Compressed Radio Transmission of Spatial Field Measurements by Virtual Sensors

Transcription

1 Compreed Radio Tranmiion of Spatial Field Meaurement y Virtual Senor Reiner Jedermann, Henning Paul and Walter Lang Atract The remote exploration or monitoring of an environment often include enor meaurement at multiple proe point and recontruction of the patial ditriution of the oerved phyical quantity y a regreion model. Epecially for long ditance etween the oerver and the environment, required data volume for tranmitting a parametric decription of the patial ditriution ecome critical. Simple phyical model or aumption of parametric ae function do not provide ufficient prediction accuracy. Statitically aed method for field recontruction uch a Kriging or Gauian Proce Regreion provide good accuracy, even if the meaurement are overlaid with noie, provided all enor data i tranmitted. The new method preented in thi paper calculate a mall et of quai optimal virtual enor poition located inetween the real enor. By tranmitting only the predicted value of thee virtual enor, the patial field can e recontructed with le tranmitted data and without ignificantly increaing the prediction error. The new approach wa verified in a imulation cenario for a temperature field caued y diffuion and advection phenomena, which yielded a data compreion y a factor of up to four. For large variation of the numer of enor and of the magnitude of meaurement noie, the prediction error wa alway lower compared with the parametric ae function model. I. ITRODUCTIO OITORIG of the patial ditriution of a phyical M property in remote environment that are difficult to acce for human can e done y wirele ytem, e.g., concentration of a chemical pollutant in a lake [], temperature and airflow in a chilled warehoue with nitrogen atmophere [], or geological or oil propertie, e.g. oil moiture in large irrigation ytem [3]. The typical information chain in thee application conit of the following element: See for official final verion. Thi reearch wa upported y the German Reearch Foundation (DFG) for the project In-network data analyi of patially ditriuted quantitie under grant Pa57/ and Je7/. For further information, viit R. Jedermann i with the Intitute for Microenor, -actuator and - ytem ( rjedermann@ima.uni-remen.de). H. Paul i with the Department of Communication Engineering ( paul@ant.uni-remen.de) W. Lang i with the Microytem Center Bremen ( wlang@ima.uni-remen.de). All three intitute are located at the Univerity of Bremen, Germany 6 IEEE The remote enor ytem collect data at multiple proe point y a wirele enor network or a moile enor or root. The meaurement are often overlaid with noie and local diturance. A gateway collect the enor data, pre-procee them, and end them over a long-ditance radio link to an oerver. The oerver carrie out a field recontruction y regreion or interpolation technique to predict the unditured, noie-free field value, for the proe location a well a for point in-etween. Epecially in extra-terretrial application, the required data volume that ha to e tranmitted over the radio link i a critical deign iue, which hould e reduced y adequate pre-proceing on the gateway. Parametric model have often een ued to reduce data from patial meaurement to a mall et of parameter y decriing the patial ditriution y a pare et of ae function. After etimating the parameter of thee function, the original meaurement data can e dicarded, reulting in a good compreion rate. The ae function can e elected according to a phyical model, e.g., a Green function for diffuion procee [4], [5], [6], [7]. Thi Green function i identical with the common Gauian Radial Bai Function (RBF) model [8], except for a time-dependent caling factor. Although thee model provide good data compreion, they introduce an additional error to the field recontruction ecaue real phyical procee cannot e reduced to imple ae function. In almot any real application, diffuion procee are overlaid with other phyical phenomena uch a the tranport of heat or chemical y advection and convection or geological procee during rock and oil formation. on-parametric model are more flexile to approximate different hape of the patial field. Thee purely data-driven approache take no aumption of the underlying phyical proce or curve hape, except for ome tatitical propertie. Kriging [9], [] and Gauian Proce Regreion (GPR) [], [] provide a etter fit to the data y etimating the field value a a linear comination of the meaurement, ut at the cot that the complete meaurement data et i required for field recontruction. Smoothing pline are another important type of nonparametric model to repreent the meaurement of a wirele network [3]. They approximate the meaurement y pline function located at a dene network of knot poition. Due to Pleae quote a: Jedermann, R.; Paul, H.; Lang, W.: Compreed radio tranmiion of patial field meaurement y virtual enor. In: 6 IEEE International Conference on Wirele for Space and Extreme Environment (WiSEE), Aachen, Germany, 6, pp (doi:.9/wisee ).

2 the high numer of required knot, they have alo limited capailitie for data compreion. In thi paper we preent a new approach that i aed on the non-parametric Kriging model to achieve accurate field recontruction, ut alo largely reduce the tranmitted data volume over the radio link y repreenting the meaurement y a mall et of o called virtual enor. The compreed decription of the field conit of the location coordinate of the virtual enor and the moothed field value at their poition. The algorithm calculate quai-optimal location of thee virtual enor to minimize the prediction error compared to field recontruction y the full data et. The term virtual enor ha een mentioned in literature occaionally [4], ut up to our knowledge, it ha not een applied for data compreion o far. Other previou work [5] employ compreion through random uperpoition of meaurement to a maller numer of value tranmitted on the cluter head communication link, ut thi approach require centralized Compreed Seningaed recontruction. In ection we hortly introduce the aic idea ehind the Kriging and RBF model, which erve a reference. Our new approach i decried in ection 3 with a focu on the trategy for etting the poition of the virtual enor. The new method i compared with the reference model y imulation tudie in term of compreion rate and prediction error in ection 4. II. MODELS FOR SPATIAL FIELD RECOSTRUCTIO For implicity, we retrict the model to a two-dimenional field, ut extenion to three dimenion i traight forward. We aume that the field f(x,y) generated y a phyical proce i overlaid with Gauian independent and identically ditriuted (i.i.d.) noie with RMS (Root mean quare) σ. The prediction g(x,y) of the model hould e a cloe a poile to the true field value f(x,y), which cannot e directly oerved. The accuracy of the model and it capaility to mooth the noie are evaluated y a quadratic error criterion (), calculated for r point, typically ditriuted on an equipaced grid. max r Re f = ( f ( xr, yr ) g( xr, yr )) r r= e () The etimation of the model ha to e aed on the enor meaurement m at max point at coordinate x and y, which erve a training data. During training, the following error criterion i minimized (): max ( m g( x, y )) ε T = () = The RBF model, for example, approximate the field with a um of RBF function with amplitude a i and width λ i at center x i, y i and a contant offet c. In total, 4 + parameter are required to decrie the field. g( x, y) = c ( x x ) + ( y y ) max i i + ai exp( ) i= λi Linear etimator uch a Kriging and GPR predict the field y a weighted average of the meaurement. The weighting factor ω depend on the coordinate of the target point at x,y (4). The numer of enor point ued y the model i often retricted to the k cloet neighoring enor of the target point to avoid high computation load. k g( x, y) = ω ( x, y) m (4) = The Kriging method calculate optimal value for the weighting factor ω, in order to minimize the expected quared prediction error for the related target point, under the condition that the oerved value reult from a econd-order tationary patial proce, i.e., the mean E{m i } i independent of location, and the correlation of any two point i,k depend only on their ditance d i,k according to the variogram function γ( ) in (5): {( m m ) } ( di, k ) = E i k γ (5) Ordinary Kriging additionally etimate E{m i }. Calculation of the ω i require inverion of a matrix with order k +. The matrix element at i,k are et to γ(d i,k ) y the ditance etween the neighoring enor point. The lat row and column of the matrix are filled with one, except for the lat diagonal element, which i zero. More detail can e found in earlier pulication [9]. For aniotropic variogram function, ee tatitical textook, uch a [6]. The crucial tep in applying the Kriging method i the etimation of the variogram y current or hitoric meaurement data. The variogram etimation y experimental data ha to e fitted y a conditionally negative definite model function. An example variogram model i given in ection 3C. The approach of GPR i imilar to Kriging, except that it decrie the patial correlation etween enor point y a covariance matrix intead of a variogram model. III. VIRTUAL SESORS Becaue the location of local field extrema i not known in advance, a high numer of enor are required to capture the patial ditriution of an unknown field. The aic idea of our approach i to identify a mall et of enor poition, which are ale to capture the field without ignificant lo of accuracy. In cae of moile enor node or meaurement device, mot enor can e witched off and only few of them can move to the identified optimal poition. In the (3)

3 more general cae of tationary enor, virtual enor can e imulated at the optimal poition. The virtual meaurement at thee poition are prediction y the regreion model. The field recontruction y the oerver i carried out in the ame way a for the tandard Kriging method, except that the weighted average in (4) i calculated for the virtual enor intead of the real enor poition. In the following, we name thi new approach a Virtual Kriging in contrat to Full Kriging with all enor poition. Thi approach overcome the diadvantage of the Full Kriging of requiring the tranmiion of all meaurement data for field recontruction. Intead only the location x v, y v and the predicted value p v at v virtual poition have to e tranmitted, with total 3 v parameter. The identification of the et poition of the virtual enor i an optimization prolem on a parameter vector Θ with ize v for the enor x v and y v coordinate. The remaining predicted value p v can e calculated y Ordinary Kriging with x v, y v and the value of the k cloet neighor a input. Sutitution of () and (4) into the optimization criterion for v virtual enor intead of k real enor reult in (6). The prolem i highly non-linear ecaue the weight ω v, a well a the p v value depend oth on a matrix inverion and the non-linear variogram function. θ = [ x y... x y ] * θ = arg minε ( θ, m ) = arg min φ φ T v m v = v= v ωv, ( xv, yv ) pv Small deviation of the virtual poition x v, y v can e tolerated without ignificantly increaing the training error ε T, a long a the predicted value for the virtual enor p v are corrected according to poition offet. See ection 4B for a enitivity analyi. Due to thi fact, it i poile to achieve at quai-optimal parameter Θ* with a heuritic earch. During an optimization loop, the parameter in Θ are optimized oney-one. A. Poition of the virtual enor For implification, we aume a quadratic hape of the oerved area. During initialization, the virtual enor were et on a quadratic piral with turn, with a ditance etween the virtual enor of the total length of the piral divided y v, ee figure for illutration. During the following optimization loop the x v, y v coordinate are adjuted one-y-one: The training error ε T i calculated for a mall hift of the elected coordinate y ± x,y. The new value for the elected parameter i given y the minimum of a parale through the ε T value for the unmodified point and the two ε T value for the hifted point. The maximum change of the elected parameter i limited to x,y to avoid intailitie. (6) After one optimization tep for all v coordinate, x,y i decreaed y % and the loop i repeated for ten time. Further loop rought only marginal improvement of ε Ref. y [m] Fig. : Example for initial etting and optimization of the coordinate of the virtual enor. Initial poition (lue quare) on a rectangular piral (green doted line). Trace of the virtual enor during optimization loop (lue line) and final poition (lue diamond). ima of the tet data for a - diminional field (red circle). B. Etimation of the RBF model A further method eide Virtual Kriging, we alo evaluated the accuracy of the RBF model, which require etimation of 4 + parameter (7). For implicity, we applied the ame optimization proce a for the virtual enor. In addition to the coordinate x i and y i, alo the width λ i and the height a i of each RBF function were adjuted oney-one per optimization loop. θ RBF = x [ c x y a y a λ λ... ] The contant offet c wa adjuted after each optimization tep to minimize the difference etween the average of the enor meaurement and the predicted field value at the ame coordinate. Alternate approache to etimate the parameter of RBF or related Green function can e found in [4], [5], [6] and [7]. C. Tet data The tet data were generated y a CFD (computational fluid dynamic) imulation of a -dimenional temperature field. In contrat to real meaurement data from field or laoratory tet, imulated data ha the advantage that full acce to the unditorted field value i provided and the numer and poition of meaurement point can e freely varied. The imulation cenario conit of three heat ource with x [m] (7)

4 a diameter of.4 m in a quare water ain of m m ize. The wall are iolated with an outide temperature of - C (Figure ). Cooling i enhanced y a mall flow of water with a peed of. mm/ at the inlet. Even at thi low flow rate, the effect of advection wa larger than that of diffuion. The influence of wall and the geometrical extent of heat ource contriute to further deviation from a imple diffuion model. Simulation were carried out with the CFD oftware COMSOL. The temperature in the water ain wa evaluated for randomly elected imulation point, a well a for a fixed grid of 5 y 5 point a a reference. For generating the required tet data, it wa ufficient to conider only the teady tate. The power of the heat ource were caled in a way that all temperature lay in the range etween - C and + C. Figure : Setup of flow and heat tranfer imulation with three heat ource inide a water ain. The variogram for the Kriging method wa etimated aed on the meaurement point. A good fit of the experimental variogram wa achieved with a model coniting of two Gauian function (8), whoe parameter depend on the noie RMS, e.g., n=.9 C, =. C, r =.5m, =.9 C and r =5m for σ =. C. 3 d 3 d r = + + r γ ( d) n ( n) e e (8) The optimization of the poition of the virtual enor wa carried out for different value of, σ and v. The average error of the model compared to the unditorted field ε Ref, wa calculated for imulation run for each parameter et. For each imulation run a new et of meaurement point wa randomly picked from the imulation data and i.i.d. noie with RMS σ wa added. The initial tep width for optimization wa et to x,y =.m. The firt point on the rectangular piral wa hifted y a random offet for each imulation run to avoid a ia y a fixed election of the initialization coordinate. IV. SIMULATIO RESULTS Our imulation were centered on a tandard etup with = enor and noie RMS σ =. C. The et poile accuracy compared with the unditorted field value wa achieved with Full Kriging with a prediction error of ε Ref =.673 C (Tale ). Virtual Kriging achieved at almot the ame error with ε Ref =.674 C with v =5 virtual enor. The error for the RBF model ε Ref =.88 C wa 3% higher for = ae function. A. Effect of the numer of enor, noie RMS and numer of virtual enor Figure 3 how the influence of the numer of enor on the prediction error. The error i for Virtual Kriging wortcae 7.8% higher than for Full Kriging at =5. For <, Virtual Kriging achieve at an even lower error. The Virtual Kriging model ha fewer degree of freedom and thu enforce a higher moothing of meaurement noie. The dependency of the prediction error from the noie RMS i hown in figure 4. Even under zero noie condition, all model entail a certain minimal error, which i.8 time higher for Virtual Kriging than for Full Kriging. Only for σ. oth error ecome almot the ame. Virtual Kriging wa in all cae etter than the RBF Model. Figure 5 how how the error decreae with the numer of virtual enor. For v hardly any difference wa oerved etween Virtual and Full Kriging. But to e afe for variation of and σ, we elected v =5. The RBF model achieved at an error of 9% higher than Full Kriging with a low numer of model parameter with =7, ut almot no improvement wa achieved y additional model parameter. [ C] Error RBF Kriging v =. C Full Kriging M a x = m a x = umer of Senor M a x Fig. 3. Influence of the numer of enor on the prediction error for RBF and Kriging model.

5 Error [ C] RBF = M a x Kriging v =5 m a x Full Kriging (full) = m a x B. Quantization of virtual enor poition In a final imulation, the poition coordinate of the virtual enor were rounded to different numer of it. The predicted value for the virtual enor poition wa calculated anew after rounding. The effect of thi quantization i hown in figure 6. Even with 6 it the error i only.9% higher compared to field recontruction aed on floating point value for the virtual enor coordinate. The lo of accuracy reduced to.4% for 8 it oie RMS [ C] Fig. 4. Influence of noie RMS on the prediction error for RBF and Kriging model. [ C] Error =. C = M a x RBF Virtual Kriging Full Kriging umer of Virtual Senor v or Bae function M a x Fig. 5. Influence of model ize on the prediction error for RBF and Kriging model. C. Compreion rate Tale how an example calculation of the achieved compreion rate for max = and σ =. C. The coordinate of the virtual enor are le enitive toward quantization and can e tranmitted in one yte. For all other model parameter we aume two yte. Full Kriging require tranmiion of all meaurement value. The variogram ha to e etimated on the gateway ide for Virtual Kriging, o 5 variogram parameter have to e added to the data volume. The required data volume to tranmit the parameter of the two reference model and of Virtual Kriging i given in tale. The RBF i ale to compre the data volume to.5% compared with Full Kriging with =, ut at the cot of a higher prediction error. Virtual Kriging reduce the prediction error to a value imilar to Full Kriging with a higher data volume of 6.5%. Tale : Prediction error and required numer of parameter for different model. Model Full Kriging Virtual Kriging RBF model Prediction error ε Ref Byte value max for m v for p v 4 max+ for p x,i, 5 for variogram p x,i, a i,,λ i and c Byte value - v for v x, v y - Typical model ize max= v = max= Data volume 4 yte 5 yte 8 yte Relative volume % 6.5%.5% [ C] Error m a x = =. C Quantization of virtual enor coordinate [Bit] Fig. 6. Effect of quantization of the poition of the virtual enor V. DISCUSSIO The patial ditriution of a phyical quantity cannot e exactly approximated y a model with a limited numer of parameter, o inaccuracie in field recontruction cannot e avoided. But in cae that the meaurement are overlaid with high noie, a it i often the cae in geological exploration project for example, the error of the field recontruction i maked y the noie. In our imulation tudie with a tandard deviation of the meaured temperature of ±.4 C, there wa hardly any difference etween the field recontruction aed on 5 virtual enor and y the full et of real enor if the meaurement noie RMS wa. C. If a higher prediction error i acceptale, the meaurement can e tranmitted y the parameter of an RBF model. Diadvantage of the RBF model i that it cannot compenate for the higher prediction error y increaing the numer of parameter. Even for 3 ae function, the error i till 5%

6 higher compared with Full Kriging a reference. But it ha the advantage that lowering the numer of parameter only lightly increaed the error until a minimum of 7 ae function with an error 3% higher than of Full Kriging. The equation of a Gauian RBF give only a perfect match for pure diffuion procee. Our imulation howed that the RBF model i capale to adapt to other phyical phenomena uch a advection y adding more ae function. In all imulation for different numer of enor and meaurement noie RMS value, the prediction error of Virtual Kriging wa alway etter compared to the RFB model. Virtual Kriging hould e therefore preferred for compreed tranmiion of the data from patial meaurement, even if the mathematical model i more complex and the required data volume wa 6% higher in our tet cenario. REFERECES [] Y. Wang, R. Tan, G. Xing, J. Wang, and X. Tan, "Accuracy-aware aquatic diffuion proce profiling uing rootic enor network," in Proceeding of the th international conference on Information Proceing in Senor etwork,, pp [] R. Jedermann,. Hartgenuch, M. Boryov, C. Lloyd, U. Praeger, M. Spuler, et al., "Spatial profiling of airflow condition in cold torage warehoue y wirele anemometer," preented at the 6th International Cold Chain Management Conference, Bonn, Germany, 6. [3] J. McCulloch, P. McCarthy, S. M. Guru, W. Peng, D. Hugo, and A. Terhort, "Wirele enor network deployment for water ue efficiency in irrigation," preented at the Proceeding of the workhop on Realworld wirele enor network, Glagow, Scotland, 8. [4] I. Dokmanic, J. Ranieri, A. Cheira, and M. Vetterli, "Senor network for diffuion field: Detection of ource in pace and time," in Communication, Control, and Computing (Allerton), 49th Annual Allerton Conference on,, pp [5] Y. M. Lu, P. L. Dragotti, and M. Vetterli, "Localizing Point Source in Diffuion Field from Spatiotemporal Sample," preented at the The 9th International Conference on Sampling Theory and Application, Singapore,. [6] J. Murray-Bruce and P. L. Dragotti, "Etimating Localized Source of Diffuion Field Uing Spatiotemporal Senor Meaurement," IEEE Tranaction on Signal Proceing, vol. 63, pp , 5. [7] H. Paul and R. Jedermann, "Spare Point Source Etimation in Senor etwork with Gauian Kernel," in th International ITG Workhop on Smart Antenna, Munich, Germany, 6. [8] T. J. Hatie, R. J. Tihirani, and J. H. Friedman, The element of tatitical learning: data mining, inference, and prediction, nd ed.: Springer, ew York, 9. [9] R. Jedermann and W. Lang, "The minimum numer of enor - Interpolation of patial temperature profile," preented at the Wirele Senor etwork, 6th European Conference, EWS 9, Lecture ote in Computer Science (LCS), Berlin/Heidelerg, 9. [] M. Umer, K. L., and E. Tanin, "Spatial interpolation in wirele enor network: localized algorithm for variogram modeling and Kriging," GeoInformatica, vol. 4, pp. -34,. [] D. Gu and H. Hu, "Spatial Gauian Proce Regreion With Moile Senor etwork," IEEE Tranaction on eural etwork and Learning Sytem, vol. 3, pp. 79-9,. [] M. Jadaliha, Y. Xu, J. Choi,. S. Johnon, and W. Li, "Gauian Proce Regreion for Senor etwork Under Localization Uncertainty," IEEE Tranaction on Signal Proceing, vol. 6, pp. 3-37, 3. [3] G. Reie, G. Matz, and K. Grochenig, "Ditriuted field recontruction in wirele enor network aed on hyrid hift-invariant pace," Signal Proceing, IEEE Tranaction on, vol. 6, pp ,. [4] A. Agarwal, "A ew Approach to Spatio-temporal Kriging and It Application," Ph.D. Thei Ph.D. Thei, The Ohio State Univerity,. [5] H. Hu and Z. Yang, "Spatial Correlation-Baed Ditriuted Compreed Sening in Wirele Senor etwork," in 6th International Conference on Wirele Communication etworking and Moile Computing (WiCOM),, pp. -4. [6] H. Wackernagel, Multivariate geotatitic - an introduction with application. Berlin: Springer, 3.