# Hybridization of Haar Wavelet Decomposition and Computational Intelligent Algorithms for the Estimation of Climate Change Behavior

Haruna Chiroma, Sameem Abdulkareem, Adamu I. Abubakar, Eka Novita

Sari, Tutut Herawan, Abdulsalam Ya'u Gital

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Haruna Chiroma, Sameem Abdulkareem, Adamu I. Abubakar, Eka Novita Sari, Tutut Herawan, et al.. Hybridization of Haar Wavelet Decomposition and Computational Intelligent Algorithms for the Estimation of Climate Change Behavior. 2nd Information and Communication Technology - EurAsia Conference (ICT-EurAsia), Apr 2014, Bali, Indonesia. pp.238-247, 10.1007/978-3-642-55032-4_23 hal-01397202

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# Hybridization of Haar Wavelet Decomposition and Computational Intelligent Algorithms for the Estimation of Climate Change Behavior

Haruna Chiroma<fnref n="1" />, Sameem Abdulkareem<fnref n="1" />, Adamu I. Abubakar<fnref n="2" />, Eka Novita Sari<fnref n="3" />, Tutut Herawan<fnref n="4" />, and Abdulsalam Ya'u Gital<fnref n="5" />

freedonchi@yahoo.com, 100adamu@gmail.com
{sameem,tutut}@um.edu.my, eka@amcs.co, asgital@yahoo.com

**Abstract.** We propose a hybrid of haar wavelet decomposition, relevance vector machine, and adaptive linear neural network (HWD-RVMALNN) for the estimation of climate change behavior. The HWD-RVMALNN is able to improve estimation accuracy of climate change more than the approaches already discussed in the literature. Comparative simulation results show that the HWD-RVMALNN outperforms cyclical weight/bias rule, Levenberg-Marquardt, resilient back-propagation, support vector machine, and learning vector quantization neural networks in both estimation accuracy and computational efficiency. The model proposes in this study can provide future knowledge of climate change behavior. The future climate change behavior can be used by policy makers in formulating policies that can drastically reduce the negative impact of climate change, and be alert on possible consequences expected to occur in the future.

**Keywords:** Haar Wavelet Decomposition, Relevance Vector Machine, Adaptive Linear Neural Network, Climate Change.

## 1 Introduction

In the last ten (10) to fifteen (15) years, the greater portion of the world experiences an unprecedented change in the temperature of sea water. For instance, the ice covering the Arctic is dramatically vanishing; the glaciers and green ice cap are observed to be significantly melting; reduction in the volume of Antarctic ice sheets; unusual increase of sea level, and the seas are getting stormier; upsurge in the intensity, and frequency of rainfall due to precipitation variability; earlier occurrence

<footnote n="1">¹Department of Artificial Intelligence</footnote>

<footnote n="4">⁴Department of Information systems
University of Malaya</footnote>

<footnote n="50603">50603 Pantai Valley, Kuala Lumpur, Malaysia</footnote>

<footnote n="2">²Department of Information system
International Islamic University
Gombak, Kuala Lumpur, Malaysia</footnote>

<footnote n="3">³AMCS Research Center
Yogyakarta, Indonesia</footnote>

<footnote n="5">⁵Department of Computer Science
University of Technology Malaysia
Kampus Skudai, Jahor Baru, Malaysia</footnote>
of springtime, and high intensity of hurricane [1]. In addition, the world is experiencing an extraordinary warming. These changes experience throughout the globe are consequences of the climate change [2]. There is a growing report of numerous natural disasters triggered by climate change [3]. Therefore, the future knowledge of climate change is required for proper planning in order avoid the negative impact of natural disasters expect to be caused by the behavior of climate change. In the literature, several studies have been conducted using neural networks to predict climate change. For example, [4] proposed a neural network to build a model for predicting weather in Jordan. Goyal et al. [5] uses neural network to develop a downscaled model for predicting maximum and minimum temperature across 14 stations in Upper Thames River Basin Ontario, Canada. Holmberg et al. [6] proposed neural network to model the concentration of organic carbon, Nitrogen and phosphorus in a runoff stream water collected from Finland. The model is then used to predict future fluxes under climatic change. Tripathi [7] applied support vector machine (SVM) for statistical downscaling of precipitation. Subsequently, the SVM model is used to predict future climate changes in India. The results obtained from the SVM model was compared to the neural network and it was found that the SVM model performs better than the neural network. However, the studies are limited to the region where the research data were conducted.

The studies in the literature for predicting climate change using computational intelligent techniques, mainly focus on individual intelligent technique (IITs) whereas it is well known that hybrid intelligent technique is superior to the IITs when properly designed. Hybrid techniques capitalized on their strengths to eliminate their weaknesses to build a synergistic model for effective prediction. Haar wavelet decomposition (HWD) has the ability to project data into time scale domain, and perform multi scale analysis to unveil useful hidden patterns in the historical data [8]. The sensitivity of the relevance vector machine (RVM) to parameters is less compared to the SVM, and RVM is faster than SVM [9]. We propose to apply RVM to different scales coefficients produce by HWD to predict global climate change in order to reduce estimation bias that might be introduced by the HWD. Adaptive linear neural network (ALNN) will be used to ensemble the results produce by each RVMs to produce more accurate ensemble result than the results produce by individual RVMs. We intend to investigate whether the proposed hybrid of HWD, RVM and ALNN can yield superior novel results than the approaches in the literature.

# 2 Essential Rudiments

## 2.1 Haar Wavelet Decomposition

The HWD can be considered on the basis of the real line (IR) [10]: Let consider $L_2(\mathbb{R})$ to be the space of complex value function (f) on $\mathbb{R}$ in such a way that

$$ \|f\| = \left( \sqrt{\int_{-\infty}^{\infty} |f(x)|^2 dx} \right) < \infty. \qquad (1) $$
Eq. (1) is donated with scalar product, as such we have

$$f(f,g) = \int_{-\infty}^{\infty} f(x)g(\bar{x})dx. \qquad (2)$$

where the complex conjugate of $g(x)$ is given by $\overline{g(x)}$. If $(f,g)=0$ then $f,g \in L_2(IR)$ can be defined to each other, the functions can be defined as

$$\{\varphi_k, k \in \mathbb{Z}\}, \varphi_k \in L_2(IR). \qquad (3)$$

Eq. (3) is referred to as an orthonormal system if Eq. (4) is satisfied

$$\int \varphi_k(x) \varphi_j(\bar{x}) dx = \delta_{jk}. \qquad (4)$$

where $\delta_{jk}$ and $\{\varphi_k, k \in \mathbb{Z}\}$ are Kronecker delta, and orthonormal basis in a subspace $V$ of $L_2(IR)$ if any function $f \in V$ has a depiction

$$f(x) = \sum C_k \varphi_k(x). \qquad (5)$$

The coefficient satisfy $\sum_k |C_k|^2 < \infty$

$$Z = \{\dots, -1, 0, 1, \dots\}, \sum_k = \sum_{k=-\infty}^{\infty}, \int_{-\infty}^{\infty} \qquad (6)$$

The subspace $V_o$ of $L_2(IR)$ is defined as $V_o = \{f, f \in L_2(IR) : f$ which is constant on $(k, k+1], k \in \mathbb{Z}\}$, therefore

$$f \in V_o \Leftrightarrow f(x) = \sum C_{k\varphi}(x-k), \qquad (7)$$

where $\sum_k |C_k|^2 < \infty$, the series converges in Eqs. (8) and (9)

$$L_2(IR), \qquad (8)$$

$$\varphi(x) = I\{x \in (0,1]\} = \begin{cases} 1, & x \in (0,1] \\ 0, & x \in (0,1] \end{cases}, \qquad (9)$$

where $\varphi_{ok}(x) = \varphi(x-k), k \in \mathbb{Z}$ in which $\{\varphi_{ok}\}$ is the orthonormal basis in $V_o$, as such a new linear subspace of $L_2(IR)$ is defined as
$$V_1 = \{h(x) = f(2x) : f \in V_o\}, \qquad (10)$$

comprises of all functions in $L_2(IR)$ that are constant within the intervals of $\left(\frac{k}{2}, \frac{k+1}{2}\right]$, $k \in Z$. The $V_o \subset V_1$, orthonormal basis in $V_1$ is given by $\{\varphi_{1k}\}$ where $\varphi_{1k}(x) = \sqrt{2}\varphi(2x-k)$, $k \in Z$. This process continues and the space is generally defined as

$$V_j = \{h(x) = f(2^j x) : f \in V_o\}. \qquad (11)$$

The $V_j$ is a linear subspace of $L_2(IR)$ with the orthonormal basis in Eqs. (12) and (13)

$$\varphi_{jk}(x) = 2^{\frac{j}{2}} \varphi(2^j x - k), k \in Z. \qquad (12)$$

$$V_o \subset V_1 \subset \dots \subset V_j \subset \dots \qquad (13)$$

In a similar manner we define $V_j$ for $j < 0$, $j \in Z$ to get the inclusions

$$\dots \subset V_{-1} \subset V_o \subset V_1 \subset \dots \qquad (14)$$

The process is infinitely continued until the entire space $L_2(IR)$ is approximated. The orthogonal sum decomposition of $V_j$ is expressed as

$$L_2(IR) = V_o \bigoplus_{j=0}^{\infty} w_j \qquad (15)$$

where $w_j = V_{j+1} \oplus V_j$ which is the orthogonal complement of $V_j$ in $V_{j+1}$. Eq. (15) indicates that each $f \in L_2(IR)$ can be represented as series in the form

$$f(x) = \sum \alpha_{ok} \varphi_{ok}(x) + \sum_{j=0}^{\infty} \sum_{k} \beta_{jk} \Psi_{jk}(x) \qquad (16)$$

where $\alpha_{ok}$, $\beta_{jk}$, and $\Psi_{jk}$ are representation of coefficients of the expansion.

## 2.2 Adaptive Linear Neural Network

The ALNN can mathematically be defined as [11]:
$$f(x) = \phi\left(\sum_{i=1}^{m} w_i x_i + b\right), \qquad (17)$$

where $x_i(i=1,2,3,...,n)$, $f(x),b,w_i(i=1,2,3,...,n),m$, and $\phi(\cdot)$ are the input variables, output, bias, weights, number of input neurons, and activation function for computation in the ALNN neurons. The architecture of the ALNN comprised of only one layer. The activation function and learning algorithm are pure linear and Widrow-Hoff, respectively. The learning algorithm minimizes mean square error (MSE) to optimized weights and bias as defined in Eq. (18)

$$MSE = \frac{1}{N} \sum_{j=1}^{N} [e_j(x)]^2 = \frac{1}{2} \sum_{j=1}^{N} [T_j(x) - p_j(x)]^2, \qquad (18)$$

where $N,e_j(x),T_j(x)$, and $p_j(x)$ are the number of data points in the samples, error, target output, and network output. An adaptation of the weights and bias for multiple neurons can be defined as

$$w(x+1) = w(x) + 2\eta e(x)T^T(x). \qquad (19)$$

$$b(x+1) = b(x) + 2\eta e(x), \qquad (20)$$

where $\eta$ is the learning rate.

## 2.3 Relevance Vector Machine

The RVM is typically used for solving classification and regression problems. In this study, we present regression relevance vector machine algorithm for our problem is a regression. Let $\{X_i, t_i\}_{i=1}^N$ be input-datasetoutput pairs of dataset. Each desired output $t_i$ assumed a normal distribution with mean $y(x_i)$ and uniform variance $\delta^2$ of the noise $\in$ define as

$$p(t/x) = N(t/y(x), \delta^2) \qquad (21)$$

The desired output is also assumed joint normal distribution expressed as $N(\mu, \Sigma)$, where $\mu$ and $\Sigma$ are unknown variables which can be determined by the RVM [13].

# 3 Experiments

## 3.1 Dataset

The global annual land-ocean temperature (GLOT) is chosen as the indicator of the climate change behavior because it significantly determines climate change more than
other indicators such as atmospheric temperature, heat emanating from the sun, among others [14]. The GLOT data are provided in degrees Celsius (degree C) extracted from the National Aeronautic and Space Administration (NASA) website (www.nasa.gov), Goddard Institute for Space, for a period from 1880 to 2012 on a yearly basis. The GLOT data is a global data that is not restricted to a particular region, and were not normalized as the data are already provided in approximately equal proportion within -1 to 1.

## 3.2 The Propose Ensemble HWD-RVMALNN Framework

The propose ensemble HWD-RVMALNN paradigm is represented in the flowchart illustrated in Fig. 1. The major components of the entire process composed of the following stages: The original GLOT time series data are decomposed using HWD into five scale coefficient. For each of the scale coefficient, the RVM is employed as an estimation technique to model the decomposed scale coefficient and estimate for each of the five scale coefficient. The estimation results produce by each of the RVMs (HWD-RVM1, HWD-RVM2, HWD-RVM3, HWD-RVM4, and HWD-RVM5) is integrated by ALNN to generate an ensemble estimate of the GLOT which can be viewed as the estimation result of the original GLOT time series data. The optimal parameters of RVM were obtained through initial experimentation. The methodology can be summarized as a hybrid ensemble approach comprising of HWD (Decomposition) -RVM (estimate) -ALNN (ensemble) referred to HWD-RVMALNN. To verify the effectiveness of the our approach, we compared the proposed HWD-RVMALNN with the popular approaches of prediction such as Levenberg-Marquardt (LMQ), resilient backpropagation (RBP), Cyclical weight/bias rule (CWB), learning vector quantization (LVQ), and SVM. These algorithms are used as the benchmarks since they are the popular approaches in the literature for the estimation of climate change, readers can refer to [4-6] for details. The algorithms are also applied to estimate the GLOT for the purpose of comparing with the results generated by our proposal.

<figure><img src="image_2.png" /><figcaption>Fig. 1. The propose ensemble HWD-RVMALNN framework</figcaption></figure>
# 4 Results and Discussion

In this section, the simulated results are presented and discussed. The original GLOT signal is depicted in Figure 2. Decomposition of the GLOT time series data that significantly influence the fluctuation of climate is shown in Figure 3 displaying the multi resolution analysis that includes fifth approximation and five ($d_1, d_2, d_3, d_4$, and $d_5$) levels of details. The five levels of detail coefficients including amplitude of the scaling coefficient are shown in Figure 4. The decomposition contributed to an understanding of the time varying patterns in the GLOT data. Some useful information is revealed from the analysis, and interpretation of the GLOT data. The original GLOT data (Figure 2) and the extracted portion (Figure 3) is in degree Celsius.

<figure><img src="image_3.png" /><figcaption>Fig. 2. The original signal of the GLOT</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 3. Haar Wavelet Decomposition of the GLOT Signal</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Fig. 4. Original details coefficients of the GLOT signal</figcaption></figure>

The optimal parameters of RVM obtained after the initial experimentation were kernel function = radial basis, relevance vectors = 241, γ = 1.7, kernel length = 4.6, minimum delta scale = 0.0002, and maximum was 1×10⁴.

<table><caption>**Table 1.** Performance metrics of the algorithms compared with the proposed ensemble HWD-RVMALNN</caption><thead><tr><th>Algorithm</th><th>MSE</th><th>MAE</th><th>SSE</th><th>R<sup>2</sup></th><th>RT(sec.)</th></tr></thead><tbody><tr><td>LMQ</td><td>0.00413</td><td>0.0071</td><td>0.422</td><td>0.7047</td><td>2</td></tr><tr><td>CBW</td><td>0.00828</td><td>0.01923</td><td>0.51193</td><td>0.87268</td><td>932</td></tr><tr><td>RBP</td><td>0.015</td><td>0.0788</td><td>1.14</td><td>0.7932</td><td>3</td></tr><tr><td>SVM</td><td>0.002911</td><td>0.019884</td><td>0.195</td><td>0.88626</td><td>5</td></tr><tr><td>LVQ</td><td>0.091</td><td>0.234</td><td>8.07</td><td>0</td><td>2</td></tr><tr><td><strong>HWD-RVMALNN</strong></td><td><strong>0.00081</strong></td><td><strong>0.00721</strong></td><td><strong>0.00916</strong></td><td><strong>0.93087</strong></td><td><strong>0</strong></td></tr></tbody></table>

Mean absolute error (MAE), Sum of square error (SSE), Runtime (RT)

It is clear shown in Table 1 that the algorithms compared with our approach estimated GLOT with a good degree of accuracy as suggested by the simulated results. The SVM performs better than the LM, CBW, RBF, and LVQ in terms of R², MSE and MAE. The results corroborate with the study conducted by [7]. The possible reason for this performance exhibited by SVM could be attributed to the use of support vectors by the SVM due to its ability to improve performance. The worst performance exhibited by LVQ in terms of R² which is merely random could likely be caused by the non-zero differentiability of the LVQ. The runtime of LM and LVQ outperform SVM, CBW, and RBP in terms of computational speed. The poorest convergence
speed is observed to be exhibited by the CBW. The likely cause of this poor
convergence can best be attributed to the cyclical nature of the CBW to present inputs
to the network which might have caused the delay. An algorithm that converges to the
optimal solution within the shortest time is considered the most efficient. The propose
HWD-RVMALNN as reported in Table 1 based on the performance metrics was able
to improve on the performance of the approaches typically use in the estimation of
climate change in the literature. They propose an approach performs better than the
commonly used approaches in both accuracy and computational efficiency. The likely
reason for the performance of HWD-RVMALNN could be attributed to the fact that
the results of several experts are ensemble to produce superior output. Estimates of
GLOT obtained from simulating the propose ensemble HWD-RVMALNN on the
independent test dataset is plotted in Figure 5.

<figure><img src="image_6.png" /><figcaption>Fig. 5. Performance of the propose HWD-RVMALNN on the test dataset</figcaption></figure>

5 Conclusion

In this research, a hybrid of wavelet transform and computational intelligent
algorithms has been proposed for the estimation of climate change behavior.
Specifically HWD, RVM, and ALNN were hybridized to propose ensemble HWD-
RVMALNN. A series of comparative simulations were performed and it was found
that the propose HWD-RVMALNN performs better in both accuracy and
computational speed. The propose HWD-RVMALNN can provide better
understanding of the future knowledge of climate change behavior than the methods
already proposed in the literature. The model can assist policy makers in formulating
global response policy in tackling the negative impact espected to be cause by the
behavior of climate change. We intend to further this research by developing the real
life application of the propose HWD-RVMALNN model to help climate change
experts and policy makers in the decision making process.
**Acknowledgments.** This work is supported by University of Malaya High Impact Research Grant no vote UM.C/625/HIR/MOHE/SC/13/2 from Ministry of Higher Education Malaysia.

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