Air Quality Assessment via FPCA
G. Agrò, F. Di Salvo*, A. Plaia, M. Ruggieri
*[email protected]
Dipartimento di Scienze Statistiche e Matematiche “Silvio Vianelli”
Università di Palermo
Bologna, July 5-9, 2009
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
Assessment via FPCA
Bologna, July 5-9, 2009
1 / 57
Outline
1
Introduction
2
The Air Pollution data
Aggregation and Standardization Steps
Exploring Data
3
The Functional Data Analysis Approach
Observed Functional Data
Multivariate Functional Principal Component Analysis
4
Further Developments
Use of the optimal empirical orthonormal basis
Methodological Perspective
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
Assessment via FPCA
Bologna, July 5-9, 2009
2 / 57
Introduction
Introduction
The present study focus on the assesment of Air Pollution in Palermo
The approach used is the Functional Principal Component Analysis
Air quality data for the municipal area are provided from a monitoring
network of 9 stations. Validated data are available for the year 2006.
Our main objective is to understand the dynamic of the spatial and
temporal variations of four monitored pollutants
The Functional Principal Component Analysis is is used to find
directions in the observation space along which the data have the
highest variability.
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
Assessment via FPCA
Bologna, July 5-9, 2009
3 / 57
Introduction
Introduction
The present study focus on the assesment of Air Pollution in Palermo
The approach used is the Functional Principal Component Analysis
Air quality data for the municipal area are provided from a monitoring
network of 9 stations. Validated data are available for the year 2006.
Our main objective is to understand the dynamic of the spatial and
temporal variations of four monitored pollutants
The Functional Principal Component Analysis is is used to find
directions in the observation space along which the data have the
highest variability.
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
Assessment via FPCA
Bologna, July 5-9, 2009
3 / 57
Introduction
Introduction
The present study focus on the assesment of Air Pollution in Palermo
The approach used is the Functional Principal Component Analysis
Air quality data for the municipal area are provided from a monitoring
network of 9 stations. Validated data are available for the year 2006.
Our main objective is to understand the dynamic of the spatial and
temporal variations of four monitored pollutants
The Functional Principal Component Analysis is is used to find
directions in the observation space along which the data have the
highest variability.
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
Assessment via FPCA
Bologna, July 5-9, 2009
3 / 57
Introduction
Air Pollution in Palermo - 2006
Air Pollution Data: The 2006 Palermo Study
The Sicilian capital Palermo is applying the European Directive on air
quality from 1996
The local council has put in a place a system to monitor air quality
and to announce the results to the public.
The main culprit of this urban pollution is the car traffic, which is
particularly chaotic in this southern city.
At present, they are studying a traffic management and public
transport development scheme.
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
Assessment via FPCA
Bologna, July 5-9, 2009
4 / 57
The Air Pollution data
Pollutants and Monitoring Sites
Time: From 1st of January to 31th of December 2006
Pollutants:
NO2
CO
PM10
O3
SO2
Monitoring Sites:
9 monitors were instrumented to continously record hourly average
concentrations
Readings of Ozone are taken only at two sites.
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
Assessment via FPCA
Bologna, July 5-9, 2009
5 / 57
The Air Pollution data
Pollutants and Monitoring Sites
Time: From 1st of January to 31th of December 2006
Pollutants:
NO2
CO
PM10
O3
SO2
Monitoring Sites:
9 monitors were instrumented to continously record hourly average
concentrations
Readings of Ozone are taken only at two sites.
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
Assessment via FPCA
Bologna, July 5-9, 2009
5 / 57
The Air Pollution data
Figure: 1 - Air Monitoring Sites
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
Assessment via FPCA
Bologna, July 5-9, 2009
6 / 57
The Air Pollution data
Aggregation and Standardization Steps
Aggregation: from hourly data to daily data
Daily air pollutants measurements has been provided by the monitoring
networks in accordance with International directive:
Table: Daily Synthesis
Pollutant
NO2
PM10
CO
O3
SO2
Daily Synthesis
Daily Maximum
Daily Average
maximum 8-hour moving average
maximum 8-hour moving average
Daily Average
Standards
200µg /m3 (2010)
50µg /m3
10mg /m3
120µg /m3
125µg /m3
For the calculation of daily values, it is required to have at least 75% of
the one hour values on that particular day.
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
Assessment via FPCA
Bologna, July 5-9, 2009
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The Air Pollution data
Aggregation and Standardization Steps
Standardization Step
Table: Breakpoints of the EPA AQI (Murena,2004)
Pollution category
Unhealthy
Unhealthy for sensitive groups
Moderate pollution
Low pollution
Good quality
AQIk
PM10 24h
NO2 1h
CO 8h
SO2 24h
O3 8h
85 - 100
70 - 85
50 - 70
25 - 50
0 - 25
238 - 500
144 - 238
50 - 144
20 - 50
0 - 20
950 - 1900
400 - 950
200 - 400
40 - 200
0 - 40
15.5 - 30
11.6 - 15.5
10 - 11.6
4 - 10
0-4
500 - 1000
250 - 500
125 - 250
20 - 125
0 - 20
223 - 500
180 - 223
120 - 180
65 - 120
0 - 65
AQIk =
IH −IL
BPH −BPL
· (Ck − BPL ) + IL
AQIk is the index for pollutant k;
Ck is the concentration (daily synthesis) of the pollutant k;
BPH is the breakpoint ≥ Ck ;
BPL is the breakpoint ≤ Ck ;
IH is the AQI value corresponding to BPH ;
IL is the AQI value corresponding to BPL .
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
Assessment via FPCA
Bologna, July 5-9, 2009
8 / 57
The Air Pollution data
Exploring Data
Methods:Functional Data Analysis
In air quality monitoring data come to us through a process naturally
described as functional .
Table: Data Structure
s,p X(t)
t = 1, 2, ..., 365
s = 1, 2, ..., 9
p = 1, 2, ..., 5
Day
Station
Pollutant
Functional datum is recorded in discrete readings:
s,p X(1) ...s,p X(365)
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
Assessment via FPCA
Bologna, July 5-9, 2009
9 / 57
The Air Pollution data
Exploring Data
What are the main ways in which the patterns vary from one to another?
Figure: Daily series of 5 Pollutants for 9 Stations
pollutant: PM10
Aug
Sep
Oct
Nov
Dec
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Day
Day
pollutant: CO
pollutant: O3
Sep
Oct
Sep
Oct
300
Jul
200
Jun
100
May
0
Apr
Nov
Dec
Nov
Dec
Jul
Aug
Sep
Oct
Nov
Dec
Jan
Feb
Mar
Apr
May
Day
Jun
Jul
Aug
300
Jun
200
May
100
Apr
0
Mar
300
Feb
200
Jan
100
0
0
10
5
20
10
15
30
20
40
25
50
30
60
Mar
300
Feb
200
Jan
100
0
0
10
10
20
20
30
30
40
40
50
50
60
70
60
pollutant: NO2
Day
6
Belgio
Boccadifalco
Castelnuovo
Cep
DiBlasi
GiulioCesare
Indipendenza
Torrelunga
UnitàItalia
Aug
Sep
Oct
Nov
Dec
10
Jul
Day
8
Jun
6
May
4
Apr
2
Mar
300
Feb
200
Jan
100
0
0
2
10
4
20
30
1:10
40
8
50
10
pollutant: SO2
1:10
The basic idea is to decompose the space of curves into principal
directions of variation.
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
Assessment via FPCA
Bologna, July 5-9, 2009
10 / 57
The Air Pollution data
Exploring Data
Exploring Data
Figure: Monthly aggregated data: Standard deviations vs Means
7.5
NO2
PM10
SO2
5
11
7.0
7
9
7
12
6
10
3
6
8
9
6.5
4
1
2
3
8
4
2
6.0
5
7
4
9
6
6
1
5
10
4
5.5
1
2
10
8
3
5
28
29
30
31
7
11
32
33
32
34
36
38
CO
3
5
5.0
12
27
8
6
40
42
7
4
9
12
11
5
6
7
8
9
10
O3
7
16
6
12
11
6
10
14
5
9
12
2
4
3
95
4
7
3
10
4
3
10
6
5
12
11
8
8
4
8
1
6
8
10
12
2
1
20
25
30
35
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
Assessment via FPCA
Bologna, July 5-9, 2009
11 / 57
The Air Pollution data
Exploring Data
Correlations of the Pollutants at the 9 Stations
NO2
PM10
CO
SO2
NO2
1.00
0.37
0.56
0.52
PM10
0.37
1.00
0.41
0.39
CO
0.56
0.41
1.00
0.37
SO2
0.52
0.39
0.37
1.00
NO2
1.00
0.31
0.56
0.43
PM10
0.31
1.00
0.30
0.53
CO
0.56
0.30
1.00
0.33
SO2
0.43
0.53
0.33
1.00
NO2
1.00
0.43
0.45
0.52
PM10
0.43
1.00
0.19
0.46
CO
0.45
0.19
1.00
-0.01
SO2
0.52
0.46
-0.01
1.00
NO2
PM10
CO
SO2
1.00
0.48
0.56
0.43
0.48
1.00
0.45
0.34
0.56
0.45
1.00
0.60
0.43
0.34
0.60
1.00
1.00
0.48
0.65
0.59
0.48
1.00
0.50
0.43
0.65
0.50
1.00
0.51
0.59
0.43
0.51
1.00
1.00
0.44
0.50
0.29
0.44
1.00
0.50
0.26
0.50
0.50
1.00
0.46
0.29
0.26
0.46
1.00
NO2
PM10
CO
SO2
1.00
0.48
0.48
0.39
0.48
1.00
0.39
0.34
0.48
0.39
1.00
0.29
0.39
0.34
0.29
1.00
1.00
0.24
0.33
0.39
0.24
1.00
0.41
0.12
0.33
0.41
1.00
0.47
0.39
0.12
0.47
1.00
1.00
0.52
0.58
0.38
0.52
1.00
0.50
0.27
0.58
0.50
1.00
0.35
0.38
0.27
0.35
1.00
Station2-O3
NO2
-0.27
PM10
0.32
CO
-0.16
SO2
0.27
Station3-O3
-0.27
- 0.18
-0.61
-0.06
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
Assessment via FPCA
Bologna, July 5-9, 2009
12 / 57
The Air Pollution data
Exploring Data
Spatial Correlations
Table: Geographic Distances between Monitoring Sites
1
2
3
4
5
6
7
8
9
1
0.00
5.80
3.68
2.39
3.12
5.54
4.83
7.94
1.86
2
3
4
5
6
7
8
9
0.00
4.65
3.04
2.54
5.31
3.94
7.57
4.69
0.00
4.21
2.29
1.86
1.58
4.28
1.83
0.00
2.30
5.78
4.61
8.31
3.05
0.00
3.56
2.33
6.09
2.19
0.00
1.37
2.55
3.70
0.00
3.80
3.15
0.00
6.08
0.00
Table: Correlation Matrix for PM10
Belgio
Boccadifalco
Castelnuovo
Cep
DiBlasi
GiulioCesare
Indipendenza
Torrelunga
UnitàItalia
Belg
1.000
0.594
0.823
0.737
0.767
0.745
0.723
0.753
0.875
Boccd
0.594
1.000
0.734
0.630
0.570
0.586
0.706
0.559
0.637
Castel
0.823
0.734
1.000
0.662
0.715
0.766
0.823
0.715
0.847
Cep
0.737
0.630
0.662
1.000
0.644
0.667
0.562
0.683
0.728
DBlsi
0.767
0.570
0.715
0.644
1.000
0.766
0.687
0.667
0.752
Cesar
0.745
0.586
0.766
0.667
0.766
1.000
0.758
0.686
0.815
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
Assessment via FPCA
Indip
0.723
0.706
0.823
0.562
0.687
0.758
1.000
0.640
0.778
Torre
0.753
0.559
0.715
0.683
0.667
0.686
0.640
1.000
0.763
UItal
0.875
0.637
0.847
0.728
0.752
0.815
0.778
0.763
1.000
Bologna, July 5-9, 2009
13 / 57
The Air Pollution data
Exploring Data
Spatial Correlations
Figure: Spatial Correlogram for each pollutant
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4
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CO
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PM10
0.0
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NO2
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7
8
2
distance(km)
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SO2
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G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
Assessment via FPCA
Bologna, July 5-9, 2009
14 / 57
The Air Pollution data
Exploring Data
Table: Distributions of the 735 overcomings
Belgio
Boccadifalco
Castelnuovo
Cep
DiBlasi
GiulioCesare
Indipendenza
Torrelunga
UnitàItalia
NO2
0
1
5
0
2
3
2
0
3
16
O3
31
0
31
PM10
96
19
63
53
223
72
38
28
95
687
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
Assessment via FPCA
SO2
0
0
1
0
0
0
0
0
0
1
Bologna, July 5-9, 2009
15 / 57
The Air Pollution data
Exploring Data
70
Figure: Exceeding values over the threshold
NO2
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PM10
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SO2
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G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
Assessment via FPCA
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Bologna, July 5-9, 2009
16 / 57
The Air Pollution data
Exploring Data
70
Figure: Exceeding values over the threshold
Belgio
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Aug
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
Assessment via FPCA
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Dec
Bologna, July 5-9, 2009
17 / 57
The Functional Data Analysis Approach
Observed Functional Data
The basic idea of Functional Analysis is that any centered process of order two can be expressed
as a combination of orthonormal functions.
s,p X(t)
=
K
X
s,p ck φk (t)
(1)
k=1
where
{φk (t)}
is the set of basis functions.
Use cubic B-spline smoothing : φk (t) = (t − τk )3
where
τk knots equally spaced
Begin with a dense set of knots
Estimate the function by minimizing the Penalized Residual Sum of Squares:
(
)2
2
Z 2
K
X
X
d x(s)
+λ
PENSSEλ (x|y ) =
X
−
c
φ
(t
)
ds
s,p (tj )
s,p k k j
ds
j
k=1
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
Assessment via FPCA
Bologna, July 5-9, 2009
18 / 57
The Functional Data Analysis Approach
Multivariate Functional Principal Component Analysis
Multivariate Functional Principal Component Analysis
The focus is on simultaneuos variability of the 4 pollutants, NO2, PM10, CO, SO2 excluding O3 -, at the 9 sites and along the year.
We apply a Multivariate version of the Functional PC at the functional data.
The functional eigenequation:
Z
V (s, t)ξ(t)dt = ρξ(s)
1
2
3
can be written out as:
Z
V
Z
V
Z
V
Z
V
NO2,NO2
PM10,NO2
CO,NO2
SO2,NO2
(s, t)ξ
(s, t)ξ
(s, t)ξ
NO2
NO2
NO2
(s, t) ξ
Z
(s)ds +
V
Z
(s)ds +
V
Z
(s) ds +
NO2
V
PM10,PM10
CO,PM10
Z
(s) ds +
NO2,PM10
V
(s, t) ξ
(s, t)ξ
(s, t) ξ
SO2,PM10
PM10
PM10
PM10
(s, t)ξ
Z
(s)ds. . . +
NO2,SO2
(s, t)ξ
SO2
(s)ds
=
ρξ
NO2
PM10,SO2
(s, t)ξ
SO2
(s)ds
=
ρξ
PM10
=
ρξ
CO
=
ρξ
SO2
V
Z
(s)ds . . . +
V
Z
(s)ds . . . +
PM10
V
CO,SO2
Z
(s)ds . . . +
V
(s, t)ξ
SO2,SO2
SO2
(s, t)ξ
(s) ds
SO2
(s)ds
(t)
(t)
(t)
(t)
The resulting mth eigenvector, named harmonic is:
NO2 PM10 CO SO2
ξm = (ξm
, ξm
, ξm , ξm )
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
Assessment via FPCA
Bologna, July 5-9, 2009
19 / 57
The Functional Data Analysis Approach
Multivariate Functional Principal Component Analysis
For the mth eigenvalue ρm , measuring the proportion of variability explained by the mth principal
component, the mth harmonic ξm has:
kξm k2 = 1
and
p 2
kξm
k
is the proportion of the variability in the mth principal component accounted for by variation in
the p th Pollutant.
The eigenvectors are orthonormal.
1
We implemented the procedure in R, and in particular we used the library fda (Ramsay,
2005) to perform some of the analyses.
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
Assessment via FPCA
Bologna, July 5-9, 2009
20 / 57
The Functional Data Analysis Approach
Multivariate Functional Principal Component Analysis
For the mth eigenvalue ρm , measuring the proportion of variability explained by the mth principal
component, the mth harmonic ξm has:
kξm k2 = 1
and
p 2
kξm
k
is the proportion of the variability in the mth principal component accounted for by variation in
the p th Pollutant.
The eigenvectors are orthonormal.
1
We implemented the procedure in R, and in particular we used the library fda (Ramsay,
2005) to perform some of the analyses.
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
Assessment via FPCA
Bologna, July 5-9, 2009
20 / 57
The Functional Data Analysis Approach
Multivariate Functional Principal Component Analysis
Some Results
Table: Main results from Multivariate Principal Component Analysis
4 P
p
ξm
2
p 2
ξm Prop. of variab.
p=1
explained by PC
PC1 0.75
PC2 0.11
PC3 0.04
0.90
NO2
0.312
0.136
0.210
PM10
0.447
0.110
0.588
CO
0.153
0.021
0.109
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
Assessment via FPCA
SO2
0.082
0.698
0.029
1
1
1
Bologna, July 5-9, 2009
21 / 57
The Functional Data Analysis Approach
Multivariate Functional Principal Component Analysis
Plots of the first three Harmonics
0.05
0.00
−0.10
−0.10
0.00
0.05
0.10
PM10
0.10
NO2
0
100
200
300
0
100
300
0.05
0.00
−0.10
−0.10
0.00
0.05
0.10
SO2
0.10
CO
200
0
100
200
300
0
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
Assessment via FPCA
100
200
300
Bologna, July 5-9, 2009
22 / 57
The Functional Data Analysis Approach
Multivariate Functional Principal Component Analysis
0.00
−0.05
Apr
May
Jun
Jul
Aug
Sep
Oct
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
300
Mar
200
Feb
100
Jan
Nov
Dec
Nov
Dec
300
200
100
0
10
20
30
40
50
60
0
−0.10
harmonics
0.05
Harmonics - PM10
2
PC1(75% ), PC2(11%), PC3(4%) - ξ1PM10 = 0.447
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
Assessment via FPCA
Bologna, July 5-9, 2009
23 / 57
The Functional Data Analysis Approach
Multivariate Functional Principal Component Analysis
0.02
0.00
Apr
May
Jun
Jul
Aug
Sep
Oct
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
300
Mar
200
Feb
100
Jan
Nov
Dec
Nov
Dec
300
200
100
0
10
20
30
40
50
0
−0.04
−0.02
harmonics
0.04
0.06
Harmonics - NO2
2
PC1(75% ), PC2(11%), PC3(4%) - ξ1NO2 = .312
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
Assessment via FPCA
Bologna, July 5-9, 2009
24 / 57
The Functional Data Analysis Approach
Multivariate Functional Principal Component Analysis
0.06
0.04
0.02
Apr
May
Jun
Jul
Aug
Sep
Oct
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
300
Mar
200
Feb
100
Jan
Nov
Dec
Nov
Dec
300
200
100
0
0
5
10
15
20
25
30
35
0
−0.02
0.00
harmonics
0.08
0.10
Harmonics - SO2
2
PC1(75% ), PC2(11%), PC3(4%) - ξ2SO2 = .698
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
Assessment via FPCA
Bologna, July 5-9, 2009
25 / 57
The Functional Data Analysis Approach
Multivariate Functional Principal Component Analysis
Monthly Averages
pollutant: PM10
20
15
20
30
25
40
30
35
50
40
pollutant: NO2
4
6
8
10
12
2
4
6
8
Month
Month
pollutant: CO
pollutant: SO2
10
12
10
12
0
0
5
5
10
10
15
15
20
20
25
2
2
4
6
8
10
12
2
Month
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
Assessment via FPCA
4
6
8
Month
Bologna, July 5-9, 2009
26 / 57
The Functional Data Analysis Approach
Multivariate Functional Principal Component Analysis
0.00
−0.05
Apr
May
Jun
Jul
Aug
Sep
Oct
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
300
Mar
200
Feb
100
Jan
Nov
Dec
Nov
Dec
300
200
100
0
10
20
30
40
50
60
0
−0.10
harmonics
0.05
Harmonics - PM10
2
PC1(75% ), PC2(11%), PC3(4%) - ξ3PM10 = .588
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
Assessment via FPCA
Bologna, July 5-9, 2009
27 / 57
The Functional Data Analysis Approach
Multivariate Functional Principal Component Analysis
0.02
0.01
0.00
Apr
May
Jun
Jul
Aug
Sep
Oct
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
300
Mar
200
Feb
100
Jan
Nov
Dec
Nov
Dec
300
200
100
0
0
5
10
15
20
25
0
−0.01
harmonics
0.03
Harmonics - CO
2
2
2
PC1(75% ), PC2(11%), PC3(4%) - ξ1C 0 = .153; ξ2C 0 = .021; ξ3C 0 = .109
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
Assessment via FPCA
Bologna, July 5-9, 2009
28 / 57
The Functional Data Analysis Approach
Multivariate Functional Principal Component Analysis
0
100
200
60
50
40
30
Harmonic 1
20
+
+ +
+
++ ++ + +
+ +
++
+ + +
++ + − + + ++
++++++ ++ + + ++ ++ + ++ ++++ ++ ++
+++++ + ++ ++++
+ +
++ +++++ + + + + +++ − +
+ ++ +++
+
+ ++ +++ + ++
−
−
++
−
− −−−+ −
− − −−− − −
+ +
−
+
− −−
−
−
−
−
−−−−−− −
−− −− − − − − −−
− −−
−−
− −
− − −−−− −
−
− − − − −−−
− − −−−
−
− −
−
−
−
−
− −−
−−−
−−− −− −− −−
−−
−
−
−
−
−
−
−
10
35
30
25
15
20
Harmonic 1
40
The effects of the first three PC - Harmonic ξ1
300
+
++++
+
+ ++
−−−
+
+
+
++
−
+ ++ +++++ + +++
−
+++ + ++ +−
+ +
+
+
+
+++
+
+
+
+
+
+
+
+
+
+
−+ + + +
+++++
+++++ ++ −+ +
+
+
+
−
−
+ +
− + − +− ++ − −++ −+ + − −−
+ ++++
−
−− +−+ +
+
−
++−+
−
−
−
−
−
−
−
−
−
+
−
+− −−
−−− −−−−−
+
−− − − −
−
−
− − − −
−
− − − −−
−−
−
−−−
+−−−− − − −
−− − −−
−
−
−
− −−− −−
−
−−
−−
−
−
−
−
+
+
−
0
x
Pollutant[ 1 ]: NO2 − PCA 1 (Percent. of variability 75.1 )
0
100
200
300
300
+
15
10
5
Harmonic 1
20
+
x
Pollutant[ 3 ]: CO − PCA 1 (Percent. of variability 75.1 )
200
x
Pollutant[ 2 ]: PM10 − PCA 1 (Percent. of variability 75.1 )
0
15
10
0
5
Harmonic 1
20
+
+
++++ ++
+ + + +
+ + ++
++
+ ++ +
++
+
+
+
++
++++
+ ++
+
+++++ +++ +
+ +
−
++
+++
++++ + −+
− −−
+++
++++++++++ ++ + + ++ + + + −
−−−−
+++ +++++ +++
−−−−
−− −−− − −−−
−
+
−
−
++ +++
− − − − − −−
−−− −−−−−−−−−− −− −− −−−−
−− −− −
−
+
−−−
−−−−− −−− −−−− −−−−−−−−−−−−−− − −− − −
−
−− −−
−
100
+++
+
+
+
+
+
+ + ++
−+
+
+
+
++ + +
+++
+ ++ +
+
+
+ −−− ++ −
+ +
+ + ++
+
+
+++ ++ ++ ++++++++ ++
++ + +++ +
+ − +
+ + +
−
++ +++
−
−
+
+
+
−
+
− +++ − −
−+ +
− − −++
− +
−−−−
+ −− − −
+++ −++−−− −− − − − −−−− +
+− − −−− −− − − −− −−− −+−−−−−−+
++ −− −−− −−− −−−−−− − −−−−
− −+−
− −− −−−− − − −
− −
−
−− −
−
−
0
100
200
300
x
Pollutant[ 4 ]: SO2 − PCA 1 (Percent. of variability 75.1 )
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
Assessment via FPCA
Bologna, July 5-9, 2009
29 / 57
The Functional Data Analysis Approach
Multivariate Functional Principal Component Analysis
The effects of the first three PC - Harmonic ξ2
0
100
200
50
40
30
Harmonic 2
20
35
30
20
25
Harmonic 2
40
+
−
+
−+
−
−− −
+−
−
+ + ++ ++
+
+ −
− −
− −
+−
+−+
− +
−− −− − +−+
−+++ +
−−+ −+− −− ++−++ −
−
+
+
−
−
−
−
+−−− −
− −
−
+ + −+ −−−+
+
−
++−
+ −−
++− + −+−−−− −+−− +−−−
+
++ −+ − − + −−
++
− +−−
+ −
−−
−
+−
+
+ +− −
++−−− − −++−− +−− ++ +
− − − −
+++ −
+−
++++ +++ − + +
+
−
+
+ +
++ − +−
−
++
+ + + +−+ ++
−
+
+
+
+
+
+
−++
+
−
+
−+ −−−
+ − −−
−
−
+
+−
−
−
+
+
−
+
+
+
+
−+− −−
−
+
+−
−−+− − +++−
+
+−+++
++
++ + −
+
−
+
−−− −
+ +−+ ++ −−
−++
+−
++−−+ −
+−+ − −
+
+
+
+
−
−
+
+
−
−− −
+
− + −−−− −+ +
−
++
−
−−
−
−
+
−
+
−
+
−
−
+ ++ − + +
++−
+− + −
−
+−−
+ −−
+
−
+
+
−
+
+
+
−
−
−
−
+
+ − − +−
++ −
+ + −
−−− −
+−
+ −+−
+
+
+
−
−−
++ +
−
+
+
+
−
+
+
−
−
300
0
0
100
200
300
x
Pollutant[ 3 ]: CO − PCA 2 (Percent. of variability 11 )
200
300
15
10
Harmonic 2
20
25
Pollutant[ 2 ]: PM10 − PCA 2 (Percent. of variability 11 )
5
−
−
+
+
−
−−
−
−
−
+
−
− − −+++ −−
−
+−
+ ++
+
+
−
−++−
−
−+ −
+−
+−
+ +++ + +−
−
+ −−−
−+
+
−− − −
−− +−−
−
+
+
−
+
−
+
+
+
+
+ −
++ −−− ++−−+
−−
−−
−− −−
−
+++−+− + +− −−− +
−+++
−+ −+
+−+ + −−− −−−
−++++ + −
+
−−−−+ −− −−
+
+
+
−
+
+
+
−
+−
+
−
+
+
+
+
+
−
+ ++ ++++− ++−
−
+−
+−
++++−
+−+ −
+ + −
+
+−
++
− +−
+
−
100
x
0
10
5
Harmonic 2
15
x
Pollutant[ 1 ]: NO2 − PCA 2 (Percent. of variability 11 )
+
+
++
++
+
+
+
++ ++ +
++
+
+
++
+ ++
+ +
+
+−
+ +
++ +
+
+
−
+
+−
−
++−+
++++
+ + ++ ++ ++++ +++++
+−− − +++ +++ + −− −−−++
+
+ −
+
++
− +
− ++−−−−
− + ++++
−−−−− −
−
− + ++ + + ++
−− +++−
− −+−− +−−−−−−−−−−−+
+−
−
−−
+
−−+−− −− −−− −−− − −−− −−−++ −−−−−−−−−−−−−−
−
−
− −−− − −−
+
−
−−−−
−
−
0
100
200
300
x
Pollutant[ 4 ]: SO2 − PCA 2 (Percent. of variability 11 )
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
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di Palermo)
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The Functional Data Analysis Approach
Multivariate Functional Principal Component Analysis
0
100
200
50
40
Harmonic 3
30
+
−
+
+
++ +
−+
−
+
−+ ++ +++
−
− −
+
++
+
−+
− +−−
+− −
− ++− −−−
−
+ −
+ − ++ + −+−
−− −
+−
++−−
−
−
−
− +−−
−
+
+−
+−
+−
++ −−
+−
+ −− ++
+−+ − −
+ ++ +++ +− +−−+−+−+ +
−
++
+
++ + ++ + ++ − −+ + +− + −+−
+−−++−+−++ −
−
−
−
−
++
+ ++
+
− − − + −− −+−+
−
−−−
+
+ − +−− −−
−
−
+
−
−
+
+−
−−
− +− + −
−
+
+
−
−
+
−
+ −
−
−
20
35
30
25
Harmonic 3
40
The effects of the first three PC - Harmonic ξ3
300
−
+−−
+
−
−
−
++
+−
− −− + − −−
+
− −
+
− +− +
−
+− +− − −
+ −− −+ + ++−−− − − +++−−
−
−
++
+−
+
++
− ++−
− +
+−
−
++ − + −
+−−++− + −+ − ++ +−−+ + +−−
+−
+ +
+−−
+ −
+
−
−
− −+
+
−
+
−−−
−
+
+ − −+ +
−++− ++− + − −
−
−
−
+
−
++
+−
− −
+ −
+ −−+++
+ + + −
++
+ −
++ −+
++−
+
+++ −−
−
+ +−−
−
+−
++ − −−−−
−
++ + + +
+
+
+
−
−
+
0
x
Pollutant[ 1 ]: NO2 − PCA 3 (Percent. of variability 4.2 )
100
200
Pollutant[ 2 ]: PM10 − PCA 3 (Percent. of variability 4.2 )
0
100
200
300
x
Pollutant[ 3 ]: CO − PCA 3 (Percent. of variability 4.2 )
+
−
15
Harmonic 3
10
5
10
5
Harmonic 3
15
+
+
−
+−
++++ ++
+
− +−
+
−−+
+−
+
+−
−
+
−
−
−
+
−
+− +
−+
+−
++
− −−−+ −−+−+
++
−
− ++−
+ +
+−
++
+ −+++ +++−+
−
+
−
−
−
+
+
+
+
−
+
+
+
+
−
−
−− −
−++
+++ +−
−−−−++ −− −+++++++++−− +−−
+
+−−+−−− − −
+
+
+
+
+
−
+
−
+
+
−
+
−
+
+
+
−
+
− −+
−
−
+
+− −
−−− −− −−−−+ −−+−−−−−+−+−
−−
+++
− − −+−+ −
−− −−
−
+
300
x
+−
−−
+
+
−
++
−−
++
−
−
−
−
+
−+
−
+ −+− −
−
−
+−
+ +
+ + +− −
+−
−
−
++−
+++−
−
+ −
+
+
−
−
+
+
+
++ +−
+ −
+
−
+
+
+
−
+ −
+ −+
++−
−−
+
++−
+
+−
−+
− − −
−
+ −
+−
+
+−
−
+−
+ +−
+
+−
−
+
−
+
+
−
+
+
++−−
+
−
−
−
+
−
+
+ +
+ +++
−−
+ −
+−++
+ −+−− +−
−
+−
− −
+−
+−
+ −
+
−
+
+
+
−
−+
−−−− + −+− − − −
+−
−
++−
+
−
−
−
+−+ −
+
−
−−
+
0
100
200
300
x
Pollutant[ 4 ]: SO2 − PCA 3 (Percent. of variability 4.2 )
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
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The Functional Data Analysis Approach
Multivariate Functional Principal Component Analysis
Examination of Scores
Table: Principal Component Scores
Belgio
Boccadifalco
Castelnuovo
Cep
DiBlasi
GiulioCesare
Indipendenza
Torrelunga
UnitàItalia
PC1
65.54
-372.96
73.35
-202.96
311.07
136.90
-32.78
-63.22
85.06
PC2
-60.40
87.69
163.64
-52.58
37.69
-23.77
-41.91
-92.20
-18.14
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
Assessment via FPCA
PC3
-24.23
12.16
9.58
-108.89
-27.52
22.49
98.25
30.68
-12.52
Bologna, July 5-9, 2009
32 / 57
The Functional Data Analysis Approach
Multivariate Functional Principal Component Analysis
Examination of Scores
Figure: Plots of the first three Principal Component Scores of the stations
150
●
Cep
100
100
150
●
Castelnuovo
50
●
Torrelunga
−50
−50
●
Cep
●
Belgio
−200
−100
−100
●
Torrelunga
−400
0
●
Castelnuovo
●
GiulioCesare
●
Boccadifalco
●
UnitàItalia
●
GiulioCesare
●
Indipendenza
●
DiBlasi
●
Belgio
●
UnitàItalia
0
PC3 (perc.var. 4%)
50
●
DiBlasi
0
PC2 (perc.var. 11%)
●
Boccadifalco
200
400
●
Indipendenza
−400
PC1 (perc.var. 75%)
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
Assessment via FPCA
−200
0
200
400
PC1 (perc.var. 75%)
Bologna, July 5-9, 2009
33 / 57
The Functional Data Analysis Approach
Multivariate Functional Principal Component Analysis
70
Figure: Threshold value overcomings
Belgio
●
Boccadifalco
●
Castelnuovo
●
●
●
DiBlasi
●
●
GiulioCesare
● ●
Indipendenza
●
UnitàItalia
●
●
●
●
●
●
55
●
● ●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●●
●
●
●
Jan
●
●
●
●
●
●
●
●
●●
●
●
●●
●
● ●
●
●
●
●
●
●● ● ●
●●
●
●
●
●
●● ●●●●
● ●
●
●
●
●
●
●
● ●●●
●
●●●● ●
●
● ●
●
●●●
●●
●
●
●
●●
●●
●
●●
● ●
●● ●
● ●●
● ●
●
●
●
●
●
●
●
●
●●●
●● ●
● ●
●
●
●
●●
●
●
●
●
●●
●● ●
● ● ● ●
● ●●
●●
●
●
●● ●
● ●●
●●
● ● ●
●
●
●●●
●●
●●●
●● ●●
●●
●●
●●●
●● ●●●●●
●●
●
●
●
●
●
● ●
50
●
●
●
●
● ●
●
●
●● ●
●
●●
● ●
●
●●
●
●
●
●
Feb
●●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
Torrelunga
60
concentrations
65
Cep
●
Mar
Apr
May
●
●
●
●●
●●
●
●●
● ●
● ●
●
●●●
●
●
●●
●
●●
●
●●
●
●●●
●
●
●
●
● ●●
●
●● ● ●●
●
●
●
●
●
●
●
● ●●
● ●
●
●
● ●
●
●
●●
● ● ●
●●●
●
●
●
●●
●
●
●
●●●●●●
●
●
● ●
●
●●
● ●
●●
●●
●
●●
●●
●
●● ●● ●●
●
●
●● ●●
● ● ●
●●● ●
●
●
●
●
●
●
● ● ●● ● ● ●
●
●●●
●●
●● ●●
●
●●
●
●
●●
●●
●
●●
Jun
Jul
●
●●
●
●
●
●
●●
●
●
●
●●
●
●●
●
●
●
●
● ●
●●●●
● ●
Aug
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
Assessment via FPCA
●
●
●●
●
●
●
●●
●
●
●
●
●
● ●●
●
●
●
●
●
●
●
●●
● ● ●
●
●●
●●
●
●
●●
●
●
●
● ●
● ●
●
●
●●
●
●
●
●● ●
●
●
●● ●
●● ●
●
●
●●
● ●
●●
●
●
●
●●●
● ●●● ●
● ●●
●●
●
●● ●● ●
●● ●● ●
● ●● ●
●●
●
● ●●●●
● ●
●●●●●
●
●
●●
●
● ●
● ●●
●●
●
●
●●
● ●
●
● ● ●● ● ●●
●
●
● ●
●
Sep
Oct
●
●
●
●●
●
●
●
●
●●
●●
●
●
● ●● ●
●●●
●
●
●
●
●
●
●
● ●
●
●
●●●● ●●●
●●
●
●
●
●
Nov
●
●
●
●
●●● ● ●●
●
●
●
●
●
● ●●●
●
●
●●
●
● ●●
●
●
● ●● ●
●
●●
●●
●
●
●●
● ●●
●●
●
●●
●●●●
● ●●●
●
●
●●●●●●● ● ●
● ●
●
●●
● ● ● ●
Dec
Bologna, July 5-9, 2009
34 / 57
The Functional Data Analysis Approach
Multivariate Functional Principal Component Analysis
Monthly Averages Values
pollutant: PM10
20
15
20
30
25
40
30
35
50
40
pollutant: NO2
4
6
8
10
12
2
4
6
8
Month
Month
pollutant: CO
pollutant: SO2
10
12
10
12
0
0
5
5
10
10
15
15
20
20
25
2
2
4
6
8
10
12
2
Month
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
Assessment via FPCA
4
6
8
Month
Bologna, July 5-9, 2009
35 / 57
The Functional Data Analysis Approach
Multivariate Functional Principal Component Analysis
Functional PCA for 5 Pollutants
1
Now we consider a quite different situation, reintroducing the OZONE (O3)
Figure: Ozone series at 2 Sites
OZONE
Boccadifalco
Boccadifalco
Boccadifalco
Castelnuovo
Castelnuovo
Castelnuovo
Jan
Mar
May
Jul Aug
Oct
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
Assessment via FPCA
Dec
Bologna, July 5-9, 2009
36 / 57
The Functional Data Analysis Approach
Multivariate Functional Principal Component Analysis
Functional PCA for 5 Pollutants
1
2
3
Now we consider a quite different situation, reintroducing the OZONE (O3)
The serie of Ozone of Castelnuovo are used as series for the other urban sites.
We repeat the Functional PCA to understanding the effects of this ”‘perturbation”’ of the
data on the results.
Figure: Ozone series at 9 Sites
OZONE
Boccadifalco
Other_8_Sites
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
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The Functional Data Analysis Approach
Multivariate Functional Principal Component Analysis
Functional PCA for 5 Pollutants
1
We have obtained the following results:
Prop. of variab.
explained by PC
PC1 0.717
PC2 0.137
PC3 0.05
0.904
NO2
0.267
0.392
0.237
p 2
ξm PM10 CO
0.369 0.125
0.131 0.011
0.070 0.165
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
Assessment via FPCA
SO2
0.178
0.054
0.093
O3
.057
.391
.396
1
1
1
Bologna, July 5-9, 2009
38 / 57
The Functional Data Analysis Approach
Multivariate Functional Principal Component Analysis
Examination of Scores - 5 Pollutants
●
Boccadifalco
●
DiBlasi
50
●
Castelnuovo
−50
0
●
GiulioCesare
●
UnitàItalia
●
Belgio
●
Indipendenza
−100
PC2(perc.var.:0.14%)
100
150
Figure: Plots of the first three principal component scores of the stations
●
Torrelunga
−150
●
Cep
−400
−200
0
200
400
PC1 (perc.var.:0.72%)
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
Assessment via FPCA
Bologna, July 5-9, 2009
39 / 57
The Functional Data Analysis Approach
Multivariate Functional Principal Component Analysis
Examination of Scores
Figure: Plots of the first three principal component scores of the stations
150
100
●
Boccadifalco
●
UnitàItalia
●
GiulioCesare
●
Indipendenza
●
Cep
●
Belgio
50
−100
●
Belgio
●
Indipendenza
●
Torrelunga
●
Cep
−150
−150
−100
●
Torrelunga
●
GiulioCesare
●
UnitàItalia
0
PC2(perc.var.:0.14%)
50
0
●
DiBlasi
−400
−200
0
200
400
●
DiBlasi
●
Castelnuovo
−50
●
Boccadifalco
−50
PC2 (perc.var. 11%)
100
150
●
Castelnuovo
−400
PC1 (perc.var. 75%)
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
Assessment via FPCA
−200
0
200
400
PC1 (perc.var.:0.72%)
Bologna, July 5-9, 2009
40 / 57
The Functional Data Analysis Approach
Multivariate Functional Principal Component Analysis
Figure: Plots of the first three Harmonics
−0.05
−0.05
0.05
PM10
0.05
NO2
0
100
200
300
0
100
300
−0.05
−0.05
0.05
O3
0.05
CO
200
0
100
200
300
0
100
200
300
−0.05
0.05
SO2
0
100
200
300
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
Assessment via FPCA
Bologna, July 5-9, 2009
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The Functional Data Analysis Approach
Multivariate Functional Principal Component Analysis
0.00
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Aug
Sep
Oct
300
Feb
200
Jan
100
0
−0.04
−0.02
XSI[, , 2]
0.02
0.04
PM10
Nov
Dec
Nov
Dec
50
40
30
20
Mar
Apr
May
Jun
Jul
1:365
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
Assessment via FPCA
300
Feb
200
Jan
100
0
10
PRED.spline1[, , 2]
60
1:365
Bologna, July 5-9, 2009
42 / 57
The Functional Data Analysis Approach
Multivariate Functional Principal Component Analysis
0.00
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Aug
Sep
Oct
300
Feb
200
Jan
100
0
−0.05
XSI[, , 5]
0.05
SO2
Nov
Dec
Nov
Dec
30
25
20
15
10
5
Mar
Apr
May
Jun
Jul
1:365
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
Assessment via FPCA
300
Feb
200
Jan
100
0
0
PRED.spline1[, , 5]
35
1:365
Bologna, July 5-9, 2009
43 / 57
The Functional Data Analysis Approach
Multivariate Functional Principal Component Analysis
0.02
0.00
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Aug
Sep
Oct
300
Feb
200
Jan
100
0
−0.04
−0.02
XSI[, , 4]
0.04
0.06
O3
Nov
Dec
Nov
Dec
50
40
30
20
Mar
Apr
May
Jun
Jul
1:365
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
Assessment via FPCA
300
Feb
200
Jan
100
0
10
PRED.spline1[, , 4]
1:365
Bologna, July 5-9, 2009
44 / 57
The Functional Data Analysis Approach
Multivariate Functional Principal Component Analysis
0.02
0.01
0.00
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Aug
Sep
Oct
300
Feb
200
Jan
100
0
−0.02
−0.01
XSI[, , 1]
0.03
0.04
NO2
Nov
Dec
Nov
Dec
40
30
20
Mar
Apr
May
Jun
Jul
1:365
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
Assessment via FPCA
300
Feb
200
Jan
100
0
10
PRED.spline1[, , 1]
50
1:365
Bologna, July 5-9, 2009
45 / 57
The Functional Data Analysis Approach
Multivariate Functional Principal Component Analysis
0.02
0.01
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Aug
Sep
Oct
300
Feb
200
Jan
100
0
0.00
XSI[, , 3]
0.03
CO
Nov
Dec
Nov
Dec
25
20
15
10
5
Mar
Apr
May
Jun
Jul
1:365
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
Assessment via FPCA
300
Feb
200
Jan
100
0
0
PRED.spline1[, , 3]
1:365
Bologna, July 5-9, 2009
46 / 57
The Functional Data Analysis Approach
Multivariate Functional Principal Component Analysis
100
200
60
40
Harmonic 1
+
+
++
+
++
+ ++
−+
−−
+ ++ ++ + ++ + + + + ++
−
+
++++ + ++ +−−
+
++
+++ + ++ + + ++ +++ +
+
++ + + ++++
+++ + + ++++++ − + +
+−
−+ ++ + − −−++++−−+ −
+
+
−
+
− −−
−
+
+
+
+
+
−+ −
+ +−
−−− − −− +− +++
−−
−−−− − − − − −−
−−
− − − −−−−− −− − +
−
−
− −−− − − − −−
−−
− −−−−
+−−− − − − −− −− −
−
−
−− − −
− − −− −
−−
−
−
−
300
0
20
15
10
5
300
− −−
−
−
− −−−−−+− −−++ −
−− − − −−
−−
−
− +
− −
−
−−
+ − + −− −− − − −−− −
− −−
−−− − + −− −−−−+− ++ + +
−
++ + − − + ++ − −−− −−
− ++ −
+
−− − −
++ + + +
−−−+ +− + ++
−−
− −−
+ −− + + +
++ + ++++− +−−− −+ + −− −− −−−
++ + +
+ ++ ++ +
+
+
−
+
−
+
+
+
+++
++ +
−− −++
++
+
++ +++
+
+ + + + ++ ++− ++
+
+
+
+ + +−
++++
+
+
100
200
300
20
30
40
+
+
+ ++++ ++
+ + +++
++++ + +
+ ++
+
+
+ ++
+
+
+
+
+
+ ++
+ +
+
−
++++ + ++ +
+++
+++
++ ++ + − +
− − −−
+++
++++++++++ ++++ + ++ +++++ + + + ++ −
− − −−
−− −
+ ++ + +++
− −− − −−− − − − −−−−−
−− + −−− +
− − −−− − −−−−−−−
−
−
−
+
−
− − −−−
+
−−−−−−−−−− −−−−−− −−−−−−−−−− − −−−−−−−−−− − − − −
−−
−
0
x
Pollutant[ 3 ]: CO − PCA 1 (Percent. of variability 71.7 )
Harmonic 1
200
x
Pollutant[ 2 ]: PM10 − PCA 1 (Percent. of variability 71.7 )
0
0
100
x
Pollutant[ 1 ]: NO2 − PCA 1 (Percent. of variability 71.7 )
Harmonic 1
20
5 10
0
Harmonic 1
0
20
+
+
+
++
++ ++ + +
+ +
+
+ + + − + + ++
+++++ + ++ + + + ++ + ++++ ++ ++ + + +++++
++ +++ + + +++ + ++ +++ ++ + ++
+
+ + +++ + + +
+ + − ++
++
+
++ + + + ++
− +
+
−− +
− −− −
− −
−
−
−
−
+
−
+
−
− −− − −
− − −−−− −−−
+ −
−
−−−−− − − −
−− − − − − − − − − −−
−− −
−− − − −−− −
− − − − −−
−
−
−
−
−
− −
−−
−
−−− −−
−−−
−−−
−
−−
−
− −
−
−
−
10
35
25
15
Harmonic 1
The effects of the first three PC
100
200
300
x
Pollutant[ 4 ]: O3 − PCA 1 (Percent. of variability 71.7 )
+
+++
+++ +
+
+−
+
+
+
+
+
+ + −−+
+ +++ +
+ ++ +
+
+−
+
−
++
+ +
+++ + +−
+++
++ + + ++++ ++++
+
++ +
−
+ − − ++ ++
++ + ++++++ −+ + −++ + + ++++
++
− − −
−
−
−
−
+
−
−
−
−
++−
+ −− −
−− − − + +− −− − − − −
++ −−
+−
−− −−+−−−−
+− − − −−−− − −−−−−−−−− − − −+
− −− − −− − −
+−
−
−
+
+−−
−
−
−
+ − −
− −− − −−
−
−−−−
−
−
−
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The Functional Data Analysis Approach
Multivariate Functional Principal Component Analysis
The effects of the first three PC
100
200
Harmonic 2
20 30 40 50
+
−++
+
+
−
+
− −−
−+
+
−
− ++
−
+−
+
+ − − −−
−
+ +−
+ −
+ +
+−
+ −
++−
+ ++ + + ++
+
−
+
+−
+−
++ + −+ −
++ ++++ −
++−
−
+
+ +−−+ − −
+ +−−−−+−
+ −+−
−
+
+
−
−
+
−
−
−
−
+
+
+ +
+ + ++− + − −
−+−
+
+−
− −+− +++
+++−−
− −−− −− − +
− ++−
+ −
++ −
−
+
+
−
−+
−
−
−
+
−
+
+
++ −−−+
− −− −+ + −− −
+−
− +−
+−
−
− − −
− −−
+
+
−
+
−
−
+
−
300
0
100
200
300
x
Pollutant[ 2 ]: PM10 − PCA 2 (Percent. of variability 13.7 )
+
−
+
−
+ ++
+
+
+ +
+ + −
−
+−
− +−− ++
− +− ++−
+ −− + −− −
++−
−
−
−
+
+ − ++−
−−
− +
+
++
+−
+
+
+
−
+
+
−
−−
−
+
+
−
−
+
−
−
+
+−
+ − ++ −−
+
+
− +−
+ +−
−
+++
+−
+−
++ −
−
−− + + −
− +−+
+
++−
++− + ++ −
− ++
+ −−++ +++++ + ++ +−−++
++−
−+
− −
+−−
+−−
−
−
−−
−−
+−−+ −
−+−−−−−+− + −− +− −
−−
++−
−
−
− +−+ −
−
−
−
+
−
+ ++
− −+
+
+ +++++ + ++ −−+
++ + ++
++
+−−−−−
+
++
+ +
++
−+−− − ++ ++ − + + − +++ + +++
+++ +−−−−++ ++++−+−
+
+ + −− − − − +−− +++− ++
+
−+
++ + +
−
−
+
−
+
−
−
−
−
−−
−
++
− −
−− −
− ++ −+ + −
+ ++
−
− −−
−− −
− −−
− − −− − +−−+++ +− −− −++−++ +++−−
−
−
−
−
++++−−+
− − − −− −−+ −−
−
− − −+−
−
−−−− −
−
−
0
100
200
300
15
25
35
x
Pollutant[ 1 ]: NO2 − PCA 2 (Percent. of variability 13.7 )
Harmonic 2
15
10
5
Harmonic 2
0
5
Harmonic 2
25 30 35 40
+
+
−
+
−
−+
−
+ + ++ + ++ +
+
−− −
− −
+++
+
+ +−
+−
++− −
− + + − +− + + ++
+ −−−
+ −
++ −−+− + − ++ −
−
+ −
+
++
+−
−− −
+ + −−
+−
++ −
++ − −+ − +− −− − +−
+
+
−−
−+ −
−−−
+−
+−
+−
−
−+
−
+
−
−
− −
−
−
− −
−
+
− +−
++
+ −
− −−
+ − −
+−
+ + ++ ++++−−
− − ++
++ −
+−
−+−
+
+
− − −−−
+−
−
−
+
−
+
−
++
−
+ − ++ −
+
++
−
+
−
−
+ −
+
−
+
+
+
0
x
Pollutant[ 3 ]: CO − PCA 2 (Percent. of variability 13.7 )
100
200
300
x
Pollutant[ 4 ]: O3 − PCA 2 (Percent. of variability 13.7 )
20
10
5
0
Harmonic 2
+
+++
+ +
+
+++ + +
+
+
++ +
+
+ ++ +
+
++ ++
+
++ +
+
++ − +
+ ++ +− +
+
−
+++
+++ +
+−−−+−
−− ++
+
−
+
+ +
−
−
+ + +++ ++ −+ ++++++ + +−++ +
+++ − −−−− −−− − − − − ++ +++++ + + +−
− − − ++−−−+−
−+−−−− +−−−−−−−−− −+−+
−
+
−
−
−
−
−
+
−
−
−− − −− − − − − −
−−−−
+− −− − −
−−
+−
+
− −− − − −−−−−−−−−
−
−
−
−−−
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
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The Functional Data Analysis Approach
Multivariate Functional Principal Component Analysis
100
200
50
40
30
Harmonic 3
300
−
++−
+
−
−
+−
−
+−
+
−
+
+
−
−
+ +−
+−
−
+
+−
+ −
+
+−
+ −
+
+−
+
−
+
+
+− −
+ +−
+−
−
++−
+ − −
+−
+++ −
−
+− + −
++
−− −
−
+
−
−
+−
+−
+−
+
+
+−
+−
+ +
− + +−
+ +−
+
+−
+ +
−
−
++
− + −
+ −−−
+ −
−
−
−
+
+ −+−
+
−
−
+
+
− + −
−
−
+
+
−
+++
+
+
−
−
−
+−
+
+ −
−
+
+−
−+
++− ++−++
+ −
−
+−
−− +−−
−
++ + −
+
+−
+
−− −−
+
−
−
−+
−
−
+
−
−
−
+
+
−
0
20
15
10
5
300
+
++
− +
−−
++−
+ + + −+
+
+ +
+−
++
++ + ++
+ −
− +−− − − −−
+−
−+ ++
−
+−
+
−
−
++ ++
+
+ +−− ++−+− − + −− +
+ +− +−
+
++−
+
− ++
−
− −− − ++−+ + ++−+− ++
+
−
−
−
+−−+−
+−
++ +
−
−
+−−+−
+ − +−
++−
+
−
−
−− −+−− − −− + −+−++− +
−
+ − −
+−
+ ++
+ − −− ++−+−
−−
−
+ +−+ −
−+
− −
+−
+
−− − +
− −−
++ −
+−−
−−−+
−
−
+
−
100
200
300
15
25
35
+
+
−
−
+ ++++ ++
+ +
−
++
+− −−−−+
+ −
−
+
+
+
+
−
−
−
++
+−
−− ++
+ ++
+ + −− + −−− + − +−−
−+ + + +
+++
+++
−
−++ ++ +
++ +++− +−+ −
−
− − −+ + + + −− +−−−++ +++++++ −−
+
+
− −− −+−+ −−
− +++++ −−−++ +++++ + ++ +−−+−−− − − +
−−−
−
+
−−−−−−−−−− −−−− − −−−−−−−−+−+++−−+++− −
−
−
−
− − −−−
−
+
0
x
Pollutant[ 3 ]: CO − PCA 3 (Percent. of variability 5 )
Harmonic 3
200
x
Pollutant[ 2 ]: PM10 − PCA 3 (Percent. of variability 5 )
0
0
100
x
Pollutant[ 1 ]: NO2 − PCA 3 (Percent. of variability 5 )
Harmonic 3
15
10
5
Harmonic 3
0
20
−
+
+
+
−
++ −+
−
−
+
+
−
+
+
++ +
−−
+ −
+
−+
+ −
+− −
+
+
+ −
+−
−
+−
+ −
− −−
−
−
++−
+
+
+
−
−
+
+
+ ++ +
+
+
+
++ +
−
−
+
+
+
−−−+−
+ −−
+ −
+
++ + −++ +
+ −− − −+−
+−
+− −
+−
− − −
−
+ −−+−
−
− −+ +− + −
+ ++−
+ ++
+ −+ + −
+ + +
−
+
−− +
−+
+
− −
+− ++
−
+
− −
−−
+ −+ + + + − + −
− − − −−− +−+
−+
+−
−
− −+ − − +−
+ −
−
−
+
−
−
+
−
+
−
−
−
−
5
35
30
25
20
Harmonic 3
40
Figure: The effect of the harmonic 3 on each pollutant
100
200
300
x
Pollutant[ 4 ]: O3 − PCA 3 (Percent. of variability 5 )
−
−−
−
−+ −
−
−
−
−− − − −
+
−−
−−−− − + − −+++ −−
+−
− −
+
−−
+−
−−
++
+
+ + +
− −+ +
+
− − − −+ − − +− −−− −−−− −
−− + +−+ + − ++ + − + −− −−−
−
++
+ +−+−
−+ +−++ + +−
+−
++ ++
+−
−
+ + ++−
+
+++++−
+ −−
−+ + − −−−− − −−+ +−
− + +−
+
+
+
−
− −−
+
+
−
+−
+
+
+−
+
+
−
+
+
++
− −−−
+
++ ++ +++++++−−
++
++ ++++
+
−
+
+
0
100
200
300
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
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Further Developments
Use of the optimal empirical orthonormal basis
Optimal Empirical Orthonormal Basis
1
Use of the optimal empirical orthonormal basis for approximating the
functional data.
As Principal Components form a orthogonal base:
s,p X(t)
=
3
X
ξkp (t)PCs,k
k=1
In matrix notation, for a fixed p (the Pollutant), let :
F3,9 = PC T
p
ξ365x3
= ξ1p , ξ2p , ξ3p
p
p
X365x9
= ξ365x3
F3,9
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Further Developments
Use of the optimal empirical orthonormal basis
Optimal Empirical Orthonormal Basis
functional data − PM10
Approximation via EOB − PM10
Figure: PM10: Plots the approximated data via Optimal EOB (left panels) and
observed functional data (right panel)
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Jan
Feb
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
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Apr
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Nov
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Further Developments
Use of the optimal empirical orthonormal basis
Optimal Empirical Orthonormal Basis
functional data − NO2
Approximation via EOB − NO2
Figure: NO2: Plots the approximated data via Optimal EOB (left panels) and
observed functional data (right panel)
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Jan
Feb
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
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Apr
May
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Nov
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Further Developments
Use of the optimal empirical orthonormal basis
Optimal Empirical Orthonormal Basis
functional data − SO2
Approximation via EOB − SO2
Figure: SO2: Plots the approximated data via Optimal EOB (left panels) and
observed functional data (right panel)
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Jan
Feb
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
Assessment via FPCA
Mar
Apr
May
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Jul
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Oct
Nov
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Further Developments
Use of the optimal empirical orthonormal basis
Optimal Empirical Orthonormal Basis
functional data − co
Approximation via EOB − CO
Figure: CO: Plots the approximated data via Optimal EOB (left panels) and
observed functional data (right panel)
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Jan
Feb
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
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Assessment via FPCA
Mar
Apr
May
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Jul
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Oct
Nov
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Further Developments
Methodological Perspective
Methodological Perspective
1
Definition of a new family of Indices
2
Use of alternative approaches to smoothing in the multivariate contest
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Further Developments
Methodological Perspective
References
Bruno F, Cocchi D. 2007. Recovering information from synthetic air quality indices.
Environmetrics 18, 345-359.
EPA (Environmental Protection Agency). 2006. Guideline for reporting of daily air quality:
air quality index (AQI). United States Environmental Protection Agency,
EPA-454/B-06-001.
European Community. 1999. CouncilDirective 1999/30/EC of 22 April 1999 relating to
limit values for sulphur dioxide, nitrogen dioxide and oxides of nitrogen, particulate matter
and lead in ambient air. Official Journal L 163, 29/06/1999: 41-60.
European Community. 2000. Directive 2000/69/EC of the European Parliament and of the
Council of 16 November 2000 relating to limit values for benzene and carbon monoxide in
ambient air. Official Journal L 313, 13/12/2000: 12-21.
European Community. 2002. Directive 2002/3/EC of the European Parliament and of the
Council of 12 February 2002 relating to ozone in ambient air. Official Journal L 67,
9/3/2002: 14-30.
Ignaccolo R, Ghigo S, Giovenali E. 2008. Analysis of air quality monitoring networks by
functional clustering. Environmetrics, 19, 672686.
G. Agrò, F. Di Salvo, A. Plaia, M. Ruggieri (Università
Air Quality
di Palermo)
Assessment via FPCA
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Further Developments
Methodological Perspective
References
Ingrassia S, Costanzo GD. 2005. Functional principal component analysis of financial time
series New Developments in Classification and Data Analysis, Vichi M, Monari P, Mignani
S, Montanari A. (Eds.), Springer-Verlag, Berlin, 351-358.
Murena F. 2004. Measuring air quality over large urban areas: development and
application of an air pollution index at the urban area of Naples. Atmospheric Environment
38, 6195-6202.
Ott WR. 1978. Environmental Indices: Theory and Practice. Ann Arbor Science
Publishers: Ann Arbor.
Pollice A, Jona Lasinio G. 2009. Spatiotemporal analysis of the PM10 concentration over
the Taranto area. Environmental Monitoring and Assessment, Published online: 11 March.
Ramsay JO, Silverman BW. 2002. Applied Functional Data Analysis. Springer-Verlag.
Ramsay JO, Silverman BW. 2005. Functional Data Analysis. Second Edition.
Springer-Verlag.
Shaddick G, Wakefield JC. 2002. Modelling multiple pollutants at multiple sites. Journal of
the Royal Statistical Society, Series C (Applied Statistics), 51, 351-372.
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