Construction of Generalized k-Bessel–Maitland Function with Its Certain Properties

The main motive of this study is to present a new class of a generalized k -Bessel–Maitland function by utilizing the k -gamma function and Pochhammer k -symbol. By this approach, we deduce a few analytical properties as usual differentiations and integral transforms (likewise, Laplace transform, Whittaker transform, beta transform, and so forth) for our presented k -Bessel–Maitland function. Also, the k -fractional integration and k -fractional differentiation of abovementioned k -Bessel–Maitland functions are also pointed out systematically.

Using Field Spectroradiometer to Estimate the Leaf N/P Ratio of Mixed Forest in a Karst Area of Southern China: A Combined Model to Overcome Overfitting

The ratio between nitrogen and phosphorus (N/P) in plant leaves has been widely used to assess the availability of nutrients. However, it is challenging to rapidly and accurately estimate the leaf N/P ratio, especially for mixed forest. In this study, we collected 301 samples from nine typical karst areas in Guangxi Province during the growing season of 2018 to 2020. We then utilized five models (partial least squares regression (PLSR), backpropagation neural network (BPNN), general regression neural network (GRNN), PLSR+BPNN, and PLSR+GRNN) to estimate the leaf N/P ratio of plants based on these samples. We also applied the fractional differentiation to extract additional information from the original spectra of each sample. The results showed that the average leaf N/P ratio of plants was 17.97. Plant growth was primarily limited by phosphorus in these karst areas. The sensitive spectra to estimate leaf N/P ratio had wavelengths ranging from 400–730 nm. The prediction capabilities of these five models can be ranked in descending order as PLSR+GRNN, PLSR+BPNN, PLSR, GRNN, and BPNN when considering both accuracy and robustness. The PLSR+GRNN model yielded high R2 and performance to deviation (RPD), and low root mean squared error (RMSE) with values of 0.91, 3.15, and 1.98, respectively, for the training test and 0.81, 2.25, and 2.46, respectively, for validation test. Compared with the PLSR model, both PLSR+BPNN and PLSR+GRNN models had higher accuracy and were more stable. Moreover, both PLSR+BPNN and PLSR+GRNN models overcame the issue of overfitting, which occurs when a single model is used to predict leaf N/P ratio. Therefore, both PLSR+BPNN and PLSR+GRNN models can be used to predict the leaf N/P ratio of plants in karst areas. Fractional differentiation is a promising spectral preprocessing technique that can improve the accuracy of models. We conclude that the leaf N/P ratio of mixed forest can be effectively estimated using combined models based on field spectroradiometer data in karst areas.

Fractional differentiation and its use in machine learning

This article covers the implementation of fractional (non-integer order) differentiation on real data of four datasets based on stock prices of main international stock indexes: WIG 20, S&P 500, DAX and Nikkei 225. This concept has been proposed by Lopez de Prado [5] to find the most appropriate balance between zero differentiation and fully differentiated time series. The aim is making time series stationary while keeping its memory and predictive power. In addition, this paper compares fractional and classical differentiation in terms of the effectiveness of artificial neural networks. Root mean square error (RMSE) and mean absolute error (MAE) are employed in this comparison. Our investigations have determined the conclusion that fractional differentiation plays an important role and leads to more accurate predictions in case of ANN.

Microwave and radio frequency fractional differentiation with a 49GHz Kerr soliton crystal microcomb source

We report a photonic radio frequency (RF) fractional differentiator based on an integrated Kerr micro-comb source. The micro-comb source has a free spectral range (FSR) of 49 GHz, generating a large number of comb lines that serve as a high-performance multi-wavelength source for the differentiator. By programming and shaping the comb lines according to calculated tap weights, arbitrary fractional orders ranging from 0.15 to 0.90 are achieved over a broad RF operation bandwidth of 15.49 GHz. We experimentally characterize the frequency-domain RF amplitude and phase response as well as the temporal response with a Gaussian pulse input. The experimental results show good agreement with theory, confirming the effectiveness of our approach towards high-performance fractional differentiators featuring broad processing bandwidth, high reconfigurability, and potentially reduced sized and cost.

Photonic RF and microwave fractional differentiation, integration, and Hilbert transforms with Kerr optical micro-combs

Integrated Kerr micro-combs are a powerful source of multiple wavelength channels for photonic radio frequency (RF) and microwave signal processing, particularly for transversal filter systems. They offer significant advantages featuring a compact device footprint, high versatility, large numbers of wavelengths, and wide Nyquist bands. We present our recent progress on photonic RF and microwave high bandwidth temporal signal processing based on Kerr micro-combs with comb spacings from 49GHz to 200GHz. We focus on integral and fractional Hilbert transforms, differentiators as well as integrators. The future potential of optical micro-combs for RF photonic applications in terms of functionality and ability to realize integrated solutions is also discussed.