Introduction

Several models have been developed predicting thermal sensation. Thermal indices can be divided in four categories: (a) indices which relate only to a few meteorological parameters and they are usually applied in specific climates (direct indices) (b) indices that correlate subjective perception with the thermal conditions (empirical indices) (c) thermo-physiological indices based on heat balance equations considering the human body in a uniform environment and simulating it with one or more cylinders in order to describe its thermoregulatory responses and the heat transfer between the human body and its environment and (d) thermo-psychological indices that consider non-uniform environments and provide local and total thermal sensation of the body dividing it into several sections. The last two categories are also called rational indices (Cheng et al. 2012b).

Epstein and Moran (2006) listed about 40 indices widely used around the world and since then others have been proposed (COST Action 730 2013; Nagano and Horikoshi 2011; Monteiro and Alucci 2008). However, many of these indices were created for indoor conditions, others are difficult to understand and apply, most refer to uniform conditions, some can be used only in certain circumstances and very few take account of psychological factors that affect thermal perception (Cheng et al. 2012b). Furthermore, their predictions differ (Blazejczyk et al. 2012; Jones 2002) while they also deviate considerably from the actual thermal sensation of the population. Monteiro and Alucci (2006) calibrated a large number of indices in order to improve their predictability for the Metropolitan area of Sao Paulo, Brazil. Similar researches were conducted in different regions. Tseliou et al. (2010) attempted to correct three indices according to the climatic mean temperature of the region of application. Cheng et al. (2012a) concluded that Predicted Mean Vote (PMV) overestimated in summer, and underestimated in winter the thermal sensation in Hong Kong, Lin and Matzarakis (2008) proposed a new range for Physiological Equivalent Temperature (PET) for Taiwan and Pantavou et al. (2013) examining how well can the subjective thermal sensation be predicted by Universal Thermal Climate Index (UTCI) values consistent with weather measured data in the case of a Mediterranean climate, found that the cold classes of the UTCI scale were significantly lower than that of the Mediterranean.

On a different approach several researchers developed empirical thermal sensation indices based on actual sensation votes (Monteiro and Alucci 2008; Nikolopoulou and Lykoudis 2006; Stathopoulos et al. 2004). In only a few of the outdoor questionnaire surveys published for Mediterranean climates models have been developed for predicting thermal sensation. Nikolopoulou et al. (2003) proposed Actual Sensation Vote (ASV) and Ghali et al. (2011) generated the formula of Thermal Sensation (TS). In both studies the statistical analysis was performed using multiple linear regression and the models included only meteorological parameters. ASV predicts thermal sensation based on a 5-point scale while TS on a 9-point scale. Although 5-point and 7-point scales are the most common in use (Dawes 2008), 7 is considered to be the ideal number of discrete categories for describing sensation (Miller 1956) and it is used traditionally for the estimation of thermal sensation (Nicol 2008), while scales of higher precision offer no additional improvement (Dawes 2008). In that context it would be interesting to develop a thermo-physiological, 7-point scaled model of thermal sensation applicable to Mediterranean climates.

As already mentioned, in most studies of empirical indices, multiple linear regression was applied in order to predict thermal sensation even though thermal sensation was indicated by the interviewees based on a point scale (Andrade et al. 2011; Metje et al. 2008; Monteiro and Alucci 2008; Nikolopoulou and Lykoudis 2006). Thus, the dependent variable was ordinal in nature rendering the usual linear regression models possibly inappropriate, since it might be argued that the simplifying assumption of linear regression, that the dependent variable is measured on an interval scale, is not satisfied (IBM 2011). Ordinal regression, on the other hand, explicitly recognizes the ordinal nature of the dependent variable avoids arbitrary assumptions about its scale and allows testing of the assumptions regarding the probability distribution of the continuous variable that underlays the observed ordinal variable (Winship and Mare 1984).

During the year 2010–2011, a field questionnaire survey was carried out in the greater urban area of Athens, to examine the effects of meteorological, personal and psychological parameters on thermal sensation. The experimental procedure and the initial results were presented in detail in Pantavou et al. (2013). The main purpose of the present study is to demonstrate a simple predictive thermo-physiological model for outdoor thermal sensation expressed in a 7-point scale, adjusted with psychological factors, as well as to examine the adequacy of two different methods in estimating thermal sensation: ordinary linear and ordinal regression. This model development procedure elucidates the significance of each variable to the determination of thermal sensation in Mediterranean climates and the applicability of each model fitting method.

Materials and methods

Study area

Athens is the capital of Greece and one of the most distinguished travel destinations in the world. The urban area of Athens extends beyond the boundaries of the municipality of Athens to Piraeus, the main port of Greece and one of the largest in the world concerning passenger traffic (PPA S.A. 2013). Around one third of the Greek population lives in the larger urban area of Athens. The climate is Mediterranean while the climatological mean minimum and maximum temperatures (1961–1990) are 22.5 °C and 32.6 °C respectively for July (the hottest month of the year), as well as 6.5 °C and 12.9 °C for January (the coldest month of the year) (NOA-IERSD 2013). The questionnaire survey was carried out at two sites in the center of Athens and in a coastal location within the larger urban area of Athens in order to capture different microclimates: Syntagma square, the most central square of the city with green areas and a water fountain in the centre; Ermou Street, a pedestrian street canyon (width 10 m, length 200 m, height 20 m) next to Syntagma square; and Flisvos coast located about 6 km south of the center of Athens and 4 km southeast of Piraeus.

Data

The field surveys involved microclimatic measurements along with structured interviews of the users of the site during a total of 16 days in July, October and February 2010–2011. Field surveys were conducted for 2 days per season at each site, one day during morning-midday and one during afternoon–evening–night in order to collect data for the entire day. Their duration changed over season for security reasons as it was getting dark earlier in autumn and winter than in summer. The surveys started at about 10:00 am and run until 22:00 in summer and 19:30 in autumn and winter. Due to low attendance, no measurements were performed in Flisvos coast during winter and autumn afternoon, whereas an additional survey was performed in Syntagma during winter to obtain a number of questionnaires closer to the other seasons. Spring was excluded as a transient season with similar thermal conditions to autumn (October). A detailed list of the surveys conducted can be found in Table 1.

Table 1 Dates and time frames of the surveys
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A Rotronic S3CO3 thermo-hygrometer equipped with a non-ventilated radiation shield of aluminum, a Second Wind C3 anemometer, two Kipp & Zonen CM3 pyranometers and an ECO pyrgeometer were used to monitor air temperature (Tair), relative humidity (RH), average and maximum wind speed (WS, WSmax), downwelling (SR↓) and reflected (SR↑) solar radiation as well as downwelling and ground total radiation (TR↓, TR↑) on a horizontal plane. A PVC sphere, 40 mm diameter painted grey, with an emissivity of 0.3 and response time less than 4 min (Nikolopoulou et al. 1999) was used to measure globe temperature (Tglobe). All sensors were placed on a mobile tripod at the height of 1.1 m above the ground. The data were stored on a CR10X Campbell Scientific data logger at 1 min intervals. Ground surface temperature (Tground) was measured using an Infrared Thermometer at three points close and around the interviewee, at the onset of each interview.

The questionnaire consisted of the main question asking for the subjective assessment of thermal sensation based on ASHRAE 7-point scale (−3 - cold, −2 - cool, −1 - slightly cool, 0 - neutral, 1 - slightly warm, 2 - warm, 3 - hot), namely thermal sensation vote (TSV). Moreover it included questions about personal characteristics of the respondents such as gender, age, main activity during the last half hour, clothing, position in relation to the sun (shading), exposure time (visit duration at the interview site - ‘For how long have you been in this site?’) and exposure history (whether or not the respondents were in an air-conditioned environment before their visit to the interview site), medical history, smoking and health status as well as questions related to psychological factors that may have an effect on thermal sensation such as season, time of the day, companionship (whether or not the respondents were alone at the interview site), recent experience (place respondents’ were before their visit to the interview site), visit purpose, overall comfort (satisfaction in relation to the overall environment), preference related to overall sensation and the way it would be improved (changing air temperature, humidity, wind, irradiation) (Table 2). The interviewees were randomly selected.

Table 2 The collected and estimated data used in the analysis divided into meteorological, personal and psychological parameters
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The recorded radiation data SR↓, SR↑, TR↓ and TR↑ were used to calculate the long-wave radiation on a horizontal plane (IR↓, IR↑) while mean radiant temperature (Tmrt) was estimated by Tair, Tglobe and WS (Thorsson et al. 2007). Furthermore thermal insulation of clothing (Icl), and metabolic rate (M) were estimated using the descriptions of the clothes and the activity stated in the questionnaire in accordance with ISO 9920 and 8996 (ISO 1993, 2004). Body surface (Adu) was calculated from the height and weight of the respondents using the formula of DuBois and DuBois (1916). The atmospheric pressure used for Syntagma and Ermou comes from the Thissio meteorological station of the National Observatory of Athens, Institute for Environmental Research and Sustainable Development, (NOA-IERSD) and from Hellinikon meteorological station of the Hellenic National Meteorological Service (HNMS) for the case of Flisvos coast.

The original 1-min data were checked for apparent inconsistencies that led to less than 0.5 % of the values being rejected and then aggregated into 3 min averages, since that was the estimated time for completing a questionnaire. The whole sample was used while the overall completeness of the data corresponded to 99.1 %.

Data analysis

Trying to find the most suitable method to predict thermal sensation vote (TSV), linear and ordinal logistic regression were applied to meteorological, personal and psychological variables separately, to produce meteorological, personal and psychological models, as well as the combination of meteorological and personal variables to produce the thermo-physiological models. Ordinal regression is an extension of generalized linear models for ordinal data which estimates the probability of observing a particular class or lower. The independent variables can be either continuous or categorical while in the case of categorical variables a reference category should be defined. The method produces a set of equations, one for each class of the dependent ordinal variable, providing the cumulative probability of occurrence up to this class. The probability of each class is calculated by the difference of the cumulative probabilities of two consecutive classes. The class predicted by the model is the one with the highest probability.

In order to establish a useful model for predicting TSV, some of the original classes -according to the questionnaire- of the independent categorical variables were merged. The criterion considered was that of separability according to ANOVA and cross-tabulation analyses as well as the nature of the parameters (Table 2).

First, the independent variables were checked for possible correlations between them, so as to avoid problems of multicollinearity. Linear and ordinal regression on TSV were applied for each independent variable individually. The independent variables that showed no significant correlation with TSV were excluded from further analysis. For the final stage of model development, selection among the remaining variables was based on avoiding multicollinearity and the physical significance of the parameters. In the case of categorical independent variables, in the model remained those with at least one class having a statistically significant correlation with the reference category and the significance level for that parameter being <0.05 in a univariate ANOVA test. Categorical predictor variables were dummy codded in order to be entered into the linear regression model. A categorical variable with k levels was transformed into k-1 variables each with two levels (0–1). The regression equation was then formed as:

$$ \mathrm{y}\prime ={\mathrm{b}}_0+{\mathrm{a}}_1\cdot {\mathrm{D}}_1+{\mathrm{a}}_2\cdot \mathrm{D}{}_2+....{\mathrm{a}}_{k-1}{\mathrm{D}}_{k-1}+{\mathrm{b}}_1\cdot {\mathrm{x}}_1+{\mathrm{b}}_2\cdot {\mathrm{x}}_2+\dots +{\mathrm{b}}_v\cdot {\mathrm{x}}_v $$
(1)

where D the categorical variable with D1 = 1 when level 1 was valid and D1 = 0 when any of the other levels were valid, a1, a2, … ak−1 coefficients of each level of the categorical variable, x1, x 2…, xv the continuous variables and b1, b2, …, bv the coefficients of the continuous variables.

Three types of models were developed: one that included only the meteorological parameters (meteo), one that included only the personal parameters (personal) and one including meteorological and personal parameters, named thermo-physiological model (T/P).

Thereafter the thermo-physiological model was corrected by the psychological factors. First, the difference between TSV and thermo-physiologically predicted TSV was calculated (DTSV = TSV-thermo-physiologically predicted TSV). The dependence of DTSV on the psychological factors was then examined producing the predicted DTSV. The final model resulted from the addition of thermo-physiologically predicted TSV and the predicted DTSV. This process yielded what were termed as ‘extensive models’, considering all the experimental parameters, one using linear and another using ordinal regression. The process was repeated selecting only those parameters theoretically expected to affect thermal sensation, namely Tair, Tmrt, WS, RH, Icl and M (Macpherson 1962; Jones 2002). This resulted in two ‘empirical’ models a linear and an ordinal one.

The statistical analysis was carried out using SPSS 17.0 (SPSS Inc, 2008). All results presented below are statistically significant at a confidence level equal to or less than 0.05 (Sig ≤ 0.05).

Results

The 1706 interviews of the survey were performed in a variety of outdoor environmental conditions (Table 3). Mean air temperature by season ranged 4–5 °C above the respective climatological means (1961–1990) (NOA-IERSD 2013) while wind speed was generally low except for some gusts. In summer the recorded wind velocities were higher than autumn and winter. Some 55 % of the interviewees were males, whereas about 90 % were between 18 and 64 years of age indicating a lower than expected percentage of children and elderly people – accounting for 35 % of the area’s population according to the latest available census (Hellenic Statistical Authority 2013). Males’ average height and weight were 1.77 m 80 kg while females’ were 1.66 m 60 kg approaching the average man and woman (ISO 8996 2004). Almost 45 % of the interviews were conducted in summer.

Table 3 Meteorological conditions during the field surveys
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Univariate linear and ordinal regression analyses between TSV and the explanatory variables showed that all meteorological parameters had statistically significant relationship with TSV. The personal parameters: gender, age, weight, height, body surface, smoking status as well as the psychological factors of opinion on RH and WS, and residence were found to be uncorrelated with TSV so they were excluded from further analysis. Although according to regression analysis Tglobe, Tmrt and Tground have an effect on TSV, these were also excluded from the modeling in order to avoid problems of multicollinearity. Pearson’s correlation test showed a strong relationship between Tair and Tglobe (0.98) as well as Tair and Tmrt (0.97), so only Tair was used in the model since it is easier to measure and with greater accuracy than Tglobe or Tmrt. Strong correlation was also observed between IR↑ and Tground (0.84). The IR↑ was chosen to be included in the model because the measurement of Tground involves greater uncertainty. Also the personal parameters of the additional clothing, sunglasses, and gloves were excluded because they were meaningful only for a specific season.

Extensive model

Linear regression analysis

Considering only the meteorological parameters, the TSV can be estimated by the following formula (standard errors of the parameters in Table 4a):

Table 4 Regression coefficients of simple linear regression prediction models for thermal sensation vote
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$$ \begin{array}{l}{\mathrm{LRP}}_{\mathrm{meteo}}^{\mathrm{ext}}=50.7+0.112\cdot {\mathrm{T}}_{\mathrm{air}}-0.018\cdot \mathrm{RH}-0.10\cdot \mathrm{WS}+0.001\cdot S{R}_{\downarrow }-0.002\cdot I{R}_{\uparrow }-0.05\cdot P\\ {}\kern29.25em \left({R}^2=0.65\right)\end{array} $$
(2)

where LRPmeteo ext is Linear Regression Prediction of TSV by meteorological parameters while the superscript ext stands for extensive model, Tair is in °C, RH in %, WS in m · s−1, SR↓, IR↑ in W · m−2 and P in hPa.

The values generated from Eq. 2 are continuous. In order to test the correspondence between TSVs and the respective predicted votes (LRPmeteo ext), the latter were classified into seven categories using simple rounding to the nearest integer and further assigning values greater than +3 to +3 and lower than −3 to −3. Figure 1a demonstrates the cross-tabulation of TSVs and ranked LRPmeteo ext. The symmetrical measure of association, Gamma was used as an index of agreement. The linear meteorological model predicts TSV classes fairly well (Gamma = 0.81) showing reduced predictability in the categories −3, −2, +2.

Fig. 1
figure 1

Reproduction of the thermal sensation vote by the linear and ordinal extensive models. Distribution of predicted in relation to thermal sensation votes (each row adds to 100 %)

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The linear personal model, LRPpersonal ext (Linear Regression Prediction of TSV by personal parameters in the case of extensive model) indicates the most important personal parameters (Table 4b). The dominant personal factors when combined with the meteorological variables to produce a thermo-physiological model were Icl, shading and exposure time. The linear thermo-physiological model (LRPT/P ext) is presented in Eq. 3 (standard errors of the parameters in Table 4c):

$$ \begin{array}{l}{\mathrm{LRP}}_{\mathrm{T}/\mathrm{P}}^{\mathrm{ext}}=29.3+0.088\cdot {T}_{air}-0.013\cdot RH-0.08\cdot WS+0.001\cdot S{R}_{\downarrow }-0.002\cdot S{R}_{\uparrow}\\ {}\kern4em -0.030\cdot P-0.4\cdot {I}_{cl}-0.50\cdot {S}_1-0.14\cdot {S}_2+0.20\cdot {E}_1+0.10\cdot {E}_2\\ {}\kern27em \left({R}^2=0.67\right)\end{array} $$
(3)

where LRPT/P ext is Linear Regression Prediction of TSV by thermo-physiological parameters for extensive model, Tair is in °C, RH in %, WS in m · s−1, SR↓, IR↑ in W · m−2 and P in hPa, Icl in clo,

$$ {\mathrm{S}}_1=\left\{\begin{array}{l}1\kern1.5em \mathrm{if}\ \mathrm{shading}=\mathrm{no}\ \mathrm{sun}\\ {}0\kern1.25em \mathrm{if}\ \mathrm{shading}=\mathrm{shade}\ \mathrm{of}\ \mathrm{a}\ \mathrm{building}/\mathrm{under}\ \mathrm{the}\ \mathrm{sun},\end{array}\right. $$
$$ {\mathrm{S}}_2=\left\{\begin{array}{l}1\kern1.5em \mathrm{if}\ \mathrm{shading}=\mathrm{shade}\ \mathrm{of}\ \mathrm{a}\ \mathrm{building}\\ {}0\kern1.25em \mathrm{if}\ \mathrm{shading}=\mathrm{no}\ \mathrm{sun}/\mathrm{under}\ \mathrm{the}\ \mathrm{sun},\end{array}\right. $$
$$ {\mathrm{E}}_1=\left\{\begin{array}{l}1\kern1.5em \mathrm{if}\ \mathrm{exposure}\ \mathrm{time}\le 5\ \min \\ {}0\kern1.5em \mathrm{if}\ \mathrm{exposure}\ \mathrm{time}>5\ \min, \end{array}\right. $$
$$ {\mathrm{E}}_2=\left\{\begin{array}{l}1\kern1.5em \mathrm{if}\kern0.5em 5\ \min <\mathrm{exposure}\ \mathrm{time}<1\mathrm{h}\\ {}0\kern1.5em \mathrm{if}\ \mathrm{exposure}\ \mathrm{time}\le 5\ \min\ \mathrm{or}\ge 1\mathrm{h}\end{array}\right. $$

The Gamma statistic (0.82, p < 0.001) suggests that the predictions of the thermo-physiological model were slightly improved in relation to those of the meteorological model. Nevertheless the cross-tabulation of the results (Fig. 1b) shows that the model classified all cases in the middle five categories, therefore it cannot be considered as successful.

Ordinal regression analysis

Applying ordinal regression to the meteorological parameters resulted that the model included the same parameters with the linear one while the cross-tabulation of the results (Fig. 1c) (Gamma = 0.80) indicated that the ordinal model produces improved predictions of thermal sensation in categories −2, +2 and +3. The ordinal regression prediction of TSV by meteorological parameters (ORPmeteo ext) can be estimated according to the method described in the Appendix. The cumulative probabilities were calculated according to Eq. (4) (standard errors of the parameters in Table 5a):

Table 5 Regression coefficients of ordinal regression prediction models for thermal sensation vote
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$$ \begin{array}{l}\mathrm{P}\left(\le -3\right)={\left\{1+ \exp \left[-\left(-101.0-\left(0.24\cdot {\mathrm{T}}_{\mathrm{air}}-0.035\cdot \mathrm{RH}-0.22\cdot \mathrm{WS}+0.002\cdot {\mathrm{SR}}_{\downarrow }-0.004\cdot {\mathrm{IR}}_{\uparrow }-0.10\cdot \mathrm{P}\right)\right.\right]\right\}}^{-1}\\ {}\mathrm{P}\left(\le -2\right)={\left\{1+ \exp \left[-\left(-99.4-\left(0.24\cdot {\mathrm{T}}_{\mathrm{air}}-0.035\cdot \mathrm{RH}-0.22\cdot \mathrm{WS}+0.002\cdot {\mathrm{SR}}_{\downarrow }-0.004\cdot {\mathrm{IR}}_{\uparrow }-0.10\cdot \mathrm{P}\right)\right.\right]\right\}}^{-1}\\ {}\mathrm{P}\left(\le -1\right)={\left\{1+ \exp \left[-\left(-97.0-\left(0.24\cdot {\mathrm{T}}_{\mathrm{air}}-0.035\cdot \mathrm{RH}-0.22\cdot \mathrm{WS}+0.002\cdot {\mathrm{SR}}_{\downarrow }-0.004\cdot {\mathrm{IR}}_{\uparrow }-0.10\cdot \mathrm{P}\right)\right.\right]\right\}}^{-1}\\ {}\mathrm{P}\left(\le 0\right)={\left\{1+ \exp \left[-\left(-94.9-\left(0.24\cdot {\mathrm{T}}_{\mathrm{air}}-0.035\cdot \mathrm{RH}-0.22\cdot \mathrm{WS}+0.002\cdot {\mathrm{SR}}_{\downarrow }-0.004\cdot {\mathrm{IR}}_{\uparrow }-0.10\cdot \mathrm{P}\right)\right.\right]\right\}}^{-1}\\ {}\mathrm{P}\left(\le 1\right)={\left\{1+ \exp \left[-\left(-93.1-\left(0.24\cdot {\mathrm{T}}_{\mathrm{air}}-0.035\cdot \mathrm{RH}-0.22\cdot \mathrm{WS}+0.002\cdot {\mathrm{SR}}_{\downarrow }-0.004\cdot {\mathrm{IR}}_{\uparrow }-0.10\cdot \mathrm{P}\right)\right.\right]\right\}}^{-1}\\ {}\mathrm{P}\left(\le 2\right)={\left\{1+ \exp \left[-\left(-91.2-\left(0.24\cdot {\mathrm{T}}_{\mathrm{air}}-0.035\cdot \mathrm{RH}\hbox{--} 0.22\cdot \mathrm{WS}+0.002\cdot {\mathrm{SR}}_{\downarrow }-0.004\cdot {\mathrm{IR}}_{\uparrow }-0.10\cdot \mathrm{P}\right)\right.\right]\right\}}^{-1}\\ {}\mathrm{P}\left(\le 3\right)=1\end{array} $$
(4)

The probability of each TSV category was computed by the subtraction of the cumulative probability per two consecutive classes (e.g. P(3) = P(≤3)-P(≤2)). The ORPmeteo ext category was the one with the maximum probability.

The parameters that remained in the ordinal personal model (ORPpersonal ext) (Table 5b) were the same with the linear personal model (LRPpersonal ext) with the addition of clothing color. The thermo-physiological ordinal model (ORPT/P ext) (Table 5c) reproduced TSVs slightly better than the ordinal meteorological model (ORPmeteo ext) (Gamma = 0.82) (Fig. 1d) predicting cases in class −3. In comparison with the respective linear, the thermo-physiological ordinal model included the same parameters and Gamma statistics, while it reproduced the extreme categories with greater success.

Final model

Concluding that the ordinal thermo-physiological model (Table 5c) provides the best overall predictions of TSVs, an attempt was made to further improve this model assuming that psychological factors may account for, at least a part of, the deviation between the predicted and the TSV. Thus, the difference of thermal sensation vote and the ordinal thermo-physiologically predicted thermal sensation vote was estimated (DTSV = TSV − ORPT/P ext). Then univariate ordinal regression analysis was performed between the DTSV and the psychological parameters. No statistically significant correlation was identified between DTSV and visit purpose, companionship, region of residence, the place that people were before they visited the experimental site (recent experience), people’s opinion about RH, weather’s suitability for activity and people’s preference in relation to RH. The model of DTSVext is presented in Table 6a.

Table 6 Regression coefficients of psychological parameters in Ordinal Regression Prediction models for the difference of thermal sensation vote and thermo-physiological ordinal regression prediction of thermal sensation vote (DTSV)
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The effect of psychological factors on TSV were taken into account in the final model (ORPfinal ext) by adding the predicted DTSV (DTSVext) to the ORPT/P ext (ORPfinal ext = ORPT/P ext + DTSVext). The cross-tabulation of the results shows that the resulting model reproduces fairly well thermal sensation vote (Gamma = 0.83) even in the extreme categories (Fig. 1e).

Empirical model

Linear regression analysis

An empirical thermo-physiological model was established using as independent variables Tair, Tmrt, RH, WS, Icl and M. According to the linear thermo-physiological model, TSV can be estimated according to the equation (standard errors of the parameters in Table 4d):

$$ \begin{array}{l}\mathrm{LR}{\mathrm{P}}_{\mathrm{T}/\mathrm{P}}^{\mathrm{emp}}=-2.3+0.120\cdot {\mathrm{T}}_{\mathrm{mrt}}-0.11\cdot \mathrm{WS}-0.4\cdot {\mathrm{I}}_{\mathrm{cl}}\\ {}\kern14em \left({\mathrm{R}}^2=0.65\right)\end{array} $$
(5)

where LRPT/P emp is Linear Regression Prediction of TSV by thermo-physiological parameters while the superscript emp stands for empirical model, Tmrt is in °C, WS in m · s−1, Icl in clo.

Interestingly, air temperature and relative humidity were not included in the model. The model indicated mean radiant temperature as the most important meteorological variable in the determination of TSV in accordance to Mayer (1993). Although the WS variation during the measurements was low, WS remained in the model. We also note that the correlation coefficient (0.65) is practically the same as the corresponding extensive model (LRPT/P ext), while the respective cross-tabulation (Fig. 2a) produced an identical Gamma measure of symmetry (0.82). Observing Fig. 2a more closely, we see that the percentage of correctly predicted TSV is higher in the warmer categories in relation to the corresponding extensive model.

Fig. 2
figure 2

Reproduction of the thermal sensation vote by the linear and ordinal empirical models. Distribution of predicted in relation to thermal sensation votes (each row adds to 100 %)

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Ordinal regression analysis

The ordinal thermo-physiological model ORPT/P emp (Table 5d) includes the same parameters as the corresponding empirical linear model (LRPT/P emp), while the cross-tabulation gives almost the same Gamma statistic (0.81) with the linear model but behaves even better in the extreme categories of thermal sensation.

However, both linear and ordinal empirical models do not produce values in the category −3.

Final model

The overall empirical model (ORPfinal emp) corrected for psychological effects, was finally composed based on the ordinal thermo-physiological empirical model. Following the same method as in the extensive model, the difference of TSV minus the ORPT/P emp (DTSV ′ = TSV − ORPT/P emp), was estimated. Ordinal regression was applied between DTSV´ and those subjective psychological factors not related with personal opinion or preference. The analysis showed that DTSV´ was related with season and purpose of visit (Table 6b). Adding the ORPT/P emp and the predicted DTSV (DTSVemp), results in the overall empirical model (ORPfinal emp = ORPT/P emp + DTSVemp). The cross tabulation of the ORPfinal emp (Fig. 2c) demonstrates an improvement in relation to the empirical thermo-physiological model (ORPT/P emp) albeit worse performance compared to the extensive overall model (ORPfinal ext) (Gamma = 0.82), notably not classifying any cases as cold (−3).

Overview of the models’ parameters

The main meteorological parameters affecting thermal sensation were Tair and WS. RH showed a negative effect on TSV in accordance to Givoni et al. (2003) and Nikolopoulou (2004) although in other studies a positive effect was identified (Monteiro and Alucci 2008). The personal models demonstrated that recent thermal experience and health status affect TSV as well highlighted the important role of adaptation and physical fitness. People previously exposed in a controlled environment or with a health status- on the day of the interview- ‘worse than usual’ felt ‘warmer’ in relation to those who were in a non air-conditioned place or their health status was ‘better than usual’. However one reason for these results could be the unbalanced distribution of questionnaires towards mild and hot season when people’s thermal discomfort may be expressed by voting higher classes of thermal sensation scale. The main personal parameters that also remained in the thermo-physiological models were Icl, shading and exposure time. The interviewees reported higher classes of TSVs as Icl decreased in agreement with the UTCI-clothing model (Havenith et al. 2012) which indicated that Icl decreased with rising air temperature. The respondents who were exposed to the sun during the interview or those who were at the site of the interview less than 5 min indicated higher classes of TSV compared to those who were not exposed to the sun or were at the site of the interview for more than 1 h. Metabolic rate was not retained in the model even though in the context of metabolic heat production it is an important factor of the heat balance equation (Fanger 1970; Katavoutas et al. 2009) and expectedly of thermal sensation. However, the context of the experiment narrowed the variation of this parameter and as a result it appears to have an insignificant effect on pedestrian TSV.

The deviation of predicted TSV by the extensive and empirical thermo-physiological models from the TSV may be justified by psychological factors. In summer and in autumn the predicted TSVs were higher than TSVs and these differences were larger than those in winter indicating that the variation may be attributed to expectation. The extensive model showed that the abovementioned deviation was biased by overall comfort at the site of the interview and preference. The dominant preference variables were those related to air temperature and irradiation. The empirical model indicated that the purpose of visit stimulated predicted TSVs higher than the respective TSVs. This could be attributed to adaptation as visitors with a purpose (usually work) remained on site for longer periods than those being there for fun or simply passing by.

Conclusions

This study demonstrates the methodology and the evaluation of modeling outdoor pedestrian thermal sensation in order to extend our knowledge in the case of the urban Mediterranean climates and to examine the appropriateness of using linear and ordinal regression for the statistical analysis. The predictive models arose using data of a field survey where people were asked to indicate their thermal sensation and were interviewed in personal and psychological inquiries while there were in situ measurements of meteorological parameters.

The interviews carried out under normal conditions for each of the three seasons, warm, cool, and transitional. The study area included three different locations, a square, a street canyon and a coastal area in the larger urban area of Athens. People who participated in the study were in the majority healthy and their activity was moderate.

The thermal sensation of the respondents’ was influenced by weather and personal factors while the deviation of the actual from the predicted thermal sensation was affected by psychological factors. The analysis showed an improvement in the prediction model based purely on meteorological parameters when personal parameters were taken into account and even greater improvement when psychological factors were considered. Both linear and ordinal regression models indicated the same parameters as determinants of thermal sensation, with only two exceptions, health status and air-conditioning. In the ordinal thermo-physiological model remained the factor of health status. Overall ordinal regression results showed an improvement in relation to linear yielding better results in the case of the extreme values of thermal sensation scale. Linear showed better reproduction of TSV in the middle categories (−1, 0, +1). Regarding only meteorological variables, thermal sensation was determined by air temperature, relative humidity, solar radiation, irradiation and ambient pressure. In the thermo-physiological model the factor of exposure time which in combination to recent experience (Place before exposure) and exposure history (Air conditioning) depicts the effect of adaptation on thermal sensation while the shading parameter highlights the importance of solar radiation. The psychological parameters that affect TSV were the season and preference that illustrate the effect of expectation and overall comfort. In the empirical model that only the objective psychological parameters were considered, the factors that showed an effect on TSV were the season and the visit purpose. People visiting the site for work tended to indicate lower thermal sensation than the predicted signifying a decisive role of adaptation in TSV. Pantavou et al. (2013) concluded that the interviewees showed better adaptation to warm than to cool environment.

The empirical model produced satisfactory results in relation to the extensive indicating that choosing the right parameters TSV could be predicted as well. According to the empirical thermo-physiological model TSV can be predicted by Tmrt, WS and Icl.

The results of this study could be useful in estimating thermal sensation in Mediterranean climates for health and touristic purposes, as well as for planning outdoor spaces. Nevertheless this research should be expanded to greater range of weather conditions and especially in heat waves since Athens is a very popular summer destination. Moreover it would be interesting to compare the thermal sensation with the already existing models.