Estimating nitrogen concentration in rape from hyperspectral data at canopy level using support vector machines

Abstract

The estimation of nitrogen concentration from remotely sensed data has been the subject of some work. However, few studies have addressed the effective model for monitoring nitrogen status at canopy level using Support Vector Machines (SVM). The present study is focused on the assessment of an estimation model for nitrogen concentration of rape canopy with hyperspectral data. Two types of estimation model, the traditional statistical method based on stepwise linear regression (SLR) and the emerging computationally powerful techniques based on support vector machines were applied The Root Mean Square Error (RMSE) and T values were used to assess their predictability. The results show that a better agreement between the observed and the predicted nitrogen concentration were obtained by using the SVM model. Compared to the SLR model, the SVM model improved the results by lowering RMSE by 11.86–21.13 %, and by increasing T by 20.00–29.41 % for different spectral transformations. The study demonstrated the potential of SVM to estimate nitrogen concentration using canopy level hyperspectral data and it was concluded that SVM may provide a useful exploratory and predictive tool when applied to canopy-level hyperspectral reflectance data for monitoring nitrogen status of rape.

Appendix: LS-SVM algorithm

In this section, the LS-SVM algorithm is discussed briefly. It was developed to solve the regression problem defined as follows: $\{ x_{i} ,y_{i} \}_{i = 1}^{N}$, where $x_{i} \in X \subseteq {\mathbb{R}}^{n}$ is the ith input data point in input space and $y_{i} \in Y \subseteq {\mathbb{R}}$ is the corresponding output value. The aim is to model the relationship between the input and output data points by a LS-SVM regression model. Equation 5 is linear in a higher dimensional feature space.

$$ y(x) = w^{T} \varphi (x) + b, $$

(5)

where w is a vector in feature space, $\varphi (x)$ is a non-linear mapping from input space to a higher dimensional feature space, b is the bias term. In LS-SVM, the following optimization problem is formulated by

$$ \mathop {\hbox{min} }\limits_{w,e} \;J(w,e) = \frac{1}{2}w^{T} w + \frac{1}{2}\gamma \sum\limits_{i = 1}^{N} {e_{i}^{2} } , $$

(6)

Subject to the equality constraints

$$ y_{i} = w^{T} \varphi (x) + b + e_{i} ,\quad i = 1, \ldots ,N, $$

(7)

where γ is the regularization factor, which balances the model’s complexity and the training errors, and $e_{i}$ is the error variable. The solution is obtained after constructing the Lagrangian equation

$$ L(w,b,e,\alpha )\; = J(w,e) - \sum\limits_{i = 1}^{N} {\alpha_{i} \left\{ {w^{T} \phi (x_{i} ) + b + e_{i} - y_{i} } \right\}} , $$

(8)

where $\alpha_{i}$ are the Lagrange multipliers. By solving this optimization problem, the coefficients $\alpha_{i}$ and b can be determined by the following solution:

$$ f(x) = \sum\limits_{i = 1}^{N} {a_{i} K(x,x_{i} ) + b} , $$

(9)

where $K(x,x_{i} )$ is the so-called kernel function with which the input vector can be mapped implicitly to a high-dimension feature space. The type of kernel and kernel function parameters selected are usually based on domain knowledge of where they are to be applied, and they should also reflect the distribution of input values of the training data (Durbha et al. 2007). Common examples of kernel functions include linear, polynomial, RBF and multi-layer perceptron.