<<

. 6
( 9)



>>




0.0010
Frequency




Density
40




2500
0.0005
20




2000
0.0000
0




2000 2500 3000 3500 1500 2500 3500
Data$KR N = 293 Bandwidth = 85.4

Fig. 7.2 Histogram, density function, and boxplot of QBER and KR of the device “Entangled”



7.2.2 Data Set for the Device “Freespace”
In the following, the data for the device “Freespace” are processed. As in Sect. 7.2.1
the measurements in the ¬ber ring network in Vienna started on October 8, 2008.
The device “Alice” was positioned in Forum (FOR), the device “Bob” was located
in ERD, and the route FOR“ERD was used. The data also consist of qubit error
rate (QBER), key rate (KR), temperature, humidity, sunshine duration, and global
radiation. In Table 7.4 a short extract of the data set is given.
Table 7.5 gives the six summary statistics. Furthermore, Fig. 7.3 shows the scat-
terplot for the data set “Freespace”. In this scatterplot a correlation between QBER
and KR can be seen. Moreover, the correlation matrix is given in Table 7.6.
The correlation matrix con¬rmed the correlation between QBER and KR. As in
Sect. 7.2.1 the scatterplot matrix and the correlation matrix show that there is no
linear dependence between air temperature, humidity, sunshine duration, and global
7 Statistical Analysis of QKD Networks in Real-life Environment 131

Table 7.4 Extract of the data set for the device “Freespace”
QBER KR Temperature Humidity Sunshine duration Global radiation
203 17503 145 89 0 0
209 14065 143 90 0 0
215 11499 140 91 0 0
221 1906 141 91 0 0
220 13119 139 91 0 0
248 7697 132 92 0 0




Table 7.5 Numerical overview of the device “Freespace”
QBER KR Temperature
Min. : 172.0 Min. : 952 Min. : 117.0
1st Qu.: 187.0 1st Qu.: 11969 1st Qu.: 133.2
Median : 208.0 Median : 14522 Median : 148.0
Mean: 218.2 Mean: 14455 Mean: 149.5
3rd Qu.: 222.0 3rd Qu.: 17428 3rd Qu.: 165.0
Max. : 849.0 Max. : 29302 Max. : 184.0
Humidity Sunshine duration Global radiation
Min. : 71.00 Min. : 0.0 Min. : 0.0
1st Qu.: 78.25 1st Qu.: 0.0 1st Qu.: 0.0
Median: 84.00 Median : 0.0 Median : 0.0
Mean : 83.94 Mean : 104.2 Mean : 100.2
3rd Qu.: 91.00 3rd Qu.: 0.0 3rd Qu.: 153.0
Max. : 93.00 Max. : 600.0 Max. : 539.00




radiation on the one hand and QBER as well as KR on the other hand. Thus, it seems
that external in¬‚uences have no effect on the quality of QKD systems of the device
“Freespace.”
For a graphical overview, Fig. 7.4 displays the histogram, the density function,
and the boxplot of QBER and KR. In the boxplot of QBER a lot of outliers can be
seen. As aforementioned, it is important to eliminate these outliers because they can
falsify the result, e.g., the prediction. It is particularly essential to eliminate them
before testing the distributions of the numerical variables of the data set.
Alternatively, one could try to model the outlier-prone distribution of QBER on
the basis of some extreme-value-distribution. This will not be considered here, how-
ever.
Again tests like the Kolmogorov“Smirnov test or the Cramer“von Mises two
sample test are used to get further information about the distribution of QBER. The
Cramer“von Mises two sample test supports the hypotheses that QBER of the data
set is gamma, normal, and log-normal distributed, respectively, because in all cases
the p-value is greater than 0.05. For the generalized linear mixed model (GLMM)
in Sect. 7.3.2, QBER is taken as gamma distributed because the gamma distribution
is more general than the normal or log-normal distribution.
132 K. Lessiak and J. Pilz

0 10000 25000 75 85 0 200 400




800
500
QBER




200
25000




KR
0 10000




180
150
temperature




120
85




humidity
75




500
sunshine duration




200
0
400




global radiation
200
0




200 500 800 120 150 180 0 200 500

Fig. 7.3 Scatterplot matrix for the data set of the device “Freespace”


Table 7.6 Correlation matrix of the device “Freespace”
QBER KR Temperature
QBER 1.0000000 “0.4637677 “0.0259311
KR “0.4637677 1.0000000 0.1122507
Temperature “0.0259311 0.1122507 1.0000000
Humidity “0.0119615 “0.0997538 “0.9519479
Sunshine duration 0.1983148 “0.0006192 0.5981148
Global radiation 0.1222088 0.0179512 0.5844248
Humidity Sunshine duration Global radiation
QBER “0.0119615 0.1983148 0.1222088
KR “0.0997538 “0.0006192 0.0179512
Temperature “0.9519479 0.5981148 0.5844248
Humidity 1.0000000 “0.6266734 “0.5776959
Sunshine duration “0.6266734 1.0000000 0.8419395
Global radiation “0.5776959 0.8419395 1.0000000
7 Statistical Analysis of QKD Networks in Real-life Environment 133

Histogram of QBER Density function of QBER Boxplot of QBER
150




800
0.015




700
100




600
Frequency




0.010
Density




500
400
50




0.005




300
200
0.000
0




200 400 600 800 200 400 600 800
Data$QBER N = 250 Bandwidth = 7.791




Histogram of key rate Density function of key rate Boxplot of key rate




30000
100




1e-04
80




8e-05




20000
4e-05 6e-05
60
Frequency




Density
40




5000 10000
0e+00 2e-05
20
0




0




0 10000 20000 30000 0 10000 25000
Data$KR N = 250 Bandwidth = 1215

Fig. 7.4 Histogram, density function, and boxplot of QBER and KR of the device “Freespace”


7.2.3 Data Set for the Device “Autocompensating Plug&Play”
The measurements for the device “Autocompensating Plug&Play” in the ¬ber ring
network in Vienna also started on October 8, 2008 where Alice was located in
Gudrunstrasse (GUD) and Bob was positioned in Breitenfurterstrasse (BRT). The
route GUD“BRT was used. In Table 7.7 a short extract of the data set is shown.

Table 7.7 Extract of the data set of the device “Autocompensating Plug&Play”
QBER KR Temperature Humidity Sunshine duration Global radiation
146 1240 148 88 0 0
126 1250 145 89 0 0
150 986 143 90 0 0
146 1280 141 91 0 0
146 977 140 91 0 0
159 944 141 91 0 0
134 K. Lessiak and J. Pilz

Table 7.8 gives the six summary statistics for QBER, KR, temperature, humidity,
sunshine duration, and global radiation. In Fig. 7.5 the scatterplot matrix of the data
set is given, similar to the data sets before a correlation between QBER and KR. For
numerical information the correlation matrix is used, which is given in Table 7.9.

Table 7.8 Numerical overview of the device “Autocompensating Plug&Play”
QBER KR Temperature
Min. : 98.0 Min. : 81.0 Min. : 117.0
1st Qu.: 133.0 1st Qu.: 434.0 1st Qu.: 135.0
Median : 148.0 Median : 512.0 Median : 148.0
Mean: 150.7 Mean: 525.2 Mean: 149.8
3rd Qu.: 165.0 3rd Qu.: 603.0 3rd Qu.: 163.5
Max. : 411.0 Max. : 1280.0 Max. : 184.0
Humidity Sunshine duration Global radiation
Min. : 71.00 Min. : 0.0 Min. : 0.0
1st Qu.: 80.00 1st Qu.: 0.0 1st Qu.: 0.0
Median: 84.00 Median : 0.0 Median : 0.0
Mean : 83.99 Mean : 97.71 Mean : 94.02
3rd Qu.: 91.00 3rd Qu.: 0.0 3rd Qu.: 146.00
Max. : 93.00 Max. : 600.0 Max. : 539.00


The correlation matrix con¬rms the assumption that there is a correlation between
QBER and KR. It can be seen that there is no correlation between QBER and tem-
perature, humidity, sunshine duration, and global radiation. So, it can be assumed
that external in¬‚uences have no effect on the quality of QKD systems of the device
type “Autocompensating Plug&Play.”
In Fig. 7.6 the histogram, the density function, and the boxplot of QBER and
KR are given. The next step is to ¬nd out the outliers in the data set and to eliminate
them because they can lead to a faulty result of the statistical analysis. In the boxplot
of the QBER three outliers can be clearly identi¬ed.
The Kolmogorov“Smirnov test declares that the QBER of the data set for the
device “Autocompensating Plug&Play” is gamma, log-normal, and normally dis-
tributed, respectively, because the p-value in all three cases is greater than 0.05.
The Cramer“von Mises two sample test con¬rmed the result of the Kolmogorov“
Smirnov test because in all three cases the p-value is greater than 0.05, again, and
for the same reason as before, the gamma distribution is assumed for QBER of the
device “Autocompensating Plug&Play” “ Sect. 7.3.2.

7.2.4 Data Set for the Device “Continuous Variables”

The measurements of the device “Continuous Variables” in the ¬ber ring network
in Vienna also started on October 8, 2008. Alice was positioned in Erdberg (ERD)
and Bob was located in Gudrunstrasse (GUD). For the measurements of the device
“Continuous Variables” the route ERD “ GUD was used. A short extract of the data
set is given in Table 7.10.
7 Statistical Analysis of QKD Networks in Real-life Environment 135

200 800 75 85 0 200 400




400
250
QBER




100
800




KR
200




180
150
temperature




120
85




humidity
75




500
sunshine duration




200
0
400




global radiation
200
0




100 250 400 120 150 180 0 200 500

Fig. 7.5 Scatterplot matrix for the data set of the device “Autocompensating Plug&Play”


Table 7.9 Correlation matrix for the device “Autocompensating Plug&Play”
QBER KR Temperature
QBER 1.0000000 “0.5974331 0.0664187
KR “0.5974331 1.0000000 “0.0854596
Temperature 0.0664187 “0.0854596 1.0000000
Humidity “0.0663922 0.1678327 “0.9487629
Sunshine duration “0.0084388 “0.0543023 0.5785442
Global radiation 0.1105632 “0.1339746 0.5516755
Humidity Sunshine duration Global radiation
QBER 0.0663922 “0.0084388 0.1105632
KR 0.1678327 “0.0543023 “0.1339747
Temperature “0.9487629 0.5785442 0.5516755
Humidity 1.0000000 “0.6176290 “0.5626842
Sunshine duration “0.6176290 1.0000000 0.8457442
Global radiation “0.5626841 0.8457442 1.0000000
136 K. Lessiak and J. Pilz

Histogram of QBER Density function of QBER Boxplot of QBER
150




0.015




400
350
100




0.010




300
Frequency




Density




250
0.005




200
50




150
0.000




100
0




100 200 300 400
100 200 300 400
N = 279 Bandwidth = 6.969
Data$QBER

Histogram of key rate Density function of key rate Boxplot of key rate
80




0.0030




1200
1000
60




0.0020
Frequency




800
Density
40




600
0.0010




400
20




200
0.0000
0




0 200 600 1000 0 400 800 1200
Data$KR N = 279 Bandwidth = 36.8

Fig. 7.6 Histogram, density function, and boxplot of QBER and KR of the device “Autocompen-
sating Plug&Play”

Table 7.11 lists the summary statistics, i.e., minimum, 1st quantile, median, mean,
3rd quantile, and maximum for the device “Continuous Variables”.
In Fig. 7.7 the scatterplot matrix for the data set of the device “Continuous Vari-
ables” can be seen. A high correlation between QBER and KR is obvious. The


Table 7.10 Extract of the data set for the device “Continuous Variables”
QBER KR Temperature Humidity Sunshine duration Global radiation
272 6070 148 88 0 0
244 9039 145 89 0 0
331 9325 143 90 0 0
372 6897 141 91 0 0
286 9952 140 91 0 0
456 9686 141 91 0 0
7 Statistical Analysis of QKD Networks in Real-life Environment 137

Table 7.11 Numerical overview for the device “Continuous Variables”
QBER KR Temperature
Min. : 17.0 Min. : 1223 Min. : 117.0
1st Qu.: 220.2 1st Qu.: 8045 1st Qu.: 135.0
Median : 327.5 Median : 9091 Median : 148.0
Mean: 407.6 Mean: 8620 Mean: 149.7
3rd Qu.: 586.2 3rd Qu.: 9764 3rd Qu.: 163.0
Max. : 1222.0 Max. : 12943 Max. : 184.0
Humidity Sunshine duration Global radiation
Min. : 71.00 Min. : 0.0 Min. : 0.0
1st Qu.: 80.00 1st Qu.: 0.0 1st Qu.: 0.0
Median: 84.00 Median : 0.0 Median : 0.0
Mean : 84.05 Mean : 92.57 Mean : 89.39
3rd Qu.: 91.00 3rd Qu.: 0.0 3rd Qu.: 131.25
Max. : 93.00 Max. : 600.0 Max. : 539.00


2000 8000 75 85 0 200 400




1000
QBER




400
0
8000




KR
2000




180
150
temperature
120
85




humidity
75




500




sunshine duration
200
0
200 400




global radiation
0




0 400 1000 120 150 180 0 200 500

Fig. 7.7 Scatterplot matrix for the data set of the device “Continuous Variables”
138 K. Lessiak and J. Pilz

Table 7.12 Correlation matrix for the device “Continuous Variables”
QBER KR Temperature
QBER 1.0000000 “0.8567638 “0.5269060
KR “0.8567639 1.0000000 0.4354725
Temperature “0.5269060 0.4354724 1.0000000
Humidity 0.4427780 “0.3793859 “0.9480146
Sunshine duration “0.1716637 0.1229877 0.5661680
Global radiation “0.1167913 0.0804763 0.5411758
Humidity Sunshine duration Global radiation
QBER 0.4427780 “0.1716637 “0.1167913
KR “0.3793859 0.1229877 0.0804763
Temperature “0.9480146 0.5661680 0.5411757
Humidity 1.0000000 “0.6013030 “0.5501430
Sunshine duration “0.6013030 1.0000000 0.8546850
Global radiation “0.5501431 0.8546851 1.0000000



correlation matrix “ Table 7.12 “ gives the numerical values for the correlations
between the observed variables.
There is a high correlation between QBER and KR and a small correlation
between QBER and temperature and also between QBER and humidity. It seems
that in this case external in¬‚uences have an effect on the quality of QKD systems of
the device type “Continuous Variables.”
The histogram, the density function, and the boxplot of QBER and KR are given
in Fig. 7.8. In the boxplot of QBER and KR the outliers can be seen. Again, for
prediction purposes, we have eliminated the outliers.
The Cramer“von Mises two sample test supports the hypothesis that QBER is
gamma distributed as well as the hypothesis that it is log-normal distributed. As
before, for the GLMM in Sect. 7.3.2, QBER is taken to be gamma distributed.


7.2.5 Data Set for the Device “One Way Weak Pulse System”

The measurements for the device “One Way Weak Pulse System” started in the
¬ber ring network in Vienna on October 8, 2008, too. Alice was positioned in
Breitenfurterstrasse (BRT) and Bob was located in Siemensstrasse (SIE). For the
measurements the route BRT “ SIE was used. In Table 7.13 a short extract can be
seen.
Table 7.14 lists the six summary statistics for the variables observed. The scatter-
plot matrix for the data set is given in Fig. 7.9. Table 7.15 gives the exact numerical
values for the correlations.
In the correlation matrix a slight correlation between QBER and KR can be seen.
On the other hand, a signi¬cant correlation between QBER and temperature is obvi-
ous. There is also a correlation between QBER and humidity, sunshine duration
and global radiation. Due to this it can be assumed that external in¬‚uences have an
effect on the quality of QKD systems of the device type “One Way Weak Pulse Sys-
7 Statistical Analysis of QKD Networks in Real-life Environment 139

Histogram of QBER Density function of QBER Boxplot of QBER




0.0020




1200
60
50




1000
0.0015
40
Frequency




800
Density
0.0010
30




600
20




0.0005




400
10




200
0.0000
0




0
0 200 600 1000 0 500 1000 1500
Data$QBER
N = 282 Bandwidth = 76.39

Histogram of key rate Density function of key rate Boxplot of key rate
100




12000
0.00030
80




0.00020




2000 4000 6000 8000
Frequency
60




Density
40




0.00010
20




0.00000
0




0 4000 8000 12000
2000 6000 10000
N = 282 Bandwidth = 373.6
Data$KR

Fig. 7.8 Histogram, density function, and boxplot of QBER and KR of the device “Continuous
Variables”

Table 7.13 Extract of the data set of the device “One Way Weak Pulse System”
QBER KR Temperature Humidity Sunshine duration Global radiation
254 2463 161 81 0 0
250 2418 161 81 0 0
257 2448 160 82 0 0
283 3203 160 82 0 0
307 1987 158 83 0 0
254 2884 158 83 0 0

tem.” The histograms, density plots, and boxplots of QBER and KR are displayed
in Fig. 7.10.
The Cramer“von Mises two sample test declares that QBER is gamma distributed
because the p-value is greater than 0.05. For the hypotheses of log-normal and nor-
mal distribution, respectively, the Cramer“von Mises two sample test statistics result
in p-values smaller than 0.05; thus, these hypotheses are rejected.
140 K. Lessiak and J. Pilz

Table 7.14 Numerical overview for the device “One Way Weak Pulse System”
QBER KR Temperature
Min. : 224.0 Min. : 1 Min. : 127.0
1st Qu.: 248.0 1st Qu.: 3007 1st Qu.: 139.0
Median : 264.0 Median : 4002 Median : 148.0
Mean: 260.9 Mean: 3515 Mean: 149.3
3rd Qu.: 269.0 3rd Qu.: 4552 3rd Qu.: 155.0
Max. : 315.0 Max. : 5482 Max. : 184.0
Humidity Sunshine duration Global radiation
Min. : 71.00 Min. : 0.0 Min. : 0.0
1st Qu.: 81.00 1st Qu.: 0.0 1st Qu.: 0.0
Median: 83.00 Median : 0.0 Median : 0.0
Mean : 83.61 Mean : 116.9 Mean : 111.4
3rd Qu.: 87.00 3rd Qu.: 0.0 3rd Qu.: 146.0
Max. : 93.00 Max. : 600.0 Max. : 539.00



0 2000 5000 75 85 0 200 400




280
QBER




240
5000




KR
2000
0




130 150 170
temperature
85




humidity
75




500




sunshine duration
200
0
200 400




global radiation
0




240 280 130 150 170 0 200 500

Fig. 7.9 Scatterplot matrix for the data set of the device “One Way Weak Pulse System”
7 Statistical Analysis of QKD Networks in Real-life Environment 141

Table 7.15 Correlation matrix for the device “One Way Weak Pulse System”
QBER KR Temperature
QBER 1.0000000 “0.2591815 “0.5015706
KR “0.2591815 1.0000000 0.1886785
temperature “0.5015706 0.1886785 1.0000000
humidity 0.3460800 “0.4721942 “0.8909994
Sunshine duration “0.2456862 “0.0034571 0.7155210
global radiation “0.2023045 0.0234804 0.6072473
Humidity Sunshine duration Global radiation
QBER 0.3460800 “0.2456861 “0.2023044
KR “0.4721943 0.0034571 0.0234804
Temperature “0.8909994 0.7155210 0.6072472
Humidity 1.0000000 “0.6756165 “0.5884132
Sunshine duration “0.6756166 1.0000000 0.9519974
Global radiation “0.5884132 0.9519974 1.0000000


Histogram of QBER Density function of QBER Boxplot of QBER
60




0.030




300
50
40




0.020




280
Frequency




Density
30




260
0.010
20




240
10




0.000
0




220 240 260 280 300 320
220 260 300
Data$QBER
N = 141 Bandwidth = 5.242

Histogram of key rate Density function of key rate Boxplot of key rate
0e+00 1e“04 2e“04 3e“04 4e“04 5e“04
50




5000
40




4000
Frequency
30




3000
Density
20




2000
10




1000
0




0




0 1000 3000 5000
0 2000 4000 6000
Data$KR
N = 141 Bandwidth = 385.7

Fig. 7.10 Histogram, density function, and boxplot of QBER and KR of the device “One Way
Weak Pulse System”
142 K. Lessiak and J. Pilz

7.3 Statistical Analysis
The statistical analysis uses the generalized linear model introduced in Sect. 7.1.1
and furthermore the generalized linear mixed model introduced in Sect. 7.1.2. The
implementations for the GLM and GLMM are based on [4] and [16].


7.3.1 Generalized Linear Model
For the illustration of the generalized linear model (GLM) only the data set of
the entangled-based device (device “Entangled”) is used. The Cramer“von Mises
two sample test to the entangled-based data processed in Sect. 7.2.1 favors the
gamma distribution of the data set. Due to this fact, the gamma generalized lin-
ear model is used which is intended for continuous, skewed responses. In this case
the log link is used because it is consistent with the linear model. Subsequently,
the gamma GLM with QBER as response, in dependency of the KR and exter-
nal in¬‚uences as predictors, is processed. In Table 7.16 the estimated regression
coef¬cients β and the estimated error terms µ from the regression output can be
ˆ ˆ
seen.

Table 7.16 The estimated regression coef¬cients β and the estimated error terms µ from the GLM
ˆ ˆ
β µ
ˆ
Factor ˆ Unit
2.791 · 10’1
Intercept 6.452 1
’2.593 · 10’4 1.777 · 10’5
KR bit/s
7.563 · 10’4 7.733 · 10’4 —¦
Temperature C/10
2.203 · 10’4 2.077 · 10’3
Humidity %
5.206 · 10’5 4.322 · 10’5
Sunshine duration s
’2.514 · 10’4 5.695 · 10’5 W/m2
Global radiation


For demonstrating the goodness of ¬t of the current model several tests and
criteria are used. One of these criteria is the Akaike Information Criterion (AIC).
Usually, criteria for the goodness of ¬t of the regression model are based on the
regression residuals. Moreover, leverage and in¬‚uence can be used as criteria for
the performance of a model. Concerning leverages it is important to know that they
are somewhat different for GLMs. Other possibilities are to look at the jacknife
residuals and the Cook statistics, respectively.
In contrast to the described residuals, leverage, and in¬‚uence measures, also
graphical methods such as a half-normal plot are used to detect points that do not ¬t
to the model and consequently in¬‚uence the ¬t unduly. In Fig. 7.11 a half-normal
plot of the jacknife residuals is given. It seems that the data values 91 and 267 are
outliers.
The half-normal plot can also be used for positive-valued diagnostics, i.e., for
leverages and the Cook statistics. In Fig. 7.12 a half-normal plot of the leverages is
shown. There is some indication that the data values 106 and 108 may have some
leverage.
7 Statistical Analysis of QKD Networks in Real-life Environment 143

91




3
267

Sorted Data
2 1
0




0.0 0.5 1.0 1.5 2.0 2.5 3.0
Half-normal quantiles

Fig. 7.11 Half-normal plot of the jacknife residuals
0.08




106
108
0.06
Sorted Data
0.04
0.02
0.00




0.0 0.5 1.0 1.5 2.0 2.5 3.0
Half-normal quantiles

Fig. 7.12 Half-normal plot of the leverages


To con¬rm the results of the jacknife residuals and the leverages a half-normal
plot of the Cook statistics is given in Fig. 7.13. It seems that the data values 78 and
91 have a higher in¬‚uence compared to the rest of the data.
Due to this results it can be assumed that the data values 78, 91, 106, 108, and
267 are outliers. To these assumptions the concerned values are omitted and the
gamma GLM analysis is repeated.
144 K. Lessiak and J. Pilz




0.10
91




0.08
0.06
Sorted Data
78
0.04 0.02
0.00




0.0 0.5 1.0 1.5 2.0 2.5 3.0
Half-normal quantiles

Fig. 7.13 Half-normal plot of the Cook statistics


Table 7.17 The estimated regression coef¬cients β from the original regression model and the
ˆ
estimated regression coef¬cients β
ˆ wO from the regression model without outliers
β βwO
ˆ ˆ
Factor Unit
Intercept 6.452 6.452 1
’2.593 · 10’4 ’2.593 · 10’4
KR bit/s
7.563 · 10’4 7.563 · 10’4 —¦
Temperature C/10
2.203 · 10’4 2.203 · 10’4
Humidity %
5.206 · 10’5 5.206 · 10’5
Sunshine duration s
’2.514 · 10’4 ’2.514 · 10’4 W/m2
Global radiation


From this regression output, the estimated regression coef¬cients βwO and the
ˆ
regression coef¬cients β from the original regression output can be seen in
ˆ
Table 7.17.
No signi¬cant differences can be observed, which disagrees with the assumption
that the data values 78, 91, 106, 108, and 267 are outliers. So, for the prediction the
original regression model can be used.
In Sect. 7.2.1 the ¬rst six data points of the data set of October, 8, 2008 are shown.
In the following the original regression model is used to predict QBER of these six
data points. Comparing real and predicted QBER gives further information about
the performance of the model. The comparison between real and predicted values
can be seen in Table 7.18.
Moreover, in Table 7.19 an extract of the data set for the entangled-based device
of October 11, 2008 is given. To predict QBER the original regression model with
KR, temperature, humidity, sunshine duration, and global radiation as parameters is
used. In Table 7.20 real and predicted QBER values are shown where a difference
is obvious.
7 Statistical Analysis of QKD Networks in Real-life Environment 145

Table 7.18 Comparison between real and predicted QBER of GLM
Original Predicted
329 355
345 362
329 355
336 317
379 377
354 325


Table 7.19 Extract of the data set for the entangled-based device of October 11, 2008
QBER KR Temperature Humidity Sunshine duration Global radiation
351 2302 165 78 600 173
365 2244 165 78 600 146
358 2606 166 78 600 120
380 2498 165 78 600 93
365 2561 165 78 600 73
414 2041 165 78 339 47


Table 7.20 Comparison between real and predicted QBER of GLM
Original Predicted
351 397
365 406
358 372
380 385
365 380
414 432



7.3.2 Generalized Linear Mixed Model

In Sect. 7.3.1 the GLM is used for prediction of the entangled-based device. There-
fore, this regression model is useless for the prediction of other devices, i.e., for
prediction of the device types “Freespace,” “Autocompensating Plug&Play,” “Con-
tinuous Variables,” and “One Way Weak Pulse System.” In the following the gen-
eralized linear mixed model (GLMM) is used to include this random effect caused
by the different devices. The idea in this case is to involve the random effect by
introducing a single random intercept for each device.
As mentioned in Sect. 7.1.2 the complex problem is to estimate the model param-
eters. Here a penalized quasi-likelihood approach is used.
In Table 7.21 the output of the GLMM is represented. There the ¬xed effects
estimates, i.e., the estimated regression coef¬cients β, and their approximated stan-
ˆ
dard errors, i.e., the estimated error terms µ, corresponding to the different ¬xed
ˆ
effects are shown. The random components of the model are also of interest. In
Table 7.22 the values for the random intercepts of the different devices are given.
In the random intercepts a clear difference between the devices is obvious, which
con¬rms the assumption that there is a random effect caused by the ¬ve device types.
146 K. Lessiak and J. Pilz

Table 7.21 The estimated regression coef¬cients β and the estimated error terms µ from the
ˆ ˆ
GLMM
β µ
ˆ
Factor ˆ Unit
’8.474 · 10’3 1.746 · 10’3
Intercept 1
2.227 · 10’7 1.130 · 10’8
KR bit/s
4.074 · 10’5 4.076 · 10’6 —¦
Temperature C/10
6.346 · 10’5 1.097 · 10’5
Humidity %
4.968 · 10’7 2.513 · 10’7
Sunshine duration s
’1.261 · 10’6 3.118 · 10’7 W/m2
Global radiation


Table 7.22 The values for the random intercepts of the different devices from the GLMM
No. Device Intercept
’6.800 · 10’4
1 Entangled
’1.195 · 10’3
2 Freespace
3.736 · 10’3
3 Autocompensating Plug&Play
’2.110 · 10’3
4 Continuous Variables
2.493 · 10’4
5 One Way Weak Pulse System



4




2
Standardized residuals




0




“2




0.002 0.004 0.006 0.008
Fitted values

Fig. 7.14 Fitted values versus standardized residuals



Moreover, to get information about the goodness of ¬t of the current model the
standardized residuals are considered. In Fig. 7.14 the ¬tted values versus the stan-
dardized residuals can be seen.
To increase information about the performance we try to predict the QBER. In
Table 7.23 the real QBER of the data set of the entangled-based device of October 8,
2008 (processed in Sect. 7.2.1) is compared with the predicted QBER of the GLM
7 Statistical Analysis of QKD Networks in Real-life Environment 147

Table 7.23 Comparison between real QBER, predicted QBER (GLM), and predicted QBER
(GLMM)
Real QBER Predicted QBER (GLM) Predicted QBER (GLMM)
329 355 325
345 362 334
329 355 334
336 317 325
379 377 347
354 325 328


and the predicted QBER of the GLMM. The prediction of the GLMM approximates
the real QBER much better than the prediction based on the GLM.
A comparison between the real QBER and the predicted QBER of the GLMM is
given in Table 7.24. There are also predictions for the other four devices “Freespace,”
“Autocompensating Plug&Play,” “Continuous Variables,” and “One Way Weak
Pulse System.” In the ¬rst column of Table 7.24, the data point of the data set can be
seen which is used for prediction. In the second column the corresponding device
is indicated. In the other two columns the real and the predicted QBER values are
opposed. Generally, the predicted values based on the GLMM approximate the real
values quite well.


7.4 Summary

In the statistical analysis, regression models such as generalized linear models
(GLM) and generalized linear mixed models (GLMM) are used to ascertain whether
external in¬‚uences like temperature, humidity, sunshine duration, and global radia-
tion effect the quality of QKD systems. Furthermore, these two regression models
are, in consequence, used to predict the qubit error rate (QBER). The advantage
of GLMs and GLMMs is that in both regression models it is possible to handle
non-normal responses. Moreover, in a GLMM, random effects can be included.
A prototype of a QKD network was implemented in Vienna where the measure-
ments of the different devices have been performed. The measurements started on
October 1, 2008 and were concluded on November 8, 2008. The measurements for
this chapter are based only on the data of October 8, 2008 to October 10, 2008 where
the data sets of ¬ve different devices are used, i.e., of the device types “Entangled,”
“Freespace,” “Autocompensating Plug&Play,” “Continuous Variables,” and “One
Way Weak Pulse System.” From these measurements the QBER and the key rate are
available. From the Central Institute for Meteorology and Geodynamics (ZAMG)
the data of external in¬‚uences, i.e., temperature, humidity, sunshine duration, and
global radiation, are obtained.
Statistical methods are used to process the data sets for the different devices.
Furthermore, GLMs and GLMMs are used to get further information and to pre-
dict QBER. The conclusion that can be drawn from the statistical analysis is that
temperature, humidity, sunshine duration, and global radiation have no in¬‚uence
148 K. Lessiak and J. Pilz

Table 7.24 Comparison between real and predicted QBER for different devices
Data point Device Real QBER Predicted QBER
1 Entangled 329 325
2 Entangled 345 334
3 Entangled 329 334
4 Entangled 336 325
5 Entangled 379 347
6 Entangled 354 328
320 Freespace 209 223

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