Monika Musia

The excited, ionized and electron attached states within the EOM-CC approach with full inclusion of connected triple excitations

Keywords: equation-of-motion CC, connected triple excitations, excitation energies, ionization potentials, electron affinity

The excited, ionized and electron attached states within the EOM-CC approach with full inclusion of connected triple excitations

Monika Agnieszka

Introduction

In this work we focus on the equation-of-motion (EOM) [1][2][3][4] coupled cluster (CC) [5][6][7][8][START_REF] Bartlett | Modern Electronic Structure Theory, Part 2[END_REF][START_REF] Bartlett | Reviews in Computational Chemistry[END_REF][START_REF] Bartlett | [END_REF][START_REF] Shavitt | Many-Body Methods in Quantum Chemistry: Many Body Perturbation Theory and Coupled Cluster Theory[END_REF] approach in the context of its application to the calculations of excitation energies (EE), ionization potentials (IP) and electron affinities (EA). The main topic of the article is connected with the triple excitation operator and its role in the theoretical evaluation of the above quantities.

The idea of extending the EOM-CC scheme to the full triples originated in the Bartlett's group in Quantum Theory Project, University of Florida. The inclusion of the full triple excitation operator into the EOM theory has been a long term project realized on the basis of previous works done in this group concerning incorporation of the higher clusters (quadruples [START_REF] Kucharski | [END_REF]14] and pentuples [15,16]) into the CC theory of the ground state. Most of the work has been done during my numerous inspiring visits to Quantum Theory Project.

The original formulation of the time-dependent CC theory goes back to the works of Monkhorst [17], Emrich [18] and Mukherjee [19] (see also paper by Paldus [20]). This approach, known also as a linear response theory (LRT) [1,21,22], is -for the full CC models -entirely equivalent to the EOM formulation. The latter scheme has been intensively studied in the Bartlett's group which resulted in the first general purpose computer code developed by Stanton and Bartlett [3] in the elegant way exploiting an Abelian symmetry of a molecule. This was the EOM-CC model with single and double excitations applied to the description of the excited states. There were also several successful attempts to include the connected triples in the partial, i.e., approximate manner [23][24][25]. The results show that the triples are important to the proper description of the transition energies in general and crucial for the correct reproduction of the energies of the states dominated by doubly excited configurations.

The CCSD model has been successfully implemented also for the IP [START_REF] Bartlett | Reviews in Computational Chemistry[END_REF] and EA [26,27] calculations and the results indicate its great usefulness also in the description of the ionized and electron attached states. Both for the EE and IP/EA quantities the EOM-CCSD model generates results with average deviation from the experimental data of ca 0.2 -0.3 eV.

The central problem in the EOM-CC calculations is a construction of the similarity transformed Hamiltonian H (≡ e -T He T ) the diagonalization of which provides the eigenvalues related to the studied process. The full inclusion of the triple excitations makes the EOM-CC approach by far more complicated as compared to the CCSD model. At the implementation stage the new features include:

• the reference (ground state) function must be constructed at the full CCSDT level;

• the H matrix includes new terms, usually much more complicated than those constructed at the CCSD level; • several H elements, present already at the CCSD level, include new terms dependent on the triple excitation amplitudes;

• the H operator must be diagonalized within the significantly larger configurational subspace (now consisting of singles, doubles and triples).

In addition the application of the EOM-CCSDT scheme [28][29][30] is connected with the significantly higher scaling than that for the EOM-CCSD method: n 8 vs. n 6 which requires very careful analysis of each coded term to make the whole method feasible.

In practical realization we benefited from the fact that the ground state program at the CCSDT level has been available for us [31,32]. In addition the new H elements -three-and four-body components -have been coded when developing the coupled cluster version including the T 5 operator in the approximate [15] and full [16] model. The latter scheme has been constructed in the so called quasi-linear formulation in which all nonlinear terms were included into the amplitude equations via H elements. Due to that the complicated amplitude equations assumed relatively simple linear form which, however, required prior evaluation of the H element (see also [START_REF] Kucharski | [END_REF]14]) and they were available at the start of the EOM-CCSDT project.

Another important factor which also had to be taken into consideration was a feasibility of the EOM-CCSDT calculations. A straightforward introduction of the three-and four-body H elements would create a prohibitive demand concerning both the disk storage as well as the cost of the calculations. In order to avoid these limitations the inclusion of all four-body elements and most of the three- body ones required reorganization of the equations to avoid explicit constructions of the latter terms -on one hand -and to keep the method fully rigorous -on the other. An efficient and precise implementation was possible when using the diagrammatic formalism to derive and code all the equations. All variants of the EOM-CC approach were introduced into the ACES II [START_REF] Stanton | ACES II program is a product of the Quantum Theory Project[END_REF] program system developed in Quantum Theory Project.

In next section we give a short description of the EOM-CC theory in the form applicable to all studied processes: EE, IP and EA.

Theory

We look for the solution of the Schrödinger equation:

H N |Ψ k = ∆E k |Ψ k k = 1, 2, ... (1)
where

|Ψ k = R(k)|Ψ o (2)
The R(k) is a linear (CI-like) excitation, electron-attachment and/or ionization operator limited in this approach to the single, double and triple excitations (see Fig. 1): or in the expanded form:

R(k) = R o (k) + R 1 (k) + R 2 (k) + R 3 (k) (3)
R(k) EE = r 0 + ia r a i (k)a † i + 1 4 ab ij r ab ij (k)a † b † ji + 1 36 abc ijl r abc ijl (k)a † b † c † lji (4) R(k) IP = i r i (k)i + 1 2 a ij r a ij (k)a † ji + 1 12 ab ijl r ab ijl (k)a † b † lji (5) R(k) EA = a r a (k)a † + 1 2 ab i r ab i (k)a † b † i + 1 12 abc ij r abc ij (k)a † b † c † ji (6)
where a, b, ... run over unoccupied levels and i, j, ... run over occupied levels in the Φ o . Inserting the |Ψ k wave function, Eq. ( 2), into the Schrödinger equation, Eq.

(1), we have:

H N R(k)|Ψ o = ∆E k R(k)|Ψ o (7)
Multiplying the Schrödinger equation for the ground state by R(k) we obtain:

R(k)H N |Ψ o = ∆E o R(k)|Ψ o (8)
and then subtracting from Eq. ( 7) gives the EOM:

(H N R(k) -R(k)H N )|Ψ o = (∆E k -∆E o )R(k)|Ψ o (9) [H N , R(k)]|Ψ o = ω k R(k)|Ψ o ( 10
)
where Multiplying from the left with e -T we obtain:

ω k = ∆E k -∆E o = E k -E o is
[H N , R(k)]e T |Φ o = ω k R(k)e T |Φ o (11)
e -T H N R(k)e T |Φ o -e -T R(k)H N e T |Φ o = ω k e -T R(k)e T |Φ o (12)
Since T and R commute:

[ HN , R(k)]|Φ o = ω k R(k)|Φ o ( 13
)
where HN is a similarity transformed Hamiltonian HN = e -T H N e T = ( HN e T ) c (14) which includes also three-, four-and, in general, higher-body elements. Expanding HN into one-body, two-body, three-body, ... etc. contributions we get:

HN = I o + 2 k=0 I 1 k + 4 k=0 I 2 k + 3 k=0 I 3 k + 3 k=0 I 4 k + • • • (15)
where I n k represents the n-body element of HN with k annihilation lines (lines below the vertex, see Fig. 2). So, the HN contains also closed diagrams (= ∆E o ) which are eliminated by the commutator. Indicating open diagrams within HN by HN,o we may get rid of commutator:

HN,o R(k) = ω k R(k) (16)
HN,o = (e -T H N e T ) o = (H N e T ) c,o (17)
In the matrix form we may write out a CI-like matrix eigenvalue equation: eigenvectors, both corresponding to the same eigenvalue. Moreover, both are needed to obtain density matrices.

HR(k) = ω k R(k) (18)
To solve the EOM-CC equations we have to diagonalize H matrix. We use the generalized Davidson diagonalization procedure [START_REF] Davidson | [END_REF]35]. A crucial step in this procedure is taking product of the amplitude vector R and the matrix to be diagonalized, i.e., x= ( HN R) c . So, the schematic EE-EOM-CCSDT equations look like:

x a i (k) = Φ a i |( HN,o R EE (k)) c |Φ o ( 19
)
x ab ij (k) = Φ ab ij |( HN,o R EE (k)) c |Φ o ( 20
)
x abc ijl (k) = Φ abc ijl |( HN,o R EE (k)) c |Φ o (21)
This is the standard form of the equations (see Fig. 3). So we may say that in practice the EOM equations are identical to the familiar configuration interaction (CI) problem with one modification which is the replacement of the Hamiltonian operator with the similarity transformed Hamiltonian and this is an important difference since H N has only one-and two-body contributions while HN consists also of higher-body contributions (see Fig. 2). Thus the form of the matrix H is:

F o r P e e r R e v i e w O n l y S d d D d d d d T d d d d d d I 1 1 I 1 2 I 2 4 S d d I 2 ′ 2 I 2 3 I 2 1 I 1 1 I 1 2 D d d d d I 3 ′ 2 I 2 2 I 2 3 I 3 3 I 3 1 I 2 1 I 1 1 T d d d d d d I 4 ′ 2 I 3 2 I 2 2 I 4 3 I 3 3
where prime indicates that some of the HN elements are not complete, e.g., in I 4′ 2 component we take only holeparticle type (for annihilation lines, see Fig. 2) for CCSDT case and we do not take particleparticle and holehole types of this element. They enter EOM-CCSDTQ variant.

As we can see the standard way of derivation of the EOM-CC equations assumes that we use in the equation all required HN elements regardless of the complexity of the considered term. This means that in the standard version we employ all required three-and four-body terms (see Fig. 2). Such a formulation of the EOM-CC problem, although the most natural one, would result in the high rank of the computational procedure (n 9 for CCSDT case). The remedy invented to solve this problem, which is the crucial point in creation of the efficient code, is a f actorization scheme [START_REF] Kucharski | [END_REF]36].

We can describe the factorization with following scheme. The X quantity, see left hand side of Eqs. ( 19)-( 21 the X via HN then we have:

X(T, R) = HN (T )R (22)
Here in the first step we construct HN (T ) while in the second one we take product of the HN and R (realized in the EOM equation).

The other possibility is presented in the equation:

X(T, R) = Z(R)T (23)
In the first step we construct the intermediate Z depending on R and in the second step we contract Z and T to obtain X. The drawback of the second-possibility is that the Z intermediate must be constructed in each iteration while the HN operator is obtained only once (as a follow up of the ground state solution).

Here we present example of such factorization for I 4 3 element of HN in EE-EOM-CCSDT model. So in order to omit construction of this element:

d d c d d ⌢ ⌢ ⌢ ⌢ ⌢ ⌢ ⌢ ⌣ ⌣ ⌣ ⌣ ⌣ ⌣ ⌣ d d = d d T © © d d d d
which requires the scaling factor equal to n 9 with the CI-type step scaling also as

n 9 : d d d d d d = ... + d d c T c d d d d ⌢ ⌢ ⌢ ⌢ ⌢ ⌢ ⌢ ⌢ ⌢ ⌣ ⌣ ⌣ ⌣ ⌣ ⌣ ⌣ ⌣
we rewrite the R 3 with the expanded form of the I 4 3 element: ).

d d d d d d = • • • + d d d d d d T
where

T = c T c
The difficult terms (i.e., those engaging the three-and four-body HN elements) are indicated in Fig. 3 by the rectangles. Their construction is omitted by the factorization procedure as described above. The factorized form of the EE-EOM-CCSDT equations is presented in Fig. 4 with intermediates shown in Fig. 5.

The factorization procedure, i.e., replacement of some HN elements contracted with the R operators with the appropriate intermediates contracted with the T operator makes the evaluation of the HN R quantities much more efficient. Moreover, the factorization does not introduce any approximation, the method is fully rigorous.

It should be explained that the HN elements as presented in Fig. 2 enter the EOM-CCSDT scheme -in a standard or factorized form -only for the EE case.

For the IP formulation only one I 4 3 element for which two out of three annihilation lines are of the hole type, denoted as I ijbc aklm , is involved. For the EA, vice versa, the of seven three-body elements one, I 3 2 with two particle lines, denoted in Fig. 2 as I aib cjk is excluded from the IP treatment, while its two-hole line counterpart, I ija klm , is excluded from the EA-EOM-CCSDT model. Note that it follows from the Fig. 3 and Fig. 6 that only two three-body elements of the HN enter the EOM equations in the standard, i.e., not factorized form, namely those which have the annihilation lines exclusively of the hole type, i.e., I iab jkl and I ija klm . This means that, e.g., for the EA case all three-and four-body terms are treated in a factorized way.

I
The factorization makes it possible to achieve for the EOM part a scaling no worse than n 8 for EE-EOM-CCSDT and n 7 for IP-EOM-CCSDT and EA-EOM-CCSDT case. Thus for the EE case the EOM part scales identically as the ground state, i.e., n 8 , while for the IP and EA cases the ground state solution is the slowest step and becomes a bottleneck of the whole procedure.

Moreover, the IP-EOM-CC formalism in its standard formulation (see Fig. 6) [29] formally parallels the EA-EOM-CCSDT theory [30]. Diagrammatically it is just reverting directions of all lines, but replacing the hole indices with particle ones changes the scaling (IP: n 3 occ n 4 vir ; EA: n 2 occ n 5 vir for factorized variants).

All the resulting equations for the IP and EA variants (standard and factorized ones) are presented in Refs. [29,30], respectively. Note, that the sine qua non condition of efficient performance of EOM-CC is a very careful implementation of the method with systematic factorization of the difficult terms.

Summing up, we may consider each of these processes as a special case of the general EOM-CC theory. However, the detailed equations are slightly different (con- 3 and6) in their form but entirely different when writing computer code.

In particular when we want to apply factorization technique, which turns out to be different for each case (see Fig. 3 and Refs. [29,30]).

It should be indicated that the most crucial quantity in all these theories is a similarity transformed Hamiltonian which plays essential role in the construction of the EOM-CC method.

Results and discussion

The incorporation of the connected triples into the EOM-CC scheme creates much more reliable method than the EOM-CCSD model. The price we pay for the higher accuracy is a significantly larger cost of calculations. Although at the EOM step we have different scaling for each type of the EOM problem, i.e., for EE we have n 8 (n 3 occ n 5 virt ), for IPn 7 (n 3 occ n 4 vir ) and for EAn 7 (n 2 occ n 5 vir ) , the general scaling for all EOM-CC schemes is assumed to be n 8 since the bottle neck of the calculations is a ground state wave function and this step scales as n 8 .

The EOM-CCSDT results were compared to the reference data in two ways: for small basis sets they were related to available full Configuration Interaction (FCI) results, and for majority of examples -to the existing experimental data.

We are not going to repeat the description of the results reported in the original papers [29,30,[37][38][39][40][41][42][START_REF]AIP Conf. Proc[END_REF][START_REF] Musial | [END_REF], but we want to reiterate the general conclusions concerning performance of the SDT variant of the EOM-CC approach. In 1 relate -for obvious reasons -to the small basis sets and we may expect that for the larger systems and basis sets we cannot keep the same proportion of the errors for the EOM-CCSD and EOM-CCSDT methods. Nevertheless it indicates that the inclusion of the connected triple excitations is an important factor reducing errors of the computed quantities.

In Table 2 we analyze the performance of the EOM-CCSDT in the description of the excited states for two molecules: N 2 and CO. We quote the mean absolute deviations from the experimental values for excitations energies and selected properties. In cases when the theoretical values are being compared with the experimental data an important factor to be eliminated are deficiencies of the basis set to be sure that the possible errors are not caused by the inadequacies of the basis set. The results cited in Table 2 were obtained for the aug-cc-pVQZ basis set (160 functions per molecule) which is relatively large to make basis set error small compared to the inaccuracies in the correlation treatment. The quantities listed in Table 2 The quality of the theoretical results is easy to assess when the reference values are obtained with the well defined accuracy. In Table 3 we compare the electron affinity of the C 2 molecule obtained with the EOM-CCSD and EOM-CCSDT methods. We quote the values extrapolated from triple and quadruple zeta to the basis set limit both for the regular and augmented Dunning basis set [45][46][47]. In both Moreover, in the paper [START_REF] Musial | [END_REF] we compare the data computed for the ozone molecule. For that system we were limited to the smaller basis sets (POL1 [48] and aug-cc-pVTZ [46]) and, consequently the larger discrepancies between computed and experimental values may be due to the basis set inadequacies. Nevertheless, what catches our attention is a dramatic change in the computed excitation energy for the 2 1 A 1 state where the inclusion of triples reduces the theoretical value by ca 5 eV. This is due to the fact that the 2 1 A 1 state is dominated by doubly excited configurations and in that case the EOM-CCSD scheme fails. However, also for the remaining EEs the effect of the connected triples is well pronounced and lowers the computed value by 0.2 -0.4 eV. In all cases it is a desired result since it brings down the deviation from the experiment.

More erratic behavior is observed for the IP values. An inclusion of triples lowers the theoretical values, however the T 3 effect is smaller, ranging from 0.2 to 0.02 eV and is more basis set sensitive, e.g., the POL1 values are in significantly larger error than those due to aug-cc-pVTZ basis set. Thus the apparent relatively larger deviations from the experiment, e.g., 2 B 2 state, can be explained by the basis set limitations. The same observation applies to the electron affinity calculations.

To make more general conclusion we presented in Fig. 7 a histogram indicating the mean absolute deviations of the computed excitation energies from the reference values for the EOM-CCSD and EOM-CCSDT models. The data considered in Fig. 7 where taken from the papers [39][40][41]. Analogous data, referring to the ionization potential calculations [29,42,[START_REF]AIP Conf. Proc[END_REF] are presented in Fig. "The excited, ionized and electron attached states within the EOM-CC approach with full inclusion of connected triple excitations" by Monika Musial.

d d R EE 1 d d d d R EE 2 d d d d d d R EE 3 (A) R EA 1 T R EA 2 d d T d d d d R EA 3 T (B) R IP 1 c R IP 2 d d c d d d d R IP 3 c (
d d = T d d ⌢ ⌢ ⌣ ⌣ + c d d s ⌢ ⌢ ⌣ ⌣ + d d T c ⌢ ⌢ ⌣ ⌣ + T c ⌢ ⌢ ⌢ ⌣ ⌣ d d + T c c ⌢ ⌢ ⌢ ⌣ ⌣ + T c T © ⌢ ⌢ ⌢ ⌣ ⌣ + T c T c d d d d d d = T d d ⌢ ⌢ ⌢ ⌣ ⌣ ⌣ d d + c d d s ⌢ ⌢ ⌢ ⌣ ⌣ ⌣ d d + c d d s ⌢ ⌢ ⌢ ⌣ ⌣ c + T d d ⌢ ⌢ ⌢ ⌣ ⌣ T © + T d d ⌢ ⌢ ⌢ ⌣ ⌣ c + T d d ⌢ ⌢ ⌢ ⌣ ⌣ d d + c d d s ⌢ ⌢ ⌢ ⌣ ⌣ d d + T c d d ⌢ ⌢ ⌢ ⌢ ⌢ ⌣ ⌣ ⌣ ⌣ ⌣ d d + a b T d d T c ⌢ ⌢ ⌢ ⌢ ⌢ ⌣ ⌣ ⌣ ⌣ d d + c c d d s T c ⌢ ⌢ ⌢ ⌢ ⌢ ⌣ ⌣ ⌣ ⌣ d d + d d d d ⌢ ⌢ ⌣ ⌣ T c + T d d T c ⌢ ⌢ ⌢ ⌣ ⌣ d d + c d d s T c ⌢ ⌢ ⌢ ⌣ ⌣ d d d d d d d d = d T d d ⌢ ⌢ ⌢ ⌢ ⌢ ⌣ ⌣ ⌣ ⌣ ⌣ d d d d + c d d s ⌢ ⌢ ⌢ ⌢ ⌢ ⌣ ⌣ ⌣ ⌣ ⌣ d d d d + e T c ⌢ ⌢ ⌢ ⌢ ⌢ ⌢ ⌢ ⌢ ⌣ ⌣ ⌣ ⌣ ⌣ ⌣ ⌣ ⌣ d d d d d d + f T d d T © ⌢ ⌢ ⌢ ⌢ ⌢ ⌢ ⌣ ⌣ ⌣ ⌣ ⌣ d d + c d d s c ⌢ ⌢ ⌢ ⌢ ⌢ ⌢ ⌣ ⌣ ⌣ ⌣ ⌣ d d + g d d T c ⌢ ⌢ ⌢ ⌢ ⌢ ⌣ ⌣ ⌣ ⌣ ⌣ d d d d + d d c d d s ⌢ ⌢ ⌢ ⌣ ⌣ d d + d d T d d ⌢ ⌢ ⌢ ⌣ ⌣ d d + h T d d T c ⌢ ⌢ ⌢ ⌢ ⌢ ⌢ ⌢ ⌢ ⌣ ⌣ ⌣ ⌣ ⌣ ⌣ ⌣ d d d d + i c d d s T c ⌢ ⌢ ⌢ ⌢ ⌢ ⌢ ⌢ ⌢ ⌣ ⌣ ⌣ ⌣ ⌣ ⌣ ⌣ d d d d + j T d d T c ⌢ ⌢ ⌢ ⌢ ⌢ ⌣ ⌣ ⌣ ⌣ ⌣ d d d d + k c d d s T c ⌢ ⌢ ⌢ ⌢ ⌢ ⌣ ⌣ ⌣ ⌣ ⌣
⌢ ⌢ ⌢ ⌢ ⌢ ⌣ ⌣ ⌣ ⌣ ⌣ d d d d c d d s c T = k d d T T c c ⌢ ⌢ ⌢ ⌢ ⌢ ⌣ ⌣ ⌣ ⌣ ⌣ d d d d T d d T T = h d d T T c d d ⌢ ⌢ ⌢ ⌢ ⌢ ⌢ ⌢ ⌣ ⌣ ⌣ ⌣ ⌣ ⌣ d d d d d d s d d c = i d d c T c d d ⌢ ⌢ ⌢ ⌢ ⌢ ⌢ ⌢ ⌣ ⌣ ⌣ ⌣ ⌣ ⌣ d d d d d d d d T = e T c ⌢ ⌢ ⌢ ⌢ ⌢ ⌢ ⌢ ⌢ ⌣ ⌣ ⌣ ⌣ ⌣ ⌣ ⌣
= c ⌢ ⌢ ⌢ ⌣ ⌣ + c ⌢ ⌢ ⌢ ⌣ ⌣ ⌣ T c + c T c ⌢ ⌢ ⌣ ⌣ + c T c T c c d d = c ⌢ ⌢ ⌢ ⌣ ⌣ d d + c ⌢ ⌢ ⌢ ⌣ ⌣ c + c T © ⌢ ⌢ ⌣ ⌣ + c ⌢ ⌢ ⌢ ⌣ ⌣ c + c ⌢ ⌢ ⌢ ⌣ ⌣ T © + a c T c ⌢ ⌢ ⌢ ⌢ ⌢ ⌣ ⌣ ⌣ ⌣ d d + c T c ⌢ ⌢ ⌢ ⌣ ⌣ T © + c T c ⌢ ⌢ ⌢ ⌣ ⌣ c + c T c ⌢ ⌢ ⌣ ⌣ d d c d d d d = b c d d d d ⌢ ⌢ ⌢ ⌢ ⌢ ⌣ ⌣ ⌣ ⌣ + c T © d d ⌢ ⌢ ⌢ ⌣ ⌣ + c c d d ⌢ ⌢ ⌢ ⌣ ⌣ + c c T © d d ⌢ ⌢ ⌢ ⌢ ⌢ ⌣ ⌣ ⌣ ⌣ ⌣ + c c d d ⌢ ⌢ ⌢ ⌢ ⌢ ⌣ ⌣ ⌣ ⌣ ⌣ + d c T c d d d d ⌢ ⌢ ⌢ ⌢ ⌢ ⌢ ⌢ ⌣ ⌣ ⌣ ⌣ ⌣ ⌣ + c c d d s d d ⌢ ⌢ ⌢ ⌣ ⌣ + c T d d d d ⌢ ⌢ ⌣ ⌣ + c c d d s c ⌢ ⌢ ⌢ ⌣ ⌣ + c T d d T © ⌢ ⌢ ⌢ ⌣ ⌣ + c T d d c ⌢ ⌢ ⌢ ⌣ ⌣ + e c T c T © ⌢ ⌢ ⌢ ⌢ ⌣ ⌣ ⌣ ⌣ d d + f c T c c F o
Dear Editors, Please, find attached the revised version of the MS.

In response to the referee comments:

1. in abstract (p. 1) we corrected sentence, i.e instead of: ...triples reduces the average error for the EE by 2 anf for the IP by 3... we have: ...triples for the EE reduces the average error twice and for the IP -three times... 2. p. 5: we added after Eq. ( 1): where H N is a normal ordered Hamiltonian with respect to the Φ o vacuum. 3. p. 6 after Eq. ( 6) we added "in the Φ o " so instead of: where a, b, ... run over unoccupied levels and i, j, ... run over occupied levels we have: where a, b, ... run over unoccupied levels and i, j, ... run over occupied levels in the Φ o . 4. p. 7 we added "lines below the vertex" so instead of: ....annihilation lines (see Fig. 2). we have: ....annihilation lines (lines below the vertex, see Fig. 2). 5. and 6. fixed notation 7. fixed 8. fixed 9. fixed 10. p. 14 instead of: ... of the results ... we have: ...of the EE results obtained upon inclusion of connected triples. 11. p. 14 insted of "discuss" we used "analyze" 12. p. 15 we added "extrapolated from triple and quadruple zeta"...

Sincerely yours,

  H -for the CCSDT case -represents matrix of the HN,o operator in the configurational subspace of single, double, and triple excitations: Since H is a non-Hermitian matrix, it has different left (L(k)) and right (R(k))

  ), depends on both T and R operators. If we evaluate

  now we factorize the above diagram, i.e., we cut the diagram along the vertical line obtaining the final contribution to the equation (scaling as n 7 ) with the new intermediate (scaling as n 5

  cases an extension of the EOM-CC model results in lowering the EA value by 0.13 -0.15 eV. Since the experimental value is measured with an accuracy of 0.1 eV it is difficult to conclude on the quality of the computed number. Nevertheless we would rather point out to the value close to 3.25 eV than to that indicated by the experiment.

Figure caption Figure 1 .

 caption1 Figure caption

Figure 2 .

 2 Figure 2. Diagrammatic representation of the HN components entering the

Figure 3 .

 3 Figure 3. Diagrammatic form of the EE-EOM-CCSDT equations in an antisym-

Figure 4 .

 4 Figure 4. Diagrammatic picture of the factorization of selected contributions to

Figure 5 .

 5 Figure 5. Diagrammatic form of the intermediates used in factorized contribu-

Figure 6 .

 6 Figure 6. Diagrammatic form of the IP-EOM-CCSDT equations in an antisym-

Figure 7 .

 7 Figure 7. (Color online) Mean absolute deviation (eV) from the experimental

Figure 8 . 22 Page 22 of 34 URL

 82234 Figure 8. (Color online) Mean absolute deviation (eV) from the experimental

  Fig. 2.

  Fig. 4.aT c ⌢ ⌢ ⌢ ⌢ ⌢ ⌢ ⌣ ⌣ ⌣ ⌣ ⌣ d d d d

  Fig. 5.

  c

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 34 : http://mc.manuscriptcentral.com/tandf/tmph Molecular Physics

  the energy change connected with the excitation, ionization or electron attachment process. Since |Ψ o = e T |Φ o we have:

Table 1

 1 and IP values[37,38]. We compare MAD for the five systems when the electronic excitations are considered (N 2 , C 2 , H 2 O, HF, Ne) and for three systems (C 2 , BH, H 2 O) in connection with IP results. The MAD were computed on the basis of 2 to 5 excited/ionized states depending on the molecule. We observe a significant improvement of the EEresults obtained upon inclusion of connected triples. For the EE values the MAD is smaller by the order of magnitude for the N 2 and 2 to 8 times for the remaining systems. The improvement for the IP values is even more spectacular with MAD for C 2 and BH lower more than the order of magnitude and five times for the H 2 O system. The results quoted in Table

	we report

  the average error for the EOM-CCSDT results amounts to 0.1 eV, for the IPit is slightly lower and goes down to 0.06 eV. The histograms show that the gain in accuracy due to the inclusion of connected triples into the EOM-CC schemes is substantial and the EOM-CCSDT method may be considered as the source of reliable reference data for the more approximate theoretical tools in case where the experimental data are scarce or unavailable.

	Page 17 of 34	Molecular Physics
	F o r F o r F o r F o r F o r F o r
	P P P P P P
	e e e e e e
	e r e r e r e r e r e r
		R R R R R R
		e e e e e e
		v i e v i e v i e v i e v i e v i e
		w w w w w w
		O n l O n l O n l O n l O n l O n l
		y y y y y y
		8. For the EE calculations

URL: http://mc.manuscriptcentral.com/tandf/tmph

Table 1 .

 1 Mean absolute deviation (eV) from the FCI values for the excitation energies and ionization potentials.

	Page 29 of 34	Molecular Physics
	Fig. 7.
	r
	P
	e
	e r
		R
		e
		v i e
		w
		O n l
		y
		29

URL: http://mc.manuscriptcentral.com/tandf/tmph URL: http://mc.manuscriptcentral.com/tandf/tmph URL: http://mc.manuscriptcentral.com/tandf/tmph

Table 2 .

 2 Mean absolute deviations from experiment. EOM-CC methods with aug-cc-pVQZ basis set and core electrons frozen (Ref.[39]).

			Molecular Physics
		V EE a) (eV )		AEE b) (eV )
	Molecule SD	SDT	Molecule SD	SDT
	N 2	0.234 0.024	N 2	0.313 0.026
	CO	0.171 0.077	CO	0.221 0.011
		R e ( Å)			ω (cm -1 )
	Molecule SD	SDT	Molecule SD	SDT
	N 2	0.022 0.003	N 2	177	42
	CO	0.024 0.002	CO	114	14

a) VEE -vertical excitation energies. b) AEE -adiabatic excitation energies. 32 Page 32 of 34 URL: http://mc.manuscriptcentral.com/tandf/tmph

Table 3 .

 3 Extrapolated electron affinity (eV) for the C 2 molecule within EOM-CC methods (Ref.[30]).

	Sym.	cc-pV∞Z	aug-cc-pV∞Z
		CCSD CCSDT	CCSD CCSDT	Exp. a)
	2 Σ + g	3.36	3.23	3.39	3.24	3.30±0.1
	a) Ref. [49].

URL: http://mc.manuscriptcentral.com/tandf/tmph

Acknowledgments

It is a great pleasure to contribute this paper to a special volume celebrating 50th Sanibel Symposium and work done in Quantum Theory Project in Gainesville.

I would like to express my gratitude for the hospitality of Professor Rodney J. Bartlett, faculty members and staff during my stay in QTP. Current position: Assistant Professor, Institute of Chemistry, University of Silesia, Katowice, Poland. Quantum Theory Project: Postdoctoral Associate (2002-2003); several 2-3 months visits in the years 2004-2010. This work has been supported by the Ministry of Science and Higher Education, Poland, under grant No. N N204 090938.


Hai Xiang Zhao
email: zhaohaix@ecp.fr

Frédéric Magoulès
email: frederic.magoules@hotmail.com

A Parallel Statistical Learning Approach to the Prediction of Building Energy Consumption Based on Large Datasets

Keywords: Support Vector Machines (SVMs), Prediction, Model, Energy Efficiency, Parallel Computing

The prediction of future energy consumption of buildings based on historical performances is an important approach to achieve energy efficiency. A simulation method is here introduced to obtain sufficient clean historical consumption data to improve the accuracy of the prediction. The widely used statistical learning method, Support Vector Machines (SVMs), is then applied to train and to evaluate the prediction model. Due to the time-consuming problem of the training process, a parallel approach is applied to improve the speed of the training of large amounts of data when considering multiple buildings. The experimental results show very good performance of this model and of the parallel approach, allowing the application of Support Vector Machines on more complex problems of energy efficiency involving large datasets.

INTRODUCTION

Building energy consumption plays an important role in the total energy consumption of end use. Energy efficiency in building management and retrofit is important for energy conservation which significantly benefits both individuals and society. For a specific building, the energy transformation is treated as a complex system with many factors involved, such as building construction and materials, ambient weather conditions, occupants' behaviors, inner facilities, etc. In fact, most of these factors vary irregularly from one building to another building and from time to time, making analysis of the energy performance extremely difficult in practice. Many works have been done in the past few decades in order to find out the empirical rules of energy performance. Some work aimed to simulate and evaluate the energy efficiency of buildings or facilities from the engineering point of view [START_REF] Al-Homoud | Computer-aided building energy analysis techniques[END_REF]2]. A large number of tools were developed such as DOE-2, Apache, AkWarm, etc. Other works took the historical energy consumption behaviors into consideration, and tried to find out the rules from those recorded data and then applied these rules to predict the performance in unknown periods or conditions. Some models and methods were proposed for this kind of data mining approach, e.g. regression model [START_REF] Ruch | A four parameter change-point model for predicting energy consumption in commercial buildings[END_REF][START_REF] Katipamula | Multivariate regression modeling[END_REF], time-series analysis [START_REF] Kimbara | On-line prediction for load profile of an air-conditioning system[END_REF], neural networks [START_REF] Mihalakakou | On the energy consumption in residential buildings[END_REF][START_REF] Kreider | Building energy use prediction and system identification using recurrent neural networks[END_REF][START_REF] Gonzàlez | Prediction of hourly energy consumption in buildings based on a feedback artificial neural network[END_REF][START_REF] Karatasou | Modeling and predicting building's energy use with artificial neural networks: Methods and results[END_REF][START_REF] Ekici | Prediction of building energy consumption by using artificial neural networks[END_REF] and statistical learning method [START_REF] Dong | Applying support vector machines to predict building energy consumption in tropical region[END_REF][START_REF] Lai | Vapnik's learning theory applied to energy consumption forecasts in residential buildings[END_REF][START_REF] Li | Applying support vector machine to predict hourly cooling load in the building[END_REF]. This last method is widely used in industry to analyze complex real-life problems due to the great generalization ability of Support Vector Machines (SVMs) in solving non-linear problems [START_REF] Vapnik | The nature of statistical learning theory[END_REF][START_REF] Brugger | Parallel support vector machines[END_REF]. Dong et al. [START_REF] Dong | Applying support vector machines to predict building energy consumption in tropical region[END_REF] first applied SVMs to predict the monthly electricity consumption of four buildings in the tropical region. Three years' data was trained and the derived model was applied to predict the landlord utility in one year. Lai et al. [START_REF] Lai | Vapnik's learning theory applied to energy consumption forecasts in residential buildings[END_REF] applied robust regression model on one year's data of one building with electricity consumption and climate involved, then they applied the model to three month's data to test the predication ability of their model. They also trained models on each daily basis datasets and then compared the obtained models to verify the stability of this method. In addition, they added some perturbations to a certain part of the data and tried to detect these perturbations by examining the change of contributing weights. Li et al. [START_REF] Li | Applying support vector machine to predict hourly cooling load in the building[END_REF] compared SVMs with the conventional back-propagation neural network in predicting the hourly cooling load of an office building. SVMs method was proved to be better than the traditional solution. The above three work has shown good performances of SVMs method in predicting hourly and monthly building energy consumption. However, some problems remain in these analyses. Firstly, the predictions are based on the historical performance of the same building, which means that we do not know the ability of SVMs in predicting the energy performance in a completely new building. Secondly, only the dependence of energy requirements on a limited number of features, such as weather conditions, is considered. Obviously this limitation leads to inaccuracy in the previous models.

To improve the accuracy of the predicting model, this paper presents a simulation method to obtain enough energy consumption data for multiple buildings. Also, it applies SVMs to predict the performance of a new building with construction parameters involved. A parallel implementation is then applied to very large dataset in order to make SVMs more practical in solving complex problems. The plan of the paper is the following. In section 2, we briefly introduce the theory of SVMs method and its parallel approach. In section 3, we present the simulation method for obtaining energy consumption data of multiple buildings. The data analysis and numerical experiments are then presented in section 4. Finally, conclusions are given in section 5.

SUPPORT VECTOR REGRESSION METHOD

Support vector machines aim at finding a decision function (model) to represent the relationship between the features and the target. Let vector i

x indicates the ith sample of the features and i y represents the corresponding target value. Therefore, all the samples can be represented as: , i K y ∈ and l is the number of samples. If y is a continuous variable with real values, then this method is called a support vector regression (SVR). The decision function we are looking for can be expressed as:

()

f xx b ω = + (
ll i i ii C ωξ ξ == ⎛⎞ ++ ⎜⎟ ⎝⎠ ∑∑ (2)
under the constraints * ()

ii i yf x ε ξ -≤ + () ii i fx y ε ξ -≤+ * ,0 , 1 , 2 , . . . , ii il ξξ≥=
where C is a regularizing constant, which determines the trade off between the capacity of ( ) f x and the number of points outside the ε -tube. To solve the problem of Eq. ( 2), we can ,,,,,,) 2

i i ii Lb C ωξ ξ α α γ γ ω ξ ξ == ⎛⎞ =+ + ⎜⎟ ⎝⎠ ∑∑ [ ] ** 11 () ( ) ll ii ii i i i i ii yx b γ ξγ ξ α ω ε ξ == -+ - -⋅ -+ + ∑∑ ** 1 () l ii i i i xb y α ω ε ξ = ⎡⎤ -⋅ + -+ + ⎣⎦ ∑ (3)
The four Lagrange multipliers satisfy the constraints * 0

α ≥ , 0 α ≥ , * 0 γ ≥ and 0 γ ≥ , i = 1, 2, ..., l. If the relations * 0 LL L L b ω ξ ξ ∂∂ ∂ ∂ == == ∂∂ ∂ ∂ occur, we can get the following conditions, * 1 () l ii i i x ωα α = =-∑ (4) * 11 ll ii ii α α == = ∑∑ (5) * 0, ii C αα ≤≤ (6) ** ,1 , 2 , . . . , ii ii Ci l αγαγ =+=+ = (7
) Putting them back into the Lagrange function, we can obtain the solution of the optimization problem through maximizing Eq. ( 3) with respect to the Lagrange multipliers. In the resulting function, there is a dot product of two vectors i x and j x . In practice, it is difficult to find out a linear function ( ) f x for problems involving large dataset. Therefore, we need to map the 1-dimensional problem into a higher dimensional feature space where it is easier to find a linear function similar to ( ) f x in a lower dimensional space. Fortunately, it is not necessary to express explicitly the mapping during the computation. Actually, the dot product can be replaced by a kernel function ( , ) ij K xx . Putting Eq. (4) into Eq. ( 1), and taking into consideration the kernel function, the decision function can be alternated as:

ll ii ii i i i ii Wy α αα α ε α α == =- - + ∑∑ ** ,1 1 () ( ) ( ) 2 l ii jj i j ij K xx αα αα = - -- ⋅ ∑ (8)
under the constraints ( 5) and [START_REF] Mihalakakou | On the energy consumption in residential buildings[END_REF]. Normally, only a certain parts of the samples satisfy the property of * 0 ii αα -≠ , these parts are called support vectors (SVs).

There are four frequently used kernel functions, the linear function, the polynomial function, the radial basis function (RBF) and the sigmod function. They represent different decision shapes in the feature space. In our work, we chose RBF in the training process because it has been tested to be proper for a couple of industrial applications [START_REF] Dong | Applying support vector machines to predict building energy consumption in tropical region[END_REF]. RBF is also called the Gaussian kernel, it has the form 2 ( , ) exp( )

ij i j Kx x x x γ =-- where 0 γ > is the kernel parameter.
Sequential minimal optimization (SMO) method was proposed in [START_REF] Platt | Fast training of support vector machines using sequential minimal optimization[END_REF] to solve Eq. ( 8) instead of standard quadratic problem (QP) techniques. The main idea of SMO is to divide the entire optimization problem into sub-problems and solve each sub-problem independently. In each step, two Lagrange multipliers are optimized analytically until the Karush-Kuhn-Tucher (KKT) conditions are satisfied. Which of the two Lagrange multipliers are selected in the current step is decided by a heuristics algorithm. The parameter b is re-computed in each step. The SMO method has been proved to be a better alternative to handle large datasets because it successfully solved memory load and numerical problems.

However, the problem solving is a time-consuming process when applied to large datasets. In [START_REF] Brugger | Parallel support vector machines[END_REF], Brugger proposed a parallel approach to solve the quadratic problem by using SMO method. By profiling the performance of the training process on different datasets, the author found that the most time-consuming part is located in the kernel evaluation. Therefore, the author parallelized the kernel evaluations and gradient updates, then combined with inner sequential QP solver and distributed storage of kernel rows, and achieved linear speedup for regression. The tool Pisvm was developed accordingly by the author. In our work, we applied this tool to train our model in order to accelerate the training process in high burden conditions.

In our work, we applied two performance evaluation methods. The first one is the mean squared error (MSE) which gives the average deviation of the predicted values to the measured one. Lower the MSE, better is the performance of the prediction. The second one is the squared correlation coefficient (SCC) which lies between [0,1] and gives the ratio of successfully predicted number of target values on total number of target values, i.e. how accurate the predicted values are compared to the measured one. Higher the SCC, stronger is the evaluating ability.

OBTAINING HISTORICAL DATA

The main energy form in a building is the electric and gas consumption for every day use and the space heating demand in cold weather. The energy requirement for a building depends on many factors, so that it would be quite different for each building and quite different upon the time for one given building. The most important influence factors are the ambient weather conditions, building constructions and materials, inner occupants' behaviors. Therefore, as indicated in [START_REF] Pérez-Lombard | A review on buildings energy consumption information[END_REF], it is difficult to obtain accurate consumption data from the real world, normally from measurement. Here we adopt a simulation method by using EnergyPlus [2]. We choose this software because it is a succession of the widely used building energy simulation software including BLAST and DOE-2. It is well tested and has comprehensive ability in calculating complex systems.

In this paper, an office located in Paris-Orly is simulated. The most important parameters for this building are given in Table 1. It is a simple building model with only one room and one thermal zone. The occupants' behaviors, e.g. the people using of facilities, air ventilation, are scheduled as normal office hours. We suppose that the space is heated by district heating, which means that the inlet temperature of the heating system keeps constant. We decided to do this simulation with really recorded weather data from November 1st to March 31th because this duration is a typical heating season in this location and we will analyze how much district heating energy would be consumed by this building. The weather conditions contain solar radiation, dry bulb temperature, relative humidity and so on. To indicate the weather data, the dry bulb air temperatures of the first 20 days in January and July are plotted in Figure 1. The output of the simulation consists of hourly recorded data related to the energy transformation of the building. These data are preprocessed in order to extract useful information and reorganized according to the format required by the analyzing tool. In our model training step, we take district heating demand as the target variable and 25 other variables as the features including day type which indicates if the current day is holiday or not, hourly weather conditions, zone mean air temperatures, infiltration volume, heat gain through each window, heat gain through lights and people, zone internal total heat gain. The energy consumption data for multiple buildings is generated based on the parameters of the above single one. An interface was created to automatically simulate multiple buildings under EnergyPlus. The input file is divided into two parts: an alterable part containing parameters which would be different for each building and a stable part containing parameters which would be kept the same for every building. For example, the building constructions are in the alterable part while weather data is in the stable part. For the alterable parameters, the values are obtained by a stochastic method in a reasonable range. The two parts are then combined together to form the final input file. After successfully simulating one building, we update the alterable part and repeat the simulation process for a new one. In order to analyze multiple buildings, it is necessary to put the output results of all buildings into one single output file in the proper format. The whole controlling flow is presented in Figure 2.

EXPERIMENTS AND NUMERICAL RESULTS

The dataset is divided into two parts, one is used for training models which indicates the dependence of the target on the features and the other one is used for testing the prediction performance of the obtained models. In the testing step, the predicted target regarding to the new values of features is compared with the measured one. Before training the data by SVR, one has to scale the values linearly into a small range in order to avoid numerical problems in the calculations. We scale the training subset into the range [0,1] and then apply the scaling function to scale the testing subset. The experiments are performed on a cluster composed by two homogeneous workstations with one Gigabit Ethernet network. Each computer has 8 processors (2.5GHz), 1333MHz FSB and 4G memory. The operating system used is Linux, the kernel version is 2.6.

Optimizing the parameters of SVR is important to improve the generalization ability of the models. The best choice will give the model well performance without over-fitting problem. In our work, RBF kernel is selected to train the model, so that the parameters need to be optimized are C , γ , ε . The estimation of γ is solved by

2 2 ,1 /( ) l ij ij lx x γ = =-∑
as indicated in [START_REF] Brugger | Parallel support vector machines[END_REF] and [START_REF] Tsang | Core vector regression for very large regression problems[END_REF]. The parameters C and ε are optimized by 5-fold cross validation on randomly selected 3000 samples from the training set.

One hundred buildings located in the same place in heating season are here simulated. The energy consumption data of the first 99 buildings is taken as training set, and the data for the last building is taken as testing set. Therefore, the number of samples for training is equal to 358776 and the number of samples for testing is 3264. The demand of district heating is taken as the target and the other 28 parameters such as weather conditions, the building structures, are taken as features in the analyzing process. The parameters for SVR and RBF kernel are set as 4 C = , 0.3179 γ = and 0.01 ε =

. As the result shows, the number of support vectors are 27501, MSE is 5.01e-5 and SCC is 0.997639. The predicted target compared with the measured one for the first 100 samples is plotted in Figure 3. The model shows very good generalization performance in the result. It proves that SVR with optimized parameters has a comprehensive ability in predicting energy consumption of a new building if we provide enough historical information to the model training. In order to improve the precision of the prediction, one has to collect more historical samples for training. However, the calculating time on processors would increase very fast while the data size is growing. It is a trade-off between the modeling time and the precision. In the above experiment, the model training process needs 31.4 hours in sequential implementation of SVMs -Libsvm [START_REF] Chang | LIBSVM: A library for support vector machines[END_REF]. It is really long for analyzing 100 buildings. To make the analyzing method more practical, it is necessary to reduce the learning time on large datasets. In next experiment, we turn to a parallel approach of SVMs to train the model and test how it performs on this problem.

We analyzed the same dataset as in the above experiment on 1, 2, 4, 6, 8 processors in Pisvm. The cache size is set as 256MB for each process of MPI. Every training process is repeated three times. The average running time and speedup are calculated and shown in Figure 4. From the curves we can see that the speedups of the parallel implementation on 2 and 4 processors are quite close to linear speedup. But the performance enhancement seems not obvious when the number of processors is increased to 6 and 8. That is because of the limitation of our testing environment. On one side, SMP structure inside one node can not make distributed cache approach achieving its full effect when the number of processes is big. On the other side, gigabit network connection is very slow compared to the high performance processors. However, from the performance under 2 and 4 processors, a better speedup can be expected on servers connected by higher speed devices such as infiniBand. The results obtained in those experiments, the number of SVs, MSE and SCC, keep stable under different number of processors. The average results of parallel implementation are compared with sequential implementation in Table 2. This table indicates that, the prediction performance of the parallel solution on large datasets is quite close to the sequential one performed on smaller datasets. The parallel solution can be applied to predict the building energy performances in more complex situations which will allow longer time duration and more features and sample buildings.

CONCLUSION

This paper introduces a simulation approach to collect enough historical time series data for multiple buildings' energy consumption. A statistical learning method is then applied to predict the energy behavior in a completely new building. A parallel implementation of support vector regression with RBF kernel is applied to analyze large amounts of energy consumption data. Experimental results demonstrate that SVR with optimized parameters show a very good generalization ability on such problems. The good performance of the parallel implementation enhances the potential application of SVR on more complex models involving large amounts of samples or features.

Figure 1 :

 1 Figure 1: Dry bulb temperature in the first 20 days of January and July.

Figure 3 :

 3 Figure 3: Measured and predicted district heating demand for the last building in heating season.

Figure 4 :

 4 Figure 4: (a) Running time of the training process using a parallel implementation of SVMs. (b) Comparison of the speedup with a theoretical optimal linear speedup.

Table 1 :

 1 Information of a single building (in metric units)

	Parameters	Values
	Location	Paris-Orly, City
	Duration	From Nov 1 to Mar 31
	Building Shape	Rectangle
	Structure	Length:11 Width:10 Ceiling Height:4 North axis: 10 o
		1IN Stucco, 0.0253m
	Walls	8IN Concrete HW, 0.2033m
		Insulation, 0.0679m
	Fenestration surface	14m 2 for each wall
	Thermal Zones	1
	Number of people	14
	Air infiltration	0.0348 m 3 /s
	Heating type	District heating
		Heating, ventilating, and air
	Cooling type	conditioning
		(windowAirCondioner)
	Other facilities	Light, Water heater

Table 2 :

 2 Comparison of parallel and sequential implementations: The number of Support Vectors (SVs), Mean Squared Error (MSE), Squared Correlation Coefficient (SCC).

	Implementations	SVs	MSE	SCC
	Sequential	27501	5.01097e-05	0.997639
	Parallel	27382	5.08532e-05	0.997571


Frédéric Hai Xiang Zhao
email: zhaohaix@ecp.fr

Magoul Ès

Parallel Support Vector Machines Applied to the Prediction of Multiple Buildings Energy Consumption

Keywords: Energy consumption, building, prediction, support vector machines, parallel computing, grid computing

Analyzing the energy performance in a building is an important task in energy conservation. To accurately predict the energy consumption is difficult in practice since the building is a complex system with many parameters involved. To obtain enough historical data of energy uses and to find out an approach to analyze them become mandatory. In this paper, we propose a simulation method with the aim of obtaining energy data for multiple buildings. Support vector machines method with Gaussian kernel is applied to obtain the prediction model. For the first time, a parallel implementation of support vector machines is used to accelerate the model training process. Our experimental results show very good performance of this approach, paving the way for further applications of support vector machines method on large energy consumption datasets.

INTRODUCTION

Nowadays, energy conservation is a critical task. It is involved in almost every aspect of energy flow including exploitation, delivery and end-use. Predicting the energy consumption of a building is an important approach in energy conservation which benefits both individual and society. With the basic knowledge of the energy performance of a building, it is possible to more wisely design a new building.

However, it is difficult to realize in practice. The first reason lies on the fact that a building is a rather complex system with a great number of influencing factors involved. Taking the heating load for example, it is influenced by the outside weather conditions, the structures, the thermal behavior of the materials, the inner facilities and the behavior of the occupants. Another important reason is that there is not enough data available because of the difficulties in gathering valuable data from the real world [START_REF] Pérez-Lombard | A review on buildings energy consumption information[END_REF]. The third reason lies in the difficulty of analyzing the historical data due to the lack of methods, especially regarding the time consuming aspects when dealing with large datasets.

In order to achieve accurate predictions, it is necessary to collect enough historical data and to find out a practical way to analyze these data. This paper generates data of multiple buildings from simulations with the EnergyPlus [2] software. A well known statistical learning theory, the support vector machines (SVMs) method [START_REF] Vapnik | The nature of statistical learning theory[END_REF], together with the Gaussian kernel (RBF) are applied to obtain the prediction model. A parallel implementation of SVMs is used for the first time to accelerate the model training process on large datasets. Our experimental results show that SVMs method has a good performance in predicting unknown buildings, and that the parallel implementation makes it possible to analyze large energy consumption datasets in practice. Section 2 of this paper briefly introduces the related work, and Section 3 introduces the learning theory and its parallel implementation. How to collect the data is introduced in Section 4. Description of the experiments and analysis of the results are presented in Section 5. Finally, the conclusions are presented in Section 6.

RELATED WORK

In recent years, the analysis of building energy performance is wildly studied. A large number of software have been developed for evaluating energy efficiency and sustainability of buildings [2,[START_REF] Al-Homoud | Computer-aided building energy analysis techniques[END_REF], such as DOE-2, AkWarm, Antherm, Apache, etc. Most of them are simulation tools which calculate, on an hourly or monthly basis, the thermal transfer or energy variation of buildings and facilities from an engineering point of view.

At the same time, researchers have developed several analyzing models to predict the energy performance based on historical data. The regression model proposed in [START_REF] Fels | Special issues devoted to measuring energy savings: the scorekeeping approach[END_REF][START_REF] Ruch | A fourth parameter change-point model for predicting energy consumption in commercial buildings[END_REF][START_REF] Katipamula | Multivariate regression modeling[END_REF] is well suited for long period energy predictions and is easy to develop. The time-series analysis [START_REF] Kimbara | On-line prediction for load profile of an air-conditioning system[END_REF] and the Fourier series model [START_REF] Dhar | Modeling hourly energy use in commercial buildings with fourier series functional form[END_REF] were proposed for analyzing historical time series data. The neural network related methods are the most frequently applied approaches [START_REF] Wong | Artificial neural networks for energy analysis of office buildings with daylighting[END_REF][START_REF] Azadeh | Annual electricity consumption forecasting by neural network in high energy consuming industrial sectors[END_REF][START_REF] Hernandez | Comparison between detailed model simulation and artificial neural network for forecasting building energy consumption[END_REF][START_REF] Karatasou | Modeling and predicting building's energy use with artificial neural networks: Methods and results[END_REF][START_REF] Ekici | Prediction of building energy consumption by using artificial neural networks[END_REF]. Some of these methods have been thoroughly tested and thus can be used as benchmark calculations for other prediction methods. Support vector machines (SVM) are a set of methods for classification and regression. It is close to multi-layer neural network and has good generalization abilities in solving non-linear problems [START_REF] Vapnik | The nature of statistical learning theory[END_REF][START_REF] Brugger | Parallel support vector machines[END_REF]. Despite it is widely used in industries, only few work has been done for applying it to analyze building energy behaviors [START_REF] Dong | Applying support vector machines to predict building energy consumption in tropical region[END_REF][START_REF] Lai | Vapnik's learning theory applied to energy consumption forecasts in residential buildings[END_REF][START_REF] Li | Applying support vector machine to predict hourly cooling load in the building[END_REF]. Dong et al. [START_REF] Dong | Applying support vector machines to predict building energy consumption in tropical region[END_REF] first applied SVMs to predict the monthly electricity consumption of four buildings in tropical region. Three years' data was trained and the derived model was applied on one year's data to predict the landlord utility in that year. The results showed good performances of SVMs on this problem.

Lai et al. [START_REF] Lai | Vapnik's learning theory applied to energy consumption forecasts in residential buildings[END_REF] applied robust regression model on one year's data of a build-ing. The dataset was recorded on a daily basis with electricity consumption and climate variation involved. In these experiments, the authors trained the model on one year's data and then applied it to three month's data to test the predication ability of this derived model. The authors also trained the models on each daily basis dataset and then compared the obtained models to verify the stability of this approach. In addition, perturbation to certain data has been added and detection of the perturbation by examining the change of the contributing weights has been investigated. Li et al. [START_REF] Li | Applying support vector machine to predict hourly cooling load in the building[END_REF] used SVMs to predict the hourly cooling load of an office building. The performance of the support vector regression was compared with the conventional back-propagation neural networks and it was proved to be better than the traditional solution.

The above three works have shown that SVMs can provide good performances in predicting hourly and monthly building energy. However, there are several shortages in these studies. Firstly, all the experiments were performed on a small number of buildings which means that we do not know the ability of SVMs in predicting the energy performance in a completely new building for instance. Secondly, the datasets for model training were not large enough, which could lead to a certain degree of limitations of these models. For instance, the relation between the energy consumption and a limited number of features such as weather conditions are considered, and the time duration is not long enough.

THEORY OF SUPPORT VECTOR MACHINES

Support vector regression

Support vector machine is a supervised learning method which aims at finding a decision function to represent the relationship between the features and the target. This function is also called a model or a pattern. The features are also called the variables. According to different types of targets, SVM is classified into two types: one is the classification in which the target has only two values, e.g. {0, 1}, and the other type is the regression in which the target has continuous real value.

Let vector x i represents the ith sample of the features and y i represent the corresponding target value, therefore, all of the samples can be represented as: (x 1 , y 1 ), (x 2 , y 2 ), ...(x l , y l ) where x i ∈ R n and means that there are n features, y i ∈ R and l is the number of samples. We are aiming at finding a decision function of the form:

f (x) = ωx + b (1)
where f (x i ) is the estimation of the corresponding y i . If such a function is found, we can use it to predict the unknown y i with the new input x i . This is defined as the prediction ability of the derived model. The process to find (1) is called model training. In the above function, ω and b are coefficients. The main task now is to find out the proper values of ω and b. We approximate them by minimizing the empirical risk with respect to the loss function:

L(y -f (x)) = l i=1 |y i -f (x i )| (2)
In this paper, the target is the total energy consumption in a building-which is a real continuous variable. Thus we only use support vector regression (SVR) to estimate the target, and we talk about SVR. To make our estimation robust, we use an ε-insensitive loss function instead of the above quadratic loss function.

L(y -f (x)) = 0 if |y -f (x)| ≤ ε |y -f (x)| -ε otherwise
This loss function corresponds to an ε-tube around the measured target values. We assume that there is no deviation of the predicted values from the measured ones if they lie inside the tube. To find out the vector ω, the problem is equivalent to the following quadratic optimization problem after introducing two slack variables ξ i and ξ * i , i = 1, 2, ..., l, and where we minimize

1 2 ω 2 + C( l i=1 ξ * i + l i=1 ξ i ) (3)
subject to the constraints

y i -f (x i ) ≤ ε + ξ * i f (x i ) -y i ≤ ε + ξ i ξ * i , ξ i ≥ 0, i = 1, 2, ..., l
where C is a regularizing constant, which determines the trade off between the capacity of f (x) and the number of points outside the ε-tube. To find the saddle point of the function (3) under the previous inequalities constraints, one can turn to the Lagrange function by introducing four Lagrange multipliers, α * , α, γ * , γ. The Lagrange function becomes:

L(ω, b, ξ * , ξ, α * ,α, γ * , γ) = 1 2 ω 2 + C( l i=1 ξ * i + l i=1 ξ i ) - l i=1 α i [y i -(ωx i ) -b + ε + ξ i ] -α * i [y i -(ωx i ) -b + ε + ξ * i ] - i=1 l(γ * i ξ * i + γξ) (4)
The four Lagrange multipliers satisfy the constraints α * ≥ 0, α ≥ 0, γ * ≥ 0 and γ ≥ 0, i = 1, 2, ..., l. If the relations

∂L ∂ω = ∂L ∂b = ∂L ∂ξ * = ∂L ∂ξ
occur, we get the following conditions,

ω = l i=1 (α * i -α i )x i l i=1 α * i = l i=1 α i (5) 0 ≤ α * i , α i ≤ C (6)
C = α * i +γ * i = α i + γ i , i = 1, 2, ..., l (7)
Putting them back into the above Lagrange function, we obtain the solution of the optimization problem which is equal to the maximum of the function ( 4) with respect to the Lagrange multipliers. The next step is to find α * i and α i in order to maximize the following function:

W (α * i , α i ) = l i=1 y i (α * i -α i )-ε l i=1 (α * i +α i )- 1 2 l i,j=1 (α * i -α i )(α * j -α j )(x i •x j ) (8)
under the constraints ( 5) and ( 6), where x i • x j stands for the dot product of two vectors x i and x j . Normally, only some of the samples satisfy the property of α * iα i = 0 , which are called support vectors (SVs). In fact, only these samples lying outside the ε-tube will contribute to determine the decision function Eq. ( 1). In practice, it is difficult to find out a linear function f (x) for real problems with a large dataset. In such cases, one often maps the 1-dimensional problem into a higher dimensional feature space where it is easier to find a linear function similar to the previous decision function. Fortunately, it is not necessary to express the mapping explicitly. Instead, the final form f

(x i ) = ϕ(x i ) + b is good enough. Thus, the last term x i • x j of (8) is changed into ϕ(x i ) • ϕ(y i )
which is called the kernel function and the term K(x i • y i ) is used to replace it. Then, combining with Eq. ( 4), the decision function goes to:

f (x) = l i=1 (α * i -α i )K(x i • x) + b (9)
where α * i and α i are determined by maximizing the quadratic function:

W (α * i , α i ) = l i=1 y i (α * i -α i ) -ε l i=1 (α * i + α i ) - 1 2 l i,j=1 (α * i -α i )(α * j -α j )K(x i • x j ) ( 10
)
under the constraints ( 5) and ( 6).

There are four frequently used kernel functions in SVMs, which are the linear function, the polynomial function, the radial basis function (RBF) and the sigmod function. They represent different decision shapes in the feature space. In this paper, we chose RBF in the training process because it has been tested to be proper for some industrial applications [START_REF] Dong | Applying support vector machines to predict building energy consumption in tropical region[END_REF]. RBF is also called Gaussian kernel, with the form K(x i , x j ) = exp(-γ x ix j

2 ) where γ > 0 is the kernel parameter.

Solving the quadratic problem

When solving Eq. ( 10) via standard quadratic problem (QP) techniques, there would be a matrix in the memory whose number of elements is equal to the square of the number of samples [START_REF] Platt | Fast training of support vector machines using sequential minimal optimization[END_REF]. If the trained data is very large, it would become difficult to load the whole matrix into the computer memory. Furthermore, there are a significant number of numerical problems when solving such a function. Therefore, sequential minimal optimization (SMO) method was proposed and has been proved to be a better alternative to handle these two difficulties when dealing with large datasets. The main idea of SMO is to divide the entire optimization problem into sub-problems and to solve each sub-problem step by step, until finally conquer the whole problem. In each step, two Lagrange multipliers are optimized analytically until the Karush-Kuhn-Tucher (KKT) conditions are satisfied. Which two Lagrange multipliers are selected in current step is decided by an heuristics algorithm. The parameter b is re-computed in each step. More details on the SMO method could be found in [START_REF] Brugger | Parallel support vector machines[END_REF][START_REF] Platt | Fast training of support vector machines using sequential minimal optimization[END_REF].

Parallel implementation

SVM training becomes a time consuming process when applied to large datasets. Brugger [START_REF] Brugger | Parallel support vector machines[END_REF] integrated SMO method to solve the quadratic problem in parallel. By profiling the performance of the optimization process in different datasets, the author found that the most time-consuming part is located in the kernel evaluation. Therefore, he parallelized the kernel evaluations and gradient updates, combined with inner sequential QP solver and distributed storage of kernel rows, achieved linear speedup for regression in his test. The tool Pisvm was developed using this idea.

In this paper, we trained our model on multiple buildings datasets with the Pisvm tool for the purpose of accelerating the training process in high burden conditions.

Performance evaluation

Two performance evaluation methods were applied in this work. One is the mean squared error (MSE) which gives the average deviation of the predicted values to the measured one. The lower the MSE, the better the performance of the prediction. The other method is the squared correlation coefficient (SCC) which lies between [0, 1] and gives the ratio of successfully predicted number of target values on the total number of target values, i.e. how certain the predicted values are compared to the measured one. The higher the SCC, the stronger the evaluating ability.

DATA PREPARATION

Buildings for office use located in Paris-Orly were simulated via EnergyPlus. We chose this tool because, as a succession of the well-known building energy simulation software BLAST and DOE-2.1E, it is well tested and comprehensive in calculating energy performance of complex systems. In our work, we generated the data of a single building in heating season initially. Then, by modifying some alterable parameters, we generated the data for multiple houses.

Before running the simulation, one has to create an input file (idf file) to give the parameters of a building for EnergyPlus. The input parameters include the weather conditions, the building structures, the inside occupants' behaviors, the schedule of light using, etc. The most important parameters are shown in Table 1 and the materials of the surfaces are given in Table 2. The description of these materials can be found in the documents of EnergyPlus [2]. We supposed that there was only one floor and one room in this building. Because it was in heating season, the energy consumed by this building mainly from three sources which are the district heating used to keep the inside space warm, the electricity lights used mostly in working days, and the hot water for office use. For the walls in each orientation, there are several construction layers for thermal reasons. That explains why there are three materials for the walls as indicated in Table 2. The same occur for the roof. Besides these parameters, we used hourly recorded weather data in Paris-Orly along with other important input parameters. The weather data includes dry bulb air temperature, relative humidity, global horizontal radiation and ground temperature. To have a glance of these data, the dry bulb air temperatures of the first 20 days in January and July are plotted in Figure 1, with the relative humidity in the same days plotted in Figure 2.

In this simulation, the output was hourly damped. There were several output files in EnergyPlus, where we extracted useful data, mainly from the eso file. As the analyzing step required, we have to reformat this data into the form required by the analyzing tools. We take district heating demand or total electricity consumption as the target. Meanwhile, we take 25 variables as the features for a single building, which are day type indications if the current day is holiday or not, weather conditions, zone mean air temperatures, infiltration volume, heat gain through each window, heat gain through lights and people, zone internal total heat gain.

In order to generate the data for multiple buildings, we developed an interface to automatically control the simulation process since there was no available user interface for EnergyPlus for this purpose. In our approach, the input file Relative Humidity (%)

Day

In January In July was divided into two parts. The first part is called the alterable part, containing the parameters which would be different for each building. The values of these parameters were obtained by stochastic methods. The second part is called stable part where the parameters are always the same for each building. After updating the alterable part for a new building, we combined this part with the stable part to create the final input file. To analyze multiple buildings, we have to put all the generated output data of each building into a single output file, which is named as output.txt. Then, the program constantly updates the alterable part to create the data for the next building. The above steps of generating the multiple buildings are shown in Figure 3.

EXPERIMENTS AND RESULTS

In scaling function. Therefore, it is possible for the scaled test values to be in a different range from that of the training values.

To choose the optimal values for the SVM parameters is really crucial in practice. The best parameters should have the ability to well predict on unknown data without the over-fitting problem. In our experiments, we selected SVR with RBF kernel to train our model. The parameters needed were C, γ, ε. The estimation of γ was solved by γ = l i,j=1 ( x ix j 2 ) as proposed in [START_REF] Tsang | Core vector regression for very large regression problems[END_REF] and [START_REF] Brugger | Parallel support vector machines[END_REF]. The SVR parameters C and ε were solved by 5-fold cross validation on randomly selected 3000 samples from the training set. The above steps necessary for our experiments are shown in Figure 4.

Three experiments have been performed on different types of datasets. The first one is to analyze energy consumption data for a single building in order to test the prediction performance of SVMs method on building energy. The second one is to analyze multiple buildings with detailed building structures involved. The third one is to test the performance of parallel SVMs on a large dataset from multiple buildings. The first and the second experiments were done in Libsvm [START_REF] Chang | LIBSVM: a library for support vector machines[END_REF] which is a widely used sequential implementation of SVMs. The third one was performed in the parallel implementation tool -Pisvm. The experimental environment is a cluster composed by two homogeneous workstations linked with Gigabit Ethernet. Each PC has 8 * 2.5GHz CPU, 1333MHz FSB and 4G memory. The operating system is Linux (kernel version 2.6).

In the first experiment, the district heating consumption data was gathered hourly from November 1st to March 31st. There were 3624 samples with 24 features in the final dataset. We took the samples of the last two days as testing set, and the rest of them for training use. The number of training samples was 3576 and the number of testing samples was 48. The parameters were set as C = 16, γ = 0.7533 and ε = 0.01. The result of the learning and predicting processes showed that the number of support vectors (SVs) was 2229, MSE was 2.3e-3 and SCC was 0.927918. The measured and predicted targets are plotted Figure 5 which shows very good prediction performance of the model.

important energy type consumed by buildings is electricity. We also trained a similar model to predict the electricity consumption through one year. To be different from the first experiment, this time we randomly selected 48 samples as testing set for the evaluation of the model. The features were the same as in the first experiment. The number of training samples was 8712. The parameters were set as C = 16, γ = 0.3043 and ε = 0.01. The result showed that, there were 2126 support vectors, MSE was 5.27e -4, and SCC was 0.959905. The measured and predicted values are plotted in Figure 6. The model performs very well in this situation, except in the hours 18 and 28.

The above two learning processes are based on a single building energy consumption data. The evaluation of the model is to predict the unknown future in the same building. In practice, it is quite useful to predict how much energy would be used in a completely new building. Therefore, in the second experiment, we tried to learn a model based on the energy data according to the building structures involved. That is to say, we trained a model from the consumption behaviors of several buildings, then applied the model to predict the behavior of a different building. In this experiment, one hundred buildings were simulated in the heating season. They were in the same weather conditions but have different properties, such as different orientations, volumes, people densities and fenestration. We chose the data of the first 99 buildings as the training set and data of the last building as the testing set. The number of features was 28. The number of training samples was 358776, and the number of testing samples was 3624. The parameters were set as C = 4, γ = 0.3179 and ε = 0.01. In the training step, the number of SVs was 27501, while in the predicting step, MSE was 5.01e -5 and SCC was 0.997639. The predicted and measured values on the first 100 samples in test dataset are plotted in Figure 7. The experimental results prove that SVR has a very good prediction performance in building energy consumption when building diversity is taken into account. It gives us the possibility to predict the energy performance of a building for designing as well as for retrofitting. To increase the accuracy of the prediction, we have to collect energy consumption data as much as possible. But the problem is that the analyzing time increases quite fast while the data size is growing. In the second experiment, the most time-consuming process is the training step which needs 31.4 hours in our experimental environment. It becomes really long for analyzing 100 buildings. It is obvious that the time would go even longer if we calculate more buildings. To make the model more practical, it is necessary to reduce the learning time. As explained in Section 4, we turn to a parallel approach of SVMs in the next experiment. To test the speedup of this parallel solution, we did a set of experiments on 1, 2, 4, 6, 8 processors with the cache set as 256MB for each process. Each experiment was repeated three times and the average training time and speedup were calculated as shown in Figure 8. From the plotting, we can see that the speedups in 2 and 4 processors is close to linear speedup. But the performance enhancement is not obvious in 6 and 8 processors. One reason is that, in our testing environment with SMP structure inside the node, the distributed cache approach could not achieve its effect when the number of processes is big. Another reason is the low speed network connection between two nodes. However, from the varying trend in 2 and 4 processors, a better speedup can be expected on servers connected by higher speed devices such as infiniBand. The prediction performances, MSE and SCC, keep stable in each experiment. The number of SVs, MSE and SCC are compared with the sequential implementation as shown in Table 3. solution on large datasets is quite close to the sequential implementation on smaller datasets. The parallel solution can be applied to predict the building energy performances in more complex situations.

CONCLUSION

In this paper, a parallel implementation of support vector machines (SVM) method is applied for the first time to the prediction of energy consumption in multiple buildings based on large time series datasets. SVM has showed good performance in the prediction of building energy performance for some related work [START_REF] Dong | Applying support vector machines to predict building energy consumption in tropical region[END_REF][START_REF] Lai | Vapnik's learning theory applied to energy consumption forecasts in residential buildings[END_REF][START_REF] Li | Applying support vector machine to predict hourly cooling load in the building[END_REF]. Since building energy problems are quite complex, we have to obtain historical data as much as possible for making our prediction more accurate. Consequently, the learning process of SVM would become very slow as the size of data increases. This work introduced an approach of collecting abundant energy consumption data of multiple buildings by simulation. The prediction of energy performance in a completely new building was performed. A parallel approach of SVM is applied on those data to accelerate the training process. SVR with carefully chosen parameters showed a good performance in these experiments. Parallel SVM has a strong potential for analyzing extremely large energy data which would contribute to the accuracy of prediction in more complex situations in practice.

However, the evaluation of our work based on simulated buildings is not good enough. The better way is to record the real energy consumption data in various types of buildings. In addition, the speedup is not as good as linear one. To improve it, there are at least two possibilities, one is to take more consideration on selecting the features, the other is to improve the parallel solution of SVMs.

Figure 1 :

 1 Figure 1: Dry bulb temperature in the first day of each month.

Figure 2 :

 2 Figure 2: Relative humidity in the first day of each month.

Figure 3 :

 3 Figure 3: Flow chart of generating energy consumption data of multiple buildings.

Figure 4 :

 4 Figure 4: Flow chart of a learning process.

Figure 5 :

 5 Figure 5: Measured and predicted district heating demand in heating season in the first experiment.

Figure 6 :

 6 Figure 6: Measured and predicted electricity consumption through one year in the first experiment.

Figure 7 :

 7 Figure 7: Measured and predicted electricity consumption in the second experiment.

Figure 8 :

 8 Figure 8: (a) Running time of the training process in parallel implementation of SVMs. (b) The speedup in comparison with linear speedup.

Table 1 :

 1 Input parameters of a single building (in metric units).

	Parameters	Values
	Location	Paris-orly, City
	Duration	From Nov 1 to Mar 31
	Building Shape	Rectangle
	Structure	Length:11 Width:10 Ceiling Height:4 North axis: 10 o
	Fenestration surface	14m 2 for each wall
	Thermal Zones	1
	People	14
	Air infiltration	0.0348 m 3 /s
	Heating type	District heating
	Cooling type	HVAC windowAirCondioner
	Other facilities	Light, Water heater

Table 2 :

 2 Building materials used for the simulation.

	Structures	Material's name	Thickness(m) Conductivity(W/mK)
		1IN Stucco	0.0253	0.6918
	Wall	8IN Concrete HW	0.2033	1.7296
		Wall Insulation	0.0679	0.0432
	Ground MAT-CC05 8 HW CONCRETE	0.2032	1.311
		Roof Membrane	0.0095	0.16
	Roof	Roof Insulation	0.1673	0.049
		Metal Decking	0.0015	45.006
	Windows	Theoretical Glass [117]	0.003	0.0185

  supervised learning theory, the experiments can be roughly divided into two steps, training and predicting. Accordingly, the data is divided into two parts, one is for training, called a training set, and the other is for predicting, called a testing set. A decision model is obtained in the training step on the training set to indicate the dependence of the target on the features. In the predicting step, the trained model is applied on the testing set to predict the target values with regard to new features. By comparing the predicted target with the measured one, it is possible to evaluate the prediction performance of the model.Before training the data by SVR, we need to scale the values linearly into a small range in order to avoid numerical problems in the training. Here we chose the range [0, 1]. Likewise, the testing values should also be scaled with the same

		Input file (idf)
		Divide
	Yes	Alterable?	No
	Alterable part
			Stable part
	Update
		Combine
		New input file
		Simulate
	Simulate next
		Ouput of energyplus
		Reformat
		Update Output.txt

Table 3

 3 indicates that the result of parallel implementation is quite close to the sequential one, which means that the prediction performance of the parallel Parallel Statistical Learning Method for Building Energy Prediction

		0.5
														measured
		0.45													predicted
		0.4
		0.35
	Scaled electricity consumption	0.1 0.15 0.2 0.25 0.3
		0.05
		0
		-0.05
		-0.1
		6	12	18	24	30	36	42	48	54	60	66	72	78	84	90	96
									Hour

Table 3 :

 3 Comparison of parallel and sequential implementations.

	Implementations	SVs	MSE	SCC
	Sequential	27501 5.01097e-05 0.997639
	Parallel	27382 5.08532e-05 0.997571