MPM: Map Based Predictive Monitoring for Wireless Sensor Networks.
ABSTRACT We present the design of a Wireless Sensor Networks (WSN) level event
prediction framework to monitor the network and its operational
environment to support proactive self* actions. For example, by
monitoring and subsequently predicting trends on network load or sensor
nodes energy levels, the WSN can proactively initiate
selfreconfiguration. We propose a Map based Predictive Monitoring (MPM)
approach where a selected WSN attribute is first profiled as WSN maps,
and based on the maps history, predicts future maps using time series
modeling. The "attribute" maps are created using a gridding technique
and predicted maps are used to detect events using our regioning
algorithm. The proposed approach is also a general framework to cover
multiple application domains. For proof of concept, we show MPM's
enhanced ability to also accurately "predict" the network partitioning,
accommodating parameters such as shape and location of the partition
with a very high accuracy and efficiency.

Conference Paper: TOM: Topology Oriented Maintenance in Sparse Wireless Sensor Networks
In Proc. of The Eighth Annual IEEE Communications Society Conference on Sensor, Mesh, and Ad Hoc Communications and Networks (SECON); 01/2011  SourceAvailable from: tudarmstadt.de
Conference Paper: Quality of Information in Wireless Sensor Networks: A Survey
In Proc. of The International Conference on Information Quality (ICIQ); 01/2010
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MPM: Map based Predictive Monitoring for
Wireless Sensor Networks?
Azad Ali, Abdelmajid Khelil, Faisal Karim Shaikh, and Neeraj Suri
Technische Universit¨ at Darmstadt, Hochschulstr. 10, 64289 Darmstadt, Germany.
{azad,khelil,fkarim,suri}@informatik.tudarmstadt.de
Abstract. We present the design of a Wireless Sensor Networks (WSN)
level event prediction framework to monitor the network and its op
erational environment to support proactive self* actions. For exam
ple, by monitoring and subsequently predicting trends on network load
or sensor nodes energy levels, the WSN can proactively initiate self
reconfiguration. We propose a Map based Predictive Monitoring (MPM)
approach where a selected WSN attribute is first profiled as WSN maps,
and based on the maps history, predicts future maps using time series
modeling. The ”attribute” maps are created using a gridding technique
and predicted maps are used to detect events using our regioning al
gorithm. The proposed approach is also a general framework to cover
multiple application domains. For proof of concept, we show MPM’s
enhanced ability to also accurately ”predict” the network partitioning,
accommodating parameters such as shape and location of the partition
with a very high accuracy and efficiency.
Key words: Predicitve Monitoring, Time Series Analysis, Wireless Sen
sor Networks, Event Prediction
1 Introduction
Wireless Sensor Networks (WSN) typically entail an aggregation of sensing/
communicating sensor nodes to result in an ad hoc network linking them to the
base station or sink. The sensor nodes typically possess limited storage and com
putational capabilities and require lowenergy operations to provide longevity of
operational time.
WSN’s are often used for monitoring of spatially distributed attributes such
as detection of physical events, e.g., fire, temperature gradients and high/low
pressure. To maintain the required WSN dependability, network events such as
partitioning of the network are also to be detected. The reporting of such events
beyond simple monitoring becomes highly useful if these events can be predicted
in advance. Consequently, appropriate autonomic actions could be taken either
to avoid or delay events from happening by triggering self* actions.
Varied works exist for event detection [1] such as fire detection [2] and network
partitioning [3–5]. Most of these efforts are specialized for specific scenarios.
?Research supported in part by HEC, MUET, EC INSPIRE, EC CoMiFiN, and DFG
GRK 1362 (TUD GKMM)
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2Azad Ali et al.
Some consider generic scenarios [6], but they suppose the event to take certain
shapes and patterns. Also most efforts focus on detecting the events after they
have already happened. In the perspective of reacting to the event, it may be
too late to detect the events after it has already taken place. It would be far
more practical if we could predict the event occurrence.
Multiple efforts exists for predictions [7,8,11–13]. However, most are either
limited to predicting specifically a certain attribute like energy or provide only
node level shortterm prediction for data compression to minimize data to be
reported from the network.
It is natural to combine prediction of physical and network events with gen
eralized event detection to take proactive actions. To the best of our knowledge
there exists no work that proposes generalized event prediction. In this paper
we develop a framework to predict the future states of the network for the at
tribute of interest such as temperature or residual energy. Based on the predicted
states of the network we develop a generalized event detection technique. In this
work we target longterm predictions that require more computational resources
than a sensor node has and sufficient history of the attribute that contains all
variation patterns. Hence, we collect many profiles of the network and perform
the modeling operations on the sink. Typically, the sink uses optimized data
collection techniques to collect such a history of the interested attributes from
the network [8]. We refer to the collection of attribute values or samples from
network as profiling.
In the WSN environment, events usually are defined as spatially correlated
attribute distributions [6]. Thus, the spatial distribution of attribute of interest
needs to be quantified in order to detect events. One such natural spatial quan
tification is the Map. For a WSN an eMap is an energy map that represents the
current residual energy of the network [9], or tMap for temperature etc. Maps
can generally be created for any attribute and provide a basic utility to detect
events using pattern matching [6]. The use of maps in our work emphasizes the
fact that there is a wide class of events in which a discrete view of a network
on node level is either not necessary or not efficient. Rather, a view at a higher
abstraction level of regions and maps, that represent a group of nodes, is needed.
For example in temperature, pressure, humidity and residual energy monitoring
a map is far more meaningful than discrete node values. This also relates to the
inherent WSN node redundancy that leads to spatially correlated node states.
On this background this paper presents three specific contributions, namely
– Design of a generalized framework for sink aided profile prediction. The frame
work is adaptable to different simple as well as complex physical and network
events.
– Development of a regioning technique to detect events from predicted profiles
to support autonomous actions in the specified regions of the network.
– As validation for our Map based Predictive Monitoring (MPM) approach, we
propose a solution of network partition prediction as a case study.
The remainder of the paper is organized as follows. Section 2 discusses the
related work. Section 3 gives the preliminaries. Section 4 details our MPM ap
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Map based Predictive Monitoring of WSN3
proach for predictive monitoring in WSN. In Section 5, the case study is given. In
Section 6, we evaluate our approach for the given case study through simulations.
Section 7 concludes our work and outlines future directions.
2 Related Work
A variety of work is available for event detection [1]. The most relevant work to
our event detection strategy is [6] that investigates map based event detection
and requires the user to (a) specify the distribution of an attribute over space
and (b) the variation of distribution over time incurred by the event. They fur
ther define three common types of events namely Pyramid, Fault and Island. Our
work is independent of event shape due to our regioning technique. Furthermore,
our framework predicts the events rather than just detecting it. In [10], Banerjee
et al. present a technique to detect multiple events simultaneously. They employ
a polynomial based scheme to detect event regions with boundaries and propose
a data aggregation scheme to perform function approximation of events using
multivariate polynomial regression. Our work in addition to the capability of de
tecting multiple events, can predict events beforehand. Various other works exist
that address specific event scenarios like partition detection [3], fire detection [2]
and others [1]. These specific solutions do not feature portability to adapt to
different application scenarios.
There is a variety of work to monitor WSN’s and to minimize the overhead
of data collection for the monitoring. Strategies in
power consumption in WSN. In [12] Mini et al. propose a network state model
and use it to predict the energy consumption rate and construct the energy map
accordingly. In [13], the authors focus on predicting energy efficiency of multi
media networks. These works concentrate on predicting specifically the energy,
also they do not provide any extension of their work to attributes other than
energy. Authors in [14] give a theoretical framework for abstracting the world
as WSN maps. Our work in this paper however presents a practical approach
not only for abstraction but also facilitates to predicts the events to take place
in future.
As we present a case study for partition prediction, hence here we discuss
the related work in this respect. In [3], this problem has been addressed for a
subclass of linearly separable partitions, i.e., cuts. Memento [4] is a health mon
itoring system that suggests to continuously collect connectivity information at
the sink to be able to detect network partitioning. The Partition Avoidance Lazy
Movement protocol for mobile sensor networks [5] is a decentralized approach,
where a sensor node can locally suspect network partitioning and move to avoid
it. A node periodically collects the position of all its neighbors and checks if at
leat one neighbor is located in a small angle towards the sink. If no neighbor is
located in this ”promising zone”, the node suspects network partitioning. Based
on our event prediction framework as an example we propose a solution that is
generalized and is not dependent on the shape, size or location of the partition.
Moreover, we provide prediction of the time, when network partitioning is going
to happen.
[7] and[11] predict the
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4Azad Ali et al.
3 Preliminaries
We now describe the system model, the requirements driving our approach and
give basic definitions.
3.1 System Model
We consider a WSN composed of N static sensor nodes and one static sink. Sen
sor nodes are battery powered and usually entail limited processing and storage
capabilities. Sensor nodes are assumed to know their geographic position either
using distributed localization methods [15] or GPS. A typical WSN deployment
may contain hundreds of sensor nodes with varying densities according to the
coverage requirements. In this work, we do not consider a particular node distri
bution. We assume all sensor nodes to be homogeneous. Hence the nodes have
the same transmission range R and same initial battery capacity. We consider
that nodes crash due to energy depletion only. We assume the events for pre
dictions to be happening over a longer period of time, for example, events that
may take hours, days or even months to develop. We consider that events are
not spontaneous, spatially correlated, do not depend discretely on a single node
and are predictable.
3.2 Requirements on the MPM
We identify the following three requirements on the MPM. First, the MPM
should be lightweight, i.e., its creation, management and usage require minimal
resources with respect to energy. Second, we desire the MPM to longterm predict
the network status and hence the events accurately. Depending on the context of
the problem, longterm may mean hours to days or even months that should be
enough for the preventive mechanism to activate a self* mechanism to support
autonomic actions. Third, we desire the framework to be generalized to adapt
to prediction of varied event types.
3.3 Definitions
Here we give some basic definitions that are necessary to develop the MPM
framework.
Definition 1 A time series is a sequence of data points xtconsidered as a sam
ple of random variable X(t), typically measured at successive times. The time
series can be modeled to predict future values based on past data points.
Definition 2 A stationary random process exhibits similar statistics in time,
characterized as constant probability distribution in time. However, it suffices to
consider the first two moments of the random process defined as weak stationary
or wide sense stationary (WSS) as follows:
1. The expected value of the process (E[X(t)]) does not depend on time. If mx(t)
is the mean of X(t) then
E[X(t)] = mx(t) = mx(t + τ) ∀τ ∈ ?
2. The autocovariance function for any lag τ is only a function of τ not time t
E[X(t1)X(t2)] = Rx(t1,t2) = Rx(τ,0) ∀τ ∈ ?
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Map based Predictive Monitoring of WSN5
Definition 3 X(t) is an Autoregressive Moving Average Process ARMA(p,q)
process of order (p,q) p,q?ℵ, if X(t) is WSS and ∀ t,
X(t) = φ1Xt−1+ ··· + φpXt−p+ θ1Zt−1+ ··· + θqZt−q
where Ztis white noise with mean zero and variance σ2, denoted as WN(0,σ2).
Definition 4 To quantify the continues space of WSN and construct the map a
grid is virtually placed over the WSN field and each grid cell represents the aggre
gated attribute of all the nodes located within the grid cell. We define the process
of network space quantification as Segmentation and resultant quantification as
Grid Map.
Definition 5 The quantification of WSN space and the conversion of a WSN
to a map level abstraction is the key to detecting generic events. The abstraction
of WSN as map, transforms the event into a region of a map as shown in Fig. 1.
For example an event of fire will be a region of a map in which the value of
temperature exceeds a given threshold. In our framework we define an event as a
region of map whose values fall in the range of attribute values for which event
is defined.
(1)
4 Predictive Monitoring: The MPM Approach
We present here our MPM Framework that can be used to support self* actions.
To keep the event prediction as generic as possible, we have proposed it as a four
phases process. In each phase we have proposed a technique that is independent
of the attribute to be monitored. It is important to highlight that the use of
proposed techniques in the framework does not imply to limit the framework
to only these, rather for a particular implementation specialized techniques can
always be easily accommodated due to its modular structure. The four phases
of the framework are summarized here. The segmentation phase specifies the
properties of the grid maps (Def. 4) such as grid cell size. In the data collection
phase, data is periodically but efficiently fetched from the network on the sink.
The prediction phase is used for predicting future status of the network in the
form of future grid maps. The event detection phase is used to detect events
(Def. 5) in the predicted grid maps. These phases are individually detailed in
the following sections.
4.1 The Segmentation Phase
In order to reach an acceptable spatial resolution with higher level abstrac
tion of network as a map, we considered virtual grids and Voronoi diagram [16]
techniques to segment (Def. 4) WSN space. Voronoibased segmentation depends
only on sensor node distribution and is static for a given node distribution. How
ever, we require a segmentation strategy that allows variable spatial sampling to
accommodate both the physical and network parameters. Such variability allows
to investigate prediction accuracy and profiling efficiency tradeoffs. Grid allows
such flexibility therefore, we base our segmentation on grid.
The virtual grid or simply grid divides the WSN area into fixed size squares
or grid cells as shown in Fig. 1. Thus nodes that fall within a cell are grouped.
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6Azad Ali et al.
For the grid maps construction, two parameters must be specified. The first
parameters is the grid cell size γ, which is a spatial sampling or resolution param
eter. The second parameter is the aggregation value ξ that a grid cell represents.
Both parameters are essential for event detection. γ defines the geographic area
covered by the grid cell. The number of nodes being grouped in a grid cell is de
pendant on γ. It can also be seen as a zooming parameter. Hence it can be used
to decide at which level the user intends to detect the event, i.e., very detailed
(zoomedin) level of node or an overview at the level of regions. Depending on
the application the appropriate value of γ is affected by (1) physical parameters
such as attribute’s spatial distribution, (2) network parameters, such as commu
nication range, (3) application requirements such as the (zoom) level at which
to detect the event. In applications such as temperature and humidity, the grid
size can be selected big enough that it represents the patches of the geographic
areas, each differing considerably in the attribute values.
The grid cell value ξ is an aggre
gate of the attribute values of the
set of nodes in a cell. The choice of
the exact function depends on the
application. For example to sense
temperature or pressure, it is most
appropriate to average the values of
the nodes in the grid cell. If ξij is
the grid cell value in the (i,j)thgrid
cell gijand vnrepresents attribute
value of node n in gijthen ξijis an aggregation function such as average, min,
max of vn
ξij= f(vn) ∀ n ∈ gij
During map construction the nodes undergo coordinate transformation. If a node
has (x, y) coordinates then the grid cell coordinates (i, j) can be calculated as
WSN Spatial
Distribution
X
Y
Regions
Predicted
Collected
Maps
Maps
Grid Cell Value
Fig. 1: Temporal stack of the grid maps
(2)
i = ?x/γ?,j = ?y/γ?
(3)
All the nodes in the area of a certain grid cell have same grid cell coordinates.
We do not impose assumptions on the selection of γ and f, highlighting the
generality of our framework (requirement on our framework). An illustration for
the selection of both parameters is given in the case study in Section. 5.
4.2 The Data Collection Phase
In this phase the profiles are obtained by collecting the attribute values from
the network for which the event is to be predicted. The grid technique also aids
in reducing the number of the reporting nodes by forming clusters of nodes.
Once the grid size has been specified (and disseminated) in the network, the
nodes calculate their corresponding grid cell using coordinate transformation.
Consequently, innetwork aggregation can be performed according to f by the
nodes within the same grid cell. Only single node should then report ξ to the
sink. This reduces the amount of data to be reported to the sink enormously.
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Map based Predictive Monitoring of WSN7
Simple criteria can be used for selection of reporting node such as, node with
maximum energy. It is not efficient even for reporting nodes to send the ξ values
periodically to the sink. To further reduce the overhead on the network as per
our design requirements, the aggregated attribute are modeled as in [8] and only
the model parameters are sent to the sink, which can be used to regenerate the
actual data. The model is recalculated only when the prediction error exceeds
a predefined threshold. With each model update the outliners of the last model
are also reported. To achieve this the reporting nodes maintain a limited history
of the aggregated attribute values to be reported to the sink and fits 3rdorder
autoregressive model (AR3) which is only a particular case of the ARMA(p,q)
model (Def. 3), when p = 3, q = 0. This data compression approach uses fixed
simple model because nodes have too low computational resources to determine
the parameters of a general ARMA model.
As per requirement on our framework we emphasize the generality of MPM
with respect to data collection, since we abstract the attribute type as a generic
time series (Def. 1). Using this technique any attribute type can be reported
to sink with a comparable efficiency, making our framework independent of the
attribute type to be reported. We fulfill our first requirement of framework to
be lightweight using the segmentation and data collection techniques. These
techniques collectively reduce the overhead on the network tremendously.
4.3 The Prediction Phase
The models received on the sink in data collection phase from each reporting
node are used to regenerate the variation patterns or attribute history through
reverse transformation. The regenerated history is essentially the grid map rep
resentation of the WSN. This forms a temporal stack of the grid maps as shown
in Fig. 1. Each grid cell in the grid maps stack can be treated and modeled
as a separate time series for prediction. Individual models of each grid cell can
then be used to predict future values by fitting a prediction model, effectively
predicting grid maps. The time series can be modeled in different ways [13]. In
this paper, we use the widely used time domain modeling because of its general
applicability.
It is important to point out that time series modeling performed in this
section is different from that done on node level for data collection, which is
only shortterm prediction to compress the data using the fixed AR3 model. In
this phase, we perform a full scale modeling of the collected data to predict the
future states using the complete history and model each component separately.
Modeling Time Series: A time series X(t) can be modeled as a process
containing following components
X(t) = Tt+ St+ Rt
(4)
where Tt is a trend, St is a function of the seasonal component with known
period, and Rtis the random noise component. To keep the notion of generality
valid for the framework we use a well known generalized technique BoxJenkins
Model to model a time series containing any of these components.
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8Azad Ali et al.
BoxJenkins (BJ) Model: BoxJenkins model predicts a time series by fitting
it an ARIMA process (Autoregressive Integrated Moving Average). The term
integrated here means differencing the series to achieve stationarity (Def. 2).
To fit an ARIMA process the model and the order of the model needs to be
specified. The BJ model provides a guideline to select the appropriate model,
i.e., either Autoregressive (AR, Eq. 5.1) or Moving Average (MA, Eq. 5.2)
X(t) = φ1Xt−1+ ··· + φpXt−p{5.1}, X(t) = θ1Zt−1+ ··· + θpZt−q{5.2} (5)
or combination of both, i.e., ARMA process as given in Eq.1. It also gives the
guideline for the model order selection. BJ modeling is a four steps procedure:
i.) Data Preparation: As BoxJenkins model requires a time series to be
stationary (Def. 2). If it contains trends and seasonal components then these
should be appropriately removed. This can be achieved by either Least Square
Polynomial Fitting (LSPF) or differencing as X(t) = X(t) − X(t + u). For a
simple linear trend, u is 1. For higher order trends or seasonal component of
period s, u equals s. This operation is repeated until stationarity is achieved.
ii.) Model Identification: At this stage runsequence plot or Autocorrelation
Function (ACF) can be used to identify the stationarity of the time series and
the order of the AR model. ACF for k lag is given by
?Xi−¯ X??Xi+k−¯ X?
N
where¯ X is the mean value. Nonstationarity is often indicated by an ACF plot
with very slow decay. Order of the AR and MA models are determined with the
help of ACF and Partial Autocorrelation Function (PACF) [17]. To automate the
model selection process either Akaike’s Information Criterion (AIC) or Akaika’s
Final Prediction Error (FPE) [18] can be used. Various models can be computed
and compared by calculating either AIC or FPE. The least value of AIC or FPE
ensures the best fit model.
iii.) Parameter Estimation: In this step the values of the ARMA model co
efficients that give the best estimate of the series are determined. Iterative tech
niques are used for model parameter estimation [18].
iv.) Prediction: Once the modeling is complete, it is simple to predict the
series values using the estimated model. It comprises of calculating the future
values at next time instances and reversing all the transformations applied to
the series in phase 1 for data preparation.
To fulfill our second requirement of longterm prediction and to accommodate
any variation patterns in the time series we use a generic time series modeling
techniques that models each component (trend, seasons, random) to facilitate
longterm predictions. We keep the generality of framework valid also in this
phase by abstracting the attribute to be predicted as time series, which makes
this phase also independent of attribute type.
ρk=
?N−k
i=1
?i=1
?Xi−¯ X?2
(6)
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Map based Predictive Monitoring of WSN9
4.4 The Event Detection Phase
From Def. 5 we know that events appears as regions in a map. We introduce
here a generalized regioning technique that can detect the regions and their
perimeters which leads to generic event detection.
The regions are formed due to the fact that the attribute values fall into cer
tain class of values. For example we normally classify the temperature as freezing,
low, normal, high or very high. These classes also contain event class (range of
values belonging to event such as temperature above 90ofor fire). This gives
us a more acceptable abstraction than the exact values themselves. Therefore
the thresholding of values into classes becomes logical representation for event
detection. Thus to detect these events we define the class maps that thresholds
the exact values of the cells in grid map with their class denominations. If we
define class map values C as c1, c2,··· for the range of the values of grid cell gij
between (ξ2,ξ1] and (ξ3,ξ2]··· respectively, then a class map value is defined by
Our regioning algorithm (Fig. 2) takes the grid map as input and determines
perimeter and regions belonging to different classes and hence events. We refer
to the resultant output as the regions map. The grid cells in the grid map that
belong to the same class are grouped to form the regions. The regioning technique
essentially needs a class map to group all the same class cells and determine the
boundary. The process of converting to class map and determining the regions
boundary are both carried out concurrently. In order to merge the cells into
regions, we define attribute classes as in Eq. 7. Neighboring cells are merged
to form the same region if they belong to the same class. The definition of
attribute classes and fusion of same class grid cells makes the regioning algorithm
independent of the shape that a region takes or the number of regions (hence
the number of events) in the map.
C =
c1 if ξ2< ξij≤ ξ1
c2 if ξ3< ξij≤ ξ2
···
(7)
15: for each neighbor in neighborsList[] do
16: if (currentCell & neighbor are in the same class)
& neighbor in regionsMap[]==1 then
17: neighbor in regionsMap[]=regionId;
18: include cell in the newRegionBorder[];
19: changeInBorder=1;
20: end if
21: end for
22: if currentCell and (1 or more) neighbors are not
in the same class then
23: add cell to newRegionBorder[];
24: end if
25: end for
26: regionBorder[]=newRegionBorder[];
27: until changeInBorder
1: (mapX, mapY)=dimensions(map);
2: regionsMap[]= 1;
3: mapBorders[][];
4: while there is cell not assigned region yet do
5: regionBorder[]= (Find cell with regionsMap=1)
6: dilateRegion(map,regionBorder[],regionsMap[],regionId)
7: mapBorders[regionId][]=regionBorder;
8: regionId++;
9: end while
10: dilateRegion(map,regionBorder[],regionsMap[],regionId)
11: repeat
12: changeInBorder=0;newRegionBorder[]=0;
13: for (currentCell=each cell in regionBorder[]) do
14: neighborsList[]= eightNeighborsOf(currentCell);
Fig. 2: Regioning algorithm
The algorithm starts by defining all the cell as not assigned a region by
initializing the variable regionsMap[]=1. The algorithm next searches a grid
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10Azad Ali et al.
cell that has not been assigned a class yet and lists it as the border of the region,
as the region itself and region border at this moment consists of a single cell (line
5, cell with 1 in regionsMap is not assigned a region yet). Algorithm then starts
expanding/dilating the region (line 6). To expand the region, the neighboring
eight cells around the this region cell are checked if they already belong to the
any class (line 1416), if not then they are also classified according to Eq. 7. If
they belong to the same region they are assigned the same region ID and the
new qualifying cells are listed as the region border, otherwise the previous cells
retain their status as region border (line 1718). To further expand the region
neighboring cells of each cell in the border cells are searched iteratively until
no change occurs in the border of the region (line 11,27), which implies the
completion of the construction of a single region with its boundary. The whole
process repeats again by searching a new cell that has not been assigned a region
yet. It keeps on repeating until all the cells in the map are classified into their
corresponding regions (line 4,9).
We maintain the generality of the framework by devising a technique that
does not depend on shape, size or the number of events occurring in the WSN.
5 Case Study: MPM Adaptation for Predicting Network
Partition
To use our framework for network partition prediction, the problem needs to be
formulated according to the abstractions (maps, classes etc.) in the framework.
5.1 Problem Formulation
Partition detection is a complex problem as a physical and a network parameter
are being coupled, i.e., energy level of the nodes and communication range neces
sary to maintain connectivity. Given that sensor nodes are resource constrained,
eventually a WSN has to consider the depletion of node batteries leading to
the partitioning of the network. The energy dissipation however, is generally
spatially correlated. Therefore, groups of nodes form hotspots that deplete to
coverage holes. A hole can be defined as a part of the network, which due to the
energy depletion is no longer covered. These holes can sometimes disconnect a
part of network from accessing the sink defined as a partition.
If the network energy state can be modeled and predicted, then we can pre
dict the occurrence of the holes and consequently the partitions. The holes and
partitions appear as regions in an energy map. Our framework has all the tools
to profile the energy dissipation patterns, predict the network future energy state
and detect the regions formed due to partitioning. Therefore, partition prediction
becomes a natural candidate problem to be solved using our framework.
We can now define the problem according to the abstraction of our frame
work. A grid cell gets disconnected from the network if it has energy below a
minimum threshold so that it can not communicate anymore. These depleted
grid cells form a region that represents a hole in an eMap. Partition however,
is a group of nondepleted grid cells that can not access sink due to the holes.
It is therefore sufficient to profile the energy status of the network during its
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Map based Predictive Monitoring of WSN11
lifetime by collecting the eMaps in order to predict network partitioning. As per
definition the adaptation of the MPM framework to predict network partitioning
consists of four phases that we discuss as follows.
5.2 The Segmentation Phase
R
2 2
R
γ<
Fig. 3: Max grid
size
The first step towards the abstraction of the WSN net
work as a grid map (eMap in this case) is the selection of
resolution ,i.e., grid cell size at which this event (network
partitioning or holes) is to be detected. From the formu
lation of the problem we know that we have two coupled
parameters, i.e., energy and communication range. There
fore, an upper bound for γ is the communication range
(R). To accommodate a worst case scenario of two nodes
lying on opposite corners of two grid cells the γ < R/2√2,
as shown in Fig. 3. The lower bound can be obtained from
the node density, it should be selected such that the network area is not over
sampled, as we show in simulation Section. 6.2.
5.3 The Data Collection Phase
Once the grid cell size is specified, the nodes determine their corresponding grid
cells using Eq. 3. As we discussed earlier energy dissipates in a spatially corre
lated manner, the hotspot energy dissipation model is considered [9]. Therefore,
the nodes in a grid cell are expected to have similar energy dissipation pattern. A
cell is connected to the network until at least a single node has enough energy to
communicate. The node having the highest energy level is selected as reporting
node and Eq. 2 becomes
ξij= max(vn) ∀ n ∈ gij
All reporting nodes start aggregating the energy values. AR3 model is fitted to
the data as per the scheme given in Section. 4.2. Model parameters are sent to
the sink. The sink regenerates the time series (data of reporting sensor node) by
applying reverse transformation. The regeneration of the data of the reporting
nodes actually generates the grid maps according to the given parameters.
(8)
5.4 The Prediction Phase
The time series of each grid cell is modeled and predicted as described in Section.
4.3. Energy dissipation is a decaying process so the time series contains trends
but no seasonal components. The trends are removed by fitting polynomials.
ARMA models are fitted to random components, selecting the best fit model
using AIC criteria. After completion of modeling the grid cell values are predicted
and hence the future grid maps.
5.5 The Event (Holes/Partition) Detection Phase
The regioning algorithm developed in Section. 4.4 is used for both partition and
hole detection. As per the given scheme we define two energy classes at 10% and
below as the partition (or hole) class, and above 10% as nonpartition class. This
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12Azad Ali et al.
definition of energy classes gives the areas that are vulnerable to partitioning
because of low energy. The regioning algorithm detects these regions along with
the perimeter. The regions with energy 10% and below are holes in the network.
To detect partitioning however, the algorithm does not need to find all the
regions, therefore it executes only two iterations. In the first iteration with two
energy classes it starts from the area connected to the sink and merges the non
partitioned area around it. In the next iteration the classes are omitted and rest
of the area is merged to find its perimeter that represents the partitioned region.
6 Evaluation  Viability of our Approach
To evaluate how well our framework meets to design requirements, we evaluate
it for the problem of partition prediction as formulated in the case study. To de
termine accuracy and efficiency of MPM we compare it with ideal case situation
in which the data from all nodes is assumed. In the ideal case we predict the
future energy states of every node separately and hence the future profiles of the
network. The future profiles are then converted to maps. We denote these maps
predicted using profiles of all the nodes as ideal grid maps Gi. We denote maps
generated through our approach as optimized grid maps Go.
6.1 Evaluation Metrics
The transformation of a value spatial distribution into a map is a three stage
process, i.e., a grid map, then a class map and finally a regions map. The regions
map is physically same as a class map with additional information of region
perimeters. Hence, we use two error criteria for the grid map and the class map.
We use two more metrics to assess the accuracy of regions and efficiency in
terms of number of fetched packets from the network. Our first metric is the
mean sum of absolute error (Eq. 9.1) between the reference grid map Gr
(the actual data generated on the nodes) and the test grid map (Gi and Go),
defined as
?
m
where (i,j) are grid cell coordinates, Ge is the mean sum of absolute error, ξrijis
the grid cell value of reference map and ξtijis the grid cell value of test grid map
and m is the number of occupied grid cells. Gr is the true data generated on the
nodes, while Gi and Go are the gathered data from network, which undergo local
modeling and hence will deviate from true data due to modeling. Ge determines
the relative accuracy of our approach against the ideal case.
The second metric misclassification cell count (Eq. 9.2) counts the num
ber of misclassified cells between the reference and the test class map. Ce is the
total count of class cells that differ between the reference Cr and test class map
Ct (Ct are ideal class map Ci and optimized class map Co). ’count’ function
returns ’1’ if the two cells do not belong to the same class else it returns ’0’. Ce
is the direct measure of correct classification of the grid cells into the classes and
indirect measure of the accuracy of area and perimeter of the detected event area.
Ge =
i
?
jabs(ξrij− ξtij)
{9.1},Ce =
?
i
?
j
count(Crij− Ctij){9.2} (9)
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Map based Predictive Monitoring of WSN 13
Our third metric is the misclassified cells percentage for each region to assess
the accuracy the framework on regions level that we call regional percentile
error.
The fourth metric message count is the efficiency metric, where we count
the messages required to profile the network.
6.2 Simulation Settings
As three phases of the framework are carried out on the sink, therefore we
performed our simulations on Matlab. It is a very well known simulation
tool and suits our work as it facilitates to model energy dissipation pat
terns of very huge number of nodes. The network that we used in our sim
ulations is generated as a random nonuniform distribution of nodes. The
node distribution, as shown in Fig. 4, was selected to cover many possible
scenarios in a real deployment. It contains some areas with high node den
sity and some with low node density. It also contains two narrow bridges
between two parts of the network that may lead to network partitioning.
For energy dissipation modeling the common
hotspot model [9] was used. The energy dissi
pates in a spatially correlated manner around
the hotspot. The nodes nearest to the hotspot
are more active and hence dissipate more en
ergy. The parts of the network that act as
the coveragebridge between two parts of the
network and around the sink show relatively
high energy dissipation rates. Subsequently
these areas are modeled as hotspots.
We used a network containing 5000 nodes that span an area of 50 × 100
unit2, with each node having a communication range R = 2 units. For R = 2
the upper bound for grid cell size is 0.7 units. We found 0.3 as the lower bound
because if we take a grid size smaller than 0.3 then we have more occupied grid
cells than the number of nodes that over samples the network area. We therefore
selected three grid sizes between upper and lower bounds 0.3, 0.5 and 0.7 units.
Energy dissipation history of 164 profiles was collected for the ideal case from
all the nodes and then using our approach 164 grid maps were collected from
a subset of reporting nodes. To evaluate the statistics we divided the history
of profiles into two parts. 139 profiles were used for modeling purposes and 25
used for validation. 164 profiles represent the network lifetime history. If we scale
164 lifetime profiles to 164 days then 139 days of network operation are used to
predict the next 25 days network status. First, we considered the ideal situation
and 139 profiles of each node were used to predict next 25 states. Each ideally
predicted profile of the network was transformed to grid map. Then using MPM
approach 139 collected grid maps were used to predict next 25 grid maps.
Bridges
High activity
hotspot
Sink
Fig. 4: Node distribution
6.3 Simulation Results
Fig. 5(a) shows the graphs for mean sum of absolute error for 25 prediction steps
for 3 different grid sizes. It shows that the optimized grid maps are almost as
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14Azad Ali et al.
accurate as ideal grid maps. The mean error ranges from 0.6 to 0.9 even after
25 prediction steps, showing the level of accuracy of the prediction model. The
lower bound of error 0.6 is actually the maximum error that was allowed in the
local models on sensor nodes. The increasing trend is natural, as an increasing
number of prediction steps makes the prediction model less accurate. In ideal
case due to the cumulative error of considerably more number of nodes, the error
is slightly more than our approach.
Fig. 5(b) shows the misclassified cells count. The results of the mean sum of
absolute error imply that we can not expect much inaccuracy in misclassification
graphs. The highest count is naturally in the case of grid size 0.3, which touches
67 at the peak. The total number of occupied grid cells at this resolution is
4146, so a worst case misclassification of 67 cells accounts to less than 2% of the
total cells. We also see a slight increasing trend in the misclassification for each
prediction step because of the increasing error between model approximation
and the actual data. The peak in the graph gives interesting insight. We have
0 5 10
15 20 25
0.6
0.65
0.7
0.75
0.8
0.85
0.9
Mean Sum of Abs: Error
Prediction Steps
0.3 Grid, Gi
0.3 Grid, Go
0.5 Grid, Gi
0.5 Grid, Go
0.7 Grid, Gi
0.7 Grid, Go
(a) Mean sum of absolute error
Fig. 5: Predictability and accuracy measures of MPM
defined two classes of energy and as soon as the grid cells cross the class threshold
(10% of energy) they are classified into the partitioned class. From Fig. 5(a) it is
obvious that there is a minimum difference of 0.6 between the reference data and
test data, which creates a lag in the value of both data sets. This peak appears
when some cells in the actual data (Cr) cross the threshold of 10% but the cells
from the modeled data (Ct), due to the lag in value do not cross the threshold
at the same time. Therefore, many cells from the reference class are classified
in the partitioned class but corresponding cells in the test class are still in the
nonpartitioned class, which increases the count of difference cells. As soon as
the modeled data crosses the threshold this peak disappears but a slight trend
in increase of error continues.
Fig. 6(a) gives account of the error in the detected regions predicted through
profiles of the nodes and Fig. 6(b) is the regional error predicted through col
lected maps. To summarize the results we have selected prediction profiles/maps
separated by five prediction steps. On first prediction step there are only two
regions with less than 0.04% max percentile error. With each next prediction
step the number of regions increases and errors distribute between the different
regions. In the worst case scenario a region has a maximum percentile error of
less than 0.06%. The results however show that each region is very accurately
0
5
15
25
35
45
55
65
0
5
10 15 20 25
Misclassified Cell Count
Prediction Steps
0.3 Grid, Gi
0.3 Grid, Go
0.5 Grid, Gi
0.5 Grid, Go
0.7 Grid, Gi
0.7 Grid, Go
Max. error count
on class transition
(b) Misclassification cell count
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Map based Predictive Monitoring of WSN15
1
Regions
5
1
10
15
20
25
Prediction step
2
3
4
0
0.02
0.04
0.06
0.08
Regional Percentile Error
(a) Profile based prediction
Fig. 6: Misclassification percentile error per region
detected. Moreover our approach of maps that uses only a subset data is almost
as good as the profiles that consists of the data from all the nodes. Misclassifi
cation per region on the average is less than 0.03% which shows the accuracy of
our approach to detect the regions and their boundaries.
Now, we summarize the results w.r.t. efficiency metric, i.e., number of packets
needed for profiling. To profile the whole network of 5000 nodes and to collect 164
profiles for the entire lifetime, requires nearly 1 million data points. This overhead
is reduced dramatically by sending the models instead of raw samples. It is
further reduced to 14214 packets with the utilization of the gridding technique.
This equates to each node sending less than 3 packets instead of 164, which is
less then 1.8% of raw data to be collected. If the network is scaled to 100 nodes
the packets to be sent scale down to 284. These results satisfy our requirement
on the framework to be lightweight.
1
Regions
5
110
15
20
25
Prediction step
2
3
4
0
0.02
0.04
0.06
0.08
Regional Percentile Error
(b) Map based prediction
6.4 Discussion
The results obtained in the evaluation are in accordance to the design require
ments for the framework. It is very lightweight as data compression and gridding
cumulatively reduce the data to less than 1.8% of the raw data needed to profile
all nodes. The prediction is very accurate, represented by max prediction error
of approximately 0.06% in misclassification of the areas of the maps for 25 pre
diction steps. The 22 days (in scaled time as explained in Section. 6.2) earlier
prediction of partition is also feasible for proactive self reconfiguration, enabling
autonomicity of WSN.
Our framework ensures reliably accurate information for proactive action and
well before the event takes place. The proactive action can be triggered using
regioning algorithm results. If there exists an event region, it will be detected
by the regioning algorithm along with the perimeter of the event regions. The
regioning algorithm determines the perimeter of the holes and the partition, it
implicitly provides the exact information about the area, location and the af
fected nodes that lie within that perimeter. As soon as the sink successfully
detects an event, it can either trigger an early warning or initiate a proactive
action like move some nodes to affected areas, redeploy new nodes etc. if applica
ble. The MPM framework gives vital information beforehand to act proactively,
but this proactive action itself is beyond the scope of this work.