The role of the parameter is also twofold as standing for scaling requirements and partially responsible for the final linear combination of basic waves. Now, we present the solution of the model equation 1 and its direction to the proposed application in dengue incidence. Proposition 3. The solution of 1 is given by , and it has the following two cases for. Case 1. Case 2. By applying the Laplace transform to 1 with the convolution property, we obtain , which provides.
Since , we arrive at. If , then we reach the solution in Case 1 and if , then we have Case 2. Next, Proposition 4 presents the inverse process of obtaining that ensures the analytical existence of for a given. Proposition 4. The result is immediate by the earlier expression:. If the Laplace inverse is not straight forward, one would try Bromwich contour integrals [ 8 , 9 ].
Definition 5. We structure the superposition upon as , where is the basis function corresponding to the influencing function for each case of. We call this as an overall transmission potential. Approximations in finite context can be obtained by , where number of harmonics are used. This finally leads to approximate incidence data in the least square sense. Since natural oscillations can be expected in epidemiological phenomena, we test with Case 1 of Proposition 3.
Weekly dengue incidence data from to reported in the Colombo Municipal area, Sri Lanka [ 10 ] are used to extract basis functions and subsequently the influencing functions.
This municipal area is highly vulnerable to dengue transmission due to the densely urbanized environment. Suppose a trigonometric polynomial of order , is taken to approximate data in a discrete sense. First, we determine the Fourier harmonics , and the intercept term i.
Here, refers to the data value at a point for. Next, we estimate initial guesses for each set of parameters of infused in. These estimations are obtained by aligning with sequential harmonics separately.
It is fulfilled by a trial and error method with a reasonable tolerance on the squared deviation between each and its Fourier counterpart. Here, near-periodic behaviors are attributed into while an exponential effect finally plays the role of fine-tuning the overall fit.
The following result guarantees that no complete restriction is employed on the exponential effect of while forcing the removal of the exponential effect in the space of. Proposition 6. The result is straightforward as we have. Next, we initialize the parameters as described above and minimize that represents the deviation between data and superposition output. By using the resultant parameter values, we can determine all and.
It is obvious that both the Fourier fit and the model fit yield better approximations of data when the number of series increases.
For a given number of series, our model fit may reach data much closer than that of the Fourier fit by the effect of in. It is assisted by the boundedness of and the possibility of any real value for. Note that the terms involving in the Fourier fit are sine and cosine terms allowing only oscillations tolerated by respective coefficients.
However, in the model fit, one may see the adaptability of coefficients as per the effect of. Options on are indirectly subjected to the condition on Proposition 3. Notwithstanding, sophisticated searching abilities in the software may reduce its burden in computational trials.
In Figure 1 , Fourier fit and model fit are illustrated. We add the intercept term into to compromise with actual data. At the trial and error stage, the tolerance. Graphs in Figure 1 visualize the reliability of the fitted curves for different , and Table 1 contains the sum of squared deviations.
It shows that fits well compared to its corresponding for each. Therefore, the basis function can further be used as a reliable intermediate tool to extract. One can observe that the model fit and the Fourier fit mainly differ at the extremes of the data set.
It is an optimistic rectification over Fourier harmonics to achieve a better fit via the model 1. This hints the effect of in as discussed in Section 2. Figure 2 further illustrates such an effect at an extreme. Here, the red curve stands for and the blue curve contains an exponential effect.
Observe that the additional fluctuations of the blue curve at the latter stage allow for reaching attributes in data, which cannot be extracted only via sinusoidal curves. Thus, the convergence of series at extremes is more equipped with exponential orders. Since improvements are expected upon near-periodic waves, it is worthwhile to see the deviations in trigonometric terms. Figure 3 depicts the relevant cases in each harmonic for.
We illustrate sine terms and cosine terms in the Fourier fit and their counterparts in the model fit, excluding the exponential effect.
Besides, harmonics are also presented without Figure 3 ,. Type Your Search Here. Identification of harmonic sources Protecting the equipments from power related problems Safety against loss and interruptions. Enquire Now. Feichtinger Copyright: Hard cover Soft cover eBook. Casey Kasso A. Okoudjou Michael Robinson Brian M. Sadler Copyright: Hard cover Soft cover eBook. View all book titles. A Fourier series and a Fourier integral relate each function f x and its spectrum for a Fourier integral a set of harmonics is the spectrum.
In some approaches for instance, in turbulence theory the study of functions by their spectra is used. The notion of Fourier transformation can be easily generalized into the functions of many variables.
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