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29TH DAAAM INTERNATIONAL SYMPOSIUM ON INTELLIGENT MANUFACTURING AND AUTOMATION
DOI: 10.2507/29th.daaam.proceedings.076
VISUALIZATION OF DATA FROM NETWORK OF SENSORS:
APPROPRIATE SPATIAL INTERPOLATION METHOD
Adnan Masic, Dzevad Bibic, Boran Pikula, Emina Dzaferovic-Masic & Faruk Razic
This Publication has to be referred as: Masic, A[dnan]; Bibic, D[zevad]; Pikula, B[oran]; Dzaferovic-Masic, E[mina]
& Razic, F[aruk] (2018). Visualization of Data from Network of Sensors: Appropriate Spatial Interpolation Method,
Proceedings of the 29th DAAAM International Symposium, pp.0529-0533, B. Katalinic (Ed.), Published by DAAAM
International, ISBN 978-3-902734-20-4, ISSN 1726-9679, Vienna, Austria
DOI: 10.2507/29th.daaam.proceedings.076
Abstract
Graphical presentation of the measured data is more popular than ever before. However, rich and colourful images are
often misrepresentation of the measured quantities. In this paper we discuss proper mathematical technique for correct
presentation of data from the network of sensors. Three common methods for spatial interpolation are demonstrated and
compared using real data from the network of sensors (for air pollution with aerosols). Finally, best procedure is
recommended and discussed in details.
Keywords: network of sensors; spatial interpolation; Hilbert space; air pollution
1. Introduction
Number of sensors of various types that are connected to the internet is increasing rapidly. This fits into trends
popularly called “Internet of Things” and “Smart Cities”. Despite the fact that number of sensors is increasing, we will
never have a sensor at every single point of interest. Therefore, some sort of spatial interpolation is needed. The main
question is: if we have N sensors over certain geographical area, which spatial interpolation method is appropriate for
visualization of measured quantity over the entire area? In this paper we will try to answer that question using the real
data from the network of sensors that measure air pollution.
The simplest approach is to use bilinear interpolation [1]:
(1)
where a0, a1, a2 and a3 are coefficients which can be calculated by knowing values of function f at four different points
around (four sensors from the network). Obviously, this is not the best method, due to the nature of sensors (inevitable
errors during the measurements) and local disturbances (very localized source of air pollution for example).
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29TH DAAAM INTERNATIONAL SYMPOSIUM ON INTELLIGENT MANUFACTURING AND AUTOMATION
More natural approach is inversed distance weighting [2]. This is a multivariate interpolation with a known
scattered set of points. The assigned values to unknown points are calculated with a weighted average of the values
available at the known points. This method gives few options of choosing parameters, so that the nature of the
measurements is represented properly.
Next candidate is the procedure called kriging [3] (name given from the master thesis of Danie Krige [4]). Kriging
is a method of interpolation for which the interpolated values are modelled by a Gaussian process governed by prior
covariance. The kriging estimation may also be seen as a (sort of) spline in a Hilbert space. When performing kriging,
the user must specify the function which describes the degree of spatial dependence of a spatial random field – this is
called variogram. Under suitable assumptions, kriging gives the best linear unbiased prediction of the intermediate
values [5].
2. Case study
Network of 13 sensors which measure the concentration of aerosols smaller than 10 m (PM10) was installed in and
around the city of Sarajevo. Daily average values of PM10 for 30/08/2018 are chosen for this study. Figure 1 illustrates
the location of sensors, while table 1 shows calculated average values of PM10 for each sensor. The method for data
acquisition is explained in [6] and [7], while the calibration of PM10 sensor was described in [8].
Fig. 1. Location of sensors
sensor
latitude
longitude
PM10 (g/m3)
1
43.89488
18.37228
33.51
2
43.82361
18.54752
26.09
3
43.86172
18.41277
34.47
4
43.84860
18.37393
39.53
5
43.85390
18.39553
37.80
6
43.85284
18.38026
41.12
7
43.84482
18.32074
41.66
8
43.85846
18.44020
32.04
9
43.86775
18.42296
33.27
10
43.95983
18.27251
47.09
11
43.94594
18.25530
46.14
12
43.85121
18.36727
41.40
13
43.82605
18.34702
40.01
Table 1. Daily average values of PM10 for 30/8/2018
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29TH DAAAM INTERNATIONAL SYMPOSIUM ON INTELLIGENT MANUFACTURING AND AUTOMATION
3. Spatial interpolation methods
Bilinear interpolation was applied in figure 2. Akima library for R programming language was used for calculations
of interpolated values. Contour lines (for values of PM10) were added, together with locations of sensors. There is no
interpolation outside the polygon of sensors. We can also see that this spatial interpolation of PM10 doesn’t look natural
because values from sensors around the city linearly propagate all the way to the city center.
Fig. 2. Bilinear interpolation
Figure 3 shows inverse distance weighting (IDW) interpolation with three different power factors (1, 2 and 3).
Calculations were performed in R programming language again. Contour lines look more natural now. However, we
don’t know a priori which power factor is correct.
Fig. 3. Inverse distance weighting interpolation with three different power coefficients
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29TH DAAAM INTERNATIONAL SYMPOSIUM ON INTELLIGENT MANUFACTURING AND AUTOMATION
Now we proceed to the kriging. As noted above, first thing we need to do is to choose the proper variogram. Figure
4 shows the dependence of semivariance on distance, for the selected dataset. We can see from the graph that the
Gaussian fit is the best option. Using the Gaussian fit from figure 4 we can finally perform kriging interpolation. Figure
5 shows the result of this operation. We can also estimate quality of interpolation by means of cross-validation: it
removes each data location, one at a time and predicts the associated data value. R programming language was used for
plotting kriging interpolation and cross-validation errors in figure 5. We would like to note that IDW and kriging
techniques can be used to extrapolate data as well. Please note the location of sensors in figure 1: we can’t have good
estimation in the region where we don’t have sensors at all!
Fig. 4. Variogram and Gaussian fit
kriging interpolation
cross-validation errors
Fig. 5. Kriging interpolation and cross-validation
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29TH DAAAM INTERNATIONAL SYMPOSIUM ON INTELLIGENT MANUFACTURING AND AUTOMATION
4. Conclusion
Three different techniques for interpolation of data from network of sensors were analysed: bilinear, IDW and
kriging interpolation. They produced significantly different contour lines of PM10 concentrations from the same
dataset. Bilinear interpolation should not be used for such cases (where we have non-trivial spatial correlation of data).
IDW interpolation gives more natural contour lines, but it requires parameters that we don’t know a priori. The final
conclusion is very clear: kriging is the preferable method. If the variogram fit is chosen properly, kriging is the best
interpolation technique among these three (and probably best of all, for most general applications).
As a suggestion for further research we propose:
- Extension of network of sensors to more different places,
- Measurement of various episodes of air pollution, especially during the winter,
- Analysis of more variables such as altitude, humidity etc.
5. Acknowledgments
This research was supported by the Ministry of Education of Sarajevo Canton and the University of Sarajevo -
Faculty of Mechanical Engineering. We would like to thank to Mr. Damir Muslić for his generous contribution to our
project by writing core part of the in-house developed software.
6. References
[1] Press, W. H.; Flannery, B. P.; Teukolsky, S. A. & Vetterling, W. T. (1992). Numerical Recipes in C: The Art of
Scientific Computing, Second Edition. Cambridge University Press. ISBN-13: 978-0521431088, New York
[2] Łukaszyk, S. (2004). A New Concept of Probability Metric and its Applications in Approximation of Scattered
Data Sets. Computational Mechanics, 33 (2004) 299–304, DOI: 10.1007/s00466-003-0532-2
[3] Cressie, N. A. C. (1990). The Origins of Kriging, Mathematical Geology, 22 (1990) 239–252
[4] Krige, D. G. (1951). A Statistical Approach to Some Mine Valuations and Allied Problems at the Witwatersrand,
Master's thesis, University of Witwatersrand, South Africa
[5] Oliver, M. A. (1990). Kriging: A Method of Interpolation for Geographical Information Systems. International
Journal of Geographic Information Systems, 4 (1990) 313–332
[6] Masic, A.; Pikula, B. & Bibic, D. (2017). Mobile Measurements of Particulate Matter Concentrations in Urban
Area, Proceedings of the 28th DAAAM International Symposium, pp.0452-0456, B. Katalinic (Ed.), Published by
DAAAM International, ISBN 978-3-902734-11-2, ISSN 1726-9679, Vienna, Austria DOI:
10.2507/28th.daaam.proceedings.063
[7] Masic, A.; Bibic, D.; Pikula, B.; Dzaferovic-Masic, E. & Musemic, R. (2018). Experimental Study of Temperature
Inversions Above Urban Area Using Unmanned Aerial Vehicle, Thermal Science (2018)
[8] Masic, A.; Pikula, B.; Bibic, D.; Musemic, R. & Halac, A. (2018). Calibration and Assessment of Low-cost Dust
sensors, Proceedings of the 29th DAAAM International Symposium, B. Katalinic (Ed.), Published by DAAAM
International, ISSN 1726-9679, Vienna, Austria, in press
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