Statistical modelling of measures of thermal comfort.

Popa, Monica ; Sirbu, Dana ; Curseu, Daniela 等

1. INTRODUCTION

The first requirement for an acceptable thermal environment is that a person feels thermally neutral as regards the whole body, i.e. the worker does not know whether he would prefer a higher or lower ambient temperature level. This is usually evaluated by the PMV index (ISO7730). The index is a function of the activity level, clothing insulation, air speed operative temperature and humidity. Thermal neutrality as described by the PMV index is not the only condition for thermal comfort. A person may feel thermally neutral as regards the body as a whole, but may not be comfortable if one part of the body is warm and other cold. It is therefore a further requirement for thermal comfort that no local warm or cold discomfort exists in any part of the body. Such local discomfort may be caused by an asymmetric radiant field, by local convective cooling (draught) and by a vertical air temperature gradient. The percentage of those who were dissatisfied (PD) due to draught in ventilated working areas is described by the PD index (Melikov et al, 1989).

The logical way of investigating how the thermal climate is perceived in buildings is to ask people to describe the thermal climate with the use of questionnaires and rating scales. Seven-point scales, such as Bedford or PMV scale have often been used. In addition, questions on air motion and indoor air quality are usually asked. Great individual differences were found in the thermal work environment between subjective estimated and calculated PMV values (Popa, 2000).When the temperature at the work site rose, the workers graded the thermal comfort according to the PMV scales as "too warm", although the air temperature was estimated to be lower than the actual value. In offices, the calculated PMV values were lower than the estimated PMV values. Then observation is in agreement with the findings of other studies (Croome et al., 2000; De Dear et al., 2001). The PD index predicts that usually considerably less than 20% of the workers should feel the velocity-related discomfort. The results of the questionnaire concerning draught indicated that there was a markedly great difference between the subjectively estimated and calculated values. The aim of the study was to develop a thermal comfort model by using statistical modeling of the measured environmental thermal variables and subjective ratings of the thermal environment given by the workers in a moderate climate.

2. MATERIAL

A thermal questionnaire was used in 80 working locations. The workers were asked to fill out at least one self-administered questionnaire inquiring thermal sensation, ventilation and whether they felt air movement, and if yes, whether the air movement felt uncomfortable. The workers also graded their thermal comfort according to the PMV scale and estimated the air temperature of the working areas. Altogether 600 workers evaluated their degree of comfort by using the PMV scale. The thermal insulation of a whole clothing ensemble ([I.sub.cl]) was estimated by summing up the values for individual garments. Activity level (M) was estimated by means of tables. The predicted mean vote (PMV index) was determined from the measured values of thermal variables and the values of M and Icl. At the same time, continuous of the thermal variables at the work sites was conducted using indoor climate analysers. The Brilel & Kjaer 1213 Indoor Climatic Analyser measured all the thermal variables including radiant temperature asymmetry. One of the faces of the asymmetry transducer was directed towards the window. The second thermal analyzer was constructed using an HP portable computer, an HP 3421 data acquisition unit, YSI 401 thermistors, Disa 54N50 low velocity flow analyzer and Humicap humidity sensors. Air temperatures were measured at the ankle and neck levels and air velocities at the neck level only. The indoor thermal climate measurements were made according to the ISO 7730 standard. The sensors were fixed on a tripod placed near the worker.

3. METHODS

We consider three response variables which measure the experience of thermal comfort of a person. The first is the subjective experience [Y.sub.0]. The response [Y.sub.0] is a random variable depending on environmental variables including biological and psychological factors of a person. The second measure [Y.sub.1] is the PMV index or PD index. In both cases the model of estimated response [Y.sub.2] of the fitted statistical model of [Y.sup.0]. The predictors x = ([x.sub.1], [x.sub.2], ... [x.sub.k]) of model are the measured environmental variables such as air temperature, air velocity, relative humidity, etc. The selected model is a regression model which is an ordinary linear one (Weissberg, 1985) if [Y.sub.0] is continuous (PMV) and logistic (Hosmer et al., 1989) if [Y.sub.0] is a dichotomous (PD) variable. The statistical analyses have been performed by the SAS software system. The ordinary regression model of the continuous response [Y.sub.0] is:

[Y.sub.0] = [[beta].sub.0] + [[beta].sub.1][x.sub.1] + [[beta].sub.2] [x.sub.2] + ... + [[beta].sub.k][x.sub.k] + [epsilon] (1)

The coefficients [[beta].sub.i], I = 1, ..., k are unknown parameters. The estimates bi of the coefficients [[beta].sub.i] are computed from the data set by the regression analysis. The statistical error [epsilon] is supposed to be normally distributed. The fitted model of [Y.sub.0] is now defined as:

[Y.sub.2] = [b.sub.0] + [b.sub.1][x.sub.1] + [b.sub.2][x.sub.2] + ... + [b.sub.k][x.sub.k] (2)

Comparison of the measures [Y.sub.1] and [Y.sub.2] can be made by the correlation coefficient R. If R ([Y.sub.2], [Y.sub.0]) > R ([Y.sub.1], [Y.sub.0]) then [Y.sub.2] will fit the data set better than PMV index [Y.sub.1]. If the response [Y.sub.0] is a dichotomous variable with two possible values: 1 = satisfied or 0 = unsatisfied, the measure [Y.sub.2] is constructed by a logistic regression model. In this case we are modeling the probability p(x) that a person is satisfied (=1) in a thermal environment x = ([x.sub.1], [x.sub.2], ..., [x.sub.k]):

p(x) = P ([Y.sub.0] = 1 | x) (3)

Thus the probability that a person is unsatisfied in the environment is:

P ([Y.sub.0] = 0 | x) = 1 - p(x) (4)

In a logistic regression model the logit transformation of the probability p(x) is modeled as follows:

g(x;[beta]) = log [p(x)/1 - p(x)] = [[beta].sub.0] + [[beta].sub.1][x.sub.i] + [[beta].sub.2][x.sub.2] + ... + [[beta].sub.k][x.sub.k] (5)

The subjective experience Y0 is the response of the model and the estimates [b.sub.i] of the coefficients [[beta].sub.i] are computed from the data set by the logistic regression analysis. The model of [Y.sub.2] will be the estimated logistic regression model of [Y.sub.0]:

p(x,b) = [e.sup.g(x;b)]/1 - [e.sup.g(x;b)], g(x;b) = [b.sup.0] + [b.sub.1][x.sub.1] + [b.sub.2][x.sub.2] + ... + [b.sub.k][x.sub.k] (6)

Thus P([Y.sub.2] = 1 | x) = p(x;b). As a measure of difference between the observed value of [Y.sub.0] and the fitted value of either [Y.sub.1] or [Y.sub.2] is the Pearson residual:

r([y.sub.j], [p.sub.j]) = [y.sub.j] - [n.sub.j][p.sub.j]/[square root of [n.sub.j][p.sub.j](1 - [p.sub.j]) (7)

Here [y.sub.j] is the observed value of [Y.sub.0], [p.sub.j] = P([Y.sub.1] = 1 | [x.sub.j]) (PD index) or [p.sub.j] = P([Y.sub.2] = 1 | [x.sub.j]), [n.sub.j] is the number of observations classified according to x. In unclassified data [n.sub.j] = 1, j = 1, 2, ... n.

The measure [Y.sub.1] or [Y.sub.2] which has the smallest [X.sup.2] is to be considered as the best estimate of [Y.sub.0]. The Pearson chi-square statistic is the summary statistics:

[Z.sup.2] = [n.summation over (j=1)] r[([y.sub.j], [p.sub.j]).sup.2] (8)

4. RESULTS

The response variable Y0 is subjective experienced PMV. A stepwise procedure has selected the following environmental variables to the estimated regression model for the measure [Y.sub.2]:

Environmental variable Parameter Standard P-
 estimate b error of b value

Constant -6.5054 0.7808 0.0001
Winter/summer 0.5184 0.1349 0.0001
Morning/afternoon -0.3456 0.0981 0.0005
Man / woman 0.2768 0.1052 0.0088
Turbulence intensity -0.0055 0.0025 0.0307
Metabolic rate 0.0063 0.0018 0.0007
Air temperature, ankle -0.0972 0.0465 0.0373
Air temperature, neck 0.3608 0.0435 0.0001
Relative humidity 0.0128 0.0060 0.0332
Temperature asymmetry -0.0347 0.0197 0.0789

The correlation coefficients are R ([Y.sub.2], [Y.sub.0]) = 0.52 and R ([Y.sub.1],[Y.sub.0]) = 0.25. This indicates that the estimated statistical regression model of [Y.sub.2] (PMV) fits the data set better than the deterministic model of [Y.sub.1] (PMV). When the variable [Y.sub.0] is the subjective experienced PD, a stepwise procedure has selected the following environmental variable to the estimated logistic regression model for the measure [Y.sub.2]

Environmental Parameter Standard P-value
variable estimate b error of b

Constant 5.4216 1.6977 0.0014
Metabolic rate -0.0124 0.0039 0.0014
Morning/afternoon -1.2140 0.6027 0.0440
Air temperature, neck -0.1959 0.0588 0.0009
Air velocity 1.7300 0.7988 0.0303

The Pearson chi-square statistic for the logistic regression model of [Y.sub.2] was 592 and for the deterministic model of [Y.sub.1] was 3199.This indicates that the estimated logistic regression model of [Y.sub.2] fits the data set better then the deterministic model of [Y.sub.1](PD).

5. DISCUSSION

The correlation coefficient of subjective experienced PMV and calculated PMV (IS0 7730) was 0.25, which was considered weak. This indicates that the calculated PMV measured poorly the workers' degree of discomfort. The linear regression model was used to get a better fit between the estimated PMV and the measured environmental thermal variables. The correlation coefficient was now 0.52, being two times higher than that obtained by the method given in ISO 7730. The workers' subjective estimation correlated with the percentage of people feeling draught to the measured environmental thermal variables. The estimated logistic regression model fits the data set better than the deterministic model of the PD according to the Pearson chi-square-statistic. These models seem to explain the actual sensations of the workers much better than the complicated index.

6. REFERENCES

Croome, D.C. & Baizhan, L (2000) Productivity and indoor environment, Proceedings of Healthy Buildings 2000, vol.1, pp 629 -634.

De Dear R. & Brager S.G. (2001) The adaptive model of thermal comfort and energy conservation in the built environment, International Journal of Biometeorology, vol.45, nr.2, pp. 100-108.

Hosmer, D.& Lemeshow, S.(1989) Applied logistic regression. John Wiley & Sons, New York.

ISO 7730 (1996) Moderate thermal environments. Determination of the PMV and PPD indices and specification of the conditions for thermal comfort. International Organization for Standardization, Geneva.

Melikov, A.K.; Fanger P.O. (1989) Draught risk in practice, Proceedings of 2nd World Congress on Heating, Ventilating, Refrigerating and Air-conditioning, vol. 3, pp. 121-126.

Popa, M. & Popa, M.S. (2000). Indoor air quality problems in Romania: monitoring and control, Proceedings of the International Conference on Advanced Metrology "Metrologia 2000, vol.1, pp. 476-482.

Weissberg, S. (1985) Applied linear regression, second edition, John Wiley & Sons, New York.