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