Prediction of wind pressure coefficients on building surfaces using Artificial Neural Networks

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.

Full text

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
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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.
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(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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