# qtree: Properties of sample quantiles from a tree population... In lmfor: Functions for Forest Biometrics

## Description

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`.

## Usage

 ```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) ```

## Arguments

 `r, r1, r2` The ranks of the sample order statistics. `r=1` means the smallest, `r=n` the largest. `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` `F` The values of the cdf that correspond to the percentiles of `xi`. The first elements should be 0 and the last 1. `x` a matrix with two columns that gives the x-values for which the joint density is computed in `qtree.jointdens`. `npts` The number of regularly placed points that is used in the integral approximation of E(X_{r1:n}X_{r2:n}) for each percentile interval in function `exy`. `obs` A data frame of observed sample quantiles, possibly from several plots. The data frame should include (at least) columns `r` (the ranks), `n` (sample size), `plot` (plot id) and `d` (observed diameter). The rows should be ordered by `r` within each plot, and all observations from same plot should follow each other. `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 `F` and `xi`. `ppi` The values of cdf for which the covariances needs to be interpolated in `interpolate.D`.

## Value

Function `qtree.moments` returns a list with elements

 `mu` The expected value of X_{r:n}. `sigma2` The variance of X_{r:n}. `x,y` y gives the values of the pdf of X_{r:n} for values given in x for plotting purposes. Try `plot(sol\$x,sol\$y,type="l")`.

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 `Ed` and the corresponding values of the cdf of X in column `pEd`. `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 `ppi`. `F` The values of cdf that correspond to the augmented matrix `D`. `D1` The variance-covariance matrix of the percentiles that correspond to the cdf values given in `ppi` `D2` The covariance matrix between the percentiles that correspond to `ppi` and `F`

## Author(s)

Lauri Mehtatalo <lauri.mehtatalo@uef.fi>

## References

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.

## Examples

 ``` 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) ```

lmfor documentation built on July 8, 2020, 5:49 p.m.