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Prediction of wind pressure coeﬃcients on building surfaces using

Artiﬁcial Neural Networks

Facundo Brea,b, Juan M. Gimeneza,c , V´ıctor D. Fachinottia

aCentro de Investigaci´on de M´etodos Computacionales (CIMEC), UNL, CONICET, Predio “Dr. Alberto Cassano”,

Colectora Ruta Nacional 168 s/n, 3000, Santa Fe, Argentina

bGrupo de Investigaci´on en Mec´anica Computacional y Estructuras (GIMCE), Facultad Regional Concepci´on del Uruguay

(FRCU), Universidad Tecnol´ogica Nacional (UTN), 3260, Concepci´on del Uruguay, Argentina

cFacultad de Ingenier´ıa y Ciencias H´ıdricas - Universidad Nacional del Litoral, Ciudad Universitaria, Paraje “El Pozo”,

Santa Fe, Argentina

Abstract

Knowing the pressure coeﬃcient on building surfaces is important for the evaluation of wind loads and

natural ventilation. The main objective of this paper is to present and to validate a computational

modeling approach to accurately predict the mean wind pressure coeﬃcient on the surfaces of ﬂat-, gable-

and hip-roofed rectangular buildings. This approach makes use of Artiﬁcial Neural Network (ANN) to

estimate the surface-average pressure coeﬃcient for each wall and roof according to the building geometry

and the wind angle. Three separate ANN models were developed, one for each roof type, and trained

using an experimental database. Applied to a wide variety of buildings, the current ANN models were

proved to be considerably more accurate than the commonly used parametric equations for the estimation

of pressure coeﬃcients. The proposed ANN-based methodology is as general and versatile as to be easily

expanded to buildings with diﬀerent shapes as well as to be coupled to building performance simulation

and airﬂow network programs.

Keywords: Pressure coeﬃcient, Natural ventilation, Building performance simulation, Artiﬁcial Neural

Network

1. Introduction

Energy consumption and indoor environment of buildings are inﬂuenced by air inﬁltration and venti-

lation [1, 2]. Wind induced pressure on the building envelope aﬀects the air inﬁltrations because of the

indoor-outdoor pressure diﬀerence. Wind pressure is also an important boundary condition for a wide

range of problems, including heat, air and moisture (HAM) transfer, airﬂow network (AFN), and building

energy simulation (BES) [3]. Generally, the wind pressure is characterized by the pressure coeﬃcient

deﬁned as:

Cp=Px−P0

ρU2

h/2,

where Pxis the static pressure at a given point on the building fa¸cade, P0is the static reference pressure

at freestream, ρU2

h/2 is the dynamic pressure at freestream, ρis the air density and Uhis the wind speed,

Preprint submitted to Elsevier November 29, 2017

which is often taken at the building height hin the upstream undisturbed ﬂow.

A review of pressure coeﬃcient data for building energy simulation and airﬂow network programs

was made by C´ostola et al. [3]. They classiﬁed the sources of Cpdata in two main groups: i) primary

sources, including full-scale measurements, reduced-scale wind-tunnel tests and computational ﬂuid dy-

namics (CFD) simulations, and ii) secondary sources, like databases and analytical models. Usually,

because of their diﬃculty and cost, full-scale and wind-tunnel scale measurements are only used for the

development of wind pressure coeﬃcient databases or the evaluation of complex high-rise buildings. Ana-

lytical models are commonly used to predict surface-average Cpon low-rise buildings, where the variation

of Cpover the surface can be neglected assuming that cracks are homogeneously distributed over the

building fa¸cades [4]. This simpliﬁed approach is widely used to include airﬂow network analysis in BES

[5].

Swami and Chandra [6] proposed simple equations for low- and high-rise buildings separately, which

were obtained using step-wise regression analysis to ﬁt some previously published studies of wind pressure

coeﬃcients. The Swami and Chandra’s equation for low-rise buildings –from now on, referred to as the

S&C equation– is a popular analytical model to predict surface average Cpthat has been implemented in

widely used BES programs like EnergyPlus [7]. It is valid for rectangular ﬂoor-plan buildings and depends

on two parameters: wind direction and side ratio. In the original work [6], the S&C equation was applied

to a broad range of data, including buildings with diﬀerent heights and roof pitch angles, yielding an

acceptable correlation coeﬃcient of 0.797. However, it needs to be improved and updated in the light of

the new high-tech measurement databases. A detailed description of the low-rise S&C equation and its

parameters can be found in the Appendix 5.1.

Grosso [8] proposed a set of complex parametric models in order to take into account sheltering

eﬀects, which cannot be estimated using the S&C equation. However, because of the lack of complete and

high-quality experimental data, he recognized that the most useful contribution of his work may be the

proposed methodology rather than the speciﬁc results.

Recently, Muehleisen and Patrizi [9] developed a new parametric equation –the so-called M&P equation

henceforth– to predict wind pressure coeﬃcient for low-rise buildings. This is a simple rational equation

calibrated on the based of the new, large and very detailed database of the Wind Engineering Information

Center at the Tokyo Polytechnic University (TPU) [10]. The M&P equation ﬁts very well the TPU

database, with a coeﬃcient of determination R2= 0.993. The low-rise M&P equation and its parameters

are detailed in the Appendix 5.2.

Compared to the S&C equation, the M&P equation better ﬁts not only the TPU database but also the

database compiled by the Air Inﬁltration and Ventilation Centre (AIVC) [11]. However, the S&C equation

performs better in the common case of buildings with unity depth-to-breadth ratio over a considerable

range of wind attack angles (90◦< θ < 165◦). So, both S&C and M&P equations have relative advan-

tages and disadvantages depending on their application. Furthermore, neither of them can be applied to

2

buildings with non-rectangular ﬂoor plans. Actually, up to the authors’ knowledge, this is the common

limitation of all the available analytical models.

To overcome such limitations, the current paper proposes a methodology to obtain a computational

model satisfying three main requirements: i) accurate prediction of Cp, ii) valid for buildings with various

ﬂoor-plan shapes (rectangular, U-shape, L-shape, etc.), and ii) easy coupling to AFN and BPS programs.

To this end, Artiﬁcial Neural Networks (ANN) are used to build an analytical model of the surface-

average Cpfor every surface of the building (walls and roofs) and for every wind attack angle. The

robustness of the method is proved through its application to three low-rise rectangular building cases: ﬂat-

roofed, gable-roofed, and hip-roofed. Data for training and testing is taken from the TPU database [12].

Finally, the results obtained using the current models are compared with both S&C and M&P equations

and the TPU experimental database, highlighting the accuracy of the proposed methodology to ﬁt the

experiments.

2. Methodology

This section deﬁnes the methodology proposed to develop analytical models of the surface-average Cp

in low-rise buildings. Section 2.1 details the method to build and calibrate of the ANN models. Section 2.2

describes the experimental database that served to calibrate the proposed model, together with the chosen

case studies.

2.1. Artiﬁcial Neural Networks (ANN)

An ANN is a massively parallel distributed processor made up of simple processing units that has

a natural propensity for storing experimental knowledge and making it available for use in analytical

way [13]. ANN is often used as a surrogate model or a response surface approximation model because of

its robustness to solve multivariate and nonlinear modeling problems, like function approximations and

classiﬁcation.

Some authors have used ANN to predict or interpolate Cpvalues. Kalogirou et al. [14] used ANN to

predict Cpacross the openings in a light weight single-sided naturally ventilated test room. Chen et al. [15]

developed an ANN to predict Cpon gable roofs according to wind direction, roof height, and normalized

roof coordinates. This work was recently extended by Gavalda et al. [16] in order to include variable plan

dimensions and roof slopes as parameters. On the other hand, Fu et al. [17] developed a Fuzzy Neural

Network (FNN) approach to predict wind loads and their power spectra on a large ﬂat roof. But, up to

the authors’ knowledge, neither ANN nor any other surrogate or response surface method can be found

in the literature to predict Cpfor every surface of a building for a wide range of input parameters.

In this work, a feed-forward multilayer ANN is used. Fig. 1 shows the general ANN architecture,

which has an input layer, a set of hidden layers and an output layer. In each hidden and output layer,

there are artiﬁcial neurons interconnected via adaptive weights. These weights are calibrated through a

3

training process with input-output data. For each artiﬁcial neuron, there is an activation function, which

can be any function with range [−1,1]; the most common activation functions are the tangent sigmoid

and the logarithmic sigmoid [13].

The deﬁnition of an ANN architecture includes determining the number of inputs, outputs, and hidden

neurons, and the number of hidden layers [18]. The universal approximation theorem [19] states that a

feed-forward network with a single hidden layer containing a ﬁnite number of neurons can approximate

continuous functions on compact subsets of Rn, being nthe number of inputs. However, this does not

mean that an ANN with one single hidden layer is optimal in terms of versatility, learning time and ease

of implementation. Indeed, given the sets of input and output data, there is not a general rule to deﬁne

the best ANN architecture (number of neurons and hidden layers). We propose here a method to calibrate

the ANN architecture based on trial & error according to the complexity of each case. It consists of two

steps: 1) a coarse calibration is made to determine the number of hidden layers, and 2) training with a

increasing number of hidden neurons (starting with a few ones) until achieving the desired performance.

The so-determined ANN architectures for the three considered case studies are detailed in Appendix 5.3.

Input 2

Input n

Input 1

Hidden layers

Input layer

i

Output layer

o

Output 1

Output n

h1h2hn

Fig. 1. Artiﬁcial neural network architecture (ANN i-h1-h2-hn-o).

The ANN training process was made using the Levenberg-Marquardt (LM) backpropagation algo-

rithm [20], considering the mean squared error (MSE) as convergence indicator and a maximum of 500

epochs. Let us remark that the LM method has second-order convergence rate and it was recommended

by Hagan and Menhaj [21] because of its eﬃciency for ANN with no more that a few hundreds weights,

as it is curently the case.

The previous guidelines to build an ANN models can be implemented on several computational plat-

form such as MATLAB [22], R [23], TensorFlow [24], among others.

Now, let us apply the proposed methodology to predict the surface-average Cpon low-rise building

surfaces. In this case, the ANN outputs are the surface-average Cps on the building surfaces, while the

4

ANN inputs deﬁne the wind direction and building characteristics, the latter being speciﬁc for each one

of the three case studies described in the Section 2.2.1, giving rise to particular ANNs. The input-output

data to train all these models is described below.

2.2. Databases

There are several databases containing measured Cpin building surfaces for various building geometries

and a wide range of wind attack angles. The NIST aerodynamic database [25] contains time series of wind

load data for low-rise gable-roofed buildings with various dimensions and terrain conditions use in the

design of low-rise buildings. The database from the ASHRAE Handbook of Fundamental [26] contains the

surface-average Cpfor low- and high-rise buildings with rectangular ﬂoor-plan, considering four diﬀerent

side ratios and wind attack angles from 0◦to 180◦every 15◦. A more detailed Cpdatabase was compiled

by the Air Inﬁltration and Ventilation Centre (AIVC) [11, 27]. It contains tables with surface-averaged

Cpfor each face of rectangular low-rise buildings for wind attack angles from 0◦to 180◦every 45◦, and

for three diﬀerent shielding levels: exposed, semi-sheltered and sheltered.

More recently, the Wind Engineering Information Center of the Tokyo Polytechnic University (TPU)

published a comprehensive Cpdatabase derived from a series of wind tunnel tests on a wide variety of

buildings, including high-rise buildings and isolated and non-isolated low-rise buildings [12]. Given its

quality and completeness, the TPU database for isolated low-rise buildings [28] will be used to train and

test the ANN-based models of surface-averaged Cpin the current work. Such database contains 4176

contours of statistical values of local Cp, 700 graphs of statistical values of area-averaged Cpon the roof

or the wall surfaces and time series data of point Cpfor 812 test cases. The surface-averaged Cpon the

building surfaces (walls and roofs) are presented as plots of mean-surface pressure for wind attack angles

from 0◦to 90◦every 15◦.

2.2.1. Case studies

The TPU database for isolated low-rise buildings [28] gives the surface-averaged Cpfor all the surfaces

(walls and roofs) of rectangular ﬂoor-plan buildings with either ﬂat, gable or hip roof, see Fig. 2. Specif-

ically, the surface-averaged Cp, say Cp, is given for some values of the depth-to-breadth (or side) ratio

D/B, the height-to-breadth (or height) ratio H/B, the pitch angle β(in case of gable and hip roofs) and

wind attack angle θ. Note that for gable- and hip-roofed buildings, His deﬁned as the mean roof height.

In this database, θranges from 0 to 90◦every 15◦. Since Cpis measured on all the surfaces of rectangular

buildings, the upper bound of θis easily extended to 180◦considering that the Cpfor θat a given surface

coincides with the Cpfor the wind attack angle 180◦−θat the opposite surface. In a similar way, results

for the side ratio D/B =S > 1 and 0◦≤θ≤90◦can be extended to D/B = 1/S ≤1 taking into account

that the Cpfor Sand wind attack angle θat a certain surface coincides with the Cpfor 1/S and wind

attack angle 90◦−θat the perpendicular surface.

5

Also, although Cpmust be identical for opposite surfaces parallel to the wind direction, this is not the

case in the TPU database because of measurements errors which may have been derived from diﬀerent

sources (e.g., sensor errors, nonuniform airﬂow through the wind tunnel, inaccuracies in the model con-

struction or in its orientation in the wind tunnel). In these cases, a unique Cpis deﬁned as the mean of

the measured values.

12

3

BD

4

H

5

4

1

2

3

Wind

Wind

q

q

B

D

8

6

7

6

8

75

12

3

BD

6

5

5

4

1

2

3

Wind

Wind

q

q

B

D

b

46

H

12

3

BD

4

H

5

5

4

1

2

3

Wind

Wind

q

q

B

H

D

(a) Flat-roofed

(b) Gable-roofed

(c) Hip-roofed

Fig. 2. Deﬁnition of the building dimensions D,B, and H, the building surfaces, the wind attack angle θ

and the pitch roof angle βfor the analyzed cases: (a) Flat-roofed buildings, (b) Gable-roofed buildings,

and (c) Hip-roofed buildings.

6

In the formulation of the S&C equation for low-rise buildings, Swami and Chandra [6] assumed Cp

independent of the height ratio (H/B) and the roof angle (β). After processing the TPU database, we

realized that Cpis low sensitive to the height ratio but the variation of Cpwith diﬀerent roof type and

roof pitch angle can not be neglected. Based on these observations, we decided to formulate a separate

ANN model for each roof type, to be trained using the following data:

•Flat-roofed buildings: For all the walls (surfaces 1 to 4) and roof (surface 5), Cpis given for side

ratios D/B = 1/1,1.5/1,2.5/1,1/1.5,1/2.5, and wind attack angles θfrom 0◦to 180◦every 15◦.

This Cpis the average of the Cps for height ratios H/B = 1/4,2/4,3/4,4/4 in the database.

•Gable-roofed buildings: For all the walls (surfaces 1 to 4) and roof (surfaces 5 and 6), Cpis given

for D/B = 1/1,1.5/1,2.5/1,1/1.5,1/2.5, θfrom 0◦to 180◦every 15◦, and β= 4.8◦,9.4◦,14◦,18.4◦,

21.8◦,26.7◦,30◦,45◦. Like in the previous case, this Cpis the average of those for H/B = 1/4,2/4,

3/4,4/4 in the database. Note that the roof pitch βhas been ﬁnely discretized, denoting how

relevant it is.

•Hip-roofed buildings: For all the walls (surfaces 1 to 4) and roof (surfaces 5 to 8), Cpis given for

D/B = 1.5/1,1/1.5, θfrom 0◦to 180◦every 15◦, and β= 26.7◦,45◦; once again, it is the average

of the Cps for H/B = 1/4,2/4,3/4,4/4 in the database. . In this case, the Cpis smaller than in a

gable-roofed building for the same D/B,H/B,θand β. This database is not complete enough to

make a reliable model upon it, but it was included in this work in order to test the robustness of

the proposed ANN-based model.

Thus, the number and the nature of inputs and outputs of the ANN models for the case studies,

depending on the type of roof, are the following:

•Flat-roofed buildings:

–2 inputs: D/B,θ.

–5 outputs: Cpfor walls (surfaces 1 to 4) and roof (surface 5).

•Gable-roofed buildings:

–3 inputs: D/B,β,θ.

–6 outputs: Cpfor walls (surfaces 1 to 4) and roof (surface 5 and 6).

•Hip-roofed buildings:

–3 inputs: D/B,β,θ.

–8 outputs: Cpfor walls (surfaces 1 to 4) and roof (surface 5 to 8).

7

Since each of these buildings have particular inputs and outputs, we formulate a separate ANN model

for each case, namely the FANN, GANN and HANN models for ﬂat-, gable- and hip-roofed buildings,

respectively. Details on the formulation of each one of these models are given Appendix 5.3.

3. Results and discussion

First, let us evaluate the ﬁtting between the Cps computed as outputs of the current ANN models

and those from the TPU experimental database, which are the ANN targets. Fig. 3 shows the correlation

between the ANN outputs and the measured Cpfor the three types of roof. The ﬁtting is excellent in all

the cases: MSE = 0.0001 and correlation coeﬃcient R= 0.9996 for ﬂat-roofed buildings, MSE = 0.0002

and R= 0.9995 for gable-roofed buildings, and MSE = 0.0004 and R= 0.9990 for hip-roofed buildings.

Next, let us compare the current ANN results with those obtained using the popular S&C and M&P

equations for each case.

3.1. Flat-roofed buildings

Fig. 4 shows Cpfor the surface 1 (a wall) of a ﬂat-roofed, square-plan building as a function of the

wind attack angle θ. The S&C and M&P equations accurately ﬁts for θ≤90◦; for higher θ, S&C

performs usually better than M&P , as already noted by Muehleisen and Patrizzi [9]. On the other hand,

FANN ﬁts the measured TPU data with high accuracy for any θ; further, the FANN interpolation of the

experimental measurements is smooth and free of overﬁtting. These two observations validate the chosen

ANN architecture and its usefulness to predict Cpover the whole range of wind attack angles.

Fig. 5 shows Cpon surface 1 for diﬀerent D/B 6= 1 as a function of the wind attack angle θ. As

pointed out by Muehleisen and Patrizzi [9] when they proposed the M&P equation, the performance of

S&C is seriously deteriorated for D/B 6= 1; M&P gives deﬁnitely better results for these cases. But once

again, FANN exhibits the best ﬁtting of the measured data for any D/B and θ; at the same time, it gives

a smooth interpolation of the measured data.

FANN also overcomes the S&C and M&P equations as it predicts Cpfor all the walls and the roof at

once. Let us remind that S&C and M&P equations determine Cpfor one surface at a time and are not

suitable for computing Cpfor the roof. As an example, Fig. 6 shows the variation of the remaining FANN

outputs, i.e., Cpwith respect to θfor the other walls (surfaces 2-4) and the roof (surface 5) for ﬂat-roofed

buildings with a square ﬂoor-plan. Again, there is an excellent agreement between the Cpfrom the TPU

experimental database and that computed using FANN for any value of the wind attack angle.

3.2. Gable-roofed buildings

Fig. 7 shows the variation of Cpwith respect to the wind attack angle θfor the surface 1 (a wall) of

square-plan building having a gable roof with pitch angle β= 45◦. In this case, S&C gives the poorest

ﬁtting of the experimental results, specially for 60◦< θ < 150◦; for instance, the diﬀerence between S&C

prediction and TPU for θ= 90◦is above 20%.

8

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

R = 0.9996

MSE = 0.0001

Ideal

(a) Flat-roofed

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

R = 0.9995

MSE = 0.0002

Ideal

(b) Gable-roofed

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

(c) Hip-roofed

R = 0.9990

MSE = 0.0004

Ideal

measured Cp

model C

p

measured Cp

model C

p

measured Cp

model C

p

Fig. 3. Fitting between the ANN outputs and the measured targets: (a) Flat-roofed buildings, (b)

Gable-roofed buildings, and (c) Hip-roofed buildings.

9

Wind angle q [ ° ]

0 30 60 90 120 150 180

S&C

FANN

M&P

TPU data

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Cp

Fig. 4. Square plan ﬂat-roofed buildings: TPU measurements vs. predictions of FANN model, the M&P

and the S&C equations of the variation of the surface-averaged pressure coeﬃcient Cpwith respect to the

wind attack angle θon the surface 1.

On the other hand, GANN gives not only an excellent ﬁtting but also a smooth interpolation of TPU

data.

Fig. 8 shows GANN results for all the surfaces of a square-plan building with 45◦-pitched gable roof.

Note that these coeﬃcients were obtained evaluating the GANN only once per incidence angle.

Let us remark that the application of S&C equation to this case carries an excessive simpliﬁcation:

Since it depends on D/B and θonly, it gives the same Cpfor walls with the same side ratio but geometri-

cally diﬀerent (as it is the case for surfaces 1 and 2, for instance). Moreover, S&C equation is not suitable

for computing Cpfor the roof surfaces.

3.3. Hip-roofed buidlings

Fig. 9 shows the variation of Cpwith respect to the wind attack angle θfor the surface 1 of buildings

with a side ratio D/B = 1.5 and a hip roof with pitch angle β= 45◦. In this case, M&P is better than

S&C but it is still highly inaccurate to ﬁt the experimental results. Meanwhile, once again, HANN gives

an excellent ﬁtting as well as a smooth interpolation of the TPU data.

Fig. 10 shows Cpas a function of θfor the surface 1 of hip-roofed buildings with side ratio D/B = 1/1.5,

considering two roof pitch angles β: 26.7◦and 45◦. Besides the good agreement between HANN and TPU

data for both roof pitch angles and for any wind attack angle θ, this ﬁgure highlights the diﬀerences in

Cpdue to the diﬀerence in the roof pitch angle, specially for θ > 90◦.

3.4. Accuracy of the method

In this section, the accuracy of the current ANN-based models is quantitatively assessed in the light

of some examples.

10

(c) D/B=1.5

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

(d) D/B=2.5

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0 30 60 90 120 150 180

(a) D/B=0.4

Wind angle q [ ° ]

(b) D/B=0.667

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0 30 60 90 120 150 180

Wind angle q [ ° ]

0 30 60 90 120 150 180

Wind angle q [ ° ]

0 30 60 90 120 150 180

Wind angle q [ ° ]

S&C

ANN-Model

M&P

TPU data

S&C

ANN-Model

M&P

TPU data

S&C

ANN-Model

M&P

TPU data

S&C

ANN-Model

M&P

TPU data

S&C

FANN

M&P

TPU data

S&C

FANN

M&P

TPU data

Cp

S&C

FANN

M&P

TPU data

S&C

FANN

M&P

TPU data

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Cp

Cp

Cp

Fig. 5. Flat-roofed buildings with side ratio D/B 6= 1: TPU measurements vs. predictions of FANN

model, the M&P and the S&C equations of the variation of the surface-averaged pressure coeﬃcient Cp

with respect to the wind attack angle θon the surface 1.

Considering ﬂat-roofed buildings with side ratio D/B = 1.5 and a wind attack angle θ= 90◦, Table 1

enables the comparison between the experimental Cpfrom the TPU database and the predictions given

by the S&C and the M&P equations and the FANN model. While the S&C and the M&P equations are

accurate enough only when they are applied to surface 2 (i.e., the windward surface), FANN is accurate

enough for all the surfaces, including the roof. Actually, the maximal error magnitude using FANN is

2.3%, while it rises up to 46.9% for the S&C equation and 21.7% for the M&P equation.

Now, let us consider gable- and hip-roofed buildings and compare the respective ANN models (GANN

and HANN) to the S&C equation. Note that the S&C equation was calibrated on the base not only of

ﬂat-, but also of gable- and hip-roofed buildings [6], and it is widely used for all these types of buildings;

it is actually included in EnergyPlus, a popular software for building performance simulation.

Table 2 shows Cpfor gable- and hip-roofed buildings having a side ratio D/B = 1.5 and a roof pitch

angle β= 45◦for a wind attack angle θ= 90◦. Let us remind that the S&C equation does not take

11

FANN

TPU data

(a) Surf 2 (b) Surf 3

ANN-Model

TPU data

(d) Surf 5

ANN-Model

TPU data

(c) Surf 4

ANN-Model

TPU data

0 30 60 90 120 150 180

Wind angle q [ ° ]

0 30 60 90 120 150 180

Wind angle q [ ° ]

0 30 60 90 120 150 180

Wind angle q [ ° ]

0 30 60 90 120 150 180

Wind angle q [ ° ]

FANN

TPU data

FANN

TPU data

FANN

TPU data

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Cp

Cp

Cp

Cp

Fig. 6. Square plan ﬂat-roofed buildings: TPU measurements vs. FANN predictions of the variation of

the surface-averaged pressure coeﬃcient Cpwith respect to the wind attack angle θfor all the surfaces,

except the surface 1.

into account the pitch angle, which seriously aﬀects its accuracy for buildings with pitched roof. Given

the magnitude of the errors in Table 2, S&C is directly useless for all the surfaces of such pitched-roof

buildings other than the windward wall. On the contrary, either GANN for gabled-roof buildings or

HANN for hip-roofed ones gives good results, with errors that are always below 4.3%.

Let us remark that the performance of the proposed ANN models for the computation of Cpis equally

good (actually, excellent) for ﬂat-, gable- and hip-roofed buildings. Furthermore, they allow to predict

the Cpon roofs, enabling the study of air inﬁltration and ventilation in case of roof apertures.

Finally, note that, just modifying the ANN architecture (number of inputs and outputs, number of

hidden layers and neurons therein, etc.), the proposed methodology is versatile enough to be applied to

12

S&C

GANN

M&P

TPU data

0 30 60 90 120 150 180

Wind angle q [ ° ]

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Cp

Fig. 7. Square-plan buildings with 45◦-pitched gable roof: TPU measurements vs. predictions of GANN

model, the M&P and the S&C equations of the variation of the surface-averaged pressure coeﬃcient Cp

with respect to the wind attack angle θon the surface 1.

Table 1. Flat-roofed buildings with side ratio D/B = 1.5: TPU measurements vs. predictions of the S&C

and the M&P equations and the FANN model of the surface-averaged pressure coeﬃcient Cpon all the

building surfaces for the wind attack angle θ= 90◦.

Surface TPU S&C M&P FANN

CpCperror [%] Cperror [%] Cperror [%]

1−0.580 −0.308 −46.906 −0.533 −8.154 −0.567 −2.327

2 0.618 0.604 −2.394 0.607 −1.779 0.621 0.437

3−0.580 −0.308 −46.906 −0.533 −8.154 −0.577 −0.534

4−0.284 −0.377 32.945 −0.345 21.728 −0.286 0.882

5−0.727 - - - - −0.738 1.389

any building shape with an available database, which should be reliable enough to be used as target for

training the ANN.

4. Conclusions

In this work, we proposed and developed a novel methodology based on artiﬁcial neural networks

(ANN) to obtain analytical models to accurately predict the surface-averaged wind pressure coeﬃcients

in all the surfaces (walls and roofs) of low-rise buildings with diﬀerent types of roofs.

Three separate ANN models were generated, the so-called FANN, GANN and HANN, addressed to

13

(b) Surf 2

ANN-Model

TPU data

0 30 60 90 120 150 180

Wind angle q [ ° ]

(a) Surf 1

GANN

TPU data

0 30 60 90 120 150 180

Wind angle q [ ° ]

(d) Surf 4

ANN-Model

TPU data

0 30 60 90 120 150 180

Wind angle q [ ° ]

(c) Side 3

ANN-Model

TPU data

0 30 60 90 120 150 180

Wind angle q [ ° ]

(f) Surf 6

ANN-Model

TPU data

0 30 60 90 120 150 180

Wind angle q [ ° ]

(e) Surf 5

ANN-Model

TPU data

0 30 60 90 120 150 180

Wind angle q [ ° ]

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

GANN

TPU data

GANN

TPU data

GANN

TPU data

GANN

TPU data

GANN

TPU data

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Cp

Cp

Cp

Cp

Cp

Cp

Cp

Cp

Fig. 8. Square-plan buildings with 45◦-pitched gable roof: TPU measurements vs. GANN predictions of

the variation of the surface-averaged pressure coeﬃcient Cpwith respect to the wind attack angle θfor

all the surfaces.

14

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

S&C

HANN

M&P

TPU data

0 30 60 90 120 150 180

Wind angle q [ ° ]

Cp

Fig. 9. Buildings with side ratio D/B = 1.5 and 45◦-pitched hip roof: TPU measurements vs. predictions

of HANN model, the M&P and the S&C equations of the variation of the surface-averaged pressure

coeﬃcient Cpwith respect to the wind attack angle θon the surface 1.

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

HANN ( b = 45º )

HANN ( b = 26.7º )

TPU data ( b = 45º )

TPU data ( b = 26.7º )

0 30 60 90 120 150 180

Wind angle q [ ° ]

Cp

Fig. 10. Hip-roofed buildings with side ratio D/B = 1/1.5 and diﬀerent roof pitch angles: TPU measure-

ments vs. HANN predictions of the variation of the surface-averaged pressure coeﬃcient Cpwith respect

to the wind attack angle θon the wall surface 1.

ﬂat-, gable- and hip-roofed rectangular-plan buildings, respectively. They were trained and tested taking

as target the experimental database of the Wind Engineering Information Center of the Tokyo Polytechnic

University (TPU).

In the lights of the results for these three types of buildings, ANN models proved to be very accu-

rate, largely overcoming the widely used parametric equations proposed by Swami and Chandra [6], and

Mueleisen and Patrizi [9]. Actually, the performance of these parametric equations, which depend on the

15

Table 2. Relative error comparison between the novel ANN-Model and S&C equation for Gable-roofed

and Hip-roofed cases with D/B=1.5, β=45◦and θ=90◦.

Roof type Surface TPU S&C ANN

CpCperror [%] Cperror [%]

Gabled

1−0.832 −0.308 −62.994 −0.82 −1.526

2 0.604 0.604 −0.099 0.597 −1.225

3−0.832 −0.308 −62.994 −0.844 1.454

4−0.571 −0.377 −34.005 −0.589 3.169

5 0.293 - - 0.301 2.594

6−0.691 - - −0.698 1.100

Hipped

1−0.667 −0.308 −53.837 −0.664 −0.513

2 0.579 0.604 4.285 0.578 −0.178

3−0.667 −0.308 −53.837 −0.676 1.314

4−0.508 −0.377 −25.749 −0.512 0.775

5−0.832 - - −0.808 −2.826

6 0.239 - - 0.243 1.628

7−0.832 - - −0.797 −4.242

8−0.625 - - −0.637 2.05

building side ratio and the wind attack angle only, was shown to be poor in general, and even very poor

for most of the surfaces of pitched-roof buildings.

Another advantage of the proposed ANN is their capacity of computing the pressure coeﬃcients over all

the building surfaces at once. Unlike the above mentioned parametric models, also the roof is considered,

enabling the use of the current models in the study of phenomena like air inﬁltration and ventilation in

presence of roof apertures.

Although it was applied here to speciﬁc buildings, the proposed methodology is amenable to be ex-

tended to buildings with an arbitrary topology (U-shape, L-shaped, etc.), including the eﬀect of surround-

ing buildings, provided that you have a reliable database for training the ANN for the desired building

shape and level of sheltering.

Furthermore, since the so-formulated ANN models are ultimately a closed, analytical expression of

the outputs as functions of the inputs, they can be easily incorporated into softwares for building energy

simulation, airﬂow network analysis, etc..

To ensure the accuracy of any ANN-based methodology, the ANN training has to be done with a

reliable Cpdatabase like the TPU database used in this work.

Note that the accuracy of the current models was proved for isolated buildings of three particular

16

types and for the range of the inputs in the database used for training, otherwise it is uncertain.

Motivated by the current ﬁndings, we aim to continue this work in three main directions: 1) to embed

the proposed models into EnergyPlus source code; 2) applications to low- and high-rise, isolated and

non-isolated buildings with diﬀerent shapes, and 3) to test the reliability of computational ﬂuid dynamic

(CFD) models to develop Cpdatabases of buildings with other shapes (U, L, etc.).

Acknowledgments

For funding this work, we would like to thank the Agency for Science, Technology and Innovation

(SECTEI) of the Province of Santa Fe (Argentina) via the Research Project 2010-022-16 “Optimization

of the Energy Eﬃciency of Buildings in the Province of Santa Fe”.

F. Bre is a doctoral student granted by the National Scientiﬁc and Technical Research Council of

Argentina (CONICET).

5. Appendix

5.1. Swami and Chandra (S&C ) equation

The S&C equation [6] deﬁnes the surface-average wind pressure coeﬃcient at a wall of a low-rise

building as

Cp(θ, D/B ) = Cp(0◦) ln 1.248 −0.703 sin(θ/2) −1.175 sin2(θ)+0.131 sin3(2Gθ)

+0.769 cos(θ/2) + 0.07G2sin2(θ/2) + 0.717 cos2(θ/2),

where θis the wind attack angle on the surface, and G= ln(D/B) is the natural logarithm of the side ratio

D/B, and Cp(0◦) is the Cpfor θ= 0◦, assumed by Swami and Chandra to be equal to 0.6 independently

of D/B.

5.2. Muehleisen and Patrizi (M&P ) equation

The M&P equation [9] deﬁnes the surface-average wind pressure coeﬃcient at a wall of a low-rise

building using the following rational function of the wind attack angle θand the side ratio D/B:

Cp(θ, D/B ) = a0+a1G+a2θ+a3θ2+a4Gθ

1 + b1G+b2θ+b3θ2+b4Gθ ,with G= ln(D/B),

where aiand biare adjustable coeﬃcients. Muehleisen and Patrizi calibrated them using non-linear curve

ﬁtting on the base of the TPU database for low-rise buildings [28], obtaining a0= 6.12 ×10−1, a1=

−1.78 ×10−1, a2=−1.15 ×10−2, a3= 3.28 ×10−5, a4= 1.67 ×10−3,b1=−3.12 ×10−1, b2=−1.59 ×

10−2, b3= 9.82 ×10−5, b4= 2.15 ×10−3. Note that in the original paper [9], b2has a typo mistake; we

ﬁxed it after communication with Muehleisen.

17

5.3. Formulation of the ANN models

In this section, is given a detailed information to implement the ANN models developed. These ANN

models are available in doi:10.17632/mj6s6x37vm.3

An ANN is a set of unit cells (or artiﬁcial neurons) arranged in an input layer, one or more hidden

layers, and an output layer. Each neuron is connected to those neurons in the neighboring layers via

adaptive weights. Fig. 11 shows the model of a generic neuron jin the hidden layer k, whose output is

deﬁned as follows:

yk

j=f n

X

i=1

wk

ij xk

i+bk

j!,

where wk

ij is the weight of the connection between the i-th neuron of the previous layer and the considered

neuron, xk

iis the input from i-th neuron of the previous layer, bk

jis the bias associated with the current

neuron, and fis the activation function. Here, we adopt the tangent sigmoid f(u) = −1+2/(1 + e−2u)

for the hidden layers, and the linear function f(u) = ufor the output later.

Activation function

1

Tangent sigmoid

Neuron j in layer k

-1

Fig. 11. Model of the neuron jin layer k.

Table 3 gives the inputs, the outputs and the architecture (i.e., the number of layers and neurons per

layer) for the three ANN models.

Table 3. Parameters of the ANN models.

Parameter FANN GANN HANN

Inputs D/B,θ D/B,β,θ D/B,β,θ

Outputs Cpon surfaces 1-5 Cpon surfaces 1-6 Cpon surfaces 1-8

Architecture (I-H-. . . -O) 2-9-8-5 3-20-20-6 3-20-8

The inputs xand the outputs yare linearly normalized as follows:

xn= 2 x−xmin

xmax −xmin

−1, yn= 2 y−ymin

ymax −ymin

−1,(1)

where xmin,xmax ,ymin and ymax for each ANN model are given in Table 4

18

Table 4. Minimum and maximum inputs and ouputs for the ANN models.

Parameter FANN GANN HANN

Min. Max. Min. Max. Min. Max.

Inputs D/B 0.4 2.5 0.4 2.5 0.667 1.5

β4.8◦45◦26.7◦45◦

θ0◦180◦0◦180◦0◦180◦

Outputs CpSurf. 1 −0.6995 0.6334 −0.9610 0.6695 −0.7892 0.6403

CpSurf. 2 −0.6939 0.6581 −0.9516 0.6687 −0.6672 0.6403

CpSurf. 3 −0.6995 0.6334 −0.961 0.6695 −0.7892 0.6403

CpSurf. 4 −0.6995 −0.2179 −0.961 −0.1934 −0.7892 −0.2116

CpSurf. 5 −0.8304 −0.3000 −1.1645 0.3315 −0.9127 0.1291

CpSurf. 6 −1.1645 0.2947 −0.8640 0.2422

CpSurf. 7 −0.9127 0.1291

CpSurf. 8 −0.8640 0.2422

Finally, Tables 5, 6 and 7 give the results of training (i.e., the weights and biases) for FANN, GANN

and HANN, respectively.

Table 5. Weights and bias values for FANN.

Layer kNeuron j bk

jwk

1jwk

2jwk

3jwk

4jwk

5jwk

6jwk

7jwk

8jwk

9j

H1 1 6.89 -5.57 1.43

2 -1.13 0.74 -1.35

3 2.02 -2.98 -1.36

4 1.85 0.33 2.01

5 0.98 -1.09 1.57

6 0.26 -0.17 -1.56

7 -1.89 0.58 2.41

8 2.24 2.34 -2.23

9 -4.96 -6.16 -0.39

H2 1 0.50 3.87 -2.42 -0.02 -3.14 -1.76 2.45 0.85 -0.42 0.24

2 0.93 -1.03 -0.81 0.01 -0.57 -0.62 -0.59 -0.67 0.18 -0.04

3 -11.47 3.78 1.05 0.78 8.02 -5.28 -6.35 -1.39 0.14 0.05

4 -4.12 1.35 -14.32 1.30 -2.19 -8.91 -1.29 1.73 0.17 0.01

5 1.86 -0.66 -0.54 -1.08 0.17 -0.40 -0.70 0.69 0.53 -0.15

6 -2.53 2.53 4.85 0.12 1.27 1.61 -1.97 0.33 -0.10 0.03

7 -0.22 2.81 3.42 -0.32 -1.29 2.07 0.77 -0.43 -0.25 0.05

8 2.01 1.21 1.33 0.07 0.76 1.10 -1.80 1.92 -0.03 0.14

O 1 1.21 -0.83 0.08 0.18 0.42 0.45 1.45 2.71 -0.07

2 -2.87 0.11 -4.20 0.15 -0.44 0.46 -5.97 -2.58 -1.00

3 0.12 0.01 -4.43 0.24 -0.57 -1.92 -4.86 -6.93 -3.51

4 2.31 -1.62 4.44 -2.19 -3.37 -2.09 6.63 -0.72 2.47

5 1.22 -2.33 -6.11 -0.98 -3.93 -3.58 -4.60 -9.43 -3.74

19

Table 6. Weights and bias values for GANN.

L. kN. j bk

jwk

1jwk

2jwk

3jwk

4jwk

5jwk

6jwk

7jwk

8jwk

9jwk

10jwk

11jwk

12jwk

13jwk

14jwk

15jwk

16jwk

17jwk

18jwk

19jwk

20j

H1 1 -2.69 1.37 -0.02 2.17

2 1.58 -0.18 1.28 -1.24

3 -1.38 0.28 -1.22 1.06

4 -0.73 1.16 0.89 -0.22

5 -1.73 0.02 -0.03 -1.92

6 -1.34 -0.13 -0.76 -3.94

7 0.99 -0.16 -0.09 1.65

8 0.07 1.07 -0.38 -0.80

9 0.33 -1.06 0.02 -0.86

10 0.71 0.22 -2.25 -1.77

11 0.11 0.24 -0.02 1.22

12 -0.76 -0.62 1.82 -1.45

13 0.49 1.46 -0.08 -1.53

14 -0.56 -0.28 -1.93 -0.48

15 -3.76 -4.13 0.02 -1.35

16 -1.44 0.17 -0.01 2.07

17 2.06 2.53 0.01 0.47

18 4.76 5.64 0.19 -1.40

19 9.32 0.23 -7.48 -6.25

20 -6.30 -6.75 -0.18 0.17

H2 1 -11.11 -0.90 -1.83 -3.12 0.20 -8.39 0.20 -1.68 -0.04 1.25 -0.28 5.93 -0.05 1.61 0.94 3.06 -7.01 -2.39 -2.28 0.06 -2.94

2 1.01 -10.77 0.09 -0.36 -0.16 -0.37 0.06 -0.15 -0.29 0.28 -0.06 1.37 -0.06 0.84 -0.08 -0.17 11.69 -0.03 -0.33 -0.04 0.07

3 2.53 -9.09 -1.14 -1.52 -0.22 -0.16 0.22 -0.28 -0.48 0.40 -0.08 2.39 -0.07 0.81 -0.09 -0.06 9.38 0.56 -0.77 -2.00 -0.16

4 -1.44 -0.06 -0.93 -1.00 0.12 0.24 0.99 -0.27 0.19 -0.22 0.03 3.86 0.09 -0.21 -0.07 0.27 0.12 -2.69 0.18 0.04 -1.04

5 0.64 -0.36 0.83 0.78 0.06 -0.25 -0.20 -0.30 0.08 -0.24 0.02 -1.54 0.02 -0.02 0.03 0.04 0.16 0.36 -0.05 0.01 -0.02

6 0.04 0.27 -1.13 -1.00 -0.16 -0.04 0.16 -0.52 -0.22 0.11 -0.02 1.21 -0.06 0.25 -0.05 -0.36 -0.08 -0.46 -0.21 -0.02 -0.19

7 0.03 0.01 0.34 0.10 0.14 1.26 -0.15 1.46 0.53 -0.27 -0.04 -0.76 0.03 -0.62 0.05 0.21 -0.34 0.45 0.05 -0.02 0.08

8 -0.24 -8.61 -2.94 0.28 2.24 -2.19 0.36 1.56 8.03 6.45 -0.99 3.05 0.45 -2.86 0.68 -4.44 7.48 -8.15 6.12 0.72 4.85

9 -1.08 0.07 -0.86 -1.18 -0.29 0.45 -0.11 1.27 -1.91 -1.20 0.14 -0.51 0.31 1.39 0.55 -0.63 0.58 -3.07 0.12 0.33 -1.12

10 7.67 -0.60 0.32 0.49 -0.12 0.77 0.10 0.84 -0.07 -0.16 -0.06 -0.97 -0.17 0.52 -0.01 0.19 0.39 -0.96 -0.19 -0.09 5.81

11 0.23 -0.51 1.03 0.95 0.11 0.96 -0.05 0.28 0.21 0.04 0.01 -0.31 0.03 -0.14 0.01 0.15 0.57 -0.06 0.17 -0.01 0.12

12 6.82 -1.56 3.69 3.88 0.26 -2.41 -0.05 -10.41 0.11 2.12 -0.02 2.94 -0.07 1.02 -0.03 -0.71 1.03 -3.19 1.59 -0.01 1.75

13 1.80 0.08 -1.76 -1.11 -0.19 2.51 0.12 0.53 1.35 0.05 -0.19 -1.23 -0.21 -0.84 -0.41 -1.05 -0.59 1.71 -0.83 -0.16 0.72

14 0.20 1.29 -3.80 -4.10 -0.26 11.65 -0.02 12.03 -0.09 -1.88 0.06 -1.74 0.17 -0.75 0.06 0.17 -1.04 2.27 -1.28 0.08 -1.34

15 -0.46 0.16 1.45 1.46 0.31 -1.01 0.28 -1.42 -0.74 0.80 0.23 2.46 0.33 0.25 0.11 1.69 -0.26 1.48 1.06 0.26 0.50

16 1.12 0.18 -2.85 -2.83 -0.05 -0.56 0.27 -1.70 0.57 0.43 -0.06 1.62 -0.04 -0.12 -0.08 -0.14 0.18 0.04 -0.41 -0.03 -0.30

17 -2.56 0.79 -1.15 -1.13 -0.09 -0.60 0.01 -0.64 0.18 -0.22 -0.04 -0.88 -0.07 0.16 -0.01 -0.46 -2.84 -0.38 -0.32 -0.13 -0.31

18 10.31 -0.83 4.72 4.80 0.28 -0.47 -1.24 1.26 -0.30 -0.38 0.24 -3.25 0.15 0.11 -0.07 -0.38 1.69 -0.46 1.99 0.10 12.10

19 1.18 -0.15 -0.51 -0.33 0.27 0.32 0.04 -0.39 1.71 0.80 -0.10 -0.14 -0.16 -1.17 -0.37 0.25 -0.22 2.21 -0.16 -0.18 0.95

20 -6.65 1.52 4.89 3.76 -1.26 -0.66 0.00 2.62 0.02 -2.10 -1.91 -3.44 0.98 -0.09 1.75 -4.99 -1.18 2.72 -3.22 0.14 1.39

O 1 2.04 -0.02 -0.42 0.05 1.22 -1.54 -2.08 -0.95 0.01 0.29 0.27 0.10 2.83 -0.04 3.30 -0.03 -0.30 0.82 -0.14 0.48 -0.32

2 -2.90 -0.04 -1.06 0.26 -2.87 -0.88 -1.32 0.49 0.00 0.19 0.56 -2.09 -0.30 -0.07 -0.26 0.02 0.17 -0.13 0.19 0.34 0.13

3 -0.04 -0.04 1.33 -0.17 -2.17 -3.05 1.57 1.67 0.01 0.26 1.50 0.82 -2.29 -0.25 -1.86 0.75 -0.70 2.59 0.37 0.70 0.16

4 9.58 0.44 -2.67 2.31 6.08 -2.20 1.07 2.18 -0.03 -0.22 0.87 0.02 0.49 0.95 0.84 1.07 -2.53 2.66 -0.19 -0.76 -0.88

5 -4.38 -0.06 -7.09 3.84 -8.06 -0.21 -5.39 -5.50 -0.12 1.77 -4.24 -0.48 -0.41 -0.30 -0.11 -2.81 0.94 -3.00 0.25 1.75 1.16

6 -0.05 -0.02 2.30 -1.03 -0.32 -8.37 -15.15 -3.04 -0.21 -5.55 -0.56 1.42 -0.67 2.51 -0.58 -1.55 2.62 -4.53 -1.19 -10.17 0.69

Table 7. Weights and bias values for HANN.

L. kN. j bk

jwk

1jwk

2jwk

3jwk

4jwk

5jwk

6jwk

7jwk

8jwk

9jwk

10jwk

11jwk

12jwk

13jwk

14jwk

15jwk

16jwk

17jwk

18jwk

19jwk

20j

H1 1 1.09 -0.10 -0.40 0.96

2 1.43 0.10 1.29 -1.64

3 0.69 -0.11 -0.01 -0.80

4 4.51 -0.07 -3.98 2.59

5 -2.18 6.59 2.97 -6.17

6 1.92 -5.49 4.37 -4.49

7 -0.34 -0.08 -1.15 -0.96

8 0.43 0.12 -0.10 -2.70

9 0.43 0.15 -0.32 -1.29

10 -0.45 -0.18 0.08 -2.74

11 -0.09 -1.22 -0.01 0.05

12 -0.36 -0.07 0.82 -0.83

13 1.59 0.11 1.21 -2.44

14 -0.74 0.11 2.81 -1.99

15 0.72 0.20 -0.70 1.25

16 2.11 -0.10 -1.87 2.03

17 -3.77 -0.13 -3.96 1.33

18 -1.34 -0.12 0.03 -1.50

19 -24.70 -21.89 0.37 -19.45

20 5.65 1.51 4.05 1.34

O 1 0.16 1.09 -0.76 0.42 1.05 -0.03 -0.03 0.05 -1.01 -0.54 1.12 -0.23 -1.34 1.12 -0.64 -0.54 -2.69 -0.54 0.49 -0.01 -0.07

2 -2.03 2.10 4.61 1.02 -1.49 0.01 0.01 1.51 0.15 -0.04 -0.02 -0.44 2.00 -2.04 1.34 0.01 1.98 3.43 -1.62 0.02 0.01

3 -1.14 -0.74 -1.21 2.99 -0.59 -0.03 -0.01 -2.31 -1.22 0.15 0.34 -0.12 -3.95 0.42 0.04 -0.61 0.62 -1.01 0.96 -0.01 -0.08

4 8.64 1.77 34.64 -9.34 -13.16 0.11 0.03 12.52 6.71 -10.02 -0.80 -1.87 28.28 -16.45 1.87 4.71 19.97 25.59 -1.23 -0.03 0.42

5 14.60 2.68 5.35 -18.81 4.22 -0.24 -0.24 26.05 -4.72 -11.23 -0.74 -1.87 36.63 4.89 -9.45 0.47 -12.66 5.30 -12.86 -0.27 -0.77

6 6.50 -6.50 -34.06 -13.17 0.79 -0.31 -0.30 6.30 -0.05 7.22 0.02 2.10 3.56 10.28 2.37 2.68 0.38 -24.81 -6.97 -0.32 -1.39

7 -12.02 -3.93 16.18 20.03 -17.94 0.01 0.05 -19.96 9.30 2.52 1.12 -0.41 -34.71 -16.58 16.37 -4.99 32.15 9.08 10.47 0.04 0.08

8 -8.92 13.89 29.56 16.98 -7.89 0.36 0.21 -9.17 2.16 -8.21 -0.77 -2.58 9.10 -9.36 -5.49 6.73 5.98 23.91 13.29 0.27 1.13

20

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