# L-moment Ratio Diagram Components of Tau4 and Tau6 This function returns a list of the L-kurtosis ($\tau_4$ and sixth L-moment ratio $\tau_6$, respectively) ordinates for construction of L-moment Ratio (L-moment diagrams) that are useful in selecting a distribution to model the data. ```r lmrdia46() ``` ## Returns An `list` is returned. - **aep4**: $\tau_4$ and $\tau_6$ of the 4-parameter Asymmetric Exponential Power (AEP4) distribution given L-skew set as $\tau_3 = 0$. This becomes then the (Symmetrical) Exponential Power. The complementary entry `pwrexp` are the effectively the same curve for the power exponential distribution based on lookup table archived in the `lmomco` package. The table stems from `inst/doc/SysDataBuilder02.R`. The `aep4` and not `pwrexp` is used in the line drawing by `plotlmrdia46`. - **gld_byt5opt**: $\tau_4$ and $\tau_6$ of the Generalized Lambda (GLD) distribution given L-skew set as $\tau_3 = 0$ and optimized by `pargld` with `pargld(..., aux="tau5")` with $\tau_5 = 0$. The table stems from `inst/doc/SysDataBuilder02.R`. The table `gld_byt5opt` is used in the line drawing by `plotlmrdia46` in relation to the argument therein of `nogld_byt5opt`. This is the trajectory of the symmetrical GLD having constant L-scale ($\lambda_2$); this is different than the structurally similar by not identical Tukey Lambda distribution. - **gld_byt6tukeylam**: $\tau_4$ and $\tau_6$ of the Generalized Lambda distribution given L-skew set as $\tau_3 = 0$ and optimized by `pargld` with `pargld(..., aux="tau6")` with $\tau_6(\tau_4)$ ($\tau_6$ as a function of $\tau_4$, see `gld_byt6tukeylam` table). The table stems from `inst/``doc/` `SysDataBuilder02.R`. The `gld_byt6tukeylam` is used in the line drawing by `plotlmrdia46` in relation to the argument therein of `notukey`. This relation between $\{\tau_4, \tau_6\}$ is that of the Tukey Lambda distribution; this is the trajectory of the symmetrical GLD having nonconstant L-scale ($\lambda_2$). - **nor**: $\tau_4$ and $\tau_6$ of the Normal distribution. The table stems from `inst/doc/` `SysDataBuilder02.R`. The `nor` is used in the point drawing by `plotlmrdia46`. - **pdq4**: $\tau_4$ and $\tau_6$ of the Polynomial Density-Quantile4 distribution, which implicitly is symmetrical, and therefore L-skew set as $\tau_3 = 0$. The table stems from `inst/doc/SysDataBuilder02.R`. The `pdq4` is used in the line drawing by `plotlmrdia46`. - **pwrexp**: $\tau_4$ and $\tau_6$ of the Power Exponential distribution of which the Asymmetric Exponential Power distribution (see also `lmomaep4`). The lookup table archive in the `lmomco` package for the Power Exponential (`PowerExponential.txt`) is confirmed to match the computation in `aep4` based on the AEP4 instead. The table stems from `inst/doc/` `SysDataBuilder02.R`. - **st2**: $\tau_4$ and $\tau_6$ of the well-known Student t distribution. The lookup table archive in the `lmomco` package for the Student t (`StudentT.txt`) is confirmed to match the computation in `st3` based on the ST3 instead. The table stems from `inst/doc/SysDataBuilder02.R`. The `st3` and not `st2` is used in the line drawing by `plotlmrdia46`. - **st3**: $\tau_4$ and $\tau_6$ of the Student 3t distribution (`lmomst3`). The table stems from `inst/doc/SysDataBuilder02.R`. The `st3` and not `st2` is used in the line drawing by `plotlmrdia46`. - **symstable**: $\tau_4$ and $\tau_6$ of the Stable distribution, which is not otherwise supported in `lmomco`. The lookup table archive in the `lmomco` package for the Symmetrical Stable distribution is `StableDistribution.txt`. The table stems from `inst/doc/SysDataBuilder02.R`. The `symstable` is used in the line drawing by `plotlmrdia46`. - **tukeylam**: (reference copy of `gld_byt6tukeylam`) $\tau_4$ and $\tau_6$ of the Tukey Lambda distribution ([https://en.wikipedia.org/wiki/Tukey_lambda_distribution](https://en.wikipedia.org/wiki/Tukey_lambda_distribution)) that is not supported per se in `lmomco` because the Generalized Lambda distribution is instead. The `SymTukeyLambda.txt` is the lookup table archive in the `lmomco` package for the Tukey Lambda distribution confirmed to match the mathematics shown herein. The measure $L-scale$ or the second L-moment is not constant for the Symmetric Tukey Lambda as formulated. So, the trajectory of this distribution is not for a constant L-scale, which is unlike that for the Generalized Lambda. The table stems from `inst/doc/SysDataBuilder02.R`. The `tukeylam` is used in the line drawing by `plotlmrdia46`. - **cau**: $\tau^{(1)}_4 = 0.34280842$ and $\tau^{(1)}_6 = 0.20274358$ (trim=1 TL-moments) of the Cauchy distribution (TL-moment [trim=1]) (see Examples `lmomcau` for source). - **sla**: $\tau^{(1)}_4 = 0.30420472$ and $\tau^{(1)}_6 = 0.18900723$ (trim=1 TL-moments) of the Slash distribution (TL-moment [trim=1]) (see Examples `lmomsla` for source). ## Details The `lmrdia46` returns a list of the tables for drawing the trajectories of the distributions by its access of `.lmomcohash$``t46list` created by the `inst/doc/SysDataBuilder02.R` script for `sysdata.rda` construction used by the `lmomco` package itself. The lookup table references below are pointing to the `inst/doc/t4t6` subdirectory of the package. A lookup table for the Exponential Power distribution is provided as `PowerExponential.txt` (`.lmomcohash$``tau46list$``pwrexp`), and this distribution is a special case of the Asymmetric Exponential Power4 (`lmomaep4`) (`.lmomcohash$``tau46list$``aep4`). A lookup table for the Symmetric Stable distribution is provided as `StableDistribution.txt` (`.lmomcohash$``tau46list$``symstable`). A lookup table for the Student t distribution is provided as `StudentT.txt` (`.lmomcohash$``tau46list$``st2`), and this distribution is the same as the Student 3t (`lmomst3`) (`.lmomcohash$``tau46list$``st3`). A lookup table for the Tukey Lamda distribution is provided as `SymTukeyLambda.txt` (`.lmomcohash$``tau46list$``tukeylam`), and this distribution is not quite the same as the Generalized Lambda distribution (`lmomgld`) (`.lmomcohash$``tau46list$``gld`). The normal distribution plots as a point in a Tau4-Tau6 L-moment ratio diagram as `.lmomcohash$``tau46list$``nor` for which $\tau_4^\mathrm{nor} = 30/\pi\times \mathrm{atan}(\sqrt{2}) - 9$ $= 0.1226017$ and $\tau_6^\mathrm{nor} = 0.04365901$ (numerical integration). Finally, the Cauchy and Slade distributions are symmetrical and can be plotted as well on Tau4-Tau6 L-moment ratio diagram if we permit their trim=1 TL-moments to be shown instead. These are inserted into the returned list as part of the operation of `lmrdia46()`. Tukey Lambda Notes ---The Tukey Lambda distribution is a simpler formulation than the Generalized Lambda. $$ Q(F) = \frac{1}{\lambda} \biggl[F^\lambda - (1-F)^\lambda \biggr]\mbox{,} $$ for nonexceedance probability $F$ and $\lambda \ne 0$ and $$ Q(F) = \mathrm{log}\biggl(\frac{F}{1-F}\biggr)\mbox{,} $$ for $\lambda = 0$ using the natural logarithm. Inspection of the distribution formulae inform us that the variation in the distribution, the scaling factor $1/\lambda$ to far left in the first definition, for instance, implies that the L-scale ($\lambda_2$) is not constant and varies with $\lambda$. The second L-moment of the Tukey Lambda (all odd order L-moments are zero) is $$ \lambda_2 = \frac{2}{\lambda}\biggl[ -\frac{1}{1+\lambda} + \frac{2}{2+\lambda}\biggr]\mbox{, and} $$ the fourth and sixth L-moments are $$ \lambda_4 = \frac{2}{\lambda}\biggl[ -\frac{1}{1+\lambda} + \frac{12}{2+\lambda} - \frac{30}{3+\lambda} + \frac{20}{4+\lambda}\biggr]\mbox{,} $$ $$ \lambda_6 = \frac{2}{\lambda}\biggl[ -\frac{1}{1+\lambda} + \frac{30}{2+\lambda} - \frac{210}{3+\lambda} + \frac{560}{4+\lambda} - \frac{630}{5+\lambda} + \frac{252}{6+\lambda}\biggr]\mbox{\, and} $$ $\tau_4 = \lambda_4 / \lambda_2$ and $\tau_6 = \lambda_6 / \lambda_2$. The Tukey Lambda is not separately implemented in the `lmomco` package. It is provided herein for theoretical completeness, but it is possible to implement the Tukey Lambda by the following example: ``` tukeylam <- .lmomcohash$tau46list$gld_byt6tukeylam lmr1 <- tukeylam[tukeylam$lambda2 == 1, ] # L-scale equal to one (for instance) lmr1 <- vec2lmom(c(0, lmr1$lambda2, 0, lmr1$tau4, 0, lmr1$tau6)) tuk1 <- pargld(lmr1, aux="tau6") print(tuk1$para, 12) # xi alpha kappa h # 2.50038766315e-04 -5.82180675380e+03 -1.71745206920e-04 -1.71702273015e-04 lambda <- mean(tuk1$para[3:4]) # remember optimization is used for parameters in # GLD parlance and so the two shape parameters are not constrained in pargld() # to be numerically identical. So, here, let us compute a mean of the two and then # use that as the Lambda in the distribution. eps <- 1/tuk1$para[2] - lambda message("EPS should be very close to zero, eps = ", eps, " !!!!!") tuk2 <- vec2par(c(0, 1/lambda, lambda, lambda), type="gld") # now Tukey Lambda lmr2 <- lmomgld(tuk2) "ofunc" <- function(lambda, lambda2=NA) { tukeyL2 <- ( 2 / lambda ) * ( -1 / (1+lambda) + 2 / (2+lambda) ) return(lambda2 - tukeyL2) } lam <- uniroot(ofunc, interval=c(-1, 1), lambda2=1)$root tuk3 <- vec2par(c(0, 20/lam, lam, lam), type="gld") lmr3 <- lmomgld(tuk3) gld5 <- pargld(lmr3, aux="tau5"); gldlmr5 <- theoLmoms(gld5, nmom=6) gld6 <- pargld(lmr3, aux="tau6"); gldlmr6 <- theoLmoms(gld6, nmom=6) plotlmrdia46(lmrdia46(), nogld_byt5opt=FALSE) points(gldlmr5$ratios[4], gldlmr5$ratios[6], pch=16, col="purple") points(gldlmr6$ratios[4], gldlmr6$ratios[6], pch=21, col="purple", bg="white") # See how GLD by tau5 optimization, which leaves Tau6 to float plots on the # "gld_byt5opt" trajectory, but GLD by tau6 optimization, plots on the Tukey # Lambda line, and gld6$para[2] / (1/gld6$para[3]) is equal to the 20 in the # parameter setting for tuk3. ``` The finally differences in the L-moments between the two `lmr` objects are all are reasonably close to zero with the recognition that `optim()` has been involved getting us close to the Tukey Lambda that we desire as a GLD with fixed shape parameters and scale factor equal to the inverse of the shape parameter. The demonstration to how to thus acquire a Tukey Lambda from GLD implementation in the `lmomco` package is thus shown. ## Author(s) W.H. Asquith ## See Also `plotlmrdia46`, `lmrdia` ## Examples ```r lratios <- lmrdia46() ```

lmrdia46 function