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Modeling spatially-correlated sensor network data
Apoorva Jindal, Konstantinos PsounisDepartment of Electrical Engineering-SystemsUniversity of Southern CaliforniaSECON 2004
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Outline
Introduction Statistical analysis of experimental data The model Model verification and validation Tools to generate large synthetic traces Conclusion
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Introduction
The sensors in sensor networks will be densely deployed and detect common phenomena.
It is expected that a high degree of spatial correlation will exist in the sensor networks data.
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Introduction
However since very few real systems have been deployed, there is hardly any experimental data available to test the proposed algorithms.
No effort has been made to propose a model which captures the spatial correlation in sensor networks.
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Introduction
We propose a mathematical model to capture the spatial correlation in sensor network data.
We present a method to generate large synthetic traces from a small experimental trace while preserving the correlation pattern, and a method to generate synthetic traces exhibiting arbitrary correlation patterns
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Statistical analysis of experimental dataA. data set description
(1)S-Pol Radar Data Set The resampled S-Pol radar data, provided by N
CAR, records the intensity of reflectivity of atmosphere in dBZ.
(2)Precipitation Data Set This data set consists of the daily rainfall precip
itation for the Pacific Northwest from 1949-1994.
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Statistical analysis of experimental datab. Statistic used to Measure Correlation in Data
Given a two dimensional stationary process X(x,y), the autocorrelation function is defined as
Another statistic often to characterize spatial correlation in data is the variogram defined as
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Statistical analysis of experimental datab. Statistic used to Measure Correlation in Data
For isotropic random process, the variogram depends only on the distance d=d1+d2 between two nodes.
For a set of samples x(xi,yj), i=1,2,…,γ(d) can be estimated as follows
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Statistical analysis of experimental datac. analysis of data using Variograms
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Statistical analysis of experimental datac. analysis of data using Variograms
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Statistical analysis of experimental datac. analysis of data using Variograms
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The model
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The model The parameters of the model are h, the αi’s, β, fY(y),
fZ(z).
For mathematical convenience, we define the three random variable:
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We can find the probability density function fX(x) as follows :
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In stationarity, have the same distribution.
Using the above and equation(6) the characteristic function of fX(x) can be written as:
Characteristic function
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Without loss of generality, we will assume that Z is a normal random variable with (0,σZ)
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Since X and Z are independent the characteristic function of fA can be written as
Hence, Equation(7) reduce to
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For mathematical convenience, we define a new random variable L having characteristic function given by
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The modelA. Parameters of the Model an Correlation
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The modelB. Inferring Model Parameters
Infer fX(x) from its empirical distribution.
Inferring σz ,αi’s and β is more involved. Using Equation(3)
leads to the following:
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The modelB. Inferring Model Parameters
Equating for 1≦ i ≦h+1 gives h+1 equations. These equations along with the equation
form a system of h+2 equations. After solving the above system, we can obtain σz ,αi’s ,β
and fY(y) through
Determining h. To start from an overestimated h and lower its value until all the αi’s are positive.
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Model verification and validationA. verification
(1)S-Pol Radar data set
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Model verification and validationA. verification
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Model verification and validationA. verification
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Model verification and validationA. verification (2)Precipitation data set:
The data inferred for the trace are h=1, α1=0.72, β=0.28 and σZ=2.61
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Model verification and validationA. verification
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Model verification and validationB. Model Validation
DIMENSIONS[6] proposes wavelet based multi-resolution summarization and drill down querying.
Spatial Correlation based Collaborative Medium Access Control (CMAC) [10]
The evaluation metric used is the query error which is defined as
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Model verification and validationB. Model Validation
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Tools to generate large synthetic traces
The tools are freely available at
http://www-scf.usc.edu/~apoorvaj generateLargeTraceFromSmall generateSyntheticTraces
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Conclusion
We have proposed a model to capture the spatial correlation, which can generate synthetic traces.
We also described a mathematical procedure to extract the parameters of the model from a real data set.
We verified and validated the model. Final, we have created two freely available tools
to enable researchers to generate data.
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