Demand prediction of rice growth stage-wise irrigation water requirement and fertilizer using Bayesian genetic algorithm and random forest for yield enhancement

Abstract

Rice cultivation is the major source of earning revenues worldwide. The productivity and yield of rice crops mainly depend on soil water balance and soil fertility. Irrigation water requirement (IWR) analysis helps to retain appropriate soil water balance and judiciously allocate water resources considering vegetative, reproductive, and ripening stages of rice growth. To restore fertility, the application of fertilizers is inevitable but most of these are squandered owing to improper fertilizer selection without evaluation of soil macro-nutrients. So, the enhancement of rice yield demands the well-balanced application of fertilizers along with specific IWR analysis in each growth stage. In this paper, eXtreme Gradient Boosting (XGBoost) is used to extract high-scoring, correlated environmental parameters with IWR. Stacking-based ensemble learning is used to predict evapotranspiration since it is a very crucial indicator of rice water demand in different growth stages. Based on selected features of XGBoost and predicted evapotranspiration, IWR specific to all the rice growth stages is predicted using the Bayesian genetic algorithm ($Bay_{GA}$) hyper-tuned random forest (RF). The parameters of the maximum and the minimum number of samples required to be at the leaf node of RF are hyper-tuned using $Bay_{GA}$ to optimize performance. Comparative results indicate that IWR prediction using $Bay_{GA}-RF$ outperforms other methods with Accuracy (86.12, 92.42, 91.24), MSE (0.182, 0.162, 0.196), RMSE (0.426, 0.402, 0.442), MAE (0.193, 0.174, 0.205) and NSE (0.911, 0.952, 0.944) in Vegetative, Reproductive and Ripening rice growth stages and accuracy of $98\%$ to predict suitable fertilizer depending on Nitrogen, Phosphorous, and Potassium soil macro-nutrients.

Appendix 1: EXtreme Gradient Boosting

In eXtreme Gradient Boosting (XGBoost), the training data $a_{i}$ has been trained to predict a target variable $b_{i}$ and an ensemble of K Classification and Regression Trees. T1 ($a_{i}$, $b_{i}$)...TN ($a_{i}$, $b_{i}$) where $a_{i}$ is the descriptor’s training set. To acquire the overall prediction, the XGBoost ensemble incorporates a gradient descent approach to minimize loss and evaluate the errors of the previous model.

The data d($a_{i}$, $b_{i}$): i = $1 \rightarrow n$ with s sample of f features, while $b_{i}$ is the predicted value expressed as:

$$\begin{aligned} b_{i}^{'}= \sum _{j=1}^{J} f_{j}(a_{i}), f_{j} \in N \end{aligned}$$

(20)

here $f_{j}$ is the regression tree, and $f_{j}(a)$ is the prediction score of the jth tree to the data sample. $N =f(a)=W_{p(a)}$ (p: ${\mathbb {R}}^{m}$ to T, $W \in {{\mathbb {R}}}^{T}$ ), the space of regression tree, where W is the leaf weight and p is the tree mapping structure to its leaf index. T is the number of leaf nodes in the tree. Learning the function $f_{j}$ is based on minimizing the objective function,

$$\begin{aligned} \phi = \sum _{i=1}^{n} l(b_{i}, b_{i}^{'}) + \sum _{j=1}^{J} \Omega (f_{k}) \end{aligned}$$

(21)

where l is the training loss and the regularization term $\Omega$ penalizes model complexity. The optimal weight of the leaf can be represented as:

$$\begin{aligned} \Omega (f_{j}) = \lambda _{1}T + \frac{1}{2\lambda _{2}} \vert \vert w_{t} \vert \vert ^{2} \end{aligned}$$

(22)

where $\lambda _{1}$ and $\lambda _{2}$ are the regularization degrees. T and $w_{t}$ are the leaf nodes and score. Considering $b_{i}^{'}(t)$ is predicted at t iteration, $f_{t}$ is added to minimize the objective,

$$\begin{aligned} \phi ^{t} = \sum _{i=1}^{n} l (b, b^{'(t-1)} + f_{t}(b)) + \Omega (f_{t}) \end{aligned}$$

(23)

The first and the second-order gradient on l are $\delta _{b^{'}(t-1)} l(b,b^{'(t-1)})$ and $\delta ^{2}_{b^{'}(t-1)} l(b,b^{'(t-1)})$ denoted by $g_{i}$ and $h_{i}$. Thus, using the second-order Taylor expansion, the above equation can be rewritten as:

$$\begin{aligned} \phi ^{t} = \sum _{i=1}^{n} [g_{i}f_{t}(a) = \frac{1}{2}h_{i}f_{t}(a^{2})] + \Omega (f_{t}) \end{aligned}$$

(24)

where $g_{i}$ and $h_{i}$ is the second-order gradient on l. It can be defined as $I_{k}$=$f_{i} \vert p(a_{i})=k$ which is the instance of leaf k. Thus, the above equation can be written as:

$$\begin{aligned} \phi (t)= & {} \sum _{i=1}^{n} [g_{i}f_{t}(a) = \frac{1}{2}h_{i}f_{t}(a^{2})] + \lambda _{1} T + \frac{1}{2} \lambda _{2} \sum _{k=1}^{T} w_{k}^{2} \end{aligned}$$

(25)

$$\begin{aligned} \phi (t)= & {} \sum _{k=1}^{T} [(\sum _{i \in I_{k}}g_{i})w_{k} + \frac{1}{2} (\sum _{i \in I_{k}} h_{i} + \lambda _{1})w_{k}^{2}] + \lambda _{2} T \end{aligned}$$

(26)

The optimal weight, $w^{*}_{k}$ of leaf k on a fixed structure q(a) is expressed as:

$$\begin{aligned} w^{*}_{k}= - \frac{G_{k}}{H_{k} + \lambda _{2}} \end{aligned}$$

(27)

whose values can be expressed as:

$$\begin{aligned} \phi ^{*}= - \frac{1}{2} \sum _{k=1}^{T} \frac{G_{k}^{2}}{H_{k} + \lambda _{2}} + \lambda _{2}T \end{aligned}$$

(28)

where $G_{k}=\sum _{i \in I_{k}} g_{i}$, $H_{k}= \sum _{i \in I_{k}} h_{i}$ and $\Phi$ is the scoring function for the tree structure where smaller value indicates better tree structure. Both the gradient and second-order gradient statistics on each leaf needs to be added to get the overall reliable score before implementing the scoring algorithm.

The optimal split finding algorithm, as well as the loss reduction are similar to the ideas in Chen and Guestrin (2016b).