Erik D. Jacobson
Tree Growth  Home
The first task of this
exploration was to plot the given data.
From the figure below, it looked like the curve is quadratic, cubic (or perhaps
exponential). It seemed plausible
that the number of board feet would grow in a quadratic manner (assuming a
fixed height was usable lumber) since the diameter of the tree would grow
linearly with respect to time, so the area of the crosssection would grow
quadratically with respect to time.
However, this model doesn't seem to take into account the growth in the
tree's height, so it may be a case of a cubic (or exponential) relationship
approximated by a quadratic function.
The first model was just 10% of the tree's age, squared. The error for this model is quite high, and the discrepancy between the model and the data is quite obvious when they are plotted together.
Age of
Tree 
100s of Board Feet 
Model 
Squared
Error 


(x*0.1)^2 

20 
1 
4 
9 
40 
6 
16 
100 
60 

36 

80 
33 
64 
961 
100 
56 
100 
1936 
120 
88 
144 
3136 
140 

196 

160 
182 
256 
5476 
180 

324 

200 
320 
400 
6400 


TOTAL: 
18018 
Subtracting 40% of the tree's age improved the model by a significant amount, but my goal was to get the total squared error under 100.
Age of
Tree 
100s of Board Feet 
Model 
Squared
Error 


(x*0.1)^2(x*0.4) 

20 
1 
4 
25 
40 
6 
0 
36 
60 

12 

80 
33 
32 
1 
100 
56 
60 
16 
120 
88 
96 
64 
140 

140 

160 
182 
192 
100 
180 

252 

200 
320 
320 
0 


TOTAL: 
242 




The next improvement of the model was to add a linear term. I noticed that many of the expected values were below what was observed.
Age of Tree 
100s of Board Feet 
Model 
Squared Error 


(x*0.1)^2(x*0.5)+10 

20 
1 
4 
9 
40 
6 
6 
0 
60 

16 

80 
33 
34 
1 
100 
56 
60 
16 
120 
88 
94 
36 
140 

136 

160 
182 
186 
16 
180 

244 

200 
320 
310 
100 


TOTAL: 
178 
Although the model was significantly improved, I thought I could do better. The last step was to fine tune each parameter by trial and error, beginning with the coefficient of the quadratic term and finishing with the linear term.
Age
of Tree 
100s of Board Feet 
Model 
Squared
Error 


(x*0.104)^2(x*0.62)+9.8 

20 
1 
1.7264 
0.52765696 
40 
6 
2.3056 
13.64859136 
60 

11.5376 

80 
33 
29.4224 
12.79922176 
100 
56 
55.96 
0.0016 
120 
88 
91.1504 
9.92502016 
140 

134.9936 

160 
182 
187.4896 
30.13570816 
180 

248.6384 

200 
320 
318.44 
2.4336 


TOTAL: 
69.4713984 
The last model achieved my goal, having a total squared error of less than 100. Since the model's purpose is to provide approximations of the number of board feet at 60, 140, and 180 years I would argue that the model's estimates for each of these may be revised. The model's values are below those observed at 40 and 80 years and so it is sensible to expect its value at 60 years will be also be low. Similarly, the model's values for 120 and 160 years are both high, so it is likely that its value for 140 years will likewise be high. Since the model's value is low for 200 years, its prediction for 180 may be high or low of the actual value.