Analysis of the model outputs

In order to investigate the impact of climate variability on malaria transmis-sion, we perform the following analysis on the model outputs.

7.3.4.1 Principal Component Analysis (PCA)

In this study, Principal Component Analysis (PCA) is used to analyse the data generated from the model. PCA is useful in identifying common modes of vari-ability between variables [83, 139, 161], and can reduce numerous number of inter-related variables to a few principal components that capture much of the variance of the original dataset [139]. PCA has been widely and success-fully used to help understand, interpret, and reconstruct large, multivariate datasets, both with spatial extent [197] and at single sites [160]. Here, PCA

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is applied to identify the meteorological variables that are coupled with the model outputs. To achieve this, Statistica software (Statsoft Inc., 2013) using the varimax rotation option to obtain a clear pattern of loadings is used for the analysis.

7.3.4.2 Wavelet Power Spectrum (WPS)

Wavelet analysis is a method of decomposing a time series into time-frequency space. This view offers interesting insights into the dominant modes of a time series and how those modes vary over time. In contrast to Fourier analysis, wavelet analysis highlights the study of signals whose spectra change with time.

In addition, the time-frequency analysis reveals further characteristics such as the periodic components with time progression [29, 152, 191]. The WPS also calculates the distribution of variance between frequency f and different time locations τ. In order to compare the WPS with classical spectral methods, the global wavelet spectrum is computed as the time average of the WPS for each frequency component [152]. For a better understanding of this method and analysis, see [30].

Here in this study, we introduce the basic approach of using wavelet anal-ysis to extract periodic components from the climate variables and our model outputs. The wavelet analysis investigates the time-scale decomposition of the signal by estimating its spectral characteristics as a function of time [191, 202].

This approach reveals how the different scales (periodic components) of the time series change over time as the wavelet function is stretched in time by varying its scale [43, 202]. We have considered here the continuous Morlet wavelet transform as the wavelet base function since it provides a good bal-ance between time and frequency localization, which is desirable for feature extraction purposes [67, 202]. The Morlet function is essentially a damped complex exponential, which can capture local (in time) cyclical fluctuations in the time series [202]. Although, the wavelet spectra are scale dependent and can produce distorted power spectra by underestimating short-period peaks

7.3. Material and methods 167

[98, 191], the problem can be corrected through normalising the power spec-tra by the corresponding scale, so that specspec-tral peaks can be compared across scales [98, 202].

7.3.4.3 Wavelet Cross-coherence Analysis (WCA)

Time-series analyses have been used to examine the dynamics of several disease epidemics, as it seemed to be the only substitute [28, 77], they are more useful in short-term analyses [11, 50]. They are typically noisy and complex [30]. For these reasons, and in order to qualitatively explore the correspondence of the wavelet spectra of rainfall and temperature on malaria incidence, we examine their cross-coherence spectrum as shown in Fig. 7.10 and Fig. 7.11 using Wavelet Cross-coherence Analysis (WCA).

Wavelet Cross-coherence Analysis is a method for analyzing the coher-ence and phase lag between two time series as a function of both time and frequency [31]. As given in Fourier analysis, the univariate WPS can be ex-tended to quantify statistical relationships between two time series x(t) and y(t) by computing the wavelet coherence

R_x,y(f, τ) = | hW_x,y(f, τ)i |

| hW_y(f, τ)i |^1/2 . | hW_y(f, τ)i |^1/2,

where hi indicates smoothing in both time and frequency; Wx(f, τ) represents the wavelet transform of series x(t); Wy(f, τ) is the wavelet transform of series y(t); and Wx,y(f, τ) = Wx(f, τ).Wy^∗(f, τ) is the cross WPS. The wavelet coher-ence provides local information about the extend to which two non-stationary signals x(t) and y(t), are linearly correlated at a particular frequency (or pe-riod). Rx,y(f, τ) is equal to 1 when there is a perfect linear relationship at a particular time and frequency between the two signals [30].

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Jan 020 Jun02 Jan 03 Jun 03 Jan 04 Jun 04 Dec 04

200 400 600 800 1000 1200

Time (Months)

Malaria cases

Modelled Cases ^{r = 0.67}

Figure 7.3: The modelled and reported cases of malaria over KwaZulu-Natal province, South Africa for 2002-2004.

7.4 Results and discussion

7.4.1 Model validation

Comparing the model output with observed data, our results produce a fairly similar curve with the observed data as shown in Fig. 7.3. However, we do notice some discrepancies between the simulated and observed data. For instance, in January 2002, our model estimates almost 600 infected humans, whereas only 300 cases were observed. This is one of the limitations of the model, it could also be as a result of some control measures implemented at that particular period which are not considered in our model. Also, our model underestimates malaria incidence as noticed in June 2003, which might be the lack of other factors affecting malaria in our model. However, the model output further indicates along with observed data that malaria transmission over the province is seasonal, and that malaria incidence in the province was higher in 2004 than in 2003. These results are consistent with the previous study of Craig et al [41].

7.4. Results and discussion 169