# Stochastic Models to Qualify Stem Tapers
## 3 Results
### Stem Taper and Volume Models  
A comparison of all fitted models was undertaken using the validation dataset. In order to be an applicable taper model, a model needs to predict diameter measurements at a particular height in a stem highly accurately. Diameter predictions at any specified height were computed for all fitted stem taper models, and the stem volume predictions were computed by integration-derived stem taper curves using the validation dataset for each species group. To compare all fitted taper and volume models, we used statistical measures (i.e., the coefficient of determination, mean prediction error (percent prediction error, %), mean absolute prediction error (percent absolute prediction error, %), and root-mean-square error (percent root-mean-square error, %), including analysis of the visual tests of the residuals and the simplicity. Quantitative assessment of models performance will rely on statistical measures including four basic features: the bias, which indicates that the predictions are systematically too high or too low (underperformance or overperformance); the absolute bias, which describes the typical magnitude of the residuals (does not indicate underperformance or overperformance); the root mean squared error of differences between observed and predicted values, which indicates standard deviation of the unexplained variance; and the coefficient of determination, which informs on the proportion of the total variability explained by the model. The root-mean-square error, in a statistical sense, normally takes precedence over the other errors, where it is known as the standard error of the regression in regression analysis or as the estimated white noise standard deviation in stochastic processes. The main problem of the comparison of the stem taper models is measuring the predictive ability to accurately predict the diameter and volume of new tree stems. Statistical measures calculated for the estimation dataset evaluate how well the stem taper model fits the estimation dataset for which it was developed, and, for the validation dataset, confirm how valid the stem taper model is in predicting diameters at any specified height and tree volumes for new dataset [37]. The decrease of statistical measures for the validation dataset in this study was small, so we can conclude that the used stem taper models can be used to predict stem diameters at any specified height, as well as stem volumes. Statistical measures for the prediction of the diameter and volume using stem diameter measurements from the validation dataset for all species group are presented in Tables 2 and 3, respectively. In most cases (models and species groups), the stem diameters and volumes have been slightly overpredicted. The black alder species group models produced the best root-mean-square error percent values, namely, 6.62% to 8.93% and 7.40% to 8.06% for the diameter and volume predictions, respectively. The birch species group models produced the worst root-mean-square error percent values, namely, 10.42% to 13.04% and 17.70% to 27.34% for the diameter and volume predictions, respectively.