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 cross-section 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.