This thesis evaluates different methods for applying the Kalman filter to correct air quality model forecasts in Hong Kong. It introduces two alternative techniques - the logarithmic ratio bias method and relative difference bias method - to model bias beyond the traditional difference method. The performance of each correction method is assessed for PM2.5, NO2 and O3 at three monitoring stations over three years of hourly data using various statistical metrics. The aim is to determine the optimal bias modeling approach for each pollutant and location.

1. Kalman filter predictor
applied to air quality forecasts
Bachelor’s Thesis
Oskar Triebe
June 20, 2015
Supervisor at ETH: Prof. Dr. Jing Wang
Supervisor at HKUST: Prof. Dr. Alexis Lau
Advisors at HKUST: Dr. Ying Li, Mr. Tong Liu
Department of Civil, Environmental and Geomatic Engineering, ETH Z¨urich

3. Abstract
The Kalman filter (KF) is a recursive post processing method gaining
popularity in the field of air quality (AQ) forecasting. It is can be used
to remove bias from model predictions. Preceding studies achieved
this by modelling the difference of predictions and observations (D-
KF). This study aims to provide a larger set of methods to correct AQ
forecasts. Two alternative bias modelling techniques are additionally
introduced: The logarithmic ratio bias (LR-KF) and the relative differ-
ence bias (RD-KF). Complimented with further variations, these were
applied on the forecasts with the aim to remove the model bias and
improve the prediction of high pollution episodes (HPE). Three years
of hourly concentrations of fine suspended particulates (PM25), nitro-
gen dioxide (NO2) and ozone (O3) at three stations in Hong Kong SAR
were utilised. An optimal method for each station and pollutant was
selected by evaluating the normalised mean square error (NMSE), un-
paired peak prediction accuracy (UPPA), fractional bias (FB), fraction
of predictions within a factor of two (FAC2) and a measure of the skill
to predict HPE (CFM). A comparison with persistence, the trivial time-
lagging of observed concentrations by one day, showed that the raw
model forecasts perform significantly worse than this trivial method.
The selected methods showed a significantly improved overall perfor-
mance compared to both, the raw forecasts and persistence. The CFM
however was not improved in all cases. The optimal method was dif-
ferent for each pollutant and station with the exception of NO2 where
the RD-KF was most skilful. It was also found that the LR-KF is best at
correcting extreme differences and peaks, whereas the D-KF and RD-
KF perform better in general cases. Further, strengths and weaknesses
of the KF in general and of the different bias modelling techniques are
explained.
i

5. Acknowledgements
I would like to acknowledge Professor Alexis Lau for his great guidance
and the valuable insights he offered me during our frequent meetings at the
Hong Kong University of Science and Technology, Doctor Ying Li and Tong
Liu for advising me in the process of creating this study. Also I would like
to thank David Yeung for his help to access the data. Finally I am grateful
for the support provided by Professor Jing Wang as our supervisor at our
home institution, ETH Zurich.

7. Contents
Contents v
List of Tables vii
List of Figures ix
1 Introduction 1
2 Methods 3
2.1 Kalman Filter Predictor . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2.1 Observations . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.2 Model Predictions . . . . . . . . . . . . . . . . . . . . . 5
2.3 Verification Statistics . . . . . . . . . . . . . . . . . . . . . . . . 6
2.4 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4.1 Modeled Bias . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4.2 Variations . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3 Results 11
3.1 Performance of Best Methods . . . . . . . . . . . . . . . . . . . 11
3.2 Sample Time Series . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2.1 Special Time Series . . . . . . . . . . . . . . . . . . . . . 15
3.3 Live Display of Actual Implementation . . . . . . . . . . . . . 17
4 Discussion and Conclusions 19
4.1 Discussion on Best Methods . . . . . . . . . . . . . . . . . . . . 19
4.2 Discussion on Time Series . . . . . . . . . . . . . . . . . . . . . 20
4.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.4 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . 22
A A short introduction to the Kalman filter 23
v

8. Contents
B Variance Computation and Initial Values 25
C Literature Review 27
D Statistical Meaures 31
D.1 Normalisation of Statistics and Selection of Best Methods . . 31
D.2 Normalised Mean Scores of All Methods and Models . . . . . 32
D.3 Statistical Measures for Corrected Model Ensemble Forecasts 33
Bibliography 43
vi

9. List of Tables
2.1 Ratio limits for LR and RD bias. The values are the maximum
ratio of Pi/Oi and Oi/Pi for each origin respectively, restricting
less than 1% of the upper and lower data points. . . . . . . . . . . 9
3.1 Performance of best methods for PM25 compared with model
performance and persistence. . . . . . . . . . . . . . . . . . . . . 12
3.2 Performance of best methods for NO2 compared with model per-
formance and persistence. . . . . . . . . . . . . . . . . . . . . . . 12
3.3 Performance of best methods for O3 compared with model per-
formance and persistence. . . . . . . . . . . . . . . . . . . . . . . 13
D.1 Normalised average scores of KF corrected forecasts for each
model. Mean taken over all methods. . . . . . . . . . . . . . . . . 32
D.2 Normalised average corrected forecast scores for each method.
Mean taken over all models. . . . . . . . . . . . . . . . . . . . . . 33
D.3 Statistical measures of corrected model ensemble forecasts for
PM25 at TW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
D.4 Statistical measures of corrected model ensemble forecasts for
PM25 at CL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
D.5 Statistical measures of corrected model ensemble forecasts for
PM25 at CB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
D.6 Statistical measures of corrected model ensemble forecasts for
NO2 at TW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
D.7 Statistical measures of corrected model ensemble forecasts for
NO2 at CL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
D.8 Statistical measures of corrected model ensemble forecasts for
NO2 at CB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
D.9 Statistical measures of corrected model ensemble forecasts for O3
at TW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
vii

10. List of Tables
D.10 Statistical measures of corrected model ensemble forecasts for O3
at CL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
D.11 Statistical measures of corrected model ensemble forecasts for O3
at CB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
viii

11. List of Figures
3.1 Hourly sample plot of PM25 at TW for the three different bias
modelling procedures. . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 Hourly sample plot of PM25 at TW for the four variants of RD. . 14
3.3 False model spike corrected by D, LR and RD. . . . . . . . . . . . 15
3.4 Peaks of observed and predicted concentrations modelled by D,
LR and RD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.5 Extreme model bias changes modelled by D, LR and RD. . . . . . 17
C.1 Flow chart describing the Kalman filter predictor process, adapted
from Delle Monache et al.[2] . . . . . . . . . . . . . . . . . . . . . 29
ix

13. Chapter 1
Introduction
Background and Motivation Air Quality is not a new concern, but more
and more people live in highly populated areas and become more affected
by the locally and regionally produced pollution. Ever since London’s great
smog of 1952, the science of air quality modelling has steadily developed.
Today these models are integrated with advanced numerical weather pre-
diction (NWP) models. Model output statistics (MOS) have also been exten-
sively researched. Traditional MOS such as different regression techniques
are mature and are a common post processing procedure for many models.
New methods such as neural networks have recently been developed and
are growing more mature. Although the Kalman filter algorithm was devel-
oped half a century ago, it has only been increasingly applied in the field of
air quality forecasting during the last decade. Some further insights can be
found in the appended literature review.
The AQ model executed for the Hong Kong SAR at the Hong Kong Univer-
sity of Science and Technology is not post processed by any MOS. The AQ
forecasts are currently affected by systematic and random errors.
The KF has also been successfully used to post process AQM forecasts and
improve the accuracy. However, a study done by Delle Monacea et al. [2]
showed that the KF decreased the forecast performance for some stations.
This suggests that the performance is dependent on the characteristics of
the model-observations relationships. All reviewed studies used the KF to
model the simple difference of observations and predictions as the model
bias. As the same modelled pollutant might show different patterns at each
analysed station, a different way of modelling the bias might be optimal for
different stations.
1

14. 1. Introduction
Scope This study aims to develop and evaluate different ways to model
and minimise the AQM forecast bias using the Kalman filter algorithm.
Other types of bias modelling such as the natural logarithm of the ratio
of forecasts versus observations or the relative difference bias will be intro-
duced. Further variations of the basic KF algorithm such as a weighted
average or different ways of modelling the variances will be examined.
In order to ensure the significance of the results, a long time period of multi-
ple pollutants and stations will be processed. The results will we evaluated
by statistical measures significant for the performance of AQM forecasts.
Objectives
1. Create different procedures utilising the KF algorithm to post process
AQM forecasts.
2. Assess the performance of the corrected forecasts and display weak-
nesses and strengths.
3. Suggest a customised procedure for each pollutant at each station.
2

15. Chapter 2
Methods
2.1 Kalman Filter Predictor
The whole Kalman filter algorithm is recursively based on the last estimated
state and its variances. In predictor mode, the latest state estimate is used
as a prediction of the next state. Jolliffe and Stephenson provide a good
description of the Kalman filter predictor (KFP): “The KFP correction is an
automatic post-processing method that uses the recent past observations
and forecasts to estimate the model bias in the future forecast, where bias
here is defined as the difference of the central location of the forecasts and
the observations”[3, 2].
In this study other methods of modelling the bias are also introduced in the
procedure section 2.4.1.
The equations of the KFP can be derived from the general KF equations.
A short introduction to the Kalman filter can be found in section A of the
appendices. In this case only one state parameter xt is modelled and no ad-
ditional system parameter µt are used. We define the signal input zt as yt to
make clear that it represents the bias of the AQM and not the concentrations
of the pollutants themselves. To further simplify the system, the matrices A,
B and C are chosen to be identity.
Now the simplified equations for the estimated (true) model bias xt can be
written as
ˆxt = xt−1 + qt−1
where qt is a normal distributed random error with zero mean and variance
σ2
q . The observed model bias yt is expected to differ from the estimated bias
by another normal distributed random error et with zero mean and variance
σ2
e .
ˆyt = ˆxt + et = xt−1 + qt−1 + et
3

16. 2. Methods
The state covariance matrix, or in this case the expected mean square error
of the state ˆRt is estimated to be
ˆRt = Rt−1 + σ2
q
Next, the Kalman gain Kt can be computed as
Kt =
ˆRt
ˆRt + σ2
e
=
Rt−1 + σ2
q
Rt−1 + σ2
q + σ2
e
Finally, the state xt and its covariance matrix Rt are updated
xt = ˆxt + Kt(yt − ˆyt) = xt−1 + Kt(yt − xt−1)
Rt = (1 − Kt) ˆRt = (1 − Kt)(Rt−1 + σ2
q )
The state estimate xt is now assumed to be the bias of the AQM forecast for
the next time step. When predicting, we can only use present observations to
correct the future prediction as the future observations are not yet available.
This is why this use of the KF is usually named KFP.
After every iteration, the whole system is time-shifted t = t + 1. In this study
a time step of one day is applied because the AQM forecasts are generated
once a day. For each day 24 independent KFP are run for each forecast hour.
The details about the computation of the variances and initial values can be
found in appendix B.
2.2 Data
The data of three full years, 2012-2014 at three locations, Tsuen Wan (TW),
Central / Western (CL) and Causeway Bay (CB) was used. Three pollutants,
fine suspended particulates of size smaller than 2.5 µm (PM25), nitrogen
dioxide (NO2) and ozone (O3), are analysed at each station.
Tsuen Wan is an ambient measuring station often used to represent Hong
Kong as it is located in the heart of the SAR HK. Central / Western is also an
ambient station in an area where many people work and live. As it is locked
in by the surrounding skyscrapers, it follows different dynamics than the
ambient station TW. Causeway Bay is a roadside station, also situated in a
densely populated area. As the station is located right next to a main road,
the patterns of AQ parameters is quite different from TW and CL. At CB the
AQM forecasts differ at times quite significantly from the observations.
The Hong Kong air quality health index (AQHI) is defined by PM25, NO2,
O3 and two further pollutants, respirable suspended particulates smaller
4

17. 2.2. Data
than 10 µm (PM10) and sulphur dioxide (SO2). As PM25 and PM10 are
strongly correlated and the control of SO2 in Hong Kong has improved, this
study is limited to PM25, NO2 and O3. These are also the three parameters
defining the Canadian AQHI.
Each station and pollutant combination will be treated as a separate data set
and will referred to from now on as a (data) origin. The statistical measures
for the raw data will be provided in the Results chapter when contrasted
with the best Kalman filtering methods. To avoid negative concentrations
and division by zero, a minimal threshold of 1.0 µgm−3 respectively 1.0 ppb
was set for all observations and model forecasts.
2.2.1 Observations
The reported concentration for each full hour is computed by the average
concentration of the last 60 minutes at that station. Missing observations
were replaced with the last reported concentration. In total, all stations
had less than 4% of missing values with the exceptions of CB, where 12.0%
of PM25 and 5.4% of O3 measurements were missing. All data sets meet
the requirement of less than 15% missing data in order to be considered
consistent.
2.2.2 Model Predictions
Air Quality forecasts computed by the U.S. EPA’s Community Multi-scale
Air Quality Model (CMAQ) and by the open-source Comprehensive Air
Quality Model with Extensions (CAMx) are evaluated. Both models are well
documented and will not be further explained in this study. The hourly con-
centrations generated by CMAQ are instant concentrations at the respective
hour, while CAMx delivers the average concentration of the past 60 minutes.
The models deliver a forecast for several future days. Theoretically, the fore-
cast of the next day would be optimal to use. However, the models usually
perform bad during their spin-up time of around 12 hours and due to the
time it takes for the model to run, the first forecast hours are already past
until the results are available. Due to the spin-up issues and the temporal
delays of delivering the results, the past days model forecast for the second
day is generally used as a prediction for the next day. Therefore this study
uses the previous days model second day forecast and post processes this
with the observations of the present day.
The AQM forecast data was complete with the exception of the first 36 hours
of CAMx in 2012. These missing values were replaced with the respective
CMAQ forecasts.
5

18. 2. Methods
Additional Models More model data was simulated by applying a weighted
average of the two models, taking the mean of the two models and of the av-
eraged models. The model ensemble forecasts are marked by an ’M-E’. Fur-
ther model data sets were obtained by combining already corrected forecasts
into a new data set. Hereby the corrected forecasts of the two original Model
inputs were averaged. This was done for each bias modelling method with
the basic KFP, the KFP with modelled variance and the KFP with modelled
variance and interconnected hours. The KFP variations and the smooth-
ing are explained in section 2.4.2. These Kalman filtered ensemble forecast
model inputs are labelled as ’Kf-E’. Lastly three means of all variations and
all bias types was taken for not smoothed model input, smoothed before and
one smoothed before and after. These data sets are marked by ’Kf-E-mean’.
This procedure lead to 18 model data sets in total.
2.3 Verification Statistics
The following statistical measures are used to evaluate the performance of
the corrected forecasts and to compare them to the corresponding raw fore-
casts. Five measures were selected to account for a balanced estimate of how
well the predicted concentrations (Pi) match the observations (Oi) in general
and at peak times. Hereby the first 10 days of 2012 are not evaluated as this
period is considered as a spin-up phase for the KF. The details about the
normalisation of the statistical measures and their combination into scores
can be found in the appendix D.2.
Normalised Mean Square Error The normalised mean square error (NMSE)
measures the overall deviation of predictions towards observations.
NMSE =
1
n
n
∑
i=1
(Oi − Pi)2
¯O · ¯P
with n being the total amount of hourly data points, the mean observed
concentration and mean predicted concentration as ¯O = 1
n ∑n
i=1 Oi and ¯P =
1
n ∑n
i=1 Pi
The optimal value of the NMSE lies at 0. It should be noted that the NMSE
is sensitive to extreme outliers as the error is squared. The NMSE was cho-
sen over the RMSE to make comparisons between the different data sets
possible.
Unpaired Peak Prediction Accuracy The unpaired peak prediction accu-
racy (UPPA) is a simple but effective measure of the models accuracy to
predict a days peak concentration. It is defined as the fraction by which the
6

19. 2.3. Verification Statistics
predicted peak concentration differs from the observed peak.
UPPA =
1
days
days
∑
d=1
|max(Od,i) − max(Pd,i)|
max(Od,i)
where the absolute difference of the maximum hourly observed and pre-
dicted concentration |max(Od,i) − max(Pd,i)| is divided by the maximum ob-
served concentration for each day. The UPPA should be as close as possible
to zero.
Fractional Bias A measure of the systematic bias is presented with the
fractional bias (FB). It displays a models tendency to over or under estimate
the concentration.
FB =
∑n
i=1 (Oi − Pi)
0.5 · ∑n
i=1 (Oi + Pi)
A value close to zero indicates the absence of systematic errors, with FB ∈
[−2, 2].
Fraction of predictions within a factor of two The FAC2 is defined as the
fraction of predictions within a factor of two of the observations.
FAC2 =
| Pi
Oi
∈ [1
2 , 2]|
n
The range of this measure is limited to FAC2 ∈ [0, 1] with 1 being optimal.
Balance of correctly predicted episodes versus false and missed episodes
The balance of correct alarms versus false and missed alarms (CFM) is a
statistical measure defined by the author of this study. It displays the mod-
els ability to correctly predict high pollution episodes and penalises falsely
predicted and missed high pollution episodes. A day was considered an
episode if it any of its three hour average concentrations was greater than a
given threshold. The CFM is defined as
CFM = 1 +
C − F − M
C + F + M
where C is the number of correctly predicted episodes, F is the amount of
falsely predicted episodes, M is the number of missed episodes (not pre-
dicted episodes) with CFM ∈ [0, 2]. A CFM value of zero signifies that no
episode was correctly predicted. A CFM value of one means that either
7

20. 2. Methods
the amount of correct predictions was identical to the total amount of false
and missed events combined or that there were no predicted and no actual
events.
The CFM is a generally robust measure. However it is sensitive to the
amount of total actual episodes. If there are no episodes, it becomes only a
measure of false predictions and is limited to maximum 1.
The thresholds were chosen so that all origins had at least 1% of days with
an episode. The thresholds for each pollutant were set identical for all three
stations. NO2 is an extreme case as, in order to reach 1% of days with an
episode at TW and CL, the threshold had to be set so high that the days
with an episode reaches 31% at CB .
For fine suspended particulates the threshold was set to 75 µgm−3, identical
to the daily average value for PM25 suggested by the Hong Kong air quality
objectives in 2014 (HKAQO). The threshold for nitrogen dioxide was set
to 100 ppb, half of the one hour average value for NO2 in the HKAQO.
For Ozone the threshold was chosen to be 50 ppb. The HKAQO suggest a
concentration of 160 ppb as the 8 hour average for O3. The episode criterion
for ozone was set lower in order to observe a sufficient amount of episodes
at CB.
2.4 Procedure
2.4.1 Modeled Bias
As the KFP approximates the true state of a linear variable under noisy
measurements, the variable modelled by the Kalman filter should be linear.
The modelled variable, the model forecast bias, is assumed to be linear. It
is defined to be optimal at zero bias with any value above zero meaning
an over prediction while a value below zero signifies an under prediction.
Three different bias modelling procedures were chosen, the difference (D),
the natural logarithm of the ratio (LR) and the relative difference (RD). These
are explained in the next subsections.
Difference In the case of modelling the direct bias (D), the difference of the
model prediction towards the observation is defined as yt:
yt = Pt − Ot
The observed bias yt is then used to compute a new predicted (true) bias xt
as described in the section 2.1. Finally the predicted bias is combined with
the new model prediction into a Kalman filter corrected prediction PKt+1 :
PKt+1 = Pt+1 − xt
8

21. 2.4. Procedure
Table 2.1: Ratio limits for LR and RD bias. The values are the maximum
ratio of Pi/Oi and Oi/Pi for each origin respectively, restricting less than 1%
of the upper and lower data points.
Ratio PM25 NO2 O3
TW CL CB TW CL CB TW CL CB
max{Pi/Oi} 10 10 10 10 10 10 20 35 45
max{Oi/Pi} 10 10 20 10 10 35 30 25 10
The corrected prediction is controlled to be at least 0.0 as negative concen-
trations do not occur in nature. If PKt+1 is smaller than zero, the absolute
value of PKt+1 is subtracted fromxt and PKt+1 is then set to zero. Like this
the next corrected forecast will be above zero if the new Pt+1 not lower than
the respective Pt. The corrected forecast needs to be controlled less than
1.5% overall for PM25 and NO2. The maximum percentages of controlled
forecasts for ozone concentrations are 7.7% at TW, 6,8% at CL and 15.9% at
CB.
Natural Logarithm of the Ratio Another method of defining the bias yt is
as the natural logarithm of the relationship of model forecasts versus obser-
vations (LR).
yt = log(
Pt
Ot
)
In the case of modelling the relationship we correct:
PKt+1 = Pt+1 · e−xt
Hereby the maximum and the minimum value of yt is controlled in order to
avoid infinite values or unreasonably high values that temporarily bias the
the KF towards an outlier. The limits were set so that less than 1% of the
data points above or below are restricted. The maximum ratios for Pi/Oi
and Oi/Pi were set to 10 and then increased until the requirement was met.
The ratio limits in table 2.1 are used to control yt to stay within the range
[−log(max{Oi/Pi}), log(max{Pi/Oi}].
Relative Difference The last method of defining the bias is the relative
difference (RD). It aims to combine the first two approaches. Here, the bias
yt is defined as
yt =
Pt − Ot
Pt + Ot
and the model forecast is again bias-corrected
PKt+1 = Pt+1 − xt · (Pt+1 + Ot)
9

22. 2. Methods
The optimal RD bias is again zero and falls naturally in the range [−1, 1].
The same ratio limits as explained in the section above and displayed in
table 2.1 are applied so that the range becomes [−max{Oi/Pi}−1
max{Oi/Pi}+1
, max{Pi/Oi}−1
max{Pi/Oi}+1
].
2.4.2 Variations
Two optional variations of the KFP were defined. These were combined into
four total variations: basic KFP, interconnected hours KFP (KF-i), Kalman
filtered variance KFP (KF-v) and both interconnected and Kalman filtered
variance combined (KF-i-v).
Interconnected Hours A simple weighted average is applied to the KFP
input parameters on each iteration to interconnect the separated hours. The
following weighted average was applied:
0.60 · ph + 0.15 · (ph−1 + ph+1) + 0.05 · (ph−2 + ph+2)
where ph is the inputed parameter for hour h. For the first and last hours of
each day the central hour was weighted more instead of using another days
data.
Kalman Filtered Variance In this variation, a Kalman filter is first used to
estimate the true state of the variance σ2
e, t before it is used within the basic
KFP to estimate the state of the forecast bias. The explanation of the detailed
equations for this KF can be found in the appendix B.
Smoothing Optionally, a weighted average of the model data was taken
before it was processed by a KFP. Another optional weighted average was
applied on the corrected forecasts. These data sets are marked by a small
’s’ to symbolise the smoothing. If the weighted average was applied on
unprocessed model data, the hours were weighted with the same coefficients
as the interconnected hours variant.
0.60 · Ph + 0.15 · (Ph−1 + Ph+1) + 0.05 · (Ph−2 + Ph+2)
Additionally a weighted average of the two past days was done:
0.75 · Pd + 0.20 · Pd−1 + 0.05 · Pd−2
The smoothing of the corrected forecasts was done by the application of the
following equations:
0.50 · PKh + 0.20 · (PKh−1 + PKh+1) + 0.05 · (PKh−2 + PKh+2)
0.850 · PKd + 0.125 · PKd−1 + 0.025 · PKd−2
10

23. Chapter 3
Results
The full statistical analysis can not be displayed in this report due to its ex-
tent ( 3 stations ·3 parameters ·18 models · (2 ·12 +3) methods ·5 statistics =
21870 statistical measures). The full analysis of the results of the corrected
model ensemble forecasts can be found in the appendix D.3. It is to be
mentioned that this is only one of the 18 different models. The normalised
average scores of KF corrected forecasts for each model and each method,
averaged over all methods or models respectively, are also included in the
appendix D.2. The most relevant results are the statistical measures for the
best methods found in the next section.
3.1 Performance of Best Methods
The tables 3.1, 3.2, and3.3 show the NMSE, UPPA, FB, FAC2 and CFM of
the selected methods for each pollutant at each station. These are compared
with the performance of the two raw forecasts, their average model ensem-
ble (M-E) and with persistence, which is the time-lagging of observed con-
centrations by one day without any further processing. If the raw model
forecasts were selected as method, the method is labelled ’M’. A model label
of ’Kf-E’ stands for a Kalman filtered ensemble, where the model data is the
average of the KF method applied on both raw forecasts. If the average of all
methods was taken, the model name contains ’Kf-E-mean’. Smoothed data
sets are marked ’s’ and the optional KF variations are ’i’ for interconnected
hours and ’v’ for filtered variance.
11

24. 3. Results
Table 3.1: Performance of best methods for PM25 compared with model
performance and persistence.
Origin Model Method NMSE UPPA FB FAC2 CFM
TW Kf-E-s-mean-s M-s 0.213 0.282 0.002 0.901 0.584
CMAQ - 0.402 0.409 -0.178 0.720 0.266
CAMX - 0.449 0.406 0.090 0.749 0.395
M-E - 0.315 0.351 -0.036 0.801 0.352
Persistence - 0.337 0.348 0.000 0.813 0.563
CL Kf-E-s-RD-v M-s 0.236 0.292 -0.010 0.889 0.662
CMAQ - 0.406 0.368 -0.358 0.656 0.394
CAMX - 0.399 0.365 0.049 0.791 0.516
M-E - 0.316 0.325 -0.135 0.796 0.449
Persistence - 0.389 0.386 0.000 0.791 0.593
CB Kf-E-s-D-v M 0.138 0.221 -0.002 0.951 0.796
CMAQ - 0.460 0.502 -0.653 0.432 0.304
CAMX - 0.335 0.416 -0.242 0.661 0.533
M-E - 0.349 0.438 -0.430 0.594 0.382
Persistence - 0.199 0.244 0.000 0.910 0.664
Table 3.2: Performance of best methods for NO2 compared with model per-
formance and persistence.
Origin Model Method NMSE UPPA FB FAC2 CFM
TW M-E-s RD-Kf-v 0.159 0.214 0.005 0.937 0.261
CMAQ - 0.324 0.274 -0.243 0.682 0.105
CAMX - 0.268 0.289 -0.038 0.817 0.000
M-E - 0.268 0.260 -0.135 0.792 0.125
Persistence - 0.183 0.255 -0.001 0.925 0.154
CL M-CAMX RD-Kf-i 0.214 0.251 -0.006 0.891 0.833
CMAQ - 0.396 0.314 -0.443 0.576 0.000
CAMX - 0.281 0.263 -0.203 0.806 0.000
M-E - 0.318 0.278 -0.317 0.727 0.000
Persistence - 0.304 0.303 -0.001 0.845 0.222
CB M-CAMX RD-Kf 0.140 0.209 0.002 0.949 0.872
CMAQ - 0.757 0.669 -1.208 0.087 0.000
CAMX - 0.595 0.605 -0.948 0.192 0.000
M-E - 0.670 0.641 -1.073 0.122 0.000
Persistence - 0.181 0.243 -0.001 0.919 0.780
12

25. 3.2. Sample Time Series
Table 3.3: Performance of best methods for O3 compared with model perfor-
mance and persistence.
Origin Model Method NMSE UPPA FB FAC2 CFM
TW Kf-E-s-D-i-v-s M 0.412 0.366 0.015 0.739 0.782
CMAQ - 0.972 0.741 0.400 0.552 0.850
CAMX - 0.629 0.448 -0.104 0.565 0.305
M-E 0.619 0.524 0.181 0.619 0.856
Persistence - 0.723 0.487 0.000 0.620 0.734
CL M-E-s D-Kf-i-v 0.367 0.364 0.002 0.763 0.906
CMAQ - 0.830 0.711 0.427 0.630 0.934
CAMX - 0.562 0.507 0.002 0.683 0.476
M-E - 0.564 0.560 0.240 0.700 0.958
Persistence - 0.657 0.450 0.000 0.627 0.834
CB Kf-E-s-LR-v D-Kf-i 0.759 0.587 -0.021 0.626 0.917
CMAQ - 10.988 3.662 1.190 0.133 0.087
CAMX - 3.910 2.359 0.848 0.296 0.203
M-E - 6.564 2.912 1.049 0.193 0.160
Persistence - 1.001 0.688 0.000 0.572 0.615
3.2 Sample Time Series
As TW is the reference station for the Hong Kong SAR, it is used for the sam-
ple time series. Here, the pollutant PM25 is selected because it is the only
data set best approximated by the mean of all KF methods and therefore
should present a balanced exemplary sample. The figures 3.1 and 3.2 are
plots of consecutive predicted and observed concentrations for each hour.
The first figure contrasts the three bias modelling methods and the second
figure compares the four variants for each bias type, here for RD.
13

26. 3. Results
Figure 3.1: Hourly sample plot of PM25 at TW for the three different bias
modelling procedures.
Figure 3.2: Hourly sample plot of PM25 at TW for the four variants of RD.
14

27. 3.2. Sample Time Series
3.2.1 Special Time Series
These figures show how the concentration of a certain hour evolves over a
series of consecutive days. Below each figure the observed bias yt and the
modelled bias xt are plotted for D-, LR- and RD-KF.
Figure 3.3: False model spike corrected by D, LR and RD.
15

28. 3. Results
Figure 3.4: Peaks of observed and predicted concentrations modelled by D,
LR and RD.
16

29. 3.3. Live Display of Actual Implementation
Figure 3.5: Extreme model bias changes modelled by D, LR and RD.
3.3 Live Display of Actual Implementation
Thanks to this study, a basic version of the D-KF was already implemented
at HKUST to automatically correct the AQ model forecasts. The corrected
PM25 forecasts of the next day and the recently passed days for TW, CL and
CB can be observed live at
http://envf.ust.hk/dataview/kfp/current/.
It is planned that the optimised methods will be implemented according to
this study.
17

31. Chapter 4
Discussion and Conclusions
The choice of defining the matrices A, B and C to be identity is a common
practice in most preceding studies. However, in a study by Cheung and
Steenburgh the matrices were all chosen to be identity with the exception of
one station where A was set to 0.75 [1]. It is possible that a value alternative
to identity would have performed better for certain stations in this study,
too. However, a sensitivity analysis of the matrix parameters is out of the
scope of this study.
4.1 Discussion on Best Methods
The overall performance of the best methods for each origin is superior to
the raw model forecasts for NMSE, UPPA, FB, FAC2 and CFM. The only
exception is the CFM for O3 at TW and CL, here the CMAQ model forecast
offers a greater CFM value. In contrast, the prediction of NO2 at all three
stations was significantly improved for CFM.
Remarkable is that even though the raw model forecasts for O3 at CB per-
form several times worse than at the other two stations, the corrected fore-
cast reaches statistical scores comparable to the other stations. This is the
only station where it is recommendable to run a KF twice over the data.
First a LR-KF-v is used to roughly correct the raw model forecasts and then
a D-KF-i is additionally applied to post process the predictions a second
time.
An interaction between NO2 and O3 is expected to be the reason for the poor
performance of the AQM forecast at CB where NO is emitted by the roadside
traffic. The locally increased NO concentration tends to react with ozone,
reducing O3 and increasing NO2 concentrations. The chemical reaction can
be simplified to
NO + O3 → NO2 + O2
19

32. 4. Discussion and Conclusions
and with help of solar radiation the reaction can also occur towards the other
side
NO2 + O2 + hv → NO + O3
Local interactions are hard to model by the AQM, as the spatial resolution
of the model is 1 km.
Further it can be observed that the performance of the KF is not only depen-
dent on the model performance but it is also correlated to the performance
of persistence. NO2 is a good example as the both models perform clearly
worse ad CB while persistence performs slightly better here than at the other
two stations. In the end, the KF corrected forecasts at CB are slightly bet-
ter than at the two other stations. A similar phenomenon can be observed
observed again at CB compared to the two other PM25 stations.
4.2 Discussion on Time Series
The time series are too short to draw reliable conclusions about the over-
all performance about the respective KF. The selected statistics measure the
general performance more reliably as a large amount of data was evaluated.
Nevertheless, an analysis of the time series is important to understand the
strengths and weaknesses of the KF. From figure 3.1 several insights can be
won. First, it can be seen that the three bias modelling techniques difer in
their peak correction performance. The LR-KF correctly decreases the false
model peak at hour 20 most compared to the D-KF which follows the model
predictions more closely. The RD-KF lies somewhere in between. At hour
45 the bias changes from a strong over prediction to a strong under pre-
diction. Now the KFP all perform worse than the raw forecast. The D-KF
even reaches its lower limit for several consecutive hours because its bias
correction is not relative to the magnitude of the forecast concentration. All
the methods converge in the following days again towards the observations.
Therefore, the second and third insights are that a sudden change of bias
leads to temporarily worse performance of the KF, but it then adapts to the
new situation and changes its bias correction in response to the new ob-
served bias. Whereas the rate of adaptation depends linearly on the degree
of discrepancy between the observed and modelled bias.
Figure 3.2 suggests that the differences between the optional variations are
rather small. Generally it can be seen that the variant filtering the variance
is more volatile than the interconnected variants.
Special Time Series As a separate KF is run for each hour, the special
time series plotting one hour over few days are most helpful to understand
how the KF works and to compare the different bias modelling techniques.
The falsely over predicted peak concentration seen in figure 3.3 is correctly
20

33. 4.3. Conclusions
downwards corrected by all KF methods, while the magnitude of correction
differs. D-KF follows the model closer while LR-KF predicts the lowest peak.
The estimated D bias state adapts slowly to the fast occurring peak at day
104 while the LR and RD bias are both estimated almost perfectly at day 104.
Nevertheless, the RD-KF corrected forecast differs from the LR correction
because its correction magnitude depends linearly on the last observation
whereas the LR correction only depends on the new model prediction and
the estimated LR bias (see bias definitions in the methods section 2.4.1).
Altering the weight of the past observation on the correction magnitude
could improve the performance. For example as: yt = Pt−Ot
Pt+0.5·Ot
and PKt =
Pt+1 + xt · (Pt+1 + 0.5 · Ot)
All methods fail to predict the first observed peak in figure 3.4 due to miss-
ing indication of a rise by the AQ model. The Kalman filter is not able to
create additional information, but can only un-bias the forecast. As a cause
of the first not predicted peak, the second peak was over predicted, partially
also because the observed concentration did not rise proportionally to the
predicted concentration. Finally, the last peak is best predicted by the LR-KF,
whereas it also most over predicted the second peak.
Figure 3.5 shows an extremely volatile observed bias. None of the Kalman
filtering methods are able to successfully adapt to the fast changing bias.
Therefore all three methods estimate the true bias to lie around zero. The
extreme bias ratio at day 166 is one of the cases where the ratio is limited
to the maximum ratio, defined in table 2.1, in order to prevent the KF from
being overly influenced by this outlier in the next time steps.
4.3 Conclusions
Each origin is best corrected by a different KF method. The performance of a
specific KF implementation depends on the characteristics of the underlying
model bias particular to each station. An exception is the prediction of NO2
where the RD bias methods are most skilful at all stations. Generally, the
LR-KF is best at correcting extreme differences and peaks, whereas is also
tends to over predict peaks. The D-KF and RD-KF perform better in cases
with a moderate AQ model bias.
For most stations a KF corrected ensemble of both raw model forecasts per-
formed best. PM25 prediction at TW is even optimally corrected by the
average of all introduced methods and variants. Post processing the fore-
casts twice is only recommendable for data sets with an extreme model bias
such as for O3 at CB. The type of correction done to the raw forecasts can be
described in a simplified manner as a time-varying additive correction for
the D-KF and as a time-varying multiplicative correction for the LR-KF. The
RD-KF is a combination of both.
21

34. 4. Discussion and Conclusions
The comparison with persistence shows that the raw AQ model forecasts
perform worse than a simple time-lagging of observations. The KF corrected
forecasts in contrary are more skilful. Therefore the AQ model provides
relevant information that can be used to generate accurate forecasts, but a
skilful MOS such as the KF is necessary to remove the bias.
The Kalman filters ability to adapt to a changing bias without any knowl-
edge of the underlying model makes it an attractive post processing method
to use with AQ data which does not always follow expected physical pat-
terns due to complex interaction. Additionally its recursive implementation
makes day to day operation simple and does not require a lot of computa-
tion power.
4.4 Recommendations
In this study a fixed ratio of variances r = 0.1 was applied. A sensitivity
analysis of the ratio r and the definition of an optimal value for each sta-
tion and pollutant is recommended. Values different from identity for the
matrices A, B and C could be tested. Further sensitivity analysis’ could be
done on the coefficients used in the different weighted averages and on the
amount of past days to compute the variance.
As the combination of the raw forecasts generally showed an improved per-
formance, an extended ensemble forecast consisting of multiple AQ model
runs with different scenarios combined into one ensemble should be consid-
ered.
As the a logarithmic ratio bias modelling technique also showed good per-
formance for certain data sets, the extended Kalman filter (EKF) for non
linear systems could promise an increased performance.
Different approaches of combining a KF with a traditional MOS such as
a multiple regression (MR) can be explored. The possibility of combining
both in one overlapping step, such as a Kalman filtering of the regression
parameters is to be explored. A spatial correlation analysis could overcome
part of the systematic bias due to the limited spatial resolution. It is to be
expected that a MOS combination with the KF applied last performs best as
it is highly adaptive and can correct even temporary systematic bias.
Finally, an intelligent AQMOS system could utilise the strengths of the differ-
ent KF methods presented in this study. This could be done by running a set
of KF methods for each station which are constantly analysed for their per-
formance. Then the forecast corrected by the recently most skilful method
would be selected. Or different situations types such as high and low pollu-
tion, volatile and steady bias could be defined and then the KF method best
performing in the current situation type selected for the forecast correction.
22

35. Appendix A
A short introduction to the Kalman
filter
The Kalman Filter is a state of the art estimator of the true state of a variable
affected by noisy measurements. It is commonly used to process signals and
can be found in a large variety of applications. A popular example would
be a GPS device estimating its true position from signals affected by random
errors.
The true (unknown) state is modelled by the KF and updated each time a
new signal is received. The magnitude of each correction depends on how
large the recent random error of the signal is.
First, the (true) state and the expected signal are predicted by
ˆxt = Axt−1 + Bµt + ηx
ˆzt = C ˆxt + ηz
where the predicted state ˆxt is dependent on the last state xt−1 and a system
variable µt and affected by a normal distributed random error ηx, with zero
mean and variance Ex. The sensor prediction differs from the state by a
normal distributed random error ηz, with zero mean and variance Ez. A, B
and C are matrices describing the linear relationship between the variables.
The state covariance matrix ˆRt, describing the expected square errors of the
modelled system parameters, is estimated to be
ˆRt = ARt−1AT
+ Ex
Next, the Kalman gain Kt can be computed as
Kt =
ˆRtCT
C ˆRtCT + Ez
23

36. A. A short introduction to the Kalman filter
Now the state xt and its covariance matrix Rt are updated
xt = ˆxt + Kt(zt − ˆzt)
Rt = (I − KtC) ˆRt
Where zt − ˆzt = zt − C ˆxt is the difference of the sensor prediction to the
actually received signal. I is the identity matrix. The Kalman gain Kt ∈ [0, 1]
is a measure for the trust the KF puts into the new signal and how much
the state is corrected towards the signal. The gain depends on the changing
variances and is different for each time step.
This recursive algorithm is then repeated for each time step. In the case of
estimating the true state of the AQ forecast bias, the measurement for the
future day is not yet available. Therefore the Kalman Filter has to be used
in predictor mode, sometimes also referred to Kalman Filter Predictor (KFP)
which will be discussed in the methods chapter in section 2.1.
24

37. Appendix B
Variance Computation and Initial
Values
Preceding studies stated that the KFP performance is independent of the
actual value of the variances but is sensitive to the ratio of the variances. A
fixed variance was used in those studies [2, 1]. However, in this study it was
chosen to use a time-varying variance as some early trials showed greater
performance for this approach. The ratio of the variances was also fixed to
r =
σ2
q
σ2
e
= 0.1
Strongly simplified, the ratio r can be thought of as a minimal value of the
Kalman gain. The variance σ2
q will be simply computed as σ2
q = r · σ2
e . the
ratio of the variances used in the KFP modelling the variance of et can be
expressed as
σ2
σ2
q
σ2
σ2
e
= r2
The variance σ2
σ2
q
is therefore be expressed as r2 · σ2
σ2
e
The basic KF’s time-varying variance σ2
e is computed from the variance of the
recent errors et = yt − xt. As the Kalman filter remembers the systematics
of the past 4 to 5 days, it was chosen to calculate the variance from the
past 5 data points. During the first five spin-up days the variance is simply
computed from the yet available data points.
The variance σ2
σ2
e
used in the Kalman filter estimating the variance σ2
e, t is
computed by calculating the variance of the Kalman filtered variance σ2
e, t of
the past five days.
It should be noted that the variance calculated for the basic KFP and the
Kalman filtered variance are not the same. The variance of the first con-
25

38. B. Variance Computation and Initial Values
tains both systematic and random components whereas the second should
exclude the random error components.
Kalman Filtered Variance Here the equations applied in the KF variant
which additionally models the variance are described. The Kalman gain
used estimate the true state of the variance σ2
e, t is computed as
Kt =
Rσ2
e , t−1 + σ2
σ2
q
Rσ2
e , t−1 + σ2
σ2
e
+ σ2
σ2
q
then the state of the variance σ2
e, t and its expected mean square error Rσ2
e , t
are updated
σ2
e, t = σ2
e, t−1 + Kt(ˆσ2
e, t − σ2
e, t−1)
Rσ2
e , t = (1 − Kt)(Rσ2
e , t−1 + σ2
σ2
q
)
where the estimated signal ˆσ2
e, t of the variance σ2
e, t is approximated using
the relationship of the variances. From the time series
zt = yt − yt−1 = xt + et − xt−1 − et−1 = qt−1 + et − et−1
follows that σ2
z = σ2
q + 2σ2
e = (2 + r)σ2
e and σ2
e = σ2
z
2+r Therefore we can
approximate
ˆσ2
e, t =
(yt − yt−1)2
2 + r
Initial Values The initial values for the estimated state xt and σ2
e, t of the
basic KFP and the KFP modelling the variance were simply set to zero. The
initial values of the variances σ2
e , σ2
σ2
e
and the expected mean square errors Rt,
Rσ2
e , 2 however needed to be set to a positive value for the KFP to converge
to an optimal solution. The initial values of σ2
e and Rt were chosen to be the
square of the first observed bias y2
1. Analogously the initial values of σ2
σ2
e
and
Rσ2
e , 2 were set to be the square of the initial variance σ4
e, init = y4
1. Hereby we
assume y1 not to be zero.
26

39. Appendix C
Literature Review
AQMOS: Air Quality Model Output Statistics from CMAQ model forecasts
In this publication of 2010, Miller et al. [7], aims to demonstrate how the
bias of air quality model output can be adjusted to generate more accurate
predictions. The demonstrated benefit of the AQMOS is the reduction of the
systematic model error (bias) for site specific predictions using observation
data.
A daily regression is computed between recent historic Air Quality Model
(AQM) predictions and observations. For each city, pollutant and model spe-
cific regression equations are employed. Further a division between normal
and high pollution days is made. A clear systematic bias can be seen on
a scatter plot of observed versus model data. Whereas high pollution days
data shows a different bias from low pollution days data. Therefore two re-
gression equations are created, one for normal days using all data and one
for high pollution days using only data above the 85th percentile of model
output.
As an alternative to regression analysis, two simple bias-adjustment strate-
gies, mean-bias and multiplicative ratio are named. As an advanced strategy,
the Kalman Filter Scheme by Delle Monache et al. 2006 [2] is pointed out.
This scheme is reviewed later.
Miller et al. measure the performance by comparing the average difference
of the raw model output and the bias adjusted data towards the observed
measurements. A particular focus is set on days that are critical air quality
days. The prediction of these improved in most cities through AQMOS.
A review on the forecasting of wind speed and generated power Wind
speeds posses in most sites a high temporal fluctuation. As the generated
power of wind turbines is directly proportional to the prevailing wind speed,
the prediction of wind speeds is crucial. If the generated power can be
27

40. C. Literature Review
predicted more accurately, the electricity generation can be optimized for
the system load without generating excess electricity.
Lei et al. give in this paper of 2009 [5] a brief survey over the most used and
some new methods to forecast wind speed and power. For long term pre-
diction physical methods perform better, whereas for short-term prediction
statistical methods are preferable. Often both methods are utilized simulta-
neously. Further, the spatial correlation model is also viewed to be relevant.
Artificial intelligence and other new methods have gained growing attention
for geophysical prediction.
As model input the physical models utilize physical and meteorological in-
formation. Statistical and spatial correlation models need NWP output and
historic data as input, whereas the later also needs the data from neigh-
bouring sites. A Spatial Correlation Predictor has been developed by Alexi-
adis, 1998 [6] to benefit from the spatial relationship of different sites wind
speeds.
To improve the predictions of physical models, accurate Digital Elevation
Models (DEMs) can be applied to the Numerical Weather Prediction (NWP)
and Model Output Statistics (MOS) are commonly used to post-process the
model output data.
Conventional statistical models include Autoregressive (AR), Moving Aver-
age (MA), and Autoregressive (integrated) Moving Average (AR(i)MA). The
equation C.1 shows the ARMA model. Note that for m = 0 the model
becomes the AR model.
xt =
n
∑
i=1
ϕixt−i + αt −
m
∑
j=1
θjαt−j (C.1)
ϕ AR parameter
θ MA parameter
α Normal distributed white noise
Other mentioned procedures cover ANN (Artificial Neural Network), MLP
(Multiple Layer Neural Perception Neural Network), LF-DFNN (Locally
Feedback Dynamic Fuzzy Neural Network) and SVM (Support Vector Ma-
chine).
Statistical post processing of model output from the air quality model
LOTOS-EUROS Annemiek Pinjappel uses R in her work of 2011 [8] to im-
plement MOS. The performance is measured by R2, RMSE as well as missing,
false and well modelled alarms. She found that for PM10 the LOTOS-EUROS
28

41. model output improved substantially. However, the division of the data into
summer and winter time did not lead to a significant improvement.
Combining deterministic and statistical approaches for PM10 forecasting
in Europe Knovalov et al. found in their publication of 2009 [4] that the
improvement is better the more variables are in the regression. The most
important meteorological parameter in summer is the temperature and in
winter the boundary layer height. The cross-validation method is used for
the performance measurement. The result of the study is that MOS lead to
a significant improvement not only for a given site but also for territories of
similar environment far away from the considered site.
Ozone ensemble forecasts: 2. A Kalman filter predictor bias correction In
2006, Luca Delle Monache et al. test the ability of the Kalman Filter in pre-
dictor mode to reduce systematic errors. Ozone forecasts are postprocessed,
using the bias of past forecasts to bias-adjust the future predictions. The
study finds the adjusted forecasts to become better considering gross error,
RMSE, correlation and to predict peaks far more accurate.
Figure C.1: Flow chart describing the Kalman filter predictor process,
adapted from Delle Monache et al.[2]
In the previous part of the study, a Ozone Ensemble Forecast System (OEFS)
was introduced. 12 forecasts were computed with different meteorological
inputs and emission scenarios. The mean of the ensemble performed better
than the individual forecasts. All forecasts exhibit a systematic error - a
typical problem in CTMs (Chemical Transport Models) [9]. In this part of
the study, the KF is applied on each forecast to minimize its systematic bias.
The KF is suited for this task. Due to its recursive and adaptive methodology
it can also include the temporal variation of the forecast error.
Figure C.1 shows how the KF is used to predict and correct. The bias is
29

42. C. Literature Review
constantly being re-estimated. The bias ( ˆxt t−∆t), estimated at the previous
timestep, is rectified by a fraction (β) of its discrepancy towards the observed
forecast error (yt) and then used to predict the next timeseteps bias ( ˆxt+∆t t).
30

43. Appendix D
Statistical Meaures
D.1 Normalisation of Statistics and Selection of Best
Methods
In order to be able to compare the performance of the different methods, the
statistical measures were normalised and combined into a single score for
each method and model combination at each origin. The score was designed
to represent the best method at a value of 1 and the worst method would
receive a score of 0. The normalised value of NMSE, UPPA and FB was
computed as
˙Si = 1 −
Si − min(Si)
max(Si) − min(Si)
and for FAC2 and CFM as
˙Si =
Si − min(Si)
max(Si) − min(Si)
where ˙Si is the normalised statistical measure Si. The Si of the FB was taken
as its absolute value |Si| because any positive or negative deviation from
zero represents a bias.
Then the average of the five normalised measures ˙Si was calculated for each
method and model combination and compared within the same dataset of
an origin. The method and model combination with the highest score was
then selected to be optimal for the respective origin. This procedure was
repeated to analyse only data points over or below a certain percentile to
gain further insight over the data. Due to the limited extent of the study,
these additional results will not be discussed.
31

44. D. Statistical Meaures
D.2 Normalised Mean Scores of All Methods and Mod-
els
Model PM25TW PM25CL PM25CB NO2TW NO2CL NO2CB O3TW O3CL O3CB
M-CMAQ 0.0 21.8 31.4 15.0 19.8 10.7 65.4 58.1 0.0
M-CAMX 29.6 39.9 62.9 68.5 81.9 65.6 66.1 62.0 42.6
M-CMAQ-s 60.6 72.8 59.8 72.6 65.4 40.6 84.3 61.5 8.3
M-CAMX-s 71.1 73.6 80.9 89.8 86.6 69.6 91.9 65.9 51.7
M-E 75.3 79.5 68.4 57.3 68.1 54.6 87.8 89.7 17.5
M-E-s 100.0 100.0 82.6 94.1 85.8 63.9 100.0 81.3 22.5
Kf-E-D 45.3 54.7 93.1 58.8 75.3 91.8 85.3 89.6 29.3
Kf-E-s-D-v 83.6 80.9 100.0 90.7 86.3 90.4 97.9 100.0 60.6
Kf-E-s-D-i-v-s 94.5 87.2 99.3 100.0 78.9 84.0 99.3 94.0 64.6
Kf-E-LR 13.9 0.0 0.0 0.0 0.0 0.0 0.0 0.0 87.1
Kf-E-s-LR-v 41.1 47.9 55.6 62.9 92.3 85.6 38.7 1.5 100.0
Kf-E-s-LR-i-v-s 93.0 85.8 66.7 81.1 89.7 78.7 89.1 52.3 98.2
Kf-E-RD 42.1 46.0 78.7 54.9 66.7 90.9 76.4 57.9 71.9
Kf-E-s-RD-v 80.0 72.3 92.0 91.2 86.8 95.9 93.3 60.7 82.4
Kf-E-s-RD-i-v-s 95.3 83.5 98.4 97.2 95.2 100.0 94.5 59.1 81.7
Kf-E-mean 46.8 51.2 70.4 49.9 63.1 86.1 75.7 79.0 75.6
Kf-E-s-mean 87.8 80.9 91.2 84.8 100.0 97.8 95.8 79.2 83.1
Kf-E-s-mean-s 98.0 86.3 94.9 96.7 94.8 98.4 95.1 71.2 86.7
Table D.1: Normalised average scores of KF corrected forecasts for each
model. Mean taken over all methods.
32

45. D.3. Statistical Measures for Corrected Model Ensemble Forecasts
Method PM25TW PM25CL PM25CB NO2TW NO2CL NO2CB O3TW O3CL O3CB
Persistence 0.0 0.0 86.9 99.8 0.0 95.4 38.4 0.0 100.0
M 34.3 76.5 0.0 0.0 9.9 1.6 21.7 29.2 1.7
M-s 65.7 83.5 2.4 21.6 11.9 0.0 36.1 23.8 0.0
D-Kf 87.6 93.3 96.9 89.9 81.7 96.2 85.1 87.4 85.0
D-Kf-s 87.6 93.3 96.9 89.9 81.7 96.2 85.1 87.4 85.0
D-Kf-i 96.0 98.2 99.8 96.5 88.9 95.6 93.9 94.9 80.2
D-Kf-i-s 96.0 98.2 99.8 96.5 88.9 95.6 93.9 94.9 80.2
D-Kf-v 89.2 93.9 98.3 94.6 84.3 97.2 89.9 91.2 81.9
D-Kf-v-s 89.2 93.9 98.3 94.6 84.3 97.2 89.9 91.2 81.9
D-Kf-i-v 97.9 100.0 100.0 100.0 93.1 96.7 100.0 100.0 78.5
D-Kf-i-v-s 97.9 100.0 100.0 100.0 93.1 96.7 100.0 100.0 78.5
LR-Kf 42.0 73.5 44.1 17.4 46.7 82.4 0.0 19.7 89.1
LR-Kf-s 42.0 73.5 44.1 17.4 46.7 82.4 0.0 19.7 89.1
LR-Kf-i 62.6 85.6 52.6 38.1 67.0 83.6 32.7 32.8 84.1
LR-Kf-i-s 62.6 85.6 52.6 38.1 67.0 83.6 32.7 32.8 84.1
LR-Kf-v 50.6 72.5 46.4 14.4 59.3 82.8 3.4 17.2 88.1
LR-Kf-v-s 50.6 72.5 46.4 14.4 59.3 82.8 3.4 17.2 88.1
LR-Kf-i-v 70.0 84.7 55.2 36.4 75.1 83.9 30.7 29.0 88.0
LR-Kf-i-v-s 70.0 84.7 55.2 36.4 75.1 83.9 30.7 29.0 88.0
RD-Kf 76.1 89.9 88.5 82.4 83.3 99.4 67.9 48.4 88.2
RD-Kf-s 76.1 89.9 88.5 82.4 83.3 99.4 67.9 48.4 88.2
RD-Kf-i 91.7 96.6 91.8 91.1 96.6 99.8 76.8 53.7 86.2
RD-Kf-i-s 91.7 96.6 91.8 91.1 96.6 99.8 76.8 53.7 86.2
RD-Kf-v 85.1 89.7 90.1 77.1 85.8 99.8 72.7 50.3 91.1
RD-Kf-v-s 85.1 89.7 90.1 77.1 85.8 99.8 72.7 50.3 91.1
RD-Kf-i-v 100.0 96.8 92.9 90.1 100.0 100.0 78.8 54.9 87.6
RD-Kf-i-v-s 100.0 96.8 92.9 90.1 100.0 100.0 78.8 54.9 87.6
Table D.2: Normalised average corrected forecast scores for each method.
Mean taken over all models.
D.3 Statistical Measures for Corrected Model Ensem-
ble Forecasts
Here the results of the corrected model ensemble forecasts are presented. It
should be noted that this only a fraction of the complete results, displaying
only one of the 18 models.
33

46. D. Statistical Meaures
Table D.3: Statistical measures of corrected model ensemble forecasts for
PM25 at TW
Method NMSE UPPA FB FAC2 CFM
Persistence 0.337 0.348 0.000 0.783 0.563
M 0.315 0.351 -0.036 0.778 0.352
M-s 0.286 0.338 -0.034 0.796 0.375
D-Kf 0.276 0.294 0.000 0.814 0.542
D-Kf-s 0.276 0.294 0.000 0.814 0.542
D-Kf-i 0.272 0.294 0.000 0.818 0.508
D-Kf-i-s 0.272 0.294 0.000 0.818 0.508
D-Kf-v 0.275 0.295 0.001 0.814 0.529
D-Kf-v-s 0.275 0.295 0.001 0.814 0.529
D-Kf-i-v 0.270 0.294 0.001 0.821 0.552
D-Kf-i-v-s 0.270 0.294 0.001 0.821 0.552
LR-Kf 0.298 0.331 0.018 0.830 0.662
LR-Kf-s 0.298 0.331 0.018 0.830 0.662
LR-Kf-i 0.291 0.322 0.015 0.834 0.642
LR-Kf-i-s 0.291 0.322 0.015 0.834 0.642
LR-Kf-v 0.299 0.334 0.014 0.830 0.615
LR-Kf-v-s 0.299 0.334 0.014 0.830 0.615
LR-Kf-i-v 0.290 0.324 0.009 0.835 0.600
LR-Kf-i-v-s 0.290 0.324 0.009 0.835 0.600
RD-Kf 0.276 0.306 0.009 0.828 0.641
RD-Kf-s 0.276 0.306 0.009 0.828 0.641
RD-Kf-i 0.272 0.303 0.008 0.832 0.661
RD-Kf-i-s 0.272 0.303 0.008 0.832 0.661
RD-Kf-v 0.274 0.307 0.006 0.829 0.605
RD-Kf-v-s 0.274 0.307 0.006 0.829 0.605
RD-Kf-i-v 0.270 0.303 0.004 0.833 0.614
RD-Kf-i-v-s 0.270 0.303 0.004 0.833 0.614
34

47. D.3. Statistical Measures for Corrected Model Ensemble Forecasts
Table D.4: Statistical measures of corrected model ensemble forecasts for
PM25 at CL
Method NMSE UPPA FB FAC2 CFM
Persistence 0.389 0.386 -0.000 0.769 0.593
M 0.316 0.325 -0.135 0.788 0.449
M-s 0.292 0.321 -0.134 0.814 0.365
D-Kf 0.281 0.293 -0.007 0.822 0.648
D-Kf-s 0.281 0.293 -0.007 0.822 0.648
D-Kf-i 0.277 0.294 -0.007 0.827 0.634
D-Kf-i-s 0.277 0.294 -0.007 0.827 0.634
D-Kf-v 0.280 0.295 -0.004 0.823 0.617
D-Kf-v-s 0.280 0.295 -0.004 0.823 0.617
D-Kf-i-v 0.277 0.295 -0.004 0.827 0.649
D-Kf-i-v-s 0.277 0.295 -0.004 0.827 0.649
LR-Kf 0.312 0.340 0.011 0.827 0.682
LR-Kf-s 0.312 0.340 0.011 0.827 0.682
LR-Kf-i 0.304 0.327 0.008 0.831 0.695
LR-Kf-i-s 0.304 0.327 0.008 0.831 0.695
LR-Kf-v 0.310 0.336 0.006 0.827 0.663
LR-Kf-v-s 0.310 0.336 0.006 0.827 0.663
LR-Kf-i-v 0.302 0.324 0.003 0.831 0.694
LR-Kf-i-v-s 0.302 0.324 0.003 0.831 0.694
RD-Kf 0.285 0.312 0.003 0.831 0.691
RD-Kf-s 0.285 0.312 0.003 0.831 0.691
RD-Kf-i 0.281 0.307 0.003 0.835 0.696
RD-Kf-i-s 0.281 0.307 0.003 0.835 0.696
RD-Kf-v 0.284 0.311 -0.001 0.832 0.675
RD-Kf-v-s 0.284 0.311 -0.001 0.832 0.675
RD-Kf-i-v 0.279 0.306 -0.001 0.836 0.696
RD-Kf-i-v-s 0.279 0.306 -0.001 0.836 0.696
35

48. D. Statistical Meaures
Table D.5: Statistical measures of corrected model ensemble forecasts for
PM25 at CB
Method NMSE UPPA FB FAC2 CFM
Persistence 0.199 0.244 -0.000 0.881 0.664
M 0.349 0.438 -0.430 0.584 0.382
M-s 0.336 0.455 -0.428 0.604 0.323
D-Kf 0.158 0.231 -0.004 0.907 0.811
D-Kf-s 0.158 0.231 -0.004 0.907 0.811
D-Kf-i 0.156 0.234 -0.004 0.911 0.814
D-Kf-i-s 0.156 0.234 -0.004 0.911 0.814
D-Kf-v 0.157 0.231 -0.001 0.908 0.823
D-Kf-v-s 0.157 0.231 -0.001 0.908 0.823
D-Kf-i-v 0.154 0.234 -0.002 0.912 0.793
D-Kf-i-v-s 0.154 0.234 -0.002 0.912 0.793
LR-Kf 0.353 0.402 0.067 0.838 0.746
LR-Kf-s 0.353 0.402 0.067 0.838 0.746
LR-Kf-i 0.336 0.376 0.062 0.842 0.766
LR-Kf-i-s 0.336 0.376 0.062 0.842 0.766
LR-Kf-v 0.345 0.397 0.064 0.841 0.744
LR-Kf-v-s 0.345 0.397 0.064 0.841 0.744
LR-Kf-i-v 0.331 0.374 0.059 0.845 0.790
LR-Kf-i-v-s 0.331 0.374 0.059 0.845 0.790
RD-Kf 0.187 0.269 0.023 0.895 0.880
RD-Kf-s 0.187 0.269 0.023 0.895 0.880
RD-Kf-i 0.184 0.265 0.021 0.897 0.869
RD-Kf-i-s 0.184 0.265 0.021 0.897 0.869
RD-Kf-v 0.184 0.267 0.020 0.897 0.880
RD-Kf-v-s 0.184 0.267 0.020 0.897 0.880
RD-Kf-i-v 0.181 0.263 0.017 0.899 0.868
RD-Kf-i-v-s 0.181 0.263 0.017 0.899 0.868
36

49. D.3. Statistical Measures for Corrected Model Ensemble Forecasts
Table D.6: Statistical measures of corrected model ensemble forecasts for
NO2 at TW
Method NMSE UPPA FB FAC2 CFM
Persistence 0.183 0.255 -0.001 0.909 0.154
M 0.268 0.260 -0.135 0.764 0.125
M-s 0.245 0.248 -0.134 0.796 0.000
D-Kf 0.180 0.231 -0.003 0.885 0.095
D-Kf-s 0.180 0.231 -0.003 0.885 0.095
D-Kf-i 0.177 0.228 -0.003 0.889 0.095
D-Kf-i-s 0.177 0.228 -0.003 0.889 0.095
D-Kf-v 0.178 0.232 0.001 0.888 0.095
D-Kf-v-s 0.178 0.232 0.001 0.888 0.095
D-Kf-i-v 0.175 0.228 0.000 0.891 0.095
D-Kf-i-v-s 0.175 0.228 0.000 0.891 0.095
LR-Kf 0.277 0.304 0.042 0.875 0.175
LR-Kf-s 0.277 0.304 0.042 0.875 0.175
LR-Kf-i 0.266 0.295 0.041 0.879 0.148
LR-Kf-i-s 0.266 0.295 0.041 0.879 0.148
LR-Kf-v 0.264 0.301 0.042 0.880 0.182
LR-Kf-v-s 0.264 0.301 0.042 0.880 0.182
LR-Kf-i-v 0.255 0.291 0.041 0.882 0.157
LR-Kf-i-v-s 0.255 0.291 0.041 0.882 0.157
RD-Kf 0.202 0.249 0.021 0.890 0.200
RD-Kf-s 0.202 0.249 0.021 0.890 0.200
RD-Kf-i 0.200 0.250 0.024 0.893 0.200
RD-Kf-i-s 0.200 0.250 0.024 0.893 0.200
RD-Kf-v 0.197 0.247 0.023 0.894 0.222
RD-Kf-v-s 0.197 0.247 0.023 0.894 0.222
RD-Kf-i-v 0.195 0.248 0.025 0.896 0.138
RD-Kf-i-v-s 0.195 0.248 0.025 0.896 0.138
37

50. D. Statistical Meaures
Table D.7: Statistical measures of corrected model ensemble forecasts for
NO2 at CL
Method NMSE UPPA FB FAC2 CFM
Persistence 0.304 0.303 -0.001 0.819 0.222
M 0.318 0.278 -0.317 0.721 0.000
M-s 0.302 0.303 -0.314 0.755 0.000
D-Kf 0.200 0.242 -0.010 0.865 0.200
D-Kf-s 0.200 0.242 -0.010 0.865 0.200
D-Kf-i 0.196 0.240 -0.009 0.869 0.200
D-Kf-i-s 0.196 0.240 -0.009 0.869 0.200
D-Kf-v 0.199 0.240 -0.003 0.867 0.200
D-Kf-v-s 0.199 0.240 -0.003 0.867 0.200
D-Kf-i-v 0.196 0.239 -0.003 0.870 0.200
D-Kf-i-v-s 0.196 0.239 -0.003 0.870 0.200
LR-Kf 0.283 0.331 0.022 0.838 0.353
LR-Kf-s 0.283 0.331 0.022 0.838 0.353
LR-Kf-i 0.272 0.318 0.019 0.841 0.323
LR-Kf-i-s 0.272 0.318 0.019 0.841 0.323
LR-Kf-v 0.276 0.323 0.020 0.842 0.333
LR-Kf-v-s 0.276 0.323 0.020 0.842 0.333
LR-Kf-i-v 0.268 0.313 0.018 0.843 0.345
LR-Kf-i-v-s 0.268 0.313 0.018 0.843 0.345
RD-Kf 0.227 0.268 0.003 0.857 0.462
RD-Kf-s 0.227 0.268 0.003 0.857 0.462
RD-Kf-i 0.223 0.266 0.004 0.861 0.462
RD-Kf-i-s 0.223 0.266 0.004 0.861 0.462
RD-Kf-v 0.224 0.264 0.002 0.860 0.462
RD-Kf-v-s 0.224 0.264 0.002 0.860 0.462
RD-Kf-i-v 0.220 0.262 0.002 0.863 0.462
RD-Kf-i-v-s 0.220 0.262 0.002 0.863 0.462
38

51. D.3. Statistical Measures for Corrected Model Ensemble Forecasts
Table D.8: Statistical measures of corrected model ensemble forecasts for
NO2 at CB
Method NMSE UPPA FB FAC2 CFM
Persistence 0.181 0.243 -0.001 0.904 0.780
M 0.670 0.641 -1.073 0.151 0.000
M-s 0.668 0.677 -1.070 0.146 0.000
D-Kf 0.126 0.215 -0.014 0.951 0.650
D-Kf-s 0.126 0.215 -0.014 0.951 0.650
D-Kf-i 0.122 0.227 -0.013 0.953 0.655
D-Kf-i-s 0.122 0.227 -0.013 0.953 0.655
D-Kf-v 0.124 0.214 -0.006 0.952 0.662
D-Kf-v-s 0.124 0.214 -0.006 0.952 0.662
D-Kf-i-v 0.121 0.225 -0.006 0.953 0.658
D-Kf-i-v-s 0.121 0.225 -0.006 0.953 0.658
LR-Kf 0.302 0.346 0.056 0.843 0.917
LR-Kf-s 0.302 0.346 0.056 0.843 0.917
LR-Kf-i 0.291 0.339 0.056 0.847 0.874
LR-Kf-i-s 0.291 0.339 0.056 0.847 0.874
LR-Kf-v 0.298 0.347 0.052 0.844 0.887
LR-Kf-v-s 0.298 0.347 0.052 0.844 0.887
LR-Kf-i-v 0.288 0.338 0.053 0.847 0.870
LR-Kf-i-v-s 0.288 0.338 0.053 0.847 0.870
RD-Kf 0.146 0.215 0.006 0.931 0.880
RD-Kf-s 0.146 0.215 0.006 0.931 0.880
RD-Kf-i 0.144 0.216 0.006 0.933 0.870
RD-Kf-i-s 0.144 0.216 0.006 0.933 0.870
RD-Kf-v 0.145 0.215 0.002 0.930 0.886
RD-Kf-v-s 0.145 0.215 0.002 0.930 0.886
RD-Kf-i-v 0.143 0.216 0.002 0.932 0.869
RD-Kf-i-v-s 0.143 0.216 0.002 0.932 0.869
39

52. D. Statistical Meaures
Table D.9: Statistical measures of corrected model ensemble forecasts for O3
at TW
Method NMSE UPPA FB FAC2 CFM
Persistence 0.723 0.487 -0.000 0.602 0.734
M 0.619 0.524 0.181 0.567 0.856
M-s 0.571 0.484 0.178 0.602 0.751
D-Kf 0.480 0.427 0.033 0.619 0.919
D-Kf-s 0.480 0.427 0.033 0.619 0.919
D-Kf-i 0.472 0.417 0.030 0.624 0.918
D-Kf-i-s 0.472 0.417 0.030 0.624 0.918
D-Kf-v 0.474 0.430 0.032 0.622 0.901
D-Kf-v-s 0.474 0.430 0.032 0.622 0.901
D-Kf-i-v 0.467 0.418 0.030 0.623 0.910
D-Kf-i-v-s 0.467 0.418 0.030 0.623 0.910
LR-Kf 0.879 0.626 0.030 0.632 0.713
LR-Kf-s 0.879 0.626 0.030 0.632 0.713
LR-Kf-i 0.744 0.525 0.013 0.637 0.717
LR-Kf-i-s 0.744 0.525 0.013 0.637 0.717
LR-Kf-v 0.853 0.609 0.018 0.633 0.705
LR-Kf-v-s 0.853 0.609 0.018 0.633 0.705
LR-Kf-i-v 0.732 0.516 0.003 0.639 0.705
LR-Kf-i-v-s 0.732 0.516 0.003 0.639 0.705
RD-Kf 0.538 0.435 -0.014 0.630 0.800
RD-Kf-s 0.538 0.435 -0.014 0.630 0.800
RD-Kf-i 0.517 0.420 -0.013 0.635 0.803
RD-Kf-i-s 0.517 0.420 -0.013 0.635 0.803
RD-Kf-v 0.522 0.428 -0.019 0.634 0.820
RD-Kf-v-s 0.522 0.428 -0.019 0.634 0.820
RD-Kf-i-v 0.507 0.414 -0.017 0.639 0.817
RD-Kf-i-v-s 0.507 0.414 -0.017 0.639 0.817
40

53. D.3. Statistical Measures for Corrected Model Ensemble Forecasts
Table D.10: Statistical measures of corrected model ensemble forecasts for
O3 at CL
Method NMSE UPPA FB FAC2 CFM
Persistence 0.657 0.450 -0.000 0.574 0.834
M 0.830 0.711 0.427 0.551 0.934
M-s 0.776 0.664 0.425 0.554 0.930
D-Kf 0.438 0.432 0.059 0.619 1.069
D-Kf-s 0.438 0.432 0.059 0.619 1.069
D-Kf-i 0.426 0.414 0.057 0.624 1.059
D-Kf-i-s 0.426 0.414 0.057 0.624 1.059
D-Kf-v 0.435 0.428 0.060 0.620 1.090
D-Kf-v-s 0.435 0.428 0.060 0.620 1.090
D-Kf-i-v 0.424 0.411 0.058 0.626 1.066
D-Kf-i-v-s 0.424 0.411 0.058 0.626 1.066
LR-Kf 0.478 0.410 -0.097 0.644 0.908
LR-Kf-s 0.478 0.410 -0.097 0.644 0.908
LR-Kf-i 0.454 0.384 -0.116 0.646 0.795
LR-Kf-i-s 0.454 0.384 -0.116 0.646 0.795
LR-Kf-v 0.486 0.408 -0.118 0.638 0.862
LR-Kf-v-s 0.486 0.408 -0.118 0.638 0.862
LR-Kf-i-v 0.460 0.386 -0.138 0.642 0.753
LR-Kf-i-v-s 0.460 0.386 -0.138 0.642 0.753
RD-Kf 0.481 0.390 -0.087 0.616 0.914
RD-Kf-s 0.481 0.390 -0.087 0.616 0.914
RD-Kf-i 0.467 0.381 -0.093 0.623 0.859
RD-Kf-i-s 0.467 0.381 -0.093 0.623 0.859
RD-Kf-v 0.470 0.389 -0.091 0.620 0.863
RD-Kf-v-s 0.470 0.389 -0.091 0.620 0.863
RD-Kf-i-v 0.459 0.381 -0.096 0.628 0.823
RD-Kf-i-v-s 0.459 0.381 -0.096 0.628 0.823
41

54. D. Statistical Meaures
Table D.11: Statistical measures of corrected model ensemble forecasts for
O3 at CB
Method NMSE UPPA FB FAC2 CFM
Persistence 1.001 0.688 -0.000 0.561 0.615
M 6.564 2.912 1.049 0.177 0.160
M-s 6.268 2.747 1.046 0.169 0.204
D-Kf 1.247 1.148 0.296 0.408 0.452
D-Kf-s 1.247 1.148 0.296 0.408 0.452
D-Kf-i 1.203 1.087 0.294 0.412 0.345
D-Kf-i-s 1.203 1.087 0.294 0.412 0.345
D-Kf-v 1.216 1.124 0.288 0.409 0.345
D-Kf-v-s 1.216 1.124 0.288 0.409 0.345
D-Kf-i-v 1.176 1.064 0.287 0.417 0.357
D-Kf-i-v-s 1.176 1.064 0.287 0.417 0.357
LR-Kf 0.695 0.547 -0.152 0.618 0.500
LR-Kf-s 0.695 0.547 -0.152 0.618 0.500
LR-Kf-i 0.678 0.514 -0.172 0.621 0.300
LR-Kf-i-s 0.678 0.514 -0.172 0.621 0.300
LR-Kf-v 0.690 0.534 -0.173 0.617 0.316
LR-Kf-v-s 0.690 0.534 -0.173 0.617 0.316
LR-Kf-i-v 0.679 0.513 -0.192 0.620 0.111
LR-Kf-i-v-s 0.679 0.513 -0.192 0.620 0.111
RD-Kf 0.858 0.647 -0.124 0.521 0.400
RD-Kf-s 0.858 0.647 -0.124 0.521 0.400
RD-Kf-i 0.827 0.615 -0.121 0.527 0.300
RD-Kf-i-s 0.827 0.615 -0.121 0.527 0.300
RD-Kf-v 0.823 0.644 -0.109 0.526 0.333
RD-Kf-v-s 0.823 0.644 -0.109 0.526 0.333
RD-Kf-i-v 0.802 0.618 -0.107 0.530 0.222
RD-Kf-i-v-s 0.802 0.618 -0.107 0.530 0.222
42

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