H.B.Gunay et al.Building and Environment 70(2013)31-47 41 response variables have non-normal distributions.In generalized opening behavior due to the indoor temperature,subsequently linear models,a linking function(e.g.logit function)of the response they incorporated both indoor and outdoor temperatures,and variable is a linear function of the predictor variables.For example, finally added relative humidity(RH)in a logistic regression model. the linear predictor function for the logistic regression can be Based on the improvement in the model fit,the significance of the expressed as follows: added predictor variable was assessed.It was reported that indoor and outdoor temperatures in tandem better accounted for the ob- logit(pi)=In( Pi servations in all seasons than either of them alone.This confirms 1-D: =f。+1X1i+f2x2.i+…+BmXmi the independent contribution of both variables.However,addition (2) of the RH resulted in a negligible improvement in the model,thus it was discarded. where logit is the linking function for logistic regression.In Fig.6,a logistic regression model is illustrated for the data scatter 3.4.Modeling the reversal of an adaptive behavior generated from Nicol [1211.Currently numerous researchers [10,29,39,42,46,47.65,75,108,121.122]accept that logistic regression Numerous studies suggested distinct reversal models for the models are appropriate for estimating the probability of an adap- adaptive behaviors such as window closing,light switch-off,or tive occupant behavior with respect to a particular predictor vari- blinds opening [10.39.42,46.65.108].This is important because the able or variables. stimulating factors that cause occupants to undertake a certain adaptive behavior can clearly be different than the stimulating 3.2.Accuracy of the models factors to reverse that adaptive action,as shown in Fig.3.The two methods utilized to predict the reversal models are the deadband A good model fit does not necessarily guarantee a valid statis- incorporated models [10,42.46,128]and the survival models tical model and the model cannot be any better than our under- 39,46.65,1081. standing of the system.For example,Zhang and Barrett 47 Deadbands are a controls terminology suggesting that the predicted logistic regression models to estimate the probability of controller does not produce a signal for a particular range in the window opening behavior with the outdoor temperature.Since process variable.Deadband incorporated models assume that an almost all of their samples were taken while the outdoor temper- event (i.e.change in state)does not take place for a specified range ature was below 20C,a conclusive model prediction could not be in the predictor variable.For example,Rijal et al.[10]suggested achieved for temperatures above 20C.Therefore,the predictive shifting cumulative probability distribution curves +2K from the accuracy of statistical models can be tested using techniques such original logistic regression curve such that window opening/closing as:a cross-validation [46,57,65,108]or an independent verification probabilities would always be 4K apart from each other,as shown 46,57,65,108,123].In cross-validation technique,subsets of the in Fig.7.It was reported that a deadband must be introduced to the dataset are spared and used to validate the model.For example, occupant models;otherwise its effect would result in instability.In Haldi and Robinson[108]carried out a cross-validation by retaining other words,the deadband concept aims to tackle the problems 10%of their dataset in which the process is repeated ten times by associated with the discrete-time simulation algorithm rather than permuting this data splitting.so every single observation is used to model a certain observed phenomenon.In fact,Rijal et al.[10] nine times for calibration and once for validation.This technique, acknowledged that the deadband concept was approximate and since relying on the same data set as the calibrated model,is prone required revisions.A major limitation of deadband models is that to misinterpretation [124].An unbiased cross-validation should they restrain the modeler to use an identical set of predictors (e.g. result in a symmetric deviation between the predictions and the indoor temperature)for both the adaptive behavior and its reversal. observations.An independent verification can be carried out on a This contradicts Haldi and Robinson [15]and Rijal et al.[42]sug- thoroughly independent dataset,thus the ability of the model can gesting window opening action to be modeled as a function of the be tested on other buildings with other occupants.This type of indoor variables,while the window closing action to be modeled as verification is typically very limited in the reviewed literature,since a function of the outdoor and the indoor variables. such independent datasets are not widely available.The Smart Haldi and Robinson 46,65]acknowledged the simulation Controls and Thermal Comfort(SCAT)dataset presents the partic- associated limitations,like Rijal et al.[10].and proposed the use of ipants'thermal sensation in ASHRAE's seven point scale [125] occupant control actions,clothing.and activity levels in Europe [126].For example,Haldi and Robinson [108]and Nicol [121]used this dataset to verify their models.Similarly,Reinhart et al.[57] used the data from the Lamparter building 80,127]as a verifica- 0.8 tion for their model and to compare this with Inoue et al.[51]. Closed 3.3.Uni/multivariate models Window Deadband Open Model prediction can be carried out using a univariate or a 0.4 multivariate relationship.In the univariate models,the probability of an adaptive behavior is estimated with respect to a single pre- Window 0.2 dictor variable such as the indoor temperature.In the multivariate models,the probability of an adaptive behavior is estimated with respect to multiple predictor variables such as indoor and outdoor temperatures.The marginal predictive value of increasing the 16 20 24 28 32 predictor variables should be assessed and predictor variables that Tg(C) weakly correlate to the response variable should be discarded [32]. For example,two studies [38.46]initially investigated the window Fig.7.Window opening and closing behavior model taken from Rijal,Tuohy et aL [41response variables have non-normal distributions. In generalized linear models, a linking function (e.g. logit function) of the response variable is a linear function of the predictor variables. For example, the linear predictor function for the logistic regression can be expressed as follows: logitðpiÞ ¼ ln
pi 1 pi ¼ bo þ b1x1;i þ b2x2;i þ / þ bmxm;i (2) where logit is the linking function for logistic regression. In Fig. 6, a logistic regression model is illustrated for the data scatter generated from Nicol [121]. Currently numerous researchers [10,29,39,42,46,47,65,75,108,121,122] accept that logistic regression models are appropriate for estimating the probability of an adap￾tive occupant behavior with respect to a particular predictor vari￾able or variables. 3.2. Accuracy of the models A good model fit does not necessarily guarantee a valid statis￾tical model and the model cannot be any better than our under￾standing of the system. For example, Zhang and Barrett [47] predicted logistic regression models to estimate the probability of window opening behavior with the outdoor temperature. Since almost all of their samples were taken while the outdoor temper￾ature was below 20 C, a conclusive model prediction could not be achieved for temperatures above 20 C. Therefore, the predictive accuracy of statistical models can be tested using techniques such as: a cross-validation [46,57,65,108] or an independent verification [46,57,65,108,123]. In cross-validation technique, subsets of the dataset are spared and used to validate the model. For example, Haldi and Robinson [108] carried out a cross-validation by retaining 10% of their dataset in which the process is repeated ten times by permuting this data splitting, so every single observation is used nine times for calibration and once for validation. This technique, since relying on the same data set as the calibrated model, is prone to misinterpretation [124]. An unbiased cross-validation should result in a symmetric deviation between the predictions and the observations. An independent verification can be carried out on a thoroughly independent dataset, thus the ability of the model can be tested on other buildings with other occupants. This type of verification is typically very limited in the reviewed literature, since such independent datasets are not widely available. The Smart Controls and Thermal Comfort (SCAT) dataset presents the partic￾ipants’ thermal sensation in ASHRAE’s seven point scale [125], occupant control actions, clothing, and activity levels in Europe [126]. For example, Haldi and Robinson [108] and Nicol [121] used this dataset to verify their models. Similarly, Reinhart et al. [57] used the data from the Lamparter building [80,127] as a verifica￾tion for their model and to compare this with Inoue et al. [51]. 3.3. Uni/multivariate models Model prediction can be carried out using a univariate or a multivariate relationship. In the univariate models, the probability of an adaptive behavior is estimated with respect to a single pre￾dictor variable such as the indoor temperature. In the multivariate models, the probability of an adaptive behavior is estimated with respect to multiple predictor variables such as indoor and outdoor temperatures. The marginal predictive value of increasing the predictor variables should be assessed and predictor variables that weakly correlate to the response variable should be discarded [32]. For example, two studies [38,46] initially investigated the window opening behavior due to the indoor temperature, subsequently they incorporated both indoor and outdoor temperatures, and finally added relative humidity (RH) in a logistic regression model. Based on the improvement in the model fit, the significance of the added predictor variable was assessed. It was reported that indoor and outdoor temperatures in tandem better accounted for the ob￾servations in all seasons than either of them alone. This confirms the independent contribution of both variables. However, addition of the RH resulted in a negligible improvement in the model, thus it was discarded. 3.4. Modeling the reversal of an adaptive behavior Numerous studies suggested distinct reversal models for the adaptive behaviors such as window closing, light switch-off, or blinds opening [10,39,42,46,65,108]. This is important because the stimulating factors that cause occupants to undertake a certain adaptive behavior can clearly be different than the stimulating factors to reverse that adaptive action, as shown in Fig. 3. The two methods utilized to predict the reversal models are the deadband incorporated models [10,42,46,128] and the survival models [39,46,65,108]. Deadbands are a controls terminology suggesting that the controller does not produce a signal for a particular range in the process variable. Deadband incorporated models assume that an event (i.e. change in state) does not take place for a specified range in the predictor variable. For example, Rijal et al. [10] suggested shifting cumulative probability distribution curves 2K from the original logistic regression curve such that window opening/closing probabilities would always be 4K apart from each other, as shown in Fig. 7. It was reported that a deadband must be introduced to the occupant models; otherwise its effect would result in instability. In other words, the deadband concept aims to tackle the problems associated with the discrete-time simulation algorithm rather than to model a certain observed phenomenon. In fact, Rijal et al. [10] acknowledged that the deadband concept was approximate and required revisions. A major limitation of deadband models is that they restrain the modeler to use an identical set of predictors (e.g. indoor temperature) for both the adaptive behavior and its reversal. This contradicts Haldi and Robinson [15] and Rijal et al. [42] sug￾gesting window opening action to be modeled as a function of the indoor variables, while the window closing action to be modeled as a function of the outdoor and the indoor variables. Haldi and Robinson [46,65] acknowledged the simulation associated limitations, like Rijal et al. [10], and proposed the use of Fig. 7. Window opening and closing behavior model taken from Rijal, Tuohy et al. [41]. H.B. Gunay et al. / Building and Environment 70 (2013) 31e47 41