Normalized Gompertz wavelets and their application

Question

1. What are the applications of the Gompertz function in modeling Covid-19 cases?

2. What are the basic properties of wavelets?

3. What is the Gompertz mother wavelet?

4. What are Bernoulli numbers?

Accepted Answer

The Gompertz function has been used to model the first waves of Covid-19 cases in 11 selected countries, including Japan, USA, Russia, Brazil, China, Italy, Indonesia, Spain, South Korea, UK, and Sweden. Researchers Ohnishi et al [12] demonstrated that the Gompertz function can effectively describe the spread of Covid-19 cases. Additionally, Dhahbi et al [2] utilized the Gompertz model to analyze the first wave of cases in Saudi Arabia. Furthermore, Kundu et al [8] proposed an automated COVID-19 detection system based on convolution neural networks using the Gompertz function. These applications highlight the versatility and effectiveness of the Gompertz function in modeling the spread of infectious diseases like Covid-19.

Accepted Answer

Wavelets are functions that satisfy the admissibility condition, which ensures their Fourier transform is well-defined. They are continuous and have a zero value at x=0. Conditions - ps(x)dx = 0 and (6) are equivalent in practice. Wavelets are square-integrable, with a norm defined as ||ps|| = (|ps(x)|^2)^(1/2). By dilating and translating a mother wavelet, a family of children wavelets can be generated. The continuous wavelet transform (CWT) uses these wavelets to analyze functions in L2 space.

Accepted Answer

The Gompertz mother wavelet, denoted as ps2(t), is defined by the equation EQUATION. It satisfies the conditions ||ps2|| = 1 and ps2(t) ∈ L1(R) L2(R). The admissibility condition (6) is fulfilled, and its Fourier transform can be expressed in terms of the Riemann Zeta function z(z). The Gompertz mother wavelet can be generated by dilating and translating ps2(t) using the formula EQUATION.

Accepted Answer

Bernoulli numbers are rational numbers that appear in various mathematical relations. They have an exponential generating function given by B0 + B1z + B2z^2/2! + * * * = z e^z - 1, |z| < 2. Bernoulli numbers vanish for all odd n >= 3. They have properties such as m-1k=1k+1n=1n+1j, k=1 1k 2n = (-1)^n+1 2^2n-1 p^2n (2n)!, and B2n for n = 1, 2, ... The first few nonzero Bernoulli numbers are B0 = 1, B1 = -1/2, B2 = 1/6, B4 = -1/30, B6 = 1/42, B8 = -1/30, B10 = 5/66. Bernoulli numbers are well described in Duren's book and have applications in various mathematical fields, including wavelet analysis.

Accepted Answer

Detecting points in a time series, where the sequence of second differences takes a value close to zero or a maximum, is significant for identifying zeros of the second or third derivative of the Gompertz function. This helps in understanding the behavior and patterns of the time series data. By detecting these points, researchers can analyze the sequence of second differences and identify critical points in the data, which can provide valuable insights into the underlying trends and patterns. This information can be used for various applications, such as forecasting, anomaly detection, and trend analysis. Overall, detecting points in a time series is crucial for gaining a deeper understanding of the data and making informed decisions based on the analysis.

Accepted Answer

The parameters for the first Gompertz wave are b = 25 and a = 8, while for the second Gompertz wave, b = 200 and a = 20. These parameters can be determined by analyzing the central second differences and the scalogram produced by the Continuous Wavelet Transform (CWT). The largest value of the Index in the scalogram indicates the points where the saturation levels can be calculated using the provided formulas. However, it's important to note that the calculated saturation levels using the second differences may not be exactly equal to the theoretical ones due to the nature of the analysis.

Accepted Answer

Dhabi et al. modeled the extensive first wave of Covid-19 cases in Saudi Arabia using the Gompertz function. They considered 264 days starting from March 12, 2020, until November 30, 2020. The time series of the total number of reported infections was smoothed with a 7-day moving average and denoted by (y n ). The scalogram showed a maximum Index of the large wave of cases, consistent with the estimates given by Dhabi et al. The Gompertz function was found to be more efficient for modeling than the logistic curve, as confirmed by the CWT analysis and scalograms.

Accepted Answer

The utility of second-order Gompertz wavelets is demonstrated in the theoretical situation where the signal consists of two exact Gompertz functions. They were used to study the spread of the Covid-19 pandemic in Saudi Arabia. In further work, they plan to apply these wavelets to real-world data from various fields such as economics, finance, or biology. Additionally, they could generalize basic S-shaped curves to define corresponding wavelets for efficient computation of successive derivatives for a large class of S-shaped functions.

Normalized Gompertz wavelets and their application

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Most frequently asked questions

1. What are the applications of the Gompertz function in modeling Covid-19 cases?

2. What are the basic properties of wavelets?

3. What is the Gompertz mother wavelet?

4. What are Bernoulli numbers?

5. What is the significance of detecting points in a time series?

6. What are the parameters for the first and second Gompertz waves?

7. How did Dhabi et al. model Covid-19 cases in Saudi Arabia?

8. What is the utility of second-order Gompertz wavelets?

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