Octavian Augustus At Apollonia

Amelia Carolina Sparavigna

The article discusses a possible link between the astrological sign of the Capricorn, that Octavian Augustus chose as his symbol, and the constellation in which the sun was at the beginning of the calendar of Julius Caesar, that is the First January of 45 BC. Augustus may have chosen this constellation as the symbol of the birth of a new age, which Caesar, his adoptive father, had established with the reform of the calendar. The astrological sign was assigned to Octavian when he was in Apollonia, in the same year of the reform, 45 BC. The "magnus gubernator" of the world was born. Under the sign of Capricorn, Octavian Augustus ruled the empire.

Natus est Augustus M. Tullio Cicerone C. Antonio conss. VIIII. Kal. Octob., paulo ante solis exortum, regione Palati, ad Capita bubulo, ubi nunc sacrarium habet, aliquanto post quam excessit constitutum. This is told by Suetonius in his Life of Caesars. That is, Octavian was born in the year 63 BC. The year is given by the names of two Roman consuls. The day was 23 September. However, we know that the astrological sign of Augustus was not the Virgo or the Libra but the Capricorn. The Capricorn is the sign that we find on Augusts' coins and the standards of his legions.

The problem is not only limited to this fact. It is much more complex. It is so complex that even Johannes Kepler was not able to solve it. When he was advisor to Emperor Rudolph II, the Emperor asked him to find Augustus birthday and his astrological sign. It was evident, to Rudolph and Kepler, that September 23 is not the true date of Augustus' birth (and we will see the reason in the following). After many calculations, the great astronomer wrote to his patron [START_REF] Barton | Augustus and Capricorn: Astrological polyvalency and imperial rhetoric[END_REF]: "Most noble lord, I have now for several weeks on end devoted my utmost efforts to the nativity of Caesar Augustus, about which your kingly majesty wishes to know, but, as I see, it has been almost in vain. For although it is no specially difficult task to calculate the configuration of the heavens for a particular time, nevertheless, what is lacking in the nativity we are now considering is that we no longer today know the correct time of it."

Rudolph had reasons for his request to Kepler. As told in [START_REF]Rudolph II and his world: A study in intellectual history 1576-1612[END_REF], in spite of being born under the sign of Cancer, Rudolph adopted the Augustus' sign of Capricorn, "depicted by a creature with the tail of a fish. This symbol was placed on many works depicting Rudolf as the new Augustus as were the world Astrum fulget caesareum / the imperial star shines" [START_REF]Rudolph II and his world: A study in intellectual history 1576-1612[END_REF]. Kepler spent a lot of time to find and combine the evidences given by the ancient authors on the date of Augustus' birth, in order to assess the role the Capricorn might have in his horoscope. As told in [START_REF] Barton | Augustus and Capricorn: Astrological polyvalency and imperial rhetoric[END_REF], he failed to find a solution.

It was impossible to Kepler to offer to the Emperor the solution given above, that is, the solution which is telling that the Capricorn was the sign when Octavian was conceived. Kepler could not fool the Emperor, who -and we can be sure of it -was well aware of the problems concerning the Roman chronology. Actually, the task was impossible, and Kepler explains the reason: "what is lacking in the nativity we are now considering is that we no longer today know the correct time of it". So we have to search for a different solution to the problem. As we will see we can find it probably in the Julian Calendar itself.

Before the reform of Julian Caesar, Rome had a lunisolar calendar, which is known as the Republican Calendar. It was managed by the priests in a manner so irregular that the calendar was often designating dates, the dates that we can find in the historical reports, which are very different from the Julian dates. The Julian dates were proposed by Joseph Scaliger, a classical scholar, in 1583 at the time of the Gregorian calendar reform. He proposed his system as a method to solve chronological problems. The main problem -the equivalence of the historical dates, given in different calendars, and the Julian dates -was not simple. In Scaliger's approach, astronomical phenomena, such as the eclipses, were fundamental for establishing a chronology of the ancient world (see [START_REF] Sparavigna | The Eclipses in Livy's Ab Urbe Condita[END_REF][START_REF] Sparavigna | On Dating the Lunar Eclipse of Alexander and the Battle of Gaugamela: Discussion of Evidence and Use of Archaeoastronomy for Chronology[END_REF] and references therein). An example: the lunar eclipse of 21 June 168 BC (Julian date) is reported in Livy's Ab Urbe Condita XLIV,37,5-9 at the date 3 September 168 BC of the Republican Roman calendar. We are at the time of the Pydna Battle.

Let us stress that, before 8 AD, the Julian dates are different of a few days from the dates of the Julian Calendar (see [START_REF] Sparavigna | The First Calends of the Julian Calendar[END_REF] and references therein).

The Julian reform was of 46 BC, and the new calendar started on the First January of 45 BC. At the time, Octavian was 18 years old. The year 46 BC was the last of the old system and included 3 intercalary months, the first inserted in February and two more-Intercalaris Prior and Posteriorbefore the Kalends of December.

With Julius Caesar, the calendar became a solar calendar, which can be easily adjusted by adding just one extra day in the calendar every four years. This intercalation is necessary to compensate for the fact that the period of 365 days is shorter than the tropical year. The difference is of about six hours. The Republican calendar required the addition of a month, the Mercedonius, after February every two years. Occasionally, the intercalation happened every three years. Sometimes the intercalation was avoided or employed by the Roman priests for their political reasons, in spite of the state of the solar year.

Due to the possible large differences between the Republican calendar and the astronomical time, the birthdays of Roman people born before the first of January of 45 BC, were subjected to large uncertainties. In a certain manner, after the reform, many people had a different date for the birthday. Many of the Romans knew for sure that their birthday fell in a season different from that in which the birthday was falling according to the Julian calendar.

Of the Republican calendar, the Julian reform did not change the method used to account days of the month, based on the Kalends, Nones and Ides, nor did it change the positions of these three dates within the months. But the length of the months was different. For this reason, after the reform of the Calendar, the Romans who were born after the Ides of a month, could decide differently about the change on their birthdays. "Mark Antony kept his birthday on 14 January, which changed its date from a.d. XVII Kal. Feb to a.d. XIX Kal. Feb, a date that had previously not existed. Livia [that became the second Augustus' wife] kept the date of her birthday unchanged at a.d. III Kal. Feb., which moved it from 28 to 30 January, a day that had previously not existed. Augustus kept his on 23 September, but both the old date (a.d. VIII Kal. Oct.) and the new (a.d. IX Kal. Oct.) were celebrated in some places." [6].

It is clear, from the irregularity of the Republican Calendar that it is highly probable that Octavian Augustus was not born on a day equivalent to September 23 of the Julian Calendar. No astronomical events, such as eclipses reported in historical records are given to find the proper equivalence, and this is the reason of Kepler's difficulties: we no longer today know the correct time of Augustus' birthday.

For what concerns Augustus' astrological sign, the Capricorn, in [START_REF]The Mysteries of Mithras: A Different Account[END_REF] it is told that "Suetonius and Manilius assert that Augustus was born under Capricorn". The astrological "domicile" of Saturn. In [START_REF]The Mysteries of Mithras: A Different Account[END_REF], it is also told that the "Capricorn was the sign of another vital moment in Octavian's life, and in the story of the mankind, as well: he received the laurel branches and the oak crown on January 13th, and the title of Augustus on January 16th, 27 BC, and in this very year the first coins with Capricorn were issued." Probably the first coin with the Capricorn on it was struck even before, about 40 BC. "The image of Capricorn is often accompanied by a cornucopia, one star, and a rudder, and sometimes it is holding the cosmic sphere. The cornucopia is not only an obvious symbol of abundance, but the peculiar symbol of a forthcoming, or newborn child." [START_REF]The Mysteries of Mithras: A Different Account[END_REF] Coin from the British Museum. Image courtesy British Museum CC BY-NC-SA 4.0.

The image of the coin from British Museum shows the Capricorn, accompanied by a cornucopia and a rudder. The Capricorn is holding a globe that can easily be referred to Augustus' power over land and sea [START_REF] Barton | Augustus and Capricorn: Astrological polyvalency and imperial rhetoric[END_REF]. The same Capricorn, half goat and half fish, is representing Octavian Augustus ruling the land and sea. The symbol of the rudder is clear. Augustus was the ruler of the world, its magnus gubernator.

"Magnus gubernator et scisso navigat velo et, si exarmavit, tamen reliquias navigii aptat ad cursum." Seneca Epistulae ad Lucilium Liber IV. Un abile nocchiero sa navigare anche con la vela squarciata e, se ha perso il sartiame, tuttavia cerca di mantenere la rotta sfruttando quello che gli resta della nave. (Translation in Italian, Stefano Maso).

In any case, it is necessary to define when Octavian received his horoscope according to his astrological sign. The date of 23 September is a problem for us, as it was for Kepler. We can assume it as a conventional date, such as a conception nine months before birth. Accordingly, the Capricorn is a conventional sign. But, it could exist also another possibility.

Let us consider Augustus' life [8].

When Octavian turned 15, donned the toga virilis. Shortly after, he began his first official business upon being elected a pontiff in the College of Pontiffs. It was Julius Caesar who had nominated Octavian for this position. While celebrating the Festival of the Latins, Caesar appointed Octavian Praefectus Urbi until his return. From 46 BC on, Octavian was very close to Caesar and attended theatres, banquets, and other social gatherings with him [8]. Octavian needed also experience with military affairs, and Caesar proposed that Octavian join him in Africa, even though the young man had fallen ill. Therefore, it was planned Octavian to join Caesar in Hispania. But Octavian again fell ill and was unable to travel. As soon as he was well, Octavian, with a few friends like Marcus Vipsanius Agrippa, sailed to Hispania. "He became shipwrecked and, after coming ashore with his companions, was forced to make it across hostile territory to reach Caesar's camp. Octavius' actions greatly impressed his great-uncle, who proceeded to teach Octavius the ways of provincial administration." [8] Caesar and Octavian stayed in Hispania until June 45 BC, after which they returned to Rome. Velleius Paterculus reports that Caesar and Octavian shared the same carriage [8]. In Rome, Caesar deposited a new will with the Vestal Virgins. He secretly named Octavian as the prime beneficiary. At the end of 45 BC, in order to improve Octavian's education, Caesar sent him, along with his friends Agrippa, Gaius Maecenas, and Quintus Salvidienus Rufus, to Apollonia in Macedonia. In Apollonia, Octavian learned academics and military doctrines. "Caesar, however, had more than just education in mind for Octavius. Macedonia was home of five legions and he hoped to use it as a launching ground for an upcoming war with Parthia in the Middle East." [8].

Apollonia in Macedonia is the location of the Suetonius' episode. "While in retirement at Apollonia, Augustus mounted with Agrippa to the studio of the astrologer Theogenes. Agrippa was the first to try his fortune, and when a great and almost incredible career was predicted for him, Augustus persisted in concealing the time of his birth and in refusing to disclose it, through diffidence and fear that he might be found to be less eminent. When he at last gave it unwillingly and hesitatingly, and only after many requests, Theogenes sprang up and threw himself at his feet. From that time on Augustus had such faith in his destiny, that he made his horoscope public and issued a silver coin stamped with the sign of the constellation Capricornus, under which he was born." Screenshot software Stellarium showing the sun rising at Rome on the first of 1 January 45 BC. The sun is in the Capricorn.

Augustus' reluctance to tell to Theogenes the date of his birth is perhaps also due to the fact that he was well aware that he could only tell a conventional date. The astrologer, with great intent -or because of a stroke of genius -convinced Octavian that his astrological sign was the Capricorn.

I am persuaded that it was a stroke of genius. The astrologer considered that the Capricorn was the constellation in which the sun was on the first day of the Julian Calendar, first of January 45 BC. As a consequence, Octavian Augustus received a symbol representing the birth of Julius Caesar's calendar as his astrological sign. The Capricorn became the symbol of the birth of a new age, an age that Caesar, Augustus' adoptive father, had established for the calendar, and Augustus was building with his empire. Moreover, the Capricorn was also the domicile of Saturn, and Saturn was the ruler of the Golden Age, a mythical time of prosperity and abundance. According to legends, during this age, the human beings lived without the need to cultivate the land, because it spontaneously gave all kinds of fruits to them. There were no wars. The climate was mild, giving to the world an eternal spring. With the advent of Jupiter, the golden age ended and the age of silver began. Therefore, the choice of Capricorn as astrological sign was perfect to represent, in Augustus' mind, the beginning of a new golden age, in the framework of his political propaganda that aimed to exalt his rule as a period of peace.

Note:

A preliminary version of this paper has been proposed in Italian.

  Sparavigna, Amelia Carolina. (2019, January 27). Il Calendario di Giulio Cesare e il Segno Zodiacale d'Augusto. Zenodo. http://doi.org/10.5281/zenodo.2550628

Zenodo, 19 August 2019 -DOI 10.5281/zenodo.3371676


Frederic Mazenc

Michael Malisoff

On Average Values of Time-Varying Delays and a New Representation of Systems with Time-Varying Delays

Keywords: time-varying, delay, stability, averaging

We show by a counterexample that the asymptotic stability of a system with a pointwise periodic time-varying delay cannot be deduced from the average value of the delay. We use this counterexample to motivate our new representation of systems with time-varying delays, which we use to develop a new state feedback stabilization method.

I. INTRODUCTION

Averaging is a very successful technique that was developed in many contributions (including [START_REF] Hale | Averaging in infinite dimensions[END_REF], [START_REF] Lehman | Averaging theory for delay difference equations with time-varying delays[END_REF], [START_REF] Mazenc | Reduction model approach for linear time-varying systems with delays[END_REF], and [START_REF] Peuteman | Exponential stability of nonlinear timevarying differential equations and partial averaging[END_REF]) and has been applied to many engineering problems [START_REF] Bullo | Averaging and vibrational control of mechanical systems[END_REF], [START_REF] Lehman | Extensions of averaging theory for power electronic systems[END_REF]. On the other hand, many systems admit time-varying delays, so several contributions are devoted to systems with timevarying delays, including [START_REF] Bresch-Pietri | Robust compensation of a chattering time-varying input delay with jumps[END_REF], [START_REF] Michiels | Stabilization of timedelay systems with a controlled time-varying delay and applications[END_REF], [START_REF] Niculescu | Robust exponential stability of uncertain systems with time-varying delays[END_REF], [START_REF] Obuz | Unknown timevarying input delay compensation for uncertain nonlinear systems[END_REF], [START_REF] Verriest | Robust stability of time varying systems with unknown bounded delays[END_REF], [START_REF] Zhou | Truncated Predictor Feedback for Time-Delay Systems[END_REF], and [START_REF] Zhou | Truncated predictor feedback for linear systems with long time-varying input delays[END_REF]. For instance, sampling can be represented as a timevarying delay. Time-varying delays increase the difficulty of analyzing the stability of, or stabilizing, control systems. Fundamental approaches that are available include classical LMIs or prediction based control. They only apply under some conditions, which basically imply that the variations of the delay can be neglected.

A stability technique based on approximation of timevarying delays is proposed in [START_REF] Michiels | Stabilization of timedelay systems with a controlled time-varying delay and applications[END_REF]. The work [START_REF] Michiels | Stabilization of timedelay systems with a controlled time-varying delay and applications[END_REF] is a notable step in the topic of systems with time-varying delays because it can handle delays which are too big to be neglected, and it uses an original approach that introduces a comparison system with a distributed delay. However, the limitation of [START_REF] Michiels | Stabilization of timedelay systems with a controlled time-varying delay and applications[END_REF] is the fact that the kernel of the distributed term of the comparison system contains a function which is not given by an explicit formula and then stability results cannot be derived when the delay is not rapidly varying. Thus, when the time-varying part of the delay has big variations or is discontinuous and not small enough to be neglectable, then stability and stabilization problems are still open.

The previous comments motivate the following question: When a system has a time-varying periodic delay, can we use the average value of the delay to infer the stability properties of the system? For instance, one may wonder whether

Ẋ(t) = -X(t -τ (υt)) (1)
with its state X valued in R is exponentially stable when τ : [0, +∞) → [0, +∞) is periodic of period T and such

Malisoff is with the Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803-4918, USA, malisoff@lsu.edu.

Malisoff was supported by NSF Grant 1711299. Mazenc is with EPI DISCO INRIA-Saclay, Laboratoire des Signaux et Systèmes (L2S, UMR CNRS 8506), CNRS, CentraleSupélec, Université Paris-Sud, 3 rue Joliot Curie, 91192, Gif-sur-Yvette, France, frederic.mazenc@l2s.centralesupelec.fr.

that 1 T T 0 τ (m)dm = 1 (2)
and the constant υ > 0 is large, because the origin of the system Ẋ(t) = -X(t -1) is exponentially stable (by [START_REF] Fridman | Introduction to Time-Delay Systems, Analysis and Control[END_REF]Example 2.3]). Answering this question is challenging because, as shown in [START_REF] Louisell | Delay differential systems with time-varying delay: New directions for stability theory[END_REF], a time-varying delay can be deceitful, insofar that [START_REF] Louisell | Delay differential systems with time-varying delay: New directions for stability theory[END_REF] proves that

Ẋ(t) = -X(t) -1.5X(t -η) with X valued in R is exponentially stable for each constant η ∈ [0, 2.05],
but that there is a piecewise continuous h such that h(t) ∈ [0, 2.05] for all t ≥ 0 and such that Ẋ(t) = -X(t) -1.5X(t -h(t)) has an unbounded solution.

In this work, we establish through an example that when a system has a pointwise periodic time-varying delay, it can be unstable even when it is rapidly varying and when it is stable when the delay is replaced by its average value. This fact is in sharp contrast with the averaging results available for linear systems with a time-varying periodic vector field.

To apply techniques available for time-varying systems with constant pointwise delays, we also propose a new representation of systems with time-varying delays which can be decomposed into a switched delay plus a small time-varying term. It involves a sum of terms with constant delays and time-varying coefficients plus an extra small term. We believe that this new representation is a valuable contribution of our paper because it sheds light on the fundamental mechanisms of stabilization and destabilization inherent in time-varying delays and is a bridge between two distinct classical types of systems, namely, systems with time-varying delays and timevarying systems with several constant delays. To the best of our knowledge, this representation is new, but it shares features with the representation of time-varying systems introduced in [11, Section 3.1] (which did not use averaging). In the last part of our work, we use our new representation to extend the reduction model approach (studied for instance in [START_REF] Mazenc | Reduction model approach for linear time-varying systems with delays[END_REF]) to a new family of systems with time-varying delays, which can be discontinuous and whose derivative, when defined, can be larger than 1. This extension is completely different from the approach of [START_REF] Bekiaris-Liberis | Compensation of time-varying input and state delays for nonlinear systems[END_REF]. The results we propose here are well adapted to systems with switched delays. For this family of systems, we believe that the reduction model or predictive approaches had not been developed.

The notation will be simplified whenever no confusion can arise from the context. The dimensions of our Euclidean spaces are arbitrary unless otherwise noted. For a constant T > 0, C in denotes the set of all continuous functions φ : [-T, 0] → R a , which we call the set of all initial functions. We define Ξ t ∈ C in by Ξ t (s) = Ξ(t + s) for all Ξ, s ≤ 0, and t ≥ 0 for which the equality is defined. We let | . | denote the usual Euclidean norm of vectors and the induced matrix norm, and N = {1, 2, . . .}.

II. PRELIMINARY RESULTS

Consider the system

ẋ(t) = f (t, x(t), x(t -τ (t))) (3)
with x valued in R n and with initial conditions in C in , where f is a nonlinear locally Lipschitz function and where the delay τ (t) is nonnegative valued and piecewise continuous and bounded from above by a known constant τ . In our first lemma, we will assume: Assumption 1: The system (3) is forward complete. Also, there exist a functional S : C in → [0, ∞) , constants s i > 0 for i = 1 to 4, and a constant T ≥ τ such that for all φ ∈ C in , we have

s 1 |φ(0)| 2 ≤ S(φ) ≤ s 2 sup m∈[-T,0] |φ(m)| 2 (4)
for all t ≥ 0, and such that the time derivative of S along all solutions of ( 3) is continuous and satisfies

Ṡ(t) ≥ s 3 S(x t ) -s 4 sup m∈[max{0,t-T },t] S(x m ) (5)
for all t ≥ 0 and s 3 > s 4 .

We prove the following: Lemma 1: If the system (3) satisfies Assumption 1, then this system admits an unbounded solution.

Proof: Consider any solution x(t) of the system (3) with a constant initial condition φ(t) = φ c = 0 for all t ∈ [-T, 0]. We first prove that for this solution, Ṡ(t) > 0 for all t ≥ 0. To this end, note that since φ(t) = φ c = 0 for all t ∈ [-T, 0], it follows from (4) that S(x 0 ) ≥ s 1 |φ c | 2 > 0. Since s 3 > s 4 , our condition (5) therefore gives Ṡ(0) ≥ s 3 S(x 0 )-s 4 S(x 0 ) > 0. Next, let us proceed by contradiction to prove that Ṡ(t) > 0 for all t ≥ 0. Let us suppose that there is t c > 0 such that for all t ∈ [0, t c ), Ṡ(t) > 0 and Ṡ(t c ) = 0.

Since for all t ∈ [0,

t c ), Ṡ(t) > 0, it follows that S(x t ) is nondecreasing over [0, t c ]. Consequently, S(x tc ) = sup m∈[max{0,tc-T },tc] S(x m ) ≥ S(x 0 ) > 0. It follows that Ṡ(t c ) ≥ s 3 S(x tc ) -s 4 S(x tc ) > 0.
This yields a contradiction with the definition of t c . We conclude that for all t ≥ 0, Ṡ(t) > 0. Therefore, S(x t ) is nondecreasing over [0, +∞). Hence, for all t ≥ 0, we have sup m∈[max{0,t-T },t] S(x m ) = S(x t ). This equality, in combination with (5), gives Ṡ(t) ≥ (s 3 -s 4 )S(x t ) for all t ≥ 0. It follows that lim t→+∞ S(x t ) = +∞, since s 3 > s 4 . We deduce from (4) that (3) admits an unbounded solution.

Remark 1: Observe that the case where the system (3) is not forward complete is trivial in the context of Lemma 1 because then the system admits an unbounded solution.

The formulas in the next lemma can be deduced easily by writing [START_REF] Lehman | Extensions of averaging theory for power electronic systems[END_REF] in the form ġ = Ag + Bu, then exponentiating the matrix A to obtain a rotation matrix, and then applying variation of parameters, so we omit its proof:

Lemma 2: Consider the system ġ1 (t) = g 2 (t) ġ2 (t) = -g 1 (t) + u(t) (6)
with g i and u valued in R and u being continuous. Then,

g 1 (t -t * ) = cos(t * )g 1 (t) -sin(t * )g 2 (t) - t t-t * sin(t -m -t * )u(m)dm g 2 (t -t * ) = sin(t * )g 1 (t) + cos(t * )g 2 (t) - t t-t * cos(t -m -t * )u(m)dm (7)
hold for all t ∈ R and all constants t * > 0.

In Appendix A, we prove: Lemma 3: There is a constant ∈ (0, 1) such that for all ∈ (0, ] the system

ẏ1 (t) = y 2 (t) ẏ2 (t) = -y 1 (t) + y 1 t -3π 4 ( 8
)
is globally exponentially stable to the origin.

III. KEY EXAMPLE

In this section, we use a switched system to establish a key result because it leads to a simple analysis, but a similar result can be proved with a C 1 time-varying delay.

Let τ 0 = 0 and τ 1 = 3π 2 and ν > 0 be a constant. Consider

ẋ1 (t) = x 2 (t) ẋ2 (t) = -x 1 (t) + x 1 t -τ σ(t) (9)
with x i valued in R for i = 1, 2, where ∈ (0, ) is a constant and is from Lemma 3, and where σ is periodic of period 2ν and satisfies (i) σ(t) = 0 for all t ∈ [0, ν) and (ii) σ(t) = 1 for all t ∈ [ν, 2ν). By the definition of σ, the function τ σ(t) is periodic of period 2ν and its average value is τ0+τ1 2 = 3π 4 . Thus, if we replace τ σ(t) by its average value in [START_REF] Mazenc | Reduction model approach for linear time-varying systems with delays[END_REF], then we obtain [START_REF] Louisell | Delay differential systems with time-varying delay: New directions for stability theory[END_REF], which is exponentially stable.

However, we prove in Appendix B that there is a constant ν > 0 such that (9) admits an unbounded solution when max{ν, } < ν, using the fact that (9) has the representation

   ẋ1 (t) = x 2 (t) ẋ2 (t) = -x 1 (t) + [1 -σ(t)]x 1 (t) + σ(t)x 1 t -3π 2 (10)
as a time-varying system that is periodic of period 2ν, with a constant delay. Thus, the average value of the delay τ σ(t)

does not indicate whether the system ( 9) is exponentially unstable or not, however fast the switches are.

IV. TIME-VARYING DELAY: NEW REPRESENTATION

The previous section suggests that systems with a switched delay can be represented as time-varying systems with several constant delays. We now describe this new representation in the general context of linear systems with time-varying delays, and we use this representation to solve a stabilization problem. Throughout the remaining part of the paper, we consider systems with time-varying delays τ that satisfy:

Assumption 2: The function τ : [0, ∞) → [0, ∞) is piecewise continuous, periodic of some constant period T > 0, and bounded above by some finite constant τ > 0.

We also use the following notation. Let q ∈ N, with q ≥ 2, and set ν = T q . We set t k = kν for all integers k ≥ 0. We define the switching signal σ : [0, +∞) → {0, ..., q -1} as follows: σ is periodic of period T and σ(t) = i for all t ∈ [iν, (i + 1)ν) and all i ∈ {0, ..., q -1}. For all i ∈ {0, ..., q -1}, we define the functions β i as follows: β i is periodic of period T and satisfies (i) β i (t) = 1 for all t ∈ [iν, (i + 1)ν) and (ii) β i (t) = 0 for all t ∈ [0, T ) that are not in [iν, (i + 1)ν). We introduce q constants h j ≥ 0 for j ∈ {0, ..., q -1} and define r by r(t) = τ (t) -h σ(t) for all t ≥ 0. Let r ≥ 0 be a constant such that |r(t)| ≤ r for all t ≥ 0. Since τ is bounded, such an r exists.

Consider the system

ẋ(t) = Ax(t) + Bu(t -τ (t)) ( 11
)
where x is valued in R n , the control u is valued in R p , and τ satisfies Assumption 2. Since τ (t) = h σ(t) + r(t) for all t ≥ 0, the system (11) can be written as

ẋ(t) = Ax(t) + Bu t -h σ(t) + B u(t -h σ(t) -r(t)) -u(t -h σ(t) ) . (12)
Then the definitions above yield the representation

ẋ(t) = Ax(t) + q-1 i=0 β i (t)Bu(t -h i ) + B u(t -h σ(t) -r(t)) -u(t -h σ(t) ) . (13)
A crucial question is how to choose the constants h i . By increasing the number of constants h i , one can decrease r, which is desirable because we will treat the delay r(t) as a disturbance in a later section. However, increasing the number of h i 's may increase the complexity of the problem one aims to solve. Thus the choice of the h i 's is a delicate issue. Note that (13) has several constant delays and a timevarying delay, while [START_REF] Mazenc | Stability and robustness analysis for switched systems with time-varying delays[END_REF] has only one time-varying delay.

Control problems for the family of systems [START_REF] Mazenc | Stability and robustness analysis for switched systems with time-varying delays[END_REF] have been studied in many contributions. However, our results are novel because of our mild Assumption 2, which does not require the condition τ (t) < 1 when τ is differentiable. We assume:

Assumption 3: There is a continuous function K that is bounded by a constant K ≥ 0 such that

Θ(t) = H(t)Θ(t) + B [K (t -τ (t)) Θ (t -τ (t)) -K t -h σ(t) Θ t -h σ(t)
, where

H(t) = A + q-1 i=0 e -Ahi Bβ i (t + h i ) K(t) (14)
is uniformly globally uniformly exponentially stable to 0.

By a Lyapunov approach, one can easily prove that Assumption 3 holds if Θ = H(t)Θ(t) is uniformly exponentially stable to 0 and r > 0 is sufficiently small. We prove:

Theorem 1: Let the system (11) satisfy Assumption 3. Then the origin of [START_REF] Mazenc | Stability and robustness analysis for switched systems with time-varying delays[END_REF] in closed-loop with

u(t) = K(t) x(t) + q-1 i=0 ζ i (t) (15)
and with the choices

ζ i (t) = t t-hi e A(t-s-hi) Bβ i (s + h i )u(s)ds (16)
for all i is uniformly globally uniformly exponentially stable.

Proof:

We apply the reduction model approach to the system [START_REF] Niculescu | Robust exponential stability of uncertain systems with time-varying delays[END_REF]. Elementary calculations give

ζi (t) = Aζ i (t) + e -Ahi Bβ i (t + h i )u(t) -Bβ i (t)u (t -h i ) (17)
for all t ≥ 0. We next use the operator

λ(t) = x(t) + q-1 i=0 ζ i (t). ( 18
)
By combining ( 13) with ( 17), we obtain

λ(t) = Aλ(t) + q-1 i=0 e -Ahi Bβ i (t + h i )u(t) + B u(t -h σ(t) -r(t)) -u(t -h σ(t) ) . ( 19
)
Selecting the control law [START_REF] Peuteman | Exponential stability of nonlinear timevarying differential equations and partial averaging[END_REF], we obtain

λ(t) = H(t)λ(t) + B [K (t -τ (t)) λ (t -τ (t)) -K t -h σ(t) λ t -h σ(t) ( 20
)
where H is from [START_REF] Obuz | Unknown timevarying input delay compensation for uncertain nonlinear systems[END_REF]. By Assumption 3, it follows that λ(t) is solution of a uniformly exponentially stable system. Since u(t) = K(t)λ(t), it follows from ( 18) and our formula [START_REF] Verriest | Robust stability of time varying systems with unknown bounded delays[END_REF] for the ζ i that the x system is exponentially stable too.

V. ILLUSTRATION

Consider the one-dimensional system

ẋ(t) = 1 8 x(t) + u(t -τ (t)) (21)
for any τ satisfying our Assumption 2. We apply Theorem 1 to find a feedback u ensuring the global exponential stability of (21) when T = 2π, q = 4, ν = π 2 , h 0 = 0, h 1 = h 3 = ν, and h 2 = 2ν, by choosing the constant r > 0 small enough.

With the notation of Section IV with A = 1 8 and B = 1, we have

H(t) = 1 8 + β 0 (t) + e -ν 8 [β 1 (t + ν) +β 3 (t + ν)] + e -2ν 8 β 2 (t + 2ν) K(t). (22)
We choose K(t) = -K * , where K * > 0 is any constant. Then H has period T , so for all t ≥ T , we obtain the average

H = 1 T t t-T H(m)dm = 1 T T 0 H(m)dm = 1 8 -K * q 1 + 2e -ν/8 + e -ν/4 , since T 0 β i ( )d = ν = T /q for all i. Next, consider the system λ(t) = H(t)λ(t)+K * -λ (t -τ (t))+λ t -h σ(t) (23)
which corresponds to ( 14) from Assumption 3. Using the Fundamental Theorem of Calculus, we can rewrite (23) as

λ(t) = H(t)λ(t) + K * t-h σ(t) t-h σ(t) -r(t) λ(m)dm = H(t)λ(t) + K * t-h σ(t) t-h σ(t) -r(t) [H(m)λ(m) +K * λ m -h σ(m) -λ (m -τ (m)) dm, (24)
where (24) and the equalities and inequalities to follow should be understood to hold for all t ≥ 2(τ + ν). Next note that the time derivative of

P(λ) = 1 2 λ 2 satisfies Ṗ(t) = H(t)λ 2 (t) + K * λ(t) t-h σ(t) t-h σ(t) -r(t) [H(m)λ(m) +K * λ m -h σ(m) -λ (m -τ (m)) dm ≤ H(t)λ 2 (t) + K * |λ(t)| t-h σ(t) t-h σ(t) -r(t) ν |λ(m)| +K * |λ m -h σ(m) | + |λ (m -τ (m)) | dm ≤ √ 2K * P(λ(t)) t-h σ(t) t-h σ(t) -r(t) K * P(λ(m)) + √ 2K * P(λ(m -h σ(m) )) + √ 2K * P(λ(m-τ (m))) dm+2H(t)P(λ(t)) ≤ 2H(t)P(λ(t)) + M * sup ∈[t-2(τ +ν),t] P(λ( ))
along all solutions of (24), where

K * = √ 2ν , ν = 1/8 + 1 + 2e -ν 8 + e -ν 4 K * , and M * = √ 2K * (K * + 2 √ 2K * )r.
By integrating this inequality, we obtain

P(λ(t)) ≤ e 2 t t-T H(m)dm P(λ(t -T )) + M * t t-T e 2 t m H( )d sup ∈[m-2(τ +ν),m] P(λ( ))dm ≤ e 2T H + M * T e 2T H S λ (t)
where H = sup{|H( )| :

∈ R} and S λ (t) = sup ∈[t-T -2(τ +ν),t] P(λ( )). Then [10, Lemma 1] ensures that (23) is uniformly globally exponentially stable to 0 if e 2T H + 2M * T e 2T H ∈ (0, 1), i.e., if r < 1 -e 2T H √ 2T K * ( √ 2ν + 2 √ 2K * )e 2T H . (25)
Hence, (25) implies that Assumption 3 is satisfied, so the desired stabilizing feedback is given by ( 15)-( 16)

VI. CONCLUSIONS

We showed that stability of a system with a time-varying delay cannot be determined from the average value of the delay. This motivated our new representation of systems with time-varying delays, which we used to solve a stabilization problem with piecewise continuous delays. It would be worth comparing the performance of the controls from Section IV with those of [START_REF] Mazenc | Stability and robustness analysis for switched systems with time-varying delays[END_REF]. We believe that our new representation eases the understanding of the effect of a time-varying delay and may be useful for solving other stability and control problems. We conjecture that extensions can be obtained for systems with several (pointwise and distributed) delays and with time-varying vector fields using the new representation we proposed. Nonlinear extensions are expected too. APPENDIX A: PROOF OF LEMMA 3 We assume that ∈ (0, 1/2]. Note that [START_REF] Lehman | Averaging theory for delay difference equations with time-varying delays[END_REF] gives By applying the triangle inequality to the quantity in curly braces and then applying Jensen's inequality, one can find a constant ∈ (0, 1/2] such that when ≤ , then This allows us to find small enough constants δ > 0 and c 0 > 0 such that the time derivative of W (Y t ) = W (Y t ) + δy 1 (t)y 2 (t) along all solutions of (8) satisfies Ẇ (Y t ) ≤ -c 0 W (Y t ) for large enough t > 0, by writing

g 1 (t -3π 4 ) = - √ 2 2 g 1 (t) - √ 2
Ẇ (t) ≤ - √ 2 8 y 2 2 (t) - √ 2 8π t t-3π
δ y 1 (t)y 1 (t -3π/4) ≤ δ y 2 1 (t) + |y 1 (t)| t t-3π/4 |y 2 (m)|dm , (A.6)
and applying the triangle inequality and then Jensen's inequality to the quantity in curly braces in (A.6). The global exponential stability of ( 8) for all ∈ (0, ) follows from the exponential decay of W and the positive definite quadratic upper and lower bounds on W .

APPENDIX B: INSTABILITY OF SWITCHED SYSTEM (9)

The system (10) can be rewritten as:

     ẋ1 (t) = x 2 (t) ẋ2 (t) = -x 1 (t) + 2 x 1 (t) + 2 x 1 t -3π 2 + 1 2 -σ(t) x 1 (t) -x 1 t -3π 2 . (B.1)
From ( 7) and ( 10), we deduce that

x 1 t -3π 2 = x 2 (t) - t t-3π 2 sin t -m -3π 2 ∆(t, m)dm, (B.2)
where

∆(t, m) = (1 -σ(m))x 1 (m) + σ(m)x 1 (m -3π 2 ), by choosing (g 1 , g 2 ) = (x 1 , x 2 ) in Lemma 2. Then, we obtain            ẋ1 (t) = x 2 (t) ẋ2 (t) = 2 -1 x 1 (t) + 2 x 2 (t) - 2 2 t t-3π 2 sin t-m-3π 2 ∆(t, m)dm + 1 2 -σ(t) x 1 (t) -x 1 t -3π 2 . (B.3)
Let us define the function and operator

γ(t) = 1 2ν t t-2ν t 1 2 -σ(m) dmd and ξ 2 (t) = x 2 (t) -γ(t) x 1 (t) -x 1 t -3π 2 , (B.4)
where ν is from the definition of σ. Then |γ(t)| ≤ ν 2 for all t ∈ R. For almost all t, we can use the formula γ(t) =

2 -σ(t) to get          ẋ1 (t) = ξ 2 (t) + γ(t) x 1 (t) -x 1 t -3π 2 ξ2 (t) = 2 -1 x 1 (t) + 2 x 2 (t) - 2 2 t t-3π 2 sin t-m-3π 2 ∆(t, m)dm -γ(t) x 2 (t) -x 2 t -3π 2 .
This gives ξ2 (t) = 2 -1 x 1 (t) + and E = 0 1

2 -1 2 (B.7)
and where S is symmetric and positive definite because ≤ 1. Hence, with the choice X (x 1 , ξ 2 ), the time derivative of U (X) = X SX along all solutions of (B.5) satisfies Since there are positive constants κ and T such that |κ i (t)| ≤ ( 2 + ν)κ|X| [t-T,t] for all t ≥ 3π with κ not depending on ν ∈ (0, 1) or ∈ (0, 1), we can apply the triangle inequality to the terms in curly braces and then Lemma 1 to (B.5) with S = U when ν and are sufficiently small positive constants. This provides an unbounded solution of (B.5) when ν and are sufficiently small positive constants. Then we can use our choice of ξ 2 from (B.4) to find an unbounded solution of (B.1).

U (t) = 2 U (X(t)) + 1 -2 x 1 (t)

1

 1

  1 2 -γ(t) (ξ 2 (t) + γ(t) x 1 (t) -x 1 t -3π ×γ(t) x 1 (t) -x 1 t -3π

			2	+ γ(t) ξ 2 t-3π 2
	+ γ t-3π 2 -2 2 t t-3π 2	x 1 t-3π 2 -x 1 (t-3π) sin t -m -3π 2 ∆(t, m)dm.
	It follows that all solutions of (10) satisfy
		ẋ1 (t) = ξ 2 (t) + κ 1 (t)
		ξ2 (t) =
			2
		+γ(t) ξ 2 t -3π 2 + γ t -3π 2	(B.6)
		× x 1 t -3π 2 -x 1 (t -3π) -2 2 t t-3π 2 sin t-m-3π 2 ∆(t, m)dm.
	Next observe that SE + E S = 2 S, where
	S =	1 2 -4 -8 1 -8 2

2 -1 x 1 (t) + 2 ξ 2 (t) + κ 2 (t) (B.5) where κ 1 (t) = γ(t) x 1 (t) -x 1 t -3π 2 and κ 2 (t) = -γ(t)ξ 2 (t) + 1 2 -γ(t)

  κ 1 (t) + ξ 2 (t)κ 2 (t) -4 x 1 (t)κ 2 (t) -4 ξ 2 (t)κ 1 (t) ≥ 2 U (X(t)) -ξ 2 (t) -4 x 1 (t) |κ 2 (t)| -1 -2 x 1 (t) -4 ξ 2 (t) |κ 1 (t)| .

			|κ 2 (t)| ≤ ν 2 |ξ 2 (t)| + + ν 2 2 4 x 1 t -3π 2 -x 1 (t -3π) + ν 2 ν(1+ν) 4 x 1 (t) -x 1 t -3π 2 2 |ξ 2 t -3π 2 | + 2 2 t t-3π 2 |x 1 (m)|+|x 1 (m-3π 2 )| dm.
	2	and	(B.8)

From (B.4), we deduce that for all t ∈ R, we have

|κ 1 (t)| ≤ ν 2 x 1 (t) -x 1 t -3π

Acknowledgment. The authors thank S.-I. Niculescu for indicating the enlightening references, notably [8].


H D Nguyen

K P Tran

X Zeng

L Koehl

Wearable Sensor Data Based Human Activity Recognition using Machine Learning: A new approach

Keywords: Human Activity Recognition, Wearable sensor, Ensemble learning method, Machine learning

Recent years have witnessed the rapid development of human activity recognition (HAR) based on werable sensor data. One can find many practical applications in this area, especially in the field of health care. Many machine learning algorithms such as Decision Trees, Support Vector Machine, Naive Bayes, K-Nearest Neighbor and Multilayer Perceptron are successfully used in HAR. Although these methods are fast and easy for implementation, they still have some limitations due to poor performance in a number of situations. In this paper, we propose a novel method based on the ensemble learning to boost the performance of these machine learning methods for HAR.

Introduction

The rapid development of advanced technologies nowadays makes the study of recognizing human activity easier and more effective. Human activity recognition (HAR) is widely used in several fields such as security surveillance, human computer interaction, military, and especially health care. For instance, the HAR is used for monitoring the activities of elderly people staying in rehabilitation centers for chronic disease management and disease prevention. It is used to encourage physical exercises for assistive correction of children's sitting posture. It is also applied in monitoring other behaviours such as abnormal conditions for cardiac patients and detection for early signs of illness. More examples of applications of HAR can be seen in [START_REF] Chin | Human activity recognition: a review[END_REF].

Wearable sensor method refers to the use of smart electronic devices that are integrated into wearable objects or directly with the body in order to measure both biological and physiological sensor signals such as heart rate, blood pressure, body temperature, accelerometers, or other attributes of interest like motion and location. These sensors are communicated with an integration device, like a cellphone, a laptop or a customized embedded system. As a result, the raw signals are sent to an application server for real time monitoring, visualization and analysis. The use of smart mobile phone containing several sensors such as gyroscope, camera, microphone, light, compass, accelerometer, proximity, and GPS can also be very effective for activity recognition ( [START_REF] Jennifer R Kwapisz | Activity recognition using cell phone accelerometers[END_REF]). However, the raw data from smart mobile phone is only effective for simple activities but not complex activities ( [START_REF] Dernbach | Simple and complex activity recognition through smart phones[END_REF]). Thus, the extra sensors or sensing devices should be used for a better performance of the recognition.

In the literature, various machine learning algorithms have been suggested to handle features extracted from raw signals to identify human activities. These machine learning methods are in general fast and easy to be implemented. However, they only bring satisfying results in a few scenarios because of relying heavily on the heuristic handcrafted feature extraction ( [START_REF] Wang | Deep learning for sensor-based activity recognition: A survey[END_REF]). Recent years have witnessed the rapid development and application of deep learning, which has also achieved remarkable efficiency in the HAR. Although the deep learning algorithms can automatically learn representative features due to its stacking structure, it has a disadvantage of requiring a large dataset for training model. In fact, numerous data are available but it is sometimes difficult to access data which are labeled. It is also inappropriate for real-time human activity detection due to the high computation load ( [START_REF] Chen | A novel ensemble elm for human activity recognition using smartphone sensors[END_REF]). This motivates us to apply in this study ensemble algorithm for machine learning based learners to improve the performance of these algorithms as well as keep the simplicity with implementation. By this new approach, we firstly conduct experiments with several machine learning classifiers such as Logistic Regression, Multilayer Perceptron, K-Nearest Neighbor, Support Vector Machine, and Random Forest. Then we apply a novel recognition model by using voting algorithm to combine the performance of these algorithms for the HAR. In fact, this algorithm has been suggested in [7], leading to impressive results compared to other traditional machine learning methods. In this study, we improve the study in [7] by using more efficient classifiers as base models. The obtained results show that our proposed method give better performances.

The rest of the paper is organized as follows. In Section 2, we present a brief of related works on HAR. Section 3 describes the methodology using in the study, including the sample generation process, the feature representation, the basic machine learning algorithms and the proposed voting classifier. The experimental results are shown in Section 4 and some concluding remarks are given in Section 5.

A brief review of human activity recognition based on wearable sensor data

Due to its widely practical applications, the werable sensors based HAR has attracted a large number of studies. A number of machine learning algorithms have been applied to deal with sensors data in the HAR such as Hidden Markov Models ( [START_REF] Lee | Activity recognition using hierarchical hidden markov models on a smartphone with 3d accelerometer[END_REF]), Support Vector Machines ( [START_REF] Anguita | Human activity recognition on smartphones using a multiclass hardware-friendly support vector machine[END_REF]), and K-Nearest Neighbor ( [START_REF] Media | A compar-Accuracy Recall F-score Proposed method Catal et al.'s method Proposed method Catal et al.'s method Proposed method Catal et al.'s method[END_REF]). Other machine learning algorithm like J48 Decision Trees and Logisitic Regression are also utilized for the HAR based on the accelerometer of smartphone. An extensive survey on werable sensor-based HAR was carried out by [START_REF] Lara | A survey on human activity recognition using wearable sensors[END_REF].

In general, when few labeled data and certain knowledge are required, the machine learning pattern of recognition approaches can give satisfying results. However, there are several limitations of using these methods because of the following arguments: (1) these regular methods rely heavily on practitioners' experience with heuristic and handcrafted ways to extract interested features; (2) the deep features are difficult to be learned; and (3) a large amount of well-labeled data to train the mode is required ( [START_REF] Wang | Deep learning for sensor-based activity recognition: A survey[END_REF]). This is the motivation for the use of deep learning in wearable sensor based HAR recently.

[12] used a 5-hidden-layer deep neural network to perform automatic feature learning and classification. [START_REF] Sun | Sequential human activity recognition based on deep convolutional network and extreme learning machine using wearable sensors[END_REF] proposed a hybrid deep framework based on convolution operations, long short-term memory recurrent units, and extreme learning machine classifier. A comprehensive survey on deep learning for sensor-based activity recognition can be seen in [START_REF] Wang | Deep learning for sensor-based activity recognition: A survey[END_REF].

Another method is also widely used for HAR is ensemble learning. This method not only improve significantly the performance of traditional machine learning algorithms and but also avoid a requirement of large dataset for training model of deep learning algorithms. [7] used a set of classifiers including J48 Decision Trees, Logistic Regression and Multilayer Perceptron, to recognize specifice human activities like walking, jogging, sitting and standing based on accelerometer sensor of a mobile phone. [START_REF] Gandhi | Hybrid ensemble of classifiers using voting[END_REF] applied a hybrid ensemble classifier that combines the representative algorithms of Instance based learner, Nave Bayes Tree and Decision Tree Algorithms using voting methodology to 28 bench mark datasets and compared their method with other machine learning algorithms. A novel ensemble extreme learning machine for HAR using smartphone sensors has been presented in [START_REF] Chen | A novel ensemble elm for human activity recognition using smartphone sensors[END_REF].

Methodology

In this Section, we explain the method of generating samples as well as extracting features of data. The individual classifiers and ensemble method using in this study is also presented.

Sample generation process for HAR

Generating the samples from raw signasl is a crucial step to perform wearable sensor data based HAR. In general, the raw signals are divided into small parts of the same size, which are called temporal windows, and are used as training and test dataset to define the model. The common process to generate the temporal windows is semi-non overlappingwindow (SNOW). [START_REF] Jordao | Human activity recognition based on wearable sensor data: A standardization of the state-of-the-art[END_REF] pointed out that it is highly biased, i.e, part of the sample's content appears both in the training and testing at the same time. Then, the authors proposed two new methods to handle this bias drawback, including full-non overlapping-window (FNOW) and leave-one-trial-out (LOTO). Although the FNOW can avoid the bias property in SNOW, it has another disadvantage that it provides a less number of samples compared to the SNOW process. Therefore, the LOTO is proposed to use. In this method, the activities from a trial are initially segmented and then 10-fold cross validation is employed. A figure illustrated the process for the example of 2-folde cross validation is shown in [START_REF] Jordao | Human activity recognition based on wearable sensor data: A standardization of the state-of-the-art[END_REF]. This LOTO process is also applied in this study.

Feature Representation for HAR

Another important step in the process of HAR before applying machine learning algorithms is to extract features from raw data. Since the raw signals from sensors are usually noisy, it is necessary to extract the robust representations, or features from these signals. Several feature representation approaches for HAR have been presented in [START_REF] Lara | A survey on human activity recognition using wearable sensors[END_REF]. In this study, we focus on a common techniques using for acceleration signals which consists of time-and frequency-domain features. Typical time domain features are mean, standard deviation, variance, root squared mean, and percentiles; while typical frequency domain features include energy, spectral entropy and dominant frequency. ( [START_REF] Sani | Learning deep features for knn-based human activity recognition[END_REF]). These measures are designed to capture the characteristics of the signal that are useful for distinguishing different classes of activities. Table 1 shows some widely used features from the literature.

The basic machine learning algorithms

Logistic Regression

Logistic regression (LR) is a well-known statistical classification method for modelling a binary response variable, which takes only two possible values. It simply models probability of the default class. The LR method is commonly used because it is easy with implementation and it provides competitive results. Although LR is not a classifier, it can still be used to make a classifier or prediction by choosing a cutoff value and classifying inputs with probability greater than the cutoff as one class and less than the cutoff as the other. More detail of the algorithm and its various applications can be seen in [START_REF] Hosmer | Applied logistic regression[END_REF].

Multilayer Perceptron

A multilayer perceptron (MLP) is a classical type of feedforward artificial neural network. It contains one or more layers of neurons. Data is fed to the input layer, there may be one or more hidden layers providing levels of abstraction, and predictions are made on the output layer. The nodes in MLP are fully connected in the sense that each node in one layer connects to every node in the following layer with a certain weight. Based on the amount of error in the output compared to the expected result, the network is trained in the perceptron by changing these connection weights after processing of data. The backpropagation learning technique is applied for the training of the network. The wide applications of MLP can be found in large fields such as speech recognition, image recognition, and machine translation software [START_REF] Philip | Neural networks. ii. what are they and why is everybody so interested in them now[END_REF].

Support Vector Machine

Support Vector Machine (SVM) is a discriminative classifier belonging to supervised learning models. It constructs a hyperplane (a decision boundary that helps to classify the data points) or set of hyperplanes in a high or infinite dimensional space, which can be used for both classification and regression problem. In practice, there are many possible hyperplanes that could separate the two classes of data points. The SVM aims to find a hyperplane that has maximum margin, i.e the maximum distance between data points of both classes. A review on the rule extraction and the main features of the algorithm from SVM can be seen in [START_REF] Barakat | Rule extraction from support vector machines: a review[END_REF].

Gaussian Naive Bayes

Gaussian Naive Bayes refers to a naive Bayes classifier as dealing with continuous data in the case that the continuous values associated with each class are distributed following a Gaussian distribution. The naive Bayes classifier is a classification technique based on Bayes' theorem with an independence assumption that the presence of a particular feature in a class is not related to the presence of any other features. The major advantage of the naive Bayes classifier is its simplicity and short computational time for training. It can also often outperform some other more sophisticated classification methods ( [START_REF] Domingos | On the optimality of the simple bayesian classifier under zero-one loss[END_REF]).

K-Nearest Neighbor

Being considered as the simplest of all machine learning algorithms, K-nearest neighbour (KNN) is an instance based classifier method. By this algorithm, a new object is classified based on the distance from its K neighbours in the training set and the corresponding weights assigned to the contributions of the neighbors, where the nearer neighbors contribute more to the average than the more distant ones. That is to say, it follows an assumption that similar things are near to each other. The Euclidean distance is commonly used for continuous variables to calculate the distance in KNN. A review of data classification using this KNN algorithm is given in [START_REF] Kataria | A review of data classification using k-nearest neighbour algorithm[END_REF].

Random Forest

Random forest is an ensemble learning method operated by constructing a multitude of decision trees at training time. Each tree in a random forest learns from a random sample of the data points when training. The samples are drawn with replacement, i.e, some samples can be used several times in an individual tree. It is trained via the bagging method. Depending on the final task, the output class could be the mode of the classes (for classification) or mean prediction (for regression) of the individual tree. The random forests can overcome the high risk of overfitting the training data of the decision trees algorithm. A method of building a forest of uncorrelated trees using a CART (Classification And Regression Tree) and several ingredients forming the basis of the modern practice of random forests has been introduced in [START_REF] Breiman | Random forests[END_REF].

Proposed voting classifier

Each machine learning for classification presented above has their own disadvantages. Ensemble method, a machine learning technique that combines several base models in order to obtain one optimal predictive model, are then proposed in this study. The main idea of ensemble learning is to aggregate multiple base learners to boost the performance. Several ensemble rules for combining the multiple classification results of different classifiers, including Vote rule, Minimum probability rule, Maximum probability rule, Product of probabilities rule, Median rule, and Sum Rule have been proposed in [START_REF] Kittler | Combining classifiers[END_REF]. In this study, we apply the voting rule, which is simple but powerful technique, for combining the aforementioned algorithms. Detail of three versions of voting, involving unanimous voting, majority voting and plurality voting, can be found in [START_REF] David | Stacked generalization[END_REF]. In unanimous voting, the final decision is approved by all the base learners. In majority voting, more than 50% vote is required for final decision and in plurality voting most of the votes decides the final result.

Experimental results

In this Section, we show the experimental results of our proposed method and compare these results with the ones obtained by using the method suggested in [7]. The two following data sets are considered: Fig. 1: The proposed framework for HAR using an ensemble algorithm oped in [START_REF] Zhang | Usc-had: a daily activity dataset for ubiquitous activity recognition using wearable sensors[END_REF]. This is specifically designed to include the most basic and common human activities in daily life from a large and diverse group of human subjects, including 12 activities and 14 subjects. The dataset is available at the website of the authors (see [START_REF] Zhang | Usc-had: a daily activity dataset for ubiquitous activity recognition using wearable sensors[END_REF]). The first step in experimental setup is temporal sliding window where the samples is split into subwindows and each subwindow is considered as an entire activity. We apply the same value of the temporal sliding window size which is t = 5 seconds as suggested in [START_REF] Jordao | Human activity recognition based on wearable sensor data: A standardization of the state-of-the-art[END_REF]. Then, the important features from these raw signals (presented in Table I) are extracted to filter relevant information and to give the input for classifiers. The output of ensemble algorithm is a recognized specific human activity. Figure 1 presents the proposed framework for HAR using ensmble method. Based on the MHEALTH and USC-HAD data set, twelve basic and common activities in people's daily lives is the target to be recognized, including walking forward, walking left, walking right, walking upstairs, walking downstairs, running forward, jumping, sitting, standing, sleeping, elevator up, and elevator down. Table 3 in [START_REF] Zhang | Usc-had: a daily activity dataset for ubiquitous activity recognition using wearable sensors[END_REF] presents a description for each activity. The performance of proposed method is evaluated by using the following widely used measures:

• Accuracy = T P +T N T P +F P +T N +F N , • Recall = T P T P +F N , • F-score = 2 × Precision×Recall Precision+Recall .
where Precision = T P/(T P + F P ); TP (True Positive) is the number of samples recognized correctly as activities, TN (True Negative) is the number of samples correctly recognized as not activities, FP (False Positive) is the number of samples incorrectly recognized as activities, and FN (False Negative) is the number of samples incorrectly diagnosed as not activities.

Our computation was performed on a platform with 2.6 GHz Intel(R) Core(TM) i7 and 32GB of RAM. We perform the experimence on the two datasets mentioned above using both our proposed method and the method suggested in [7]. The experimental results are presented in Table II. It can be seen that our method leads to higher values of all the measures being considered. For example, for the MHEALTH data set, our method lead to Accuracy = 94.72% compared to Accuracy = 93.87% from the Catal et al.'s method. Similarly, for the USCHAD data set, Recall is equal to 83.20% corresponding to our propsed method, which is relatively larger than the value Recall = 81.74% corresponding to the Catal et al.'s method. That is to say, the proposed method in this study outperforms the method used in [7]. Moreover, the obtained results show that the performance of ensemble learning algorithm can be significantly improved by combining better machine learning algorithms. This should be considered in design a new ensemble method for HAR.

Conclusion and future work

Activity recognition increasingly plays an important role in many practical applications, especially in health care. Increasing the performance of recognition algorithms is a major concern for researchers. In this paper, we have proposed a new method for wearable sensor data based HAR using the ensemble algorithm. By combining better classifiers, our proposed method improves significantly the previous study ( [7]) in all measures of comparision.

In the future, we would like to address the problem of wearable sensor data based HAR using Convolutional Neural Networks (CNN) and Long Short-Term Memory (LSTM) networks. The motivation for using these methods is because the CNN offers advantages in selecting good features and the LSTM has been proven to be good at learning sequential data.

TABLE I :

 I Hand-crafted features for both time and frequency domains.

• MHEALTH: This dataset is based on a new framework for agile development of mobile health applications suggested by [4]. Four types of signals are provided in this dataset, involving accelerometer signals, gyroscope signals, magnetometer signals and electrocardiogram signals. Similar to [14], we only consider the first three signals in our study. • USC-HAD: The USC-HAD, standing for University of Southern California Human Activity Dataset, is devel-


Zhenglei He
email: zhenglei.he@ensait.fr

Kim-Phuc Tran

Sébastien Thomassey

Xianyi Zeng

Changhai Yi

Figure 1. The reactor of ozonation

Keywords: Artificial intelligence, Modeling, Color Fading, Ozonation, Textile

Textile products with faded effect achieved via ozonation are increasingly popular recently. In this study, the effect of ozonation in terms of pH, temperature, water pick-up, time and applied colors on the color fading performance of reactive-dyed cotton are modeled using Extreme Learning Machine (ELM), Support Vector Regression (SVR) and Random Forest Regression (RF) respectively. It is found that RF and SVR perform better than ELM in this issue, but SVR is more recommended to be sued in the real application due to its balance predicting performance and less training time.

Introduction

Increasingly, textile products with faded effect, worn look or vintage style have gained considerable share of fashion market and are attracting a growing number of consumers' attention [1]. However the faded effect conventionally was achieved by chemical methods which are not only highly water and power consuming, but also leave a heavy burden to the environment.

Ozone is an excellent gaseous oxidant with environmentally friendly property that can be decomposed rapidly to O 2 after using without emitting secondary pollution. In recent years, the study of color fading dyed textile using ozone instead of conventional method has been increasingly reported because of the advancement of ozonation. However the complicated and nonlinear relationship between the applied parameter in ozonation and color fading effects are still not clearly known, in other words, a model that can predict the engineering application effect is rather necessary. Thus, models for predicting color properties of ozone fading dyed textile using different artificial intelligent techniques of ELM, SVR and RF is comparatively researched in this study.

Experimental

Material

Cotton fabrics ( 3 / 1 twill, 325.7g/m 2 , Shunfu, China) dyed by reactive dyes of Blue-RN, Red-2BL and Yellow-2RN (provided by Color Root, China) were used in the experiment.

Methods

Ozone employed in this work was generated by a corona discharge ozone generator, CF-G50 (Guolin, China, fed by (≥99.9% pure and compressed dry oxygen (1Mpa, 12L/min) from oxygen cylinder) and supplied into the reactor (as shown in Figure 1. which is made of glass). Ozone was continuously fed in and flowed over the ozone gas (used gas would be outputted as exhaust) with 137±3 mg/min at 2L/min. Dyed fabrics were wetted and placed evenly in the reactor before treating by ozone. The fading performance of three colors (blue, red, yellow) of dyed fabrics at different pH (1,4,7,10,13) and temperatures (0, 20, 40, 60, 80℃) with different water pick-up(0, 75%, 150%) over varied period (0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60 min) were investigated respectively.

According to Kubelka-Munk theory, it is known that K/S value indicates the color depth of samples. While L* a* b* values (or CIELab), an international standard widely used for color measurements, is capable of illustrating the color variation of textile samples. Thus, these values tested by Datacolor 110 spectrophotometer (Datacolor, USA) were used to characterize the color variation of dyed textile in the color fading ozonation.

Artificial Intelligence techniques

As introduced above, it is clear that expected models would be inputting by the color of fabric as well as ozonation parameters of pH, temperature, water content and time, meanwhile outputting four outputs of K/S, L * , a * , b * values.

ELM

ELM is an algorithm for single-hidden layer feedforward neural networks (SLFNs) which randomly chooses hidden nodes and analytically determines the output weights of SLFNs. It tends to provide good generalization performance at extremely fast learning speed [2]. In this study, activation functions of Sigmoid, Sine and Hardim are comparatively analyzed with different number of nodes in the use of ELM.

SVR

Support vector machine (SVM) analysis is a popular machine learning tool for classification and regression, first identified by Vladimir Vapnik and his colleagues in 1992. In particular, multi-output least-squares support vector regression machines (MLS-SVR) toolbox [3] was used in this study with a range of different kernel functions such as Linear, Sigmoid, Polynomial and Radial basis function (RBF).

RF

Random Forest is a predictive model composed of a weighted combination of multiple regression trees. In general, combining multiple regression trees increases predictive performance. Multivariate Random Forest (MRF) developed by Raziur Rahman et al [4] were used in this study with an optimization of three parameters in terms of the number of trees in the forest and minimum number of samples in the leaf node as well as the randomly selected features considered for a split in each regression tree node.

Process modeling

The real samples treated in the ozonation at pH7, 20℃ with 150% pick-up for different time from 0 to 60min are demonstrated in Figure 2. Corresponding K/S, L * , a * , b * values of these samples are provided in

RB-RN

K/S 8.12 3.70 2.17 1.63 1.24 1.17 0.89 0.85 0.66 8.12 L* 6.13 14.89 21.90 26.12 29.50 29.93 33.44 33.80 36.66 6.13 a* -2. 86 -4.47 -5.18 -5.68 -5.57 -5.27 -5.25 -5.19 -4.76 Ⅱ illustrates the variation including the minimum, maximum, average and standard deviation of inputs as well as outputs in the expected model. K-fold cross validation (k=10) based on 459 experimental datasets (75%) were distributed to train and validate all the models above, while 153 datasets (25%) were used to test the models. K-fold cross validation is a popular statistical approach for estimating the predictive models. As k=10 in present study, 459 sets of data would be divided randomly and equally into 10 disjoint folds, 9 folds of it would be split into training subset while the rest 1-fold of dataset would be used as validating subset. This method would be repeated 10 times with varied training and testing dataset at each time. Performance of models would be evaluated by Mean Square Error (MSE) based on the equation ( 1).

( 1)

where e i is the real experimental results, whereas p i is the predicted output of specific model.

In present study, the models' construction were carried out using MATLAB R2015b for ELM and MLS-SVR but R studio for MRF respectively on a personal computer (Core i7-4710, 2.5GHz, 16GB RAM).

Results and discussion

ELM models

ELM models with hidden nodes 1-136 activated by Sigmoid, Sine and Hardim functions are investigated respectively (the validated MSE shown in Figure 3.). It is worth to note that Sigmoid trained ELM models performed similarly to the ones trained by Sine that their MSE both dropped and minimized at with around 50 nodes (MSE ≈ 0.052) but enhanced dramatically after achieving the minimum MSE. By contrast, validation accuracy of Hardim activated models are minimized at the one with 97nodes (MSE ≈ 0.069). Meanwhile it is generally stable when the number of node further increased.

The use of activation functions in neural network is to convert an input signal of node to an output signal. As known that the used activation functions were , and Among which, the graph of Sigmoid and Sine has something in common which may result in their similar proximity in the ELM models. Hardim performed worse as it is over-saturated.

SVR models

Multi output SVR with kernel functions of Linear, Sigmoid, Polynomial and RBF are trained and comparatively investigated using MLS-SVR toolbox [3]. The corresponding validation results of minimum MSE are 0.0539, 0.0095, 0.084, and 0.0047 respectively. In particular, SVR model with kernel function of RBF is the best one performed in this issue when parameters are optimized as gamma = 32768, lambda = 9.7656e -4 and p = 0.125.

Kernel function is to transform the data as input into the required form to facilitate a non-linear decision surface to be a linear equation in higher dimensions. It is well known that RBF kernel is recommended to be tried first in a SVR model as the fact that it not only possesses certain similar parameters and behaviors of Linear and Sigmoid, but also has less hyperparameters than Polynomial to complex the model. RBF gained the lowest MSE is more suitable in this case because we have not too much features in model but with comparatively large numbers of observations.

RF models

RF models with different features from1 to 5 and leaves from 1 to 10 as well as trees from 1 to 100 were trained, and the validated MSE are demonstrated in Figure 4. The number of randomly selected features in each regression tree node is found that plays a very vital role in affecting the models' predicting accuracy in this issue. This is owing to that the constructed model possesses five inputs, namely color of dyed cotton couple with pH, temperature, pick-up and time in ozonation, and the independency of these five factors are obvious clear. As a result, RF models with five randomly selected features optimized in the performance comparison. It is also found that RF models perform steadily when there are more than 30 regression trees in RF. But taking training time and cost into account, the 10 trees forest is sufficient and more recommended to be used in the color fading ozonation of dyed textile predicting model. However, different from the selected features and trees, minimum number of samples in the leaf node seems to be preferable to be less (apart from 1 leaf) in this investigation as the average MSE of achieved RF models linearly enhanced when the number of leaf is increased from 2 (0.0081) to 10 (0.0087) though it is relatively uninfluential.

Prediction performance

According to the analysis above, it is known that the ELM models trained with activation function of Sine and Sigmoid with around 50 nodes, and SVR models worked with RBF kernel function as well as the RF with 5 randomly selected features, 10 trees and 2 minimum number of samples in the leaf node are the optimized models in the application of modeling color fading ozonation of dyed textile.

To estimate and compare the optimized models, the prediction test using testing dataset (which has not been used in the training and validation process) is carried out. The prediction performance results are demonstrated in Table Ⅲ. Generally, ELM models using different training function of Sigmoid (MSE=0.0172) and Sine (MSE=0.0173) does not make any big difference in regard to their prediction performance, but both of it are slightly poorer as compared with SVR and RF models. However, ELM models are the fastest-trained ones in this specific comparison, which means that ELM model is still worth to be used in certain resource limited cases especially while training time is concerned. The most accurate model in this issue is constructed by RF, which can be observed in Table Ⅲ that leads the minimum testing MSE as only 0.0036. But it is also noted that training RF model requires much longer time than the others (21 seconds, which is over 22 times more than SVR). Therefore, it is important to take the SVR models into account as it achieved the second lowest MSE (0.0043) and it is more acceptable with the correspondingly shorter training time (0.9360s).

The constructed models possess four outputs, namely the chromatic values of K/S, L * , a * , b * values of ozone faded cotton fabrics. In order to reflect the real prediction performance of each trained model with respect to each single output specifically, the predicted data of models (using testing data) versing real experimental data is illustrated in Figure 5 In Figure 5, it is found that the predicted values of models roughly agree with the actual values, although the predictive error maybe comparatively large in certain cases. The difference of models' prediction performance in terms of MSE and average error are not that significant as shown in Table Ⅲ (Sigmoid activated ELM=0.0172, Sine activated ELM=0.0173, SVR=0.0043, RF=0.0036), while the distribution of single output prediction is observed that may have a big gap in the real application.

As shown in Figure 5 (a) and (c), certain predicted data of ELM models can be clearly seen that are far different from the real target data, which may result in big mistake in certain single prediction application though the average of four outputs may alleviate the single wrong output. According to the correlation coefficients of predicting data versing real experimental data (demonstrated in Figure 5) listed in Table Ⅳ, the testing result reveals that SVR (R 2 =0.9505) model and RF model (R 2 =0.9555) are actually more suitable than ELM models (R 2 =0.8025 and 0.8007 for Sigmoid and Sine activated respectively) in modeling color fading ozonation of dyed textile.

The finding of this investigation indicates that both of SVR and RF are good candidates for making predictions on the color quality of dyed textile in color fading ozonation as they both have a good agreement between the actual and predicted chromatic values of the textile samples.

Conclusion

Color fading ozonation for dyed textile products have specific vintage effect and are going to be increasingly popular in recent years. The effects of ozonation in terms of pH, temperature, water pick-up, time and applied colors on the color fading performance of reactive-dyed cotton are modeled using ELM, SVR and RF respectively in this study. The investigation results show that both of SVR and RF are good to be used in this issue compared with ELM, however, taking the training time as a consideration, SVR model would be more recommended to be applied in the real use.

Acknowledgement

The first author would like to express his gratitude to China Scholarship Council for supporting this study (CSC, Project NO. 201708420166).

Referencess

[1] Z He, M Li, D Zuo, J Xu, C Yi. "Effects of color fading ozonation on the color yield of reactive-dyed cotton." Dyes and Pigments (2019).

[2] Huang, Guang-Bin, Qin-Yu Zhu, and Chee-Kheong Siew. "Extreme learning machine: theory and applications." Neurocomputing 70. 1-3 (2006): 489-501.

[3] Xu, Shuo, et al

Figure 2 .

 2 Figure 2. The illustration of real ozone faded cotton samples.

Figure 3 .

 3 Figure 3. Validation MSE of different trained ELM models.

Figure 4 .

 4 Figure 4. MSE of RF models with varied number of features, leaves and trees

  (a) (b) (c) (d) with respect to each single output of the model.

Figure 5 .

 5 Figure 5. Predicted data outputted by ELM (trained by Sigmoid and Sine respectively), SVR and RF versus experimental data

TABLE Ⅰ .

 Ⅰ It is clear that each sample has obvious difference in terms of color performance with different ozonation parameters.

TABLE I .

 I

K/S, L * , A * , B * VALUES OF SAMPLES SHOWN IN FIGURE 2.

Time(

min) 0 5 10 15 20 25 30 40 50 60

TABLE II .

 II THE MAXIMUM, MINIMUM, AVERAGE AND STANDARD DEVIATION OF OZONATION PARAMETERS.

					b *	-90.53	88.03545	3.875674	42.34
	Parameters	Minimum	Maximum	Average	Std. dev.
	Color	0(Blue)	1(Yellow)	0.5(Red)	-
	pH	1	13	7	3.463
	Temperature	0	80℃	40℃	24.91
	Pick-up	0	150%	75%	58.78
	Time	0	60min	30min	20.77
	K/S	0.0975	22.82	7.177406	7.94
	L *	0.99	65.27	33.52368	17.61
	a *	-58.99	53.68333	-1.81221	32.75

TABLE

TABLE III .

 III PREDICTION PERFORMANCE OF OPTIMIZED MODELS.

	Parameters	ELM (Sigmoid)	ELM (Sine)	SVR	RF
	Training time(s)	0.0312	0.0936	0.9360	21
	Average error (%)	0.1527	0.1549	0.1530	0.149
	Maximum error (%)	0.3752	0.3700	0.3503	0.302
	Minimum error (%)	0.0300	0.0029	0.0304	0.038
	Testing MSE	0.0172	0.0173	0.0043	0.0036

TABLE IV .

 IV CORRELATION COEFFICIENTS OF DATA IN FIGURE 5.

	R 2	ELM (Sigmoid)	ELM (Sine)	SVR	RF
	Target 1 -K/S	0.8474	0.8596	0.9683	0.9954
	Target 2 -L *	0.7944	0.7517	0.9442	0.8816
	Target 3 -a *	0.7903	0.8048	0.9380	0.9719
	Target 4 -b *	0.7778	0.7868	0.9513	0.9731
	Average	0.8025	0.8007	0.9505	0.9555

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