Description Usage Arguments Value Author(s) References Examples

Function `qtree.moments`

finds the expected value and variance for *X_{r:n}*;
the `r`

:th smallest observation in an iid sample of size `n`

from a population with a percentile-based distribution.

Function `qtree.jointdens`

computes the bivariate pdf for two quantiles *(X_{r1:n},X_{r2:n})* from the same sample, where *r1<r2*.

Function `qtree.exy`

approximates expected value of the product *X_{r1:n}X_{r2:n}*, i.e. the
integral of function *x_{r1:n}x_{r2:n}f_{r1:n,r2:n}(\bm x)* over the two-dimensional range of *\bm x*
by computing for each percentile interval the function mean in a regular npts*npts grid and multiplying the mean by the area.

Function `qtree.varcov`

returns the expected valuers, cumulative percentage values and the variance-covariance matrices
that correspond to given sample quantiles and underlying percentile-based distribution of the population.

Function `interpolate.D`

does a bilinear interpolation of the variance-covariance matrix of percentiles
that correspond to values `F`

of the cdf to values that correspond to values
`ppi`

.

1 2 3 4 5 | ```
qtree.moments(r,n,xi,F)
qtree.jointdens(x,r1,r2,n,xi,F)
qtree.exy(r1,r2,n,xi,F,npts=100)
qtree.varcov(obs,xi,F)
interpolate.D(D,ppi,F)
``` |

`r, r1, r2` |
The ranks of the sample order statistics. |

`n` |
The sample size |

`xi` |
The percentiles that specify the cdf in increasing order. The first element should be the population minimum and the last element should be the population maximum. A vector of same length as |

`F` |
The values of the cdf that correspond to the percentiles of |

`x` |
a matrix with two columns that gives the x-values for which the joint density is computed in |

`npts` |
The number of regularly placed points that is used in the integral approximation of |

`obs` |
A data frame of observed sample quantiles, possibly from several plots. The data frame should include
(at least) columns |

`D` |
The variance-covariance matrix of the residual errors (plot effects) of percentile models. The number of columns and rows should equal to the length of |

`ppi` |
The values of cdf for which the covariances needs to be interpolated in |

Function `qtree.moments`

returns a list with elements

`mu` |
The expected value of |

`sigma2` |
The variance of |

`x,y` |
y gives the values of the pdf of |

Function `qtree.jointdens`

returns a vector with length equal to the `nrow(x)`

, including the values of the joint pdf of *({X_{r1:n}},X_{r2:n})* in these points.

Function `qtree.exy`

returns a scalar, the approximate of *E(X_{r1:n}X_{r2:n})*.

Function `qtree.varcov`

returns a list with elements

`obs` |
The original input data frame, augmented with the expected values in column |

`R` |
The variance-covariance matrix of the sample quantiles. |

Function `interpolate.D`

returns a list with elements

`D` |
The original variance-covariance matrix, augmented with the variances and covariances
that correspond to the cdf values |

`F` |
The values of cdf that correspond to the augmented matrix |

`D1` |
The variance-covariance matrix of the percentiles that correspond to the cdf values given in |

`D2` |
The covariance matrix between the percentiles that correspond to |

Lauri Mehtatalo <lauri.mehtatalo@uef.fi>

Mehtatalo, L. 2005. Localizing a predicted diameter distribution using sample information. Forest Science 51(4): 292–302.

Mehtatalo, Lauri and Lappi, Juha 2020a. Biometry for Forestry and Environmental Data: with examples in R. New York: Chapman and Hall/CRC. 426 p. doi: 10.1201/9780429173462

Mehtatalo, Lauri and Lappi, Juha 2020b. Biometry for Forestry and Environmental Data: with examples in R. Full Versions of The Web Examples. Available at http://www.biombook.org.

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 36 37 38 39 40 | ```
F<-c(0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,0.95,1)
# Predictions of logarithmic percentiles
xi<-c(1.638,2.352,2.646,2.792,2.91,2.996,3.079,3.151,3.234,3.349,3.417,3.593)
# The variance of their prediction errors
D<-matrix(c(0.161652909,0.050118692,0.022268974,0.010707222,0.006888751,0,
0.000209963,-0.002739361,-0.005478838,-0.00655718,-0.006718843,-0.009819052,
0.050118692,0.074627668,0.03492943,0.01564454,0.008771398,0,
-0.002691651,-0.005102312,-0.007290366,-0.008136685,-0.00817717,-0.009026883,
0.022268974,0.03492943,0.029281808,0.014958206,0.009351904,0,
-0.002646641,-0.003949305,-0.00592412,-0.006556639,-0.006993025,-0.007742731,
0.010707222,0.01564454,0.014958206,0.014182608,0.009328299,0,
-0.001525745,-0.002448765,-0.003571811,-0.004470387,-0.004791053,-0.005410252,
0.006888751,0.008771398,0.009351904,0.009328299,0.009799233,0,
-0.000925308,-0.001331631,-0.002491679,-0.003277911,-0.003514961,-0.003663479,
rep(0,12),
0.000209963,-0.002691651,-0.002646641,-0.001525745,-0.000925308,0,
0.003186033,0.003014887,0.002961818,0.003112953,0.003050486,0.002810937,
-0.002739361,-0.005102312,-0.003949305,-0.002448765,-0.001331631,0,
0.003014887,0.00592428,0.005843888,0.005793879,0.005971638,0.006247869,
-0.005478838,-0.007290366,-0.00592412,-0.003571811,-0.002491679,0,
0.002961818,0.005843888,0.00868157,0.008348973,0.008368812,0.008633202,
-0.00655718,-0.008136685,-0.006556639,-0.004470387,-0.003277911,0,
0.003112953,0.005793879,0.008348973,0.011040791,0.010962609,0.010906917,
-0.006718843,-0.00817717,-0.006993025,-0.004791053,-0.003514961,0,
0.003050486,0.005971638,0.008368812,0.010962609,0.013546621,0.013753718,
-0.009819052,-0.009026883,-0.007742731,-0.005410252,-0.003663479,0,
0.002810937,0.006247869,0.008633202,0.010906917,0.013753718,0.02496596),ncol=12)
# observed tree data, 5 trees from 2 plots
obs<-data.frame(r=c(1,3,6,1,2),n=c(7,7,7,9,9),plot=c(1,1,1,2,2),d=c(10,11,27,8,12))
# See Example 11.33 in Mehtatalo and Lappi 2020b
qtrees<-qtree.varcov(obs,xi,F)
obs<-qtrees$obs
mustar<-obs$Ed
ystar<-log(obs$d)
R<-qtrees$R
Dtayd<-interpolate.D(D,obs$pEd)
``` |

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