An error correction system for sea surface temperature prediction

Abstract

One of the main indicators for detecting changes in climate and marine ecosystems around the world is the sea surface temperature (SST). Even with several models presented in the literature, it is still a challenging task when only a single model is considered for SST forecasting. In this context, hybrid approaches that combine statistical models (to estimate linear dependencies) and machine learning models (to estimate nonlinear dependencies from residuals) have attained highlighted accuracy in several time series forecasting problems. In this way, this paper proposes a hybrid system that combines the autoregressive integrated moving average (ARIMA) model with a deep morphological neural network model, which employs dilation-erosion operators to estimate ARIMA’s model residuals. In this context, the proposed hybrid system is composed of three stages: (1) time series forecast using the ARIMA model, (2) residuals forecast using the deep morphological neural network and (3) a linear combination of the stages (1) and (2). Three SST time series are used in the experimental analysis. Moreover, the achieved results are evaluated using three relevant statistical measures showing that the proposed hybrid system attains higher accuracy when compared to other hybrid models in the literature.

Appendix: A partial derivatives equations

From Eq. (10), $\frac{\partial u^{(j)}_i}{\partial {\textbf{y}}^{(j-1)}}$ can be estimated by:

$$\begin{aligned} \frac{\partial u^{(j)}_i}{\partial {\textbf{y}}^{(j-1)}} = \frac{\partial u^{(j)}_i}{\partial {\textbf{a}}^{(j)}_i} + \frac{\partial u^{(j)}_i}{\partial {\textbf{b}}^{(j)}_i}. \end{aligned}$$

(31)

From Eqs. (13), $\dot{f }(u^{(j)}_i)$ can be estimated by:

$$\begin{aligned} \dot{f }\left( u^{(j)}_i\right) = f \left( u^{(j)}_i\right) \left[ 1- f \left( u^{(j)}_i\right) \right] . \end{aligned}$$

(32)

From Eq. (10), $\frac{\partial u^{(j)}_i}{\partial \lambda ^{(j)}_i}$ can be estimated by:

$$\begin{aligned} \frac{\partial u^{(j)}_i}{\partial \lambda ^{(j)}_i} = \delta ^{(j)}_i - \varepsilon ^{(j)}_i. \end{aligned}$$

(33)

At this point, we have a problem to evaluate derivatives $\frac{\partial u^{(j)}_i}{\partial {\textbf{a}}^{(j)}_i}$ and $\frac{\partial u^{(j)}_i}{\partial {\textbf{b}}^{(j)}_i}$ since dilation and erosion operations are not differentiable. In order to overcome such drawback, we employ the approach proposed by Pessoa and Maragos [69] and extended by Araújo et al. [68], using the concept of smoothed rank indicator vector to approximate morphological operators in terms of differentiable operations. Next, we show how to evaluate both derivatives.

The partial derivative $\frac{\partial u^{(j)}_i}{\partial {\textbf{a}}^{(j)}_i}$ is given by:

$$\begin{aligned} \frac{\partial u^{(j)}_i}{\partial {\textbf{a}}^{(j)}_i} = \frac{\partial u^{(j)}_i}{\partial \delta ^{(j)}_i} \frac{\partial \delta ^{(j)}_i}{\partial {\textbf{a}}^{(j)}_i} = \lambda ^{(j)}_i \frac{\partial \delta ^{(j)}_i}{\partial {\textbf{a}}^{(j)}_i}, \end{aligned}$$

(34)

in which

$$\begin{aligned} \frac{\partial \delta ^{(j)}_i}{\partial {\textbf{a}}^{(j)}_i} = \frac{Q_\sigma \left( \delta ^{(j)}_i\cdot \!{\textbf {1}}-\left[ {\textbf {y}}^{(j-1)}+{\textbf {a}}^{(j)}_i\right] \right) }{Q_\sigma \left( \delta ^{(j)}_i\cdot \!{\textbf {1}}-\left[ {\textbf {y}}^{(j-1)}+{\textbf {a}}^{(j)}_i\right] \right) \cdot {\textbf {1}}^T}, \end{aligned}$$

(35)

where ${\textbf {1}} = (1, \ldots , 1)$ and operation $\cdot ^T$ denotes transposition.

Note that $Q_\sigma ({\textbf{z}}) = [q_\sigma (z_1), q_\sigma (z_2), \ldots , q_\sigma (z_n)]$, in which the term $q_\sigma (z_i)$ is given by:

$$\begin{aligned} q_\sigma (z_i) = \text {exp}\left[ {\frac{1}{2}\left( \frac{z_i}{\sigma }\right) ^2}\right] \,, \, \forall \, i = 1, \ldots , n , \end{aligned}$$

(36)

where $\sigma$ is a smoothing factor and n is the vector dimensionality.

The partial derivative $\frac{\partial u^{(j)}_i}{\partial {\textbf{b}}^{(j)}_i}$ is estimated by:

$$\begin{aligned} \frac{\partial u^{(j)}_i}{\partial {\textbf{b}}^{(j)}_i} = \frac{\partial u^{(j)}_i}{\partial \varepsilon ^{(j)}_i} \frac{\partial \varepsilon ^{(j)}_i}{\partial {\textbf{b}}^{(j)}_i} = \left( 1-\lambda ^{(j)}_i\right) \frac{\partial \varepsilon ^{(j)}_i}{\partial {\textbf{b}}^{(j)}_i}, \end{aligned}$$

(37)

in which

$$\begin{aligned} \frac{\partial \varepsilon ^{(j)}_i}{\partial {\textbf{b}}^{(j)}_i} = \frac{Q_\sigma \left( \varepsilon ^{(j)}_i\cdot \!{\textbf {1}}-\left[ {\textbf {y}}^{(j-1)}+'{} {\textbf {b}}^{(j)}_i\right] \right) }{Q_\sigma \left( \varepsilon ^{(j)}_i\cdot \!{\textbf {1}}-\left[ {\textbf {y}}^{(j-1)}+'{} {\textbf {b}}^{(j)}_i\right] \right) \cdot {\textbf {1}}^T}. \end{aligned}$$

(38)