Spline Interpolation Technique Research Articles

Numerical Algorithms for Approximation of Fractional Integrals and Derivatives Based on Quintic Spline Interpolation

Numerical algorithms for calculating the left- and right-sided Riemann–Liouville fractional integrals and the left- and right-sided fractional derivatives in the Caputo sense using spline interpolation techniques are derived. The spline of the fifth degree (the so-called quintic spline) is mainly taken into account, but the linear and cubic splines are also considered to compare the quality of the developed method and numerical calculations. The estimation of errors for the derived approximation algorithms is presented. Examples of the numerical evaluation of the fractional integrals and derivatives are executed using 128-bit floating-point numbers and arithmetic routines. For each derived algorithm, the experimental orders of convergence are calculated. Also, an illustrative computational example showing the action of the considered fractional operators on the symmetric function in the interval is presented.

Investigation of chemical change in water quality and spatial modeling with GIS mapping: A case study from Beşgöz Lake in Konya Province of Türkiye

In this study, Beşgöz lake in Sarayönü district of Konya province, located in the Central Anatolia Region of Türkiye, was chosen as the study area. Rehabilitation work was carried out in 2020 with the resources of the State Hydraulic Works. The average depth of Beşgöz Lake was increased from 4 m to 8.5 m, and its surface area was increased from 86.000 m2 to 230.000 m2. Beşgöz Lake, located in Sarayönü district, provides water from underground sources at the bottom of the lake. These underground sources come from the gaps in the limestone in the lower parts of the lake. When looking at the geological map, it is thought that there is a connection between Beyşehir Lake and Beşgöz Lake because the land has a calcareous structure. The changes in the water quality parameters of Beşgöz Lake in different periods were examined and spatially modeled using spline interpolation techniques with Arc GIS 10.3.1 program, one of the Geography Information Systems (GIS) software. In order to determine some water quality parameters of Beşgöz Lake, water samples were taken from eight different points representing the lake in 4 different periods (Spring, Summer, Autumn, Winter). Seasonally in the water of Beşgöz Lake, water quality parameters such as pH, Electrical Conductivity (EC), Temperature, Chloride (Cl), Nitrite (NO2), Nitrate (NO3), Sulphate (SO4) Chemical Oxygen Demand (COD), Suspended Solids (TSS), Boron (B) were investigated. As a result of the research, it was found that pH values varied between 7.77–8.42, EC: 268–470 µS/cm, water temperature varied between 6.8–20 °C, Cl varied between 0.139 - 3.9 mg/L, NO2 varied between < 0 00–0.33 mg/L, NO3 varied between < 0.00 - 4.13 mg/L, SO4 varied between 0.057 − 18.4 mg/L, COD varied between 5–32, TSS: 0.60 - 31.8 and B varied between 0.05–0.093 mg/L. As a result of the study, it was seen that the water quality of Beşgöz Lake, which is the subject of the research, is class I and potable water quality.

Peat depth mapping using grid interpolation from terrestrial peat drilling points data: A case study in Zamrud National Park

Zamrud National Park is the peak of the peat dome in the Kampar Peninsula. Peat depth data is needed as a baseline to maintain the balance of the water reservoir and carbon stock. Currently, available peat depth data is predictive photogrammetric data. Soil survey activities need to be carried out to provide accurate and up-to-date data. This research aims to map the peat depth using the best grid interpolation technique. Peat depth measurements were carried out terrestrially, while peat depth mapping used inverse distance weighted, kriging, and spline interpolation techniques. Drilling observation points are arranged based on a system of longitudinal transects and transverse transects in a northeast-southwest direction. The distance between longitudinal transects is 500 m and between transects is 1,000 m. The results showed that the Zamrud National Park area has diverse peat depths ranging from 5.3 to 16.7 m. The Kriging interpolation technique gives the best mapping accuracy results. The standard deviation value is 1.9 is smaller, the accuracy value is 98.82% higher, and the thickness of the dominating peat is 8.4-9.4 m for 7,093.47 ha (24.42% of the total park area).

Comparative Study of Geospatial Techniques for Interpolating Groundwater Quality Data in Agricultural Areas of Punjab, Pakistan

Groundwater Arsenic (As) data are often sparse and location-specific, making them insufficient to represent the heterogeneity in groundwater quality status at unsampled locations. Interpolation techniques have been used to map groundwater As data at unsampled locations. However, the results obtained from these techniques are affected by various inherent and external factors, which lead to uncertainties in the interpolated data. This study was designed to determine the best technique to interpolate groundwater As data. We selected ten interpolation techniques to predict the As concentration in the groundwater resources of Punjab, Pakistan. Two external factors, the spatial extent of the study area and data density, were considered to assess their impact on the performance of interpolation techniques. Our results show that the Inverse Distance Weighting (IDW) and Spline interpolation techniques demonstrate the highest accuracy with the lowest RMSE (13.5 ppb and 16.7 ppb) and MAE (87.8 ppb and 89.5 ppb), respectively, while the Natural Neighbor technique shows the lowest accuracy with the highest RMSE (2508.7 ppb) and MAE (712.1 ppb) to interpolate groundwater As data. When the study area’s extent was modified, IDW showed the best performance, with errors within ±1.5 ppb for 95% of the wells across the study area. While data density has a positive correlation with interpolation accuracy among all techniques, the IDW remained the best method for interpolation. It is therefore concluded that IDW should be used to interpolate groundwater quality data when observed data are sparse and randomly distributed. The utilization of IDW can be useful for As monitoring and management in groundwater resources.

Calibration of the Ångström–Prescott Model for Accurately Estimating Solar Radiation Spatial Distribution in Areas with Few Global Solar Radiation Stations: A Case Study of the China Tropical Zone

The Ångström–Prescott formula is commonly used in climatological calculation methods of solar radiation simulation. Aiming at the characteristics of a vast area, few meteorological stations, and uneven distribution in the tropical regions of China, in order to obtain the optimal parameters of the global solar radiation calculation model, this study proposes a suitable monthly global solar radiation model based on the single-station approach and the between-groups linkage of the A–P model, which utilizes monthly measured meteorological data from 80 meteorological stations spanning the period from 1996 to 2016 in the tropical zone of China, considering the similarity in changes of monthly sunshine percentage between stations. The applicability and accuracy of the correction parameters (a and b coefficients) were tested and evaluated, and then the modified parameters were extended to conventional meteorological stations through Thiessen polygons. Finally, the spatial distribution of solar radiation in the tropical region of China was simulated by kriging, IDW, and spline interpolation techniques. The results show the following: (1) The single-station model exhibited the highest accuracy in simulating the average annual global solar radiation, followed by the model based on the between-groups linkage. After optimizing the a and b coefficients, the simulation accuracy of the average annual global solar radiation increased by 5.3%, 8.1%, and 4.4% for the whole year, dry season, and wet season, respectively. (2) Through cross-validation, the most suitable spatial interpolation methods for the whole year, dry season, and wet season in the tropical zone of China were IDW, Kriging, and Spline, respectively. This research has positive implications for improving the accuracy of solar radiation prediction and guiding regional agricultural production.

Numerical Approximations of the Riemann–Liouville and Riesz Fractional Integrals

In this paper, the numerical algorithms for calculating the values of the left- and right-sided Riemann–Liouville fractional integrals and the Riesz fractional integral using spline interpolation techniques are derived. The linear, quadratic and three variants of cubic splines are taken into account. The estimation of errors using analytical methods are derived. We show four examples of numerical evaluation of the mentioned fractional integrals and determine the experimental rate of convergence for each derived algorithm. The high-precision calculations are executed using the 128-bit floating-point numbers and arithmetic routines.

Look Up Table Method for Five Hole Probe Data Reduction

A lookup method for extracting flow field information from five hole probe measurements is presented. The look up table method consists of two step interpolation. In the first step, the calibration space is divided into a large number of small intervals of yaw and pitch coefficients (0.1, 0.05 or 0.01) and the required quantities are interpolated using local cubic spline interpolation technique. The interpolated values are stored in a large square matrix. In the second step, the measured values of yaw and pitch coefficients are used to determine yaw and pitch angles by linear interpolation. Same procedure is used to determine total and static pressure coefficients. Additional data taken during the calibration is used for validating lookup table method and spline interpolation method. The errors from the look up table method, particularly with the interval of 0.01 are about the same or lower than those obtained from the spline interpolation method. The computational times for lookup table method are found to be lower than that for the spline interpolation method. The lookup table method seems to be attractive when a large amount of five hole probe data is to be processed and in cases where online data processing is required.

State of Health Prediction for Lithium-Ion Batteries through Curve Compression and CatBoost

This study proposes a novel approach for predicting the State of Health (SoH) and Remaining Useful Life (RUL) of lithium-ion batteries. The low accuracy of SoH and RUL is due to the challenges of establishing effective feature engineering for battery attributes. To address this issue, a SoH and RUL prediction model based on curve compression and CatBoost is proposed. Firstly, an improved threshold selection method based on curvature analysis is introduced to enhance the compression performance of battery attributes under different cycles. Secondly, to ensure that the extracted feature sequences have the same length, spline interpolation and local anomaly factor detection techniques are utilized to fill or eliminate feature points for feature length normalization. Finally, a dynamic time regularization algorithm is applied to calculate the shortest distance between the feature sequence and the original curve to determine the optimal feature length for input into the CatBoost prediction model. The experimental results demonstrate that the proposed approach outperforms other prediction models in the research object dataset, achieving R2 values higher than 0.98 and MSE values around 1 × 10−5. The proposed approach also achieves better prediction results in the validation object dataset, indicating its strong generalization capability. Additionally, the proposed model shows significant robustness by accurately predicting SoH and RUL under noisy environmental conditions. Overall, the proposed model shows significant potential to accurately predict SoH and RUL by efficiently addressing the challenges associated with feature engineering for battery attributes, reducing the impact of background noise on prediction results, and exhibiting strong robustness.

Optimal Trajectory Planning for Manipulators with Efficiency and Smoothness Constraint

Path planning to generate an appropriate time sequence of positions for a complex trajectory is an open challenge in robotics. This paper proposes an optimization method with the integration of an improved ant colony algorithm and a high-order spline interpolation technique. The optimization process can be modelled as the travelling salesman problem. The greatest features of this method include: (1) automatic generation for complex trajectory and a new idea of selecting the nearest start point instead of using the traditional way of human operation; (2) an optimized motion sequence of the manipulator with the shortest length of the free-load path improves efficiency by nearly 65% and (3) trajectories both in Cartesian space and joint space are interpolated with good smoothness to reduce shocks and vibrations. Simulations and experiments are conducted to demonstrate the good properties of this method.

Review of solution Fractal Differential Equations via the Riemann-Liouville

In this paper, the definitions, concepts, and basics of linear and nonlinear fractal differential equations of both homogeneous and heterogeneous types were studied, and some methods for solving fractal differential equations were presented with an explanation of each method and how to solve the equation in it An example was taken for each method that was presented to show how to solve in this way. Among the methods that were presented are (the special method, the Laplace transform Method, the Inverse Fractional Shehu Transform Method, The Power Series Method, The generalized Mittag-Leffler method, Spline Interpolation Techniques for Approximating the Solution of Fractional Differential Equations.

Prediction of Complex Stock Market Data Using an Improved Hybrid EMD-LSTM Model

Because of the complexity, nonlinearity, and volatility, stock market forecasting is either highly difficult or yields very unsatisfactory outcomes when utilizing traditional time series or machine learning techniques. To cope with this problem and improve the complex stock market’s prediction accuracy, we propose a new hybrid novel method that is based on a new version of EMD and a deep learning technique known as long-short memory (LSTM) network. The forecasting precision of the proposed hybrid ensemble method is evaluated using the KSE-100 index of the Pakistan Stock Exchange. Using a new version of EMD that uses the Akima spline interpolation technique instead of cubic spline interpolation, the noisy stock data are first divided into multiple components technically known as intrinsic mode functions (IMFs) varying from high to low frequency and a single monotone residue. The highly correlated sub-components are then used to build the LSTM network. By comparing the proposed hybrid model with a single LSTM and other ensemble models such as the support vector machine (SVM), Random Forest, and Decision Tree, its prediction performance is thoroughly evaluated. Three alternative statistical metrics, namely root means square error (RMSE), mean absolute error (MAE) and mean absolute percentage error (MAPE), are used to compare the aforementioned techniques. The empirical results show that the suggested hybrid Akima-EMD-LSTM model beats all other models taken into consideration for this study and is therefore recommended as an effective model for the prediction of non-stationary and nonlinear complex financial time series data.

A New Approach to Empirical Mode Decomposition Based on Akima Spline Interpolation Technique

The objective of this research work is to extend the scope of empirical mode decomposition (EMD) algorithm, as an efficient tool to decompose the nonlinear and non-stationary time series. For EMD to be widely applicable, the extension utilizes both clean and noisy data sets. When constructing upper and lower envelopes, the proposed algorithm utilizes the Akima spline interpolation technique rather than a cubic spline. The proposed EMD is called Akima-EMD, which is used to identify non-informative fluctuations in the signal, such as noise, outliers, and ultra-high frequency components, and to breakdown the clean and chaotic data into various components avoiding distortion. It has been shown through the synthetic as well as real-world time series data analysis that the proposed method successfully extracts noise in the form of the first IMF from the data.

Galactic Wormhole under Lovelock Gravity

We explore wormhole geometry in spiral galaxies under the third order Lovelock gravity. Using the cubic spline interpolation technique, we find the rotational velocity of test particles in the halo region of our spiral galaxy from observed values of radial distances and rotational velocities. Taking this value of the rotational velocity, we are able to show that it is possible to present a mathematical model regarding viable existence of wormholes in the galactic halo region of the Milky Way under the Lovelock gravity. A very important result that we obtain from the present investigation is that galactic wormhole in the halo region can exist with normal matter as well as exotic matter.

ASSESSMENT OF THE SPLINE INTERPOLATION METHOD'S PERFORMANCE IN PREDICTING AND MAPPING THE PHOSPHATE OVER THE COASTAL WATERS OF PULAU TUBA, LANGKAWI, KEDAH

To date, there has been limited knowledge on the applicability of the spline interpolation method to estimate coastal water quality parameters. This study aims to assess the performance of the regularised and tension spline interpolation techniques for predicting and mapping the phosphate concentration in the surface water of Pulau Tuba, Langkawi, Kedah. Sampling points were set up randomly along the coastal water of Pulau Tuba in November 2018. Samples were obtained using the Niskin Water Sampler at 1 meter below the surface and immediately transported into the marine technology laboratory. Phosphate levels were determined using an Ultraviolet Visible spectrophotometer and the ascorbic acid technique. The Geographic Positioning System (GPS) was used to record the geolocation of each sampling point. The training set (50%) and the testing set (50%) were chosen randomly based on 20 sampling points. The regularized and tension spline interpolation methods were used for this study. Evaluation of prediction of models was carried out using the Root Mean Square Error (RMSE) statistics. The study found that the tension spline interpolation method better predicted phosphate concentration. The Root Mean Square Error (RMSE) was reported at 0.106 and 0.094 for the regularized and tension spline interpolation methods, respectively. Concerned parties can utilize the study's findings as guidelines for monitoring, managing, and developing strategic strategies for the long-term development of the coastal waters of Pulau Tuba, Langkawi, Kedah.

The Comparison of Spatial Models in Peak Ground Acceleration (PGA) Study

This study was conducted to compare the performance of three different spatial analysis models: Inverse Distance Weighted (IDW), Ordinary Kriging,­­ and Regularized Spline interpolation technique to determine the best fit model representing Peak Ground Acceleration (PGA) in West Java Province, Indonesia. The three models are commonly used in spatial visualization, but have different calculation methods. The calculations were performed using available formulas while the spatial modeling was conducted using the algorithms in GIS software. Meanwhile, the accuracy of the spatial model and factual calculation was determined through the Root Mean Square Error (RMSE). The results showed differences for both spatial distribution and maximum and minimum values for each model. However, IDW was observed to be the model which approaches the factual value of the PGA calculation as indicated by its RMSE value of 0.772352 in comparison with the 7.169879 (Ordinary Kriging) and 1.140802 (Regularized Spline).

Estimation Algorithm for a Hybrid PDE–ODE Model Inspired by Immunocompetent Cancer-on-Chip Experiment

The present work is motivated by the development of a mathematical model mimicking the mechanisms observed in lab-on-chip experiments, made to reproduce on microfluidic chips the in vivo reality. Here we consider the Cancer-on-Chip experiment where tumor cells are treated with chemotherapy drug and secrete chemical signals in the environment attracting multiple immune cell species. The in silico model here proposed goes towards the construction of a “digital twin” of the experimental immune cells in the chip environment to better understand the complex mechanisms of immunosurveillance. To this aim, we develop a tumor-immune microfluidic hybrid PDE–ODE model to describe the concentration of chemicals in the Cancer-on-Chip environment and immune cells migration. The development of a trustable simulation algorithm, able to reproduce the immunocompetent dynamics observed in the chip, requires an efficient tool for the calibration of the model parameters. In this respect, the present paper represents a first methodological work to test the feasibility and the soundness of the calibration technique here proposed, based on a multidimensional spline interpolation technique for the time-varying velocity field surfaces obtained from cell trajectories.

Development of high spatial resolution weather data using daily meteorological observations over Indian region

Finer spatial resolution interpolated weather data is essential to enable utilization of satellite-based images in studies related to crop growth dynamics, etc. Satellite data are available daily at 1 × 1 km or at the most within 5 × 5 km grid. To make the weather data timely available at the same spatial scale, the procedure has been developed to generate the spatially interpolated weather data product over India. Daily weather data (minimum &amp; maximum temperature and rainfall) available at point scale on India Meteorology Department web site have been used in this study. A semi-automated user interactive Graphical User Interface (GUI) has been developed which quality checks the temperature data sets by filling the missing data sets as well as providing a platform to correct erroneous data which have been identified using statistical methods taking spatial as well as temporal incompatibility into account. Daily spatially interpolated product is generated in image form using thin plate spline interpolation technique that uses the quality checked weather data as well as elevation information from CARTODEM data in order to account for effect of elevation on temperature. The validation was performed using “Jack-knife testing method” for three different seasons i.e., monsoon, summer and winter. The mean absolute errors for decadal averaged products were found to vary within 1.2-1.5 °C for maximum temperature, 1.1-1.7 °C for minimum temperature, 1.0-7.0 mm for rainfall considering all seasons with higher error observed in monsoon for maximum temperature and rainfall and in winter for minimum temperature. It was found that errors were close to 1 °C for stations with elevation less than 550 m whereas in central portion of India, mean absolute errors were found to be less than 1 °C.

Influence of Interface trap distributions over the device characteristics of AlGaN/GaN/AlInN MOS‐HEMT using Cubic Spline Interpolation technique

AbstractTraps present at the interface of oxide and barrier layers influence the device's DC and RF parameters. An investigation is carried out on the distribution of interface traps present in the AlGaN/GaN/AlInN Metal Oxide Semiconductor‐High Electron Mobility Transistor (MOS‐HEMT) structures. By incorporating Hafnium Oxide (HfO2) as a dielectric layer, the DC and RF parameters like current drive and Power‐Added Efficiency (PAE) for the device are evaluated. The Capacitance‐Voltage (C–V) curves showed a shift in its values upon increase in trap density, which attributes to the position of fermi level once the threshold voltage is reached. A theoretical estimation of interface trap densities revealed distributions in the magnitude of 9.52 × 1012 cm−2 eV−1 near the mid‐gap and around 3.4 × 1013 cm−2 eV−1 at the Conduction Band (CB) edge. Carrier concentration at the Two‐Dimensional Electron Gas (2DEG) influences the distribution of traps at the interface, which in turn affects the on‐resistance of the device. Cubic Spline Interpolation (CSI) technique is employed here to model the device parameters precisely. The higk‐K dielectric material causes a high cut‐off of 201 GHz for the MOS‐HEMT. There is about 42% improvement in the PAE compared to the conventional MOS‐HEMT device. The comparably low trap density values (1012 cm−2 eV−1) than the conventional ones, prove the high quality gate passivation features of the HfO2/AlGaN interface and could serve as a potential candidate for high power and microwave applications.

Comparison of Steganography Using the Discrete Cosine Transform Method on Image Based Bilinear, Nearest Neighbor and Spline Interpolation

The research was conducted in the field of steganography. Discrete Cosine Transform (DCT) is a method used in the insertion technique. The results of steganography have problems if they look blurry, have low levels of similarity and high error values. One way to solve this problem is by proposing image interpolation. The interpolation method consists of various kinds and gives each other advantages. This study intends to compare three kinds of interpolation techniques to find the best one. The three interpolation techniques are bilinear, nearest neighbor, and spline. The method used in this research is experimental. Images with extension formats * .tif, * .png, and * .bmp with dimensions of 512x512 px are interpolated by scaling 1.5, 2, and 4. The results of the interpolation process are used to insert messages in * .txt format of 157 bytes with discrete cosines transform (DCT). The image quality of the message insertion is measured by the MSE and PSNR values. The result of the message insertion test shows that the value of the image quality is directly proportional, meaning that if the condition of the message size is fixed and the cover dimensions are greater, the MSE value will be smaller and the PSNR value will be greater. Images with * .tif and * .bmp extension formats have good stability, * .png images vary in size. The smallest error value test results were obtained in the spline interpolation technique and this method when compared to the other two techniques had the lowest average MSE value of 8.221 and the PSNR value of 40,301 dB.

A system for detection of moving caption text in videos: a news use case

Extraction of news text captions aims at a digital understanding of what is happening in a specific region during a certain period that helps in better communication between different nations because we can easily translate the plain text from one language to another. Moving text captions causes blurry effects that are a significant cause of text quality impairments in the news channels. Most of the existing text caption detection models do not address this problem in a way that captures the different dynamic motion of captions, gathers a full news story among several frames in the sequence, resolves the blurring effect of text motion, offers a language-independent model, or provides it as an end-to-end solution for the community to use. We process the frames coming in sequence and extract edge features using either the Hough transform or our color-based technique. We verify text existence using a Convolutional Neural Network (CNN) text detection pre-trained model. We analyze the caption motion status using hybrid pre-trained Recurrent Neural Network (RNN) of Long Short-Term Memory (LSTM) type model and correlation-based model. In case the motion is determined to be horizontal rotation, there are two problems. First, it means that text keeps rotating with no stop resulting in a high blurring effect that affects the text quality and consequently resulting in low character recognition accuracy. Second, there are successive news stories which are separated by the channel logo or long spaces. We managed to solve the first problem by deblurring the text image using either Bicubic Spline Interpolation (BSI) technique or the Denoising Autoencoder Neural Network (DANN). We solved the second problem using a Point Feature Matching (PFM) technique to match the existing channel logo with the channels’ logo database (ground truth). We evaluate our framework using Abbyy® SDK as a standalone tool used for text recognition supporting different languages.