birnbaum-Saunders distribution. Details
The additional parameters introduced by the Kumaraswamy generalization are sought as a manner to furnish a more flexible distribution. The Kw-BS distribution is expected to have immediate application in reliability and survival studies. The BS distribution arises as the basic exemplar for a = b = 1. The exponentiated Birnbaum-Saunders (EBS) distribution corresponds to b = 1. The Kw-BS is a particular case of the Mc-BS distribution when a = c; see Cordeiro et al. (2011).
F (x; α, β, a, b) = 1 − {1 − Φ(ν)a}b
where β > 0 is a scale parameter and the other positive parameters α, a, and b are shape parameters in that ν = α−1ρ(x/β), ρ(z) = z1/2− z−1/2and Φ(· ) denotes the standard normal distribution
function. Author(s)
Pedro Rafael Diniz Marinho <pedro.rafael.marinho@gmail.com> Marcelo Bourguignon <m.p.bourguignon@gmail.com>
References
Saulo, H.; Leao, J. and Bourguignon, M. The Kumaraswamy Birnbaum-Saunders Distribution. Journal of Statistical Theory and Practice 6, 745-759.
Cordeiro, G. M., and M. Castro. 2011. A new family of generalized distributions. J. Stat. Comput. Simulation, 81(7), 883-898.
Cordeiro, G.M. Lemonte, A.J. and Ortega, E.M.M. (2013). An extended fatigue life distribution. Statistics 47, 626-653. Examples a = 2.0 b = 2.5 alpha = 1.0 beta = 1.0 rkwbs(5, alpha, beta, a, b)
kwp 9
kwp Pseudo-Random Numbers - Kumaraswamy Pareto
Description
Generates pseudorandom numbers from Kumaraswamy Pareto Distribution. Usage
rkwp(n, k, beta, a, b) Arguments
n Amount of generated numbers;
k Shape parameter of the Kumaraswamy Pareto distribution; beta Scale parameter Kumaraswamy Pareto distribution; a Shape parameter Kumaraswamy Pareto distribution; b Shape parameter Kumaraswamy Pareto distribution. Details
The Kw-P (Kumaraswamy Pareto) distribution is not in fact very tractable. However, its heavy tail can adjust skewed data that cannot be properly fitted by existing distributions. Furthermore, the cumulative and hazard rate functions are simple. The Pareto distribution function Kumaraswamy function is given by F (x; β, k, a, b) = 1 − ( 1 − " 1 − β x k#a)b ,
where β > 0 (x >= β) is a scale parameter and the other positive parameters k, a and b are shape parameters.
Author(s)
Pedro Rafael Diniz Marinho <pedro.rafael.marinho@gmail.com> Marcelo Bourguignon <m.p.bourguignon@gmail.com>
References
Bourguinon, M.; Silva, R.B.; Zea, L.M. and Cordeiro, G.M. (2013). The Kumaraswamy Pareto distribution. Journal of Statistical Theory and Applications 12, 129-144.
See Also rweibull
10 rew Examples beta = 1.5 k = 1.5 a = 1.5 b = 3.5 x<-seq(beta, 10, 0.01) pdfKWP <- function(k,bet,a,b,x){ a*b*k*bet^(k)*(1-(beta/x)^k)^(a-1)*(1-(1-(beta/x)^k)^(a))^(b-1)/x^(k+1) }
hist(rkwp(100,k,beta,a,b), freq = FALSE,main="",xlab="x", ylab="Density",xlim=c(1.5,7),ylim=c(0,1.5),lwd=1) lines(x,pdfKWP(k,beta,a,b,x),lty=1,col="black",
lwd=1,type="l",xlim=c(1.5,7),ylim=c(0,1.5)) box()
rew Pseudo-Random Numbers - Exponentiated Weibull
Description
Generates pseudorandom numbers from Exponentiated Weibull Distribution.
Usage
rew(n, beta, c, a)
Arguments
n Amount of generated numbers;
beta Scale parameter of the Weibull distribution exponentiated; c Shape parameter Exponentiated Weibull distribution; a Shape parameter exponentiated Weibull distribution.
Details
The exponentiated Weibull family of probability distributions was introduced by Mudholkar and Srivastava (1993) as an extension of the Weibull family obtained by adding a second shape param-eter. The function is defined in positive real, ie x > 0.
F (x, β, c, a) = (1 − e−(β∗x)c)a, wherein β > 0, c > 0 and a > 0.
rkwweibull 11 Note
Making a = 1, we have the Weibull distribution. Making a = 1, we have the Weibull distribution. For k = 1 we have the exponentiated exponential distribution.
Author(s)
Pedro Rafael Diniz Marinho <pedro.rafael.marinho@gmail.com> References
Mudholkar, G.S.; Hutson, A.D. (1996). "The exponentiated Weibull family: some properties and a flood data application". Commun Stat Theory Meth 25: 3059-3083.
Mudholkar, G.S.; Srivastava, D.K. (1993). "Exponentiated Weibull family for analyzing bathtub failure-ratedata". IEEE Transactions on Reliability 42 (2): 299-302.
Mudholkar, G.S.; Srivastava, D.K.; Freimer, M. (1995). "The exponentiated Weibull family; a reanalysis of the bus motor failure data".
See Also rweibull
Examples
rew(20,7,2,1.7)
rkwweibull Pseudo-Random Numbers - Kumaraswamy Weibull
Description
Generates pseudorandom numbers from Kumaraswamy Weibull Distribution. Usage
rkwweibull(n, a, b, c, beta)
Arguments
n Amount of generated numbers;
a Shape parameter of the Kumaraswamy Weibull Distribution; b Shape parameter of the Kumaraswamy Weibull Distribution; c Shape parameter of the Kumaraswamy Weibull Distribution; beta Scale parameter of the Kumaraswamy Weibull Distribution.
12 TTT Details
The Kumaraswamy Weibull Distribution function is given by:
F (x, a, b, c) = 1 − {1 − [1 − e−(β∗x)c]a}b
The distribution was proposed by Cordeiro, Ortega and Nadarajah in 2010 in the Journal of the Franklin Institute. The distribution is defined on the positive real (x > 0). All parameters are also defined in the positive reals.
Author(s)
Pedro Rafael Diniz Marinho <pedro.rafael.marinho@gmail.com> References
Cordeiro, G.M.; Ortega, E.M.M., Nadarajah, S. (2010). "The Kumaraswamy Weibull distribution with application to failure data". Journal of the Franklin Institute, 347, 1399-1429.
Examples
rkwweibull(20,2,2.3,1.5,2)
TTT TTT function
Description
There are several behaviors that the failure rate function of a random variable T can take. In this context, the graph of total test time (TTT curve) proposed by Aarset (1987) may be used for obtain-ing empirical behavior of the function failure rate.
Usage
TTT(x, lwd = 2, lty = 2, col = "black", grid = TRUE, ...) Arguments
x Data vector;
lwd Thickness of the TTT curve. The argument lwd must be a nonnegative real number;
lty The argument lty modifies the style of the diagonal line chart TTT. Possible values are: 0 [blank], 1 [solid (default)], 2 [dashed], three [dotted], 4 [dotdash], 5 [longdash], 6 [twodash];
col Color used in the TTT curve;
grid If grid = FALSE graphic appears without the grid;
... Other arguments passed by the user and available for the function plot. More details inpar.
TTT 13 Note
The graphic TTT may have various forms. Aarset (1987) showed that if the curve approaches a straight diagonal function constant failure rate is adequate. When the curve is convex or concave the failure rate function is monotonically increasing or decescente respectively is adequate. If the failure rate function is convex and concave, the failure rate function in format U is adequate, otherwise the failure rate function unimodal is more appropriate.
The TTT curve is constructed by values r/n and G(r/n), wherein G(r/n) = [ Pn i=1Ti:n+ (n − r)Tr:n] Pn i=1Ti:n , r = 1, . . . , n, T1:n= 1, . . . , n. Author(s)
Pedro Rafael Diniz Marinho (pedro.rafael.marinho@gmail.com); Marcelo Bourguignon (m.p.bourguignon@gmail.com).
References
Aarset, M. V. (1987). How to identify bathtub hazard rate. IEEE Transactions Reliability, 36, 106-108.
Examples
data(carbone)
Index
∗Topic
