Keywords: inline filters, coaxial resonators, transmission zeros, prototype synthesis

This work deals with the design of true inline filters with two symmetric transmission zeros. A new filters category named "path filters" is introduced (including the considered filters), his general features are discussed and analyzed and a general method for the synthesis of the low-pass prototype network is proposed. To validate the proposed design approach, a coaxial-cavity filter with a true inline topology exhibiting two symmetrical transmission zeros has been designed and fabricated. The realized prototype shows a response in good agreement with the theory.

I. INTRODUCTION

OST applications of microwave filters in recent years are requiring increasingly compact, lightweight, and selective devices. For this reason, the research has focused on filters with simple topologies allowing the introduction of transmission zeros, even very close to the passband. As a result, many examples of filter realizations featuring the above characteristics have appeared in the literature, implemented in the most popular fabrication technologies [START_REF] Amari | Synthesis of In-Line Filters with Arbitrarily Placed Attenuation Poles by Using Non-Resonating Nodes[END_REF]- [START_REF] Cogollos | Design procedure of low-cost substrate microstrip filters based on nonresonating nodes[END_REF].

A very convenient topological solution widely adopted since a long time is the inline configuration. As well know, this topology allows basically an all-pole characteristic, i.e., no transmission zeros can be introduced in the response. However, several variants of the basic configuration have been recently proposed, to allow the introduction of transmission zeros. It can be mentioned the configurations with frequency dependent couplings [START_REF] He | Advanced direct synthesis approach for high selectivity in-line topology filters comprising N-1 adjacent frequency-variant couplings[END_REF], some implementations of extracted-pole topology [START_REF] Macchiarella | Design of Waveguide Filters with Cascaded Singlets Through a Synthesis-Based Approach[END_REF] and the use of strongly-coupled resonators [START_REF] Macchiarella | Design of In-line Filters with Transmission Zeros Using Strongly-Coupled Resonators Pairs[END_REF]- [START_REF] Bastioli | Design of In-Line Filters with Strongly Coupled Resonator Triplet[END_REF]. However, all these solutions require a more or less marked deviation from the "true" inline configuration.

In this work we introduce a new class of filters, that we have called "path filters" (for the analogies with the family of path graphs [START_REF] Diestel | Graph Theory[END_REF]), featuring the true inline topology but with the source and load placed inside (instead of being the first and last node as in all pole filters). In general, this topology cannot be synthesized from the usual characteristic polynomials producing the generalized Chebycheff characteristic, unless further constraints on the derived polynomials are imposed. We have here defined the constraints allowing up to two imaginary S. Tamiazzo is with Commscope, Via Archimede, Agrate Brianza (Mi) -Italy (e-mail: stefano.tamiazzo@commscope.com). transmission zeros. Moreover, a procedure for the synthesis of the lowpass prototype has been developed, starting with the synthesis of the folded or transversal forms, followed by suitable matrix transformations.

Using the novel approach, the design and fabrication of a true inline filter of order 10 with two symmetric transmission zeros, has been carried out in coaxial technology. The realized prototype has been used to validate the design approach for path filters here proposed.

II. THE "PATH FILTERS"

Fig. 1 shows the general configuration of path filters. This configuration is composed of three blocks of resonators, separated by source and load. The number of resonators before the source and after the load are NS and NL respectively, while the internal ones are D. The overall number of resonators is then N=N S +D+N L . We can observe that, for the minimum path rule [START_REF] Amari | On the maximum number of finite transmission zeros of coupled resonator filters with a given topology[END_REF], path filters allow introducing exactly M=NS+NL transmission zeros in the response.

In general, we can approach the synthesis of this filter class starting with the evaluation of the characteristic polynomials (F(s), P(s), E(s)) determining the Generalized Chebycheff response in the normalized frequency domain s [START_REF] Cameron | Microwave Filters for Communication Systems[END_REF]. However, it can be shown that an additional constraint must be imposed on these polynomials. Assuming the filter response symmetric (i.e. S11=S22) this constraint requires that each imposed transmission zero is also a double eigenvalue of the synthesized network. To demonstrate this requirement, let's start by the elements of ABCD and Z matrices of the filter, that can be expressed as polynomials ratios [START_REF] Cameron | Microwave Filters for Communication Systems[END_REF]:

A=D=a(s)/P(s), B=b(s)/P(s), C=c(s)/P(s) Z11=Z22=a(s)/c(s), Z12=P(s)/c(s) (1)
Assume now that j z is a root of P(s) (i.e., is a transmission zero). Due to the topology of the path filters, at the frequency G. Macchiarella is with Dipartimento di Elettronica, Informazione e Bioingneria, Politecnico di Milano, 20133 Milano, Italy (e-mail: giuseppe.macchiarella@polimi.it).

F. Seyfert is with INRIA, 06902 Sophia-Antipolis, France (e-mail: fabien.seyfert@inria.fr)

A True Inline Coaxial-Cavity Filter with Two Symmetric Zeros

Stefano Tamiazzo, Giuseppe Macchiarella, Fellow, IEEE, Fabien Seyfert M of the zero a short circuit is produced in parallel to source and load (Z 11 = Z 22 =0). Consequently, a(j z )=0 and c(j z )≠0. Moreover, the assumed lossless condition requires the determinant of ABCD matrix equal to 1, that is:

        2 2 a s P s b s c s   (2)
From ( 2) descends that jz must also be a double root of b(s). As the roots of b(s) are also the eigenvalues of the filter shunt model [START_REF] Cameron | Microwave Filters for Communication Systems[END_REF], we have demonstrated that every transmission zero of a symmetric path filter, synthesized with shunt equivalent resonators, must also be a double eigenvalue of the filter equivalent circuit. Note that, being the eigenvalues of a lossless filter imaginary, also the transmission zeros introduced by path filters must be imaginary, as above assumed. Moreover, being N z = N S +N L the number of double eigenvalues, N>2N z implies that D≥NS+NL.

From what has been said above, the synthesis of path filters with Chebycheff (equiripple) response requires an additional constraint (for each transmission zero), in addition to those defining the usual filter specifications. Unfortunately, no degree of freedom is available after the requirements have been set, so we must give up assigning one of them to make the synthesis of path filters possible. In this work we have used the return loss parameter to get the characteristic polynomials of path filters exhibiting a symmetric response with two zeros (±jz). Note that, in this case, N S +N L ≤2 (the zeros can be extracted both at source (load) side or one at source and one at load).

A. Polynomials Evaluation

As pointed out above, the condition for synthesizing a path prototype is to have eigenvalues of multiplicity 2 in correspondence of each assigned pair of symmetric transmission zero. However, this condition is in general not satisfied if the characteristic polynomials are computed with the classical methods [START_REF] Diestel | Graph Theory[END_REF]. We have then developed an empirical procedure that, although not formally proven, has been assessed on the base of several examples up to N S +N L =2. It consists of the following steps:

-Assign the filter order and the desired transmission zeros (imaginary) -Evaluate the reflection characteristic polynomial F(s) -Compute the filter eigenvalues (represented by the roots of the denominator polynomial of the filter admittance matrix) as function of the leading coefficient p0 of the transmission polynomial P(s) -Select the value of p0 producing two coincident eigenvalues at the frequencies of the transmission zeros. Once p0 has been determined, all the characteristic polynomials are defined. However, as observed before, the return loss cannot be assigned freely because it depends on p 0 . The not assignable return loss level represents the main drawback of the proposed procedure. We note however that in case the assigned transmission zeros are relatively close to the passband, the resulting level of return loss is generally acceptable. Fig. 2 gives an idea of the return loss obtained for three values of the assigned transmission zeros and N varying from 5 to 15. It can be observed that RL is little dependent on N and increases by moving the transmission zeros away from the passband.

B. Synthesis of the path prototype

In line of principle, the synthesis of the path prototype could be approached with the evaluation of the transversal prototype, followed by suitable matrix transformations. Unfortunately, the transversal form of filters with double eigenvalues cannot be synthesized using the classical method given in the literature [START_REF] Cameron | Advanced Coupling Matrix Synthesis Techniques for Microwave Filers[END_REF], which holds only for networks with simple eigenvalues. We have then used the folded canonical form as initial prototype [START_REF] Cameron | Microwave Filters for Communication Systems[END_REF], which can be synthesized using the extractions method [START_REF] Cameron | General prototype network synthesis methods for microwave filters[END_REF], not based on the eigenvalues computation. The coupling matrix of the folded prototype is then suitably reconfigured to get the coupling matrix of the path filter in Fig. 1. To this end, first the conversion to the Wheel form is carried out using the algorithm in [START_REF] Tamiazzo | An analytical technique for the synthesis of cascaded N-tuplets cross-coupled resonators microwave filters using matrix rotations[END_REF]. Then a sequence of matrix rotations is applied, to get out a triplet form from the load and moving it up to the source (the same technique used in [START_REF] Tamiazzo | An analytical technique for the synthesis of cascaded N-tuplets cross-coupled resonators microwave filters using matrix rotations[END_REF]). When the triplet arrives in the final position, all cross couplings disappear, and the "path" topology is obtained.

III. FILTER DESIGN

The filter to be designed has the following electrical requirements:

Passband: 3394-3806 MHz, Return Loss ≥18 dB Order (number of resonators): 10 Stopband attenuation: >30 dB for |f-f0|>230 MHz To fit the attenuation requirements two transmission zeros must be placed symmetrically above and below the passband at 3835 MHz and 3368.3 MHZ. Note that the rejection actually presented by the synthesized path filter depends on the resulting return loss, that cannot be assigned a priori. At the end of the polynomials evaluation, the fulfillment of the above requirements must then be verified before proceeding with the synthesis of the prototype.

The reflection zeros defining the monic polynomial F(s) are first evaluated from the normalized transmission zeros (±j1.13268): zF=[0.9917, 0.9194, 0.7562, 0.4994, 0.1747, -0.9917, -0.9194, -0.7562, -0.4994, -0.1747] The polynomial P(s) is defined as function of the unknown highest degree coefficient p 0 : P(s)=p 0 (s 2 +1.283). Finally, the polynomial E(s) defining the filter poles can be computed by spectral factorization from the Feldtkeller equation (lossless condition) [START_REF] Cameron | Microwave Filters for Communication Systems[END_REF]: E(s)E * (s)=P(s)P * (s)+F(s)F * (s). The filter eigenvalues are the roots of polynomial b(s) which can be evaluated from the characteristic polynomials as described in [START_REF] Cameron | Microwave Filters for Communication Systems[END_REF]. We can then represent the eigenvalues as

Number of resonators

Return Loss (dB)

 z =±1.15  z =±1.1  z =±1.05
function of the real parameter p 0 (Fig. 3). It can be observed that two pairs of symmetrical eigenvalues cross for p0=0.041803. Moreover, at the crossing points the eigenvalues coincide with the assigned transmission zeros (±1.13268). For the computed value of p0 the return loss is 23.4407 dB and the condition for implementing the filter with the path topology is satisfied. The response in the normalized domain is shown in Fig. 4. It can be observed that return loss is larger than 18 dB while attenuation in the stopbands exceeds 30 dB. The filter requirements are then satisfied, and it is possible to proceed with the synthesis of the path filter. We have assumed NS=2 and NL=0. The synthesis of the prototype is carried out with the procedure described before obtaining the normalized coupling matrix of the desired path filter: M i,i =0, M s2 =0. The last step of the synthesis is the de-normalization of the normalized prototype defined by the computed coupling matrix. In this step we determine the coupling coefficients and the resonating frequencies of the filter in the bandpass domain (Fig. 5), that are used for dimensioning the physical structure implementing the filter. For the physical realization of the filter (Fig. 6), a comb structure inside a box of height 19 mm has been employed. The coupled resonators are square cylinders of size 3.5 mm x 3.5 mm and of various heights (averaging 18 mm). The quality factor of the fundamental resonant mode (3.6 GHz) is about 2500 and adjacent rods are coupled by proximity and possibly connected by a thin metal bar in places where strong couplings are required. All input and output couplings (three in this case, two at the source end and one at the load end) are realized through suitable lengths of transmission (strip) lines soldered to the rods 2, 3 and 10 (Fig. 6), at a suitable distance from the base (ground plane). Fig. 7 shows the measured response of the realized prototype Although the agreement with the polynomial response of fig. 4 is good, the limited range and resolution of the adopted tuning elements prevented from a better 'shaping' of the return loss curve.

IV. CONCLUSION

In this paper an introduction to the design of a class of true in line filters with up two symmetric transmission zeros has been given, along with a practical realization in the 3.4 -3.8 GHz band. This class of filters appears to be promising for the realization of symmetrical frequency responses usually required in TDD radios for cellular networks operating in that range of frequencies. Compared to the canonical cross coupled alternatives, there is no need in path filters to implement negative couplings, usually associated with spurious resonances and tolerance issues. Finally, it must be emphasized that the design approach proposed here represents the first analytical solution for the class of filters considered. Indeed, previous works dealing with this topology [START_REF] Zhang | Realization of Even Transmission Zeros for Filter Without Cross-Couplings[END_REF] address the synthesis of the low-pass prototype using optimization techniques. We are currently working on the extension of the path filters synthesis to a larger number of transmission zeros (not necessarily symmetric).

Fig. 1 .

 1 Fig. 1. General configuration of Path Filters. Each black node represents a resonator, the lines are the coupling inverters, the white nodes are the source and load.

Fig. 2 .

 2 Fig. 2. Return Loss obtained from the computed polynomials for three different assignment of the transmission zeros and N ranging from 5 to 15.

Fig. 3 .

 3 Fig. 3. Computed filter eigenvalues as function of p0.

  67, M s3 =0.802, M 12 =1.133, M 23 =0, M34=0.604, M45=0.526, M56=0.525, M67=0.531, M78=0.545, M89=0.596, M910=0.862, M10L=1.046.

Fig. 4 .

 4 Fig. 4. Computed polynomial response of the normalized prototype with p0=0.041803 (RL=23.4407 dB)

Fig. 5 .

 5 Fig. 5. De-normalized equivalent circuit of the path filter. Black circles represent the resonators (all resonating at f0= 3594.1 MHz). The numbers above or below the solid lines are the coupling coefficients between the resonators.

Fig. 6 .

 6 Fig. 6. Fabricated structure of the path filter. The resonators are the numbered square cylinders (1 and 2 are those realizing the transmission zeros).

Fig. 7 .

 7 Fig. 7. Measured response of the fabricated prototype.


Ben Samir

Alexandre Hariz

Chunhao Brouste

AND Marius Cai

Soltane

FAST AND ASYMPTOTICALLY-EFFICIENT ESTIMATION IN A FRACTIONAL AUTOREGRESSIVE PROCESS

Keywords: Geweke Porter-Hudak estimator, LAN property, Log periodogram, Maximum Likelihood estimator, One-step estimator, Quadratic form of Gaussian process, Semi-parametric estimation, Toeplitz matrix, Whittle estimator

This paper considers the joint estimation of the parameters of a first-order fractional autoregressive model by constructing an initial estimator with convergence speed lower than √ n and singular asymptotic joint distribution. The one-step procedure is then used in order to obtain an asymptoticallyefficient estimator. This estimator is computed faster than the maximum likelihood or Whittle estimator and therefore allows for faster inference on large samples. The paper illustrates the performance of this method on finite-size samples via Monte Carlo simulations.

Introduction

Parametric estimation in fractional Gaussian processes has been widely studied over recent decades. The asymptotic properties of the maximum likelihood estimator (MLE) have been derived under conditions of regularity on the spectral density of the process [START_REF] Fox | Large-sample properties of parameter estimates for strongly dependent stationary gaussian time series[END_REF], Dahlhaus, 1989, Dahlhaus, 2006, Lieberman et al., 2012]. Moreover, for a stationary Gaussian process observed in the large sample scheme, the local asymptotic normality (LAN) of the likelihood ratio was derived by [START_REF] Cohen | LAN property for some fractional type Brownian motion[END_REF]. In this setting, the MLE is asymptotically efficient but is not in a closed form, such that numerical optimization of the likelihood is then necessary to compute the estimate for an observation sample. This step is particularly time consuming and numerically unstable. It is therefore worth seeking an alternative estimator which can be computed faster and keeps efficient asymptotic properties.

To achieve this, we extend the method presented in [Le Cam, 1956] for independent and identicallydistributed random variables. Under this procedure, a single Fisher scoring step on the loglikelihood is applied, starting from an initial √ n-consistent estimator. This is performed in order to obtain a new estimator (called the one-step estimator) whose asymptotic variance is optimal in the Fisher sense. This procedure has already been successfully extended for diffusion processes [Kamatani andMasayuki, 2015, Gloter andYoshida, 2021], ergodic Markov chains [START_REF] Kutoyants | On multi-step mle-process for markov sequences[END_REF], fractional Gaussian noise observed at high frequency [START_REF] Brouste | One-step estimation for the fractional Gaussian noise model at high-frequency[END_REF] and stable noise [START_REF] Brouste | Efficient estimation of stable lévy process with symmetric jumps[END_REF], for instance. Generally, we derive the asymptotic properties of the one-step estimator under the Sweeting conditions [Sweeting, 1980] and the properties of the initial guess estimator depend on the statistical experiment which is considered. It should also be noted that the one-step procedure has only recently been extended to n δ 2 -consistent initial guess estimators by [START_REF] Kamatani | Hybrid multi-step estimators for stochastic differential equations based on sampled data[END_REF].

In this paper, we consider a special case of fractional Gaussian process (X t ), satisfying for any t ∈ Z the recursive relation (1.1)

X t = αX t-1 + √ σ 2 ε H t
where |α| < 1, σ 2 > 0 and (ε H t ) is the fractional Gaussian noise of the Hurst exponent, H ∈]0; 1[. The process (ε H t ) is a stationary Gaussian sequence with autocovariance function

(1.2) ρ (k) = 1 2 |k + 1| 2H -2|k| 2H + |k -1| 2H .
The estimation of the parameter α in the autoregressive process directed by a fractional Gaussian noise (1.1) has been studied by [START_REF] Brouste | Asymptotic properties of the MLE for the autoregressive process coefficients under stationary Gaussian noise[END_REF], Soltane, 2018]. In these papers, the Hurst exponent, H, is assumed to be known and the asymptotic properties of the MLE are studied. Generalized least square estimators (GLSEs) of α were studied in [Esstafa, 2019] and the Hurst exponent was again assumed to be known. This type of process has therefore been widely used to model realized volatility in stochastic volatility models (see [START_REF] Gatheral | Volatility is rough[END_REF] and the reference therein). In this application, large datasets are treated conducting to the impossibility of carrying out numerical optimization for the MLE. This study therefore extends the one-step procedure in this context.

As is demonstrated in Section 2, the covariance function of the process X is not in a closed form, such that the computation of the MLE is time consuming and the MLE emerges numerically unstable. In this paper, an initial guess semi-parametric estimator of H is considered, as inspired by the work of [START_REF] Hurvich | The mean squared error of Geweke and Porter-Hudak's estimator of the memory parameter of a long-memory time series[END_REF] for fractional integrated Gaussian processes. The paper then considers an estimator of α by plugging-in the estimate of H into the GLSE presented in [Esstafa, 2019]. It then estimates σ 2 via the residual process in the same spirit as a standard AR model by the same plug-in. The unexpected result of this method is that the asymptotic joint law of the initial estimators is a singular Gaussian vector. It is then evidenced that this derives from the statistical error of the last two estimators (of α and σ 2 ), being a function of the statistical error of H modulo a negligible remainder term.

As the aforementioned initial estimator has a slow convergence speed and as it is difficult to construct a joint asymptotic confidence region for this initial estimate because of the singularity, the one-step procedure is applied on the initial estimate in order to build asymptotically-efficient estimator. To obtain the asymptotic properties of an the one-step estimator with rate improvement, we need stronger conditions than those of Sweeting [Sweeting, 1980], which in our case are estimates of the convergence rate of Toeplitz's matrix trace, as described in [START_REF] Lieberman | Asymptotic theory for maximum likelihood estimation of the memory parameter in stationary gaussian processes[END_REF]. We formulate a generic condition on the spectral density to apply the results of [START_REF] Cohen | LAN property for some fractional type Brownian motion[END_REF] and [START_REF] Lieberman | Asymptotic theory for maximum likelihood estimation of the memory parameter in stationary gaussian processes[END_REF] in the proof of the asymptotic properties of the one-step estimator.

The paper is organized as follows. Section 2 presents the main results, divided into two subsections: the first presents the initial estimator and states its asymptotic properties, and the second presents the onestep procedure and states its asymptotic properties. Section 3 displays the performance of the estimators on finite-size samples via the Monte Carlo simulations presented. Section 4 concludes and considers the perspectives raised by this research. 5 provides the numerous technical lemmas necessary to prove the main results. Section 6 lists proofs of the results presented in Section 2.

Main results

2.1. Initial estimator. The spectral density of the fractional Gaussian noise ε H t is given by

(2.1) f H (λ) = C H,σ2 2 (1 -cos λ) k∈Z 1 |λ + 2kπ| 2H+1 ,
where C H,σ2 = σ2 2π Γ(2H + 1) sin(πH) and λ ∈ [-π; π]. Thus, we consider the following parametric space (2.2) where [a, b] is any compact set in ]0; 1[. Therefore, the spectral density of the process (X t ) is given by the following proposition.

ϑ = (H, α, σ 2 ) ∈ Θ = [a; b]×] -1; 1[×]0; +∞[,
Proposition 2.1. Letting g H,α,σ2 be the spectral density of (X t ), then

(2.3) g ϑ (λ) = g H,α,σ2 (λ) = 1 -2α cos(λ) + α 2 -1 f H (λ)
Proof. See Section 6.1.

We estimate H via the log-periodogram method presented in [Geweke andPorter-Hudak, 1983, Hurvich et al., 1998] for general integrated Gaussian series (GPH estimator). The choice of this method is motivated by Theorem 1, as provided in [START_REF] Geweke | The estimation and application of long memory time series models[END_REF], which states that a general integrated Gaussian series of memory parameter d = H -1 2 can be represented (via its spectral density) by a general fractional Gaussian noise of Hurst exponent H and vice versa. The addition of the autoregressive part will not disturb this method, as the spectral density of a standard autoregressive process is regular and bounded. The spectral density of (X t ) will be the product of that of ε H t by the autoregressive part.

Letting (X 1 , X 2 , . . . , X n ) be an observation sample generated via the relation (1.1) and considering an integer m satisfying m < n., we define (2.4)

I (λ) = 1 2πn n t=1 X t exp (itλ) 2 (2.5) λ j = 2πj n
where j ∈ {1, . . . , m}

a j = log 2 sin λ j 2 , a m = 1 m m j=1 a j , S m = m j=1 (a j -a m ) 2 .
We estimate d by regressing log I (λ j ) with respect to a j , such that (2.6)

d n = - 1 2S m m j=1 (a j -a m ) log I (λ j ) .
The estimator H n of H is defined by (2.7)

H n = d n + 1 2 .
Remark 2.1. Another semi-parametric method based on the log-periodogram regression is proposed by [START_REF] Robinson | Log-periodogram regression of time series with long-range dependence[END_REF]. This method does not take into account Fourier frequencies close to 0, which induces a stronger bias in the estimation of H than the GPH estimator.

Remark 2.2. For the discretely observed fractional Ornstein-Uhlenbeck process (which is similar to our model), an estimator of H was studied using the variogram of (X t ) (see [START_REF] Brouste | Parameter estimation for the discretely observed fractional Ornstein Uhlenbeck process and the Yuima R package[END_REF]). However, we cannot use this type of approach as we are working on a large-sample statistical experiment.

We now return to estimate α and σ 2 by first considering a GLSE of α and then building the residual process to estimate σ 2 as in a classical autoregressive model. We let Φ j i = (X i , . . . , X j ) * for i j, and

(2.8) Γ n (H) = ρ (|i -j|) {1 i,j n} .
The estimators α n and σ 2,n are defined by (2.9)

α n = Φ n 2 * Γ -1 n-1 H n Φ n-1 1 Φ n-1 1 * Γ -1 n-1 H n Φ n-1 1 ,

and

(2.10)

σ 2,n = 1 n -1 Φ n 2 -α n Φ n-1
Theorem 2.1. Letting m = n δ for some 1 2 < δ < 2 3 , the initial estimator ϑ n is weakly consistent. Moreover, when n -→ ∞, we have,

(2.11) √ m   H n -H α n -α σ 2,n -σ 2   L ----→ n→∞ N (0; Σ ϑ ) .
The covariance matrix Σ ϑ is of the form

(2.12) Σ ϑ = V H Σ ϑ ,
where V H is the asymptotic variance of √ m( H n -H), π 2 24 and Σ ϑ is a built-in singular matrix (6.29). Proof. See Section 6.2 Remark 2.3. It is possible to estimate H via the local Whittle method presented in [Robinson, 1995a, Shimotsu andPhillips, 2005]. These methods lead us to an estimator which does not adopt an explicit form but is less sensitive to the parameters and more suitable for small samples.

Remark 2.4. Theorem 2.1 remains valid for any asymptotic Gaussian estimator of H whose rate satisfies the conditions of this Theorem. The limiting covariance matrix is then obtained by replacing V H with the asymptotic variance of the estimator of H considered here.

Remark 2.5. These results can be extended to the fractional AR(p) case using the causal representation of an AR(p) process in the vectorial case.

Remark 2.6. Theorem 2.1 remains valid if the initial estimate of α and σ 2 is carried out on a subsample of size n β with β > δ. The convergence in law towards the singular Gaussian vector will be in this case slower than when the estimation is carried out on the whole sample.

Remark 2.7. We can also use the method used in [START_REF] Brouste | Asymptotic properties of the MLE for the autoregressive process coefficients under stationary Gaussian noise[END_REF], Soltane, 2018] to estimate α.

2.2. One-step estimator. We assume in this subsection that (Y n ) is a stationary centred Gaussian process with spectral density f ϑ for some unknown parameter ϑ ∈ R d . We consider f ϑ to satisfy regularity conditions if the following conditions are satisfied.

We let Θ be an open subset of R d ,

• For any ϑ ∈ Θ, f ϑ (λ) is three-times continuously differentiable on Θ. In addition, for any 0 3 and j 1 , . . . , j m , the partial derivate

∂ ∂ϑ j1 . . . ∂ϑ j f ϑ (λ) , is continuous on Θ × [-π; π]\{0}
, is continuously differentiable with respect to λ and its partial derivate

∂ +1 ∂λ∂ϑ j1 . . . ∂ϑ j f ϑ (λ) ,
and is continuous on Θ × [-π; π]\{0}. • There also exists a continuous function α : Θ -→] -1, 1[, such that for any compact set Θ * ⊂ Θ and δ > 0, the following conditions hold for every (ϑ, λ)

∈ Θ * × [-π; π]\{0}. c 1,δ,Θ * |λ| -α(ϑ)+δ f ϑ (λ) c 2,δ,Θ * |λ| -α(ϑ)-δ and ∂ ∂λ f ϑ (λ) c 2,δ,Θ * |λ| -α(ϑ)-1-δ .
For any ∈ {1, 2, 3} and any j ∈ (1, . . . , m) ,

∂ ∂ϑ j1 . . . ∂ j f ϑ (λ) c 2,δ,Θ * |λ| -α(ϑ)-δ ,
where c i,δ,Θ * is some positive finite constant which only depends upon δ and Θ * . We now consider the local asymptotic normality property of the likelihood ratio.

Proposition 2.2. We let n be the log-likelihood function of a stationary Gaussian process (which can be expressed using spectral density, like in Proposition 2.1 from [START_REF] Cohen | LAN property for some fractional type Brownian motion[END_REF]). We assume that f ϑ satisfies the regularity conditions and let B (ϑ, R) (open ball of centre ϑ and radius R) for some R > 0. For any t ∈ B (ϑ, R) ,

n ϑ + t √ n -n (ϑ) = t ∇ n (ϑ) √ n - tI (ϑ) t * 2 + r n,ϑ (t) ,
where, under

P (n) ϑ , the score function ∇ (• • • ) satisfies (2.13) ∇ n (ϑ) √ n L ----→ n→∞ N (0; I (ϑ)), and
(2.14) r n,ϑ (t)
a.s.

----→ n→∞ 0 uniformly on each compact set. The Fisher information matrix is given in our case by

(2.15)

I (ϑ) = 1 4π π -π ∂ log g ϑ (λ) ∂ϑ k ∂ log g ϑ (λ) ∂ϑ j dλ 1 k,j d .
Proof. See Section 6.3.

Remark 2.8. When the LAN property is verified, it is possible to define a notion of asymptotic efficiency for the estimators (see [Ibragimov and Has'minskii, 1981]). It is also possible in a such a statistical experiment to determine the estimators maximum speed of convergence ( √ n in our case).

Remark 2.9. Proposition 2.2 is derived from Theorem 3.4, as proposed by [START_REF] Cohen | LAN property for some fractional type Brownian motion[END_REF], in which weaker conditions are given for the application α. In our case, we employ this Theorem in order to derive the asymptotic properties of the one-step estimator.

We now build an estimator whose asymptotic properties are similar to the maximum likelihood (in terms of convergence rate and asymptotic variance).

Proposition 2.3. Fist, we assume

• ϑ (1)
n is an initial estimator of ϑ, such that, for some 1 2 < δ 1,

n δ 2 ϑ (1) n -ϑ = O P (1) ,
• f ϑ satisfies the regularity conditions. We then consider the estimator

(2.16) ϑ (1) n = ϑ (1) n + I ϑ (1) n -1 1 n ∇ n ϑ (1)
n where ∇ n (•) and I (•) are defined in Proposition 2.2. Then,

(2.17) √ n ϑ (1) n -ϑ L ----→ n→∞ N 0, I (ϑ) -1 .
Proof. See Section 6.4.

Remark 2.10. To apply Proposition 2.3, it is not necessary to know the asymptotic distribution of the initial estimator but only a convergence rate of statistical errors. When the convergence rate of the initial estimator is too slow, it is possible to apply a multi-step procedure to obtain an asymptotically-efficient estimator at optimal speed.

Remark 2.11. The one-step estimator can be easily extended to other classes of stationary Gaussian processes, satisfying the assumptions of Proposition 2.3.

Remark 2.12. In the Gaussian setting, we can use a similar one-step procedure, replacing the classical score by the Whittle score. This procedure is faster to compute and does not require the inversion of the covariance matrix. However, we leave analysis of this method for future studies.

Theorem 2.2. The initial estimator ϑ n satisfies the conditions of Theorem 2.3 and the spectral density g ϑ meets the regularity conditions. Proposition 2.3 allows us to deduce that the one-step estimator of ϑ n is asymptotically efficient with speed √ n.

Simulations study

The log-likelihood function produced by the sample X (n) = (X 1 , X 2 , . . . , X n ) * which satisfies the recursive relation (1.1) is given by

n (ϑ) = - 1 2 log (det (Γ n (ϑ))) - 1 2 X (n) * Γ -1 n (ϑ) X (n) ,
where Γ n (ϑ) is the covariance matrix of X (n) . For any k ∈ N,

E (X 0 X k ) = π -π exp (ikλ) g ϑ (λ) dλ
The score function with respect to ϑ for i ∈ {1, 2, 3} is given by

∂ n (ϑ) ∂ϑ i = - 1 2 T r Γ -1 n (ϑ) ∂ ϑ i Γ n (ϑ) + 1 2 X (n) * Γ -1 n (ϑ) ∂ ∂ϑ i Γ n (ϑ) Γ -1 n (ϑ) X (n) ,
where T r is the trace operator and the Fisher information matrix is defined by (2.15). To compute the score, as well as the Fisher information, we numerically evaluate the integral of the spectral density (and its derivatives) using the Paxson method described in [START_REF] Fukasawa | Asymptotically efficient estimators for self-similar stationary Gaussian noises under high frequency observations[END_REF].

For each set of parameters, we perform M = 10000 Monte Carlo simulations for samples of size n = 1000. The number of Fourier frequencies for the initial estimation is fixed at m = n 0.6 . √ n speed, where ϑ = (0.7, -0.6, 1) . The first line corresponds to the initial estimator and the second line to the one-step estimator. The blue curves correspond to the density of the centred normal distribution where the theoretical variance is approximated by the empirical variance. The curves in red correspond to the limit of the one-step estimator.

Conclusion and perspectives

In this paper, we propose a simple and easy-to-implement estimator for the parameters of an AR(1) process with dependent errors. We prove the joint asymptotic normality of the vector of estimators to a Gaussian degenerate law. Furthermore, we observe that the Hurst parameter estimators drive and dominate other estimators.

Using a gradient descent approach, we improve on the first estimator with one iteration. The resulting estimator, called one step, achieves the √ n rate with optimal variance. Hence, we reproduce the performance of the maximum likelihood estimator. As such, not only does the one-step process improve upon the rate of the initial estimator, but it also achieves the inverse of the Fisher information as a limiting variance. This is especially relevant when dealing with large sample sizes.

Our result can be extended to other process classes, including ARFIMA models. Furthermore, it is notable that the initial poor estimator could be replaced by anyone with a sufficient speed of convergence.

In addition, a procedure for detecting breaks in parameters, as achieved by [Brouste et al., 2020a], could be developed for the autoregressive parameter. On the application level, we could implement functions in R that perform this estimation, as managed by the OneStep package described in [Brouste et al., ].

This research benefited from the support of the Chair Risques Emergents ou Atypiques en Assurance, under the sponsorship of Fondation du Risque, a joint initiative by Le Mans University, Ecole Polytechnique and MMA, a member of the Covea Group.

Auxiliary results

For clarity, we separate the technical results into two subsections. The first details the technical lemmas related to the initial estimator and the second details those related to the one-step estimator.

5.1. Technical results related to Section 2.1. Lemma 5.1. For any t ∈ Z, the process (5.1)

X t = +∞ j=0 α j ε H t-j , a.s.
is a stationary and ergodic process.

Proof. As we know, for any time series {Y t } under the monotone convergence theorem, we have

(5.2) E ∞ t=-∞ |Y t | = lim n→∞ E n t=-n |Y t | = +∞ t=-∞ E|Y t |.
With this equation and the Schwarz inequality we have

E   +∞ j=0 α j ε H t-j   = ∞ t=0 |α j |E|ε H t-j | ≤ C ∞ t=0 |α j | < ∞.
This demonstrates that the right side of Equation 5.1 converges absolutely and thus converges a.s.. Considering that

n j=0 α j ε H t-j ≤ ∞ j=0 α j ε H t-j ,
with the dominated convergence theorem,

E (X t ) = lim n→∞ E   n j=0 α j ε H t-j   = 0.
Now, for t, s ∈ Z we define

ξ n = n j=0 α j ε H t-j , η n = n k=0 α k ε H s-k .
From the previous conclusion, we have ξ n η n → X t X s a.s. and

|ξ n η n | ≤ V , where V is defined by V = ∞ j=0 ∞ k=0 α j α k ε H t-j ε H s-k .
Now, from Equation (5.2) we know that

E (V ) = ∞ j=0 ∞ k=0 |α j α k |E|ε H t-j ε H s-k | ≤ σ 2   ∞ j=0 |α j |   2 < ∞.
Finally, Theorem 5.3.8 in [Stout, 1974] ensures that (X t ) is stationary and ergodic.

Lemma 5.2. The process {X t } defined in (5.1) is a Gaussian process.

Proof. Here, we only need to prove that for any m ∈ N + we have the vector

(5.3) X = (X 1 , X 2 , • • • , X m ) T ∼ N (0, Σ m ),
where Σ m = (γ j-k ) m×m and γ k is the auto-covariance of X t and X t+k . In fact, if we define

η k (n) = n j=0 α j ε H k-j , then it follows that E|η k (n) -X k | → 0, n → ∞. Now, for any real vector b = (b 1 , b 2 , • • • , b m ) T , we define Y = b T X = m k=1 b k X k , η n = m k=1 b k η k (n).
When n → ∞, we have

E(|Y -η n |) = E m k=1 b k (X k -η k (n)) ≤ m k=1 |b k |E[|X k -η k (n)|] → 0,
where L 1 convergence implies convergence in distribution, such that η n converges in distribution to Y . As η n satisfies the normal distribution and Y ∼ N (E(Y ), Var(Y )), where E(Y ) = 0 and Var(Y ) = b T Σ m b, we have the conclusion of (5.3).

Lemma 5.3. Letting Y n ∈ R n be a centred Gaussian vector with covariance matrix Σ n and any symmetric matrix

A n of size n × n, then φ(Y n ) = Y * n A n Y n . As a result, E (φ(Y n )) = T r (A n Σ n ) and Var (φ(Y n )) = 2T r (A n Σ n ) 2 .
Proof. See Lemma A1 of [START_REF] Cohen | LAN property for some fractional type Brownian motion[END_REF].

Lemma 5.4. Under the hypothesis on the parametric space we have the following conclusions:

1) For any H ∈ [a; b] and j ∈ {0, 1, 2, 3},

∂ ∂λ ∂ j ∂ j H f H (λ) are continuous functions on [ a; b] ×[-π; π], \{0} .
2) For any j ∈ {0, 1, 2, 3} the functions ∂ j ∂ j H f H (λ) are symmetric with respect to λ. 3) For any δ > 0 and all (H, λ)

∈ [a; b] × [-π; π]\{0}, a) K 1,δ |λ| 1-2H+δ f H (λ) K 2,δ |λ| 1-2H-δ , b) ∂ ∂λ f H (λ) K 3,δ |λ| -2H-δ
c) and for any j ∈ {1, 2, 3},

∂ j ∂ j H f H (λ) K 4,δ |λ| 1-2H-δ .
Where K i,δ are some finite positive constants which only depend upon δ and the parametric space of H, namely

[ a; b] ., 4) ∂ j ρ(k) ∂ j H = π -π exp (ikλ) ∂ j f (λ) ∂ j H dλ Proof.
We start by proving the assertion 3) a). We write

f H (λ) = C H,σ2 (1 -cos λ) 1 |λ| 2H+1 + k∈Z * f k (λ) , where f k (λ) = 1 |λ + 2kπ| 2H+1 . We observe that f k (λ) C H |k| -1-2H
and so k∈Z * f k (λ) converge to a continuous bounded function uniformly in [-π; π]. Then, for some C > 0

C H,σ2 (1 -cos λ) 1 |λ| 2H+1 -C f H (λ) C H,σ2 (1 -cos λ) 1 |λ| 2H+1 + C .
As λ 2 = O |λ| 1-2H , we have the conclusion of 3)a).

Assertion 3)b) is demonstrated in the same way. Now we prove assertion 3) c). For all λ ∈ [-π; π], f k is continuous with respect to λ and H with

∂ j ∂ j H f k (λ) = j =1 (-2 log (|λ + 2kπ|)) |λ + 2kπ| 2H+1 .
For all δ 1 > 0 there exists a positive constant C δ1 , such that, for a sufficiently large k,

∂ j ∂ j H f k (λ) C δ1 k δ1 k 2a+1 = O k -1-2a+δ1 . Choosing δ 1 such that δ 1 -2a < 0, then k∈Z * ∂ j f k (λ)
∂ j H converge uniformly in (λ, H) to a continuous, bounded function, and

(1 -cos λ) ∂ j ∂ j H k∈Z * f k (λ) = (1 -cos λ) k∈Z * ∂ j ∂ j H f k (λ) = O(λ 2 ).
In the same way, for almost all λ,

∂ j ∂ j H 1 -cos λ |λ| 2H+1 = (1 -cos λ) j =1 -2 log(|λ|) |λ| 2H+1 = O |λ| 1-2b-δ2
for any δ 2 > 0 and sufficiently small λ. Note that the constant C H,σ2 depends on H very regularly and therefore does not affect the next estimates. We now choose δ 2 such that 2b + δ 2 < 2 in order to produce a positive function, h. As such, h is integrable with respect to λ and, for any j ∈ {1, 2, 3},

∂ j ∂ j H f H (λ) h (λ) and ∂ j ∂ j H f H (λ) K 4,δ |λ| 1-2H-δ for any δ > 0.
Assertion 1) is demonstrated using uniform convergence, as in the proof of 3)a) and 3) c).

Assertion 2) is demonstrated by nothing that, for any j ∈ {0, 1, 2, 3},

∂ j ∂ j H f k (-λ) = ∂ j ∂ j H f -k (λ) .
Assertion 4) is demonstrated using the derivation theorem under the integral, which is possible because of the last estimation on the proof of 3)c).

Lemma 5.5. We let g be a function defined on [-π; π], such that

(5.4) g (λ) = sin λ 2 1-2H g * (λ) .
We assume that 1)

g * (λ) > 0 ∀λ ∈ [-π; π].
2) g * is continuous and bounded in a neighbourhood of 0.

3) ∂ ∂λ g * is well defined, continuous and bounded in a neighbourhood of 0.

4) m -→ ∞, n -→ ∞ with m n -→ 0 and m log m n -→ 0. Then, (5.5) - 1 2S m m j=1 (a j -a m ) log g * (λ j ) = O m n .
Proof. This proof is based on that of Lemma 1 in [START_REF] Hurvich | The mean squared error of Geweke and Porter-Hudak's estimator of the memory parameter of a long-memory time series[END_REF]]. We have

- 1 2S m m j=1 (a j -a m ) log g * (λ j ) = - 1 2S m m j=1 (a j -a m ) ∂ ∂λ g * (ω j ) g * (ω j ) λ j ,
where for all j ∈ {1, . . . , m}, 0 < ω j < λ j . We can find a finite constant, such that

∂ ∂λ g * (ωj ) g * (ωj )
C for all j. The Cauchy-Schwarz inequality implies that

1 2S m m j=1 (a j -a m ) Cλ j C 2S m   m j=1 (a j -a m ) 2   1 2   m j=1 λ 2 j   1 2
.

It is shown in Lemma 1 of [START_REF] Hurvich | Automatic semiparametric estimation of the memomy parameter of a long memory time series[END_REF] that

S m = m (1 + o(1)) ; then C 2S m   m j=1 (a j -a m ) 2   1 2   m j=1 λ 2 j   1 2 = C 2 m j=1 λ 2 j S m 1 2 = C m j=1 j 2 m(1 + o(1))n 2 1 2 = C (m + 1)(2m + 1 n 2 (1 + o(1)) 1 2 = O m n .
5.2. Technical results related to Section 2.2. The following lemmas are needed to derive the asymptotic properties of the one-step estimator in a general centred stationary Gaussian process. The lemmas of this subsection are valid under the regularity conditions.

Lemma 5.6. We let ϑ 0 ∈ Θ * and δ > 0, such that B (ϑ 0 , δ) ⊂ Θ * . For any ϑ ∈ B (ϑ 0 , δ) ,

I (ϑ) -I (ϑ 0 ) K ϑ -ϑ 0
for some constant K.

Proof. We verified Conditions(A.1) and (A.2) of [START_REF] Cohen | LAN property for some fractional type Brownian motion[END_REF] during the proof of Lemma 5.4. These imply that, for any k, j ∈ {1, 2, . . . , d}, with the mean value inequality,

1 4π π -π ∂ log g ϑ (λ) ∂ϑ k ∂ log g ϑ (λ) ∂ϑ j dλ - 1 4π π -π ∂ log g ϑ0 (λ) ∂ϑ 0,k ∂ log g ϑ0 (λ) ∂ϑ 0,j dλ K {k,j} ϑ -ϑ 0 ,
where

K {k,j} = sup ϑ∈B(ϑ0,δ) ∂ ∂ϑ i π -π ∂ log g ϑ (λ) ∂ϑ k ∂ log g ϑ (λ) ∂ϑ j dλ 1 i d .
Note that B (ϑ 0 , δ) is a convex subset of R 3 . Owing to Conditions (A.1) and (A.2) of [START_REF] Cohen | LAN property for some fractional type Brownian motion[END_REF] that for any k, j ∈ {1, 2, . . . , d}, K {k,j} < ∞, the proof is finished.

Lemma 5.7. For any ϑ in Θ, under the law induced by ϑ,

∆ n (ϑ) √ n + √ nI (ϑ) = O P (1) .
Proof. The proof of Lemma 3.6 in [START_REF] Cohen | LAN property for some fractional type Brownian motion[END_REF] shows that, under the law induced by ϑ,

E ∆ n (ϑ) n ----→ n→∞ -I (ϑ) .
The above convergence is obtained from Theorem 2.3 in [START_REF] Cohen | LAN property for some fractional type Brownian motion[END_REF] and we need a convergence rate in this Theorem to prove this Lemma. Lemmas 3 and 4 in [START_REF] Lieberman | Asymptotic theory for maximum likelihood estimation of the memory parameter in stationary gaussian processes[END_REF] lead us to state that

E ∆ n (ϑ) n + I (ϑ) = O n -1+δ
for any δ > 0, and hence

E ∆ n (ϑ) √ n + √ nI (ϑ) = O n -1 2 +δ .
Again using the proof of Lemma 3.6 in [START_REF] Cohen | LAN property for some fractional type Brownian motion[END_REF], we have

Var ∆ n (ϑ) √ n = O (1) ,
which completes the proof.

Lemma 5.8. We let ϑ n n be a random sequence, such that, under the law induced by ϑ, ϑ n -ϑ = o P (1) .

Then, under the law induced by ϑ, for any κ > 0,

∆ n ϑ n n - ∆ n (ϑ) n = O P n κ ϑ n -ϑ .
Proof. We let κ > 0 and C ϑ,κ be a convex compact set which depends on κ and contains ϑ. Using the proof of Lemma 3.7 in [START_REF] Cohen | LAN property for some fractional type Brownian motion[END_REF], we can state that, conditional upon

ϑ n ∈ C ϑ,κ , sup ϑn∈C ϑ,κ ∂ 3 ∂ i1 ϑ 1 ∂ i2 ϑ 2 . . . ∂ i d ϑ d n ϑ n n 1+κ = O(1) a.s.. For any (i 1 , i 2 , . . . , i d ) ∈ {0, 1, 2, 3} d , such that i 1 + i 2 + • • • + i d = 3.
Using the mean value inequality, for some positive finite random value K, we produce

P ∆ n ϑ n n - ∆ n (ϑ) n Kn κ ϑ n -ϑ P ϑ n ∈ C ϑ,κ ,
which finishes the proof.

6. Proofs of the main results

6.1. Proof of proposition 2.1. We use Theorem 4.4.1 in [START_REF] Brockwell | Time series: Theory and methods[END_REF] together with Lemma 5.1 to directly prove Proposition 2.1.

6.2. Proof of theorem 2.1. Consistency of H n We start by giving the process (X n ) a representation of spectral density identical to that of Equation ( 1) in [START_REF] Hurvich | The mean squared error of Geweke and Porter-Hudak's estimator of the memory parameter of a long-memory time series[END_REF]]. We note that 1-cos λ = 2 sin λ 2 2 and we rewrite g H,α,σ2 in the form (6.1)

g H,α,σ2 = 2 sin λ 2 -2d g * H,α,σ2 (λ)
,

where d = H -1 2 and (6.2) g * H,α,σ2 (λ) = 2C H,σ2 1 -2α cos(λ) + α 2 × sin λ 2 λ 2 2H+1 × 1 + k∈Z * |λ| 2H+1 |λ + 2kπ| 2H+1 .
We note that the first two terms of (6.2) are very regular in a neighbourhood of 0 and the last term is differentiable at first order with a bounded derivative in the neighbourhood of 0 (with respect to λ). This can be shown using uniform convergence of the derivate as in the proof of Lemma 5.4. Unfortunately, g * H,θ,σ2

does not satisfy Condition 2 in [START_REF] Hurvich | The mean squared error of Geweke and Porter-Hudak's estimator of the memory parameter of a long-memory time series[END_REF]], as the function |λ| 2H+1 is not second order derivable for any H. The bias terms convergence rate of 0 is therefore slower than in [START_REF] Hurvich | The mean squared error of Geweke and Porter-Hudak's estimator of the memory parameter of a long-memory time series[END_REF]] and the choice of m is consequently restricted. It follows that (6.3)

d n -d = - 1 2S m m j=1 (a j -a m ) log g * H,α,σ2 (λ j ) - 1 2S m m j=1 (a j -a m ) ε j ,
where (ε j ) is the error process as defined in Equation (3) of [START_REF] Hurvich | The mean squared error of Geweke and Porter-Hudak's estimator of the memory parameter of a long-memory time series[END_REF]. Condition 2 in [START_REF] Hurvich | The mean squared error of Geweke and Porter-Hudak's estimator of the memory parameter of a long-memory time series[END_REF]] is required to treat only the behaviour of the first term in the decomposition of d n -d. The others are treated by only making assumptions about the behaviour for m and n. Lemma 5.5 and Theorem 1 in [START_REF] Hurvich | The mean squared error of Geweke and Porter-Hudak's estimator of the memory parameter of a long-memory time series[END_REF] We also require an estimate of convergence rate for H n (in probability) for the rest of the proof. We therefore apply Theorem 2 from [START_REF] Hurvich | The mean squared error of Geweke and Porter-Hudak's estimator of the memory parameter of a long-memory time series[END_REF]] by treating the last term of Relation (A13) in [START_REF] Hurvich | The mean squared error of Geweke and Porter-Hudak's estimator of the memory parameter of a long-memory time series[END_REF] as in the Lemma from 5.4) in order to deduce that

d n -d = O P m -1 2 and hence H n -H = O P m -1 2 . Consistency of α n Letting Φ j i = ε H i , . . . , ε H j * for i j, we have (6.4) α n -α = Φ n * 2 Γ -1 n-1 H n Φ n-1 1 Φ n-1 1 * Γ -1 n-1 H n Φ n-1 1 .
We now focus on the asymptotic behaviour of the numerator. A Taylor expansion of Γ

n-1 H n -1 at H leads to Φ n * 2 Γ -1 n-1 H n Φ n-1 1 = Φ n * 2 Γ -1 n-1 (H) Φ n-1 1 + Φ n * 2 A (1) n (H) Φ n-1 1 H n -H + 1 2 Φ n * 2 A (2) n (H) Φ n-1 1 H n -H 2 + 1 6 Φ n * 2 A (3) n H n Φ n-1 1 H n -H 3 , where
A (1) n (H) = -Γ -1 n-1 (H) ∂Γ n-1 (H) ∂H Γ -1 n-1 (H) , A (2) n (H) =Γ -1 n-1 (H) ∂ 2 Γ n-1 (H) ∂ 2 H Γ -1 n-1 (H) + 2Γ -1 n-1 (H) ∂Γ n-1 (H) ∂H Γ -1 n-1 (H) ∂Γ n-1 (H) ∂H Γ -1 n-1 (H) , A (3) n (H) = -6Γ -1 n-1 (H) ∂Γ n-1 (H) ∂H Γ -1 n-1 (H) ∂Γ n-1 (H) ∂H Γ -1 n-1 (H) ∂Γ n-1 (H) ∂H Γ -1 n-1 (H) -3Γ -1 n-1 (H) ∂ 2 Γ n-1 (H) ∂ 2 H Γ -1 n-1 (H) ∂Γ n-1 (H) ∂H Γ -1 n-1 (H) -3Γ -1 n-1 (H) ∂Γ n-1 (H) ∂H Γ -1 n-1 (H) ∂ 2 Γ n-1 (H) ∂ 2 H Γ -1 n-1 (H) -Γ -1 n-1 (H) ∂ 3 Γ n-1 (H) ∂ 3 H Γ -1 n-1 (H) ,
and H n ∈ B H, | H n -H| . We now use the generic notation

φ A (i) n X (n-1) , Y (n-1) = X (n-1) * A (i) n Y (n-1)
for any X (n-1) , Y (n-1) ∈ R n-1 × R n-1 and i ∈ {1, 2, 3}. Then, (6.5) φ

A (i) n Φ n 2 , Φ n-1 1 = 1 2 φ A (i) n Φ n-1 1 + Φ n 2 , Φ n-1 1 + Φ n 2 ) -φ A (i) n Φ n-1 1 , Φ n-1 1 -φ A (i) n Φ n 2 , Φ n 2 ) 1 n 2 T r A (2) n (H) σ 2 Γ n-1 (H) 2
In view of Remarks 6.1, 6.2 and Lemma 5.4, we can apply Theorem 2.3 in [START_REF] Cohen | LAN property for some fractional type Brownian motion[END_REF] and deduce that

1 n T r A (2) n (H) Γ n-1 (H, α, σ 2 ) ----→ n→∞ 1 2π π -π ∂ 2 ∂ 2 H f H (λ) f H,θ,σ2 (λ) f H (λ) 2 + 2 ∂ ∂H f H (λ) 2 f H,α,σ2 (λ)
f H (λ) 3 dλ and that

1 n T r A (2) n (H) Γ n-1 (H, α, σ 2 ) 2 ----→ m→∞ 1 2π π -π ∂ 2 ∂ 2 H f H (λ) f H,α,σ2 (λ) f H (λ) 2 + 2 ∂ ∂H f H (λ) 2 f H,θ,σ2 (λ) f H (λ) 3 2 dλ,
where f H,α,σ2 is the spectral density of the process (X n-1 + ε H n ). Consequently, (6.9) converge to 0 and 1 n φ A (2)

n Φ n-1 1 + Φ m 2 , Φ m-1 1 + Φ n 2 ) P ----→ m→∞ 1 2π π -π ∂ 2 ∂ 2 H f H (λ) f H,α,σ2 (λ) f H (λ) 2 + 2 ∂ ∂H f H (λ) 2 f H,α,σ2 (λ)
f H (λ) 3 dλ.

The generalization of this reasoning leads to (6.12) Φ m * 2 A

(1)

n (H) Φ n-1

1 n P ----→ n→∞ K
(1) H,α,σ2 and (6.13) Φ n * 2 A

(2)

n (H) Φ n-1

1 n P ----→ n→∞ K (2) H,α,σ2
where K

(1)

H,θ,σ2 and K

(2)

H,θ,σ2 are constants which can be expressed in integral form from Theorem 2.3 in [START_REF] Cohen | LAN property for some fractional type Brownian motion[END_REF]. We now turn to the most delicate term, which is

Φ n * 2 A (3) n H n Φ n-1 1 = Φ n * 2 Γ n-1 (H) -1 2 Γ n-1 (H) 1 2 A (3) n H n Γ n-1 (H) 1 2 Γ n-1 (H) -1 2 Φ n-1 1 .
The Cauchy-Schwarz inequality implies that

Φ n * 2 A (3) n H n Φ m-1 1 Γ -1 2 n-1 (H) Φ n * 2 Γ n-1 (H) 1 2 A (3) n H n Γ 1 2 n-1 (H) Γ -1 2 n-1 (H)Φ n-1 1 .
The quantities Γ n-1 (H)

-1 2 Φ n * 2 2
and Γ n-1 (H)

-1 2 Φ n-1 1 2
are quadratic forms of Gaussian process. Using Theorem 2.3 in [START_REF] Cohen | LAN property for some fractional type Brownian motion[END_REF], we produce 1

√ n Γ n-1 (H) -1 2 Φ n * 2 = O P (1) and 1 √ n Γ n-1 (H) -1 2 Φ n-1 1 = O P (1) . Now, Γ n-1 (H) 1 2 A (3) n H n Γ m-1 (H) 1 2 T r Γ n-1 (H) 1 2 A (3) n H n Γ n-1 (H) 1 2 2 = T r A (3) m H n Γ n-1 (H) A (3) m H n Γ n-1 (H) .
We let δ > 0 and A δ = {h ∈ R h ∈ [H -δ; H + δ]}. Fixing K > 0, then

P 1 n T r A (3) n H n Γ m-1 (H) A (3) n H n Γ n-1 (H) K 1 1 n T r A (3)
n (Hn)Γn-1(H)A (3) n (Hn)Γn-1(H) K × P H n ∈ A δ Note that A δ is a compact set, so we can choose a δ small enough to verify the third hypothesis of Theorem 2.3 in [START_REF] Cohen | LAN property for some fractional type Brownian motion[END_REF] which works for uniform convergence. Owing to this Theorem, we have

lim n→∞ sup Hn∈A δ 1 n T r A (3) n H n Γ n-1 (H) A (3) n H n Γ n-1 (H) K Θ * ,
where K Θ * is a constant depending on the set A δ . This constant is finite because it expresses itself as a supremum of an integral (i.e. it is distributed over a compact set) with respect to the λ of continuous function with respect to H for almost any λ bounded by an integrable function not depending on H. We deduce that

lim n→∞ P 1 n T r A (3) n H n Γ n-1 (H) A (3) n H n Γ n-1 (H) K Θ * = 1
and hence (6.14)

n -3 2 Φ n * 2 A (3) n H n Φ n-1 1 = O P (1) .
It is shown in [Esstafa, 2019] (see Chapter 4, Lemma 4.4) that (6.15)

1 n Φ n * 2 Γ -1 n-1 (H) Φ n-1 1 P ----→ n→∞ 0.
The combination of (6.12),(6.13), (6.14) and (6.15) allows us to deduce that

Figure 1 .Figure 2 .Figure 3 .Figure 4 .

 1234 Figure 1. Statistical error of the estimators multiplied by√n speed, where ϑ = (0.3, 0.3, 1). The first line corresponds to the initial estimator and the second line to the one-step estimator. The blue curves correspond to the density of the centred normal distribution where the theoretical variance is approximated by the empirical variance. The curves in red correspond to the limit law of the one-step estimator.

Acknowledgments

We observe that each term of the right-hand side of the above equality is a quadratic form of a Gaussian process whose matrix is expressed as a product of Toeplitz matrices (inverse of Toeplitz matrices) in which the coefficients are Fourier coefficients of a certain function owing to Assertion 4) of Lemma 5.4. We use Lemmas 5.3 and 5.4 with Theorem 2.3 of [START_REF] Cohen | LAN property for some fractional type Brownian motion[END_REF] to obtain the asymptotic behaviour of these terms (suitably renormalized). Remark 6.1. Owing to the almost perfect linear representation given by Lemma 5.1, the process (X n-1 +ε H n ), (X n ) has a spectral density which exhibits exactly the same behaviour as that of the fractional Gaussian noise in the neighbourhood of 0. Indeed, the spectral density of (X n-1 + ε H n ), (X n ) satisfies Conditions (1) and (2) of Lemma 5.4 (although only j = 0 is necessary) and satisfies the estimates given in Condition (3c) of Lemma 5.4. Remark 6.2. With the notations of Theorem 2.3 in [START_REF] Cohen | LAN property for some fractional type Brownian motion[END_REF], it is evident that, in our case, the third assumption of this Theorem is satisfied for all p ∈ N * owing to Estimation (3) of Lemma 5.4 and Remark 6.1.

To simplify the proof, we detail the treatment of one term, with the others following the same reasoning. We let

and let Γ n (H, θ, σ 2 ) (respectively Γ n (H, θ, σ 2 )) be the covariance matrix of the Gaussian process (X j ) {1 j n} (respectively (X j + ε H j+1 ) {1 j n} ). From Lemma 5.3, we have

We now consider the asymptotic behaviour of the denominator. We know that

The same reasoning which made it possible to obtain (6.12) and (6.13) also makes it possible to show that (6.17)

where K

(3) H,α,σ2 is a positive constant. In the same manner as for the treatment of (6.14), we have

It is shown in [Esstafa, 2019] (see Chapter 4, Lemma 4.3) that

Equations (6.17), (6.18) and (6.19) lead us to (6.20)

Finally, Equations (6.16) and (6.20) give us the consistency of α n .

Consistency of σ 2,n . We know that

Using similar arguments as the proof of the consistency of α n (i.e. by expanding Γ -1 n-1 H n at H to the third order), we produce

Combining (6.21), (6.22) and (6.23) with the consistency of α n , we find that σ 2,n -σ 2 converges in probability to 0.

Asymptotic joint distribution of the estimators. We show that the statistical error of each component is a function of the statistical error of H modulo a remainder term which is asymptotically negligible. The first estimator is satisfied (owing to Theorem 2 in [START_REF] Hurvich | The mean squared error of Geweke and Porter-Hudak's estimator of the memory parameter of a long-memory time series[END_REF]) as

The second estimator is shown in [Esstafa, 2019] (see Chapter 4, Lemma 4.5) as

The combination of (6.25) and the decomposition of Φ

To treat the third estimator, we show via a cumulant method (like in Theorem 2 of [Avram, 1988]) that (6.27)

for V ϑ > 0, which can be expressed in integral form as in Theorem 2.3 of [START_REF] Cohen | LAN property for some fractional type Brownian motion[END_REF]. It only remains to apply (6.27) and ( 6.22) together with the expansion of σ 2,n -σ 2 to deduce that

(1) ϑ

for some finite constant C

(1)

and (2.11) is proved.

6.3. Proof of proposition 2.2. This result is a direct consequence of Theorem 3.4 from [START_REF] Cohen | LAN property for some fractional type Brownian motion[END_REF].

6.4. Proof of proposition 2.3. Direct computations lead to (6.30)

The mean value theorem yields

and, substituting in (6.30), we produce

where

n -ϑ) for some 0 < v < 1. We then consider the first term, as follows.

Then,

The first subterm needs more regularity than the usual uniform continuity of the Sweeting conditions [Sweeting, 1980]. As in [START_REF] Kutoyants | On multi-step mle-process for markov sequences[END_REF], we impose that the Fisher information matrix is locally Lipschitz, as shown in Lemma 5.6. Consequently,

for any κ > 0. Then, combining with the previous result,

As

√ n n δ -→ 0, we get that A n → 0 in probability. The second term is

We show that the second subterm tends to 0 in probability as I n (•) is supposed to be non-degenerate and uniformly continuous. The central limit theorem for the first subterm gives the result.

Laboratoire Manceau de Mathématiques, Le Mans Université, Avenue O. Messiaen, 72085 Le Mans Cedex 9, France.

Email address: samir.ben hariz@univ-lemans.fr

Laboratoire Manceau de Mathématiques, Le Mans Université, Avenue O. Messiaen, 72085 Le Mans Cedex 9, France.

Email address: Alexandre.Brouste@univ-lemans.fr School of Mathematics, Shanghai University of Finance and Economics, No.777 Guoding Road, Yangpu District, Shanghai, China.

Email address: caichunhao@mail.shufe.edu.cn Laboratoire Manceau de Mathématiques, Le Mans Université, Avenue O. Messiaen, 72085 Le Mans Cedex 9, France.

Email address: marius.soltane@univ-lemans.fr


Kia Dashtipour
email: kia.dashtipour@glasgow.ac.uk

William Taylor

Shuja Ansari

Mandar Gogate

Adnan Zahid

Yusuf Sambo

Amir Hussain

Qammer H Abbasi

Muhammad Ali Imran

Public Perception towards fifth generation of cellular networks (5G) on social media

Keywords: Sentiment Analysis, 5G, Mobile Network Quality, Machine Learning, Opinion Mining

With advancement of social media network, there are lots of unlabelled reviews available online, therefore its necessarily to develop an automatic tools to classify these types of reviews. For utilising these reviews for user perception, there is a need for automated tools that can process online user data for optimising user perception. In this paper, a sentiment analysis framework has been proposed to identify people's perception towards mobile networks. The proposed framework consists of three basic steps: preprocessing, feature selection and applying different machine learning algorithms. The performance of the framework has taken into account different feature combinations. The simulation results show that the best performance is by integrating unigram, bigram and trigram features.

INTRODUCTION

The 5th generation mobile network is new global wireless standard after 1G, 2G, 3G and 4G network. The 5G is new types of network which is designed and developed to connect virtually everyone and everything together and consists of different machines, objectives and devices. In addition, the fifth generation of cellular networks is required to be more efficient and economical in terms of Key Performance Indicators (KPI). KPIs are of interest to stakeholders and different applications. These KPIs from an operator perspective, consist of capacity, reliability and quality of service. From the user perspective, the KPIs include uninterrupted connection, infinite capacity and zero latency. However, there no technology that can offer infinite capacity or zero latency. In the past few years, there has been lots of research carried out in next generation mobile network (5G), which consists of different opportunities and challenges. There are different challenges in fifth generation of cellular networks (5G) that have been discussed in different literature. The most significant of these are ultra-dense networks and millimetre waves, however, there are other technologies which are significant for the next generation of network such as two-layer architecture and cognitive radio based architectures which have great performance in the next generation of the network.

In fifth generation of cellular networks (5G), network data analytic and machine learning systems can perform a key role. The technique to understand people's behaviour towards the 5G network is vital and it helps to improve the performance of the network communication [START_REF] Sharma | Twitter sentiment analysis for brand reputation of smart phone companies in india[END_REF][START_REF] Opincariu | Education in the 5g and the ai context[END_REF].

With the advent of social media and e-commerce, websites allow users to share opinions and feedback about different products and services. Customers can make important decisions by reading other people's experiences. In addition, the customer feedback can be classified to make improvements on the service or product. For example, if a person wants to buy a mobile phone and the reviews provide negative information related to the battery, operation speed or camera, this can influence the consumers decision. In addition, this can assist in providing better mobile quality by taking into account the complaints made by past customers and making informed improvements to products. As another example, if someone wants to book a hotel, the buyer can look over the online reviews to understand previous customer experience such as cleanliness and services for the hotel. However, there are billions of bytes of data generated per day consisting of user feedback which cannot be manually labelled and analysed for individual organisations and companies [START_REF] Yadav | Sentiment analysis using deep learning architectures: a review[END_REF]. Sentiment analysis is the process of automatically understanding and classifying the data into positive and negative information from the source material such as reviews and comments. The main task of sentiment analysis is to assign polarity into sentences (positive or negative).

However, the online review is a mixture of positive and negative comments about different aspects of the products or services instead of expressing positive or negative opinions. For example, "the 5G mobile network is extremely fast, however I do not feel secure while I am using the 5G network". The sentence expresses positive sentiment towards the speed of the mobile and negative sentiment towards the security of the mobile network [START_REF] Kumar | Systematic literature review of sentiment analysis on twitter using soft computing techniques[END_REF], [START_REF] Dashtipour | A hybrid persian sentiment analysis framework: Integrating dependency grammar based rules and deep neural networks[END_REF].

Most of the current sentiment analysis approaches are focused on analysing products and movie reviews and there is less work been carried out in different fields such as people's perception towards fifth generation of cellular networks (5G). However, most of the current approaches consider a small corpus, which makes the task difficult for machine learning approaches to identify the overall polarity of the sentence. Furthermore, the current sentences consist of lots of sarcastic and ironic words which is difficult to determine the overall polarity for these types of sentences. For example, "Tell me something that I do not know". Most of the current approaches for sentiment analysis fail to understand the real noisy text consists of sarcasm, idioms, informal words and sentences with spelling mistakes. In addition, there are scarce availability of tools and resources.lexicon and labelled corpus are some of the tools which are available for sentiment analysis. This limited range of available tools is the main bottleneck in the design of the sentiment analysis approaches [START_REF] Kaity | An automatic non-english sentiment lexicon builder using unannotated corpus[END_REF]. One of the main issue for sentiment analysis approaches is lack of labelled dataset. However, it is worth to mention that, there are lots of unlabelled dataset available online, but it is time consuming for users to manually label dataset.

In order to address the aforementioned issues and limitations, framework that exploits n-gram features has been proposed to identify the polarity of sentence. This proposed approach demonstrated that the overall performance and effectiveness of polarity detection in real noisy data. The ngram features are based on the linguistic text rules that allow to extract text features from sentences. As a result, the ngram features take into account the relation between keywords and the word order and individual word polarity to determine the underlying polarity of the sentence. We perform an extensive and comprehensive set of experiments using novel corpus and compare the performance of the approach with different selected features. The Support Vector Machine(SVM), logistic regression, Naive Bayes and Multi Layer Perceptron (MLP) used to evaluate the performance of the approach. The comparative simulation results show that the proposed approach achieved better performance as compared to the state-of-art approach.

The rest of the paper is organised as follows: In Section 2 related work is presented, Section 3 presented the proposed framework, Section 4 presented the experimental results, Section 5 presents discussion and finally Section 6 concludes the work and presents the future work.

RELATED WORK

The extensive research in the current literature shows that the machine learning has been used in different fields such as sentiment analysis [START_REF] Hussain | Artificial intelligence-enabled analysis of public attitudes on facebook and twitter toward covid-19 vaccines in the united kingdom and the united states: Observational study[END_REF]; [START_REF] Dashtipour | A novel context-aware multimodal framework for persian sentiment analysis[END_REF][START_REF] Dashtipour | Multilingual sentiment analysis: state of the art and independent comparison of techniques[END_REF]Dashtipour et al. ( ,a, 2017c,b,a),b,a); [START_REF] Jiang | Robust visual saliency optimization based on bidirectional markov chains[END_REF]; [START_REF] Dashtipour | Exploiting deep learning for persian sentiment analysis[END_REF][START_REF] Dashtipour | A hybrid persian sentiment analysis framework: Integrating dependency grammar based rules and deep neural networks[END_REF]; [START_REF] Guellil | A semisupervised approach for sentiment analysis of arab (ic+ izi) messages: Application to the algerian dialect[END_REF]; [START_REF] Dashtipour | Persent 2.0: Persian sentiment lexicon enriched with domain-specific words[END_REF], cyber-security Adeel et al. (2019a); [START_REF] Jiang | A survey on deep learning for the routing layer of computer network[END_REF]; [START_REF] Shiva | Complex-valued computational model of hippocampal ca3 recurrent collaterals[END_REF]; [START_REF] Gogate | Random features and random neurons for brain-inspired big data analytics[END_REF], hand-written recognition Ahmed et al. (2021), speech enhancement Gogate et al. (2020aGogate et al. ( ,b, 2019aGogate et al. ( , 2017a[START_REF] Gogate | Dnn driven speaker independent audio-visual mask estimation for speech separation[END_REF]; [START_REF] Ozturk | A novel deep learning driven, low-cost mobility prediction approach for 5g cellular networks: The case of the control/data separation architecture (cdsa)[END_REF]; [START_REF] Adeel | Contextual deep learning-based audio-visual switching for speech enhancement in real-world environments[END_REF][START_REF] Adeel | Deep neural network-based contextual recognition of arabic handwritten scripts[END_REF]; [START_REF] Gogate | A novel brain-inspired compression-based optimised multimodal fusion for emotion recognition[END_REF], posture detection [START_REF] Taylor | An intelligent non-invasive real-time human activity recognition system for next-generation healthcare[END_REF]; [START_REF] Liaqat | A hybrid posture detection framework: Integrating machine learning and deep neural networks[END_REF][START_REF] Liaqat | Non invasive skin hydration level detection using machine learning[END_REF] and etc. However, there is not any research carried out to detect sentiment polarity for tweets related to 5G.

In the literature, extensive research has been carried out to implement different sentiment analysis approaches. The microblogging websites are the biggest platform that allows users to share their thoughts and opinions in the public domain. Twitter is the most well-known microblogging website that allows people to express their feelings and emotions in the form of "tweets" with a character range of 280. There are over 250 million tweets expressing the feelings and emotions of people with different opinions and situations [START_REF] Duong | Ultra-dense networks for 5G and beyond: modelling, analysis, and applications[END_REF] [START_REF] Mamgain | Sentiment analysis of top colleges in india using twitter data[END_REF].

The twitter users vary from politicians to everyday people that provide different types of reviews from different points of views. This is the the main reason a dataset collected from twitter is used in this paper.

There is lots of research carried out in this domain to determine people's perception towards different products such as Sony mobile. For example. [START_REF] Sharma | Smartphone-based fundus camera device (mii ret cam) and technique with ability to image peripheral retina[END_REF] focused to find the most well known smartphones in India, the tweets are collected and then machine learning is applied to determine a brand reputation score. This will help customers to find the most branded smartphones in India. [START_REF] Somula | Twitter sentiment analysis based on us presidential election 2016[END_REF] proposed an approach to perform a sentiment analysis to determine the winner of the US election in 2016, the tweets are collected with respect to Donald Trump and Hillary Clinton, the tweets analysing results reveals that Donald Trump received more positive scores as compared to Hillary Clinton. [START_REF] Mehta | Sentiment analysis of tweets using supervised learning algorithms[END_REF] proposed a sentiment analysis approach to identify the best Indian airlines through twitter, the analysis of results reveals that the customers are happier with airIndia services compared to spicejet. [START_REF] Kumari | Sentiment analysis on aadhaar for twitter data-a hybrid classification approach[END_REF] proposed an approach using twitter API to collect corpus, after pre-processing and use of natural language processing, the hybrid classifier utilised machine learning and long short-term memory to improve the performance of the approach. The experimental results reveal that the proposed model achieved better performance as compared with state-of-the-art approaches. [START_REF] Usama | Deep learning based weighted feature fusion approach for sentiment analysis[END_REF] introduced a novel model multi-level feature extraction and feature combination by using convolutional neural network (CNN) and recurrent neural network (RNN) to identify the sentiment in movie reviews.

The CNN and RNN received sentiment text as input and learn different features to network architecture.

The word embedding fed into CNN and learn multilevel contextual features from every layer of CNN and perform multilevel features fusion. Finally, the multilevel and multitype features are combined and softmax classifier is used to identify final polarity of sentences. Most of the aforementioned studies implement a lexicon to determine the polarity of the text. However, the use of lexicon fails to identify the polarity of word order in the sentence. In addition, most of the current studies use rules to detect negation in a sentence which cannot be directly applied to English language. However, we need a framework to identify the polarity of the sentence without developing a lexicon which is time consuming. Therefore, we implemented

METHODOLOGY

This section describes our proposed novel context-aware framework for fifth generation of cellular networks (5G) sentiment analysis. The proposed framework exploits the polarity of sentence more accurately as compared to traditional word occurrence frequency-based approaches. 1 shows the related keywords for fifth generation of cellular networks (5G).

Pre-processing: The tweets corpus has been collected using twitter API and it has been labelled into positive and negative using SentiWordNet. The corpus is divided into a training set (60%), test set (30%)

and validation set (10%) to apply machine learning algorithms including SVM, Naive Bayes and MLP. The corpus is tokenized and normalised. The tokenisation technique is used to break sentences into words. For example, "I really like mobile" will be converted into words such as "I", "really", "like" and "mobile".

Afterwards, the normalisation technique is used to normalise the tweets. For example, "The speed for 5G mobile is gr8" will be converted into "The speed for 5G mobile is great".

Ngram: N-gram features are widely used in different approaches of sentiment analysis. When there is one term is taken as feature is called unigram, for two term is called bigram and three terms is called trigram. In our proposed approach we have been using unigram, bigram and trigram and we used the combination of n-gram features.

SentiWordNet: In order to assign sentiment polarity into sentence we used SentiWordNet widely avaiable online lexicon to assign sentiment polarity (-1, 0, 1) into sentences.

Machine learning classifiers:

In order to evaluate the performance of the approach, the machine learning classifier is used to evaluate the performance of the approach. The scikit-learn python package is used to developed multi-layer perceptron, logistic regression, linear SVM, RBF SVM and naive bayes is used to train the model. The MLP consist of one hidden layer which is able to apply to supervised problem, the MLP are set of input and put and it learns to model the correlation between input and output. However, the main issue with MLP it consists of many parameters which is fully connected and each nodes is connected to other node which can result in redundancy and inefficiency. The main advantage of Naïve Bayes is performs fast and save lots of time however, the prediction can be wrong. In addition, the main advantage of logistic regression is easy to implement and train however, the number of observation is less than number of features. Finally, the main advantages of SVM is work well with unstructured data such as text, as the tweets are unstructured, therefore, the SVM performs well of tweets data. However, the main issue with SVM, the choosing correct kernel is not easy.

EXPERIMENTAL RESULTS

In order to calculate the sentiment polarity of the sentence, the SentiWordNet has been used to calculate overall polarity of the tweets. The tweets extracted using different keywords such as 5G, next generation mobile network, fifth generation of technology and 5g devices. The tweets are collected. After tweets collection, the punctation and stop words are removed and sentence are normalised, and then its been converted into BOW and finally machine learning classifiers trained. In the pre-processing stage, the sentence are normalised, for example, the word "fishing" are converted into "fish".

In order to evaluate the performance of the proposed approach, the tweets are converted into bag of words (BOW). These BOWs are sent to machine learning algorithms including Linear and RBF SVM, naive bayes, logistic regression and MLP to evaluate the performance of the approach. The intial experimental results demonstrate that the combination of unigram and bigram achieved accuracy of 86.71%.

Dataset: In order to evaluate the performance of the approach the tweets have been collected from twitter, the 5G hashtag has been used to collect tweets, more than fifty thousand tweets have been collected and the positive and negative polarity has been assigned into tweets using SentiWordNet lexicon. The neural tweets have been eliminated. The experimental results show that the combination of unigram and bigram achieved better performance as compared to other approaches. In order to evaluate the performance of the proposed approach, there are different evaluation metrics including accuracy, precision, recall and f-measure are used:

P recision = T P T P + F P

(1)

Recall = T P T P + F N (2)
F measure = 2 * P recision * Recall P recision + Recall (3) where TP denotes true positive, TN presents true negative, FP is false positive, and FN represents false negative respectively. In addition, Table 2 shows the parameters that are used to trained the machine learning methods. The scikit-learn package is used to train the machine learning classifiers. In addition, the training time for each models has been presented in Table 2.

Accuracy = T P + T N T P + T N + F P + F N (4)
Table 3 shows the results of different N-gram features. The unigram (Uni), bigram (Bi) and trigram (Tri) is extracted from the sentence, as the comparitive experimental result shows the unigram (Uni) achieved better performance as compared to other features.

Table 4 shows the comparison of different N-gram features. The empirical results shows that the combination unigram (Uni) and bigram (Bi) is achieved better performance as compared to other features. which it might cause cancer, the price of the 5G mobile which is very expensive and also the users concerns about the performance of 5G mobiles. In order to find the most negative keywords towards 5G the word frequency of the words negative sentences is calculated. The word frequency shows that the most discussed keywords are radiation, price and performance. As Figure 3 (a) shows the most twitter users who located in England, Scotland believe that the 5G can cause cancer. It is to be noted that, 31% of tweets in Scotland and 31% in England has negative concern about the cause of radiation using 5G. In addition, the Figure In contrast, the Figure 4 displays the overall trends for the twitter users. As the Figure 4 the most discussed trend in agriculture (29%), healthcare (28%) and smart home (26%).

The Figure 5 displays the most discussed trends towards 5G technology, the radiation, price and below technology towards 5G. For example, most users concerns towards 5G signal which it might cause cancer, the price of the 5G mobile which is very expensive and also the users concerns about the low technology of 5G mobiles. As shown in graph 5, England (a) the most discussed trends are agriculture (31%), healthcare (28%) and smart-home(16%). In addition, Scotland (b) the most discussed trends are agriculture (41%), healthcare (32%) and transport (12%). Additionally, in Wales the most discussed trends are agriculture (29%), healthcare (28%) and transport(18%). Finally, North Ireland, the most discussed trends are agriculture (29%), healthcare (28%) and transport (18%).

The graph 6 displays the occupation of the twitter users who had the most positive comments towards fifth generation of cellular networks (5G).

The graph 7 displays the occupation of the twitter users who had the most negative comments towards fifth generation of cellular networks (5G). As shown in Table 5, the top 10 positive and negative bigrams keywords related to next generation mobile network (5G) are presented. As the Table III shows the tweets towards price of 5G network generation is positive and towards EU countries are negative.

As shown in Table 6, the top 10 positive and negative trigram keywords related to next generation mobile network (5G) are presented. As the Table IV shows the tweets towards price of 5G network generation is positive and towards EU countries are negative.

DISCUSSION

In this study, the tweets related to the next generation mobile network (5G) were analysed and their sentiment polarity were identified. The sentiment analysis of 5G keywords tweets considered because these tweets carried lots of information related to next generation of communication network. Only English tweets were considered because it has large coverage and the most widely common language used in the world. In the current study, more than 10,000 tweets were retrieved. There were many duplicate tweets in the retrieved data which were removed from the database. However, we did not limit our dataset to tweet Moreover, as part of the data pre-processing we have removed the emoji characters from the tweets.

However, it is worth mentioning that emoji characters are commonly used in tweets and they are useful for identifying the overall polarity of the tweets. In future work, we intend to consider emoji characters as part of the process to identify the overall polarity of tweets as the emoji characters can provide more accurate sentiment scores. In contrast, the time frame to analyse the specific tweets related to fifth generation of cellular networks (5G) might be associated with events which increases the number of tweets in certain dates, weeks or months. It is worth to mention that, there are lots of tweets related to conspiracy theories, for example some of tweets believed that the 5G can cause covid-19 in people or it can cause cancer.

Herein we explain the most topics discussed which can change the overall sentiment polarity of the tweets into positive: The most discussed topics related to 5G was speed of the current network. Most of the tweets were positive towards the speed of 5G. However, it is to be noted that, there are large number of tweets were negative about the bandwidth and speech of the 5G mobile network.

Security: In addition, another most discussed topic related to 5G was security of the 5G services. Most of the twitter users were positive towards the current security towards the services for 5G. For example, "5G is more secure than 4G". Performance: Additionally, the performance of the 5G mobile network were discussed in most of the tweets. Most of the tweets were positive towards the performance of the 5G network communication. For example, "I am really excited with current performance of 5G mobile".

Herein we explain the most topics discussed which can change the overall sentiment polarity of the tweets into negative:

5G expose cancer: Our finding shows that, most of the tweets were discussed about next generation mobile network can cause cancer. For example, "I won't buy 5G mobile, because it can cause cancer".

There is strong negative comments towards the health issue towards 5G network communication. Most of the Twitter users believe that the 5G can cause damage to their health.

5G price: Our analysis shows that, most of the Twitter users have negative comments towards the price of the 5G mobiles. For example, one tweet clearly mentioned that "How can I afford to buy such expensive mobiles".

CONCLUSION

In this study, we perform a series of sentiment analysis on data retrieved from twitter. The twitter data under investigation was related to the fifth generation of cellular networks (5G). We have collected relevant tweets in the English language. Therefore, we proposed framework for mobile networks (such as 5G)

based on different feature combinations. The performance of the proposed framework is evaluated using different feature combination in terms of different evaluation metrics such as accuracy, precision, recall and f-measure. In addition, we have compared the proposed method with different machine learning algorithms such as Naïve Bayes, MLP, SVM, etc. In addition, we have analysed the tweets to understnad user perception towards 5G. As part of our future work, we intend to extend the current framework for multilingual sentiment analysis and integration of a closed loop self organising network algorithm with the proposed user sentiment analysis framework.

ADDITIONAL REQUIREMENTS

For additional requirements for specific article types and further information please refer to Author Guidelines.

Figure 1 .

 1 Figure 1. Proposed Framework for Sentiment Analysis for people perception towards fifth generation of cellular networks (5G)

Figure 2

 2 Figure 2 displays the positive sentiment towards fifth generation of cellular networks (5G) technology, the speed, security and performance towards 5G. For example, the user has a positive opinion towards speed, security and performance. In order to find the most positive keywords towards fifth generation of cellular networks (5G) the word frequency of the words positive sentences is calculated. The word frequency shows that the most discussed keywords are speed, security and performance. As shown in Figure 2 (a) Scotland and England have the most positive discussion towards speed. For example, 45% of tweets in Scotland and 30% in England has positive opinion about the speed of 5G. In the other hand, Figure 2 (b) demonstrate that England and Scotland have most positive discussion toward security trend. 41% of tweets in England and 36% in Scotland has positive opinion about the security of 5G. Figure 2 (c) shows Scotland and England have most positive discussion toward performance trend. 39% of tweets in Scotland and 28% in England has positive opinion about the performance of 5G.

3

  (b) most twitter users who located in Scotland believe that the price of 5G mobile is very expensive. It is worth to mention that, 32% of tweets in Scotland and 27% of tweets in Wales has negative concerns about the expensive price of 5G technology. Additionally, the Figure3(c) shows the most tweets users in North-Ireland are less satisfied with current performance of the 5G mobiles. It is to be noted that, 27% in North-Ireland and 25% in England has negative concern about current performance of 5G.

Figure 2 .

 2 Figure 2. Positive trends towards fifth generation of cellular networks (5G)

Figure 3 .

 3 Figure 3. Negative trends towards fifth generation of cellular networks (5G)

Figure 5 .

 5 Figure 5. Most discussed trends for fifth generation of cellular networks (5G)

Figure 6 .Figure 7 .

 67 Figure 6. Occupation for most positive twitter users towards fifth generation of cellular networks (5G)

Table 1 .

 1 Next Generation Mobile Network (5G)

	Keywords
	5G
	Next generation mobile network
	fifth generation of technology
	5G devices
	a framework for twitter sentiment analysis that integrates feature engineering and machine learning to
	improve the performance and robustness of polarity detection in real noisy data.

Table 2 .

 2 Paramets of ML algorithms

	Algorithm	Parameter		Time
	SVM		RBF Kernal		4 mins and 21 s
	Naive Bayes	sample weight = none 2 mins and 12 s
	MLP		activation = relu	3 mins and 31 s
	Logistic regression Penalty = l2		3 mins 42 s
	Table 3. Results of N-gram Features
	Feature Classifier	Accuracy Precision Recall F-score
	Uni	MLP	85.92	0.86	0.86	0.86
	Uni	LR	86.14	0.86	0.86	0.86
	Uni	Linear SVM 84.79	0.84	0.84	0.84
	Uni	RBF SVM	62.61	0.72	0.63	0.54
	Uni	NB	86.48	0.87	0.86	0.87
	Bi	MLP	79.16	0.8	0.79	0.78
	Bi	LR	78.37	0.79	0.78	0.78
	Bi	Linear SVM 73.98	0.78	0.74	0.72
	Bi	RBF SVM	62.61	0.72	0.63	0.54
	Bi	NB	78.82	0.8	0.79	0.78
	Tri	MLP	73.42	0.77	0.73	0.71
	Tri	LR	72.52	0.77	0.73	0.7
	Tri	Linear SVM 69.48	0.77	0.69	0.65
	Tri	RBF SVM	62.61	0.72	0.63	0.54
	Tri	NB	71.39	0.77	0.71	0.68

Table 4 .

 4 Comparison of combination of N-gram Features

	Feature Classifier	Accuracy Precision Recall F-score
	Uni + Bi MLP	86.71	0.87	0.87	0.87
	Uni + Bi LR	86.14	0.86	0.86	0.86
	Uni + Bi Linear SVM 85.92	0.86	0.86	0.86
	Uni + Bi RBF SVM	62.61	0.72	0.63	0.54
	Uni + Bi NB	85.81	0.86	0.86	0.86
	Uni + Tri MLP	85.13	0.85	0.85	0.85
	Uni + Tri LR	86.59	0.87	0.87	0.87
	Uni + Tri Linear SVM 86.48	0.87	0.86	0.87
	Uni + Tri RBF SVM	62.61	0.72	0.63	0.54
	Uni + Tri NB	85.47	0.85	0.85	0.85
	Bi + Tri	MLP	76.91	0.79	0.77	0.76
	Bi + Tri	LR	77.02	0.79	0.77	0.76
	Bi + Tri	Linear SVM 77.02	0.8	0.77	0.76
	Bi + Tri	RBF SVM	62.01	0.6	0.59	0.6
	Bi + Tri	NB	71.28	0.7	0.69	0.7
	Figure 3 displays the negative sentiment towards fifth generation of cellular networks (5G) technology,
	the radiation, price and below technology towards 5G. For example, most users concerns towards 5G signal

Table 5 .

 5 Most frequent positive bigrams for next generation mobile network (5G)

	Use cases
		Agriculture
	Healthcare	29.0%
	28.0%
	26.0%	6.0% 5.0% 6.0%	Transport VR/AR
		Factory
	Smarthome
	Figure 4. Trend for UK towards fifth generation of cellular networks (5G)
	Positive Bigram	Negative Bigram
	5G cheap	Low performance
	Good coverage	Awful services
	Good supply	blow Huawei
	Great performance	Hate 5G
	High security	Low speed
	Fast speed	Low coverage
	Great system	Low frequency
	communication networks 5G Crap
	Nokia performs	Slow 5G
	great satisfaction	Expensive technology

Table 6 .

 6 Most frequent positive Trigrams for next generation mobile network (5G)

	Positive Trigram	Negative Trigram
	cheap 5G phones	Slow coverage 5G
	Good coverage phone	Low suppliers services
	best 5G phones	blow Huawei China
	Good Supply 5G	5G slow connection
	Great areas coverage	5G low frequency
	Good test zones	About 5G dangers
	5G fast speed	Low frequency 5G
	5G good communication New 5G Crap
	great 5G signal	Expect 5G Slow
	Nokia performs well	Expensive New technology
	Speed:

Table 7 .

 7 Table 7 shows the example of positive and negative tweets towards the fifth generation of cellular networks Example of Positive and Negative tweets towards Next Generation Mobile Network(5G) Positive Negative Currently, The 5G phones are the best that you can buy right now

	with 5G as dangerous
	to our health

257 (5G).

258

Frontiers

ACKNOWLEDGMENTS

This research work was funded by EPSRC DTG EP/N509668/1 Eng.

FUNDING

This research was funded under EPSRC DTA studentship (EPSRC DTG EP/N509668/1 Eng

CONFLICT OF INTEREST STATEMENT

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

AUTHOR CONTRIBUTIONS