Human outdoor thermal comfort analysis for the Qatar 2022 FIFA World Cup’s climate

It is explored, in this work; some well-known classic methods to calculate thermal comfort, contrasting them with a method proposed here that is based on the Principal Components Analysis for the Doha Metropolitan Region (DohaMR) in Qatar. The Principal Components Analysis takes into account the natural outdoor space, which is influenced by the external atmosphere variables. The purpose of the comfort index is to measure the atmospheric variability and the result shows whether thermal comfort increases or decreases from one month to the next or seasonally. Considering the predominant climate characteristics of Qatar, it was possible to identify that among classical and canonical urban thermal comfort indices investigated, the Principal Components Index provides convenient evidence to be also appropriate. The overall vision of the final results of the study is related to the equivalence between the classical climate-dependent thermal comfort indices and the proposal of a self-explanatory index by the linear combination of the atmospheric variables, which captures the greatest joint variability, without a pre-defined equation, but rather by an empirical equation. The observed atmospheric variables determine, locally, the thermal comfort experienced by humans. The main conclusion of this research is the simplicity, and equiprobability, of calculating thermal comfort using the characteristic history of the atmospheric variables that can be used. Based on the principle of Principal Components construction, which captures the largest source of variability through an empirical linear combination.


Introduction
The studies related to urban thermal comfort are important to help in the planning and management of urban space, contributing to the development of more thermally pleasant environments, especially in cities affected by a marked combination of air temperature, air relative humidity, wind speed and solar radiation. It is also worth mentioning that these indices express exact measures for the feeling of thermal comfort that tries to cover most of the human population, based on some meteorological variables. One knows that thermal comfort indices are vital tools in assessing outdoor thermal comfort in hot, semi-arid (warm), arid and desert environments. There are several studies that introduce and evaluate thermal comfort indices. However, the selection of a thermal comfort index, which portrays the real human sensation, is a challenge, especially for the evaluation of outdoor thermal comfort in environments with high temperatures and low relative humidity.
It is important to point out the gaps in research and the significance of this study, the motivation: The changes that have occurred in the urban climate system of large cities are processes related to urban densification, in this context some works indicate that the classification range of classical thermal comfort indices is not appropriate for regions located under a certain climatic condition, such as humid tropical, semi-arid or desert. In this way, it is proposed in this work the construction of a self-explanatory index that can be used, dare one say without parsimony, in any empirically defined climate.
The urban open spaces provide a variety of activities such as sports, recreational activities and outdoor walking for a variety of purposes. When using these spaces, people are exposed to atmospheric conditions that interfere with their sensation of thermal comfort. In order to evaluate thermal comfort in these environments, indices are developed from the combination of influential factors, seeking to translate, through objective measurements, the feeling of comfort perceived by humans. One believes in the need to explain, not too much but enough, about thermal comfort because it is known to known to peers but many times unknown by the peculiar ones, but unknown to the even ones who may be interested in this research. One recognizes that it is confusing especially in cities affected by a marked combination of air temperature, relative humidity, wind speed and solar radiation.
The human well-being can be directly affected by urban climate. This influence can occur through variables such as solar radiation intensity, wind speed, and the amount of water available in the atmosphere, estimated through air temperature and relative humidity. Studies on the analysis of thermal comfort in open spaces (outdoors) consider the rate of metabolism, clothing, solar radiation, and physiological responses to the combined effects between climatic or meteorological factors and the activity being performed, in addition to the inclusion of psychological factors in the analysis. Bioclimatic comfort is a state of complete physical, mental, and social well-being of the individual in relation to his environment, which generates the feeling of comfort, favorable to rest and to the development of various outdoor activities. Saud Ghani et al. [1] provides a comprehensive and rich literature review on human thermal comfort indices, where they mention [2,3], reference works widely cited in scientific articles. These studies assess the outdoor thermal comfort conditions using several classic thermal comfort indices. Besides, these studies mentioned that comfortable and healthy outdoor microclimates are beneficial to sustainable urban development.
Some elements of the atmosphere of some environments are directly related to thermal comfort [4] and, consequently, to the quality of life of the human population. Outdoor thermal comfort is the thermal neutrality perceived by people. Besides environmental aspects, thermal comfort is also influenced by behavioral and physiological aspects. Thermal comfort studies aim to analyze and establish the conditions necessary for human satisfaction, allowing them to feel thermally comfortable in order to increase their physical and/or intellectual performance. Experts define the importance of thermal comfort studies, since studies show a clear trend that discomfort caused by heat or cold, reduces human performance in intellectual, manual and perceptual activities [5].
It is well known that the sensation of thermal comfort takes place when heat exchanges between the environment and people take place without further effort, allowing the maximum physical and intellectual capacity of individuals. However, if the thermal environmental conditions cause sensation of heat or cold, additional effort will be required from the individual, which may be reflected in metabolic overload. Since the sensation of thermal comfort derives from the interaction of subjective environmental and personal variables, one way to express this sensation is through calculations that aggregate atmospheric variables and convert them into indicators of thermal comfort. The more meteorological variables are considered for the calculation of a given thermal comfort index, the more representative its results will be. In fact, the selection contemplates the adaptation, to arid and desert regions, of indexes applied in urban thermal comfort evaluations in tropical regions, since their adequacy to the context to be analyzed and evaluated is pointed out as an important aspect in decision-making process.
It is well conceived, by specialists in the field of human thermal comfort, the idea that the evaluation of thermal comfort implies the definition of indices whose results are grouped according to classes that seek to reflect the level of comfort perceived by the human being as a consequence of thermal conditions (thermal neutrality) acceptable to the body. One knows that the thermal neutrality is a necessary, but not sufficient, condition for a person to be in thermal comfort. It is worth mentioning that an individual who is exposed to an asymmetric radiation field may very well be in thermal neutrality, but will not be in thermal comfort. Thermal neutrality is the condition in which a person prefers neither more heat nor more cold in the surrounding environment, which provides the mind with information that expresses satisfaction with the temperature of the body as a whole.
The analysis are done according to the standards and very well established thermal comfort indices as the Thermal Discomfort Index (TDI), the Temperature and Humidity Index (THI), the HUMIDEX, the Standard Effective Temperature (ETI) and the Effective Temperature as a function of Wind (ETWI); and to conclude, as an alternative to the Universal Thermal Comfort Index (UTCI)-widely addressed in the literature mainly concerning climate change, one proposes, a parsimonious robust alternative: The determination of a climate-based composite index by means of Principal Component, hereafter referred to as Principal Component Thermal Comfort Index (PCTCI). One of the inconveniences of the classical indices is that they do not consider solar radiation conditions, due to the scarcity of data the solar radiation is neglected -indices without solar charge information.
The Temperature and Humidity Index (THI) and Thermal Discomfort Index (TDI) are based on the conditions of the atmospheric variables: air temperature and relative humidity and are easier to be evaluated, especially at a comparative level (Thom, 1959). The other thermal comfort indices used in this study-Effective Temperature (ETI) and the Effective Temperature as a Function of Wind (ETWI)-were chosen according to [6], The ETWI has the same theoretical basis and application as the ETI, as can be seen in [6], differing from it only by taking into account the action of the wind. HUMIDEX, suggested by [7], also employs these variables for warm climates [8]. One would likes to note that the R-script has been coded faithfully considering the proposed equations, and is available.
The differential of the work, proposed in this paper, is the proposal to apply a methodology widely used in statistical methods with the evaluation of equivalence with thermal comfort indexes already established in the outdoor thermal comfort indices' literature. Considering the diversity of methodologies employed in the definition of these indices, the differences in climate conditions prevailing in the environments in which the research were developed, and the possibility of metabolic changes as a function of adaptation to these climate conditions, this study sought after to verify the existence of divergences in classification obtained by different indices for the 2022 FIFA Qatar World Cup host cities. This will take place in the transition period to the beginning of the most pleasant season to visit the country and leisurely walk through the streets.
This study uses Qatar as an illustration of a pilot study to calculate various thermal comfort indices, and introduce an alternative index, in order to inform the best time of year that is most appropriate for overland travel and to make the movement of passersby fluid and as practical as possible, so that people can move around easily and in a condition to experience pleasant thermal comfort.
The key-question about the study purpose is the insertion of the idea based on the empirical construction of an outdoor human thermal comfort index by means a linear combination of meteorological variables (of one's own choice), whose coefficients are determined by methods that capture the greatest variability among them. One knows that the thermal comfort derives from the interaction between environmental and personal variables; one way to express this feeling is through calculations that aggregate meteorological variables and convert them into thermal comfort indices.

Material
Among several features, the FIFA World Cup Qatar 2022 will also be known as the mobility cup (people's ability to move around within the urban space), that takes place in Doha Metropolitan Region (DohaMR): Lusail, Al Wakrah, Al Rayyan, Al Khor and Doha. The idea, of this manuscript, is to inform the season time of the year that is best appropriate for overland displacement and to make this movement fluid and as practical as possible, so that people can move around easily and safely. In a small country and with the proximity of the host cities, it will be possible to attend more than one game in different cities on the same day. Some pedestrians or cyclists may choose to go from one stadium to another moving in an open environment exposed to the weather, enjoying the landscape and the local culture. It is worth reminder that FIFA planning has guided the staging of the 2022 Qatar World Cup, to be held in November-December, to avoid exposure temperatures near 50 °C.
Major cities such as Lusail, Al Wakrah and Al Rayyan are within 20 km of the center of Doha. The exception is Al Khor Stadium, located 50 km north of Doha. The historical data used is presented at a resolution of 0.5° × 0.5° (50 × 50 km). Thus, in this context, one point would be representative and sufficient to be analyzed portraying the climate variability of the region. Qatar has a desert climate, located in the Middle East is bathed by the waters of the Persian Gulf; Qatar is a desert country, having a desert climate, the country experiences long summers from May to September characterized by intense warm and dry weather with temperatures above 45 °C. Nevertheless winter temperatures are mild, but can drop below 5 °C.
In this paper one analysis the thermal comfort characteristic, based on the Climate Normal, in the host cities: Doha, Al Wakrah, Ar Rayyān and Al Khor, over the seasons, based on climatological analysis of historical reports and model reconstructions of the most recent 1991-2020 climatological normal; extracted from observed historical data produced by the Climatic Research Unit (CRU) at the University of East Anglia. About Qatar's climatology 1991-2020, one can consult the available web site: https:// | https://doi.org/10.1007/s42452-022-05257-9 clima tekno wledg eport al. world bank. org/ count ry/ qatar/ clima te-data-histo rical! [9].
The data are presented at a resolution of 0.5° × 0.5° (50 × 50 km), for the most recent climatology, 1991-2020, the dataset is from the high resolution grid datasets. The typical climate of the Doha Metropolitan Region is illustrated as the Climatology calculated from the 1991-2020 Climatological Normal (Fig. 1), according to the International Standard Atmosphere model and the Climatic Research Unit (CRU) dataset [10] of the University of East Anglia. As mentioned above, the original datasets are presented at a 0.5° × 0.5° (50 × 50 km or 50 km grid) resolution.
In this work, to calculate the classical thermal comfort indices, the input atmosphere variables are air minima temperature (°C), air maxima temperature (°C), air relative humidity (%) and air wind speed (km/h), which can easily be transformed into m/s, to be the input variable for the calculation of the classical indices of thermal comfort according to the literature. Figure 1 illustrates the Doha Metropolitan Climatology (Climate Normal from 1991 to 2020) of the canonical variables employed to calculate the thermal comfort index.
Additionally, in this proposition, one also considered as input variable, the monthly mean air temperature (°C), air relative humidity (%), wind speed (km/h), solar daylight length (hs) and solar energy (kWh), these last two atmospheric variables are rarely used in thermal comfort index calculations. By way of clarification basically, there are four variables needed to calculate human thermal comfort indices: 2 m air temperature, 2 m dew point temperature (or relative humidity), wind speed at 10 m above ground level, and mean radiant temperature (MRT). In this paper solar insolation and solar irradiance are introduced.

Exploratory descriptive analysis of the DohaMR climate
This session was done in order to dissociate the input atmospheric variables from the output latent variables of the numerical algorithms, but with an association in the form of logical mathematical modeling. The climate in Qatar is desert, BWh climate classification according to Köppen and Geiger. Considering the climatic characteristics of the FIFA World Cup 2022 host cities considered and the distance between them, one decided to designate the regional climate, accounting for the arithmetic mean as a representative of the local geographical climatology and small spatial variability: Doha Metropolitan Region (DohaMR). The four classical indices of outdoor thermal comfort are evaluated for the DohaMR and an alternative index is constructed, in this scientific work, through Principal Components Analysis. All the thermal comfort indices, cited here, were calculated by means of an algorithmic implementation designed using the computer language R [11].
Thus, analyzing the monthly statistical behavior of weather variables, in Doha metropolitan area (DohaMR), summer is long, scorching, hot (warm)t, dry and partly cloudy; winter is pleasant, dry, windy and almost cloudless.
Throughout the year, in general the temperature ranges from 14 to 42 °C and is rarely below 10 °C or above 45 °C. The hot season remains for 7 months, from May to September, with an average daily maximum temperature above 38 °C. The hottest month of the year in Doha is July, with a maximum of 41 °C and a minimum of 31 °C on average. The warm (not cold) season remains for 4 months, from December to March, with an average daily maximum temperature below 26 °C. The coldest month of the year in the Doha metropolitan area is January, with a low (minimum) temperature of 14 °C and a high (maximum) temperature of 22 °C on average. DohaMR has extreme seasonal variation in the humidity sensation.
The hottest period of the year lasts 8 months, from April to November, in which the comfort level is stuffy, humid, or extremely humid at least 23% of the time. The month with the wettest days in the Doha metropolitan area is August. The wind sensation at a given location is highly dependent on the local topography and other factors. The wind speed and direction at an instant vary much more than the hourly averages. The hourly average wind speed in the Doha metropolitan area undergoes significant seasonal variations Fig. 1 The Doha Metropolitan Region current climatology, 1991-2020. The blue line is the ensemble average and the red dashed line is the standard deviation interval (mean centered rangeupper level plus the standard deviation and lower level minus the standard deviation) [11] throughout the year. The windiest time of the year lasts for 5 months, from November to April, with average wind speeds exceeding 14.9 km/h. The windiest month in the Doha metropolitan area is February, with average wind speeds above 17.5 km/h. The calmest time of the year is 7 months, from April to November. The calmest wind month in the Doha metropolitan area is September, with 12.2 km/h average wind speeds per hour. Day length and solar radiation in the Doha metropolitan area varies throughout the year, but, it is important to note, these are high levels.

Methodology
More recently, the interest in thermal control of outdoor environments had increased. The indices investigated, in this pilot study, include various parameters, which are either directly measured or given by international standards. For outdoor thermal comfort assessments, it is not recommended to adopt an analysis of the individual's thermal balance alone. In fact, outdoor environments, human thermal comfort depends on the prevailing environmental factors, such as ambient temperature, humidity, wind speed and solar radiation.

Temperature and humidity thermal comfort index (THTCI)
The classic THTCI establishes, basically, three levels of comfort for the external environment. The THTCI is an index that combines air temperature and relative humidity. The range of the THTCI, empirically established in the literature (Table 1). This index is appropriate for regions located in the tropics and evaluates the stress in the urban environment. However, there is no classification for values below 21(°C), which is rare regarding climate as in Qatar.

The humidity index (HUMIDEX)
The HUMIDEX is an index to describe how hot or warm the weather feels to the average person, by combining the effect of heat and humidity [12]. The term HUMIDEX was first coined in 1965 [13]. HUMIDEX differs from the THI used in the United States in being derived from the dew point rather than the relative humidity, though both dew point and relative humidity (when used in conjunction with air temperature) are directly related to atmospheric moisture. The calibration equation or transformation is applied in order to determine the relationship between the dew point temperature and the minimum temperature. The range of HUMIDEX, already empirically established in the literature ( Table 2).

Thermal discomfort index (TDI)
The TDI also establishes a relationship between average temperature and humidity relative air humidity; however it has different comfort levels. The TDI does not consider discomfort caused by cold. Distribution of TDI classes according to [14,15]. Range of TDI, established in the literature (Table 3).
Besides these indices, Missenard [16,17] found the following relation, a function of air temperature and relative humidity, to designate the effective temperature [18].

The standard effective temperature thermal comfort index (SETTCI)
The temperature calculated as a function of the dry bulb temperature and the wet bulb temperature. The SETTCI is defined as the temperature of a stable, saturated atmosphere that, in the absence of radiation, would produce the same effect as the conditions of the regular exposure to atmosphere conditions. It indicates the combined effects of relative humidity, air speed, air temperature, and clothing. This is a modified method for assessing the impact of air humidity on air temperature conditions acceptable to and by humans [19]. The SETTCI range, established empirically in the literature by expert researchers (Table 4).   An index that depends, besides the air temperature and relative humidity, the wind speed wind speed was also used. Being found in [12] and called effective temperature as a function of wind.

The effective temperature as a function of wind thermal comfort index (ETWTCI)
Effective Temperature as a function of wind is the temperature calculated as a function of dry bulb temperature, wet bulb temperature (relative humidity) and air speed. ETWTCI, besides also establishing a relationship between average temperature and relative air humidity, considers wind speed data, presenting eleven distinct ranges of thermal comfort levels.  [20,21]. The range of ETWTCI (a detailed index) like Universal Thermal Comfort Index (UTCI) [22,23], empirically established in the literature ( Table 5).
The recently released UTCI, will be respectfully mentioned here, but not studied, as the index proposed in this paper will be thoroughly compared in another article, in which one will details, in mathematical form, its conception and modeling. So, the Universal Thermal Comfort Index (UTCI): Is a temperature equivalent (°C), based on the measure of the human physiological response to the thermal environment. The UTCI describes the synergistic heat exchanges between the thermal environment and the human body, namely its energy budget, physiology, and clothing. The UTCI takes into account the adaptation of the human's clothing in response to the tangible ambient temperature. There are four variables needed to calculate the UTCI: 2 m air temperature, 2 m dew point temperature (or relative humidity), wind speed at 10 m above ground level, and mean radiant temperature (MRT). The Mean Radiant Temperature (MRT) is equivalent to the air temperature when related to an activity of a person moving with a speed of 4 km/h. One knows, but will not discuss in this paper, that there are 10 UTCI thermal stress categories that correspond to specific human physiological responses to the thermal environment. The categories relate to UTCI(°C) values as follows: above + 46: extreme heat stress; + 38 to + 46: very strong heat stress; + 32 to + 38: strong heat stress; + 26 to + 32: moderate heat stress; + 9 to + 26: no thermal stress; + 9 to 0: slight cold stress; 0 to −13: moderate cold stress; −13 to −27: strong cold stress; −27 to −40: very strong cold stress; below −40: extreme cold stress [22][23][24].
The proposal, presented in this scientific work, is based on a simple index, of rational calculation, that considers a first degree polynomial, whose coefficients capture the highest feasible variability existing between meteorological variables. Thus one must considers other indexes to evaluate overall, in a subjective way, the taking into account the degree of concordance.

The principal component thermal comfort index (PCTCI)
As an alternative to the compound polynomial equation to calculate the Universal Thermal Comfort Index (UTCI), widely addressed in the literature, one were encouraged and inspired by the scientific work developed by [25] that proposes alternatives for the determination of a composite indices, here one proposes an index by means of a linear combination of climate variables, taking into account the Principal Component Analysis. The Principal Component Analysis or PCA is a multivariate analysis technique that can be used to analyze interrelationships among variables and capture variability in terms of variance and its inherent dimensions, called components [26].
In summary: Principal Component Analysis, or PCA, is a statistical procedure that allows you to summarize the information content of large data tables into a smaller set of "summary indices" that can be more easily visualized and analyzed. Principal Components are empirical indices based on the construction of weighted linear combinations of the original variables, in the case of this study, climate based on atmosphere characteristics or measurements. In practice, they describe the variance and covariance structure of correlated variables in terms of a set of new uncorrelated variables. PCA purposes: Transform the variables into new uncorrelated variables; extract the signal contained in the data (eliminate or reduce the noise present in the data and; construction of indexes, as latent variables. The goal is to find a way to synthesize the information contained in several original variables into a smaller set of synthetic variables (components) with minimal loss of information. The number of possible principal components becomes the number of variables considered in the analysis, but generally the first components are the most important since they capture most of the total variance.
The Principal Components in general are extracted via covariance matrix, but also can be extracted via correlation matrix, as in this study, due to the fact that the climate variables are different in nature or genesis (measurement unit). Once the components are characterized, they can then be used as latent variables to create an information score. Basic ideas in Principal Components Analysis using the Matrix Spectral Decomposition Theorem, in terms of eigenvalues and related eigenvectors matrix decomposition. First, one determines the number of principal components that account for most of the variation in your data, using the proportion of variance that the components capture (get almost hold explain). Use the cumulative proportion to determine how much variance the principal components get it. Retain the principal components that capture an acceptable level of variance [26].
The intrinsic nature of the variables considered, together in a linear combination, based on the temporal support, will define the probabilistic thresholds of thermal comfort. PCTCI are empirical indices, which consider standard sources of variability, seasonal or intra-annual, related to local meteorological attributes behavior.
The standard deviation range is important to indicate the margin of uncertainty (or inaccuracy) regarding a calculation that has been made. However, in this study, the standard deviation dispersion range is used to determine the thermal comfort threshold values. The standard deviation (stdev) measures the amount of variability, or dispersion, from the individual data values to the arithmetic average (mean). Besides the correct interpretation of the confidence interval is probably the most challenging aspect of this statistical concept. The proposed range of the PCTCI (Table 6) is based on the standard deviation (stdev) range, as a associated threshold, and assessed with reference to the traditional and classical indices previously presented.
Alternatively to the interval defined by the standard deviation, the interquartile range (IQR) was developed in the field of statistics in order to assess the degree of data scattering (dispersion) around the centrality measure (median). While the standard deviation calculates the measure of dispersion without taking into account the order of the data, the IQR evaluates the dispersion of data only after sorting them in ascending order. The IQR is calculated based on the calculation of quartiles, being the first (lower) quartile (25%), the middle (median) quartile (50%), and the third (upper) quartile (75%), which is linked to the concept of quantile. The difference between the upper quartile and the lower quartile determines the IQR.  The concept of quartile is important for the definition of interquartile range. For this formal definition the concept of median is used to determine the quartiles (Q1, Q2, Q3) and it is necessary to understand the calculations both for data sets with odd number of elements and for data sets with even number of elements. One can determine the position of the quartiles only if the elements of a finite data set are ordered. So, the range of the PCTCI based on the IQR (IQR = Upper Quartile − Lower Quartile), proposed in this research and assessed with reference to the traditional classical indices previously presented: (Table 7). Principal component analysis, or PCA, is a dimensionality-reduction method that is often used to reduce the dimensionality of large data sets, by transforming a large set of variables into a smaller one that still contains most of the information in the large set. The goal of this paper, is to provide logical explanations of what PCA is and to simplify mathematical concepts such as standardization, covariance, correlation, eigenvectors and eigenvalues, neither of the Spectral Decomposition Theorem, without focusing on how to compute them. But everything is computed using R software. Without further argument, it is the eigenvectors and eigenvalues that are behind all the magic explained above, because the eigenvectors of the Covariance matrix are actually the directions of the axes where there is more variance (more information) and which one calls Principal Components. The eigenvalues are simply the coefficients attached to the eigenvectors, which give the amount of variance carried in each Principal Component. In summary: Principal Component Analysis, or PCA, is a statistical procedure that allows you to summarize the information content of large data tables into a smaller set of "summary indices" that can be more easily analyzed.

Results
Concerning the confirmatory analysis of the DohaMR Thermal Comfort Indices, taking into account each index previously presented. Figure 2 illustrates the Temperature and Humidity Index (THI) for the Doha metropolitan region (DohaMR), also called in this manuscript DMRTHI, throughout the year. Above the highest value for this index indicates a condition of extreme discomfort condition, and this was observed in the month of August, in which the season is summer season is summer and the highest temperatures are expected. In addition, it is observed that in the first quarter of the year, the temperature index follows below 15 °C, between the months of May and July it has a linear increasing trend, reaching its peak in August, after which the temperature and humidity index decreases with the end of the year. Observing the seasonal behavior of the DMRTHI it is possible to observe statistical significance for the initial months of the year (January-February-March),  The blue line is the ensemble average and the red dashed line is the standard deviation-based interval, and the solid lines are the thresholds established in the literature in the period of huge discomfort (July-August-September) and in the month of December, with August being the month of maximum value. Through the descriptive measures it is possible to verify maximum amplitude of (33.74-14.83)°C, which is expected for the study region. It is also noted that the median (25.25 °C) has higher values than the average (24.46 °C), which shows that the values at the top of the distribution are not far from the center, compared to the values at the bottom of the distribution. Figure 3 illustrates the HUMIDEX for the Doha metropolitan area, throughout the year. It is observed that for values lower than 20, the risks are of dangerous hypothermia, it is also sensitive to lower temperatures. In the months from June to August the values for the index are higher and this presents a risk of dangerous hyperthermia because temperatures are generally the highest, but outside a 95% confidence interval. Furthermore, HUMIDEX has a parabolic annual behavior, it is understood as downward trend starting in August and continues until February in decline, from March starts to increase and continues until June, reaching its quantitative peak of about 45 °C in July. Analyzing the climatology of the period from 1991 to 2020 via HUMIDEX, there is similarity with the seasonality found in the temperature and humidity index, in relation to the months with statistical significance. However, higher values are observed in the months of July and August. By the illustration one observes that the descriptive values and the maximum amplitude (27.91 °C) were higher than the previous index. Figure 4 evidences the Thermal Discomfort Index (TDI) for the Doha metropolitan area throughout the year.
Unlike the HUMIDEX index, it ignores the discomfort caused by cold sensation. However, just like the other indices so far, it has confirmed lower values between the months from December to February. As well can be seen an extreme with the highest values of the respective index between the months of July to September. This can lead to a risk of severe discomfort. Moreover, the discomfort index (TDI) indicates that it is increasing from February to June. It means that from February to June, the discomfort increases as the months go by, only reaching minor values in the last months of the year. The climatology via TDI indicates August the one with the highest value. It presents lower maximum amplitude of 17.25 °C when compared to the indices mentioned above and presents the mean value higher than the median which indicates that the values at the top of the distribution are far from the center, compared to the values that are at the bottom of the distribution. Figure 5 illustrates the Standard Effective Temperature (ETI) index for the Doha metropolitan region, throughout the year. It indicates a peak for the month of August, in which temperature values are also higher. This indicates a behavior of the index similar to the other indices. Additionally, the ETI has a continuous growth until the month of August, with linearity in some months it reaches its highest value in August and after that a slight decrease. It has regular discomfort values between June and September and inferior comfort values in the interval December to February. The climatology via ETI is similar to that found by TDI, with respect to the months with statistical significance and the fact that the mean values are also higher than the average, although it presents higher maximum amplitude than the TDI. Figure 6 give us an idea about the Effective Temperature as a function of Wind Index (ETWI) throughout the year. This index has values that start to increase from February to July and decreases from July to December. And so, like the other indices, the behavior of the curve follows a seasonal pattern, in which the higher temperatures are directly related to the highest values for this index. Also, the Effective Wind Temperature Index decreases in AUG-DEC with a peak in July and increases from April, the highest values are in the range of about 30 °C-33 °C in the months of June-August. The climatology via ETWI is similar to that found by HUMIDEX, regarding the months with statistical significance, however it presents smaller maximum amplitude (19.18 °C) and the median values are higher than the mean, indicating a major asymmetry.  Regarding the Principal Components construction, the factor loadings scheme shows the results of the first two components (Fig. 7). The wind speed, in Doha Metropolitan Region (DohaMR), has large positive factorial loadings on Component 1 (Comp.1 or PC1), so this component measures long-term wind stability or the intra-annual variability. Temperature, Relative Humidity and the solar exposition have large negative factorial loadings in Component 1 (Comp.1 or PC1), so this component primarily measures human thermal comfort sensation taking into account the heat feeling (about 75% of the variance captured). In these results, the first principal component (PC1) has large positive associations with Wind, so this component mainly measures long-term wellness in hot desert climate-thermal relief sensation. The second component (PC2) has large negative associations between Temperature related variables and Relative Humidity, so this component mainly measures the actual state of the local climate over the seasons (about 15% of the variance captured). Concerning to the interpretation of the main results of Principal Component Analysis (presented on Fig. 7), the first two principal components have eigenvalues greater than 1. These two components have explained about 95% of the total variability.
In multivariate statistics the "scree plot" is a graphic tool widely used to determine the number of factors to retain in an exploratory factor analysis or principal components to keep in a principal component analysis (PCA), shows that the eigenvalues start to form a straight line after and in the third one principal component. If 95% (80% with respect to PC1 and 15% regarding to PC2) is an appropriate amount of the variation explained in the data, one should use the first two principal components. To interpret each principal component, examine the magnitude and direction of the coefficients on the original variables. The larger the absolute value of the coefficient, the more important  the corresponding variable will be when calculating the component. How large the absolute value of a coefficient needs to be in order for its importance to be considered subjective. Use the specialist expertise to determine at what level the value of the correlation is important.
One point out that in the Fig. 7, the annotations are may not comprehensible or readable, apparently, there is an overlap of multiple annotations. However, this is usual when it comes to correlation analysis and the contribution of variables in the construction of linear combinations! However, this didactic and elucidative figure can also be suppressed! The figure is an illustrative part of the analysis. Biplots (Fig. 7) are a type of exploratory graph used in statistics, a generalization of the scatter plot of two variables. Biplot allows information on both the samples and variables in the data matrix to be displayed graphically. Samples are presented as points, while variables are presented as vectors and the linear orthogonal axes are a Principal Components [27].
It is very helpful to have the insight of the correlation between atmosphere variables of the same nature, forming cohesive groups; and also the contribution of each group in the construction of each Principal Components: PC1 is basically composed of the air temperature group versus the wind speed group, and PC2 is formed by the relative humidity group versus the solar information.
The interesting, and the charm of this informative Biplot graph (Fig. 7), is with regard to the interlaced information of temperatures and solar radiation with a certain inverse but significant correlation with wind speed, which form the PC1 (Comp.1), the first principal component. In addition to with respect to PC2 (Comp.2), component two, which by construction is orthogonal (independent of interpretation) of PC1, one observes and opposition between relative humidity and the variables associated with the air temperature. Figure 8 illustrates the Principal Component Thermal Comfort Index (PCTCI) throughout the year, based on the PC1 (at the top). This index has values that start to increase from February to July and decreases from July to December. And so, like the other indices, the behavior of the curve follows a seasonal pattern, in which the higher temperatures are directly related to the highest values for this index. Also, the Effective Wind Temperature Index decreases from August to December with a peak in July and increases from April, the highest values are in the range of about 30 °C-33 °C in the months between June and August. The climatology via PCTCI is similar to that found by both, HUMIDEX and ETWI, regarding the months with statistical significance, however it presents smaller maximum amplitude (19.18 °C) and the median values are higher than the mean, indicating a major asymmetry. The viability and suitability of the index proposed in this study, the PCTCI, was checked by contrasting it with the classic indexes and considering a correlation coefficient greater than 95%. Besides, the PC2 (at the bottom) is time-lagged information for the beginning of the season of best thermal comfort. As an early warning for action! As already bring up, as very well established method by data science methods, the PCA is used in exploratory data analysis and for decision making in predictive models. PCA is typically used for dimensionality reduction, using each data point only in the first principal components (in most cases first and second dimensions) to obtain lower dimensional data while maintaining as much of the variance of the data as possible. The first principal component can be equivalently as a direction that maximizes the variance of the projected data. Principal components are often analyzed by eigen decomposing the correlation matrix of the variables.

Conclusion
As point out, this work was motivated by the discussion in [28,29], using the R language [30], with the intention to support [29].
In this work one has considered to investigate the objective thermal comfort indexes, which only consider environmental factors related to atmospheric variables. That is, the climate state of the atmosphere from the 1991-2220 Climatological Normal. Further studies should be done with the objective of assess the PCIDEX for other temporal support, period, location and climate. In addition, other atmospheric attributes not mentioned in this report may also be considered.
Principal Components are latent variables that are constructed as linear combinations of the initial variables. These combinations are constructed such that the new variables are uncorrelated and most of the information within the initial variables set is compressed into the first component. PCA tries to put as much information as possible into the first component, then as much remaining information into the second, and so on. One important thing to realize here is that Principal Components are hard to be interpreted and have the possibility of no real meaning, since they are constructed as linear combinations of the initial variables. Since there are as many principal components as there are variables in the data, the principal components are constructed such that the first principal (PC1) component represents the greatest possible variance in the data set, and in this and in this work the second component (PC2) was also considered, since given the time lag, this can be interpreted as an early warning to the period of the year of best thermal comfort sensation. | https://doi.org/10.1007/s42452-022-05257-9 Case Studies As proposed in the justification and motivation of the research undertaken, in this paper one compares the applications of different indices of thermal comfort, considering data of temperature humidity and wind speed data. The results show that there was no difference in classification, regarding the region under study. This fact portrays the low climatic variability observed in the region. The 2022 FIFA World Cup will be held in the Best Time of Year to Visit, characterizing how pleasant the weather is in Doha Metropolitan Region (DohaMR) throughout the year, based on the computed thermal comfort index scores. The results highlight that, even in a very subtle way, distinct classifications of thermal comfort were obtained according to each index of thermal comfort, according to each index.
It was possible to observe that the THI and TDI provided classifications closer to each other, and predominantly more extreme than the indexes ETI and ETWI. It is interesting to note that for the ETI and ETWI indices, in general, it was possible to perceive a change in the classification of measurements taken in the sun and shade, in the same periods, revealing their greater sensitivity to variations in the parameters. Furthermore, the index obtained by means of Principal Component Analysis (PCAI) showed the same temporal pattern as the TEWI, but with a better refinement of classification. This fact makes this index a viable alternative to the classic indices of thermal comfort.
There is no difference that can be considered, of results between the indexes! What one wants to emphasize is