Very Rough Draft of Documentation for the Drying Schedule Program

This program applies to hardwood species for which no drying schedules have yet been established.

We have investigated two approaches to obtaining a recommended drying schedule. Both approaches predict schedules from specific gravity. These approaches are based on a data set that contains 268 species. There are 40 distinct drying schedules that are currently recommended for these species. (See the USDA Dry Kiln Operator's Manual.) Some of them are used quite frequently. For example, the 6 (temperature) 4 (moisture content) 2 (wet bulb depression) schedule, T6-D2, was used for 63 of the 268 species. Others are used infrequently. Eighteen of the schedules were used only once.

The classification approach to obtaining a recommended schedule takes into account the prior probability of a schedule (e.g., 63 of the 268 schedules were 6 4 2 while only 1 of the 268 was 5 2 2) and the distance in normalized specific gravity units of a specimen to be classified from the mean specific gravity for a particular schedule. (For example, the 63 species for which the recommended schedule was 6 4 2 had a mean specific gravity of .575 and associated standard deviation of .107. The 20 species for which the recommended schedule was 10 4 4 had a mean specific gravity of .449 and associated standard deviation of .115. Thus if a new specimen had specific gravity x, its distance in normalized specific gravity units from the 6 4 2 schedule would be (x - .575)/.107, and its distance from the 10 4 4 schedule would be (x - .449)/.115.) The prior probability and distance measurements are combined to yield a posterior probability that a specimen belongs to a particular schedule. The details of this combination are provided in the Appendix. The ``predicted'' schedule is taken to be the one with the highest posterior probability.

The regression approach is based on three regressions --- temperature schedule number on specific gravity, moisture content schedule number on specific gravity, and wet bulb depression schedule number on specific gravity. Given the specific gravity associated with a species, the regression equations are used to predict appropriate temperature (temp), moisture content (mc), and wet bulb depression (depr) schedule numbers. The predicted numbers are rounded to the nearest integers. The resulting ``schedules'' are then again ``rounded'' to the nearest of the 40 schedules that has actually been observed. We applied the two methods to the 268 species for which we knew the "correct" schedule. In Table 1 we report the numbers of ``matches'' achieved by the two methods. For the classification approach, a match was ``exact'' if the recommended schedule (based on the classification algorithm) was the same as that typically used. For the regression approach, a match was ``exact'' if the recommended schedule obtained after the initial rounding to the nearest integer was the same as that typically used. For the regression approach, a match was ``quasi-exact'' if the recommended schedule obtained after both the initial rounding to the nearest integer and the subsequent ``rounding'' to the nearest of the standard 40 schedules was the same as that typically used. For the classification approach, there was a ``harshness match'' if the recommended schedule and the schedule typically used lay in the same "harshness class." Bill Simpson created these classes based on his experience. They are presented in Table 4. For the regression approach, there was a harshness match if the recommended schedule obtained after both the initial rounding to the nearest integer and the subsequent ``rounding'' to the nearest of the standard 40 schedules lay in the same harshness class as the schedule actually used.

Table 1: matches
approach exact matches quasi-exact matches harshness matches
regression 13 45 88
classification 81 -- 120

Table 2 contains the average absolute differences between the the schedules recommended by the regression and classification approaches and the schedules actually used.

Table 2: average absolute differences
approach temp mc depr harshness
regression 1.81 .425 .828 .993
classification 1.80 .422 .784 .985

Table 3 contains the maximum absolute differences between the the schedules recommended by the regression and classification approaches and the schedules actually used.

Table 3: maximum absolute differences
approach temp mc depr harshness
regression 6 3 3 6
classification 7 3 3 6

Table 1 suggests that the classification approach might be superior. Also, in 179 of the 268 cases, after the first rounding, the schedule recommended by the regression approach was not one of the 40 schedules actually used. Tables 2 and 3 do not detect much of a difference between the two approaches. When the two approaches yield different schedules, it might be wise to choose the milder of the two.

Table 4: The 40 schedules
harshness # temp mc depr sg average sg sd number of species
1 1 1 2 1 0.91450 0.08273 2
1 2 2 2 2 0.65900 0.06921 1
1 3 2 3 2 0.75412 0.10557 25
1 4 2 4 2 0.53200 0.06921 1
1 5 2 4 4 0.67453 0.17456 19
2 6 3 3 1 0.86250 0.06718 2
2 7 3 4 1 0.61950 0.07000 2
2 8 3 3 2 0.64773 0.11637 40
2 9 3 4 2 0.68100 0.07114 9
2 10 4 2 2 0.58000 0.06921 1
3 11 5 2 2 0.74000 0.06921 1
3 12 5 4 2 0.53000 0.06921 1
3 13 5 3 3 0.41000 0.06921 1
3 14 6 4 2 0.57454 0.10735 63
3 15 6 1 3 0.66000 0.06921 1
3 16 6 2 3 0.40000 0.06921 1
3 17 6 3 3 0.62500 0.02121 2
3 18 6 6 3 0.40000 0.06921 1
3 19 6 1 4 0.51000 0.06921 1
3 20 6 4 4 0.50856 0.10029 18
4 21 7 2 3 0.75000 0.04243 2
4 22 7 2 6 0.36000 0.06921 1
4 23 8 3 2 0.56000 0.06921 1
4 24 8 2 3 0.61000 0.14000 3
4 25 8 4 3 0.60000 0.06921 1
4 26 8 2 4 0.51000 0.05657 2
4 27 8 3 4 0.49333 0.05508 3
4 28 8 4 4 0.47000 0.06782 4
5 29 10 3 4 0.48000 0.06921 1
5 30 10 4 4 0.44905 0.11489 20
5 31 10 5 4 0.39800 0.03704 3
5 32 10 6 4 0.34500 0.02121 2
5 33 10 4 5 0.44806 0.08762 16
5 34 10 6 5 0.34000 0.04243 2
6 35 11 4 4 0.40000 0.06921 1
7 36 12 4 5 0.22600 0.06921 1
7 37 12 5 5 0.46000 0.06921 1
8 38 12 5 7 0.33500 0.02121 2
9 39 13 3 4 0.35011 0.13640 9
10 40 14 3 6 0.33400 0.06921 1

Appendix

Here we use LaTeX notation (roughly).
We have (this is not completely rigorous as we are really dealing with
density functions rather than probabilities, but the argument can be
made rigorous)
\[
P(group_j | sg = x) = P(group_j and sg = x)/P(sg = x)
= P(sg = x | group_j)P(group_j)/\sum_{i=1}^n P(sg = x | group_i)P(group_i)
\]
The item is placed in the group_j that maximizes P(group_j | sg = x)
which is the group_j that maximizes
P(sg = x | group_j)P(group_j).  We have
\[
P(sg = x | group_j) = 1/\sqrt{2 \pi} 1/\sigma_j 
\times \exp(-(x - \mu_j)^2/(2 \sigma_j^2))
\]
and
\[
P(group_j) = n_j/268
\]
where n_j is the number of species in the data set for which
the jth schedule is the typical schedule.
Taking logs, we see that we can maximize the posterior probability by
maximizing
\[
-\ln(\sigma_j) - (x - \mu_j)^2/(2\sigma_j^2) + \ln(n_j/268)
\]