Power function-based Gini indices: New sparsity measures using power function-based quasi-arithmetic means for bearing condition monitoring

Abstract

The Gini index (GI), GI II, and GI III are proven to be effective sparsity measures in the fields of machine condition monitoring and fault diagnosis, and they can be reformulated as the ratio of different quasi-arithmetic means (RQAM). Under this framework, generalized Gini indices (GGIs) have been developed for sparse quantification by applying nonlinear weights to GI, and another generalized form of GI, referred to here as power function-based Gini indices I (PFGI1s), has been introduced by using power function as the generator of quasi-arithmetic means. The GGIs with different weight parameters exhibit reliable sparse quantization capability for repetitive transient features, while their repetitive transient discriminability is lower than kurtosis and negentropy under noise contamination. PFGI1 achieves enhanced repetitive transient discriminability with increasing power exponent, showing the advantage of the generalization approach. In this paper, based on RQAM, a single-parameter generalization method for generating PFGI1s is introduced into GI II and GI III from the perspective of the quasi-arithmetic mean generator, which leads to the power function-based Gini indices II and III (PFGI2s and PFGI3s) constructed from GI II and GI III, respectively. Mathematical derivation proves that PFGI2s and PFGI3s satisfy at least five of six typical attributes of sparsity measures and are two new families of sparsity measures. Simulation analysis shows that, similar to PFGI1s, PFGI2s and PFGI3s can monotonically estimate the sparsity of the data sequence and can simultaneously achieve strong random transient resistibility and high repetitive transient discriminability compared with traditional sparsity measures. The experimental results of bearing run-to-failure demonstrate that PFGI1s, PFGI2s, and PFGI3s with appropriate power exponents can effectively quantify the repetitive transient features caused by bearing faults and can accurately characterize the bearing degradation status compared with the state-of-the-art sparsity measures.

Keywords

Gini index generalized Gini indices power function-based Gini indices quasi-arithmetic means sparsity measures sparse quantification bearing condition monitoring

Highlights

A single-parameter generalization method is developed for GI II and GI III.

PFGIs II and III are proposed as new sparsity measures for sparse quantification.

Sparse attributes and three key performances of PFGIs II and III are investigated.

PFGIs II and III show excellent performance in characterizing repetitive transients.

PFGIs I, II, and III can effectively characterize bearing fault features and degradation status.

Introduction

Mechanical equipment is widely used in transportation, energy, chemical, manufacturing, and other industries. Rolling bearings have always been core components of rotating machinery such as rail trains, automobiles, aircraft, wind turbines, and gas turbines. Defects and failures of bearings have a major influence on the overall operating performance of mechanical equipment, reducing economic benefits and endangering personnel safety in severe cases. Therefore, condition monitoring and fault diagnosis of rolling bearings are receiving increasing attention from academia and industry.^1–5 Bearing defects mainly include fatigue spalling, wear, corrosion and cracks, which usually cause repetitive shock vibrations and make the bearing vibration signal exhibit impulsive features and sparse structure.⁶ Repetitive impulse features are often regarded as typical symptoms of bearing faults and have become an important basis for health monitoring and diagnostics.^7–9 The health monitoring indicators based on signal sparsity have been deeply developed, and the constructed impulse feature extraction method can effectively realize fault diagnosis.^10–12 Sparsity measures, such as kurtosis, negentropy (NE), ratio of L2 norm to L1 norm (L2/L1), and Gini index (GI), are generally recognized sparse quantification tools¹³ and have been widely employed in condition monitoring and fault diagnosis of rolling bearings.

Envelope analysis is a well-established technique for bearing fault diagnosis, where the identification of a resonance frequency band containing rich fault information is decisive. Antoni¹⁴ used the kurtosis of the envelope of the band-pass filtered signal to identify the resonance frequency band and proposed the fast kurtogram, which is a typical application of sparsity measure in machinery fault diagnosis. To overcome the sensitivity of time-domain kurtosis to strong random transients, the kurtosis of the envelope spectrum,¹⁵ the kurtosis of the power spectrum¹⁶ and the kurtosis of the autocorrelation sequence¹⁷ were proposed for identifying resonance frequency band, respectively. In addition, other sparsity measures have been introduced into the envelope analysis methods, such as L2/L1 norm,¹⁸ NE,¹⁹ GI,²⁰ reciprocal of smoothness index²¹ and Hoyer index (HI).²² The works in^22–25 comparatively investigated the performance of typical sparsity measures in discriminating resonance frequency bands. Another well-established fault diagnosis method based on sparsity measures is blind deconvolution. A representative method is minimum entropy deconvolution²⁶ with kurtosis as the objective function, which is dedicated to recovering the transient features hidden in the signal by designing an inverse filter²⁷. However, similar to the fast kurtogram, minimum entropy deconvolution is also vulnerable to random transients in the signal and tends to recover a single transient feature. To overcome this limitation, the blind deconvolution methods with generalized Lp/Lq norm,^28,29 NE, L2/L1 and HI³⁰, and GI³¹ as objective functions were developed for bearing diagnostics. A recent work³² reported the application of different blind deconvolution methods in fault diagnosis of rotating machinery. Hou et al.³³ compared the performance of seven impulsiveness-based health indicators including kurtosis, skewness, smoothness index,³⁴ NE, GI, HI,³⁵ and L2/L1 for machine condition monitoring. The above analysis shows that the sparsity measures have achieved successful applications in the fault diagnosis and condition monitoring of rolling bearings. However, the experimental results under complex operating conditions show that it is difficult for traditional sparsity measures to have strong random transient resistibility and repetitive transient discriminability simultaneously. For example, kurtosis and L2/L1 norm have strong capability to identify repetitive transients, but are easily affected by high-intensity random transients; GI and the reciprocal of smoothness index are robust to outliers, but show insufficient repetitive transient discriminability under low signal-to-noise ratio (SNR).^22,24

Recently, the development of robust health indicators and sparsity measures for characterizing repetitive transients caused by machine faults has received increasing attention. In study,³⁶ spectral kurtosis, spectral NE, spectral GI and spectral smoothness index were found to fall into the sum of weighted normalized squared envelope (SWNSE) and the main difference among them is that different weight sequences are applied to the normalized squared envelope. Based on SWNSE and Box-Cox transformation, Box-Cox sparsity measures (BCSM) were proposed by Wang et al.³⁷ as the generalization of kurtosis and NE for machine condition monitoring, and have been proved to satisfy six typical attributes of sparsity measures. The experimental results showed that BCSM can achieve the sparse quantization performance between kurtosis and NE by adjusting the transformation parameter,^37,38 which means that BCSM is also vulnerable to large random transients. Chen et al.³⁹ improved SWNSE for bearing condition monitoring by generalizing the norm order, but the construction of health indicators needs to meet strict parameter-setting requirements. In study,⁴⁰ typical sparsity measures were discovered to be reformulated as the ratio of different quasi-arithmetic means (RQAM), and a series of health indicators based on RQAM were proposed for machine condition monitoring. The experimental results showed that several health indicators are robust to outliers and can exhibit monotonic machine degradation trends. Nevertheless, these health indicators have not been proven to satisfy the typical attributes of sparsity measures. To develop sparsity measures for characterizing repetitive transients, GI II (GI2) and GI III (GI3) were proposed based on RQAM and GI.⁴¹ GI2 and GI3 satisfy the six attributes of sparsity measures and exhibit similar performance to GI in quantifying repetitive transient features. Furthermore, based on RQAM, Hou et al.⁴² proposed generalized Gini indices (GGIs) by applying the nonlinear weights to GI and proved that GGIs are a new family of sparsity measures. The experimental results showed that GGIs are robust to random transients compared with BCSMs and their sparse quantization capability is tunable through a weight parameter. Nevertheless, similar to GI, the repetitive transient discriminability of GGIs under noise interference is lower than that of kurtosis and NE.³⁹

More recently, to construct new indicators to improve the quantification performance of repetitive transient features under complex disturbances, Chen et al.⁴³ proposed another single-parameter generalization of GI by adopting power function as the quasi-arithmetic mean generator, which is called the power function-based Gini indices I (PFGI1s) in this paper. The results show that increasing the power exponent can improve the repetitive transient discriminability of PFGI1s under noise interference, which demonstrates the effectiveness of using the power function as the quasi-arithmetic mean generator. Inspired by this, based on RQAM, GI2, and GI3, the sparsity measures with excellent sparse quantification capabilities are explored and constructed for transient feature characterization and bearing condition monitoring. The main novelties and contributions of this paper are described as follows:

From the perspective of the quasi-arithmetic mean generator, a novel single-parameter generalization method is introduced into GI2 and GI3, which allows the power function-based Gini indices II and III (PFGI2s and PFGI3s) to be proposed using the power function-based quasi-arithmetic means.

PFGI2s and PFGI3s are theoretically proven to satisfy at least five of six typical attributes of sparsity measures, confirming that they are the new families of sparsity measures for repetitive transient quantification.

Similar to PFGI1s, PFGI2s and PFGI3s with different power exponents are proven to monotonically characterize the sparsity of data sequences and can achieve both strong random transient resistibility and high repetitive transient discriminability.

Run-to-failure experiments verify that PFGI1s, PFGI2s, and PFGI3s with appropriate power exponents can accurately quantify bearing fault-induced transient features and characterize bearing degradation status, confirming the advantages over traditional sparsity measures in bearing condition monitoring.

The rest of this paper is organized as follows. In Section “Definitions of the power function-based Gini indices,” the formal definitions of PFGI2s and PFGI3s are proposed after a brief review of RQAM-based health indicators. Section “Attributes of the power function-based Gini indices” presents the proofs of the sparse attributes of PFGI2s and PFGI3s. In Section “Performance analysis via numerical simulation,” the important properties of PFGI2s and PFGI3s are investigated through numerical simulations accompanied by comparisons with GGIs and PFGI1s. Section “Rolling bearing condition monitoring” validates the performance of PFGI1s, PFGI2s, and PFGI3s in characterizing repetitive transient features and bearing degradation states using two different run-to-failure experiments. Finally, the important conclusions are summarized in Section “Conclusions.”

Definitions of the power function-based Gini indices

In this section, the typical health indicators based on RQAM are first reviewed, then a health indicator framework constructed by the power function-based quasi-arithmetic means is introduced, and the formal definitions of PFGI1s, PFGI2s, and PFGI3s are provided.

Health indicators based on RQAM

In a recent work,⁴⁰ typical sparsity measures were found to be restructured as RQAM, and a generalized framework of health indicators based on RQAM was proposed for machine condition monitoring, as follows:

GHI = 1 - \frac{M_{g (x) | w_{1}}}{M_{f (x) | w_{2}}} = 1 - \frac{g^{- 1} (\sum_{n = 1}^{N} w_{1, n} g (x_{n}) / \sum_{n = 1}^{N} w_{1, n})}{f^{- 1} (\sum_{n = 1}^{N} w_{2, n} f (x_{n}) / \sum_{n = 1}^{N} w_{2, n})}

(1)

where $x = [x_{1}, x_{2}, \dots, x_{N}]$ is a discrete signal sequence with length N; $M_{g (x) | w_{1}}$ is a quasi-arithmetic mean constructed by monotonic function $g (x)$ and weight vector $w_{1} = [w_{1, 1}, w_{1, 2}, \dots, w_{1, N}]$ ; $M_{f (x) | w_{2}}$ is a quasi-arithmetic mean constructed by monotonic function $f (x)$ and weight vector $w_{2} = [w_{2, 1}, w_{2, 2}, \dots, w_{2, N}]$ ; $g^{- 1} (\cdot)$ and $f^{- 1} (\cdot)$ are the inverse functions of $g (\cdot)$ and $f (\cdot)$ , respectively. Note that the indices constructed by Equation (1) have a value range of [0, 1] when $M_{g (x) | w_{1}} \leq M_{f (x) | w_{2}}$ is satisfied.

GI, originally used in the field of economics,⁴⁴ has been experimentally proven to be an efficient sparsity measure for characterizing repetitive transients caused by machine defects.^21,33 GI can be reformulated as the ratio of a line function-based quasi-arithmetic mean with linear weights to a line function-based quasi-arithmetic mean with equal weights, as follows⁴²:

GI = 1 - \frac{M_{x_{asc} | w'}}{M_{x}} = 1 - \sum_{n = 1}^{N} \frac{2 (N - n) + 1}{N^{2}} x_{(n)} / \sum_{n = 1}^{N} \frac{1}{N} x_{n}

(2)

where $x_{asc} = [x_{(1)}, x_{(2)}, . . ., x_{(N)}]$ is the permutation of the non-negative sequence $x = [x_{1}, x_{2}, . . ., x_{N}]$ in ascending order, that is, $x_{(1)} \leq x_{(2)} \leq \dots \leq x_{(N)}$ ; $w^{'} = [{w^{'}}_{1}, {w^{'}}_{2}, \dots, {w^{'}}_{N}]$ and ${w'}_{n} = [2 (N - n) + 1] / N^{2}$ .

Based on RQAM, GI2 and GI3 were proposed for machine condition monitoring, as follows⁴¹:

GI 2 = 1 - \frac{M_{x_{asc} | w'}}{M_{x_{asc} | w ″}} = 1 - \sum_{n = 1}^{N} \frac{2 (N - n) + 1}{N^{2}} x_{(n)} / \sum_{n = 1}^{N} \frac{2 n - 1}{N^{2}} x_{(n)}

(3)

GI 3 = 1 - \frac{M_{x}}{M_{x_{asc} | w ″}} = 1 - \sum_{n = 1}^{N} \frac{1}{N} x_{n} / \sum_{n = 1}^{N} \frac{2 n - 1}{N^{2}} x_{(n)}

(4)

where $w^{″} = [{w^{″}}_{1}, {w^{″}}_{2}, \dots, {w^{″}}_{N}]$ and ${w^{″}}_{n} = (2 n - 1) / N^{2}$ .

Recently, by taking into account the nonlinear weights rather than the liner weights, GGIs were proposed by Hou et al.⁴² as a new family of sparsity measures for sparse quantification, as follows:

GGI = 1 - \frac{M_{x_{asc} | w ″'}}{M_{x}} = 1 - \frac{\sum_{n = 1}^{N} {w ″'}_{n} x_{(n)}}{\sum_{n = 1}^{N} x_{n} / N}

(5)

where $w^{‴} = [{w^{‴}}_{1}, {w^{‴}}_{2}, \dots, {w^{‴}}_{N}]$ , $w ″'_{n} = {[2 (N - n) + 1]}^{a} / \sum_{n = 1}^{N} {[2 (N - n) + 1]}^{a}$ and the weight parameter $a > 0$ .

The proposal of RQAM lays an important foundation for the construction of new health indicators or sparsity measures. The proposals of GI2, GI3, and GGIs provide new tools for sparse quantification and enrich the family of sparsity measures.

Power function-based Gini indices I, II, and III

In GI, GI2, GI3, and GGIs, only a linear function (i.e., a power function with an exponent of 1) is employed as the generator of the quasi-arithmetic means. In view of the more excellent properties of nonlinear functions, the quasi-arithmetic mean based on nonlinear function $x^{p}$ (where $p > 0$ denotes the exponent of the power function) is very valuable to be investigated. Therefore, a health indicator framework constructed by the power function-based quasi-arithmetic means (PHI) is defined as follows:

PHI = 1 - \frac{M_{x^{p} | w_{1}}}{M_{x^{p} | w_{2}}} = 1 - \sqrt[p]{\frac{\sum_{n = 1}^{N} w_{1, n} x_{n}^{p} / \sum_{n = 1}^{N} w_{1, n}}{\sum_{n = 1}^{N} w_{2, n} x_{n}^{p} / \sum_{n = 1}^{N} w_{2, n}}}

(6)

where $M_{x^{p} | w_{1}}$ and $M_{x^{p} | w_{2}}$ are two different power function-based quasi-arithmetic means. Note that the indices constructed by Equation (6) have a value range of [0, 1] when $M_{x^{p} | w_{1}} \leq M_{x^{p} | w_{2}}$ is satisfied. Thus, new health indicators can be uniquely constructed based on Equation (6) when the two weight sequences $w_{1}$ and $w_{2}$ are appropriately designed.

Using the two weight sequences of GI and employing the power function as the generator of quasi-arithmetic means, PFGI1s have been proposed as a generalization of GI for quantifying transient features in bearing fault diagnosis, as follows⁴³:

PFGI 1 = 1 - \frac{M_{x_{asc}^{p} | w'}}{M_{x^{p}}} = 1 - \sqrt[p]{\frac{\sum_{n = 1}^{N} w'_{n} x_{(n)}^{p}}{\sum_{n = 1}^{N} x_{n}^{p} / N}}

(7)

where $w'_{n} = [2 (N - n) + 1] / N^{2}$ .

The results in the work⁴³ preliminarily demonstrate the effectiveness and advantages of employing the power function as the quasi-arithmetic mean generator in constructing transient quantitative indicators. Therefore, the generalization method used to generate PFGI1s can be reasonably applied to other sparsity measures. In this paper, based on PHI and the weight sequences used in GI2, GI3, PFGI2s, and PFGI3s are proposed as two new families of machine health indicators, which are formally defined as follows:

PFGI 2 = 1 - \frac{M_{x_{asc}^{p} | w'}}{M_{x_{asc}^{p} | w ″}} = 1 - \sqrt[p]{\frac{\sum_{n = 1}^{N} w'_{n} x_{(n)}^{p}}{\sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p}}}

(8)

PFGI 3 = 1 - \frac{M_{x^{p}}}{M_{x_{asc}^{p} | w ″}} = 1 - \sqrt[p]{\frac{\sum_{n = 1}^{N} x_{n}^{p} / N}{\sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p}}}

(9)

where ${w ″}_{n} = (2 n - 1) / N^{2}$ .

PFGI1s, PFGI2s, and PFGI3s are the single-parameter generalizations of GI, GI2, and GI3, respectively, and have a tunable parameter p. PFGI1s, PFGI2s, and PFGI3s are reduced to GI, GI2, and GI3, respectively, when power exponent p = 1. A schematic diagram of the newly proposed health indicators is shown in Figure 1. In addition, like GI, GI2, and GI3, PFGI1s, PFGI2s, and PFGI3s also require the input sequence to be non-negative and also have a limited magnitude range of [0, 1]. The relevant mathematical proofs are given as follows:

(1) Proof of the magnitude range of PFGI1s

Since the sequence $x$ and weights $w'_{n} = [2 (N - n) + 1] / N^{2}$ are non-negative, $(\sum_{n = 1}^{N} w'_{n} x_{(n)}^{p}) / (\sum_{n = 1}^{N} x_{n}^{p} / N) \geq 0$ is satisfied. In addition, the following can be obtained:

\begin{matrix} \sum_{n = 1}^{N} w'_{n} x_{(n)}^{p} - \sum_{n = 1}^{N} x_{n}^{p} / N = \sum_{n = 1}^{N} x_{(n)}^{p} \frac{2 (N - n) + 1}{N^{2}} \\ - \sum_{n = 1}^{N} x_{n}^{p} \frac{1}{N} = \sum_{n = 1}^{N} x_{(n)}^{p} \frac{2 (N - n) + 1 - N}{N^{2}} \\ = {\begin{matrix} \sum_{n = 1}^{N / 2} (x_{(n)}^{p} - x_{(N + 1 - n)}^{p}) \frac{2 (N - n) + 1 - N}{N^{2}} \leq 0, N is even \\ \sum_{n = 1}^{(N - 1) / 2} (x_{(n)}^{p} - x_{(N + 1 - n)}^{p}) \frac{2 (N - n) + 1 - N}{N^{2}} \leq 0, N is odd \end{matrix} \end{matrix}

(10)

Therefore, $0 \leq (\sum_{n = 1}^{N} {w^{'}}_{n} x_{(n)}^{p}) / (\sum_{n = 1}^{N} x_{n}^{p} / N) \leq 1$ holds. Thus $0 \leq PFGI 1 \leq 1$ also holds.

(2) Proof of the magnitude range of PFGI2s

Since the sequence $x$ and weights $w'_{n} = [2 (N - n) + 1] / N^{2}$ and ${w ″}_{n} = (2 n - 1) / N^{2}$ are non-negative, $(\sum_{n = 1}^{N} w'_{n} x_{(n)}^{p}) / (\sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p}) \geq 0$ is satisfied. In addition, the following can be obtained:

\begin{array}{l} \sum_{n = 1}^{N} {w^{'}}_{n} x_{(n)}^{p} - \sum_{n = 1}^{N} {w^{″}}_{n} x_{(n)}^{p} = \sum_{n = 1}^{N} x_{(n)}^{p} \frac{2 (N - n) + 1}{N^{2}} \\ - \sum_{n = 1}^{N} x_{(n)}^{p} \frac{2 n - 1}{N^{2}} \\ = \sum_{n = 1}^{N} x_{(n)}^{p} \frac{2 (N - n) + 1 - (2 n - 1)}{N^{2}} \\ = {\begin{array}{l} \sum_{n = 1}^{N / 2} (x_{(n)}^{p} - x_{(N + 1 - n)}^{p}) \frac{2 (N - n) + 1 - (2 n - 1)}{N^{2}} \leq 0, & N is even \\ \sum_{n = 1}^{(N - 1) / 2} (x_{(n)}^{p} - x_{(N + 1 - n)}^{p}) \frac{2 (N - n) + 1 - (2 n - 1)}{N^{2}} \leq 0, & N is odd \end{array} \end{array}

(11)

Therefore, $0 \leq (\sum_{n = 1}^{N} w'_{n} x_{(n)}^{p}) / (\sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p}) \leq 1$ holds. Thus $0 \leq PFGI 2 \leq 1$ also holds.

(3) Proof of the magnitude range of PFGI3s

Since the sequence $x$ and weight ${w ″}_{n} = (2 n - 1) / N^{2}$ are non-negative, $(\sum_{n = 1}^{N} x_{n}^{p} / N) / (\sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p}) \geq 0$ is satisfied. In addition, the following can be obtained:

\begin{matrix} \sum_{n = 1}^{N} x_{n}^{p} / N - \sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p} = \sum_{n = 1}^{N} x_{n}^{p} \frac{1}{N} - \sum_{n = 1}^{N} x_{(n)}^{p} \frac{2 n - 1}{N^{2}} \\ = \sum_{n = 1}^{N} x_{(n)}^{p} \frac{N - (2 n - 1)}{N^{2}} \\ = {\begin{matrix} \sum_{n = 1}^{N / 2} (x_{(n)}^{p} - x_{(N + 1 - n)}^{p}) \frac{N - (2 n - 1)}{N^{2}} \leq 0, N is even \\ \sum_{n = 1}^{(N - 1) / 2} (x_{(n)}^{p} - x_{(N + 1 - n)}^{p}) \frac{N - (2 n - 1)}{N^{2}} \leq 0, N is odd \end{matrix} \end{matrix}

(12)

Therefore, $0 \leq (\sum_{n = 1}^{N} x_{n}^{p} / N) / (\sum_{n = 1}^{N} {w^{″}}_{n} x_{(n)}^{p}) \leq 1$ holds. Thus $0 \leq PFGI 3 \leq 1$ also holds.

Attributes of the power function-based Gini indices

In this section, six typical attributes of sparsity measures are first reviewed, and then the proofs of the sparse attributes satisfied by PFGI2s and PFGI3s are given to prove that they are sparsity measures.

Six typical attributes of sparsity measures

The six typical sparse attributes proposed in Hurley and Rickard¹³ are considered as important criteria for evaluating sparsity measures, including Robin Hood, Scaling, Rising Tide, Cloning, Bill Gates, and Babies. A good sparsity measure should satisfy as many typical sparse attributes as possible. Knowing the sparse attributes satisfied by health indicators can provide important references for their applications.

Let $x = [x_{1}, x_{2}, \dots, x_{N}]$ denote a discrete sequence with length N and $S (\cdot)$ indicate a sparsity measure, the six typical attributes of sparsity measures are described as follows:

Robin Hood: $S ([x_{1}, \dots, x_{i} - α, \dots, x_{j} + α, \dots])$ $< S ([x_{1}, \dots, x_{i}, \dots, x_{j}, \dots])$ for all $α$ , $x_{i}$ , $x_{j}$ when $x_{i} > x_{j}$ and $0 < α < (x_{i} - x_{j}) / 2$ . Robin Hood operation decreases sparsity, which means that subtracting a small value from an element with a large magnitude to an element with a small magnitude can reduce the sparsity of the sequence.

Scaling: $S (α x) = S (x)$ , $\forall α \in R$ , $α > 0$ . Scaling operation means that the sparsity is scale-invariant and multiplying the sequence by a constant factor does not change the sparsity.

Rising Tide: $S (x + α) < S (x)$ , $\forall α \in R$ , $α > 0$ , where $x = [x_{1}, x_{2}, \dots, x_{N}]$ and $x + α = [x_{1} + α, x_{2} + α, \dots, x_{N} + α]$ . Rising Tide operation means that adding a constant to each element decreases the sparsity of the sequence.

Cloning: $S (x) = S (x ‖ x) = S (x ‖ x ‖ x) = S (x ‖ x ‖ \dots ‖ x)$ , where $\cdot ‖ \cdot$ is the concatenation operation. Sparsity is invariant under cloning operation, which means that repeated splicing of a sequence does not change sparsity.

Bill Gates: $S ([x_{1}, \dots, x_{i} + β, \dots]) > S ([x_{1}, \dots, x_{i}, \dots])$ , $\forall i$ , $\exists β = β_{i} > 0$ . Bill Gates operation increases sparsity, which means that the sequence becomes sparse as an element gets larger.

Babies: $S (x ‖ 0) > S (x)$ , where $0 = [0, 0, \dots, 0]$ is a zero vector with length $M$ . Babies operation increases sparsity, which means that adding zero-magnitude elements to a sequence whose total magnitude is non-zero increases the sparsity of the sequence.

Attributes of the PFGI1s, PFGI2s, and PFGI3s

The typical sparse attributes satisfied by PFGI1s, PFGI2s, and PFGI3s are summarized in Table 1. The proofs of the “Robin Hood” attribute of PFGI1s, PFGI2s, and PFGI3s are presented in Appendix A. Proofs of Proposition 1, Proposition 2, Proposition 3, Proposition 4, and Proposition 5 are given in Appendix B. Proofs of Proposition 6, Proposition 7, Proposition 8, Proposition 9, and Proposition 10 are given in Appendix C. Since this paper mainly focuses on PFGI2s and PFGI3s, only proofs of their typical sparse attributes are presented. Using a similar derivation method, proofs of the sparse attributes satisfied by PFGI1s can be easily obtained.

Table 1.

Typical sparse attributes satisfied by PFGI1s, PFGI2s, and PFGI3s.

Attribute	PFGI1s	PFGI2s	PFGI3s
Robin Hood	Partially satisfied (Appendix A)	Partially satisfied (Appendix A)	Partially satisfied (Appendix A)
Scaling	Satisfied	Satisfied (Proposition 1)	Satisfied (Proposition 6)
Rising Tide	Satisfied	Satisfied (Proposition 2)	Satisfied (Proposition 7)
Cloning	Satisfied	Satisfied (Proposition 3)	Satisfied (Proposition 8)
Bill Gates	Satisfied	Satisfied (Proposition 4)	Satisfied (Proposition 9)
Babies	Satisfied	Satisfied (Proposition 5)	Satisfied (Proposition 10)

The above results indicate that, similar to PFGI1s, PFGI2s and PFGI3s also satisfy at least five of the six typical attributes of sparsity measures. Therefore, the proposed PFGI2s and PFGI3s are new sparsity measures.

Performance analysis via numerical simulation

In this section, the important properties of PFGI2s and PFGI3s, including sparse quantization capability, random transient resistibility and repetitive transient discriminability, are investigated by numerical simulation signals and compared with existing typical sparsity measures, including traditional sparsity measures, GGIs and PFGI1s.

Sparse quantization capability and random transient resistibility

Data sequences from Bernoulli distributions contain only numbers 0 and 1, and data sequences generated by Bernoulli distributions with different success probabilities can simulate different sparsity.^13,38,42,43 Thus, data sequences from Bernoulli distributions with different success probabilities are generated to investigate the sparse quantization capability of PFGI2s and PFGI3s. Traditional sparsity measures, GGIs and PFGI1s are used for comparison. Figure 2 displays the GGIs and seven typical sparsity measures (including kurtosis, NE, L2/L1, HI, GI, GI2, and GI3) of data sequences generated by Bernoulli distribution when success probability increases from 0.005 to 1 in increments of 0.005. All data sequences have a length of 20,000 samples. The weight parameters of GGIs include increasing from 0.1 to 0.9 in increments of 0.1 and from 1.5 to 10 in increments of 0.5. Similarly, Figures 3, 4, and 5 display the PFGI1s, PFGI2s, and PFGI3s of the data sequences generated by Bernoulli distribution with different success probabilities, respectively. The power exponents of PFGI1s, PFGI2s, and PFGI3s include increasing from 0.1 to 0.9 in increments of 0.1 and from 1.5 to 10 in increments of 0.5. The seven typical sparsity measures are also depicted in Figures 3, 4, and 5 for comparison. Note that the values of all sparsity measures are normalized between 0 and 1 for an intuitive comparison.

Similar to seven typical sparsity measures, GGIs and PFGI1s, PFGI2s and PFGI3s with different power exponents monotonically increase as the data sequence gradually becomes sparser, that is, the success probability of the Bernoulli distribution decreases from 1 to 0, as shown in Figures 4 and 5. These results demonstrate that PFGI2s and PFGI3s can monotonically quantify the sparsity of data sequences and can be employed to characterize the repetitive transient features caused by machine defects in the vibration signal. Furthermore, it can be observed that the sparse quantization curves of PFGI1, PFGI2, and PFGI3 transform from convex to concave as the power exponent increases from 0.1 to 10. Differently, as the weight parameter of GGI increases from 0.1 to 10, the sparse quantization curve of GGI transforms from concave to convex. This shows that the sparse quantization capability of PFGI1s, PFGI2s, and PFGI3s can be tuned by changing the power exponent. In addition, it can be observed from Figure 2 that the sparse quantization curve of GGI with a = 0.5 coincides with that of HI, and the sparse quantization curve of GI2 coincides with that of GI3. In Figure 3, the sparse quantization curve of PFGI1 with p = 2 is observed to coincide with the sparse quantization curve of HI. These results indicate that these sparsity measures have very similar or even identical sparse quantization capabilities.

Random transient resistibility is an important property to reveal the sparse quantization performance of a sparsity measure when encountering random transients or outliers. Coincidentally, the descending gradient of the sparse quantization curve reflects the sensitivity of the sparsity measure to transient features, which can reveal the random transient resistibility of the sparsity measure. The smaller the descending gradient, the stronger the random transient resistibility of the sparsity measure; on the contrary, the weaker the random transient resistibility of the sparsity measure. This indicates that transient feature sensitivity and random transient resistibility are a pair of opposite properties. According to the results shown in Figures 2, 3, 4, and 5, the following important findings can be drawn:

Similar to GGIs and PFGI1s, the random transient resistibility of PFGI2s and PFGI3s can be tuned by the power exponent, and decreasing the power exponent can improve the random transient resistibility of PFGI2s and PFGI3s;

The random transient resistibility of PFGI1s, PFGI2s, and PFGI3s is stronger than that of GI, GI2, and GI3, respectively, when the power exponent is less than 1;

The random transient resistibility of PFGI1s, PFGI2s, and PFGI3s is weaker than that of GI, GI2, and GI3, respectively, when the power exponent is greater than 1;

Compared with PFGI3s, changing the power exponent p has a stronger effect on the random transient resistibility of PFGI1s and PFGI2s;

Similar to PFGI1s, the random transient resistibility of PFGI2s and PFGI3s is stronger than that of kurtosis, NE and L2/L1 in characterizing the sparseness of the data sequence.

Repetitive transient discriminability

The repetitive transient discriminability is an important property to reveal the performance of sparsity measures to quantify repetitive transient features under the contamination of interfering noise. Because of the repetition of transient impulses caused by bearing faults, the mixed signal of periodic impulses and white Gaussian noise is generated to investigate the repetitive transient discriminability of PFGI2s and PFGI3s accompanied by comparisons with traditional sparsity measures, GGIs and PFGI1s. The periodic impulse signal contains 100 evenly distributed impulses (the amplitude is set to 2) and has a length of 20000 data points. The SNR of the mixed signal is increased from −20 dB to 40 dB in increments of 0.1 dB. Note that the adjustment of SNR is achieved by changing the intensity of the added white Gaussian noise. Since GGIs, PFGI1s, PFGI2s, and PFGI3s require that the input sequence is non-negative and the envelope of the signal can more effectively highlight the fault features compared to the original signal, the sparsity measures are computed on the envelope of the signal in this section. Figure 6 displays the GGIs and seven typical sparsity measures of the envelopes of the mixed signals with different SNRs. The weight parameters of GGIs include increasing from 0.1 to 0.9 in increments of 0.1 and from 1.5 to 10 in increments of 0.5. Similarly, Figures 7, 8, and 9 exhibit PFGI1s, PFGI2s, and PFGI3s of the envelopes of the mixed signals with different SNRs, respectively. The power exponents of PFGI1s, PFGI2s, and PFGI3s include increasing from 0.1 to 0.9 in increments of 0.1 and from 1.5 to 10 in increments of 0.5. The seven typical sparsity measures are also depicted in Figures 7, 8, and 9 for comparison. Note that all sparsity measures are normalized using Equation (13)³⁹ for intuitive comparison.

v_{norm} (m) = \frac{v_{mixed} (m) - v_{noise} (m)}{v_{impulse} - v_{noise} (m)}

(13)

where $v_{impulse}$ is the sparsity measure value of the envelope of periodic impulses; $v_{noise} (m)$ and $v_{mixed} (m)$ are the sparsity measure values of the envelopes of white Gaussian noise and mixed signal under SNR of m, respectively; $v_{norm} (m)$ is the normalized sparsity measure value under SNR of m.

As shown in Figures 6, 7, 8, and 9, all scaled sparsity measures gradually tend from 0 to 1 as SNR increases from −20 dB to 40 dB. The increasing gradient of the curve of the scaled sparsity measure reflects the sensitivity of the sparsity measure to transient features under noise interference, which can reveal the repetitive transient discriminability of the sparsity measure. The larger the increasing gradient, the stronger the repetitive transient discriminability of the sparsity measure. The results in Figure 6 lead to the following important findings about GGIs:

Decreasing the weight parameter a can improve the repetitive transient discriminability of GGIs, but it does not improve much compared with GI;

The repetitive transient discriminability of GGIs is inferior to kurtosis, NE, HI, GI2, and GI3.

In contrast, PFGI1s, PFGI2s, and PFGI3s obtain more excellent and richer repetitive transient discriminability than seven typical sparsity measures. According to the results in Figures 7, 8, and 9, the following important findings can be drawn:

Similar to PFGI1s, PFGI2s, and PFGI3s can achieve rich repetitive transient discriminability by tuning the power exponent p;

Increasing the power exponent p can significantly improve the repetitive transient discriminability of PFGI1s, PFGI2s, and PFGI3s, which proves the rationality and effectiveness of using the power function;

The effect of changing the power exponent p on the repetitive transient discriminability of PFGI1s and PFGI3s is stronger than that of PFGI2s;

Similar to PFGI1s, PFGI2s, and PFGI3s can achieve stronger repetitive transient discriminability compared with traditional sparsity measures and GGIs;

The effect of the power exponent p on the repetitive transient discriminability of the generalizations of GI is significantly greater than that of the weight parameter a.

A summary of performance analysis

A refined summary can be drawn from the performance analysis of PFGI2s and PFGI3s and comparisons with typical sparsity measures, as follows:

Similar to PFGI1s, PFGI2s, and PFGI3s can monotonically quantify the sparseness of data sequences, and their sparse quantization capability and random transient resistibility can be tuned by the power exponent, but too large power exponent will weaken the random transient resistibility of PFGI1s and PFGI2s compared with PFGI3s.

Similar to PFGI1s, rich repetitive transient discriminability can be achieved by adjusting the power exponent of PFGI2s and PFGI3s and outperform traditional sparsity measures, but too high power exponent makes PFGI1s and PFGI3s too sensitive to transient features and susceptible to interfering noise compared with PFGI2s.

Compared with traditional sparsity measures and GGIs, PFGI1s, PFGI2s, and PFGI3s with appropriate power exponents deliver better capabilities, that is, strong random transient resistibility and repetitive transient discriminability, in characterizing repetitive transient features.

Rolling bearing condition monitoring

In this section, two vibration datasets collected from different bearing run-to-failure experiments are used to validate the performance of the proposed and existing health indicators in bearing condition monitoring. The performance of PFGI1s for bearing condition monitoring has not been explored in previous work,⁴³ which will be discussed in this section. PFGI1s, PFGI2s, and PFGI3s with different power exponents p = 0.1, 0.3, 0.5, 0.8, 1, 2, 3, 4, 5, and 6 are investigated, and NE, HI, and GGIs with different weight parameters a = 0.1, 0.3, 0.5, 0.8, 2, 3, 4, and 6 are selected for comparison. In addition, because of the advantage of the squared envelope spectrum in revealing the cyclostationarity of the repetitive transient features caused by bearing faults,⁴⁵ the sparsity measure of the squared envelope spectrum of the monitored vibration signal is employed to characterize the degradation state of the rolling bearings.

Case 1: CIMS bearing condition monitoring

In the first case, the bearing run-to-failure dataset provided by the NSF I/UCR Center for Intelligent Maintenance Systems (CIMS)⁴⁶ is analyzed. The bearing degradation test rig mainly consists of a shaft and four Rexnord ZA-2115 double-row bearings. The shaft was driven by a motor through a belt and rotated at a constant speed of 2000 rpm. A radial load of 6000 lbs was applied to the shaft and tested bearings. Accelerometers were mounted on the bearing housings to acquire the vibration acceleration signal of each tested bearing with a sampling frequency of 20,000 Hz. At the end of the run-to-failure experiment, an outer race failure occurred in bearing 1. A total of 984 data files were recorded, and each file contained 20,480 data points. The vibration acceleration signal of bearing 1 run-to-failure is displayed in Figure 10. It can be observed that the magnitude of the vibration signal is basically stable before the bearing fault occurs. After the bearing fault occurs, the magnitude of the vibration signal gradually increases until the bearing fails. However, the time-domain waveform of the vibration signal cannot accurately reveal the occurrence time of the early bearing fault.

To evaluate the degradation status of the tested bearing, Figure 11 exhibits the bearing degradation curves characterized by NE, HI, and GGIs with weight parameters a = 0.1, 0.3, 0.5, 0.8, 2, 3, 4, and 6 of the squared envelope spectrum of the raw vibration signal. Figures 12, 13, and 14 display the bearing degradation curves characterized by PFGI1s, PFGI2s, and PFGI3s with power exponents p = 0.1, 0.3, 0.5, 0.8, 1, 2, 3, 4, 5, and 6 of the squared envelope spectrum of the raw vibration signal, respectively. It can be observed that NE, HI, GGIs, PFGI1s, PFGI2s, and PFGI3s remain basically stable when the bearing is in the normal state, but they all exhibit an increasing trend after the occurrence of the early bearing fault. These results show that NE, HI, GGIs, PFGI1s, PFGI2s, and PFGI3s effectively detect the occurrence of the early bearing fault. However, GGIs with weight parameter a ≥ 0.8 exhibit large fluctuations during the fault evolution process, and similar phenomena occur for PFGI1s with power exponent p ≤ 1, PFGI2s with power exponent p ≤ 1 and PFGI3s with power exponent p ≤ 1. Compared with GGIs with a = 0.1, 0.3, and 0.5, the bearing degradation curves characterized by NE and HI fluctuate less, showing relatively better condition monitoring effect. The bearing degradation curves characterized by PFGI1 with p = 2, PFGI2 with p = 2 and PFGI3 with p = 2 are similar to those of NE and HI, indicating that they achieve similar bearing condition monitoring performance. In the process of bearing fault evolution, the bearing degradation curves described by PFGI1s with p ≥ 3, PFGI2s with p ≥ 3 and PFGI3s with p ≥ 3 show smaller fluctuations than those of NE and HI, and exhibit better monitoring capability of bearing degradation state. However, when the power exponent is too large, PFGI1s, PFGI2s, and PFGI3s are too sensitive to transient features and are easily affected by interference noise, resulting in relatively large fluctuations in the normal state of the bearing. Bearing degradation is irreversible and an approximately monotonic degradation trend is expected (e.g., remaining useful life prediction⁴⁷), therefore, in this case, PFGI1s with p = 3 and p = 4, PFGI2s with p = 3 and p = 4 and PFGI3 with p = 3 deliver excellent capability to characterize bearing degradation state compared with other explored sparsity measures.

Case 2: XJTU-SY bearing condition monitoring

In the second case, the bearing run-to-failure dataset provided by the Institute of Design Science and Basic Component at Xi’an Jiaotong University (XJTU), Shaanxi, China and the Changxing Sumyoung Technology Co., Ltd. (SY), Zhejiang, China⁴⁸ is employed to further validate the performance of the new sparsity measures. The test rig mainly consists of an alternating current motor, a motor speed controller, a support shaft, two support bearings, a tested bearing, and a hydraulic loader. The load was applied horizontally on the housing of the tested bearing. Accelerometers mounted on the bearing housing were used to collect vibration signals in the horizontal and vertical directions at a sampling frequency of 25,600 Hz.

The dataset of bearing 3-3 tested under the condition of shaft speed of 2400 rpm and load of 10 kN is analyzed in this section. The inner race of the bearing failed at the end of the experiment. A total of 371 data files were recorded, and each data file contained 32,768 data points. Figure 15 shows the vibration acceleration signal (horizontal direction) of bearing 3-3 run-to-failure. The magnitude of the vibration signal remains stable after the start of the experiment, but the magnitude of the vibration signal sharply increases at the end of the experiment, indicating that the bearing failure is relatively rapid.

To reveal the bearing degradation process, the bearing degradation curves characterized by NE, HI, and GGIs with weight parameters a = 0.1, 0.3, 0.5, 0.8, 2, 3, 4, and 6 of the squared envelope spectrum of the raw vibration signal are depicted in Figure 16. Similarly, Figures 17, 18, and 19 display the bearing degradation curves characterized by PFGI1s, PFGI2s, and PFGI3s with power exponents p = 0.1, 0.3, 0.5, 0.8, 1, 2, 3, 4, 5, and 6 of the squared envelope spectrum of the raw vibration signal, respectively. It can be observed that NE, HI, GGIs, PFGI1s, PFGI2s, and PFGI3s increase rapidly after the occurrence of bearing fault (approximately the acquisition time of the 340th data file), showing that they effectively characterize the fault evolution of test bearing. However, before the occurrence of bearing fault, GGIs show a downward trend and deviate from the baseline, which became more obvious with the increase of the weight parameter, and similar results also appear for PFGI1s with power exponentp ≤ 1, PFGI2s with power exponent p ≤ 1 and PFGI3s with power exponent p ≤ 1. Such a phenomenon is not conducive to the condition monitoring of the bearing because it may lead to misjudgment of the health status of the bearing. In contrast, NE, HI, PFGI1s with power exponent p ≥ 2, PFGI2s with power exponent p ≥ 2 and PFGI3s with power exponent p ≥ 2 remain stable around the baseline before bearing fault occurs, delivering better bearing condition monitoring performance. NE, HI, PFGI1 with p = 2, PFGI2 with p = 2 and PFGI3 with p = 2 show similar bearing degradation curves, exhibiting a sharp drop before bearing failure, which is not consistent with the magnitude evolution of the bearing vibration signal shown in Figure 15. The degradation curves characterized by PFGI1s with p ≥ 4, PFGI2s with p ≥ 4 and PFGI3s with p ≥ 4 have a small downward trend before bearing failure, but they show large fluctuations before bearing fault, which further indicates that the power exponent of PFGI1s, PFGI2s and PFGI3s should not be chosen too large. Therefore, in this case, PFGI1 with p = 3, PFGI2 with p = 3 and PFGI3 with p = 3 exhibit excellent capability to characterize bearing degradation states compared with other explored sparsity measures.

Conclusions

In this paper, PFGI2s and PFGI3s are proposed as single-parameter generalizations of GI2 and GI3, respectively, for sparse quantization of transient features by using power function-based quasi-arithmetic means. Typical sparse attributes of PFGI2s and PFGI3s are revealed through mathematical derivation. The performance and characteristics of PFGI1s, PFGI2s, and PFGI3s in transient feature quantification and bearing condition monitoring are investigated by simulation analysis and experimental analysis and compared with traditional and state-of-the-art sparsity measures. The main conclusions drawn from the results of this paper are as follows:

Similar to PFGI1s, PFGI2s, and PFGI3s are capable of monotonically quantifying the sparseness of a data sequence (i.e., PFGI2s and PFGI3s gradually increase as the data sequence becomes sparser), and satisfy at least five of the six typical attributes of sparsity measures, which are two new families of sparsity measures.

Compared with seven typical sparsity measures and GGIs, PFGI1s, PFGI2s, and PFGI3s with appropriate power exponents can simultaneously achieve strong random transient resistibility and strong repetitive transient discriminability, thus delivering better performance in quantifying repetitive transient features.

In two bearing experimental cases including gradual failure and sudden failure, PFGI1s, PFGI2s, and PFGI3s with power exponents p = 2 and p = 3 can effectively characterize the bearing degradation state and detect the occurrence of early bearing faults. PFGI1, PFGI2, and PFGI3 with power exponent p = 3 can accurately characterize the bearing health state in both the normal state and fault evolution process of the bearing, therefore they are preferentially recommended for bearing condition monitoring.

This paper focuses on the elucidation and validation of the sparse quantification capabilities of PFGI1s, PFGI2s, and PFGI3s and their performance in bearing condition monitoring. In the experimental analysis, only part of the power exponent values are applied, the feature characterization and condition monitoring performance of PFGI1s, PFGI2s, and PFGI3s with more power exponents are worth investigating to achieve a deep understanding. The sensitivity of sparsity measures to impulsive signals is discovered to have an important influence on their performance in machine condition monitoring,⁴⁹ indicating that the robustness of the proposed sparsity measures to random transient noise requires a detailed exploration. The transient feature enhancement methods based on new sparsity measures are worth developing to improve the accuracy of fault diagnosis of rotating machinery components. In addition, to obtain the monotonic bearing degradation trend for remaining useful life prediction, an adaptive weighted signal preprocessing technique⁵⁰ can be applied to PFGI1s, PFGI2s, and PFGI3s to improve their performance.

Footnotes

Appendix A. Example of PFGI1s,PFGI2s and PFGI3s partially satisfying attribute “Robin Hood”

Two data sequences $s_{1} = [0.3, 1, 2]$ and $s_{2} = [0.31, 0.99, 2]$ ¹³ are used to illustrate PFGI1s, PFGI2s and PFGI3s partially satisfying the sparse attribute “Robin Hood.” Note that $s_{2} = [0.31, 0.99, 2]$ is the resulting sequence after performing the Robin Hood operation on $s_{1} = [0.3, 1, 2]$ . The difference value between the sparsity measure of the data sequence $s_{2}$ and the sparsity measure of $s_{1}$ is calculated. As pointed out in article,¹³ if the difference is less than zero, the sparsity measure satisfies Robin Hood; otherwise, the sparsity measure does not satisfy Robin Hood.

Figure 20(a) to (c) display the difference values of PFGI1s, PFGI2s, and PFGI3s with different power exponents before and after performing the Robin Hood operation on the data sequence, respectively. The right column is a partial enlargement of the left column. As shown in Figure 20, as the power exponent p increases, PFGI1, PFGI2, and PFGI3 first increase and exceed zero, and then gradually decrease but are always greater than zero, indicating that PFGI1s, PFGI2s, and PFGI3s partially satisfy the sparse property “Robin Hood.”

Appendix B. Proof of the typical sparse attributes satisfied by PFGI2s

Proposition 1: PFGI2s satisfy Scaling.

PFGI 2 (α x) = PFGI 2 (x), \forall α \in R, α > 0

Proof of Proposition 1: The following can be obtained:

\begin{matrix} PFGI 2 (α x) = 1 - \sqrt[p]{\frac{\sum_{n = 1}^{N} w'_{n} {(α x_{(n)})}^{p}}{\sum_{n = 1}^{N} {w ″}_{n} {(α x_{(n)})}^{p}}} \\ = 1 - \sqrt[p]{\frac{\sum_{n = 1}^{N} w'_{n} x_{(n)}^{p}}{\sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p}}} = PFGI 2 (x) \end{matrix}

where $w'_{n} = [2 (N - n) + 1] / N^{2}$ and ${w ″}_{n} = (2 n - 1) / N^{2}$ . This is true for PFGI2s.

Proposition 2: PFGI2s satisfy Rising Tide.

PFGI 2 (x + α) < PFGI 2 (x), \forall α \in R, α > 0

where $x = [x_{(1)}, x_{(2)}, \dots, x_{(N)}]$ , $x + α = [x_{(1)} + α, x_{(2)} + α, \dots, x_{(N)} + α]$ , and $x_{(1)} \leq x_{(2)} \leq \dots \leq x_{(N)}$ .

Proof of Proposition 2: The following can be obtained:

\begin{matrix} PFGI 2 (x + α) - PFGI 2 (x) = \sqrt[p]{\frac{\sum_{n = 1}^{N} w'_{n} x_{(n)}^{p}}{\sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p}}} \\ - \sqrt[p]{\frac{\sum_{n = 1}^{N} w'_{n} {(x_{(n)} + α)}^{p}}{\sum_{n = 1}^{N} {w ″}_{n} {(x_{(n)} + α)}^{p}}}, p > 0 \end{matrix}

where $w'_{n} = [2 (N - n) + 1] / N^{2}$ and ${w ″}_{n} = (2 n - 1) / N^{2}$ . A simplified form can be obtained:

\begin{matrix} \frac{\sum_{n = 1}^{N} w'_{n} x_{(n)}^{p}}{\sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p}} - \frac{\sum_{n = 1}^{N} w'_{n} {(x_{(n)} + α)}^{p}}{\sum_{n = 1}^{N} {w ″}_{n} {(x_{(n)} + α)}^{p}} = \frac{\sum_{n = 1}^{N} w'_{n} x_{(n)}^{p}}{\sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p}} - \frac{\sum_{n = 1}^{N} w'_{n} (x_{(n)}^{p} + \sum_{k = 1}^{p} C_{p}^{k} x_{(n)}^{p - k} α^{k})}{\sum_{n = 1}^{N} {w ″}_{n} {(x_{(n)} + α)}^{p}} \\ = \frac{(\sum_{i = 1}^{N} w'_{i} x_{(i)}^{p}) (\sum_{j = 1}^{N} {w ″}_{j} (x_{(j)}^{p} + \sum_{k = 1}^{p} C_{p}^{k} x_{(j)}^{p - k} α^{k})) - (\sum_{i = 1}^{N} w'_{i} (x_{(i)}^{p} + \sum_{k = 1}^{p} C_{p}^{k} x_{(i)}^{p - k} α^{k})) (\sum_{j = 1}^{N} {w ″}_{j} x_{(j)}^{p})}{(\sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p}) (\sum_{n = 1}^{N} {w ″}_{n} {(x_{(n)} + α)}^{p})} \\ = \frac{(\sum_{i = 1}^{N} w'_{i} x_{(i)}^{p}) (\sum_{j = 1}^{N} {w ″}_{j} (\sum_{k = 1}^{p} C_{p}^{k} x_{(j)}^{p - k} α^{k})) - (\sum_{i = 1}^{N} w'_{i} (\sum_{k = 1}^{p} C_{p}^{k} x_{(i)}^{p - k} α^{k})) (\sum_{j = 1}^{N} {w ″}_{j} x_{(j)}^{p})}{(\sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p}) (\sum_{n = 1}^{N} {w ″}_{n} {(x_{(n)} + α)}^{p})} \\ = \frac{\sum_{i = 1}^{N} x_{(i)}^{p} [w'_{i} (\sum_{j = 1}^{N} {w ″}_{j} (\sum_{k = 1}^{p} C_{p}^{k} x_{(j)}^{p - k} α^{k})) - {w ″}_{i} (\sum_{j = 1}^{N} w'_{j} (\sum_{k = 1}^{p} C_{p}^{k} x_{(j)}^{p - k} α^{k}))]}{(\sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p}) (\sum_{n = 1}^{N} {w ″}_{n} {(x_{(n)} + α)}^{p})} \\ = \frac{\sum_{i = 1}^{N} x_{(i)}^{p} [\sum_{j = 1}^{N} (w'_{i} {w ″}_{j} - w'_{j} {w ″}_{i}) (\sum_{k = 1}^{p} C_{p}^{k} x_{(j)}^{p - k} α^{k})]}{(\sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p}) (\sum_{n = 1}^{N} {w ″}_{n} {(x_{(n)} + α)}^{p})} \\ = \frac{\sum_{i = 1}^{N} (\sum_{j = i + 1}^{N} (w'_{i} {w ″}_{j} - w'_{j} {w ″}_{i}) [x_{(i)}^{p} (\sum_{k = 1}^{p} C_{p}^{k} x_{(j)}^{p - k} α^{k}) - x_{(j)}^{p} (\sum_{k = 1}^{p} C_{p}^{k} x_{(i)}^{p - k} α^{k})])}{(\sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p}) (\sum_{n = 1}^{N} {w ″}_{n} {(x_{(n)} + α)}^{p})} \\ = \frac{\sum_{i = 1}^{N} (\sum_{j = i + 1}^{N} x_{(i)}^{p} x_{(j)}^{p} (w'_{i} {w ″}_{j} - w'_{j} {w ″}_{i}) [\frac{\sum_{k = 1}^{p} C_{p}^{k} x_{(j)}^{p - k} α^{k}}{x_{(j)}^{p}} - \frac{\sum_{k = 1}^{p} C_{p}^{k} x_{(i)}^{p - k} α^{k}}{x_{(i)}^{p}}])}{(\sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p}) (\sum_{n = 1}^{N} {w ″}_{n} {(x_{(n)} + α)}^{p})} \end{matrix}

Define the following function:

h (x) = (\sum_{k = 1}^{p} C_{p}^{k} x^{p - k} α^{k}) / x^{p}, α > 0

Then we can get:

\begin{matrix} \frac{d h (x)}{d x} = & \frac{x^{p} (\sum_{k = 1}^{p - 1} (p - k) C_{p}^{k} x^{p - k - 1} α^{k}) - p x^{p - 1} (\sum_{k = 1}^{p} C_{p}^{k} x^{p - k} α^{k})}{x^{2 p}} \\ = & \frac{(\sum_{k = 1}^{p - 1} (p - k) C_{p}^{k} x^{2 p - k - 1} α^{k}) - p (\sum_{k = 1}^{p} C_{p}^{k} x^{2 p - k - 1} α^{k})}{x^{2 p}} = \frac{- \sum_{k = 1}^{p} {kC}_{p}^{k} α^{k} x^{2 p - k - 1}}{x^{2 p}} < 0, \forall x > 0 \end{matrix}

This means that the function $h (x) = (\sum_{k = 1}^{p} C_{p}^{k} x^{p - k} α^{k}) / x^{p}$ decreases monotonically. In addition, $0 < x_{(i)} < x_{(j)}$ , $w'_{i} > w'_{j}$ and ${w ″}_{j} > {w ″}_{i}$ are satisfied, therefore, $PFGI 2 (x + α) - PFGI 2 (x) < 0$ is true for PFGI2s.

Proposition 3: PFGI2s satisfy Cloning.

PFGI 2 (x) = PFGI 2 (x ‖ x) = PFGI 2 (x ‖ x ‖ x) = PFGI 2 (x ‖ x ‖ \dots ‖ x)

Proof of Proposition 3: Let $y$ represent the resulting sequence after sequence $x$ cloned $M$ times and have a length of $MN$ . The following can be obtained:

\begin{matrix} PFGI 2 (y) = PFGI 2 (\underset{M}{\underset{︸}{x ‖ \dots ‖ x}}) = 1 - \sqrt[p]{\frac{\sum_{n = 1}^{MN} y_{(n)}^{p} [2 (MN - n) + 1]}{\sum_{n = 1}^{MN} y_{(n)}^{p} (2 n - 1)}} \\ = 1 - \sqrt[p]{\frac{\sum_{n = 1}^{N} x_{(n)}^{p} \sum_{m = 1}^{M} 2 [MN - (M (n - 1) + m) + 1]}{\sum_{n = 1}^{N} x_{(n)}^{p} \sum_{m = 1}^{M} [2 (M (n - 1) + m) - 1]}} \\ = 1 - \sqrt[p]{\frac{\sum_{n = 1}^{N} x_{(n)}^{p} (2 M^{2} N - 2 M^{2} n + M^{2})}{\sum_{n = 1}^{N} x_{(n)}^{p} (2 M^{2} n - M^{2})}} = 1 - \sqrt[p]{\frac{\sum_{n = 1}^{N} x_{(n)}^{p} [2 (N - n) + 1]}{\sum_{n = 1}^{N} x_{(n)}^{p} (2 n - 1)}} = PFGI 2 (x) \end{matrix}

This is true for PFGI2s.

Proposition 4: PFGI2s satisfy Bill Gates.

PFGI 2 ([x_{(1)}, \dots, x_{(i)} + β, \dots, x_{(N)}]) > PFGI 2 ([x_{(1)}, \dots, x_{(i)}, \dots, x_{(N)}]), \forall i, \exists β = β_{i} > 0

Proof of Proposition 4: Without loss of generality, Bill Gates operation is chosen to be executed on $x_{(N)}$ because the additional value $β$ can change any $x_{(i)}$ to $x_{(N)}$ . The resulting sequence after performing Bill Gates operation on $x = [x_{(1)}, x_{(2)}, \dots, x_{(N)}]$ can be denoted as $y = [x_{(1)}, x_{(2)}, \dots, x_{(N)} + β]$ . The following can be obtained:

PFGI 2 (y) - PFGI 2 (x) = \sqrt[p]{\frac{\sum_{n = 1}^{N} w'_{n} x_{(n)}^{p}}{\sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p}}} - \sqrt[p]{\frac{\sum_{n = 1}^{N} w'_{n} y_{(n)}^{p}}{\sum_{n = 1}^{N} {w ″}_{n} y_{(n)}^{p}}}, p > 0

where $w'_{n} = [2 (N - i) + 1] / N^{2}$ and ${w ″}_{n} = (2 n - 1) / N^{2}$ . Then, we can get:

\begin{matrix} \frac{\sum_{n = 1}^{N} w'_{n} x_{(n)}^{p}}{\sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p}} - \frac{\sum_{n = 1}^{N} w'_{n} y_{(n)}^{p}}{\sum_{n = 1}^{N} {w ″}_{n} y_{(n)}^{p}} = \frac{\sum_{n = 1}^{N} w'_{n} x_{(n)}^{p}}{\sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p}} - \frac{\sum_{n = 1}^{N} w'_{n} x_{(n)}^{p} + w'_{N} \sum_{k = 1}^{p} C_{p}^{k} x_{(N)}^{p - k} β^{k}}{\sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p} + {w ″}_{N} \sum_{k = 1}^{p} C_{p}^{k} x_{(N)}^{p - k} β^{k}} \\ = \frac{\sum_{n = 1}^{N} w'_{n} x_{(n)}^{p}}{\sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p}} - \frac{\sum_{n = 1}^{N} w'_{n} x_{(n)}^{p}}{\sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p} + {w ″}_{N} \sum_{k = 1}^{p} C_{p}^{k} x_{(N)}^{p - k} β^{k}} - \frac{w'_{N} \sum_{k = 1}^{p} C_{p}^{k} x_{(N)}^{p - k} β^{k}}{\sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p} + {w ″}_{N} \sum_{k = 1}^{p} C_{p}^{k} x_{(N)}^{p - k} β^{k}} \\ = \sum_{n = 1}^{N} w'_{n} x_{(n)}^{p} (\frac{1}{\sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p}} - \frac{1}{\sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p} + {w ″}_{N} \sum_{k = 1}^{p} C_{p}^{k} x_{(N)}^{p - k} β^{k}}) - \frac{w'_{N} \sum_{k = 1}^{p} C_{p}^{k} x_{(N)}^{p - k} β^{k}}{\sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p} + {w ″}_{N} \sum_{k = 1}^{p} C_{p}^{k} x_{(N)}^{p - k} β^{k}} \\ = \frac{\sum_{n = 1}^{N} w'_{n} x_{(n)}^{p} ({w ″}_{N} \sum_{k = 1}^{p} C_{p}^{k} x_{(N)}^{p - k} β^{k})}{(\sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p}) (\sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p} + {w ″}_{N} \sum_{k = 1}^{p} C_{p}^{k} x_{(N)}^{p - k} β^{k})} - \frac{w'_{N} \sum_{k = 1}^{p} C_{p}^{k} x_{(N)}^{p - k} β^{k}}{\sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p} + {w ″}_{N} \sum_{k = 1}^{p} C_{p}^{k} x_{(N)}^{p - k} β^{k}} \\ = \frac{\sum_{k = 1}^{p} C_{p}^{k} x_{(N)}^{p - k} β^{k}}{\sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p} + {w ″}_{N} \sum_{k = 1}^{p} C_{p}^{k} x_{(N)}^{p - k} β^{k}} (\frac{\sum_{n = 1}^{N} w'_{n} x_{(n)}^{p}}{\sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p}} - w'_{N}) \\ = \frac{\sum_{k = 1}^{p} C_{p}^{k} x_{(N)}^{p - k} β^{k}}{\sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p} + {w ″}_{N} \sum_{k = 1}^{p} C_{p}^{k} x_{(N)}^{p - k} β^{k}} \frac{\sum_{n = 1}^{N} x_{(n)}^{p} (w'_{n} - w'_{N} {w ″}_{n})}{\sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p}} > 0 \end{matrix}

Since the weight decreases monotonically with the increase of rank, this is true for $N > 1$ .

Proposition 5: PFGI2s satisfy Babies.

PFGI 2 (x ‖ 0) > PFGI 2 (x)

Proof of Proposition 5: The resulting sequence after performing Babies operation on $x = [x_{(1)}, x_{(2)}, \dots, x_{(N)}]$ can be expressed as

y = 0 ‖ x = [\begin{matrix} \underset{M}{\underset{︸}{\begin{matrix} 0 & \dots & 0 \end{matrix}}} & x_{(1)} & x_{(2)} & \dots & x_{(N)} \end{matrix}]

Then, the following can be obtained:

\begin{matrix} PFGI 2 (y) - PFGI 2 (x) = \sqrt[p]{\frac{\sum_{n = 1}^{N} x_{(n)}^{p} [2 (N - n) + 1]}{\sum_{n = 1}^{N} x_{(n)}^{p} (2 n - 1)}} - \sqrt[p]{\frac{\sum_{n = 1}^{N + M} y_{(n)}^{p} [2 (N + M - n) + 1]}{\sum_{n = 1}^{N + M} y_{(n)}^{p} (2 n - 1)}} \\ = \sqrt[p]{\frac{\sum_{n = 1}^{N} x_{(n)}^{p} [2 (N - n) + 1]}{\sum_{n = 1}^{N} x_{(n)}^{p} (2 n - 1)}} - \sqrt[p]{\frac{\sum_{n = 1}^{N} x_{(n)}^{p} [2 (N - n) + 1]}{\sum_{n = 1}^{N} x_{(n)}^{p} (2 M + 2 n - 1)}} \\ = \sqrt[p]{\frac{\sum_{n = 1}^{N} x_{(n)}^{p} [2 (N - n) + 1]}{\sum_{n = 1}^{N} x_{(n)}^{p} (2 n - 1)}} - \sqrt[p]{\frac{\sum_{n = 1}^{N} x_{(n)}^{p} [2 (N - n) + 1]}{\sum_{n = 1}^{N} x_{(n)}^{p} (2 n - 1) + 2 M \sum_{n = 1}^{N} x_{(n)}^{p}}} > 0 \end{matrix}

This is true for PFGI2s.

Appendix C. Proofs of the typical sparse attributes satisfied by PFGI3s

Proposition 6: PFGI3s satisfy Scaling.

PFGI 3 (α x) = PFGI 3 (x), \forall α \in R, α > 0

Proof of Proposition 6: The following can be obtained:

PFGI 3 (α x) = 1 - \sqrt[p]{\frac{\sum_{n = 1}^{N} {(α x_{n})}^{p} / N}{\sum_{n = 1}^{N} {w ″}_{n} {(α x_{(n)})}^{p}}} = 1 - \sqrt[p]{\frac{\sum_{n = 1}^{N} x_{n}^{p} / N}{\sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p}}} = PFGI 3 (x)

where ${w ″}_{n} = (2 n - 1) / N^{2}$ . This is true for PFGI3s.

Proposition 7: PFGI3s satisfy Rising Tide.

PFGI 3 (x + α) < PFGI 3 (x), \forall α \in R, α > 0

where $x = [x_{(1)}, x_{(2)}, \dots, x_{(N)}]$ , $x + α = [x_{(1)} + α, x_{(2)} + α, \dots, x_{(N)} + α]$ , and $x_{(1)} \leq x_{(2)} \leq \dots \leq x_{(N)}$ .

Proof of Proposition 7: The following can be obtained:

PFGI 3 (x + α) - PFGI 3 (x) = \sqrt[p]{\frac{\sum_{n = 1}^{N} x_{(n)}^{p}}{N \sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p}}} - \sqrt[p]{\frac{\sum_{n = 1}^{N} {(x_{(n)} + α)}^{p}}{N \sum_{n = 1}^{N} {w ″}_{n} {(x_{(n)} + α)}^{p}}}, p > 0

where ${w ″}_{n} = (2 n - 1) / N^{2}$ . A simplified form can be obtained:

\begin{matrix} \frac{\sum_{n = 1}^{N} x_{(n)}^{p}}{N \sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p}} - \frac{\sum_{n = 1}^{N} {(x_{(n)} + α)}^{p}}{N \sum_{n = 1}^{N} {w ″}_{n} {(x_{(n)} + α)}^{p}} = \frac{\sum_{n = 1}^{N} x_{(n)}^{p}}{N \sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p}} - \frac{\sum_{n = 1}^{N} (x_{(n)}^{p} + \sum_{k = 1}^{p} C_{p}^{k} x_{(n)}^{p - k} α^{k})}{N \sum_{n = 1}^{N} {w ″}_{n} {(x_{(n)} + α)}^{p}} \\ = \frac{(\sum_{i = 1}^{N} x_{(i)}^{p}) (\sum_{j = 1}^{N} {w ″}_{j} (x_{(j)}^{p} + \sum_{k = 1}^{p} C_{p}^{k} x_{(j)}^{p - k} α^{k})) - (\sum_{i = 1}^{N} (x_{(i)}^{p} + \sum_{k = 1}^{p} C_{p}^{k} x_{(i)}^{p - k} α^{k})) (\sum_{j = 1}^{N} {w ″}_{j} x_{(j)}^{p})}{N (\sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p}) (\sum_{n = 1}^{N} {w ″}_{n} {(x_{(n)} + α)}^{p})} \\ = \frac{(\sum_{i = 1}^{N} x_{(i)}^{p}) (\sum_{j = 1}^{N} {w ″}_{j} (\sum_{k = 1}^{p} C_{p}^{k} x_{(j)}^{p - k} α^{k})) - (\sum_{i = 1}^{N} (\sum_{k = 1}^{p} C_{p}^{k} x_{(i)}^{p - k} α^{k})) (\sum_{j = 1}^{N} {w ″}_{j} x_{(j)}^{p})}{N (\sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p}) (\sum_{n = 1}^{N} {w ″}_{n} {(x_{(n)} + α)}^{p})} \\ = \frac{\sum_{i = 1}^{N} x_{(i)}^{p} [(\sum_{j = 1}^{N} {w ″}_{j} (\sum_{k = 1}^{p} C_{p}^{k} x_{(j)}^{p - k} α^{k})) - {w ″}_{i} (\sum_{j = 1}^{N} (\sum_{k = 1}^{p} C_{p}^{k} x_{(j)}^{p - k} α^{k}))]}{N (\sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p}) (\sum_{n = 1}^{N} {w ″}_{n} {(x_{(n)} + α)}^{p})} \\ = \frac{\sum_{i = 1}^{N} x_{(i)}^{p} [\sum_{j = 1}^{N} ({w ″}_{j} - {w ″}_{i}) (\sum_{k = 1}^{p} C_{p}^{k} x_{(j)}^{p - k} α^{k})]}{N (\sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p}) (\sum_{n = 1}^{N} {w ″}_{n} {(x_{(n)} + α)}^{p})} \\ = \frac{\sum_{i = 1}^{N} (\sum_{j = i + 1}^{N} ({w ″}_{j} - {w ″}_{i}) [x_{(i)}^{p} (\sum_{k = 1}^{p} C_{p}^{k} x_{(j)}^{p - k} α^{k}) - x_{(j)}^{p} (\sum_{k = 1}^{p} C_{p}^{k} x_{(i)}^{p - k} α^{k})])}{N (\sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p}) (\sum_{n = 1}^{N} {w ″}_{n} {(x_{(n)} + α)}^{p})} \\ = \frac{\sum_{i = 1}^{N} (\sum_{j = i + 1}^{N} x_{(i)}^{p} x_{(j)}^{p} ({w ″}_{j} - {w ″}_{i}) [\frac{\sum_{k = 1}^{p} C_{p}^{k} x_{(j)}^{p - k} α^{k}}{x_{(j)}^{p}} - \frac{\sum_{k = 1}^{p} C_{p}^{k} x_{(i)}^{p - k} α^{k}}{x_{(i)}^{p}}])}{N (\sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p}) (\sum_{n = 1}^{N} {w ″}_{n} {(x_{(n)} + α)}^{p})} \end{matrix}

Define the following function:

h (x) = (\sum_{k = 1}^{p} C_{p}^{k} x^{p - k} α^{k}) / x^{p}, α > 0

Then we can get:

\begin{matrix} \frac{d h (x)}{d x} = \frac{x^{p} (\sum_{k = 1}^{p - 1} (p - k) C_{p}^{k} x^{p - k - 1} α^{k}) - p x^{p - 1} (\sum_{k = 1}^{p} C_{p}^{k} x^{p - k} α^{k})}{x^{2 p}} \\ = \frac{(\sum_{k = 1}^{p - 1} (p - k) C_{p}^{k} x^{2 p - k - 1} α^{k}) - p (\sum_{k = 1}^{p} C_{p}^{k} x^{2 p - k - 1} α^{k})}{x^{2 p}} = \frac{- \sum_{k = 1}^{p} {kC}_{p}^{k} α^{k} x^{2 p - k - 1}}{x^{2 p}} < 0, \forall x > 0 \end{matrix}

This shows that the function $h (x) = (\sum_{k = 1}^{p} C_{p}^{k} x^{p - k} α^{k}) / x^{p}$ decreases monotonically. In addition, $0 < x_{(i)} < x_{(j)}$ and ${w ″}_{j} > {w ″}_{i}$ are satisfied, therefore, $PFGI 3 (x + α) - PFGI 3 (x) < 0$ is true for PFGI3s.

Proposition 8: PFGI3s satisfy Cloning.

PFGI 3 (x) = PFGI 3 (x ‖ x) = PFGI 3 (x ‖ x ‖ x) = PFGI 3 (x ‖ x ‖ \dots ‖ x)

Proof of Proposition 8: Let $y$ represent the resulting sequence after sequence $x$ cloned $M$ times and have a length of $MN$ . The following can be obtained:

\begin{matrix} PFGI 3 (y) = PFGI 3 (\underset{M}{\underset{︸}{x ‖ \dots ‖ x}}) = 1 - \sqrt[p]{\frac{MN \sum_{n = 1}^{MN} y_{n}^{p}}{\sum_{n = 1}^{MN} y_{(n)}^{p} (2 n - 1)}} = 1 - \sqrt[p]{\frac{M^{2} N \sum_{n = 1}^{N} x_{(n)}^{p}}{\sum_{n = 1}^{N} x_{(n)}^{p} \sum_{m = 1}^{M} [2 (M (n - 1) + m) - 1]}} \\ = 1 - \sqrt[p]{\frac{\sum_{n = 1}^{N} x_{n}^{p} (M^{2} N)}{\sum_{n = 1}^{N} x_{(n)}^{p} (2 M^{2} n - M^{2})}} = 1 - \sqrt[p]{\frac{N \sum_{n = 1}^{N} x_{n}^{p}}{\sum_{n = 1}^{N} x_{(n)}^{p} (2 n - 1)}} = PFGI 3 (x) \end{matrix}

This is true for PFGI3s.

Proposition 9: PFGI3s satisfy Bill Gates.

PFGI 3 ([x_{(1)}, \dots, x_{(i)} + β, \dots, x_{(N)}]) > PFGI 3 ([x_{(1)}, \dots, x_{(i)}, \dots, x_{(N)}]), \forall i, \exists β = β_{i} > 0

Proof of Proposition 9: Without loss of generality, Bill Gates operation is chosen to be executed on $x_{(N)}$ because the additional value $β$ can change any $x_{(i)}$ to $x_{(N)}$ . The resulting sequence after performing Bill Gates operation on $x = [x_{(1)}, x_{(2)}, \dots, x_{(N)}]$ can be denoted as $y = [x_{(1)}, x_{(2)}, \dots, x_{(N)} + β]$ . The following can be obtained:

PFGI 3 (y) - PFGI 3 (x) = \sqrt[p]{\frac{\sum_{n = 1}^{N} x_{(n)}^{p} / N}{\sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p}}} - \sqrt[p]{\frac{\sum_{n = 1}^{N} y_{(n)}^{p} / N}{\sum_{n = 1}^{N} {w ″}_{n} y_{(n)}^{p}}}, p > 0

where ${w ″}_{n} = (2 n - 1) / N^{2}$ . Then, we can get:

\begin{matrix} \frac{\sum_{n = 1}^{N} x_{(n)}^{p} / N}{\sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p}} - \frac{\sum_{n = 1}^{N} y_{(n)}^{p} / N}{\sum_{n = 1}^{N} {w ″}_{n} y_{(n)}^{p}} = \frac{\sum_{n = 1}^{N} x_{(n)}^{p}}{N \sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p}} - \frac{\sum_{n = 1}^{N} x_{(n)}^{p} + \sum_{k = 1}^{p} C_{p}^{k} x_{(N)}^{p - k} β^{k}}{N \sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p} + {Nw ″}_{N} \sum_{k = 1}^{p} C_{p}^{k} x_{(N)}^{p - k} β^{k}} \\ = \frac{\sum_{n = 1}^{N} x_{(n)}^{p}}{N \sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p}} - \frac{\sum_{n = 1}^{N} x_{(n)}^{p}}{N \sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p} + {Nw ″}_{N} \sum_{k = 1}^{p} C_{p}^{k} x_{(N)}^{p - k} β^{k}} - \frac{\sum_{k = 1}^{p} C_{p}^{k} x_{(N)}^{p - k} β^{k}}{N \sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p} + {Nw ″}_{N} \sum_{k = 1}^{p} C_{p}^{k} x_{(N)}^{p - k} β^{k}} \\ = \sum_{n = 1}^{N} \frac{x_{(n)}^{p}}{N} (\frac{1}{\sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p}} - \frac{1}{\sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p} + {w ″}_{N} \sum_{k = 1}^{p} C_{p}^{k} x_{(N)}^{p - k} β^{k}}) - \frac{\sum_{k = 1}^{p} C_{p}^{k} x_{(N)}^{p - k} β^{k}}{N \sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p} + {Nw ″}_{N} \sum_{k = 1}^{p} C_{p}^{k} x_{(N)}^{p - k} β^{k}} \end{matrix}

\begin{matrix} = \sum_{n = 1}^{N} \frac{x_{(n)}^{p}}{N} (\frac{{w ″}_{N} \sum_{k = 1}^{p} C_{p}^{k} x_{(N)}^{p - k} β^{k}}{(\sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p}) (\sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p} + {w ″}_{N} \sum_{k = 1}^{p} C_{p}^{k} x_{(N)}^{p - k} β^{k})}) - \frac{\sum_{k = 1}^{p} C_{p}^{k} x_{(N)}^{p - k} β^{k}}{N \sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p} + {Nw ″}_{N} \sum_{k = 1}^{p} C_{p}^{k} x_{(N)}^{p - k} β^{k}} \\ = \frac{\sum_{k = 1}^{p} C_{p}^{k} x_{(N)}^{p - k} β^{k}}{N \sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p} + {Nw ″}_{N} \sum_{k = 1}^{p} C_{p}^{k} x_{(N)}^{p - k} β^{k}} (\frac{\sum_{n = 1}^{N} {w ″}_{N} x_{(n)}^{p}}{\sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p}} - 1) \\ = \frac{\sum_{k = 1}^{p} C_{p}^{k} x_{(N)}^{p - k} β^{k}}{N \sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p} + {Nw ″}_{N} \sum_{k = 1}^{p} C_{p}^{k} x_{(N)}^{p - k} β^{k}} \frac{\sum_{n = 1}^{N} x_{(n)}^{p} ({w ″}_{N} - {w ″}_{n})}{\sum_{n = 1}^{N} {w ″}_{n} x_{(n)}^{p}} > 0 \end{matrix}

Since the weight increases monotonically with the increase of rank, this is true for $N > 1$ .

Proposition 10: PFGI3s satisfy Babies.

PFGI 3 (x ‖ 0) > PFGI 3 (x)

Proof of Proposition 10: The resulting sequence after performing Babies operation on $x = [x_{(1)}, x_{(2)}, \dots, x_{(N)}]$ can be expressed as

y = 0 ‖ x = [\begin{matrix} \underset{M}{\underset{︸}{\begin{matrix} 0 & \dots & 0 \end{matrix}}} & x_{(1)} & x_{(2)} & \dots & x_{(N)} \end{matrix}]

Then, the following can be obtained:

\begin{matrix} PFGI 3 (y) - PFGI 3 (x) = \sqrt[p]{\frac{N \sum_{n = 1}^{N} x_{n}^{p}}{\sum_{n = 1}^{N} x_{(n)}^{p} (2 n - 1)}} - \sqrt[p]{\frac{(N + M) \sum_{n = 1}^{N + M} y_{n}^{p}}{\sum_{n = 1}^{N + M} y_{(n)}^{p} (2 n - 1)}} \\ = \sqrt[p]{\frac{N \sum_{n = 1}^{N} x_{n}^{p}}{\sum_{n = 1}^{N} x_{(n)}^{p} (2 n - 1)}} - \sqrt[p]{\frac{(N + M) \sum_{n = 1}^{N} x_{n}^{p}}{\sum_{n = 1}^{N} x_{(n)}^{p} (2 M + 2 n - 1)}} \end{matrix}

Then, we can get:

\begin{matrix} \frac{N \sum_{n = 1}^{N} x_{n}^{p}}{\sum_{n = 1}^{N} x_{(n)}^{p} (2 n - 1)} - \frac{(N + M) \sum_{n = 1}^{N} x_{n}^{p}}{\sum_{n = 1}^{N} x_{(n)}^{p} (2 M + 2 n - 1)} = \frac{(\sum_{n = 1}^{N} x_{(n)}^{p}) [2 NM \sum_{n = 1}^{N} x_{(n)}^{p} - M \sum_{n = 1}^{N} x_{(n)}^{p} (2 n - 1)]}{(\sum_{n = 1}^{N} x_{(n)}^{p} (2 n - 1)) (\sum_{n = 1}^{N} x_{(n)}^{p} (2 M + 2 n - 1))} \\ = \frac{(\sum_{n = 1}^{N} x_{(n)}^{p}) [M \sum_{n = 1}^{N} x_{(n)}^{p} (2 N - 2 n + 1)]}{(\sum_{n = 1}^{N} x_{(n)}^{p} (2 n - 1)) (\sum_{n = 1}^{N} x_{(n)}^{p} (2 M + 2 n - 1))} > 0 \end{matrix}

Therefore, $PFGI 3 (y) - PFGI 3 (x) > 0$ is true for PFGI3s.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Key Research and Development Program of China (Grant No. 2021YFB3400704-02), the Fundamental Research Funds for the Central Universities of China (Grant No. 2682021CG003, Grant No. 2682021CX090), the independent project of State Key Laboratory of Traction Power, Southwest Jiaotong University, China (Grant No. 2021TPL-T11, Grant No. 2020TPL-T08), the open project of State Key Laboratory of Traction Power, Southwest Jiaotong University, China (Grant No. TPL2210), the National Natural Science Foundation of China (Grant No. 52275133) and the China Scholarship Council (Grant No. 202107000033).

ORCID iD

Bingyan Chen

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