A modular ridge randomized neural network with differential evolutionary distributor applied to the estimation of sea ice thickness

Abstract

In this paper, a sequential intelligent methodology is implemented to estimate the sea-ice thickness along the Labrador coast of Canada based on spatio-temporal information from the moderate resolution imaging spectro-radiometer, and the advanced microwave scanning radiometer-earth sensors. The proposed intelligent model comprises two separate sub-systems. In the first part of the model, clustering is performed to divide the studied region into a set of sub-regions, based on a number of features. Thereafter, this learning system serves as a distributor to dispatch the proper information to a set of estimation modules. The estimation modules utilize ridge randomized neural network to create a map between a set of features and sea-ice thickness. The proposed modular intelligent system is best suited for the considered case study as the amount of collected spatio-temporal information is large. To ascertain the veracity of the proposed technique, two different spatio-temporal databases are considered, which include the remotely sensed brightness temperature data at two different frequencies (low frequency, 6.9 GHz, and high frequency, 36.5 GHz) in addition to both atmospheric and oceanic variables coming from validated forecasting models. To numerically prove the accuracy and computational robustness of the designed sequential learning system, two different sets of comparative tests are conducted. In the first phase, the emphasis is put on evaluating the efficacy of the proposed modular framework using different clustering methods and using different types of estimators at the heart of the estimation modules. Thereafter, the modular estimator is prepared with standard neural identifiers to prove to what extent the modular estimator can increase the accuracy and robustness of the estimation.

Appendices

Appendix 1

Algorithm 1. Golden sectioning local search

• 1. Verify the upper and lower bounds of searching domain ($a=-1$ and $b=1$).

• 2. Generate two intermediate values, i.e. $F^{1}(i)$ and $F^{2}(i)$, as below:

  $$\begin{aligned}&F^{1}(i)=b-\frac{b-a}{\phi },\\&F^{2}(i)=a+\frac{b-a}{\phi }, \end{aligned}$$

  where $\phi =1.6181$.

• 3. Proceed with the mutation operator to find two different mutant solutions:

  $$\begin{aligned} \mathbf {v}^{j}(i)=\mathbf {s}_1(i)+F^{j}(i)\left( \mathbf {s}_{3}(i)-\mathbf {s}_{2}(i)\right) , \quad j=1,2. \end{aligned}$$

• 4. Proceed with the crossover mechanism to determine the final solutions:

  $$\begin{aligned} u_{k}^{j}(i)={\left\{ \begin{array}{ll} s_{k}^{j}(i) &{}\quad \! \text {if }\text {rand}(0,1) < \text {CR}(i) \\ v_{k}^{j}(i) &{}\quad \! \text {otherwise} \end{array}\right. } \end{aligned}$$

  for $j=1,2$ and $k=1,\dots ,d$, where d represents the dimensionality of the optimization problem.

• 5. Evaluate the fitness of each of the two obtained off-springs.

• 6. If $\mathrm{fitness}(\mathbf {u}^{1}(i)){>}\mathrm{fitness}(\mathbf {u}^{2}(i))$, then $\mathbf {u}(i)=\mathbf {u}^{1}(i),F(i)=F^{2}(i)$ and $b=F^{2}(i)$; else, $\mathbf {u}(i)=\mathbf {u}^{2}(i),F(i)=F^{2}(i)$ and $a=F^{1}(i)$.

• 7. If the termination criteria are satisfied stop the process, otherwise return to Step 2.

Appendix 2

Algorithm 2. Hill climbing local search

• 1. Define an initial value for climbing h.

• 2. Produce three intermediate values, i.e. $F^{1}(i), F^{2}(i)$ and $F^{3}(i)$, as below:

  $$\begin{aligned} F^{1}(i)&=F(i)-h\\ F^{2}(i)&=F(i)\\ F^{3}(i)&=F(i)+h. \end{aligned}$$

• 3. Perform the mutation operator using the obtained scale factors to find the mutant solutions:

  $$\begin{aligned} \mathbf {v}^{j}(i)=\mathbf {s}_1(i)+F^{j}(i)\left( \mathbf {s}_{3}(i)-\mathbf {s}_{2}(i)\right) , \quad j=1,2,3. \end{aligned}$$

• 4. Proceed with the crossover mechanism to calculate the final solutions:

  $$\begin{aligned} u_{k}^{j}(i)={\left\{ \begin{array}{ll} s_{k}^{j}(i) &{}\quad \! \text {if }\text {rand}(0,1) < \text {CR}(i) \\ v_{k}^{j}(i) &{}\quad \! \text {otherwise} \end{array}\right. } \end{aligned}$$

  for $j=1,2,3$ and $k=1,\dots ,d$.

• 5. Calculate the fitness of each of the three obtained off-springs and extract the scale factor that yields the best fitness $F^{*}(i)$.

• 6. If $F^{*}(i)=F(i)$ then $h = h/2$; else, $\mathbf {u}(i)=\mathbf {u}^{*}(i)$ and $F(i)=F^{*}(i)$.

• 7. If the termination criteria are satisfied stop the process; otherwise, return to Step 2.

Appendix 3

Algorithm 3. Standard operators

• 1. Let $F_\ell$ and $F_u$ be the minimum and maximum scale factor values in Table 2, respectively. Update the scale factor using the following formula:

  $$\begin{aligned} F(i)={\left\{ \begin{array}{ll} F_{\ell }+F_{u}u_{1} &{}\quad \! \text {if }u_{2} < \tau _{1}\\ F(i) &{}\quad \! \text {otherwise}. \end{array}\right. } \end{aligned}$$

• 2. Update the crossover rate factor using the following formula:

  $$\begin{aligned} \text {CR}(i)={\left\{ \begin{array}{ll} u_{3} &{}\quad \! \text {if }u_{4}<\tau _{2}\\ \text {CR}(i) &{}\quad \! \text {otherwise}. \end{array}\right. } \end{aligned}$$

• 3. Proceed with the mutation operator to find the mutant solutions:

  $$\begin{aligned} \mathbf {v}(i)=\mathbf {s}_1(i)+F(i)\left( \mathbf {s}_{3}(i)-\mathbf {s}_{2}(i)\right) . \end{aligned}$$

• 4. Proceed with the crossover mechanism to calculate the final solutions:

  $$\begin{aligned} u_k(i)={\left\{ \begin{array}{ll} s_k(i) &{}\quad \! \text {if rand}(0,1)<\text {CR}(i)\\ v_k(i) &{}\quad \! \text {otherwise}. \end{array}\right. } \end{aligned}$$

  for $k=1,\dots ,d$.