qtree function

Properties of sample quantiles from a tree population described by the percentile-based diameter distribution.

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.

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.

Returns

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

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. tools:::Rd_expr_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.

Author(s)

Lauri Mehtatalo <lauri.mehtatalo@uef.fi >

Examples

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)