Classification of CIE standard skies using probabilistic neural networks

Transcription

1 INTERNATIONAL JOURNAL OF CLIMATOLOGY Int. J. Climatol. 30: (2010) Published online 16 March 2009 in Wiley InterScience ( DOI: /joc.1891 Classification of CIE standard skies using probabilistic neural networks Danny H. W. Li, a * H. L. Tang, a Eric W. M. Lee a and Tariq Muneer b a Department of Building and Construction, City University of Hong Kong, Kowloon, Hong Kong SAR, China b School of Engineering and the Built Environment, Napier University, Edinburgh, UK ABSTRACT: In 2003, the International Commission on Illumination (CIE) adopted 15 standard skies that cover a broad spectrum of the usual skies found in the world. Each sky represents a unique sky luminance distribution, which is the most effective way to classify the 15 CIE Standard Skies. However, luminance distributions for the whole sky vault are far from being widely available. Alternatively, the standard skies can be categorized by various climatic parameters but the criteria to distinguish individual skies are not always clear-cut and may lead to ambiguous results. The artificial neural networks (ANNs) represent a powerful tool for pattern recognition. This paper presents the work on the classification of the standard skies using a new form of neural network architecture, namely the probabilistic neural network (PNN), which is particularly apposite in classification problems. Five meteorological variables, viz. zenith luminance, global, direct-beam and sky-diffuse illuminance, and solar altitude are employed as input data. Totally, 9000 samples covering the time span between 1999 and 2005 are shuffled. The findings suggest that the PNN is an appropriate tool for sky classification. Copyright 2009 Royal Meteorological Society KEY WORDS artificial neural networks; probabilistic neural network; sky distribution; zenith luminance; diffuse illuminance Received 24 September 2008; Revised 22 January 2009; Accepted 9 February Introduction Daylighting is an effective sustainable development strategy to alleviating the problems in energy, climate and the environment and improving the qualities for visual comfort and health (Smiley, 1996; Aoki et al. 1997; Kambezidis et al., 1998; Kittler et al., 1999). The daylight availability is mainly influenced by the luminance levels and patterns of the sky (Tregenza and Waters, 1983). The sky luminance distributions are caused by various factors including the solar position, the atmospheric turbidity and the air pollution, and the cloud amount, type and pattern that can affect unpredictably sunlight and skylight (Darula and Kittler, 2004). In 2003, the International Commission on Illumination (CIE) adopted a range of 15 standard skies covering probably the whole spectrum of the usual skies in the world (CIE, 2003). Sky conditions of the same category would have similar sky luminance distributions and the corresponding climatic parameters and indices would be within certain ranges. Once the skies have been identified, the basic daylight illuminance at various inclined surfaces and the prevailing values of some important meteorological elements can be obtained for subsequent examinations (Li et al., 2005). The essential issue would be the frequency of occurrence for individual sky standards appearing in a given location (Li * Correspondence to: Danny H. W. Li, Department of Building and Construction, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, China. bcdanny@cityu.edu.hk et al., 2004). However, luminance distributions for the whole sky are far less available. There are a number of appropriate climatic parameters to classify the daylight climates (Kittler and Darula, 2002). For instance, the ratio of the zenith luminance to the horizontal diffuse illuminance (L z /D v ) can characterize the momentary sky brightness and theoretically can classify the measurement into one of the 15 standard sky patterns (Markou et al., 2005). Nonetheless, the L z /D v theoretical curves for the 15 standard skies are not parallel but they intersect with each other at the solar altitude of 35 or higher. Using L z /D v for a place, where solar altitudes often exceed 35, can lead to vague results (Bartzokas et al., 2003, 2005; Li et al., 2003). It seems that no single climatic variable can efficiently interpret all sky standards. The artificial neural networks (ANNs) have emerged as a useful tool for classification, optimization, multivariate function approximation and forecasting. The networks attempt to imitate the characteristics of the human brain and nerve system which learn from experiences. By providing both input and output exemplars, the networks learn by processing the input exemplars and compare the results with the desired response. In most applications, ANNs can handle large and complex datasets with many interrelated parameters running much faster than dynamic simulation programs and give accurate results as compared with the traditional statistical models (Hamkmeer and Basheer, 2002). The applications include engineering, climatology studies, medicine, etc. (Michaelides Copyright 2009 Royal Meteorological Society

2 306 DANNY H. W. LI ET AL. et al., 2001; Lee et al., 2004; López and Gueymard, 2007) Previously, we classified the 15 CIE Standard Skies using various appropriate climatic parameters (Li and Tang, 2008) but the criteria to distinguish individual skies are not always clear-cut and may lead to ambiguous findings. This paper presents the work on identification of the 15 CIE Standard Skies using a special form of neural network architecture, namely the probabilistic neural network (PNN) approach. Apposite daylight variables including zenith luminance, global, direct-beam and skydiffuse illuminance data, and solar altitude are employed for the categorization. Characteristics of the findings are elaborated and discussed. 2. Measuring station In 1991, a measuring station was established at the City University of Hong Kong to systematically record the daylight illuminance. Initially, the measurements concerned global and diffuse illuminances on a horizontal surface. In 1996, the measurements were extended to record vertical global components on the four principal orientations, i.e. facing north, east, south and west. The measuring station was upgraded in 1999 with the installation of a sky scanner to record the luminance patterns of the whole sky dome. The data collection started just before sunrise and ended after sunset each day. All measurements were recorded in true solar time. The measurements of outdoor illuminance are made by means of illuminance sensors (T-10M) manufactured and calibrated by Minolta of Japan. The silicon photocells with cosine and colour corrections measure illuminances up to about 300 klx with an accuracy of ±2%. A multipoint illuminance measurement system is used. The diffuse illuminance sensor is fitted with a ring shading the thermopile from the sun. A data-measurement package is used to capture the measured results from the main body adapter and the data are then fed into a microcomputer for storage. The sky luminance distribution is measured by means of a sky scanner (EKO MS-300LR), which was manufactured and calibrated by the Japanese company EKO. The sensor head rotates in altitude and azimuth to measure the luminance at 145 circular sky patches by scanning the sky hemisphere. The sequence of the 145 sky patches is shown in Figure 1. The sky grid pattern illustrated in Figure 1 (not the measurement pattern) was suggested by Tregenza and Sharples (1995) such that the whole sky dome can be considered for further analysis. The important parts of the sky scanner are housed in a weatherproof case allowing continuous outdoor operation. The data from the scanner are recorded on a microcomputer placed inside the laboratory on the top floor. To safeguard the sensor, the scanner does not record luminance data greater than 35 kcd/m 2 by using an automatic shutter. Each scanning time is about 4 min and measurements are taken every 10 min. To eliminate spurious data and erroneous measurements, the following quality-control tests are employed: Figure 1. The scanning sequence of the 145 sky patches. Each patch shows its number together with the solar altitude and azimuth. 1. Rejecting all diffuse data greater than the corresponding global values. 2. Rejecting all global data greater than the corresponding extraterrestrial illuminance constant. 3. Rejecting all data measured at solar altitudes (α s )less than Rejecting all data of the direct-beam values [i.e. (global diffuse)/sin α s ] being greater than the corresponding extraterrestrial illuminance constant. 5. Rejecting all diffuse data greater than half of the corresponding extraterrestrial illuminance constant (because of improper shadow-ring adjustment). 6. Applying a shadow-ring correction to the observed horizontal diffuse data according to the method described by Lebaron et al. (1990). 7. Removing all sky scan data within a circle of 11 around the sun s position. 8. Rejecting all sky luminance data when the difference between the corrected horizontal diffuse illuminance and the corresponding integrated horizontal diffuse illuminance based on the 145 scanned sky luminance points is greater than 30%. 3. CIE standard sky models The set of the 15 CIE Standard Skies includes five clear, five intermediate and five overcast sky types, which probably cover the whole spectrum of skies found in nature. The distributions are described by continuous mathematical expressions that change smoothly in luminance from the horizon to the zenith and with the angular distance from the sun. The standard formula defining the relative luminance (l v ) on any standard sky can be considered as a combination of the gradation ϕ(z) and the indicatrix function f(χ): l v = L L Z = f(χ)ϕ(z) f(z S )ϕ(0 ) (1)

3 CLASSIFICATION OF CIE STANDARD SKIES 307 where L is the sky luminance in an arbitrary sky element (kcd/m 2 ), L z is the sky luminance at the zenith (kcd/m 2 ), Z is the zenith angle of a sky element (rad), Z s is the zenith angle of the sun (rad) and χ is the angular distance between the sun and a sky element (rad). The standardized gradation is defined by appropriate a and b variables as: ϕ(z) 1 + a exp (b/ cos Z) = ϕ(0 ) 1 + a exp b (2) The relative scattering indicatrix function can be modelled by an exponential function with adjustable coefficients c, d and e as: f(χ) f(z S ) = 1 + c[exp(dχ) exp(dπ/2)] + e cos2 χ 1 + c[exp(dz S ) exp(dπ/2)] + e cos 2 Z S (3) The exponential term exp(dχ) represents the effect of Mie scattering, which decreases rapidly with distance from the sun. The cos 2 χ term is due to the Rayleigh scattering and is zero at 90 to the direction of the sun (Littlefair, 1994). Both gradation and indicatrix functions are of six types including the usual range of homogenous cases from heavy overcast to cloudless skies. The combinations can form a large number of skies but only 15 relevant types were selected to be the standard set. Table I summarizes the details of the 15 standard skies (Kittler et al., 1998). It has been reported that the standard set can provide a good overall framework for categorizing actual sky conditions (Tregenza, 1999) 4. Sky classification To identify the prevailing set of standard skies in Hong Kong, the luminance distributions of individual standard skies were modelled and compared with the scanned sky luminance readings. The modelled sky luminance is normalized to the horizontal diffuse illuminance by multiplying all luminance values with the normalization ratio (NR) as: NR = L mea cos θ sin θ dθ dβ l mod cos θ sin θ dθ dβ (4) where L mea is the measured sky patch luminance (kcd/m 2 ), l mod is the modelled sky patch luminance (dimensionless), θ is the altitude (rad) and β the azimuth (rad) of the sky patch. An alternative way would be to divide all sky patch luminance values by the zenith luminance, but this can cause huge measuring error when the sun is near to the zenith (Tregenza, 2004). For low-latitude climates (e.g. Hong Kong) when the sun is frequently within a small angular distance from zenith, a normalization with respect to the diffuse horizontal illuminance would, therefore, be more appropriate. Once normalized, the performance of each standard sky luminance model is assessed by the root mean square error (RMSE), which is obtained by subtracting the measured sky patch luminances from the modelled ones of the 15 CIE Standard Skies, adding together the squares of these values, and dividing the total by the number of sky patches, and then taking the square root. The best-fit standard is the one with the lowest RMSE. Outdoor illuminance and sky luminance data Table I. Description of the 15 standard skies. No (code) Type of sky For gradation For indicatrix a b c d e 1 (I1) CIE standard overcast sky, steep luminance gradation towards zenith, azimuthal uniformity 2 (I2) Overcast, with steep luminance gradation and slight brightening towards the sun 3 (II1) Overcast, moderately graded with azimuthal uniformity (II2) Overcast, moderately graded and slight brightening towards the sun 5 (III1) Sky of uniform luminance (III2) Partly cloudy sky, no gradation towards zenith, slight brightening towards the sun 7 (III3) Partly cloudy sky, no gradation towards zenith, brighter circumsolar region 8 (III4) Partly cloudy sky, no gradation towards zenith, distinct solar corona 9 (IV2) Partly cloudy, with the obscured sun (IV3) Partly cloudy, with brighter circumsolar region (IV4) White blue sky with distinct solar corona (V4) CIE standard clear sky, low luminance turbidity (V5) CIE standard clear sky, polluted atmosphere (VI5) Cloudless turbid sky with broad solar corona (VI6) White blue turbid sky with broad solar corona

4 308 DANNY H. W. LI ET AL. 5. Probabilistic neural networks A neural network called PNN (Specht, 1990), which is appropriate for modelling specific type of classification problems, was used for the analysis. PNN is a model based on Baye s strategy (Mood and Graybill, 1962) for decision making as well as Parzen s theorem (Parzen, 1962) for estimating the probability density function (PDF). The Baye s strategy identifies the unknown sample into a population class i if h i c i f i (x) > h j c j f j (x) (5) Figure 2. (a) Frequency of occurrence of the 15 sky standards and (b) RMSE of the luminances of the actual skies relative to the 15 sky standard distributions in Hong Kong. recorded during the 84-month period between January 1999 and December 2005 were gathered for the analysis. Accordingly, (a) the frequency of occurrence of the 15 standard skies using the best-fitting approach and (b) the corresponding RMSE based on the complete standard set are presented in Figure 2. Large variations can be seen for individual sky types. Sky nos. 1 and 3 dominate the overcast condition contributing around 30% to all cases. Skies nos. 6, 7 and 8 are the main sky patterns for the partly cloudy conditions, although the sum of them is below 20%. For clear sky condition, it is matched by sky no. 13, which represents 14%. The overcast and clear skies (i.e. overcast sky nos. 1 5 and clear sky nos ) account for 78% of the Hong Kong sky conditions. The intermediate skies (i.e. sky nos. 6 10) represent the remaining 22%. In general, the best-fitting standard sky with a high frequency of occurrence results in a low RMSE and vice versa. It can be observed that sky nos. 2, 5 and 9 have a frequency of occurrence of less than 5%, but the corresponding RMSEs can be over 32%. With the frequency of occurrence over 12.5%, the RMSEs for sky nos. 1, 3 and 13 are not more than 24%. A subset of the CIE Standard Skies including overcast, partly cloudy and clear conditions should be sufficient to describe the daylight climates of Hong Kong. The RMSE for all the 15 CIE Standard Skies is 23.8%. for all populations j i, where h i is the prior probability of the sample belonging to class i, c i is the cost of misclassification and f i is the PDF. The Baye s strategy can be easily implemented once the PDF in each population is known. However, the PDF, which defines the boundaries for each data class, is not always known. Parzen s PDF estimator is a common tool to estimate a univariate PDF from the sample. In multivariate case, Cacoullos (1966) suggested that the PDF can be expressed as: [ 1 n f(x)= (2π) p/2 σ p exp (x x ] ij ) T (x x ij ) n 2σ 2 j=1 (6) where n is the number of the training set, σ is the smoothing parameter, p is the dimension of the pattern vector and x ij is the jth training vector from the class i. The smoothing parameter, σ, estimated from the training set, is the most important element to determine the PDF (Chen and Hsu, 2007). Different values of the smoothing parameters may cause different effects on the degree of interpolation that occur between adjacent pattern vectors. The influence of the choice of smoothing parameter has been shown by Specht (1990). The basic architecture of PNNs used in the study is shown in Figure 3. It consists of an input layer, a pattern layer, a summation layer and an output layer. The number of neurons in the input layer equals the number of the input variables of the problem under investigation. The neurons in the pattern layer are used to store the training samples and each neuron contains one training sample. The summation layer has one neuron for each class and each summation neuron is used to collect and sum the outcomes from all pattern neurons of the same category. By applying the Baye s decision strategy, final results are then generated via the neuron in the output layer. 6. Daylight variables Sky luminance distributions particularly for the whole sky are scarcely available or exist for very limited periods. In interpreting sky conditions, climatic parameters are often used as weighting factors to indicate the degree

5 CLASSIFICATION OF CIE STANDARD SKIES 309 Figure 3. The basic configuration of PNN. of sky clearness such that the sky luminance distributions can be categorized. Sky conditions interpreted on the basis of climatic parameters widely obtainable would be more appropriate. An earlier work (Li and Tang, 2008) used a number of climatic parameters, namely the ratio of zenith luminance to horizontal diffuse illuminance (L z /D v ), the solar altitude (α S ), the ratio of horizontal global or diffuse illuminance to horizontal extraterrestrial illuminance (G v /E v, D v /E v ) and the luminous turbidity (P v ) for classifying the 15 CIE skies. Their strengths and limitations for the sky identification are discussed. As reported by Kittler et al. (1998), L z /D v can characterize the momentary sky brightness and theoretically can represent the pattern of the 15 CIE Standard Skies. The integration of the luminance of each sky patch over the whole sky vault gives D v. As zenith is in a dominant position in the sky with respect to (i.e. normal to) the horizontal surface, D v is substantially influenced by L z. A high probability of agreement (i.e. the zenith luminance is the major component contributing to D v ) does exist between these two daylight parameters. However, the values of L z /D v for sky luminance classification intersect each other at α S of 35 or more. This could introduce errors in sky categorization for high solar altitudes (Bartzokas et al., 2003, 2005). The criteria for determining sky clearness using G v usually take the form of G v /E v. At ground level, E v is reduced and split due to air attenuation, turbidity and clouds to form G v. The measurement of G v using illuminance meters is quite straightforward. When the atmosphere is clear, a small fraction of daylight illuminance is scattered, resulting in a predominant direct component with a large amount of G v /E v. For an overcast day, a large amount of outdoor illuminance is scattered, indicating a high portion of diffuse component with a low G v /E v reading. As suggested by Kittler and Darula (2002), the pair L z /D v G v /E v hybrid daylight variable is appropriate to classify the three typical sky conditions (i.e. overcast, partly cloudy and clear). However, there are no clear-cut values for L z /D v G v /E v to represent various sky conditions. Previously, we proposed the L z /D v G v /E v ranges for the interpretation of the three typical skies (Li and Lau, 2007) based on measured Hong Kong data. The diffuse illuminance (D v ) is often measured using shadow ring to eliminate direct sunlight. Occasionally, the shadow ring may be misplaced, resulting in diffuse output being closed to that of global sensor under nonovercast skies. Moreover, the shadow ring does block off a significant portion of the sky-diffuse illuminance and correction procedures must be applied to obtain the true values. Because of the anisotropy of diffuse sky, the approaches to compute the correction factors are fairly complex. Large D v /E v values often occur in partly sky cloudiness. For low D v /E v values, they may indicate two extreme sky conditions (i.e. overcast skies and clear skies). Once the three typical sky conditions (i.e. overcast, partly cloudy and clear) are identified, D v /E v can be employed to further differentiate the five overcast and five clear sky standards (Li and Tang, 2008). The attenuation of luminous solar energy by air molecules, water vapor, dust and aerosols gives an indication of the luminous turbidity (T v ),whichisanimportant parameter in analyzing the daylight illuminance and atmospheric clearness under non-overcast skies. The general formula for T v is given as follows (Kittler et al., 1998): T v = ln(e v/b v ) (7) a v m v where B v = direct beam daylight illuminance (lx) a v = luminous ideal extinction = (Navvab et al., 1984), m v = optical air mass = m v 1 sin α S (α S ) (Kasten and Young, 1989) When there is no B v, T v tends to be infinite. It means that T v cannot be used to distinguish individual overcast skies. However, T v is a good criterion to differentiate between low turbidity and polluted sky conditions. The usual T v values are between 10 and 20 under partly cloudy skies and below 10 for clear skies. Various T v criteria can be applied to classify individual partly cloudy and clear skies (Li and Lau, 2007; Li and Tang, 2008). The luminance distributions for most sky distributions depend on the α s in particular under clear skies. Under heavy overcast skies (sky standards 1, 3 and 5), the distribution of the sky luminance is symmetrical about the zenith and sensitive to changes in the sky elevation above the horizon. It means that the sky luminance distributions for sky standards 1, 3 and 5 are independent of sky

6 310 DANNY H. W. LI ET AL. azimuth and solar position. For other sky conditions, particularly clear skies, the peak luminance would appear near to the solar position, decreasing with the distance from the sun. As indicated in Equation 3, Z s which equates to π/2 minus α s is included in the relative scattering indicatrix function. The α s alone may not be a good factor to single out individual standard skies. However, a number of climatic parameters do depend on α S and such features are useful for sky classification. 7. Methodology Daylight illuminance and sky luminance data measured during the 84-month period from January 1999 to December 2005 were used to develop and train the PNN models for sky classification. In reality, the occurrences of the standard skies were not evenly distributed as shown in Figure 2. Sky no. 1 contributed about 18% of the Hong Kong sky conditions, whereas just over 1% for sky no. 9. However, it is important that the training set should contain nearly the same number of training vectors from each class (Swingler, 1996). To avoid bias in the over represented classes, only 600 datasets for each identified sky standard were randomly selected from the database. By doing so, the extra data sets in the over represented classes were removed to ensure that the databases are not bias (Basheer and Hajmeer, 2000). Totally, 9000 datasets were gathered and used in this study. Five climatic parameters as described in Section 6 were used as the input parameters of the PNNs to discriminate the 15 standard skies. To develop PNN as a classifier for recognizing different sky types, the database was firstly divided into three subsets, namely training set, validation set and test set. The training set was used to build the architecture of PNN while the validation set to tune the smoothing parameter. The test set was hidden in the phase of network training. Upon completion of the network training, the test set was used to evaluate the performance of the trained PNN. Although the dimension of each subset has significant impacts on the classification accuracy, there are no practical rules or theories to determine the size of each subset; just few empirical rules of thumb exist (Basheer and Hajmeer, 2000). A large test set may improve the generalization capability of the network, but the remaining small training set may not provide sufficient information to train the network and vice versa. To balance the effects, 4500 and 1800 out of 9000 datasets (i.e. 50 and 20% of the database) were randomly extracted to represent the training and validation sets, respectively. Both the training set and validation set were involved in the training of the PNN. The remaining 2700 datasets comprised the test set, which was never used in the PNN development. The PDFs for each class were defined by all the training samples, while the optimal value of the smoothing factor was estimated by the validation set. For the present study, a software package NeuroShell 2 (NeuroShell 2, 2000) was used to train and develop the neural networks for sky classification. To determine the optimal value of the smoothing factor for the PNNs, the genetic algorithm (GA) which provides a useful strategy for selecting smoothing factors was adopted to improve the generalization capability of the network even though the training duration will be much longer. The details about the smoothing factor selection using GA can be referred to Mao et al. (2000). Hansen et al. (1992) and Kalatzis et al. (2005) reported that the accuracy as well as generalization capability of neural network could be improved significantly by the ensembling techniques (i.e. determine the predicted result from a set of results predicted by a number of different models). For classification problems, majority voting and weighted voting are two commonly used ensemble approaches (Osowski et al., 2008). The majority voting scheme was employed in this study since each PNN model has the same influence on the ultimate classification results. The basic idea of majority voting is to train a set of models and allow them to vote (Ravi et al., 2008). To undertake the majority voting scheme, 30 different pairs of training and validation sets were randomly extracted from the available samples to train the PNNs and hence 30 trained PNN models were built. As random process (i.e. data extraction) is involved in the network training, the 30 trained PNN models were different from each other. For the purpose of clarity, PNN1 represents the PNN model with the combination of the first training and test sets, PNN2 represents the PNN model with the combination of the second training and test sets and PNN3 represents the PNN model with the combination of the third training and test sets etc. All the 30 trained PNN models were tested independently by the same test set. The results collected from each of the PNN models were combined to form an ensembling system and the ultimate predicted class of a particular test data was obtained after majority voting. Figure 4 shows the overview of PNNs based on the majority voting scheme. 8. Results In the present study, PNN models trained by 30 pairs of training and test sets were employed firstly to categorize the three typical sky conditions. The results of the PNN models were then integrated by applying the majority voting scheme. Afterwards, the investigations were extended to the 15 CIE Standard Skies. For each analysis, a confusion matrix (i.e. Tables III and IV) was constructed to evaluate the performance of the models such that the hit percentage for PNN models can be easily found. The number of rows and columns of the matrix is equal to the number of classes in each study. The rows and columns of the table actually represent the actual sky type and the predicted sky type, respectively. The number of correct classifications appears only in the diagonal positions, whereas the number of misclassifications indicates in the off-diagonal

7 CLASSIFICATION OF CIE STANDARD SKIES 311 Figure 4. Probabilistic neural network with majority voting technique. positions. From the confusion matrix, the hit percentage, denoted as the ratio of the number of samples correctly classified to the total number of samples to be classified, can be computed accordingly Three typical skies Table II shows sky classification results over the 30 PNN models. It can be seen that the variations of the successful rate among the 30 PNN models for the training, validation and test sets are quite small. The hit percentage lies between just below 91 and 96% using the training and validation sets, and from just over 87 to 89% based on the test set. As expected, the classification results of the training and validation sets were always better than the corresponding test set, because the training set and validation set were used to train the model. To further evaluate the performance of PNNs, the classification results based on 2700 test data under the 30 PNN models were gathered and the results of the test set are presented in the confusion matrix shown in Table III. Totally, 899, 908 and 893 cases were classified by the best-fit approach as clear, partly cloudy and overcast sky conditions. The sky classification interpreted by the PNN models is in good agreement with those using the best-fitting approach. For instance, 856 out of 899 Table II. Classification results of the 30 PNN models for the three typical sky conditions. PNN Hit percentage (%) PNN Hit percentage (%) Training and validation sets Test set Training and validation sets Test set clear skies are correctly classified. Only 1 and 42 skies were identified as overcast and partly cloudy conditions,

8 312 DANNY H. W. LI ET AL. Table III. Confusion matrix of the PNNs for the three typical sky conditions based on the test set (percentage of each sky type). Actual sky type Overcast sky Predicted sky type Partly cloudy sky Clear sky Overcast Sky (90.6%) (9.3%) (0.1%) Partly cloudy sky (9.7%) (81.9%) (8.4%) Clear sky (0.1%) (4.7%) (95.2%) respectively. Similar findings were obtained for overcast skies. For the partly cloudy skies, 88 and 76 out of 908 cases were respectively categorized as overcast and clear skies. In general, the PNN models show the best performance for distinguishing clear skies, afterwards overcast skies and then the partly cloudy skies. The hit percentage of the PNN models is 89.2%. Table IV. Classification results of the 30 PNN models for the 15 standard skies. PNN Hit percentage (%) PNN Hit percentage (%) Training and validation sets Test set Training and validation sets Test set The CIE 15 Standard Skies Likewise, the same 9000 samples were used for the classification of the 15 CIE Standard Skies. Totally, 15 output nodes were set. Again, 30 PNN models were carried out and Table IV presents the findings. As expected, the successful rates were less than those for identifying the three typical skies. The variation of the accuracy ranges between 87.6 and 89.1% for the training and validation sets and from 64.3 to 67.1% for the test set. Again, classification results based on the 30 PNN models for the 15 CIE Standard Skies were got together to examine the performance of the PNN techniques; Table V illustrates the number of the test data successfully interpreted as individual sky types. The PNN models can effectively single out sky standard 11 with the correct classification percentage of 86.7% while the lowest correct classification percentage of 49.1% was found for sky standard 6. Totally, 1911 out of 2700 cases were correctly classified. This represents 70.8% of the hit percentage, which is 18.4% lower than the classification of the three general sky conditions. Subsequently, the frequency of occurrence for the 15 standard skies and the corresponding RMSE of the luminance distribution of actual skies to the 15 standard skies using the best-fitting and PNN approaches based on the 2700 test data were determined. Figure 5 demonstrates the findings. The frequency of occurrence ranges between 6.2% for sky standard 1 and 7.6% for sky standard 7 via the best-fitting model and varies from 4.8% for sky standard 9 to 7.8% for sky standard 11 with PNN techniques. As expected, the RMSE values adopting PNN are generally more than those based on the bestfitting technique. The peak difference of 8.3% appears in sky standard 5. The RMSE for all the 15 standard skies Figure 5. (a) Frequency of occurrence of the 15 sky standards and (b) RMSE of the luminances of the actual skies relative to the 15 sky standard distributions in Hong Kong using best-fitting and PNN approaches for sky classification. by best-fitting and PNN approaches are 24.9 and 28.2%, respectively.

9 CLASSIFICATION OF CIE STANDARD SKIES 313 Table V. Classification results of the test set for the 15 sky standards(percentage of each sky type). Actual sky type Predicted sky type (76.0%) (5.4%) (8.4%) (1.2%) (7.2%) (1.2%) (0.0%) (0.0%) (0.0%) (0.0%) (0.0%) (0.0%) (0.0%) (0.0%) (0.6%) (6.7%) (65.3%) (5.2%) (8.8%) (7.3%) (4.7%) (0.0%) (0.0%) (0.6%) (0.6%) (0.0%) (0.0%) (0.0%) (0.0%) (1.0%) (10.3%) (2.9%) (76.0%) (5.1%) (4.0%) (0.6%) (1.1%) (0.0%) (0.0%) (0.0%) (0.0%) (0.0%) (0.0%) (0.0%) (0.0%) (2.8%) (3.9%) (4.4%) (64.4%) (9.4%) (8.3%) (2.2%) (1.1%) (1.1%) (1.7%) (0.0%) (0.0%) (0.0%) (0.0%) (0.6%) (4.5%) (4.5%) (6.2%) (6.7%) (64.0%) (6.7%) (1.1%) (0.6%) (2.2%) (1.7%) (0.0%) (0.6%) (0.0%) (0.0%) (1.1%) (1.1%) (6.3%) (2.3%) (5.1%) (9.1%) (49.1%) (12.6%) (4.6%) (2.9%) (5.7%) (0.6%) (0.0%) (0.0%) (0.0%) (0.6%) (0.0%) (1.0%) (0.0%) (3.9%) (5.4%) (7.4%) (60.3%) (6.4%) (6.4%) (5.4%) (2.0%) (0.0%) (0.0%) (0.0%) (2.0%) (0.0%) (0.5%) (0.0%) (2.1%) (1.0%) (3.1%) (7.9%) (75.4%) (1.6%) (3.1%) (3.1%) (0.0%) (1.6%) (0.5%) (0.0%) (0.7%) (1.3%) (0.0%) (2.0%) (2.6%) (9.9%) (2.6%) (5.3%) (50.0%) (11.2%) (3.3%) (3.9%) (0.7%) (1.3%) (5.3%) (0.0%) (0.0%) (1.6%) (1.1%) (2.2%) (3.8%) (7.0%) (4.3%) (7.5%) (56.5%) (5.9%) (1.1%) (2.2%) (0.5%) (6.5%) (0.0%) (0.0%) (0.0%) (0.0%) (0.0%) (1.0%) (1.5%) (1.5%) (1.5%) (2.6%) (86.7%) (3.1%) (1.0%) (0.5%) (0.5%) (0.0%) (0.0%) (0.0%) (0.0%) (0.0%) (0.0%) (0.0%) (0.6%) (1.1%) (1.7%) (1.1%) (84.8%) (6.2%) (3.9%) (0.6%) (0.0%) (0.0%) (0.0%) (0.0%) (0.0%) (0.0%) (0.0%) (1.1%) (0.0%) (1.1%) (7.1%) (4.9%) (82.0%) (2.7) (1.1%) (0.0%) (0.0%) (0.0%) (0.0%) (0.0%) (0.0%) (0.0%) (0.0%) (0.0%) (0.0%) (0.0%) (4.1%) (3.5%) (86.5%) (5.8%) (0.6%) (0.0%) (0.6%) (0.0%) (0.0%) (1.2%) (0.0%) (0.0%) (3.5%) (4.1%) (0.0%) (0.6%) (1.2%) (5.2%) (83.1%)

10 314 DANNY H. W. LI ET AL. 9. Conclusions An evaluation of the performance of PNN for sky classification was elaborated. By using the five climatic variables, namely L v /D v, G v /E v, D v /E v, T v and α s as the input parameters to the PNN, the three general sky conditions (i.e. overcast, partly cloudy and clear skies) and the 15 CIE Standard Skies were classified. Totally, 9000 data sets were randomly selected for sky classification. Using the PNN with majority voting techniques, about 89% of the test set was correctly identified for the three typical skies. For categorizing the 15 CIE Standard Skies, the correct classification percentage reduced ranging between 49.1% for sky standard 6 and 86.7% for sky standard 11. In general, the PNNs show the best performance for interpreting clear skies, secondly the overcast skies and then the partly cloudy skies. Based on the 2700 test data (i.e. 30% of the total data), the frequency of occurrence of the 15 CIE Standard Skies, and the corresponding RMSE of the luminance distribution of actual skies to the 15 Standard Skies using the best-fit and PNN techniques were calculated. The frequency distributions were quite even and similar patterns were found using the two methods. The RMSE for the 15 standard skies based on the PNN approach was 28.2%, which was 3.3% higher than that via the best-fitting approach. Acknowledgements The work described in this paper was fully supported by a Competitive Earmarked Research Grant from the Research Grants Council of the Hong Kong Special Administrative Region, China [Project No (CityU )]. H. L. Tang was supported by a City University of Hong Kong studentship. References Aoki Y, Taniguchi T, Irikura T Analysis of space distribution of scattered daylight by the Monte Carlo Method giving consideration to atmospheric particles. 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