This work focuses on radial basis functions containing no parameters with the main objective being to comparatively explore more of their effectiveness. For this, a total of sixteen forms of shapeless radial basis functions are gathered and investigated under the context of the pattern recognition problem through the structure of radial basis function neural networks, with the use of the Representational Capability (RC) algorithm. Different sizes of datasets are disturbed with noise before being imported into the algorithm as ‘training/testing’ datasets. Each shapeless radial basis function is monitored carefully with effectiveness criteria including accuracy, condition number (of the interpolation matrix), CPU time, CPU-storage requirement, underfitting and overfitting aspects, and the number of centres being generated. For the sake of comparison, the well-known Multiquadric-radial basis function is included as a representative of shape-contained radial basis functions. The numerical results have revealed that some forms of shapeless radial basis functions show good potential and are even better than Multiquadric itself indicating strongly that the future use of radial basis function may no longer face the pain of choosing a proper shape when shapeless forms may be equally (or even better) effective.

Under the framework of intelligent machines, the ability to classify or recognise patterns through the structure of neural networks is highly essential and challenging in a wide range of applications including biomedical and biology, social medial intelligence (SMI), video surveillance, intelligent retail environment and digital cultural heritage [

RBF neural networks were originally introduced in [

In 2019, Krowiak et al. [

Alternatively to shape-contained RBFs and to avoid the demerits mentioned so far, RBFs containing no shapes, referred to as ‘shapefree’ or ‘shapeless’ have recently been paid attention to. Recent work can be found in [

Three main contributions this work is aimed to make are;

To demonstrate the application of the Representational Capability (RC) algorithm in RBF neural networks for pattern recognition problems.

To provide more information on how reliable shapeless RBFs can be for pattern recognition problems, as compared to one of the popular choices of shape-contained RBFs.

To compare the effectiveness of sixteen selected shapeless RBFs under the same problem configuration.

It is hoped that with a good choice of shapeless RBF, one should be able to achieve high-quality results without suffering the pain caused by choosing the shape. Section 2 provides the problem statement with all crucial components before the computational algorithm with all the criteria being declared in Section 3. Section 4 then reports the essential results obtained and the main conclusions are drawn in Section 5.

For the task of pattern recognition, the so-called ‘training datasets’ undergo a process designed to construct a model or a mapping (if any) that is hoped to best represent the rest of the data or ‘testing datasets’. The process occurring in the middle is the main challenge as it is typically unknown. Both sets of data can be of the following form.

The model structure of a typical radial basis function network is depicted in

Sixteen forms of shapeless RBFs are paid attention to in this work and they are listed in

No. | Name of RBF | Abbreviation | Definition |
---|---|---|---|

1 | Multiquadric | MQ | |

2 | Polyharmonic spline | PS | |

3 | Thin plate splines | TPS | |

4 | Wu’s CS-RBFs [ |
WU1 | |

5 | Wu’s CS-RBFs [ |
WU2 | |

6 | Wu’s CS-RBFs [ |
WU3 | |

7 | Wu’s CS-RBFs [ |
WU4 | |

8 | Wendland’s CS-RBFs [ |
WL1 | |

9 | Wendland’s CS-RBFs [ |
WL2 | |

10 | Wendland’s CS-RBFs [ |
WL3 | |

11 | Wendland’s CS-RBFs [ |
WL4 | |

12 | Wendland’s CS-RBFs [ |
WL5 | |

13 | Wendland’s CS-RBFs [ |
WL6 | |

14 | Wendland’s CS-RBFs [ |
WL7 | |

15 | Buhmann [ |
BUH1 | |

16 | Buhmann [ |
BUH2 | |

17 | Buhmann [ |
BUH3 |

The Representational Capability (RC) algorithm as proposed by Shin et al. (2000) [

Next, generate matrix

Compute the

As previously mentioned, all sixteen shapeless RBFs are comparatively and numerically investigated. Therefore, proper and all-around effectiveness criteria are required and they are listed below.

The trending behaviour of this number is also recorded throughout this experiment, providing information on the solvability, [

For the data-partitioning process, the suggestion provided by Lazzaro et al. (2002) [

Two datasets with the size of

To monitor and record the effectiveness of each RBF type, three large datasets are generated within the same domain in both manners uniformly (Ufm.) and randomly (Rdm.), and they contain 10000, 20164, and 30276 nodes [

In this section, two well-known benchmarking test cases are numerically experimented with using all the RBF forms mentioned above. Nevertheless, with the limitation of the space, only those significant and relevant results are being mentioned, illustrated, and discussed.

This first experiment is concerned with one of the most well-known testing functions invented by Franke in 1982 [

The results validation for all RBFs for this first case is to be numerically investigated at two different noise levels (

As can be seen in

In terms of the number of basis functions (

Uniform node distribution | Random node distribution | |||||||
---|---|---|---|---|---|---|---|---|

Types of RBF | ||||||||

Training | Testing | Training | Testing | Training | Testing | Training | Testing | |

MQ + Carlson | 1.77E-01 | 1.55E-01 | 3.57E-02 | 3.18E-02 | 1.99E-01 | 2.19E-01 | 3.78E-02 | 3.58E-02 |

PS | 5.50E-02 | 7.60E-01 | 1.15E-02 | 8.39E-02 | 5.75E-02 | 2.60E+00 | 1.24E-02 | 1.20E-01 |

TPS | 5.48E-02 | 3.54E-02 | 1.51E-02 | 6.50E-03 | 5.73E-02 | 1.52E-01 | 1.49E-02 | 7.60E-03 |

WU1 | 1.07E-01 | 8.19E-02 | 2.04E-02 | 1.33E-02 | 9.10E-02 | 8.66E-02 | 2.08E-02 | 1.45E-02 |

WU2 | 8.03E-02 | 5.80E-02 | 1.84E-02 | 1.00E-02 | 6.03E-02 | 5.00E-02 | 1.72E-02 | 8.20E-03 |

WU3 | 5.80E-02 | 2.58E-02 | 1.57E-02 | 3.90E-03 | 5.75E-02 | 2.26E-02 | 1.54E-02 | 3.80E-03 |

WU4 | 5.65E-02 | 3.07E-02 | 1.34E-02 | 7.10E-03 | 4.63E-02 | 3.08E-02 | 1.31E-02 | 7.10E-03 |

WL1 | 6.26E-02 | 5.13E-02 | 1.53E-02 | 6.00E-03 | 6.72E-02 | 7.88E-02 | 1.53E-02 | 6.60E-03 |

WL2 | 6.19E-02 | 3.72E-02 | 1.65E-02 | 6.20E-03 | 5.84E-02 | 3.09E-02 | 1.58E-02 | 4.90E-03 |

WL3 | 6.84E-02 | 4.64E-02 | 1.70E-02 | 7.50E-03 | 5.92E-02 | 3.61E-02 | 1.62E-02 | 5.40E-03 |

WL4 | 5.93E-02 | 2.98E-02 | 1.38E-02 | 6.70E-03 | 5.70E-02 | 3.90E-02 | 1.37E-02 | 6.30E-03 |

WL5 | 5.64E-02 | 1.76E-02 | 1.55E-02 | 3.70E-03 | 5.74E-02 | 2.01E-02 | 1.54E-02 | 3.90E-03 |

WL6 | 5.83E-02 | 2.45E-02 | 1.59E-02 | 4.40E-03 | 6.09E-02 | 3.71E-02 | 1.60E-02 | 5.70E-03 |

WL7 | 6.14E-02 | 3.72E-02 | 1.63E-02 | 5.80E-03 | 6.12E-02 | 3.74E-02 | 1.60E-02 | 5.60E-03 |

BUH1 | 5.74E-02 | 2.00E-02 | 1.55E-02 | 4.00E-03 | 5.56E-02 | 2.78E-02 | 1.52E-02 | 3.70E-03 |

BUH2 | 5.66E-02 | 1.95E-02 | 1.56E-02 | 3.60E-03 | 5.74E-02 | 2.39E-02 | 1.54E-02 | 3.50E-03 |

BUH3 | 5.68E-02 | 1.45E-02 | 1.51E-02 | 4.10E-03 | 5.63E-02 | 1.80E-02 | 1.50E-02 | 4.30E-03 |

Uniform node distribution | Random node distribution | |||||||
---|---|---|---|---|---|---|---|---|

Types of RBF | ||||||||

Training | Testing | Training | Testing | Training | Testing | Training | Testing | |

MQ + Carlson | 3.50E-01 | 2.55E-01 | 9.84E-02 | 3.88E-02 | 4.02E-01 | 2.02E-01 | 9.30E-02 | 3.75E-02 |

PS | 2.90E-01 | 4.27E+00 | 6.36E-02 | 3.28E-01 | 3.02E-01 | 3.92E+00 | 6.55E-02 | 3.34E-01 |

TPS | 2.76E-01 | 1.56E-01 | 8.40E-02 | 3.18E-02 | 2.90E-01 | 1.92E-01 | 8.15E-02 | 3.19E-02 |

WU1 | 3.56E-01 | 8.46E-02 | 8.97E-02 | 1.76E-02 | 3.09E-01 | 1.06E-01 | 8.63E-02 | 1.91E-02 |

WU2 | 3.39E-01 | 6.11E-02 | 8.93E-02 | 1.56E-02 | 3.00E-01 | 7.43E-02 | 8.54E-02 | 1.55E-02 |

WU3 | 3.26E-01 | 6.36E-02 | 8.82E-02 | 1.81E-02 | 2.75E-01 | 6.31E-02 | 8.44E-02 | 1.82E-02 |

WU4 | 3.36E-01 | 2.02E-01 | 7.60E-02 | 4.27E-02 | 2.62E-01 | 1.92E-01 | 7.35E-02 | 3.82E-02 |

WL1 | 3.17E-01 | 1.03E-01 | 8.51E-02 | 2.91E-02 | 3.02E-01 | 1.11E-01 | 8.22E-02 | 2.53E-02 |

WL2 | 3.30E-01 | 6.13E-02 | 8.89E-02 | 1.66E-02 | 2.99E-01 | 6.40E-02 | 8.51E-02 | 1.57E-02 |

WL3 | 3.44E-01 | 5.93E-02 | 8.93E-02 | 1.49E-02 | 2.91E-01 | 6.91E-02 | 8.52E-02 | 1.52E-02 |

WL4 | 3.52E-01 | 1.68E-01 | 7.94E-02 | 3.87E-02 | 2.70E-01 | 1.61E-01 | 7.66E-02 | 3.42E-02 |

WL5 | 3.22E-01 | 6.49E-02 | 8.81E-02 | 1.86E-02 | 2.72E-01 | 5.87E-02 | 8.44E-02 | 1.83E-02 |

WL6 | 3.32E-01 | 7.15E-02 | 8.87E-02 | 1.64E-02 | 2.77E-01 | 7.17E-02 | 8.49E-02 | 1.73E-02 |

WL7 | 3.29E-01 | 7.20E-02 | 8.89E-02 | 1.63E-02 | 2.83E-01 | 7.05E-02 | 8.50E-02 | 1.68E-02 |

BUH1 | 3.25E-01 | 7.75E-02 | 8.77E-02 | 2.04E-02 | 2.77E-01 | 8.06E-02 | 8.39E-02 | 2.06E-02 |

BUH2 | 3.20E-01 | 8.29E-02 | 8.83E-02 | 1.77E-02 | 2.73E-01 | 5.88E-02 | 8.46E-02 | 1.73E-02 |

BUH3 | 3.20E-01 | 1.03E-01 | 8.68E-02 | 2.34E-02 | 2.85E-01 | 9.31E-02 | 8.34E-02 | 2.25E-02 |

Types of RBF | ||||||||
---|---|---|---|---|---|---|---|---|

Ufm. | Rdm. | Ufm. | Rdm. | Ufm. | Rdm. | Ufm. | Rdm. | |

MQ + Carlson | 19 | 19 | 5.41E+03 | 6.44E+03 | 19 | 19 | 5.84E+03 | 6.02E+03 |

PS | 1239 | 996 | 2.92E+04 | 2.39E+04 | 1239 | 996 | 2.92E+04 | 2.39E+04 |

TPS | 295 | 243 | 1.40E+04 | 1.32E+04 | 295 | 243 | 1.40E+04 | 1.32E+04 |

WU1 | 41 | 39 | 8.26E+03 | 6.62E+03 | 41 | 39 | 8.26E+03 | 6.62E+03 |

WU2 | 47 | 47 | 6.99E+03 | 9.16E+03 | 47 | 47 | 6.99E+03 | 9.16E+03 |

WU3 | 87 | 83 | 9.01E+03 | 9.35E+03 | 87 | 83 | 9.01E+03 | 9.35E+03 |

WU4 | 689 | 662 | 9.29E+03 | 8.32E+03 | 689 | 662 | 9.29E+03 | 8.32E+03 |

WL1 | 245 | 234 | 9.62E+03 | 9.77E+03 | 245 | 234 | 9.62E+03 | 9.77E+03 |

WL2 | 64 | 62 | 9.28E+03 | 8.53E+03 | 64 | 62 | 9.28E+03 | 8.53E+03 |

WL3 | 51 | 50 | 7.02E+03 | 7.86E+03 | 51 | 50 | 7.02E+03 | 7.86E+03 |

WL4 | 511 | 504 | 9.25E+03 | 7.95E+03 | 511 | 504 | 9.25E+03 | 7.95E+03 |

WL5 | 100 | 94 | 1.23E+04 | 1.01E+04 | 100 | 94 | 1.23E+04 | 1.01E+04 |

WL6 | 69 | 65 | 1.07E+04 | 8.81E+03 | 69 | 65 | 1.07E+04 | 8.81E+03 |

WL7 | 64 | 62 | 8.71E+03 | 8.57E+03 | 64 | 62 | 8.71E+03 | 8.57E+03 |

BUH1 | 117 | 113 | 1.13E+04 | 1.01E+04 | 117 | 113 | 1.13E+04 | 1.01E+04 |

BUH2 | 83 | 78 | 1.22E+04 | 9.22E+03 | 83 | 78 | 1.22E+04 | 9.22E+03 |

BUH3 | 164 | 154 | 1.10E+04 | 1.24E+04 | 164 | 154 | 1.10E+04 | 1.24E+04 |

For the second test case, we study one of the functions called F7 in the investigation nicely carried out by Renka et al. [

Similar to the first experiment, two different noise levels or

With the results obtained in these cases, shown in

Uniform node distribution | Random node distribution | |||||||
---|---|---|---|---|---|---|---|---|

Types of RBF | ||||||||

Training | Testing | Training | Testing | Training | Testing | Training | Testing | |

MQ + Carlson | 4.93E-01 | 4.38E-01 | 1.03E-01 | 9.38E-02 | 5.84E-01 | 6.33E-01 | 1.06E-01 | 1.10E-01 |

PS | 2.90E-01 | 1.26E+01 | 6.65E-02 | 1.15E +00 | 2.33E-01 | 2.28E+01 | 4.78E-02 | 1.18E+00 |

TPS | 3.88E-01 | 4.48E-01 | 6.88E-02 | 5.98E-02 | 4.55E-01 | 1.14E+00 | 6.24E-02 | 8.27E-02 |

WU1 | 7.16E-01 | 6.96E-01 | 1.49E-01 | 1.40E-01 | 1.12E+00 | 1.78E+00 | 1.71E-01 | 1.83E-01 |

WU2 | 4.84E-01 | 4.74E-01 | 9.45E-02 | 8.42E-02 | 5.00E-01 | 5.72E-01 | 8.64E-02 | 8.66E-02 |

WU3 | 1.93E-01 | 1.94E-01 | 4.65E-02 | 3.06E-02 | 2.14E-01 | 2.72E-01 | 4.46E-02 | 3.50E-02 |

WU4 | 1.14E-01 | 7.67E-02 | 3.03E-02 | 1.63E-02 | 1.39E-01 | 7.50E-01 | 2.94E-02 | 2.67E-02 |

WL1 | 1.08E+00 | 1.08E+00 | 7.74E-02 | 6.27E-02 | 8.93E-01 | 2.80E+00 | 7.54E-02 | 9.60E-02 |

WL2 | 3.99E-01 | 4.03E-01 | 7.21E-02 | 6.05E-02 | 6.38E-01 | 7.21E-01 | 7.25E-02 | 7.09E-02 |

WL3 | 4.92E-01 | 4.77E-01 | 9.19E-02 | 8.18E-02 | 2.31E-01 | 3.27E-01 | 5.80E-02 | 5.27E-02 |

WL4 | 1.29E-01 | 7.42E-02 | 3.20E-02 | 1.54E-02 | 1.69E-01 | 8.44E-01 | 3.20E-02 | 3.03E-02 |

WL5 | 1.86E-01 | 1.43E-01 | 3.94E-02 | 2.08E-02 | 1.98E-01 | 2.62E-01 | 4.20E-02 | 3.20E-02 |

WL6 | 2.00E-01 | 2.17E-01 | 4.17E-02 | 2.45E-02 | 2.18E-01 | 2.22E-01 | 4.33E-02 | 3.14E-02 |

WL7 | 1.89E-01 | 1.72E-01 | 4.41E-02 | 2.68E-02 | 1.95E-01 | 2.13E-01 | 4.23E-02 | 2.99E-02 |

BUH1 | 1.27E-01 | 8.67E-02 | 3.62E-02 | 1.60E-02 | 1.89E-01 | 5.10E-01 | 3.85E-02 | 2.76E-02 |

BUH2 | 1.66E-01 | 1.36E-01 | 4.10E-02 | 2.36E-02 | 2.17E-01 | 2.37E-01 | 4.55E-02 | 3.57E-02 |

BUH3 | 1.47E-01 | 8.09E-02 | 3.45E-02 | 1.30E-02 | 1.23E-01 | 4.18E-01 | 3.47E-02 | 1.88E-02 |

Uniform node distribution | Random node distribution | |||||||
---|---|---|---|---|---|---|---|---|

Types of RBF | ||||||||

Training | Testing | Training | Testing | Training | Testing | Training | Testing | |

MQ + Carlson | 1.12E+00 | 7.32E-01 | 2.37E-01 | 1.38E-01 | 9.91E-01 | 7.55E-01 | 2.15E-01 | 1.16E-01 |

PS | 6.88E-01 | 1.58E+01 | 1.52E-01 | 1.35E+00 | 6.82E-01 | 2.56E+01 | 1.50E-01 | 1.35E+00 |

TPS | 7.51E-01 | 4.71E-01 | 1.92E-01 | 8.70E-02 | 7.27E-01 | 1.28E+00 | 1.88E-01 | 1.10E-01 |

WU 1 | 9.39E-01 | 7.39E-01 | 2.40E-01 | 1.41E-01 | 1.26E+00 | 1.96E+00 | 2.54E-01 | 1.86E-01 |

WU 2 | 8.34E-01 | 5.10E-01 | 2.10E-01 | 8.69E-02 | 9.69E-01 | 5.20E-01 | 2.05E-01 | 8.92E-02 |

WU 3 | 7.31E-01 | 1.91E-01 | 1.93E-01 | 4.60E-02 | 7.85E-01 | 2.83E-01 | 1.90E-01 | 4.88E-02 |

WU 4 | 7.00E-01 | 4.72E-01 | 1.65E-01 | 8.91E-02 | 6.37E-01 | 8.65E-01 | 1.64E-01 | 8.95E-02 |

WL 1 | 1.15E+00 | 1.13E+00 | 1.96E-01 | 8.31E-02 | 8.27E-01 | 3.24E+00 | 1.93E-01 | 1.15E-01 |

WL 2 | 7.43E-01 | 3.51E-01 | 2.01E-01 | 6.72E-02 | 9.69E-01 | 6.64E-01 | 2.02E-01 | 7.77E-02 |

WL 3 | 8.41E-01 | 4.51E-01 | 2.10E-01 | 8.38E-02 | 9.16E-01 | 3.31E-01 | 1.94E-01 | 5.86E-02 |

WL 4 | 7.23E-01 | 3.70E-01 | 1.72E-01 | 8.04E-02 | 6.43E-01 | 8.41E-01 | 1.71E-01 | 7.98E-02 |

WL 5 | 7.08E-01 | 1.51E-01 | 1.90E-01 | 4.33E-02 | 8.01E-01 | 2.79E-01 | 1.89E-01 | 5.04E-02 |

WL 6 | 7.05E-01 | 1.57E-01 | 1.92E-01 | 3.88E-02 | 7.92E-01 | 2.69E-01 | 1.90E-01 | 4.36E-02 |

WL 7 | 7.05E-01 | 1.89E-01 | 1.92E-01 | 3.85E-02 | 7.88E-01 | 2.06E-01 | 1.90E-01 | 4.24E-02 |

BUH 1 | 7.05E-01 | 1.55E-01 | 1.89E-01 | 4.40E-02 | 7.40E-01 | 4.54E-01 | 1.87E-01 | 4.98E-02 |

BUH 2 | 7.14E-01 | 1.67E-01 | 1.91E-01 | 4.16E-02 | 8.34E-01 | 2.82E-01 | 1.91E-01 | 4.96E-02 |

BUH 3 | 7.20E-01 | 2.23E-01 | 1.87E-01 | 5.05E-02 | 7.82E-01 | 3.79E-01 | 1.84E-01 | 5.21E-02 |

Types of RBF | ||||||||
---|---|---|---|---|---|---|---|---|

Ufm. | Rdm. | Ufm. | Rdm. | Ufm. | Rdm. | Ufm. | Rdm. | |

MQ + Carlson | 49 | 47 | 9.93E+03 | 1.20E+04 | 43 | 46 | 1.06E+04 | 8.56E+03 |

PS | 1239 | 996 | 2.92E+04 | 2.39E+04 | 1239 | 996 | 2.92E+04 | 2.39E+04 |

TPS | 295 | 243 | 1.40E+04 | 1.32E+04 | 295 | 243 | 1.40E+04 | 1.32E+04 |

WU1 | 41 | 39 | 8.26E+03 | 6.62E+03 | 41 | 39 | 8.26E+03 | 6.62E+03 |

WU2 | 47 | 47 | 6.99E+03 | 9.16E+03 | 47 | 47 | 6.99E+03 | 9.16E+03 |

WU3 | 87 | 83 | 9.01E+03 | 9.35E+03 | 87 | 83 | 9.01E+03 | 9.35E+03 |

WU4 | 689 | 662 | 9.29E+03 | 8.32E+03 | 689 | 662 | 9.29E+03 | 8.32E+03 |

WL1 | 245 | 234 | 9.62E+03 | 9.77E+03 | 245 | 234 | 9.62E+03 | 9.77E+03 |

WL2 | 64 | 62 | 9.28E+03 | 8.53E+03 | 64 | 62 | 9.28E+03 | 8.53E+03 |

WL3 | 51 | 50 | 7.02E+03 | 7.86E+03 | 51 | 50 | 7.02E+03 | 7.86E+03 |

WL4 | 511 | 504 | 9.25E+03 | 7.95E+03 | 511 | 504 | 9.25E+03 | 7.95E+03 |

WL5 | 100 | 94 | 1.23E+04 | 1.01E+04 | 100 | 94 | 1.23E+04 | 1.01E+04 |

WL6 | 69 | 65 | 1.07E+04 | 8.81E+03 | 69 | 65 | 1.07E+04 | 8.81E+03 |

WL7 | 64 | 62 | 8.71E+03 | 8.57E+03 | 64 | 62 | 8.71E+03 | 8.57E+03 |

BUH1 | 117 | 113 | 1.13E+04 | 1.01E+04 | 117 | 113 | 1.13E+04 | 1.01E+04 |

BUH2 | 83 | 78 | 1.22E+04 | 9.22E+03 | 83 | 78 | 1.22E+04 | 9.22E+03 |

BUH3 | 164 | 154 | 1.10E+04 | 1.24E+04 | 164 | 154 | 1.10E+04 | 1.24E+04 |

From all the numerical results obtained so far and in addition to the criteria stated, it has to be acknowledged that ‘underfitting and overfitting’ is another crucial figure to be considered. Regarding all cases, a significant reduction in accuracy, measured by both error norms, produced by PS type of RBF strongly indicates its overfitting nature and shall not be recommended for practical uses. On the other hand, WL5, BUH1, WU3, and BUH2 are seen to be slightly underfitting for both numerical demonstrations. The best types of RBF in terms of this aspect are WU1, WU2, WL1, WL4, and MQ.

This work aims to provide insights into the use of sixteen forms of radial basis function (RBF) containing no shape parameter, so they are referred to as ‘shapeless RBFs’. The challenge under the main experiment is the problem of pattern recognition through neural networks computation process and architecture. The ability to deal with large and noised datasets of each shapeless RBF is measured under several criteria against the well-known shape-containing RBF called multiquadric (MQ). Two testing functions are tackled numerically and important findings observed (based on each criterion) are listed below.

1) Accuracy (

2) The condition number (

3) In terms of CPU (time and storage) and also the number of basis functions (

4) User’s Interference: It is obvious that as long as no parameter-turning process is required, there is no need to interfere with the algorithm and this is one desirable aspect of using shapeless RBF. This is also the case for the sensitivity-to-parameter criteria.

5) Ease of implementation: It is observed that all shapeless RBF types under this work are equally simple when it comes to implementing the scheme. For MQ, on the other hand, an additional coding routine may be needed for the process of a reliable shape searching procedure, as is the case in this investigation for the Carlson algorithm.

Together with the ‘overfitting and underfitting’ aspect, this work suggests that shapeless RBFs from Wu’s and Wendland’s families are highly promising, whereas the rest forms are still skeptical for practical uses. Apart from this useful piece of information for pattern recognition application, it might be considered a weakness of the work that the figure discovered so far may change when dealing with other kinds of applications such as direct interpolation, function approximation, and recovery, as well as solving different equations (DEs) under the concepts of meshless or meshfree methods. Furthermore, other kinds of node distribution of sizes may result in different aspects. This is all set as our future research direction and is highly recommended for interested researchers to further explore.