155
Measurements were carried out in 10 rented apartments and 5 privately owned single family houses. Five of the 156
apartments were naturally ventilated (apart from an exhaust hood in the kitchen) while the other five were equipped 157
with constantly running exhaust ventilation from the kitchen and bathroom. Three of the single family houses were 158
naturally ventilated while the other two were equipped with exhaust ventilation. 159
With the exception of one (located 60 km from Copenhagen) all dwellings were located less than 25 km from 160
Copenhagen. 161
Features of the dwellings are described in Table 2. 162
All dwellings used waterborne radiators/convectors and natural gas boilers as a primary means of heating and two of the 163
dwellings (number 10 and 16) had a wood burning stove. 164
8
The indoor environment sensors were placed on internal walls at a height of roughly 1.8 m above the floor. We 167
attempted to place the sensors so that they would not be hit by direct sunlight, but due to acceptance of the occupants in 168
the dwellings and other practicalities this was not always possible. In the cases when direct sunlight fell on the sensors 169
the temperature measurements were corrected for the heating of the sensor. This was done in periods when the 170
measured illuminance was larger than 1000 lux. In these cases the temperature was corrected by linear interpolation 171
between temperature measurements 30 minutes prior to and one hour after direct sunlight fell on the sensor. 172
The CO2 concentration was used as an indicator of the occupancy of the rooms where the measurements took place. If 173
the CO2 concentration was below 420 ppm and the window was closed the room was classified as being unoccupied. 174
Furthermore, if the CO2 concentration was higher than 420 ppm, but decreased and continued to decrease until reaching 175
values below 420 ppm and the window was closed in the entire period, the room was classified as unoccupied during 176
the period of concentration decay. 177
The value of 420 ppm was chosen since earlier observations had shown that the outdoor concentrations might reach 178
levels of up to 400 ppm. To ensure that long unoccupied periods were not classified as occupied an uncertainty range of 179
20 ppm was added to the highest observed outdoor concentration. 180
The room was classified as occupied if the window was open. This classification was based on a questionnaire survey 181
conducted by Andersen et al. [30] who found that the statement “I had to leave the dwelling” was mentioned amongst 182
the most common reasons for closing windows. 183
If the bedroom and the living room were both unoccupied, the dwelling was classified as unoccupied. Periods when the 184
dwelling was unoccupied were not taken into consideration in the analysis. 185
When analysing the window opening data the database was divided depending on the state of the window (open/closed) 186
to infer the probability of opening and closing the window (change from one state to another) separately. The 15 187
dwellings were divided into four groups on the basis of the ownership (owner-occupied or rental) and the ventilation 188
type (natural ventilation or mechanical ventilation) (table 1). 189 190 191 192 193 194
9 1 Owner-occupied Natural 3, 4, 16 2 Owner-occupied Mechanical 1, 10 3 Rental Natural 6, 8, 9, 11, 12 4 Rental Mechanical 5, 7, 13, 14, 15 196
Table 2 shows the relation in each of the four groups between inhabitants (age and number), dwelling characteristics 197
(building construction or renovation years and the dwelling size) and the frequency of openings. 198
199
Table 2. Description of residents and characteristics of the dwellings 200 Group Dwelling number Number of openings in period
Average age of the residents Number of residents Year of construction (and renovation) Floor area ( m²) 1 3 82 57 2 1928 145 4 235 70 2 1956 (1976) 130 16 153 26 2 1967 139 2 1 334 65 1 1994 126 10 65 59 2 1901 (1957) 190 3 6 337 78 2 1945 86 8 258 55 2 1945 109 9 25 35 3 1945 87 11 82 71 2 1945 77 12 1 64 1 1945 109 4 5 73 76 2 1981 (2001) 83 7 718 63 1 1981 (2001) 83 13 341 60 3 1981 (2001) 80 14 241 28 2 1981 (2001) 85 15 166 60 4 1981 (2001) 84
3.1 Statistical Analysis
201Multivariate logistic regression with interactions between selected variables was used to infer the probability of a 202
window opening and closing event. The method relies on the probability function described in formula 1. 203 204 log �1−pp � = a + b1∙ x1+ b2∙ x2+ ⋯ + bn∙ xn (1) 205 Where, 206
p is the probability of an opening/closing event 207
a is the intercept
208
b1-n are coefficients 209
x1-n are variables such as temperature, CO2 concentration etc. 210
211
However, the probability might depend differently on x1 at one level of x2 as compared to another level of x2 (e.g. an 212
increase in temperature might increase the probability of opening a window at high CO2 levels, whereas the same 213
10
well described by a model based on equation 1. Equation 2 deals with interactions between variables by adding 215
interaction terms to the model. 216
217
𝑙𝑜𝑔 �1−𝑝𝑝 � = 𝑎 + 𝑏1∙ 𝑥1+ 𝑏2∙ 𝑥2+ ⋯ + 𝑏𝑛∙ 𝑥𝑛+ 𝑐12∙ 𝑥1∙ 𝑥2+ 𝑐13∙ 𝑥1∙ 𝑥3+ ⋯ (2) 218
Equation 2 was used to infer the probability of windows being opened or closed. The Akaike information criterion 219
(AIC) was used as a basis for forward and backward selection of variables in the regression models [31]. Each 220
individual variable was first fitted to the measured window opening data and then AIC was calculated for each fit. The 221
variable with the lowest AIC was selected and the remaining variables were then tested one by one on a bivariate level, 222
to see if any of the bivariate models resulted in a lower AIC. If this was the case, the remaining variables were tested in 223
a model with three variables and so on (forward selection). At each step, the AIC was also calculated for models, where 224
each of the selected variables was removed from the models (backward selection). In this way, the final model included 225
variables and interaction terms that resulted in the lowest AIC. To limit the complexity of the model, only interaction 226
terms between continuous and nominal variables, e.g. indoor temperature and day of week were included in the 227
analyses. 228
The statistical analyses were conducted using the statistical software “R” and the models were inferred using the ‘step’ 229
function in R. [32] 230
In the interpretation of the coefficients, the sign, the size and the scale of the corresponding variable have to be taken 231
into account. For example, a coefficient for solar hours of 0.057 might seem to impact the probability more than an 232
outdoor relative humidity coefficient of 0.029 (group 4, opening model). However, the scales of the two variables (solar 233
hours: 0 to 16.1, outdoor RH: 28% to 100%) should be taken into account: Schweiker et al. [33] suggested to multiply 234
the scale of the variable with the coefficient, to get an indication of the magnitude of the impact from each variable. In 235
the example described above the magnitude of the impact was 0.057 · (16.1-0) = 0.91 and 0.029 · (100-28) = 2.08 for 236
the solar hours and the outdoor relative humidity respectively, revealing that the outdoor RH had a higher impact on the 237
probability than the solar hours. 238
When using logistic regression, it is required that all variables are independent. Since the data was obtained in 15 239
dwellings with different physical properties and different inhabitants, all variables could not be assumed a priori to be 240
independent of the dwelling it was obtained from. Variable independency was tested by assigning an index to each of 241
the dwellings, which was used as a factor in the analyses. If an interaction term between a variable and the dwelling 242
11
model. All variables which did not interact with the dwelling number were assumed to be independent of the individual 244
dwelling. 245
Correlations between explanatory variables may result in inflation of the estimated variance of the inferred coefficient, 246
which in turn will result in too wide confidence intervals. To estimate the size of the inflation due to correlations 247
between all explanatory variables (multicolinearity), generalized variance inflation factors (GVIF) were calculated for 248
coefficients of all continuous explanatory variables. The GVIF estimates the inflation of the variance, due to 249
multicolinearity as compared to no multicolinearity. Since the GVIF is an estimate of the inflation of the variance, the 250
GVIF2∙DF1 is an estimate of the factor by which the standard error and confidence interval is inflated due to 251
multicolinearity between explanatory variables. 252
Prior to the regression analyses, four variables were transformed to obtain a better distribution. Table 3 describes how 253
the variables were transformed. 254
255
Table 3: Variable transformations 256
Variable Transformed variable
CO2 concentration [ppm] Log(CO2) [Log(ppm)]
Illumination [Lux] Log(Illumination) [Log(Lux)] Wind speed [m/s] Log(Wind speed+1) [Log(m/s)] Solar radiation [w/m²] Log(Solar radiation+1) [Log(W/m²)]
257
12
In this section the main results of the statistical analysis are presented. Table 4 presents descriptive statistics of all 260
measured variables in each of the four groups. 261
262
Table 4. Descriptive statistics of the monitored variables 263 Indoor temperature Indoor R.H. CO2 Outdoor temperature Outdoor R.H. Lux Wind Solar Radiation Solar Hours GROUP 1 wi n d o ws cl o sed Max 30.3 69 3065 26.9 100 16063 13.2 918 16.1 Min 17.1 24 355 -6.9 24 4 0.0 0 0.0 Mean 22.1 46 862 9.6 76 159 2.8 199 8.5 Median 21.8 45 773 9.3 76 51 2.5 63 8.1 St. Dev. 2.0 7 369 6.2 18 458 2.1 252 5.0 wi n d o ws o p en Max 29.2 67 2229 25.5 100 8077 9.1 904 16.1 Min 17.2 26 328 -1.4 30 4 0.0 0 0.0 Mean 22.9 38 520 13.5 61 278 3.1 413 10.8 Median 22.8 38 464 13.6 58 99 3.0 437 13.0 St. Dev. 1.8 6 175 5.1 18 447 1.7 272 4.7 GROUP 2 wi n d o ws cl o sed Max 27.3 49 4453 24.0 100 1494 17.3 904 14.9 Min 13.5 24 377 -6.0 25 4 0.0 0 0.0 Mean 22.3 36 722 7.5 75 111 3.3 165 6.9 Median 22.6 35 648 7.0 78 36 2.7 23 6.1 St. Dev. 2.0 4 310 5.1 18 183 2.6 234 4.8 w in d ow s o pe n Max 27.3 53 1959 24.0 100 32280 17.3 883 14.9 Min 12.0 25 363 -6.0 25 4 0.0 0 0.0 Mean 18.1 40 516 8.0 74 295 4.3 203 6.7 Median 17.2 40 468 7.0 78 43 3.7 91 6.3 St. Dev. 3.2 5 142 5.4 19 1500 2.9 240 4.8 GROUP 3 wi n d o ws cl o sed Max 31.2 63 3634 26.3 100 32280 13.0 904 15.3 Min 14.1 21 338 -5.8 24 4 0.0 0 0.0 Mean 22.3 37 780 7.4 73 179 3.3 164 6.1 Median 22.3 37 612 6.8 76 43 2.9 36 5.5 St. Dev. 2.0 5 462 5.2 18 888 2.2 230 5.0 w in d ow s o pe n Max 27.7 54 3295 26.3 100 2456 13.0 883 15.2 Min 11.5 22 333 -5.8 25 4 0.0 0 0.0 Mean 19.9 38 590 7.9 75 80 3.3 141 6.7 Median 19.1 38 520 6.3 80 43 2.9 6 5.7 St. Dev. 3.5 5 232 6.0 19 130 2.2 229 5.4 GROUP 4 wi n d o ws cl o sed Max 28.8 73 4636 28.6 100 23442 13.5 918 16.1 Min 9.8 21 333 -7.7 28 4 0.0 0 0.0 Mean 20.9 42 702 7.9 80 85 3.0 138 6.6 Median 20.8 42 628 7.1 84 36 2.5 11 6.2 St. Dev. 2.2 8 292 5.9 17 206 2.3 208 4.7 w in d ow s o pe n Max 29.1 69 3530 29.4 100 13935 13.5 918 16.1 Min 11.9 22 328 -7.2 28 4 0.0 0 0.0 Mean 22.0 44 492 14.1 71 132 3.1 293 9.2 Median 22.2 43 437 14.6 71 59 2.7 238 9.4 St. Dev. 2.4 8 142 5.9 19 229 2.1 276 5.0 264
The number of the dwelling affected the impact of some of the explanatory variables as concerns the probability of 265
opening and closing a window. This indicates different habits in the different dwellings included in the four groups, 266
which were not described by the measured variables. For example, the CO2 concentration interacted with the dwelling 267
13
concentrations of CO2 in each dwelling. The variables that interacted with the dwelling number were removed from the 269
models where the interaction occurred. In the further analyses, the number of the dwelling was not included, since we 270
were not interested in the behaviour in each single dwelling, but in the overall behaviour in all of the surveyed 271
dwellings. 272
Table 5 shows a list of variables that were removed from the models due to interactions with the dwelling number. 273
274
Table 5. A list of variables that interacted with the dwelling number indicating that they were not independent of the 275
dwelling where they were measured. The table states in which models (Open and/or close) the interactions were found. 276 Model Indoor temperature Outdoor temperature Solar radiation CO2 concentration
Time of day Illumination
Group 1 None None None None None None
Group 2 Open and Close Open Open None None None
Group 3 None None Close Close None None
Group 4 Close None None Close Open and Close Close
277