Ronghua Ji

Lijun Qi

Zicheng Huo

Design of fuzzy control algorithm for precious irrigation system in greenhouse

Keywords: soil humidity, fuzzy control algorithm, irrigation, greenhouse

The precious irrigation is great significance for arid and semiarid area. According to the special environment of greenhouse, a fuzzy control algorithm was proposed to make an optimal irrigation strategy based on the actual measured soil humidity during the whole plant growth process. The fuzzy control system had two inputs (soil humidity error and its rate) and one output (water level difference). In this paper, the fuzzy control algorithm was introduced in detail which included the setting of input and output, the selection of membership function and the setting of fuzzy rules. The fuzzy control system was meaningful to the smart water-saving irrigation in greenhouse.

Introduction

Water is one of our precious natural resources. Water-saving irrigation is great significance because there are many arid and semiarid areas in the word. The key in water-saving irrigation is striking to get balance for optimal plant grown with optimal usage of water. Many water-saving technologies were proposed by the researchers [START_REF] Tal | Seeking Sustainability: Israel's Evolving Water Management Strategy[END_REF], such as drip irrigation [START_REF] Caldwell | Frequency of irrigation for subsurface drip irrigation corn[END_REF] ,subsurface drip irrigation [START_REF] Ayars | Subsurface drip irrigation of row crops: a review of 15 years of research at the Water Management Research Laboratory[END_REF] and subsurface negative irrigation [START_REF] Wang | Determination of comprehensive quality index for tomato and its response to different irrigation treatments[END_REF]. There were many research works which focused on the affection of different irrigation methods to crop. In order to find better irrigation scheduling with the compromise between high yield and great quality of greenhouse-grown tomato under limit water supply, two experiments of different irrigation treatments were conducted in the arid region of northwest China during spring to summer in 2008 (2008 season)and winter in 2008 to summer in 2009 [START_REF] Wang | Determination of comprehensive quality index for tomato and its response to different irrigation treatments[END_REF]. Because the irrigation water use efficiency was very importance, the technical efficiency of unheated greenhouse farms in Tunisia was measured, and proposed a measure for irrigation water use efficiency (WUE) using an alternative form of the data envelopment analysis [START_REF] Frija | Water use and technical efficiencies in horticultural greenhouses in Tunisia[END_REF]. Application of the temporal and spatial deficit irrigation in field-grown crops had greater potential in saving water, maintaining economic yield and improving WUE [START_REF] Du | An improved water use efficiency of cereals under temporal and spatial deficit irrigation in north China[END_REF].

Many conventional methods for controlling greenhouse irrigation were not effective since they were either based on on-off control methods or proportional control methods which resulted in a loss of energy and productivity. A solution for an irrigation controller based on the fuzzy-logic methodology was presented [START_REF] Javadi Kia | Intelligent Control Based Fuzzy Logic for Automation of Greenhouse Irrigation Systemand Evaluation in Relation to Conventional Systems[END_REF]. The developed fuzzy logic controller could effectively estimate the amount of water uptake of plants in distinct depth using the reliable irrigation model, evaporation functions, environmentally conditions of the greenhouse, soil type, plant type and other factors affecting the irrigation of greenhouse. A fuzzy irrigation decisionmaking system was established by using virtual instrumentation platform [START_REF] Zhengjun | Irrigation decision-making system based on the fuuzy-control theory and virtual instrument[END_REF]. The experiment results showed that the system had friendly interface and was easy to it could give comprehensive judgment and decision-making, and provide scientific basis for water saving irrigation. In our case, the irrigation system was very different.

The optimal water volume for plants was affected by many physical parameters such as air temperature, soil humidity, wind and stage of plant grown. Soil humidity was an important irrigation indicator to ensure the crop growing well. Because environment was controllable in greenhouse, some outside parameters were constant such as air temperature and wind. As for the irrigation system which was applied in greenhouse, the change of soil humidity was the most importance and direct parameter for irrigation strategy which included irrigation time and the irrigation volume. The fuzzy control algorithm was proposed for greenhouse precision irrigation system in this paper.

Materials and Methods

The soil water movement and distribution in soil was seriously affected by the irrigation strategy. In this paper, a precision irrigation system was used. The precision irrigation system consisted of five parts: a plant container, a built-in reservoir, a level control tank, a reservoir and a height adjuster. The scheme of precision irrigation system was shown in Fig. 1. The soil humidity of plant container was affected by the water level difference between built-in reservoir and the level control tank. The water level difference was adjusted by the height adjuster which was controlled by the fuzzy controller in PC. Fuzzy controller had been widely applied as a powerful methodology for complicated nonlinear delay systems which were difficult to set up accurate mathematic models. The control strategy in controller was selected as fuzzy control because soil humidity

① ③ ② ④ ⑤ ⑥
was large delay, great inertial, time-varying and non-linearity.

The fuzzy controller was determined by the height of height adjuster based on the soil humidity error and its rate. The principle diagram of fuzzy control in this system was shown as in Fig. 2 Fig. 2. The principle diagram of fuzzy controller was the soil humidity error which was defined as desired soil humidity minus actual soil humidity.

was the rate of soil humidity error which was defined in equation ( 2).

(1)

Where was desired soil humidity, was actual soil humidity. The best range of soil humidity in every crop growing stage was ascertained according to the previously experimental data. The medium value of the best range of soil humidity was taken as the desired soil humidity.

The output of fuzzy control system was which was used adjusting the water level difference between built-in reservoir and the level control tank and changing soil humidity.

The soil humidity was always lower than desired value. The range of was [-6%, 0] and the range of [-1%, 0]. The input variables ( and ) had 4 fuzzy states which was {IL, IM, IS, IZ}. The output variable had 4 fuzzy states which was {OL, OM, OS, OZ}. The ranges of input/output variables were following:

Because the input variables of fuzzy controller (E and DE) were measured as accurate, they should be fuzzed firstly according to the equation ( 3).

(

) 3
Where was the accurate input variable and was the fuzzed value. The accurate input variables changed from a to b. y should be taken as integer.

The membership function of variables was importance to the fuzzy controller. Every fuzzy variable's membership functions should be defined respectively. There were many kinds of membership functions to meet the requirement of various variables. The general rule to select membership function was following: the higher sensitivity membership for little error and the fine stability membership for larger error. According to the changing regularity of soil humidity error and the membership selection rule, the membership functions of variables (IL and IM) were taken as triangle function, the membership functions of variables (IS and IZ) were taken as Gaussian function and the membership functions of (DE and h) were triangle function. The membership function assignment table was shown in Table 1 to 3 According to the experimental data and irrigation experience, the fuzzy control rules were shown in Table 4.

IZ OZ OZ OZ OZ

There were 16 fuzzy control rules in this system.

That was

If E= and DE= Then h= (i, j=3, 2, 1, 0) Where the value of Ai was set according to Table 1 and the value of was set according to Table 2.

= × In this paper, the fuzzy control algorithm was simulated by the fuzzy control toolbox of Matlab. The fuzzy reasoning system surface was shown in Fig 3.

Conclusion

In the greenhouse, the plant growth environment parameters were controllable, such as air temperature, wind and air humidity. The precious irrigation should be made the suitable irrigation strategy according to the requirements of plant growth. The irrigation strategy based on soil humidity and plant growth was an optimal irrigation to meet the actual need of plant growth.

The fuzzy control algorithm was effective for the dynamic characteristics of soil humidity which was large delay, great inertial, time-varying and non-linearity. The fuzzy control system was meaningful with the water-saving irrigation in greenhouse.

Fig. 1 .

 1 Fig. 1. The scheme of precision irrigation system ①the plant container ;②the built-in reservoir; ③ the level control tank; ④ the reservoir; ⑤ the height adjuster; ⑥ the fuzzy controller in computer.

Fig. 3 .

 3 Fig. 3. The fuzzy reason system surface

Table 1 .

 1 . The soil humidity error's membership function assignment table

	Fuzzy			The soil humidity error (E)
	variable	-6	-5	-4	-3	-2	-1	0
	IL	1.0	0.6	0.2	0	0	0	0
	IM	0.2	0.6	1	0.6	0.2	0	0
	IS	0	0	0	0.5	1	0.5	0
	IZ	0	0	0	0	0	0.5	1

Table 2 .

 2 The soil humidity error rate of membership function assignment table

	Fuzzy		The error rate of soil humidity (DE)
	variable	-6	-5	-4	-3	-2	-1	0
	IL	1.0	0.5	0	0	0	0	0
	IM	0	0.5	1	0.5	0	0	0
	IS	0	0	0	0.5	1	0.5	0
	IZ	0	0	0	0	0	0.5	1

Table 3 .

 3 The output membership function assignment table

	Fuzzy	The error rate of soil humidity (h)
	variable	0	1	2	3
	OL	0	0	0.5	1
	OM	0	0.5	1	0.5
	OS	0.5	1	0.5	0
	OZ	1	0.5	0	0

Table 4 .

 4 Fuzzy control rules table

	Input		IL	DE IM	IS	IZ
		IL	OL OL	OM	OZ
	E	IM IS	OM OM OS OS OS OZ	OZ OZ

Acknowledgments

The research is supported by the National Natural Science Funds ( Project No. 31101609) and Chinese Universities Scientific Fund(Project No. 2011JS144).


Abdelkader Mokkadem

Mariane Pelletier
email: pelletier@math.uvsq.fr

Baba Thiam
email: thiam@math.uvsq.fr

Large and moderate deviations principles for kernel estimators of the multivariate regression

Keywords: AMS Subj. Classification: 62G08, 60F10 Nadaraya-Watson estimator, Recursive kernel estimator, Large deviations principle, Moderate deviations principle

published or not. The documents may come

Large and moderate deviations principles for kernel estimators of the multivariate regression

Introduction

Let (X, Y ), (X 1 , Y 1 ), . . . , (X n , Y n ) be a sequence of independent and identically distributed R d ×R qvalued random variables with probability density f (x, y) with E|Y | < ∞. Moreover, let g(x) be the marginal density of X and r(x) = E (Y |X = x) = m(x)/g(x) the regression of Y on X. The purpose of this paper is to establish large and moderate deviations principles for the Nadaraya-Watson estimator and for the semi-recursive kernel estimator of the regression.

Let us first recall the concept of large and moderate deviations. A speed is a sequence (ν n ) of positive numbers going to infinity. A good rate function on R m is a lower semicontinuous function I : R m → [0, ∞] such that, for each α < ∞, the level set {x ∈ R m , I(x) ≤ α} is a compact set. If the level sets of I are only closed, then I is said to be a rate function. A sequence (Z n ) n≥1 of R m -valued random variables is said to satisfy a large deviations principle (LDP) with speed (ν n ) and rate function I if:

lim inf n→∞ ν -1 n log P [Z n ∈ U ] ≥ -inf x∈U I(x) for every open subset U of R m , lim sup n→∞ ν -1 n log P [Z n ∈ V ] ≤ -inf x∈V I(x) for every closed subset V of R m .
Moreover, let (v n ) be a nonrandom sequence that goes to infinity; if (v n Z n ) satisfies a LDP, then (Z n ) is said to satisfy a moderate deviations principle (MDP).

The Nadaraya-Watson estimator ( [START_REF] Nadaraya | On estimating regression[END_REF], [START_REF] Watson | Smooth regression analysis[END_REF]) of the regression function r(x) is defined by

r n (x) =    m n (x) g n (x) if g n (x) = 0 0 otherwise, (1)
with

m n (x) = 1 nh d n n i=1 Y i K x -X i h n and g n (x) = 1 nh d n n i=1 K x -X i h n ,
where the bandwidth (h n ) is a positive sequence such that lim n→∞ h n = 0 and lim

n→∞ nh d n = ∞, (2)
and the kernel K a continuous function such that lim x →∞ K(x) = 0 and R d K(x)dx = 1. The weak and strong consistency of r n has been widely discussed by many authors; let us cite, among many others, Collomb [START_REF] Collomb | Proprietés de convergence presque-complète du prédicateur à noyau[END_REF], Collomb and Härdle [START_REF] Collomb | Strong uniform convergence rates in robust nonparametric time series analysis and prediction: kernel regression estimation from dependent observations[END_REF], Devroye [START_REF] Devroye | The uniform convergence of the Nadaraya-Watson regression function estimate[END_REF], Mack and Silverman [START_REF] Mack | Weak and strong uniform consistency of kernel regression estimates[END_REF] and Senoussi [START_REF] Senoussi | Loi du log itéré et identification[END_REF]. For other works on the consistency of r n , the reader is refered to the monographs of Bosq [START_REF] Bosq | Nonparametric statistics for stochastic processes[END_REF] and Prakasa Rao [START_REF] Rao | Nonparametric functionnal estimation[END_REF]. The large deviations behaviour of r n has been studied at first by Louani [START_REF] Louani | Some large deviations limit theorems in conditionnal nonparametric statistics[END_REF], and then by Joutard [START_REF] Joutard | Sharp large deviations in nonparametric estimation[END_REF] in the univariate framework. Moderate deviations principles have been obtained by Worms [START_REF] Worms | Principes de déviations modérées pour des martingales et applications statistiques[END_REF] in the particular case Y = r(X) + ε with ε and X independent.

The first aim of this paper is to generalize these large and moderate deviations results.

The approach used by Louani [START_REF] Louani | Some large deviations limit theorems in conditionnal nonparametric statistics[END_REF] and Joutard [START_REF] Joutard | Sharp large deviations in nonparametric estimation[END_REF] to study the large deviations behaviour of r n is to note that, if d = q = 1 and if the kernel is positive, then, for all δ > 0,

P [r n (x) -r(x) ≥ δ] = P   1 nh n n j=1 [Y j -r(x) -δ] K x -X j h n ≥ 0   .
Obviously, their approach can not be extended to the multivariate framework. Thus, to study the large deviations behaviour of r n , our approach is totally different. We first establish a large deviations principle for the sequence (m n (x), g n (x)), and then show how the large deviations behaviour of r n can be deduced. More precisely, for x ∈ R d , let Ψ x be the function defined for any (u, v) ∈ R q × R by

Ψ x (u, v) = R d ×R q
e ( u,y +v)K(z) -1 f (x, y)dzdy, (where u, y denotes the scalar product of u and y) and let I x be the Fenchel-Legendre transform of Ψ x . We give conditions ensuring that the sequence (r n (x)) satisfies a LDP with speed (nh d n ) and good rate function J defined, for any s ∈ R q , by J(s) = inf t∈R I x (st, t).

Concerning the moderate deviations behaviour of the Nadaraya-Watson estimator, we prove that, for any positive sequence (v n ) such that

lim n→∞ v n = ∞, lim n→∞ v 2 n nh d n = 0, and lim n→∞ v n h p n = 0, (3)
(where p denotes the order of the kernel K) the sequence (v n [r n (x)r(x)]) satisfies a LDP with speed nh d n /v 2 n and good rate function G x defined for all v ∈ R q by

G x (v) = g(x) 2 R d K 2 (z)dz v T Σ -1 x v, (4)
where Σ x denotes the q × q covariance matrix V (Y |X = x). Let us note that, in the case the model Y = r(X) + ε (with X and ε independent) is considered, the matrix Σ x is the covariance matrix of ε and does depend on x; we then find the MDP proved in Worms [START_REF] Worms | Principes de déviations modérées pour des martingales et applications statistiques[END_REF] again.

A semi-recursive version of the Nadaraya-Watson estimator (1) is defined as

rn (x) =    mn (x) gn (x) if gn (x) = 0 0 otherwise, (5)
where

mn (x) = 1 n n i=1 Y i h d i K x -X i h i and gn (x) = 1 n n i=1 1 h d i K x -X i h i .
Weak conditions for various forms of consistency of rn have been obtained by Ahmad and Lin [START_REF] Ahmad | Nonparametric sequential estimation of a multiple regression function[END_REF] and Devroye and Wagner [START_REF] Devroye | On the L 1 convergence of kernel estimators of regression function with applications in discrimination[END_REF]. Roussas [START_REF] Roussas | Exact rates of almost sure convergence of a recursive kernel estimate of a probability density function: Application to regression and hazard rate estimate[END_REF] studied its almost sure convergence rate. The second aim of this paper is to establish the large and moderate deviations behaviour of rn .

It turns out that the rate function that appears in the LDP is much more complex to explicit in the case the semi-recursive kernel regression estimator is considered than in the case the Nadaraya-Watson estimator is used. That is the reason why we only consider bandwidths defined as (h n ) = (cn -a ) with c > 0 and a ∈]0, 1/d[ (instead of bandwidths satisfying (2)). For x ∈ R d , let Ψa,x be the function defined for all (u, v

) ∈ R q × R by Ψa,x (u, v) = [0,1]×R d ×R q
s -ad e s ad ( u,y +v)K(z) -1 f (x, y)dsdzdy, and let Ĩa,x be the Fenchel-Legendre transform of Ψa,x . We give conditions ensuring that the sequence (r n (x)) satisfies a LDP with speed (nh d n ) and good rate function Ja defined, for any s ∈ R q , by Ja (s) = inf t∈R Ĩa,x (st, t).

To establish the moderate deviations behaviour of rn , we consider bandwidths (h n ) which vary regularly with exponent (-a), a ∈]0, 1/d[. We prove that, for any positive sequence (v n ) satisfying (3), the sequence (v n [r n (x)r(x)]) satisfies a LDP with speed nh d n /v 2 n and good rate function defined for all v ∈ R q by Ga,

x (v) = (1 + ad)g(x) 2 R d K 2 (z)dz v T Σ -1 x v. (6)
Let us underline that, because of the factor (1 + ad) which is present in [START_REF] Dembo | Large deviations techniques and applications[END_REF] but not in (4), the rate function obtained in the MDP in the case the semi-recursive estimator is used is larger than the one which appears in the case the Nadaraya-Watson kernel estimator is considered; this means that the semi-recursive estimator rn (x) is more concentrated around r(x) than the Nadaraya-Watson estimator.

Our main results are stated in Section 2, whereas Section 3 is devoted to the proofs.

Assumptions and Main Results

We shall use the following notations.

• D(F) = {x, F(x) < ∞} denotes the domain of a function F and • D(F) is the interior domain of F.

• x is the euclidean norm of x.

• λ is the Lebesgue measure.

• a ∧ b = min{a, b}.

• 0 = (0, . . . , 0) ∈ R q .
The large and moderate deviations behaviours of the Nadaraya-Watson estimator r n are given in Section 2.1, whereas the ones of the semi-recursive kernel estimator rn are stated in Section 2.2.

Large and moderate deviations principles for the Nadaraya-Watson estimator

The assumptions required for the LDP of the Nadaraya-Watson estimator are the following.

(A1) K : R d → R is a bounded and integrable function, R d K(z)dz = 1 and lim z →∞ K(z) = 0.

(A2) For any u ∈ R q , t → R q e u,y f (t, y)dy is continuous at x and bounded.

Comments

• Notice that (A2) implies that the density g is continuous at x and bounded.

• In the model Y = r(X) + ε with ε and X independent, let h be the probability density of ε. Then

f (t, y) = g(t)h (y -r(t)) R q y f (t, y)dy = g(t) R q y + r(t) h(y)dy R q e u,y f (t, y)dy = g(t)e u,r(t)
R q e u,y h(y)dy.

Thus, (A2) can be translated as assumptions on g and r and on the moments of ε.

• As it can be seen from the proofs, the boundness assumption in (A2) is useless if K has a compact support.

• The boundness of the function t → R q e u,y f (t, y)dy for any u ∈ R q implies that ∀m ≥ 0, ∀ρ ≥ 0 the function t → R q y m e ρ y f (t, y)dy is bounded.

Proof It suffices to prove that the function t → R q e ρ y f (t, y)dy is bounded for any ρ > 0.

Set y = (y 1 , . . . , y q ), we first note that R q e qρ|y j | f (t, y)dy ≤ {y j ≥0} e qρy j f (t, y)dy +

{y j <0}
e -qρy j f (t, y)dy

≤

R q e qρy j f (t, y)dy + R q e -qρy j f (t, y)dy.

Now, we have

R q e ρ y f (t, y)dy

≤ R q e ρ|y 1 |+•••+ρ|yq| f (t, y)dy ≤ R q e qρ|y 1 | f (t, y)dy . . . R q
e qρ|yq| f (t, y)dy

1 q
by the generalized Hölder inequality.

≤

R q e qρy 1 f (t, y)dy + R q e -qρy 1 f (t, y)dy . . .

R q
e qρyq f (t, y)dy + R q e -qρyq f (t, y)dy

1 q
which is bounded.

Before stating our results, we need to introduce the rate function for the LDP of the Nadaraya-Watson estimator. Let Ψ x : R q × R → R and I x , Îx : R q × R → R be the functions defined as follows:

Ψ x (u, v) = R d ×R q e ( u,y +v)K(z) -1 f (x, y)dzdy, (8)
I x (t 1 , t 2 ) = sup (u,v)∈R q ×R { u, t 1 + vt 2 -Ψ x (u, v)} , (9)
Îx (s, t) = I x (st, t).

Moreover, for any s ∈ R q , set

J * (s) = inf t∈R * I x (st, t) = inf t∈R * Îx (s, t), J(s) = J * (s) ∧ I x ( 0, 0) = inf t∈R Îx (s, t).
To prove that J is a rate function, we need to assume that the following condition (C) is fulfilled.

(C) inf s∈R q I x (s, 0) = I x ( 0, 0). Before stating the properties of the function J, let us give some cases when Condition (C) is satisfied (under Assumptions (A1) and (A2)).

Example 1: Nonnegative kernel Condition (C) is satisfied when K is nonnegative since, in this case, I x (s, 0) = +∞ for any s = 0, (this is stated in Proposition 3 of Section 3).

Example 2: Model with symmetry Condition (C) holds when f is symmetric in each coordinate of the second variable y ∈ R q . As a matter of fact, for a diagonal q × q matrix A such that A ii = ±1, observe that

Ψ x (Au, v) = R d ×R q e ( Au,y +v)K(z) -1 f (x, y)dzdy = R d ×R q e ( u,Ay +v)K(z) -1 f (x, y)dzdy = R d ×R q e ( u,y ′ +v)K(z) -1 f (x, A -1 y ′ )dzdy ′ = Ψ x (u, v).
For any given s ∈ R q , set

U s = {u ∈ R q , u, s ≥ 0} . We have, sup u,v (-Ψ x (u, v)) = sup u∈Us, v∈R (-Ψ x (u, v)) . Now, for any u ∈ U s and v ∈ R, u, s -Ψ x (u, v) ≥ -Ψ x (u, v), so that sup u,v { u, s -Ψ x (u, v)} ≥ sup u∈Us, v∈R (-Ψ x (u, v)) ,
and thus,

I x ( 0, 0) ≤ I x (s, 0) ∀s ∈ R q ,
so that Condition (C) follows. Now, let M x denote the Laplace transform of f (x, •), then

Ψ x (u, v) = e v λ(D)M x (u) + e -v λ(D ′ )M x (-u) -λ(D) + λ(D ′ ) g(x).
For any given u, it can easily be seen that the infimum of Ψ x (u, •) is reached at

v 0 = log λ(D ′ )M x (-u) λ(D)M x (u) ,
and

Ψ x (u, v 0 ) = 2 λ(D)λ(D ′ ) M x (u)M x (-u) -λ(D) + λ(D ′ ) g(x).
Observe that

Ψ x (u, v 0 ) = Ψ x (-u, v 0 ),
and thus

sup u (-Ψ x (u, v 0 )) = sup u≥0 (-Ψ x (u, v 0 )) = sup u≤0 (-Ψ x (u, v 0 )) = I x (0, 0). Now, if s ≥ 0, we have for any u ≥ 0 us -Ψ x (u, v 0 ) ≥ -Ψ x (u, v 0 ),
and thus

I x (s, 0) ≥ I x (0, 0) ∀s ≥ 0.
Proceeding in the same way for s < 0, we obtain Condition (C).

Such an example of a four order kernel is

K = ½ [-a,a] -½ [-b,-a[∪]a,b] , with a = 1 6 3 √ 2 + 1 12 3 √ 2 2 + 1 3 b = 1 3 3 √ 2 + 1 6 3 √ 2 2 + 1 6 .
Let us now give the properties of the function J.

Proposition 1 Assume that (A1), (A2) and (C) hold. Then, (i) J is a rate function on R q . More precisely, for α ∈ R,

• if α < I x ( 0, 0), then {J(s) ≤ α} is compact.

• if α ≥ I x ( 0, 0), then {J(s) ≤ α} = R q .

(ii) If I x ( 0, 0) = ∞, then J is a good rate function on R q and J = J * .

(iii) If J * (s) < ∞, then J(s) = J * (s).

(iv) If α < I x ( 0, 0), then {J * (s) ≤ α} = {J(s) ≤ α}.

Remark 1 In view of the definition of J and J * , and of Proposition 1 (iii), we have:

J(s) = J * (s) if J * (s) < ∞ I x ( 0, 0) if J * (s) = ∞.
Let us now state the LDP for the Nadaraya-Watson estimator.

Theorem 1 (Pointwise LDP for the Nadaraya-Watson estimator) Assume that (A1), (A2) and (C) hold, and that (h n ) satisfies the conditions in (2). Then, for any open subset

U of R q , lim inf n→∞ 1 nh d n log P [r n (x) ∈ U ] ≥ -inf s∈U J * (s),
and for any closed subset

V of R q , lim sup n→∞ 1 nh d n log P [r n (x) ∈ V ] ≤ -inf s∈V J(s).
Comments.

1) Set E = {J * (s) < ∞}. For any open subset U of R q such that U ∩ E = ∅, we have lim inf n→∞ 1 nh d n log P [r n (x) ∈ U ] ≥ -inf s∈U J(s).
2) If I x is finite in a neighbourhood of ( 0, 0), then J * is finite everywhere and by Proposition 1 (iii), J(s) = J * (s) < ∞ ∀s ; thus (r n ) satisfies a LDP with speed (nh d n ) and rate function J. Of course, this does not hold for nonnegative kernel since in this case I x (s, 0) = +∞ for any s (see Proposition 3 in Section 3). However, it can hold for kernels which take negative values. For example, consider the previous Example 3, and assume f (x, y) is symmetric in y ; in this case M x (u) = M x (-u). The equation

∂Ψ x ∂v (u, v) = λ(D)e v -λ(D ′ )e -v M x (u) = 0 has solution v 0 = log λ(D ′ ) λ(D) independent from u.
Moreover, M ′ is continuous and has range R, thus, there exists u 0 such that M ′ (u 0 ) = 0. This implies that the equation

∂Ψ x ∂u (u, v) = λ(D)e v + λ(D ′ )e -v M ′ x (u) = 0
has a solution u 0 independent from v. Thus (0, 0) is in the range of ∇Ψ x . It follows from Proposition 3 Section 3 that I x is finite in a neighbourhood of (0, 0).

3) When I x ( 0, 0) = ∞, it follows from Proposition 1 and Theorem 1 that (r n ) satisfies a LDP with speed (nh d n ) and good rate function J.

In the case K is a nonnegative kernel whose support has an infinity measure, we will show in Proposition 3 that I x ( 0, 0) = ∞. We have thus the following corollary.

Corollary 1 Let the assumptions of Theorem 1 hold. If K is a nonnegative kernel such that λ x ∈ R d , K(x) > 0 = ∞, then the sequence (r n ) satisfies a LDP with speed (nh d n ) and good rate function J. This corollary is an extension of the results of Louani [START_REF] Louani | Some large deviations limit theorems in conditionnal nonparametric statistics[END_REF] and Joutard [START_REF] Joutard | Sharp large deviations in nonparametric estimation[END_REF] to the multivariate framework (and to the case the kernel K may vanish). Moreover, it proves that the rate function that appears in their large deviations results is in fact a good rate function.

To establish pointwise MDP for the Nadaraya-Watson estimator, we need the following additionnal assumptions.

(A3) For any u ∈ R q , t → R q u, y 2 f (t, y)dy and t → R q u, y f (t, y)dy are continuous at x and

g(x) = 0. (A4) lim n→∞ v n = ∞ and lim n→∞ nh d n v 2 n = ∞.
(A5) i) There exists an integer p ≥ 2 such that ∀s ∈ {1, . . . , p -1}, ∀j ∈ {1, . . . , d}, R d y s j K(y)dy j = 0, and

R d y p j K(y) dy < ∞. ii) lim n→∞ v n h p n = 0.
iii) m and g are p-times differentiable on R d , and their differentials of order p are bounded and continuous at x.

We can now state the MDP for the Nadaraya-Watson estimator.

Theorem 2 (Pointwise MDP for the Nadaraya-Watson kernel estimator of the regression) Assume that (A1)-(A5) hold. Then, the sequence (v n (r n (x)r(x))) satisfies a LDP with speed

nh d n v 2 n
and good rate function G x defined in (4).

Large and moderate deviations principles for the semi-recursive estimator

For a ∈]0, 1/d[, let Ψa,x : R q × R → R and Ĩa,x : R q × R → R be the functions defined as follows:

Ψa,x (u, v) = [0,1]×R d ×R q s -ad e s ad ( u,y +v)K(z) -1 f (x, y)dsdzdy, (11)
Ĩa,x (t 1 , t 2 ) = sup

(u,v)∈R q ×R u, t 1 + vt 2 -Ψa,x (u, v) . (12)
Moreover, let Ja and J * a be defined as follows: for any

s ∈ R q , J * a (s) = inf t∈R * Ĩa,x (st, t) (13)
Ja (s) = J * a (s) ∧ Ĩa,x ( 0, 0). ( 14
)
Let us give the following additionnal hypotheses.

(A'1) For any u ∈ R q , t → R q e α u,y f (t, y)dy is continuous at x uniformly with respect to

α ∈ [0, 1].
Condition (C) above is substituted by the following one, (C') inf s∈R q Ĩa,x (s, 0) = Ĩa,x ( 0, 0).

Examples for which Condition (C') holds are Examples 1 and 2 given for (C). The following proposition gives the properties of the function Ja .

Proposition 2 Assume that (A1), (A2), (A'1) and (C') hold. Then, (i) Ja is a rate function on R q . More precisely, for α ∈ R,

• if α < Ĩa,x ( 0, 0), then Ja (s) ≤ α is compact.

• if α ≥ Ĩa,x ( 0, 0), then Ja (s) ≤ α = R q .
(ii) If Ĩa,x ( 0, 0) = ∞, then Ja is a good rate function on R q and Ja = J * a .

(iii

) If J * a (s) < ∞, then Ja (s) = J * a (s). (iv) If α < Ĩa,x ( 0, 0), then J * a (s) ≤ α = Ja (s) ≤ α .
Notice that, like for J and J * , we have

Ja (s) = J * a (s) if J * a (s) < ∞ Ĩa,x ( 0, 0) if J * a (s) = ∞.
We can now state the LDP for the semi-recursive kernel estimator of the regression.

Theorem 3 (Pointwise LDP for the semi-recursive estimator of the regression) Set (h n ) = (cn -a ) with c > 0 and 0 < a < 1/d, and let (A1), (A2), (A'1) and (C') hold. Then, for any open subset

U of R q , lim inf n→∞ 1 nh d n log P [r n (x) ∈ U ] ≥ -inf s∈U J * a (s),
and for any closed subset

V of R q , lim sup n→∞ 1 nh d n log P [r n (x) ∈ V ] ≤ -inf s∈V Ja (s).
The comments made for Theorem 1 are valid for Theorem 3. In particular, we have the following corollary.

Corollary 2 Let the assumptions of Theorem 3 hold. If K is a nonnegative kernel such that λ x ∈ R d , K(x) > 0 = ∞, then the sequence (r n ) satisfies a LDP with speed (nh d n ) and good rate function Ja . Before stating pointwise MDP for the semi-recursive estimator of the regression, let us recall that a sequence (u n ) is said to vary regularly with exponent α if there exists a function u which varies regularly with exponent α and such that u n = u(n) for all n (see, for example, Feller [START_REF] Feller | An introduction to probability theory and its applications[END_REF] page 275). We will use in the sequel the following property (see Bingham et al. [START_REF] Bingham | Regular variation[END_REF] page 26). If (h n ) varies regularly with exponent (-a) and if βa < 1, then

lim n→∞ 1 nh β n n i=1 h β i = 1 1 -aβ . ( 15
)
We also consider the following condition.

sup i≤n h n h i < ∞. (16)
(For example, this condition holds when h n is nonincreasing).

Proofs

The proofs of the results for the Nadaraya-Watson kernel estimator are in many cases similar to those of the semi-recursive kernel estimator of the regression, so we omit some details of the proofs for this last one.

First, let us state the following propositions which give the properties of the functions Ψ x , Ψa,x , I x and Ĩa,x . Set

S + = x ∈ R d , K(x) > 0 and S -= x ∈ R d , K(x) < 0 .
Proposition 3 (Properties of Ψ x and I x ) Let Assumptions (A1) and (A2) hold. Then, i) Ψ x is strictly convex, continuously differentiable on R q × R, and I x is a good rate function on R q × R.

ii) ∇Ψ

x is an open map and the range of

Ψ x is • D(I x ). I x is strictly convex on • D(I x ) and for any t ∈ • D(I x ) ⊂ R q × R, I x (t) = (∇Ψ x ) -1 (t), t -Ψ x (∇Ψ x ) -1 (t) . ( 17
)
iii) If λ(S -) = 0, then I x ( 0, 0) = g(x)λ(S + ), and for any t 1 = 0, I x (t 1 , 0) = +∞.

Proposition 4 (Properties of Ψa,x and Ĩa,x ) Let Assumptions (A1) and (A2) hold. Then, i) Ψa,x is strictly convex, continuously differentiable on R q × R, and Ĩa,x is a good rate function on R q × R.

ii) ∇ Ψa,x is an open map and the range of Ψa,x is

• D( Ĩa,x ). Ĩa,x is strictly convex on • D( Ĩa,x ), and for any t ∈ • D( Ĩa,x ) ⊂ R q × R, Ĩa,x (t) = (∇ Ψa,x ) -1 (t), t -Ψa,x (∇ Ψa,x ) -1 (t) . (18)
iii) If λ(S -) = 0, then Ĩa,x ( 0, 0) = g(x)λ(S + )/(1ad), and for any t 1 = 0, Ĩa,x (t 1 , 0) = +∞.

The two following lemmas are used for the proofs of Theorems 1 and 3.

Lemma 1 (Pointwise LDP for the sequence (m n (x), g n (x))) Let Assumptions (A1) and (A2) hold. Then, the sequence (m n (x), g n (x)) satisfies a LDP with speed nh d n and rate function I x defined in (9).

Lemma 2 (Pointwise LDP for the sequence ( mn (x), gn (x))) Set h n = cn -a with c > 0 and a ∈]0, 1/d[, and let Assumptions (A1), (A2) and (A'1) hold. Then, the sequence ( mn (x), gn (x)) satisfies a LDP with speed nh d n and rate function Ĩa,x defined in [START_REF] Mack | Weak and strong uniform consistency of kernel regression estimates[END_REF].

Our proofs are now organized as follows. Lemmas 1 and 2 are proved in Section 3.1, Theorems 1 and 3 in Section 3.2, Theorem 2 in Section 3.3, Theorem 4 is proved in Section 3.4. Section 3.5 is devoted to the proof of Propositions 3 and 4 on the rate functions I x and I a,x . Propositions 1 and 2 are proved in Section 3.6.

Proof of Lemmas 1 and 2

Proof of Lemma 1

For any w = (u, v) ∈ R q × R, set

Ψ n (x) = (m n (x), g n (x)) , Λ n,x (w) = 1 nh d n log E exp nh d n w, Ψ n (x) .
Let us at first assume that the following lemma holds.

Lemma 3 (Convergence of Λ n,x ) Assume that (A1) and (A2) hold, then

lim n→∞ Λ n,x (u, v) = Ψ x (u, v), (19)
where Ψ x is defined in (8).

To prove Lemma 1, we apply Proposition 3, Lemma 3 and the Gärtner-Ellis Theorem (see Dembo and Zeitouni [START_REF] Dembo | Large deviations techniques and applications[END_REF]). Proposition 3 ensures that Ψ x is essentially smooth, lower semicontinuous function so that Lemma 1 follows from the Gärtner-Ellis Theorem.

Let us now prove Lemma 3. Set

Z i = [ u, Y i + v] K x -X i h n .
For any (u, v) ∈ R q × R, we have

Λ n,x (u, v) = 1 nh d n log E exp n i=1 Z i ,
and, since the random vectors (X i , Y i ), i = 1, . . . , n are independent and identically distributed, we get

Λ n,x (u, v) = 1 h d n log E e Zn .
A Taylor's expansion implies that there exists c n between 1 and E e Zn such that

Λ n,x (u, v) = 1 h d n E e Zn -1 - 1 2c 2 n h d n E e Zn -1 2 = 1 h d n R d ×R q e ( u,y +v)K x-s hn -1 f (s, y)dsdy -R (1) n,x (u, v) = Ψ x (u, v) -R (1) n,x (u, v) + R (2) n,x (u, v), with R (1) n,x (u, v) = 1 2c 2 n h d n E e Zn -1 2 , R (2) n,x (u, v) = R d ×R q e ( u,y +v)K(z) -1 [f (x -h n z, y) -f (x, y)] dzdy.
Let us prove that

lim n→∞ R (2) n,x (u, v) = 0. (20)
Set A > 0 and ǫ > 0; we then have

R (2) n,x (u, v) = { z ≤A}×R q e ( u,y +v)K(z) -1 [f (x -h n z, y) -f (x, y)] dzdy + { z >A}×R q e ( u,y +v)K(z) -1 [f (x -h n z, y) -f (x, y)] dzdy. (21)
Next, since for any t ∈ R, e t -1 ≤ |t|e |t| , we have

{ z >A}×R q e ( u,y +v)K(z) -1 |f (x -h n z, y) -f (x, y)| dzdy ≤ { z >A}×R q | u, y + v| |K (z)| e | u,y +v||K(z)| |f (x -h n z, y) -f (x, y)| dzdy ≤ { z >A}×R q | u, y + v| |K (z)| e | u,y +v||K(z)| f (x -h n z, y)dzdy + { z >A}×R q | u, y + v| |K (z)| e | u,y +v||K(z)| f (x, y)dzdy ≤ e |v| K ∞ u { z >A} |K (z)| R q y e K ∞ u y f (x -h n z, y)dy dz +e |v| K ∞ |v| { z >A} |K (z)| R q e K ∞ u y f (x -h n z, y)dy dz +e |v| K ∞ u { z >A} |K (z)| dz R q y e K ∞ u y f (x, y)dy +e |v| K ∞ |v| { z >A} |K (z)| dz R q e K ∞ u y f (x, y)dy ≤ B { z >A} |K (z)| dz, ( 22
)
where B is a constant ; this last inequality follows from [START_REF] Devroye | The uniform convergence of the Nadaraya-Watson regression function estimate[END_REF] and from the fact that K is bounded. Now, since K is integrable, we can choose A such that

{ z >A}×R q e ( u,y +v)K(z) -1 |f (x -h n z, y) -f (x, y)| dzdy ≤ ǫ 2 . ( 23
)
Now, observe that

{ z ≤A}×R q e ( u,y +v)K(z) -1 [f (x -h n z, y) -f (x, y)] dzdy = { z ≤A} e vK(z) R q e u,y K(z) (f (x -h n z, y) -f (x, y)) dy dz (24)
-

{ z ≤A} R q (f (x -h n z, y) -f (x, y)) dy dz. (25)
Assumption (A2) together with [START_REF] Devroye | The uniform convergence of the Nadaraya-Watson regression function estimate[END_REF], and the dominated convergence theorem ensure that both integrals in ( 24) and (25) converge to 0. We deduce that for n large enough,

{ z ≤A}×R q e ( u,y +v)K(z) -1 [f (x -h n z, y) -f (x, y)] dzdy ≤ ǫ 2 , (26)
so that (20) follows from ( 23) and ( 26).

Let us now consider R

n,x ; since c n is between 1 and E e Zn , we get 1

c n ≤ max 1, 1 E (e Zn )
.

By Jensen's inequality, we obtain

1 E [e Zn ] ≤ 1 e E[Zn] .
Observe that

|E (Z n )| = E ( u, Y n + v) K x -X n h n ≤ R d ×R q u y K x -s h n f (s, y)dsdy + |v| R d ×R q K x -s h n f (s, y)dsdy ≤ h d n u R d |K(z)| R q y f (x -h n z, y)dy dz + |v| R d |K(z)| R q f (x -h n z, y)dy dz ,
which goes to 0 in view of ( 7) and since lim n→∞ h n = 0. We deduce that there exists

c ∈ R * + such that 1 c 2 n ≤ c.
Noting that by [START_REF] Devroye | The uniform convergence of the Nadaraya-Watson regression function estimate[END_REF],

E e Zn -1 ≤ h d n R d ×R q | u, y + v| |K(z)|e | u,y +v||K(z)| f (x -h n z, y)dydz ≤ Bh d n ,
where B is a constant. It follows that lim n→∞ R (1) n,x (u, v) = 0, which proves Lemma 3.

Proof of Lemma 2

Similarly as the proof of Lemma 1, for any w = (u, v) ∈ R q × R, set Ψn (x) = ( mn (x), gn (x)) ,

Λn,x (w) = 1 nh d n log E exp nh d n w, Ψn (x) .
When h n = cn -a , c > 0 and 0 < a < 1/d, assume for the moment that

lim n→∞ Λn,x (u, w) = Ψa,x (u, v), (27)
where Ψa,x is defined in [START_REF] Louani | Some large deviations limit theorems in conditionnal nonparametric statistics[END_REF]. The conclusion of Lemma 2 follows from Proposition 4 and again the Gärtner-Ellis Theorem.

Let us now prove (27). Set

M i = [ u, Y i + v] K x -X i h i , then, for (u, v) ∈ R q × R, Λn,x (u, v) = 1 nh d n log E exp n i=1 M i h d n h d i = 1 nh d n n i=1 log E exp M i h d n h d i .
By Taylor expansion, there exists b i,n between 1 and

E exp M i h d n h d i such that log E exp M i h d n h d i = E exp M i h d n h d i -1 - 1 2b 2 i,n E exp M i h d n h d i -1 2 .
Noting that h n = cn -a with c > 0 and a ∈]0, 1/d[, Λn,x can be rewritten as

Λn,x (u, v) = 1 h d n n i=1 E exp M i h d n h d i -1 - 1 2nh d n n i=1 1 b 2 i,n E exp M i h d n h d i -1 2 = 1 n n i=1 i n -ad R d ×R q e ( i n ) ad ( u,y +v)K(z) -1 f (x, y)dzdy -R (1) n,x (u, v) + R (2) n,x (u, v), with R(1) n,x (u, v) = 1 2nh d n n i=1 1 b 2 i,n E exp M i h d n h d i -1 2 R(2) n,x (u, v) = 1 nh d n n i=1 h d i R d ×R q e h d n h d i ( u,y +v)K(z) -1 [f (x -h i z, y) -f (x, y)] dzdy. Since b i,n is between 1 and E exp M i h d n h d i , we have 1 b i,n ≤ max            1, 1 E e M i h d n h d i           
. By Jensen's inequality, we obtain

1

E e M i h d n h d i ≤ 1 e E M i h d n h d i .
Observe that

E M i h d n h d i ≤ h d n h d i R d | u, y + v| K x -z h i f (z, y)dzdy ≤ h d n u R d ×R q |K(z)| R q y f (x -h i z, y)dy dz + |v| R d |K(z)| R q f (x -h i z, y)dy dz ,
which goes to 0 in view of ( 7) and since lim n→∞ h n = 0. We deduce that the sequence

E exp M i h d n h d i is bounded, so that there exists c > 0 such that 1 b 2 i,n ≤ c, and thus R(1) n,x (u, v) ≤ c 2nh d n n i=1 E e M i h d n h d i -1 2 .
Now, in view of [START_REF] Devroye | The uniform convergence of the Nadaraya-Watson regression function estimate[END_REF], and since K is bounded integrable, we have

E e M i h d n h d i -1 ≤ E M i h d n h d i e M i h d n h d i ≤ h d n h d i R d | u, y + v| K x -s h i e | u,y +v| K x-s h i f (s, y)dsdy ≤ h d n R d ×R q | u, y + v| |K (z)| e | u,y +v||K(z)| f (x -h i z, y)dzdy ≤ Bh d n , where B is a constant. Thus R(1) n,x (u, v) ≤ cB 2 2 h d n ,
and

lim n→∞ R(1) n,x (u, v) = 0.
Let us now consider R

n,x . Set A > 0 and ǫ > 0 ; we then have R [START_REF] Bingham | Regular variation[END_REF] n,x (u, v)

= 1 nh d n n i=1 h d i { z ≤A}×R q e h d n h d i ( u,y +v)K(z) -1 [f (x -h i z, y) -f (x, y)] dzdy + 1 nh d n n i=1 h d i { z >A}×R q e h d n h d i ( u,y +v)K(z) -1 [f (x -h i z, y) -f (x, y)] dzdy = I + II. Since |e t -1| ≤ |t|e |t| , it follows that |II| ≤ 1 n n i=1 { z >A}×R q | u, y + v| |K (z) |e | u,y +v||K(z)| |f (x -h i z, y) -f (x, y)| dzdy.
Using the same argument as in ( 22), it holds that

|II| ≤ ǫ 2 .
Now, for I, we write

I = 1 nh d n n i=1 h d i { z ≤A}×R q e h d n h d i ( u,y +v)K(z) [f (x -h i z, y) -f (x, y)] dzdy - 1 nh d n n i=1 h d i { z ≤A}×R q [f (x -h i z, y) -f (x, y)] dzdy.
On the one hand, Assumption (A2) with u = 0 ensures that lim

i→∞ { z ≤A}×R q [f (x -h i z, y) -f (x, y)] dzdy = 0.
Moreover, since ad < 1, (15) ensures that

lim n→∞ 1 nh d n n i=1 h d i { z ≤A}×R q [f (x -h i z, y) -f (x, y)] dzdy = 0,
so that for n large enough,

1 nh d n n i=1 h d i { z ≤A}×R q [f (x -h i z, y) -f (x, y)] dzdy ≤ ǫ 4 . (28)
On the other hand, since for i ≤ n, 0 ≤ h d n h d i ≤ 1, by Assumption (A'1), there exists n 0 ∈ N such that for any i > n 0 ,

R q e h d n h d i ( u,y +v)K(z) [f (x -h i z, y) -f (x, y)] dy ≤ ǫ 8(2 -ad) z ≤A dz ∀n > i.
Noting that by [START_REF] Devroye | The uniform convergence of the Nadaraya-Watson regression function estimate[END_REF], for any α ∈ [0, 1],

sup t R q e α u,y f (t, y)dy ≤ sup t R q e u y f (t, y)dy < ∞.
Since ad < 1, by [START_REF] Nadaraya | On estimating regression[END_REF], we get for n sufficiently large

1 nh d n n i=n 0 +1 h d i { z ≤A}×R q e h d n h d i ( u,y +v)K(z) [f (x -h i z, y) -f (x, y)] dydz ≤ ǫ 8 .
Now, for n large enough, in view of ( 7),

1 nh d n n 0 i=1 h d i { z ≤A}×R q e h d n h d i ( u,y +v)K(z) [f (x -h i z, y) -f (x, y)] dydz ≤ 1 nh d n n 0 i=1 h d i { z ≤A}×R q e | u,y +v||K(z)| |f (x -h i z, y) -f (x, y)| dydz ≤ ǫ 8 .
It follows that for n large enough,

1 nh d n n i=1 h d i { z ≤A}×R q e h d n h d i ( u,y +v)K(z) [f (x -h i z, y) -f (x, y)] dydz ≤ ǫ 4 . ( 29
)
The combination of ( 28) and ( 29) ensures that |I| ≤ ǫ 2 , which ensures that

lim n→∞ R(2) n,x (u) = 0.
Hence, (27) follows from analysis considerations.

Proof of Theorems 1 and 3

Let us consider the following functions defined as:

H 1 : R q × R * → R q (α, β) → α β ,
and

H 2 : R q × R → R q (α, β) → α β if β = 0 0 otherwise.

Proof of Theorem 1

i) Let U be an open subset of R q , we have

1 nh d n log P [r n (x) ∈ U ] = 1 nh d n log P (m n (x), g n (x)) ∈ H -1 2 (U ) . ( 30
)
Observe that

H -1 1 (U ) ⊂ H -1 2 (U ) and H -1 1 (U ) is an open subset on R q × R * which is open, it follows that H -1 1 (U ) is an open subset on R q × R. We deduce from (30) that 1 nh d n log P [r n (x) ∈ U ] ≥ 1 nh d n log P (m n (x), g n (x)) ∈ H -1 1 (U ) .
The application of Lemma 1 ensures that lim inf

n→∞ 1 nh d n log P [r n (x) ∈ U ] ≥ lim inf n→∞ 1 nh d n log P (m n (x), g n (x)) ∈ H -1 1 (U ) ≥ - inf (t 1 ,t 2 )∈H -1 1 (U ) I x (t 1 , t 2 ) = -inf s∈U J * (s),
and the first part of Theorem 1 is proved.

ii) Let V be a closed subset of R q , we have

1 nh d n log P [r n (x) ∈ V ] = 1 nh d n log P (m n (x), g n (x)) ∈ H -1 2 (V ) ≤ 1 nh d n log P (m n (x), g n (x)) ∈ H -1 2 (V ) . Now, observe that H -1 2 (V ) = H -1 1 (V ) ∪ A where A ⊂ R q × {0}
and 0, 0 ∈ A (since for any s ∈ R q , ( 0, 0) ∈ H -1 2 (s)). The application of Lemma 1 again ensures that lim sup

n→∞ 1 nh d n log P [r n (x) ∈ V ] ≤ - inf (s,t)∈H -1 1 (V )∪A I x (s, t) ≤ - inf (s,t)∈H -1 1 (V )∪{( 0,0)} I x (s, t) ≤ -inf s∈V, t∈R I x (st, t) ≤ -inf s∈V, t∈R
Îx (s, t)

≤ -inf s∈V J(s),
where the second inequality comes from Condition (C); this concludes the proof of Theorem 1.

Proof of Theorem 3

Applying Lemma 2, Theorem 3 is proved by following the same approach as for the proof of Theorem 1 with replacing m n , g n , J * and J by mn , gn , J * a and Ja respectively.

Proof of Theorem 2

Set

B n (x) = 1 g(x) (m n (x) -m(x)) - r(x) g(x) (g n (x) -g(x)) .
Let us at first state the two following lemmas.

lim n→∞ v n E(B n (x)) = 0. ( 31
)
We first show that how Theorem 2 can be deduced from the application of Lemmas 4 and 5, and then prove Lemmas 4 and 5 successively. Let us first note that, for x such that g n (x) = 0, we have:

r n (x) -r(x) = m n (x) g n (x) - m(x) g(x) = (m n (x) -m(x)) g(x) + (g(x) -g n (x)) m(x) g n (x)g(x) = B n (x) g(x) g n (x) .
It follows that, for any δ > 0, we have

P v n (r n (x) -r(x)) -B n (x) > δ ≤ P v n B n (x) g(x) g n (x) -1 > δ and g n (x) = 0 + P [g n (x) = 0] ≤ P [ √ v n B n (x) > δ] + P [ √ v n |g(x) -g n (x)| > δ |g n (x)|] + P |g(x) -g n (x)| > g(x) 2 ≤ P [ √ v n B n (x) > δ] + P √ v n |g(x) -g n (x)| > δ |g n (x)| and g n (x) g(x) > 1 2 +P g n (x) g(x) ≤ 1 2 + P |g n (x) -g(x)| > g(x) 2
≤ P [ √ v n B n (x) > δ] + P √ v n |g(x) -g n (x)| > δ g(x) 2 + P g(x) -g n (x) ≥ g(x) 2 +P |g n (x) -g(x)| > g(x) 2 .
Since lim n→∞ v n = ∞, it follows that, for n large enough,

P v n (r n (x) -r(x)) -B n (x) > δ ≤ 4 max P [ √ v n B n (x) > δ] ; P √ v n |g(x) -g n (x)| > δ g(x) 2 ,
and thus

v 2 n nh d n log P v n (r n (x) -r(x)) -B n (x) > δ ≤ v 2 n nh d n log 4 + max v 2 n nh d n log P [ √ v n B n (x) > δ] ; v 2 n nh d n log P √ v n |g(x) -g n (x)| > δ g(x) 2 .
Now, since the sequence (v n B n (x)) satisfies a LDP with speed

nh d n v 2 n
and good rate function G x , there exists c 1 > 0 such that lim sup

n→∞ v n nh d n log P [ √ v n B n (x) > δ] ≤ -c 1 .
Moreover, the application of Theorem 1 in Mokkadem et al. [START_REF] Mokkadem | Large and moderate deviations principles for kernel estimation of a multivariate density and its partial derivatives[END_REF] guarantees the existence of c 2 > 0 such that lim sup

n→∞ v n nh d n log P √ v n |g(x) -g n (x)| > δ g(x) 2 < -c 2 .
We thus deduce that

lim n→∞ v 2 n nh d n log P v n (r n (x) -r(x)) -B n (x) > δ = -∞,
which means that the sequences (v n (r n (x)r(x))) and (v n B n (x)) are exponentially contiguous. Theorem 2 thus follows.

Proof of Lemma 4

For any u ∈ R q , set

Γ n,x (u) = v 2 n nh d n log E exp nh d n v n u, B n (x) -E (B n (x)) , Φ x (u) = 1 2g 2 (x) R d ×R q u, y -r(x) 2 K 2 (z)f (x, y)dzdy = u T Σ x u 2g(x) R d K 2 (z)dz.
To prove Lemma 4, it suffices to show that, for all u ∈ R q , lim n→∞

Γ n,x (u) = Φ x (u).
As a matter of fact, since Φ x is a quadratic function, Lemma 4 then follows from the application of the Gärtner-Ellis Theorem. For

u ∈ R q , set Ẑi = u, Y i -r(x) K x -X i h n ,
and note that

Γ n,x (u) = v 2 n nh d n log E exp 1 v n g(x) n i=1 Ẑi -E( Ẑi )
.

Since (X i , Y i ), i = 1, . . . , n are independent and identically distributed, it holds that

Γ n,x (u) = v 2 n h d n log E e Ẑn vng(x) - v n h d n g(x)
E( Ẑn ). Now, we follow the same lines as in the proof of Lemma 3. A Taylor's expansion ensures that there exists ĉn between 1 and E e Ẑn vng(x)

such that Γ n,x (u) = v 2 n h d n E e Ẑn vng(x) -1 - Ẑn v n g(x) -R(1) n,x (u) = v 2 n R d ×R q e 1 vng(x) u,y-r(x) K(z) -1 - 1 v n g(x) u, y -r(x) K (z) f (x, y)dzdy -R(1) n,x (u) + R(2) n,x (u), with R(1) n,x (u) = v 2 n 2ĉ 2 n h d n E e Ẑn vng(x) -1 2 R(2) n,x (u) = v 2 n R d ×R q e 1 vng(x) u,y-r(x) K(z) -1 - 1 v n g(x) u, y -r(x) K (z) f (x -h n z, y) -f (x, y) dzdy, and
1 ĉn ≤ max        1, 1 E e Ẑn vng(x)        . Noting that E Ẑn v n g(x) ≤ h d n u v n g(x) R d |K(z)| R q y f (x -h n z, y)dy dz + h d n u r(x) v n g(x) R d |K(z)| R q f (x -h n z, y)dy dz. (32)
It follows from [START_REF] Devroye | The uniform convergence of the Nadaraya-Watson regression function estimate[END_REF] that,

E Ẑn v n g(x) → 0.
We deduce that there exists

c ′ ∈ R * + such that 1 ĉ2 n ≤ c ′ ,
and thus, in view of [START_REF] Devroye | The uniform convergence of the Nadaraya-Watson regression function estimate[END_REF],

R(1) n,x (u) ≤ c ′ 2 v 2 n h d n R d ×R q e 1 vng(x) u,y-r(x) K x-s hn -1 f (s, y)dsdy 2 ≤ c ′ 2g 2 (x) h d n R d ×R q | u, y -r(x) K(z)| e 1 g(x) u,y-r(x) K(z) f (x -h n z, y)dzdy 2 ≤ c ′ e 2 K ∞ g(x) u r(x) 2g 2 (x) h d n R d ×R q | u, y -r(x) K(z)| e K ∞ g(x) u y f (x -h n z, y)dzdy 2 ≤ Bh d n , where B is a constant, so that lim n→∞ R(1)
n,x (u) = 0.

On the other hand, since ∀x ∈ R, e x -1

-x = x 2 2 + x 3 6 d(x), with d(x) ≤ e |x| , we get R(2) n,x (u) = 1 2g 2 (x) R d ×R q u, y -r(x) 2 K 2 (z) [f (x -h n z, y) -f (x, y)] dzdy + R n,x (u), (33)
with

|R n,x (u)| ≤ 1 6v n g 3 (x) R d ×R q u, y -r(x) 3 K 3 (z) e 1 g(x) | u,y-r(x) K(z)| |f (x -h n z, y) -f (x, y)| dzdy.
It follows from ( 7) that R n,x converges to 0. Applying then (A2) and (A3), we find

lim n→∞ R(2) n,x (u) = 0.
Finally, we have

Γ n,x (u) = v 2 n R d ×R q e 1 vng(x) u,y-r(x) K(z) -1 - 1 v n g(x) u, y -r(x) K (z) f (x, y)dzdy -R(1) n,x (u) + R(2) n,x (u) = Φ x (u) -R(1) n,x (u) + R(2) n,x (u) + R(3) n,x (u), with R(3) n,x (u) = v 2 n R d ×R q e 1 vng(x) u,y-r(x) K(z) -1 - u, y -r(x) K (z) v n g(x) - u, y -r(x) 2 K 2 (z) 2v 2 n g 2 (x)
f (x, y)dzdy.

By the majoration e x -1x -x 2 2 ≤ x 3 6 d(x) , we get

R(3) n,x (u) ≤ 1 6v n g 3 (x) R d ×R q u, y -r(x) 3 K 3 (z) e

Proof of Lemma 5

Observe that

E (B n (x)) = 1 g(x) [E (m n (x)) -m(x)] - r(x) g(x) [E (g n (x)) -g(x)] . (34)
Since

E (m n (x)) -m(x) = 1 h d n E Y 1 K x -X 1 h n -m(x) = 1 h d n R d ×R q yK x -z h n f (z, y)dzdy -m(x) = 1 h d n R d m(z)K x -z h n dz -m(x) = R d K(y) [m(x -h n y) -m(x)] dy,
Assumptions (A5)i), (A5)iii) and a Taylor's expansion of m of order p ensure that

E (m n (x)) -m(x) = O (h p n ) . (35)
Similarly, we have

E (g n (x)) -g(x) = O (h p n ) . (36)
We deduce from (34), (35), and (36) that

E (B n (x)) = O (h p n ) ,
and thus Lemma 5 follows from Assumption (A5)ii).

Proof of Theorem 4

Set Bn (x) = 1 g(x)

( mn (x)m(x)) -r(x) g(x) (g n (x)g(x)) , and, for any

u ∈ R q , Γn,x (u) = v 2 n nh d n log E exp nh d n v n u, Bn (x) -E Bn (x) , Φa,x (u) = 1 2(1 + ad)g 2 (x) R d ×R q u, y -r(x) 2 K 2 (z)f (x, y)dzdy = 1 1 + ad u T Σ x u 2g(x) R d K 2 (z)dz.
By following the steps of the proof of Lemma 4 and by using the property [START_REF] Nadaraya | On estimating regression[END_REF]

Now, following the proof of Lemma 5, we have

E ( mn (x)) -m(x) = 1 n n i=1 1 h d i R d ×R q yK x -z h i f (z, y)dzdy -m(x) = 1 n n i=1 R d K(y) [m(x -h i y) -m(x)] dy.
Here again, Assumptions (A5)i), (A5)iii) and a Taylor's expansion of m of order p ensure that

E ( mn (x)) -m(x) = O 1 n n i=1 h p i .
and similarly,

E (g n (x)) -g(x) = O 1 n n i=1 h p i , thus v n E Bn (x) = O v n n n i=1 h p i .
• If ap < 1, since (h n ) varies regularly with exponent (-a), we have, in view of [START_REF] Nadaraya | On estimating regression[END_REF] and Assumption (A5)ii),

v n n n i=1 h p i = O v n n [nh p n ] = o(1).
• If ap > 1, we have i h p i < ∞ and thus, since

v n = o(nh d n ), we get v n n n i=1 h p i = O v n n = o(1).
• In the case ap = 1, let L be the slowly varying function such that h n = n -a L(n), and set ε > 0 small enough. Since a(pε) < 1, we have h p n = o(h p-ε n ), and in view of ( 15) and (A4),

v n n n i=1 h p i = o v n h p-ε n = o nh d+p-ε n = o n 1-a(d+p-ε) [L(n)] d+p-ε = o n -a(d-ε) [L(n)] d+p-ε = o(1).
We thus deduce that

lim n→∞ v n E( Bn (x)) = 0. (39)
To conclude the proof of Theorem 4, we follow the same lines as for the proof of Theorem 2 (see Section 3.3), except that we apply (38) instead of Lemma 4, (39) instead of Lemma 5, and Theorem 1 in Mokkadem et al. [START_REF] Mokkadem | Large and moderate deviations principles for recursive kernel estimators of a multivariate density and its partial derivatives[END_REF] instead of Theorem 1 in Mokkadem et al. [START_REF] Mokkadem | Large and moderate deviations principles for kernel estimation of a multivariate density and its partial derivatives[END_REF]. Let us now prove (37). For u ∈ R q , set

T i = u, Y i -r(x) K x -X i h i , and note that Γn,x (u) = v 2 n nh d n log E exp h d n v n g(x) n i=1 1 h d i [T i -E(T i )] .
Since (X i , Y i ), i = 1, . . . , n are independent and identically distributed, it holds that

Γn,x (u) = v 2 n h d n n i=1 log E e h d n T i vng(x)h d i - v n ng(x) n i=1 1 h d i E(T i ).
By Taylor expansion, there exists c i,n between 1 and E e

h d n T i vng(x) h d i such that log E e h d n T i vng(x) h d i = E e h d n T i vng(x) h d i -1 - 1 2c 2 i,n E e h d n T i vng(x) h d i -1 2 ,
and Γn,x can be rewritten as

Γn,x (u) = v 2 n nh d n n i=1 E e T i h d n vng(x)h d i -1 - v 2 n 2nh d n n i=1 1 c 2 i,n E e T i h d n vng(x)h d i -1 2 - v n ng(x) n i=1 1 h d i E(T i ).
A Taylor expansion implies again that there exists c ′ i,n between 0 and

T i h d n vng(x)h d i such that E e T i h d n vng(x)h d i -1 = h d n v n g(x)h d i E(T i ) + 1 2 h d n v n g(x)h d i 2 E(T 2 i ) + 1 6 h d n v n g(x)h d i 3 E(e c ′ i,n T 3 i ). Therefore, Γn,x (u) = 1 2g 2 (x) 1 nh -d n n i=1 1 h d i R d ×R q u, y -r(x) 2 K 2 (z)f (x, y)dzdy + R(1) n,x (u) + R(2) n,x (u), ( 40
) with R(1) n,x (u) = 1 6 h 2d n v n g 3 (x) 1 n n i=1 1 h 2d i E(e c ′ i,n T 3 i ) - v 2 n 2nh d n n i=1 1 c 2 i,n E e T i h d n vng(x)h d i -1 2 , R(2) n,x (u) = h d n 2g 2 (x) 1 n n i=1 1 h d i R d ×R q u, y -r(x) 2 K 2 (z) [f (x -h i z, y) -f (x, y)] dzdy.
In view of [START_REF] Nadaraya | On estimating regression[END_REF], the first term in the right-hand-side of (40) converges to Φa,x . It remains to prove that R(1) n,x and R(2) n,x converge to 0. We have

E h d n T i v n g(x)h d i ≤ h d n v n g(x)h d i R d ×R q u, y -r(x) K x -s h i f (s, y)dsdy ≤ h d n v n g(x) R d ×R q | u, y -r(x) K (z)| f (x -h i z, y)dzdy.
In view of ( 7), the integral is bounded, thus

lim n→∞ sup i≤n E h d n T i v n g(x)h d i = 0, so that, there exists c > 0 such that 1 c 2 i,n ≤ c.
Now, on the one hand, since |e t -1| ≤ |t|e |t| , and in view of ( 7) and ( 16), we have

E e h d n T i vng(x)h d i -1 ≤ h d n v n g(x) R d ×R q | u, y -r(x) K(z)| e c 1 g(x) u,y-r(x) K(z) f (x -h i z, y)dzdy ≤ B 1 h d n v n g(x) ,
where B 1 and c are constants. We deduce that

lim n→∞ v 2 n 2nh d n n i=1 1 c 2 i,n E e T i h d n vng(x)h d i -1 2 = 0.
On the other hand,

E T 3 i e c ′ i,n ≤ E |T i | 3 e |c ′ i,n | ≤ h d i R d ×R q | u, y -r(x) K(z)| 3 e c g(x) | u,y-r(x) K(z)| f (x -h i z, y)dzdy ≤ B 2 h d i ,
where B 2 is a constant. Thus,

h 2d n 6nv n g 3 (x) n i=1 1 h 2d i E(e c ′ i,n T 3 i ) ≤ h d n 6v n g 3 (x) B 2 nh -d n n i=1 h -d i . Since lim n→∞ h d n v n = 0, (15) ensures that lim n→∞ h 2d n 6v n g 3 (x) 1 n n i=1 1 h 2d i E(e c ′ i,n T 3 i ) = 0, which proves that lim n→∞ R(1) n,x (u) = 0.
Finally, using ( 15), (A2) and (A3), we have

lim n→∞ R(2) n,x (u) = lim n→∞ 1 2g 2 (x) n i=1 h -d i nh -d n 1 n i=1 h -d i n i=1 h -d i R d ×R q u, y -r(x) 2 K 2 (z) [f (x -h i z, y) -f (x, y)] dzdy = 0,
which proves (37).

3.5 Proof of Propositions 3 and 4

Proof of Proposition 3

• The strict convexity of Ψ x follows from its definition, since for any γ ∈]0, 1[, and (u

, v) = (u ′ , v ′ ), Ψ x γ (u, v) + (1 -γ) u ′ , v ′ = Ψ x γu + (1 -γ)u ′ , γv + (1 -γ)v ′ = R d ×R q e [ γu+(1-γ)u ′ ,y +γv+(1-γ)v ′ ]K(z) -1 f (x, y)dzdy < γ R d ×R q
e ( u,y +v)K(z) -1 f (x, y)dzdy

+(1 -γ) R d ×R q e ( u ′ ,y +v ′ )K(z) -1 f (x, y)dzdy,
where the last inequality follows from the fact that x → e x is strictly convex. Since |e t -1| ≤ |t|e |t| ∀t ∈ R and K is bounded and integrable, [START_REF] Devroye | The uniform convergence of the Nadaraya-Watson regression function estimate[END_REF] imply that

R d ×R q e ( u,y +v)K(z) -1 f (x, y) dzdy ≤ R d ×R q |( u, y + v) K(z)| e |( u,y +v)K(z)| f (x, y)dzdy ≤ e |v| K ∞ u R d |K(z)| dz R q y e u y K ∞ f (x, y)dy +e |v| K ∞ |v| R d |K(z)| dz R q e u y K ∞ f (x, y)dy < ∞,
which ensures the existence of Ψ x . Next, set

h x (u, v, y, z) = e ( u,y +v)K(z) -1 f (x, y).
Since h x is differentiable with respect to (u, v) and ∇h x (u, v, y, z) = ye ( u,y +v)K(z) K(z)f (x, y) e ( u,y +v)K(z) K(z)f (x, y) ,

using Assumption (A1) and ( 7), it can be seen that Ψ x is differentiable on R q × R. Since Ψ x is a smooth convex on R q × R, it follows that Ψ x is essentially smooth so that I x is a good rate function on R q × R (see Dembo and Zeitouni [START_REF] Dembo | Large deviations techniques and applications[END_REF]), which proves the first part of Proposition 3. Now, observe that [START_REF] Rockafellar | Convex analysis[END_REF]). Thus, Part 2 of Proposition 3 follows from Theorem 26.5 of Rockafellar [START_REF] Rockafellar | Convex analysis[END_REF].

• D(Ψ x ) = R q × R,
• Let us now assume that λ(S -) = 0. Thus

Ψ x (u, v) = R d ×R q e u T yK(z) e vK(z) -1 ½ S + (z)f (x, y)dzdy.
For each u ∈ R q , the function v → e u T yK(z) e vK(z) -1 ½ S + (z)f (x, y) is increasing in v and goes to -f (x, y) when v → -∞. Thus lim v→-∞ Ψ x (u, v) = -g(x)λ(S + ) and I x ( 0, 0) = g(x)λ(S + ). Now, when t 1 = 0, let us show that

I x (t 1 , 0) = +∞.
Let M > 0, ǫ > 0 and set u = (M + ǫ)t 1 / t 1 2 . Let v ∈ R such that

-Ψ x (u, v) > g(x)λ(S + ) -ǫ if λ(S + ) < ∞ -Ψ x (u, v) > M if λ(S + ) = ∞.
Then, on the one hand, when λ(S + ) < ∞, we have u T t 1 -Ψ x (u, v) ≥ M + ǫ + g(x)λ(S + )ǫ > M.

On the other hand, when λ(S + ) = ∞, we get

u T t 1 -Ψ x (u, v) ≥ M + ǫ + M > M.
It follows that sup u,v u T t 1 -Ψ x (u, v) = +∞.

Proof of Proposition 4

Following the same lines of the proof of Proposition 3, we prove Proposition 4. When λ(S -) = 0, for each u ∈ R q and s ∈]0, 1], the map v → s -ad e s ad u T yK(z) e vK(z) -1 ½ S + (z)f (x, y) is in- creasing in v and goes to -s -ad f (x, y) when v → -∞. We deduce that lim v→-∞ Ψa,x (u, v) = -g(x)λ(S + ) 1 0 s -ad = -g(x)λ(S + )/(1ad) and I x ( 0, 0) = g(x)λ(S + )/(1ad). We first show that Ĝ is a compact subset of R q × R. First, observe that since I x is a good rate function, G is a compact subset of R q × R.

Let us define the following function

F : R q × R → R q × R (s, t) → (st, t).
Observe that F is continuous and Ĝ = F -1 (G). We deduce that Ĝ is a closed subset of R q × R. Now, let (s n , t n ) be a sequence of real numbers of Ĝ, there exists (x n , y n ) ∈ G such that (x n , y n ) = F (s n , t n ) = (s n t n , t n ) ∈ G.

The compactness of G on R q × R ensures that there exists a sequence of real numbers (x n k , y n k ) ∈ G such that (x n k , y n k ) → (x 0 , y 0 ) as k → ∞, where (x 0 , y 0 ) ∈ G. Therefore, (s n k t n k , t n k ) → (x 0 , y 0 ) as k → ∞.

Noting that Condition (C) ensures that ∀s ∈ R q , I x (s, 0) ≥ I x ( 0, 0) > α so that (s, 0) / ∈ G. It follows that y 0 = 0, and thus t n k → y 0 and s n k → s 0 as k → ∞, where s 0 = x 0 /y 0 . We deduce that (s n k , t n k ) → (s 0 , y 0 ) as k → ∞, so that (s 0 , y 0 ) ∈ Ĝ. Thus Ĝ is a compact set. Now we claim that the set A = {s, J(s) ≤ α} is the image of Ĝ by the continuous map π : (s, t) → s, and thus it is a compact. Indeed, clearly π( Ĝ) ⊂ A. For the opposite inclusion, consider α < α ′ < I x 0, 0 ; the set Ĝ′ = Îx (s, t) ≤ α ′ is compact. Let s 0 ∈ A, since J(s 0 ) ≤ α, we have J(s 0 ) = inf (s 0 ,t)∈ Ĝ′ Îx (s 0 , t); by compacity, there exists t 0 such that J(s 0 ) = Îx (s 0 , t 0 ); (s 0 , t 0 ) ∈ Ĝ and π(s 0 , t 0 ) = s 0 , thus A ⊂ π( Ĝ).

• If α ≥ I x ( 0, 0), let s ∈ R q , we have J(s) ≤ I x (st, t) ∀t ≤ I x ( 0, 0) ≤ α.

We deduce that R q ⊆ {J(s) ≤ α} and the second part of Proposition 1 (i) follows.

(ii) It is an obvious consequence of (i) and the definitions of J and J * .

(iii) Assume that J * (s) < ∞.

• If I x ( 0, 0) > inf t I x (st, t), then inf t I x (st, t) = inf t =0 I x (st, t), so that J(s) = J * (s).

• If I x ( 0, 0) = inf t I x (st, t), since J * (s) < ∞, there exists t 0 = 0 such that I x (st 0 , t 0 ) < ∞.

By the convexity of I x , we have for any ν ∈]0, 1], I x (st 0 ν, t 0 ν) < ∞ and I x (st 0 ν, t 0 ν) ≤ νI x (st 0 , t 0 ) + (1ν)I x ( 0, 0).

We deduce that 0 ≤ I x (st 0 ν, t 0 ν) -I x ( 0, 0) ≤ ν I x (st 0 , t 0 ) -I x ( 0, 0) , and if we take ν → 0, the third part of Proposition 1 follows.

(iv) Let us suppose that α < I x ( 0, 0) and let s ∈ {J * (s) ≤ α}, then we have J * (s) < ∞. We deduce from (iii) that J(s) = J * (s). It follows that J(s) ≤ α, which ensures that s ∈ {J(s) ≤ α}. Conversely, if s ∈ {J(s) ≤ α}, then

I x ( 0, 0) > inf t I x (st, t), so that inf t I x (st, t) = inf t =0 I x (st, t).
That is J(s) = J * (s). Therefore, J * (s) ≤ α, which ensures that s ∈ {J * (s) ≤ α}, and thus Proposition 1 is proved.

Proof of Proposition 2

Proposition 2 is proved by following the same approach as for the proof of Proposition 1 with replacing I x , J and J * by Ĩa,x , Ja and J * a respectively.

Example 3 :

 3 A negative kernel without symmetry assumption on f , and for d = q = 1 If the kernel K can be written as K = ½ D -½ D ′ where D and D ′ are two subsets of R such that D ∩ D ′ = ∅ and λ(D)λ(D ′ ) = 1, then Condition (C) holds. As a matter of fact, we then have Ψ x (u, v) = R×R e (uy+v)K(z) -1 f (x, y)dzdy = D×R e uy+v -1 f (x, y)dzdy + D ′ ×R e -uy-v -1 f (x, y)dzdy = e v λ(D) R e uy f (x, y)dyλ(D) + λ(D ′ ) g(x) + e -v λ(D ′ ) R e -uy f (x, y)dy.

Lemma 4 2 nLemma 5

 425 Under the assumptions of Theorem 2, the sequence (v n (B n (x) -E (B n (x)))) satisfies a LDP with speed nh d n v and good rate function G x . Under the assumptions of Theorem 2,

3. 3 . 1 d n v 2 n

 312 Proof of Theorem 2 Lemmas 4 and 5 imply that the sequence (v n B n (x)) satisfies a LDP with speed nh and good rate function G x . To prove Theorem 2, we show that (v n (r nr)) and (v n B n ) are exponentially contiguous.

1

  vng(x) | u,y-r(x) K(z)| f (x, y)dzdy, and (7) ensures that lim n→∞ R(3) n,x (u) = 0, which concludes the proof of Lemma 4.

3. 6 2 3. 6 . 1 •

 6261 Proof of Propositions 1 and Proof of Proposition 1 (i) Let us prove the first part of Proposition 1. If α < I x ( 0, 0), setG = {(a, b) ∈ R q × R, I x (a, b) ≤ α} and Ĝ = {(s, t) ∈ R q × R, Îx (s, t) ≤ α}.

  Theorem 4 (Pointwise MDP for the semi-recursive kernel estimator of the regression) Assume that (h n ) varies regularly with exponent (-a) with a ∈]0, 1/d[, and satisfies[START_REF] Rao | Nonparametric functionnal estimation[END_REF]. Let (A1)-

(A5) hold. Then, the sequence (v n (r n (x)r(x))) satisfies a LDP with speed

nh d n v 2 n

and good rate function Ga,x defined in (6).

  and since Ψ x is strictly convex, it holds that the pair

• D(Ψ x ), Ψ x is a convex function of Legendre type. It follows that • D(I x ), I x is a convex function of Legendre type (See Rockafellar


Xingang Xu

Jihua Wang

Wenjiang Huang

Cunjun Li

Xiaoyu Song

Xiaodong Yang

Hao Yang

Winter wheat yield estimation coupling weight optimization combination method with remote sensing data from Landsat5 TM

Keywords: Weight optimization combination, remote sensing, yield estimation, winter wheat, Landsat5 TM

Crop yield models using different VIs (vegetation index) from remote sensing data show the various precision, but each of them can provide useful information related with yield. So it is very significant how to integrate the useful information of these models. In this study, a few of typical VIs, such as NDVI (Normalized Difference Vegetation Index), SR (Simple Ratio index), TCARI/OSAVI (Trans-formed Chlorophyll Absorption Ratio Index (TCARI), and Optimized Soil-Adjusted Vegetation Index (OSAVI)), NDWI (Normalized Difference Water Index) extracted from Landsat5 TM image covering Beijing region, are used to build yield modes of winter wheat, respectively. And then the Weight Optimization Combination (WOC) method is utilized to integrate the models by calculating optimized weights to form the combining model. It is proved that the combining model with WOC exhibits better performance with the slightly higher determination coefficient R 2 in comparison with each single yield models with four different VIs, respectively. The analysis of comparing the weights in the combining model with WOC indicates that the two VIs, SR and NDWI are more sensitive to winter wheat yield than the other two during the winter wheat jointing stage. The preliminary results of coupling the WOC method with remote sensing imply that WOC can be used to improve the accuracy of yield estimation based on remote sensing.

Introduction

Crop yield are impacted by a wide range of factors, not only natural environment conditions such as light, temperature, water, soil quality, terrain and so on, but also social factors, for instance, field management, agricultural machine, fertilizer, etc [START_REF] Rieger | Growth, yield, and yield components of winter wheat and the effects of tillage intensity, preceding crops, and N fertilisation[END_REF][START_REF] Xu | Research advances in crop yield estimation models based on remote sensing[END_REF]. So it is difficult, and even impossible to establish the yield estimating models which can include all impact factors of crop yield. Although the factors influencing crop yield are many and complex, these factors are not independent and affect each other. In different stage, there are different dominant factors impacting crop growth, the estimating models of crop yield can be set up by using the dominating factors according to some principals and methods, Therefore, all kinds of models can all provide various useful-information more or less to some extent. If the scattered models are synthetically utilized to form the combining model and estimate crop yield, the precision of crop yield estimation will improve, Weight Optimization Combination (WOC) is just an algorithm for solving the type of problem. WOC aims at making best use of useful information from each of single models by giving the optimal weights to the single models to form the combining model with LSSE (Least sum of square error), and is often used in economic prediction and decision analysis fields [START_REF] Tang | Study of computing method of combination forecasting[END_REF][START_REF] Tang | Study of combination forecasting error information matrix[END_REF][START_REF] Tang | An iterative algorithm for optimal combination forecasting of non-negative weights[END_REF], and seldom applied for yield estimation with remote sensing.

Remote sensing plays a unique role in estimating crop yield by its nondestructive, fast, and large-area-measurement characteristic. The typical vegetation indices (VIs) with different bands combination from remote sensing, such as NDVI [START_REF] Rouse | Monitoring the vernal advancements and retrogradation of natural vegetation[END_REF][START_REF] Gitelson | Spectral reflectance changes associated with autumn senescence of Aesculus hippocastanum L and Acer platanoides L leaves-spectral features and relation to chlorophyll estimation[END_REF], SR [START_REF] Jordan | Derivation of leaf area index form quality of light on the forest floor[END_REF], TCARI/OSAVI [START_REF] Daughtry | Estimating corn leaf chlorophyll concentration from leaf and canopy reflectance[END_REF][START_REF] Rondeaux | Optimization of soil-adjusted vegetation indices[END_REF][START_REF] Haboudane | Integrated narrow-band vegetation indices for prediction of crop chlorophyll content for application to precision agriculture[END_REF], NDWI [START_REF] Gao | NDWI-A Normalized Difference Water Index for Remote Sensing of Vegetation Liquid Water from Space[END_REF] can monitor crop growth status in different stages, indirectly provide useful information on crop yield, and be used to estimate crop yield in this study. Since different VI can detect crop growth status from different views, crop yield models with different VIs can also provide some useful information from different ways, so if we integrate the scattered models together, and form the combining model in order to make full use of the useful information from each of the single scattered models, the precision of crop yield estimation can improve more or less.

The objective of the study is to compare the performance of the above VIs for estimating yield of winter wheat, and then the WOC principle is applied to integrate the single yield estimating models with the VIs to form the combining models, finally assess the performance of the WOC method with remote sensing for estimating yield of winter wheat.

Material and Method

Study area and data acquisition

The study area covers Shunyi & Tongzhou District in Beijing, China. The plant area of winter wheat in the two districts accounts for about 50% of the whole Beijing. Fgure 1 showed the location of the study area (116.45~117.01°E, 40.0~40.3°N), the triangles in Fig. 1 represent the 30 measured yield plots in winter wheat field. Among the plots, there are 15 plots in Shunyi, and the other 15 in Tongzhou District.

Remote sensing data and preprocessing

In the study, a satellite image from Landsat5 TM which passed over the study area on April 15, 2009 is acquired to estimate yield of winter wheat for assessing the performance of WOC. Winter wheat is during the jointing stage. In order to eliminate some noises and improve data precision, the image is preprocessed by radiant calibration, atmospheric and geometrical corrections with ENVI software.

Vegetation Indices used in the study

In this analysis, the four vegetation indices, NDVI, SR, TCARI/OSAVI (the ratio of TCARI and OSAVI as a VI), and NDWI are utilized to estimate yield of .winter wheat in the study area in 2009. Table 1 showed the summary of the four vegetation indices, R is the land surface reflectance, and the subscripts green, red, nir, and swir were respectively band2, band3, band4 and band5 from the Landsat5 TM image.

NDVI (Normalized Difference Vegetation Index) should be the most extensively used VI which couples the reflection in the infrared with that in the red, and responds to changes in amount of biomass, chlorophyll and LAI. SR (Simple Ratio index) is directly formulated with the reflection in the infrared and that of the absorption in the red, and can enhance the contrast between soil and vegetation while minimizing the effect of the illumination conditions (Barnet and Guyot, 1991).

TCARI (Transformed Chlorophyll Absorption Ratio Index) is proposed to eliminate reflectance effect of background matters (soil and non-photosynthetic components) and to increase the sensitivity at chlorophyll content, and it is often formulated with 550,670, and 700nm in hyperspectral applications, the detailed description of the TCARI can be found in [START_REF] Daughtry | Estimating corn leaf chlorophyll concentration from leaf and canopy reflectance[END_REF]. In the study, in order to use the multispectral wide-band image, the calculation of TCARI is modified with bands from Landsat5 TM as the formula in Table 1. OSAVI (Optimized Soil-Adjusted Vegetation Index) belongs to SAVI family [START_REF] Huete | A soil-adjusted vegetation index (SAVI)[END_REF], and is developed to further reduce background soils. The ratio of TCARI and OSAVI as a new index is proposed by [START_REF] Haboudane | Integrated narrow-band vegetation indices for prediction of crop chlorophyll content for application to precision agriculture[END_REF], which integrate advantages of indices minimizing soil background effects and indices that are sensitive to chlorophyll concentration. So the index of TCARI/OSAVI can monitor crop nutritional status, and further assess crop yield.

NDWI (Normalized Difference Water Index) is based on liquid water absorption band near 1240nm, and a relative non-absorption reference band near 860nm [START_REF] Gao | NDWI-A Normalized Difference Water Index for Remote Sensing of Vegetation Liquid Water from Space[END_REF], so it can detect crop water stress. In the analysis, NDWI is modified by the combination of the reflectance in the Nir with that in the Swir (short infrared) from Landsat5 TM image bands.

In the present study, the four indices are relatively related with crop growth, nutrition and water, all of which interact with crop yield, so it is possible that integrating the single models with these indices establishes the combining model to estimates crop yield with higher precision.

R R R R R R R R R R           

Weight optimization combination method

Weight optimization combination (WOC) is a method which computationally gives optimal weights of different models solving the same problem to form the combination model with the aim of the least errors [START_REF] Xu | Estimation of crop yield based on weight optimization combination and multi-temporal remote sensing data[END_REF]. The principle of WOC is as the following: There are different N models constructed with n samples for settling the same problem, where both N and n are the same as the following mentioned, respectively. The estimated value of the combination model formed by N models for j sample is defined as the following:

1 N j i ij i f k f    (1)
Here, i k are weights of N single models, and k is constrained by the following conditions:

0 1 i i k k       (2)
The error of the combination model for j sample is formulated with the following:

1 N j j j ij i i e y f e k      (3)
For determining k i , e j is usually looked on as independent variable of the objective function, and the mathematic framework of WOC is commonly expressed as the following: 12 min ( , , , )

1 0 i i i E E k k k k k          (4)
Where, minE is the objective function that can be the minimum error square sum, or minimum absolute error sum, or the other cost function. The process solving the formula (4) for the acquisition of k i is a little complicated, and there are various solving algorithms for the formula (4) with different effectiveness. In the study, the iterative optimization algorithm based on dual optimal combination is selected to calculate the optimal weights, whose detailed description can be found in [START_REF] Tang | An iterative algorithm for optimal combination forecasting of non-negative weights[END_REF] 3.

Result and Discussion

Yield estimation with single vegetation index

To set up the combining model of winter wheat yield estimation base on WOC, the above four VIs is firstly used to establish the yield models with the field measured yield data and the preprocessed remote image from Landsat5 TM, respectively. The LLST(Linear Least Squares Fit) is the main modeling method in the study; and the equations of the models with different index are as the following ( 5) -( 8): 6879 1970

NDVI f NDVI    (5)

3868

SR f SR    (6)
/ -1462 / 6143

TCARI OSAVI f TCARI OSAVI    (7)
6801 4764

NDWI f NDWI    (8)
In the equation ( 5) ~ ( 8), f is yield calculated by the individual model from different vegetation index, and the unit is kg/hm. Figure 2 shows the correlation coefficients between the measured and estimated yield with four VIs models of winter wheat yield. From the figure, it is proved that the four models with different Vis have the approximate precisions, but the precisions of the models with three VIs, SR, TCAI/OSAVI and NDWI are more proximal, and the determinant coefficients R 2 are 0.368, 0.365 and 0.364, respectively.

Estimating yield with the combining model

In order to use WOC to build the combining model for estimating yield of winter wheat, the above four models from the equations ( 5) ~ ( 8) are input into WOC, and then the iterative optimization algorithm above mentioned is applied to calculate the weights. The result is as Table 2 shows. From Table 2, it can been seen that the combining model has better performance in comparison with the four individual models with different VIs, and its determinant coefficient R 2 arrives at 0.379 (also see Fig. 3) and are slightly higher than the four single models. In addition, Table 2 also shows that the two models with NDVI and TCARI/OSAVI obtain the zero weights, which indicates that the two models have no contribution to the combining model. In fact, in the process of using WOC with the iterative optimization algorithm, the more useful information the single model contributes to, the bigger weight it is given, and vice versa. That is to say, WOC has the function of judging the redundant information [START_REF] Tang | Study of combination forecasting error information matrix[END_REF]. As far as the single model providing little or useless information is concerned, WOC will give the model less or zero weight. Therefore, WOC can be employed to determine which of the VIs are more probably sensitive to winter wheat yield according to the weights when the four models as constructed separately by the above four VIs are input into WOC at the same time. In this study, the analysis demonstrates that the two yield models with the VIs SR and NDWI are enough, and the other two are redundant, which is significant for how to select the VIs sensitive to crop yield based on WOC. In the analysis, the two VIs SR and NDWI are probably more sensitive to winter wheat yield than the others two during the winter wheat jointing stage.

According to the result from table 2, the combining model can be described by the following equation ( 9 6) and ( 8), respectively. Figure 3 displays the correlation between the measured and estimated yield with the combining model based on WOC by plotting scatter diagram.

Conclusion

In this study, the weight optimization combination method is introduced to the yield estimation of winter wheat with remote sensing data from Landsat5 TM. Firstly, the four typical vegetation indices, NDVI, SR, TCARI/OSAVI and NDWI are adopted to establish the yield estimating models, respectively, and then the four models with different VIs are input into WOC to calculate the optimum weights and form the combining model, and there are conclusions as the followings:

(1) In comparison with the single yield models of winter wheat with different vegetation indices from remote sensing data, the combining model based on WOC has slightly higher precision for estimating winter wheat yield. (2) In the combining model using WOC with the four VIs yield models, the two models with SR and NDWI are given the non-zero weights, which indicate that the two VIs are probably more sensitive to winter wheat yield than the other two during the winter wheat jointing stage.

Fig. 1 .

 1 Fig. 1. Study area covering Shunyi & Tongzhou District(right), in Beijing(left), China.

  estimated result for j sample with i model( the error for j sample with i model;

Fig. 2 .

 2 Fig. 2. Coefficients of correlation between the measured and estimated yield with different VI models of winter wheat

Y

  is yield calculated by the combining model coupling the vegetation index SR with NDWI in terms of WOC, and the unit is kg/hm. both SR f and NDWI f are corresponding to dependent variables in the equations (

Fig. 3 .

 3 Fig. 3. Correlation between the measured and estimated yield with the combining model

Table 1

 1 Summary of the four vegetation indices for this analysis

	VI	Full name			Definition	Described by
	NDVI	Normalized Difference Vegetation Index	(	nir R R red 	) / (	nir R R red 	)	Rouse et al., 1974; Gitelson et al., 1994
	SR	Simple Ratio Index			/ RR		Jordan, 1969
					nir		red
		Transformed
		Chlorophyll
			3 ((	) 0.2 (				) ( / ))
	TCARI/	Absorption Ratio	nir	red			nir	green	nir	red
	OSAVI	Index / Optimized	(1 0.16) (	nir	red	) / (	nir	red	0.16)
		Soil-Adjusted
		Vegetation Index

Table 2

 2 Optimum weights of the estimating models with different VIs based on WOC

	Model with different VIs	Weights	R 2 of yield estimation
	SR	0.54	0.368
	NDVI	0.00	0.327
	TCARI/OSAVI	0.00	0.365
	NDWI	0.46	0.364
	Combining model	--	0.379

The study only discusses the ability of the WOC method for estimating winter wheat yield in one wheat growth period, and the following research work would focus on multi-periods with multi-temporal remote sensing data.

Funding for this research is supported by 863 National High Technology R&D Program (No.2008AA10Z214), National Natural Science Foundation of China (No.40701120), and Beijing Natural Science Foundation (No.4092016 & 4092017) (P. R. China)


Xiangfei-Meng

Changming - Wang

Application of Data Fusion Technology in Greenhouse Environment Monitoring and Control System

Keywords: greenhouse environment, multisensor data fusion, D-S evidence theory, agricultural expert system

Multisensor data fusion technology is a new frontier technology. According to the demand of greenhouse environment monitoring, the paper puts forward a new kind of data fusion method based on Dempster-Shafer (D-S) evidence theory and agricultural expert system. The experimental results show that the method can improve the veracity of decision of measuring and controlling greenhouse environment parameters, and can significantly improve the control effect of greenhouse environment.

Introduction

The environmental conditions in the greenhouse play an important role in crop growth and development. It is a key point of the development of greenhouse technology that how to make the most suitable greenhouse environment for crop growth. Now a single sensor is mainly used to monitor the greenhouse environment, such as temperature monitoring, humidity monitoring, CO 2 monitoring, soil moisture monitoring, illumination monitoring, etc, since the lack of collaboration processing and comprehensive utilizing the multi-source and multidimensional information, there are varied defects in accuracy, reliability, practicability, etc. By better utilizing the information resources, data fusion technology is applied to the greenhouse environment monitoring and control system to incorporate and corresponding process multidimensional information from various sensors [START_REF] Fengxia-Xu | The Study of the Development of Establishment Agriculture in China[END_REF] . So more accurately reflection about the greenhouse environment condition can be got and more accurately and reliably greenhouse environment monitor can be carried out.

Data fusion technology 2.1 Data Fusion process

Data fusion is the technology which makes full use of multisensor resources. It can combine the information, which is redundant and complementary in time and space, according to certain standards, by using the multisensor and observation information properly to get the consistent explanation or description of the tested object. The processes of Data fusion mainly includes multisensor (signal acquisition), data preprocessing, data fusion center (feature extraction, data fusion Calculation) and result output [2] .

Fusion algorithm

As a data processing technology, data fusion is in fact the integration and application of traditional science and new technologies. Traditional multiple sensor fusion mainly includes simple filtering method, the weighted averages method, production rules, the Bayesian estimation, D-S evidence reasoning method, etc. In recent years, many new fusion methods have also been put forward, including expert system, neural network and fuzzy theory, etc. This paper put forward a kind of fusion method based on the evidence theory and the expert system to monitor the greenhouse environment [START_REF]Data Fusion Theory and Application[END_REF] . The biggest characteristic of D-S evidence theory structure is introducing the uncertainty into evidence and establishing the axioms. For example, basic probability assignment function (BPAF), believe function (BEL), plausibility function (PL), etc. The most basic concept of D-S evidence theory is the recognition framework, recorded as  .  represents the set of all possible proposition in a domain, Every proposition is represented as a subset of  . 2  is  Power set and makes up a set of proposition () . If the elements meet the condition that incompatible with each other in a discernment frame , the m (A), values A assigns to function m, is a mapping form power set to [0,1]. If the function m meets the following conditions:

( ) 0 m  (1) 2 2 , ( ) 0, ( ) 1 A A m A and m A         (2)
m (A) can be called as a basic probability assignment function of A. Among them Ф represents empty set, m (A) represents the precise degree of believe in proposition A and the directly support of A.

For a given basic probability assignment function m and 2

A   , the corresponding belief function is defined as:

( ) ( ) BA Bel A m B    (3)
Plausibility function is defined as:

( ) 1 ( ) ( ) BA Pl A Bel A m B       ( 4
)
Bel function is called as lower limit function, expressed the degree of believe in proposition A, including Bel (Ф) = 0, Bel (Ω) = 1. Pl function also known as the upper limit function or not negative function, it is a sum of all the set of basic probability assignment functions that meet A. As the evidence refused to A, Pl (A) = 0, when there is no evidence against A, Pl (A) = 1, it is easy to prove Pl (A) ≥ Bel (A).

[Bel (A), Pl (A)] represents the belief range of A. If A is a subset in discernment frame and m (A)>0, A is called as a focal of belief function Bel. In this way, trust degree and plausibility have summarized the relations evidence of proposition A, the relations between them as shown in figure 1; this constitutes a complete range of evidence. and Bel 1 ,Bel 2 respectively, for subset A, the D-S rules composed of probability assignment of the two reasoning system:

1 2 1 2 1 1 2 2 1 1 2 2 1 1 2 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) A A A AA m A m A m A m A m A m A m A         (5)
The corresponding Bel of m called synthesis or sum value of Bel 1 and Bel 2 , recorded as Bel=Bel 1 ⊕Bel 2 .

1 2 1 2 1 1 2 2 1 1 2 2 ( ) ( ) 1 ( ) ( ) 1 A A A A m A m A m A m A k           (6)
1 k  is a correction factor. In fact, the introduction of 1 k  is in order to avoid assigning nonzero probabilities to empty sets when combining the evidences, so that the abandoned belief assignment of empty sets can be added to nonvoid set proportionally. The k objectively reflects the degree of conflict among evidences during the date fusion process. The range of k is 0 1 k  , the bigger k is the greater conflict will be. If k close to 1, it will probably produces unreasonable results and leads to the fusion decisions against intuition; If k=1, D-S theory cannot be use to fuse.

Expert system

Expert system is a branch of artificial intelligence. Its main purpose is to make the computers play the role of human experts in each field. It is a kind of intelligence program subsystems, which has a large number of domain knowledge and experiences on expert level. The expert system can solve the problems in the field by using the knowledge and methods that human experts will adopted. It is a computer

1 Pl (A) Bel (A) Plausibility Support evidence

Belief

Refuse evidence 0 program, which can finish general or humanoid simulation problem-solving strategies on expert level and combine the problem with operational knowledge and experience knowledge [START_REF] Zhuo-Zhao | The Countermeasures of Developing Intellective Agriculture Expert System[END_REF] .

Expert system generally consists of knowledge base, database, inference engine, explain part and the part of the knowledge acquisition. Its core is the knowledge base and Inference engine. The ways that expert system works can be simply boiled down to two parts: knowledge using and reasoning. The basic framework as shown in figure 2:

Fig. 2 The basic framework of expert system

3

The greenhouse environment monitoring system based on multisensor data fusion Environmental monitoring technology is the core of greenhouse technology. In a modern large greenhouse, all the environmental factors such as indoor temperature, light, wet and heat detecting, sensing and adjusting are managed by the computer comprehensively and controlled automatically [START_REF] Jingyu-Liang | The Research and Design of Intelligent Greenhouse Environment Temperature and Humidity Measurement and Control System[END_REF] . The purpose of greenhouse environment controlling is overcome some or entire climate environment in outside world and the objective restrictions, according to the physiological property of different crop, to build a best environment for different crop to growth. At present, the greenhouse environment controlling mostly uses industrial control system. On the one hand, the industrial control system costs more, can't meet the low-cost requirement of establishment agriculture; on the other hand, it focuses on the computer technology during the development process, without the participation of agricultural expert system. It also lacks the expert system that includes physiological information of crops. Because of these reasons, the greenhouse environment controlling has some limitations in the management application [6] [7] .

This paper adopts a method, which combined the D-S evidence theory with expert system, to do the fusion processing on the three main environment parameter, temperature, humidity and light intensity in happen in the greenhouse environment, a definition of the rational recognition framework can be worked out form expert knowledge, and the assignment on basic probability assignment can be down on the date which is fused locally. And then initial conditions of D-S theory can be acquired. After D-S evidence theory combination rules fusion, all the fusion results will be sent into the knowledge base of expert system to combine with the experiences of domain experts as the basis of inference in the expert system so that a accurate judgment will be worked out, and appropriate measures will be applied to adjust environmental parameters. Finally, the best environment conditions for crop to growth will be acquired. System structure fusion as shown in figure 3:

Fig.3 Structure diagram of system fusion

The experiment and application study for greenhouse cultivation

The simulation experiment of fusion algorithm has been down on specific environmental conditions of strawberry cultivation in a greenhouse trellis. Strawberry belongs to the thermophilic plants. A group of environmental data of greenhouse strawberry cultivation in March is collected for experiment. There is only one or two of the environmental factors play a leading role under certain conditions in the greenhouse. We call them the leading factor. During fruiting period of strawberry, temperature is the leading factor, which affects the yield and quality of strawberry, in It can be inferred from the above two tables that m (Θ) was decreased. That means data fusion reduce the uncertainty of the system. At the same time, the fusion of the basic possibility is better separability than before fusion. Table 1 shows that after merging the temperature and humidity, basic probability function value of A is the largest one, and is more than before fusion temperature and humidity their basic possibility function value is bigger. Table 2 shows that after merging the light intensity and temperature, basic possibility function value of D is the largest one. Comprehensively considering the fusion results of the two tables, it can be judged that temperature, humidity is appropriate in the greenhouse now, but light intensity is insufficient. So it is necessary to do light supplement by using dysprosium lamp. Compared to high voltage sodium lamp and halide lamp, dysprosium lamp can make the light increased from 1000 lx to 4000 lx. Because of this, dysprosium lamp can be used in raining days, fog days and evening.

Conclusions

At present, Multisensor data fusion technology has been successfully applied to the automatic control, the military and aviation navigation. It is not a single technology, but an interdisciplinary comprehensive theory and method. We should actively learn the research results home and abroad. The research in the agricultural engineering area should be vigorously promoted. This paper through two-step data fusion, the fusion results will be sent into the knowledge base of expert system as the basis of inference in the expert system and then adds the experiences of agricultural expert, more accurate decision results can be got.

The combination of data fusion technology and agriculture expert system is the development direction of precision agriculture. With the continuous improvement and development of the data fusion technology, it is believed that technical support will be offered to informationization and better educated the agricultural production in the near future. The agricultural production will become more and more automated and precision. The pace of development of modern agricultural production in China will be speedup.

Fig. 1

 1 Fig.1 Evidence range schemes Basic rules of D-S evidence combination: Hypothesizing that there are two reasoning system, their basic probability assignment and belief function is m 1 ,m 2

  C =﹛low temperature, humidity is on the high side﹜; D =﹛low temperature, light intensity is insufficient﹜; E =﹛low temperature, humidity is on the high side, light intensity is insufficient﹜; Take a group of greenhouse parameters got form 16:30 to 18:30 in the afternoon ﹛ temperature 18℃, humidity 65%, light intensity 1500 lx﹜to do the evidence theory fusion experiments. The fusion results as shown in table1 and table 2:

	sunlight greenhouse. So recognition framework can be built with the knowledge and
	experiences of domain expert. The recognition framework is as follows:
			, , , , A B C D E		;
	A =﹛appropriate, temperature 20℃~25℃, humidity 60%~70%, light intensity 2500
			~ 3000 lx﹜;
	B =﹛low temperature﹜;
			Humidity sensor	Temperature sensor	Light sensor
				Local fusion center	Local fusion center
						Tab.1 Temperature and Humidity fusion
			k=0.1812		Global fusion center ( D-S reasoning algorithm ) A B C D	E	m(Θ)
			Temperature		0.645	0.103	0.025	0.148	0.075	0.018
			Humidity		0.382	0.067	0.460	0.042	0.115	0.027
			Fusion result		0.844	fusion result 0.018 0.116	0.016	0.015	0.009
						Tab.2 Temperature and Light intensity fusion
			Expert experience k=0.1125	A	B	Knowledge base C	D	Inference engine E	m(Θ)
			Temperature		0.645	0.103	0.025	0.148	0.075	0.018
	Light intensity		0.053	0.104	0.019	0.702	0.058	0.001
			Fusion result		0.126	Agricultural expert system 0.056 0.002	0.743	0.002	0.006

Author introduction:

Xiangfei-Meng (1977-), the Doctor of test and measurement professional direction, E-Mail: mxf0316@163.com.