Bayesian Iterative Method Using Parameter Scheduling for Predicting Optimal Condition on Thermal Index Due to Air Conditioning with Minimized Power

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1 Bayeian Iterative Method Uing Parameter Scheduling for Predicting Optimal Condition on Thermal Index Due to Air Conditioning with Minimized Power Yohei Saika ational Intitute of Technology, Maebahi, Japan Maahiro akagawa agaoka Univerity of Technology, agaoka, Japan unwrapping for interferometric SAR data [4] and timeerie analyi with the principal component analyi for environmental data [5]. ow, reearcher have tudied information technology uing the pare modeling from the tatitical-mechanical viewpoint uing the loopy belief propagation [6]. On the other hand, due to the occurrence of the globalcale energy and environmental problem [7], reearcher and engineer have been contructing a lot of ytem for power aving. Some of them [8] have been contructing ytem for effective ue of electric power by uing the mart grid and enor network connecting power plant with variou ytem conuming power, uch a each reidence and indutry with variou cale. Then, the Minitry of Economic, Technology and Indutry (METI) [9] in Japan ha pointed out that power aving at each reidence doe not proceed until 030 due to an increae in load of nuring for aging population, and that it delay in each mall-cale indutry due to the delay in introducing the ytem for power aving. Alo, global warming ha become one of the eriou problem all over the world. Therefore, it ha become of importance to realize many mall-cale comfortable pace on the bai of the appropriately aumed thermal index by making ue of the air conditioner both uing air cooling and dehumidifier with minimized power for the power aving ytem. Previouly, we contructed one of information technique for thi problem, though the optimal condition i not derived for any cae [0], []. Therefore, in thi tudy, on the bai of the Bayeian inference uing the expected a poterior (EAP) etimation, we contruct an information technique to provide an optimal cheduling of air conditioning due to the air cooling and the dehumidifier realizing comfortable mallcale pace with minimized power. For our purpoe, we propoe an information technique baed on the Bayeian iterative method erving the environmental variable at each time a expectation averaged over the poterior probability predicted by the Bayeian iterative method Abtract On the bai of the Bayeian iterative method via parameter cheduling, we invetigate the prediction of a et of optimal environmental variable of mall-cale pace by uing air conditioning with minimized power. umerical calculation clarify dynamic propertie of the environmental variable, uch a temperature and relative humidity at each ampling point in the preent method for everal realitic cae in evere ummer eaon. We find the optimal parameter cheduling realizing the optimal environment with minimized power of the air conditioning both uing air cooling and dehumidifier. Index Term Bayeian inference, expected a poterior etimation, indoor environment, parameter cheduling I. ITRODUCTIO Reearcher have invetigated data-driven information technique [], [] due to the development of the computer technology. Reearcher have developed variou information method extracting ome valuable information among large-cale data oberved by variou enor, uch a the principal component analyi []. Alo, reearcher have developed information technique to predict ome variable on the bai of Bayeian inference [3], [4] in many reearch field, uch a the time-erie analyi [5]-[7]. On the other hand, for three or four decade, phyicit have tudied information baed on the analogy between Bayeian inference and tatitical mechanic [8]. They have firt applied tatitical-mechanical method [9], uch a the mean-field theory and the replica theory to fundamental topic, uch a error-correcting code [0] and image retoration []. Alo, one of the preent author (Y. S.) have tudied the application to cience and technology, uch a image retoration uing plane rotator model [], invere halftoning [3], phae Manucript received February 6, 08; revied June 9, 08. doi: 0.878/ijoee

2 uing the EAP etimation. In thi method, we etimate the poterior probability baed on the Baye-formula uing the model of the true prior and the likelihood expreing the model of the tranition probability. Here, we ue the model prior which enhance comfortable condition baed on the thermal index uing the Temperature-Humidity Index (THI). Alo, we ue the likelihood which expree the model of the air conditioning both uing the air cooling and the dehumidifier. Further, we here introduce the model of cheduling for air conditioning. Then, in order to clarify dynamical propertie of the preent method, we etimate the time evolution of everal variable, uch a the et of the environmental variable, the thermal index uing the THI and the power conumption of the air conditioning for everal realitic cae. Due to numerical calculation for the cae, we find that the preent method ucceed in realizing the optimal condition due to the air conditioning with the minimized power, if we appropriately ue the cheduling of the model of air conditioning. Thi paper i organized a follow. Firt, we outline the control of air conditioning via the Bayeian iterative method. Then, we how our formulation for predicting the optimal parameter cheduling to realize the optimal condition via the air conditioning with minimized power. ext, we examine dynamic propertie of the Bayeian iterative method for everal realitic cae. Lat chapter i devoted to ummary and dicuion. ome enor, uch a data logger and power enor. ext, baed on the Bayeian iterative method with the parameter cheduling, we extract the optimal cheduling to obtain the comfortable indoor environment at the mall pace with the minimized power of air conditioning. Further, by making ue of we control the air conditioning uing the predicted variable. We repeat the above procedure until we obtain the optimal environment. III. BAYESIA IFERECE AD STATISTICAL MECHAICS In thi chapter, we how the analogy between the Bayeian inference uing the EAP etimation and tatitical mechanic of the Q-Iing model. Firt, the Bayeian inference uing the probability theory predict ome quantitie a the expectation averaged over the poterior probability etimated baed on the Baye-formula via the aumed model of the true prior and the noie probability from each original to it oberved information. On the other hand, in the field of tatitical mechanic, we infer the macrocopic variable averaged over the Boltzmann factor of the many-body ytem, uch a the Q-Iing model. A een from Fig., both framework i contructed baed on the probability theory. Then, Fig. how that predicting ome quantitie in the Bayeian inference i equivalent to etimating ome macrocopic variable in tatitical phyic. From the correpondence in Fig., we ee that technique in tatitical phyic, uch a the mean-field theory and the replica theory etablihed in the theory of pin glae, can be applied to predicting ome variable and to etimating tatitical performance of ome technique in the Bayeian inference. II. OUR PROJECT Figure. Framework of predicting the environmental variable baed on the Bayeian inference uing the EAP etimation with the ue of the extracted information due to perceptron Figure. Correpondence between Bayeian inference and tatitical phyic A hown in Fig., we briefly how main objective of thi tudy on controlling air conditioning for realizing mot comfortable circumtance in mall area with the minimized power due to air conditioning. Here, we tudy thi problem for a laboratory of the ational Intitute of Technology, Gunma College, a a typical example of the mall-cale area. Then, we ue the Temperature-Humidity Index (THI) a the thermal index to etimate body enation of human being at the area. A hown in Fig., we firt oberve large-cale data on everal indoor environmental variable, uch a temperature and relative humidity at each ampling point and the electric power conumed by the air conditioning at the objective ytem. Then, by uing the perceptron learning, we extract valuable information among large-cale data oberved by IV. OUR FORMULATIO Firt, we conider a et of original tate with mot comfortable condition {T(i,t), H(i,t)} (i=,,, t=,, ) which provide the optimal thermal index via the THI. Here, T(i,t)(H(i,t)) i the temperature (relative humidity) of the original tate at i-th point at t. Here, Tmin<T(i,t)<Tmax, Hmin<H(i,t)<Hmax. Each tate {T(i,t), H(i,t)} i generated by the aumed true prior expreed a ΓC f (T (i, t ), H (i, t )) - f () Pr({T (i, t ), H (i, t )}) exp Z f opt opt Then, we alo conider the model of the true prior: 33

3 Pr({T (i, t ), H (i, t )}) H ({ T },{ H } {T },{H }) J Γ C exp f (T (i, t ), H (i, t )) - f opt (V k (i, t ) Vk (i, t )) i j, k Z f opt R T (i, t ) () (i, t ) ( R ) H (i, t ) T (i, t ) (6) here, we hould et the parameter C, R appropriately. ext, uing the et of oberved variable {τt(i,t), H τ (i,t)}, we derive the optimal parameter cheduling of the air conditioning with air cooling and dehumidifier baed on the Bayeian iterative method uing the EAP etimation which provide the et of the temperature and the relative humidity and the electric power conumed by air conditioning. For our purpoe, we ue a et of model variable {T(i,t), H(i,t)}. Here, Tmin<T(i,t)<Tmax, Hmin<H(i,t)<Hmax, i=,,, t=0,,... In thi method, in order to derive the optimal parameter cheduling, we infer the time evolution of the environmental variable, the THI at each ampling point and the power conumed by the air conditioning a follow. Firt, we conider an initial tate which i expreed a (T*(i,0), H*(i,0))=(τT(i, 0)), τh(i, 0)) at t=0. Then, we predict the et of the variable expreed a (T*(i,t+),H*(i,t+)) (i=,,, t=0,,,.,-) uing the etimated one (T*(i,t), H*(i,t)) a (3) here, β, Γ, C and J are parameter et appropriately and fopt= 60 a the optimal value of the THI. Then, f(t(i,t), H(i,t)) i the thermal index uing the THI which expree the degree of body enation of human being. Further, the definition of the THI i expreed a f (T, H ) 0.8 T 0.0 H (0.99 T 4.3) 46.3 T j where T (i, t ) (k ) Vk (i, t ) H (i, t ) (k ) (4) TABLE I. THERMAL IDEX VIA THE TEMPERATURE-HUMIDITY IDEX AD BODILY SESATIO T * (i, t ) Pr({T (i, t ), H (i, t )} {T * (i, t ), H * (i, t )}) T (i, t ) {T ( i,t )} { H ( i,t )} (7) H (i, t ) * Pr({T (i, t ), H (i, t )} {T * (i, t ), H * (i, t )}) H (i, t ) {T ( i,t )} { H ( i,t )} (8) A een from (7), (8), thee two variable at t+ are obtained a expectation which are averaged over the poterior etimated baed on the Baye-formula via the model of the true prior and the model of the tranition probability from original to oberved tate. The explicit form of the model prior i expreed a Figure 3. Curve with THI=60, 65, 70, 75, 80 and 85 decribed on the T-H plane Pr(T (i, t ), H (i, t )}) ΓC f (T (i, t ), H (i, t )) - f opt exp Zm f opt Then, Table I denote the correpondence between the thermal index uing the THI and the body enation of human being. A een from thi table, we find that the human being ha no feeling in 60<f(T,H)<65, and then that he/he feel comfort in 65<f(T,H)<70. The degree of dicomfort become more eriou, if f(t,h) increae in 70<f(T,H) and if f(t,h) decreae in f(t,h)<60. Then, in Fig. 3, we how curve with everal THI on the T-H plane. Thi mean that the comfortable environment with f(t,h)=60 realized even at high temperature, if the relative humidity i low. ext, each original tate {T(i,t), H(i,t)} i converted into it oberved one {τt(i,t), τh(i,t)} by introducing realitic environment into the original ideal one. Thi mean that the oberved tate {τt(i,t), τh(i,t)} expre realitic environment, if no air conditioning i ued. We aume tranition probability from original to oberved tate a Pr({ },{ } {T },{H }) T Alo, we ue another model prior whoe explicit form i expreed a Pr({T (i, t ), H (i, t )}) ΓC J exp f (T (i, t ), H (i, t )) - f opt (V k (i, t ) V k (i, t )) i j,k Z f opt (0) where T (i, t ) (k ) V k (i, t ) H (i, t ) (k ) () If we conider the correlation between each pair of the environmental variable. Then, the explicit form of the model of the noie probability i expreed a Pr( T (i, t ), H (i, t ) T (i, t ), H (i, t )) exp - ( C ) {R(T (i, t ) T * (i, t )) ( R)( H (i, t ) T * (i, t )) } () here, β, Γ, C and J are the parameter we hould appropriately. Then, we et the parameter in eq. (9) a fopt=60 throughout of thi reearch. Further, R(t)/R i a H exp ( C ) H ({ T },{ H } {T },{H }) (9) (5) 34

4 parameter controlled o a to minimize the electric power of the air conditioning via the air cooling and the dehumidifier. We repeat the procedure () until the optimal olution with THI=60 i approximated. If we obtain the optimal one, we top the iteration. At each tep, we etimate the degree of the comfort human being feel baed on the thermal index uing the THI whoe explicit form i given in (). Then, we aue the conumed power via air conditioning a Pow(T (i, t ), H (i, t )) n { T T (i, t ) T * (i, t ) H H (i, t ) H * (i, t ) } Here, we etimate the dynamic property of the Bayeian iterative method from time evolution of the environmental variable, if R(t) i et to be teady. In thi cae, we find that the thermal index via the THI and the conumed power of the air conditioning depend on the parameter R(t). Then, the reult ugget that tuning the cheduling of R(t) i important for realizing the optimum of the thermal index via the THI at each ampling point and for minimizing the conumed power of the air conditioning. So, we earch the optimal cheduling R(t) of air conditioning for realizing the optimal value of the THI at each ampling point. A hown in Fig. 5, 6(a), (b), 7, the numerical calculation clarify that the optimal value of the thermal index via the THI i realized mot moothly by mean of the air conditioning with the minimized power, if we tune the parameter {R(t)} a {0,,,,, } among the candidate: {R(t)}={,,,, },{0,,,,, }, {0,0,,,, },{0,0,0,,, },{0,0,0,0,, }. Thee reult indicate that the Bayeian iterative method ucceed in extracting optimal parameter cheduling realizing mot comfortable environment at each ampling point by mean of the air conditioning with the minimized power. (3) i In thi tudy, we et to εt=5.66 and εh=0.53. V. PERFORMACE Here, we etimate the tatic property of the Bayeian inference uing the EAP etimation for clarifying the role of the model of the true prior and the tranition from original to oberved environmental variable and the dynamic property of the Bayeian iterative method for the realitic application. Then, we etimate the performance for the everal cae, uch a (τt, τh)=(35.0[ ], 80.0[%])], (40.0[ ], 60.0[%]), (30.0[ ], 00.0[%]) at the laboratory of the ational Intitute of Technology, Gunma College in evere ummer both with high temperature and high relative humidity. Firt, we etimate the role of the model of the true prior and the tranition probability in the Bayeian inference uing the EAP etimation via the numerical calculation for the realitic cae, if we et to β=, Γ=, R(t)=0.3 and J=0. Alo, due to the rigorou proof via an inequality, we clarify that the lower bound of the mean quare error between the original and predicted environmental variable i realized, if we have information on the aumed true prior and the noie probability from the original to oberved variable. Then, the numerical calculation find that the Bayeian inference via the EAP etimation predict mot comfortable condition baed on the thermal index uing the THI at C=, and that the method predict the oberved variable (τt, τh) at C=0. Alo, we find that the Bayeian inference uing the EAP etimation. Then, it i located between the poition of the oberved tate (τt, τh) and the curve with f(t,h)=fopt, if the parameter C i et between null and unity. (Fig. 4) Figure 5. Time evolution of the thermal index uing the temperaturehumidity index at a ampling point due to the Bayeian iterative method uing parameter cheduling. (a) Figure 7. Total power conumption due to the air conditioning via the air conditioning and the dehumidifier depending on the parameter cheduling Figure 4. Location of the predicted tate obtained by the Bayeian inference uing the EAP etimation for (τt, τh). (b) Figure 6. Time evolution of the power conumption due to the model of air conditioning via air cooling and dehumidifier 35

5 [3] Y. Saika, J. Inoue, H. Tanaka, and M. Okada, Baye-optimal olution to invere halftoning baed on tatitical mechanic of the Q-Iing model, Central European Journal of Phyic, vol. 7, pp , 009. [4] Y. Saika and T. Uezu, Statitical mechanical approache to phae unwrapping in remote ening via ynthetic aperture interferometry, Interdiciplinary Information Science, vol. 7, no., pp , 03. [5] A. Aami and Y. Saika, Forecat of meteorological data utilizing tate-pace model utilizing metric-multidimenional caling, International Journal Reearch in Engineering and Technology, vol. 3, no. 7, pp. 0-6, 04. [6] Y. Kabahima, T. Wadayama, and T. Tanaka, A typical recontruction limit for compreed ening baed on Lp-norm minimization, Journal of Statitical Mechanic: Theory and Experiment, pp. L09033-L0904, 009. [7] Energy Roadmap, Smart Grid, International Energy Agency, 0. [8] Outline of the FY06 Annual Report on Energy (Energy White Paper 07), Minitry of Economic, Technology and Indutry, 07. [9] Japan Energy White Paper, Minitry of Economy, Trade and Indutry, 07. [0] Y. Saika and M. akagawa, Bayeian inference uing the expected a poterior etimation via environmental factor for predicting thermal comfort and conumed power, in Proc. International Conference on Big Data Analyi, 07, pp [] Y. Saika and M. akagawa, Dynamic of predicting temperaturehumidity index and power conumption in mall-cale ytem utilizing Bayeian inference via the EAP etimation, in Proc. 7th International Conference on Control, Automation and Sytem, 07, pp VI. SUMMARY AD DISCUSSIO In above chapter, baed on the analogy between the Bayeian inference and tatitical mechanic, we have contructed the information technique for controlling the air conditioning with air cooling and dehumidifier by uing the Bayeian iterative method with the parameter cheduling. In thi tudy, we have contructed the preent method baed on tatitical mechanic of the Q-tate Iing model to predict the model of the parameter cheduling leading the optimal value of the thermal index via the THI with the power conumption due to the air conditioning via the air cooling and the dehumidifier. Here, we have conidered the model of the true prior enhancing the optimal value of the thermal index via the THI at the ampling point and the model of the tranition probability from the original to the oberved environment expreed a the temperature and the relative humidity. Then, we have here examined the time evolution of the thermal index uing the THI and the power conumption due to the air conditioning. We have obtained the reult that the preent method ucceed in predicting the optimal parameter cheduling which realize the optimal value of the thermal index via the THI mot moothly with the minimized value of the electric power conumed by the air conditioning. For intance, we have found that the preent method ucceed in predicting the parameter cheduling of air conditioning which realize the optimal variable of the THI mot moothly with the minimized power with the minimized power of the air conditioning, if we carry out air cooling at firt tep and then dehumidifier at the following tep. A future problem, it i important to apply the preent ytem to realitic application for variou mall-cale ytem. Yohei Saika received B. Sc. from Science Univerity of Tokyo in 989. Then, he received M. Sc. from Tokyo Intitute of Technology in 99. Alo, he received Ph. degree from Department of Phyic, Tokyo Intitute of Technology in 995. Hi major field are tatitical phyic, information cience and technology, and control and meaurement. He ha been a member of Department of Electrical Engineering, Wakayama ational College of Technology in Japan from 995 to 0. He ha been a viiting reearcher at the Graduate School of Condened Matter Phyic, Tokyo Intitute of Technology in Japan from 998 to 999. He ha been at the Department of Information and Computer Engineering, ational Intitute of Technology, Gunma College from 0. He i a profeor of the ational Intitute of Technology, Gunma College in Japan. Then, hi recent activitie are concerned with the interdiciplinary field related to tatitical phyic, information cience and technology, electrical engineering, control and meaurement, and environmental engineering and o on. Dr. Saika i a member of the Phyical Society of Japan. He work a a member of the organizing committee of the th area of Condened Matter Phyic from REFERECES [] EMC Education Service, Data Science and Big Data Analytic: Dicovering, Analyzing Viualizing and Preenting Data, USA: Wiley, 05. [] B. Mirkin, Core Concept in Data Analyi: Summarization, Correlation and Viualization (Undergraduate Topic in Computer Science), London: Springer, 0. [3] D. J. C. Mackay, Information Theory, Inference and Learning Algorithm, Cambridge: Cambridge Univerity Pre, 00. [4] C. Bihop, Pattern Recognition and Machine Learning, London, U.K.: Springer, 00. [5] A. Gelman, J. B. Carlin, H. S. Stern, D. B. Dunon, A. Vehtari, and D. B. Rubin, Bayeian Data Analyi, third ed., Boca Raton: CRC Pre, 03. [6] H. Maden, Time-erie Analyi, USA: Chapman & Hall, 03. [7] J. D. Hamilton, Time-erie Analyi, USA: Princeton Univ. Pre, 994. [8] H. ihimori, Statitical Phyic of Spin Glae and Information Proceing: An Introduction, Oxford Univ. Pre, 00. [9] K. Tanaka, Statitical mechanical approache to image proceing (topical review), Journal of Phyic A: Mathematical and General, vol. 35, no. 37, pp. R8-R50, 00. [0]. Sourla, Spin-gla model a error-correcting code, ature, vol. 339, pp , 989. [] J. M. Pryce and A. D. Bruce, Statitical mechanic of image retoration, J. Phy., vol. 8, pp. 5-54, 995. [] Y. Saika and H. ihimori, Statitical mechanic of image retoration by the plane rotator model, The Journal of the Phyical Society of Japan, vol. 7, pp , 00. Maahiro akagawa received the B.Sc. and M. Sc. degree in the Department of Electrical Engineering from agaoka Univerity of Technology in 980 and 98, repectively. He alo received Ph. degree in the Department of Applied Phyic from agoya Univerity in 988. He ha been a faculty member of engineering at the agaoka Univerity of Technology ince 98, and became an aociate profeor and a Profeor in 989 and 00, repectively. He i a faculty member of department of cience of technology innovation ince 04. He wa a viiting reearcher, upported from Minitry of Education, Culture, Sport, Science and Technology of Japan, at the Department of Mathematic, Univerity of Strathclyde in Scotland from 988 to 989. Hi recent activitie are concerned with the chao and fractal and in cience and technology, e.g. the chao in phyiological ignal a EEG and vocal ound et al, the chao neural network with a periodic activation function, the iterated function ytem for image coding etc. Moreover, he i intereted in the phyic of liquid crytal involving the molecular theory and the vicoelatic continuum theory. Dr. akagawa i a member of the Japanee eural etwork Society, the Phyical Society of Japan, Japanee Society of Applied Phyic et al. He ha authored book including "Chao and Fractal in Engineering (World Scientific, 999)", "Liquid Crytalline Phae Tranition (Springer-Verlag, 00)" and o on. 36