Description Usage Arguments Value Note Author(s) References See Also Examples

This function fits a distribution to available quantiles (or irregular quantiles) through *n*-dimensional minimization using the `optim`

function. The objective function forms are either root mean-square error (RMSE) or mean absolute deviation (MAD), and the objective functions are expected to result in slightly different estimates of distribution parameters. The RMSE form (*σ_{\mathrm{RMSE}}*) is defined as

*σ_{\mathrm{RMSE}} = \biggl[ \frac{1}{m}\,∑_{i=1}^m [x_o(f_i) - \hat{x}(f_i)]^2\biggr]^{1/2}\mbox{,}*

where *m* is the length of the vector of *o*bserved quantiles *x_o(f_i)* for nonexceedance probability *f_i* for *i \in 1, 2, \cdots, m*, and *\hat{x}(f_i)* for *i \in 1, 2, \cdots, m* are quantile estimates based on the “current” iteration of the parameters for the selected distribution having *n* parameters for *n ≤ m*. Similarly, the MAD form (*σ_{\mathrm{MAD}}*) is defined as

*σ_{\mathrm{MAD}} = \frac{1}{m}\,∑_{i=1}^m | x_o(f_i) - \hat{x}(f_i) | \mbox{.}*

The `disfitqua`

function is not intended to be an implementation of the *method of percentiles* but rather is intended for circumstances in which the available quantiles are restricted to either the left or right tails of the distribution. It is evident that a form of the method of percentiles however could be pursued by `disfitqua`

when the length of *x(f)* is equal to the number of distribution parameters (*n = m*). The situation of *n < m* however is thought to be the most common application.

The right-tail restriction is the general case in flood-peak hydrology in which the median and select quantiles greater than the median can be available from empirical studies (e.g. Asquith and Roussel, 2009) or rainfall-runoff models. The available quantiles suit engineering needs and thus left-tail quantiles simply are not available. This circumstance might appear quite unusual to users from most statistical disciplines but quantile estimates can exist from regional study of observed data. The **Examples** section provides further motivation and discussion.

1 2 |

`x` |
The quantiles |

`f` |
The nonexceedance probabilities |

`objfun` |
The form of the objective function as previously described. |

`init.lmr` |
Optional initial values for the L-moments from which the initial starting parameters for the optimization will be determined. The optimizations by this function are not performed on the L-moments during the optimization. The form of |

`init.para` |
Optional initial values for the parameters used for starting values for the |

`type` |
The distribution type specified by the abbreviations listed under |

`verbose` |
A logical switch on the verbosity of output. |

`...` |
Additional arguments to pass to the |

An **R** `list`

is returned, and this `list`

contains at least the following items:

`type` |
The type of distribution in character format (see |

`para` |
The parameters of the distribution. |

`source` |
Attribute specifying source of the parameters—“disfitqua”. |

`init.para` |
A vector of the initial parameters actually passed to the |

`disfitqua` |
The returned |

The `disfitqua`

function is likely more difficult to apply for *n > 3* (high parameter) distributions because of the inherent complexity of the mathematics of such distributions and their applicable parameter (and thus valid L-moment ranges). The complex interplay between parameters and L-moments can make identification of suitable initial parameters `init.para`

or initial L-moments `init.lmr`

more difficult than is the case for *n ≤ 3* distributions. The default initial parameters are computed from an assumed condition that the L-moments ratios *τ_r = 0* for *r ≥ 3*. This is not ideal, however, and the **Examples** show how to move into high parameter distributions using the results from a previous fit.

W.H. Asquith

Asquith, W.H., and Roussel, M.C., 2009, Regression equations for estimation of annual peak-streamflow frequency for undeveloped watersheds in Texas using an L-moment-based, PRESS-minimized, residual-adjusted approach: U.S. Geological Survey Scientific Investigations Report 2009–5087, 48 p., http://pubs.usgs.gov/sir/2009/5087

`dist.list`

, `lmoms`

, `lmom2vec`

, `par2lmom`

, `par2qua`

, `vec2lmom`

, `vec2par`

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 | ```
# Suppose the following quantiles are estimated using eight equations provided by
# Asquith and Roussel (2009) for some watershed in Texas:
Q <- c(1480, 3230, 4670, 6750, 8700, 11000, 13600, 17500)
# These are real estimates from a suite of watershed properties but the watershed
# itself and location are not germane to demonstrate this function.
LQ <- log10(Q) # transform to logarithms of cubic feet per second
# Convert the averge annual return periods for the quantiles into probability
P <- T2prob(c(2, 5, 10, 25, 50, 100, 200, 500)); qP <- qnorm(P) # std norm variates
# The log-Pearson Type III (LPIII) is immensely popular for flood-risk computations.
# Let us compute LPIII parameters to the available quantiles and probabilities for
# the watershed. The log-Pearson Type III is "pe3" in the lmomco with logarithms.
par1 <- disfitqua(LQ, P, type="pe3", objfun="rmse") # root mean square error
par2 <- disfitqua(LQ, P, type="pe3", objfun="mad" ) # mean absolute deviation
# Now express the fitted distributions in forms of an LPIII.
LQfit1 <- qlmomco(P, par1); LQfit2 <- qlmomco(P, par2)
plot( qP, LQ, pch=5, xlab="STANDARD NORMAL VARIATES",
ylab="FLOOD QUANTILES, CUBIC FEET PER SECOND")
lines(qP, LQfit1, col=2); lines(qP, LQfit2, col=4) # red and blue lines
## Not run:
# Now demonstrate how a Wakeby distribution can be fit. This is an example of how a
# three parameter distribution might be fit and then the general L-moments secured for
# an alternative fit using a far more complicated distribution. The Wakeby for the
# above situation does not fit "out of the box." The types "gld", "aep4", and "kap"
# all with four parameters work with some serious CPUs burned for gld.
lmr1 <- theoLmoms(par1) # need five L-moments but lmompe3() only gives four,
# therefore must compute the L-moment by numerical integration provided by theoLmoms().
par3 <- disfitqua(LQ, P, type="wak", objfun="rmse", init.lmr=lmr1)
lines(qP, par2qua(P, par3), col=6, lty=2) # dashed line, par2qua alternative to qlmomco
# Finally, the initial L-moment equivalents and then the L-moments of the fitted
# distribution can be computed and compared.
par2lmom(vec2par(par3$init.para, type="wak"))$ratios # initial L-moments
par2lmom(vec2par(par3$para, type="wak"))$ratios # final L-moments
## End(Not run)
``` |

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