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Full Research Article
Predictive Modelling of Height and Diameter Relationships of Himalayan Chir Pine
First Online 23-07-2022|
MATERIALS AND METHODS
In view of the above various height and diameter statistical models were used to study the tree height and diameter relationships of chir trees in each division. The functional form of the models used in this study is as follows:
Hi = ith observation of the response variable tree height (m).
Di = ith observation of the predictor variable diameter at breast height (cm).
b = A vector of model parameters.
ei = Unexplained error, that it is assumed to be independent and normally distributed with mean zero and a constant variance.
A constant, 1.3 (which is bh in models below) is added to the model to avoid prediction of a Hi less than 1.3 m when Di approaches zero.
Based on the above functional form eleven height diameter models were used in this study, whose description is given in Table 2.
Then adequacy of the fitted models was tested by using different selection criteria like AIC, BIC, RMSE, (Adj. R2), etc. which is given below:
Adjusted R2 (Adj. R2)
R2 = Coefficient of determination.
n = Number ofobservation and k is number of parameters.
Root mean square error (RMSE)
n = Total number of observations.
yi = Actual observation.
= Predicted value.
= Mean of observed value.
Mean absolute error (MAE)
n = Total number of observations.
yi = Actual observation.
= Predicted value
= Mean of observed value, Raghav et al., (2022).
Akaike information criterion (AIC)
AIC = n × In(RMSE) + 2k
n = Number of observations.
k = Number of parameters.
Bayesian information criterion (BIC)
BIC = -2In(l) + In(n) × k
l = Maximum likelihood value of the model.
k = Number of degrees of freedom (or independently adjusted parameters) in the model.
n = Number of observations (total sample size).
Residual standard error (RSE)
y = The observed value.
= The predicted value.
df = The degrees of freedom.
Further, the assumption of normality of errors of fitted models was accessed by Shapiro Wilk test of significance. Then to check the predictive performance of the fitted models cross validation method was used and data was randomly partitioned into training and testing sets, where 80 per cent f data was used for training and the remaining 20 per cent of the data for testing. Statistical analysis of above models was carried out in R studio (version 3.5.1, 2018) utilising its various libraries. Further the function geom_line and ggtitle of R studio was used for creating these GG-plots.
RESULTS AND DISCUSSION
In order to access the predictive performance of fitted models, cross validation method (80:20) was adopted, utilizing 80 per cent data as training set and remaining 20 per cent as testing set. Table 5 reveal the performance criteria of models under training and testing data set in case of Jammu forest division. A perusal of Table 5 revealed that MG (Manfred N3) and MJ (Michaelis-Menten2) performed better across both data sets in Jammu forest division as both the models resulted in lower values prediction error rates (PER).
Apart from this all the models were ranked based on prediction error rates (PER), a perusal of which is given in Table 6.
The evaluation of height diameter models with respect to training and testing data sets were also visualized graphically with the help of library (ggplot2) in R studio. A perusal of GGplots across Jammu forest division with respect to various performance criteria are given in Fig 1, 2 and 3.
At primary stage data on height and diameter variables of 500 Chir Pine trees were takenand accordingly eleven height diameter models were fitted on the data. Initial analysis revealed that almost all the parameters across the models were found out to be significant. At second stage various selection criteria’s like RMSE, MAE and BIAS etc. were used to study the predictive performance of fitted height diameter models. Various functions of R were utilized with the help of various libraries like minpack.lm, Metrics, caret, tidyverse and nlme of R to generate the results of various selection criteria’s. It was revealed that RSE ranged from 4.23 to 6.74, while as AIC and BIC ranged from 438.34 to 670.28 and 443.55 to 678.09. Further MAE and RMSE ranged from 2.48 to 4.78 and 4.19 to 6.70, while as Adj.R2 and R2 from 0.66 to 0.85 and 0.66 to 0.88. Based on these results MG (Manfered N3) and MJ (Michaelis-Menten2) height diameter models performed well in Jammu forest division among all the models used in this study. Further the prediction error rate (PER) ranged from 0.29 to 0.77 and 0.25 to 0.65 in case of training and testing data sets, which revealed that results were in favour of MG (Manfered N3) and MJ (Michaelis-Menten2) model as both models resulted in lower values of PER, also have lowest values of RMSE, MAE, AIC, BIC and higher values of Adj.R2. Further the function geom_line and ggtitle of R studio was used for creating these GG-plots. Predicted height on Jammu forest data sets based on the fitted models were also plotted using library (ggplot2).
Conflict of interest
- Curtis, R.O. (1967). Height-diameter and height-diameter-age equations for second growth Douglas fir. Forest Science. 13: 365-375.
- Harsh, M., Pawan, K., Avinash C.R., Kaushal, R. and Anand, K.G. (2022). Evaluation of Grewia optiva clones for fodder yield under North Western Himalayas conditions. Journal of Sustainable Forestry. DOI: 10.1080/1054981 1.2022.2045502.
- Hasanzad, N.I., Alavi, S.J., Ahmadi, M.K. and Radkarmi, M. (2016). Comparison of different non-linear models for prediction of the relationship between diameter and height of velvet maple trees in natural forests (Case study: Asalem Forests, Iran). Journal of Forest Science. 62: 65-71.
- Huang, S., Titus, S.J. and Wiens, D.P. (1992). Comparison of nonlinear height-diameter functiond for major Alberta tree species. Can. J. For. Res. 22: 1297-1304. DOI: 10.1139/ x92-172.
- Jeelani, F., Sharma, M.K., Rizvi, S.E.H. and Jeelani, M.I. (2017). Predictive modelling and validation for estimating fodder yield of Grewia optiva. Malaysian Journal of Science. 36: 103-115.
- Jeelani, F., Sharma, M.K., Rizvi, S.E.H. and Jeelani, M.I. (2018). A study on cross validation for model selection and estimation. International Journal of Agricultural Sciences. 14: 165-172.
- Jeelani, M.I., Mir, S.A., Khan, I., Nazir, N. and Jeelani, F. (2015). Rank set sampling in improving the estimates of simple regression model. Pakistan Journal of Statistics and Operation Research. 11: 39-49.
- Larson, S. (1931). The shrinkage of the coefficient of multiple correlations. Journal of Educational Psychology. 22: 45- 55.
- Kumar, R.R., Chauhan, J. and Joshi, U. (2021). Social Economical and Medicinal Importanc of Grewia optiva. Agriculture and Food e-Newsletter. 3: 252-254.
- Menten, L. and Michaelis, M.I. (1913). Die kinetik der invertinwirkung. Biochem. Z. 49: 333-369.
- Meyer, H.A. (1940). A mathematical expression for height curves. Journal of Forestry. 38: 415-420.
- Míchal, I. (1992). Obnovaekologické stability lesù.Praha, Academia. 169. (in Czech).
- Mosteller, F. and Turkey, J.W. (1968). Data Analysis, Including Statistics.In Handbook of Social Psychology. Addison- Wesley. pp. 601-720.
- Naslund, M. (1937). Forest Research Institute’s Thinning Experiments in Scots Pine Forests. 29: 7-9.
- R Development Core Team, (2016). R: A Language and Environment for Statistical Computing. The R Foundation for Statistical Computing. Vienna, Austria. R version 3.2.4 (2016-03-10). https://www.Rproject.org.
- Raghav, Y.S., Mishra, P., Alakkari, K.M., Singh, M., Al Khatib, A.M.G. and Balloo, R. (2022). Modelling and forecasting of pulses production in South Asian countries and its role in nutritional security. Legume Research. 45: 454-461. DOI: 10.18805/LRF-645.
- Wykoff, W.R., Crookston, N.L. and Stage, A.R. (1982). User’s guide to the stand prognosis model. USDA For. Serv. Gen. Tech. Rep. 133.
- Zeide, B. (1993). Analysis of growth equations [J]. Forest Science. 39: 594-616.
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