Going Beyond Nodal Aggregates: Spatial Average of a Continuous Physical Process in Sensor Networks

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Going Beyond Nodal Aggregates Spatial Average of
a Continuous Physical Process in Sensor Networks

• Simon Han, Ganeriwal Saurabh,
• Mani Srivastava
• simonhan, saurabh, mbs _at_ee.ucla.edu

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Aggregation

• Communication is expensive
• Compress data near source to reduce communication
  (e.g. max, average, etc)

MICA mote Berkeley
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Aggregation in Sensor network

• Strong coupling with the physical world
• Underlying physical process is going to be mainly
  continuous.
• Temperature, CO gas content, Precipitation etc.
• Deployment is going to be random and generally
  unknown to the end user.
• Typical Query will be of form
• Give me the average temperature in this room
• And not
• Give me the average temperature over the nodes
  N1, N2,, Nk.

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Nodal v/s Spatial Aggregation

• Problem Calculate the average reading over
  region R.
• Proposed solution of nodal aggregation
  Calculate it over the discrete set of nodes lying
  in region R.
• Results in inaccurate answer

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• Deficiency Missing notion of space

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Approaches of Spatial Aggregation

• What we need?
• The value of the underlying process throughout
  region R.
• What we have?
• A few samples of it in the form of sensor values
• Moreover they are non uniform in space.
• What can we do?
• Use interpolation to estimate values at virtual
  nodes on a uniform grid.
• Use classical results of generating the process
  from non-uniform samples.
• Fit a surface to existing available samples and
  calculate aggregate on that surface.

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Design Space

• Localized distributed algorithm
• Energy efficient
• Involve distance factor
• Physical phenomena
• Light weight
• Limited computation resource

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Spatial Interpolation Methods

• By Scope
• Global use all data points
• Local use limited set of data points
• By Fit
• Exact observed data points predicted exactly
• Approximated even observed data points predicted
  with error
• By Model
• Deterministic math model
• Stochastic probabilistic model

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Spatial Interpolation I

• IDW (Inverse Distance Weighting)
• zx Si wizi / Si wi , wi 1/dix2
• Local, exact, deterministic
• Distributed?
• Point association problem

• Trend Surface
• approximated by least-squares regression fit to
  data points
• Global, approximated, stochastic
• Aggregation not possible

Picture from http//skagit.meas.ncsu.edu/helena/g
mslab/viz/sinter.html
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Spatial Interpolation II

• Spline
• Fits a minimum-curvature surface through input
  points with piecewise polynomials
• Local, exact, deterministic
• Moderate computation

• Kriging
• uses a semivariogram, a measure of spatial
  correlation between two points, so the weights
  change according to the spatial arrangement of
  the samples.
• Local/global, exact, stochastic.
• Need to know the spatial correlation of physical
  process
• Computational intensive!

Picture from http//skagit.meas.ncsu.edu/helena/g
mslab/viz/sinter.html
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Spatial Interpolation III

• Triangulated Irregular Network
• Also known as Delaunay triangulation
• Models surface as a set of contiguous,
  non-overlapping triangle planes.
• Local, exact, deterministic
• May require global knowledge

• Voronoi Polygons
• Collection of all points that are closer to known
  point than any other points
• Local, exact, deterministic
• May require global knowledge
• Light computation

Picture from http//skagit.meas.ncsu.edu/helena/g
mslab/viz/sinter.html
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Spatial Average Voronoi Approach

• Use the area of Voronoi polygon as weight of the
  node and apply weighted average
• ?i Si(t) Ri(t) / ? Ri(t)
• Si(t) sensor reading in node i at time t
• Ri(t) the area of Voronoi polygon at node i
• Boundary condition?
• Use polygon clipping algorithm
• Localized and Distributed?
• Polygon clipping on Square Inscribed in the radio
  circle

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Spatial Average (Centralized)

• Get locations and sensor readings of nodes
• Construct Voronoi diagram
• Polygon clipping on the boundary
• Compute weighted average

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Spatial Average (Distributed)

• For each node
• Do neighbor discovery
• Construct Voronoi diagram
• Polygon clipping on inscribed square
• Compute polygon area
• Compute partial weighted sum and partial sum of
  areas

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Modes of Spatial Aggregates

• Centralized Snapshot (Centralized-s)
• When the frequency of query gt average sensor node
  life time
• Centralized Periodic (Centralized-p)
• When the frequency of query lt average sensor node
  life time
• Distributed
• Depends on the size of network and the number of
  sensor nodes in the network
• Depends on the height of aggregation tree and
  number of nodes as well!

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Energy Analysis Aggregation tree model

• LH(k) number of nodes in k level of aggregation
  tree
• L size of the field
• N number of nodes in the network
• r radio range
• LH(1) ?r2/(L2/N)
• LH(2) ?((2r)2 r2)/(L2/N)
• LH(k) ?((kr)2 ((k-1)r)2)/(L2/N)
• LH(k) (N?r2/L2)(2k 1)

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Energy Analysis Centralized-p

• Upon receiving query
• Every sensor node sends its node id and location
  to the user node. The user node calculates the
  Voronoi polygon area for every node.
• Ecp1 Ebit(Nsize(hdr)?k1L/2r
  size(idloc)kLH(k))
• The user node sends back the area of Voronoi
  polygon area to every node.
• Ecp2 Ebit(Nsize(hdr)?k1L/2r
  size(idarea)kLH(k))
• Aggregation
• Ecp3 Ebit(Nsize(hdridpartial_state))

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Energy Analysis Centralized-s

• Upon receiving query
• Every sensor node sends its location and data to
  the user node.
• Ecs1 Ebit(Nsize(hdr)?k1L/2r
  size(idlocvalue)kLH(k))
• The user node calculates spatial average using
  Voronoi approach
• Ecs2 0 (for centralized server)

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Energy Analysis Distributed

• Upon receiving query
• Compute the area of Voronoi polygon for the
  weight
• Ed1 t(Nneighbor)Pcpu
• Node sends partial weighted sum and partial sum
  of Voronoi area
• Ed2 Ebit(Nsize(hdridpartial_state))

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Centralized or Distributed?

• Snapshot
• Ecs1 gtgt Ed2
• If Ed1 is small, distributed version is better
• Periodic
• Ecp3 Ed2
• If Ed1 lt Ecp1 Ecp2, distributed version is
  better

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Simulation Model

• V(p,t) ?isources (kdist(i)1)-a V(p(i),t)
• V(p,t) the value of sensed quantity at point p
  at time t
• dist(i) the distance between point p and source
  i at time t
• p(i) the position of source i at time t
• Network size of 100 x 100
• Average over 100 random topologies and 200 data
  points for single topology.
• Simulates relative error between ground truth
  average and nodal/spatial average

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Varying diffusion model

• One fixed source and varying number of nodes and
  diffusion models
• Node density? Error?
• Always better than nodal average under quadratic
  and cubic models!

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Varying number of sources

• Quadratic diffusion model with fixed sources
• Outperform nodal average under different number
  of sources!
• Worst case lt 10 for sparse networks

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Varying mobility of sources

• One mobile source
• Clearly outperform nodal average with low
  standard deviation in relative errors

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Robustness to link failures

• One fixed source with quadratic diffusion model
• Using max distance between neighbors as radio
  range
• Robust to link failures!

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Simulation on precipitation data set

• Precipitation data from University of Delaware
  (Thank you, Deepak!)
• Randomly pick 50-300 points out of 2152 girded
  data set
• Result verifies simulation on physical process
  model

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Implementation

• On Berkeley Motes
• Time efficient algorithm
• Steven Fortune Voronoi
• Polygon clipping
• Greens Polygon area
• 22k flash, 4k RAM for 10 neighbors
• Memory efficient algorithm
• Dummy nodes placements to handle clipping
• Bisector node with each neighbor
• Greens Polygon area
• 11k flash, 1.5k RAM for 25 neighbors

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Performance Measurement

• Average over 100 set of random neighbors
• Memory efficient O(N2), but O(N logN) exist!
• Time efficient O(N), but no practical to Motes
• Voronoi computation is heavy in motes
• Floating point operations
• High memory constraint

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Energy Analysis Parameters

• Assume SMAC and Manchester coding
• Energy per bit include both RFM and
  microcontroller

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Energy Analysis L 100, Height 5
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Centralized vs. Distributed

• Under dense network or large aggregation tree,
  Centralized-p is energy efficient
• The curve can be used for query-time decision!
• Require number of nodes and height of aggregation
  tree

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Centralized-p vs. Centralized-s

• Periodic query can be done with Centralized-s
  mode.
• If frequency of the query is low (once per day)
• Or number of queries are small

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Integrating with Tiny DB

• Existing frameworks are for doing nodal
  aggregation.
• Calculating spatial aggregate
• Find the subset of nodes lying in region R
  (already available).
• Invoke the spatial_weight API.
• Send back the weighted sensor value.
• Flexible
• With the simple Voronoi based approach, spatial
  aggregation reduces to nodal aggregation!
• Not possible with other approaches like virtual
  estimation on uniform grid, non-sampling theory
  or even with other interpolation techniques like
  Kriging etc.

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Going beyond spatial average

• Problem Isobar mapping
• Proposed solution
• Divide terrain into regions with just one node
  per grid.
• Define partial state as this polygon.
• Use polygon combination algorithm.
• Deficiency
• How to form this grid?
• Requires every node to sends its coordinates to
  central entity.
• What if no nodes in a grid?
• Define them as holes and neglect them.
• Thus problem is redefined as finding regions of
  contiguous space with equal temperature and where
  sensor node exists
• Why does the answer has to depend on deployment

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Isobar mapping (spatial)

• Define polygon as voronoi cell.
• Integrates easily into existing framework
• We are just redefining the polygon
• Solves all deficiency
• Distributed and requires no communication.
• No holes.
• More accurate.

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Histogram

• Problem Forming the histogram of values of the
  process
• Proposed solution
• Redefine it to be histogram of sensor node
  values.
• Use simple aggregation algorithms.
• Deficiency
• Neglects density completely.
• Each sensor node value does not contribute
  equally (refer to earlier figure).
• Similar to spatial average
• Just weight the sensor values by their voronoi
  cell area.

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Is spatial aggregation valid always

• Sadly speaking NO!
• Process might be discrete
• Sensors deployed on doors to count the number of
  persons leaving entering the room.
• Monitoring system properties like battery life,
  connectivity etc.
• Depends on aggregates
• Max and min are almost similar in both the
  domains.
• Some aggregates might not be defined in spatial
  aggregation.
• There is no kth max of a continuous process.
• Have to work with percentiles.