ostiv2005, 16-18 september, istanbul/türkiye · ostiv2005, 16-18 september, istanbul/türkiye 3....

OSTIV2005, 16-18 September, Istanbul/Türkiye


ABSTRACT

1. INTRODUCTION2. METHODOLOGY

2.1. NEURAL NETWORKS (ANN)2.2. STUDY AREA AND DATA

3.DETERMINATION OF SED VALUES BY USING ANN4. RESULTS AND DISCUSSION 5. WAVELET ANALYSİS AND COMPARİSON WİTH ANN MODEL6. CONCLUSION

ACKNOWLEDGEMENTS

REFERENCES


1. Introduction The radiative effects combine with latent heating to modulate boundarylayer dynamics, turbulence generation, and evolution. The analysisboundary layer structure shows a strong influence of the underlyingterrain, (Tokgozlu et al. , 2000). To determine sed (dry static energy)values is a complex process mainly because of the measurement dataand analytical functions required. In this study a new model for predictingof sed values are developed. ANN model was used for predicting sed values depending on pressure, temperature, humidity, surfacetemperature. Simple equations were developed by using ANN, (Şencanet al. , 2005). Although the concept of artificial neural network (ANN) analysis has beendiscovered nearly 50 years ago it is only in the last two decades that application software has been developed to handle practical problems.ANNs are good for some tasks while lacking in some others. Specifically,they are good for tasks involving incomplete data sets, fuzzy orincomplete information, and for highly complex and ill-defined problems, where humans usually decide on an intuitional basis [Kalogirou S.A.,(1999a)].


A N N s h a v e b e e n a p p lie d s u c c e s s fu lly in v a r io u s f ie ld s o fm a th e m a tic s , e n g in e e r in g , m e d ic in e , e c o n o m ic s , m e te o ro lo g y ,p s y c h o lo g y a n d n e u ro lo g y . T h e y h a v e a ls o b e in g u s e d inw e a th e r a n d m a rk e t t re n d s fo re c a s tin g , in th e p re d ic tio n o fm in e ra l e x p lo ra tio n s i te s , in e le c tr ic a l a n d th e rm a l lo a dp re d ic tio n , in a d a p tiv e a n d ro b o tic c o n tro l a n d o th e rin te rd is c ip lin a ry to p ic s [K a lo g iro u S .A .,(2 0 0 0 b )] .

A r tif ic ia l n e u ra l n e tw o rk s a re s y s te m s o f w e ig h t v e c to rs , w h o s ec o m p o n e n t v a lu e s a re e s ta b lis h e d th ro u g h v a r io u s m a c h in e -le a rn in g a lg o r ith m s , w h ic h ta k e a s in p u t a l in e a r s e t o f p a tte rnin p u ts a n d p ro d u c e a s o u tp u t a n u m e r ic a l p a t te rn re p re s e n tin gth e a c tu a l o u tp u t. A N N s m im ic s o m e w h a t th e le a rn in g p ro c e s so f a h u m a n b ra in . In s te a d o f c o m p le x ru le s a n d m a th e m a tic a lro u tin e s A N N s a re a b le to le a rn k e y in fo rm a tio n p a tte rn s w ith ina m u lti- in fo rm a tio n d o m a in . In a d d it io n , th e in h e re n tly n o is yd a ta d o n o t s e e m to p re s e n t a p ro b le m , a s A N N s a re to le ra n t inn o is e v a r ia tio n s [K a lo g iro u S .A . ,(2 0 0 0 c ) ] .


Artificial neural networks differ from the traditionalmodelling approaches in that they are trained to learnsolutions rather than being programmed to model a specificproblem in the normal way. They are usually used to addressproblems that are intractable or cumbersome to solve withtraditional methods. They can learn from examples and faulttolerant in the sense that they are able to handle noisy andincomplete data. They are able to deal with non-linear problems, and once trained can perform predictions at very high speed. ANNs have been used in many engineeringapplications such as in control systems, in classification, andin modelling complex process transformations [KalogirouS.A., (2000d)]. The advantages of ANN compared to classical methods arespeed, simplicity and capacity to learn from examples. In thelast decade, some works about the use of ANN in energysystems have been published [Kalogirou S.A., (1999a, 2000b,2000c, 2000d, 2004e), Kalogirou S.A, et al., (1999), Kalogirou S.A, Bojic M., (2000), Chouai A., et al., (2002), Pacheco-Vega A., et al., (2001), Bechtler H., et al., (2001) ]. This techniquecan be used in the modelling of complex physicalphenomena. So, engineering effort can be reduced. Wavelet analysis can provide information aboutdiscontinuities of data on different scales. In the second partof this paper, evaporation values are analysed by using 1Dcontinuous wavelet and wavelet packets, (Siddiqi et al. , 2002).


2. Methodology2.1. Neural Networks (ANN)

Neural networks are composed of simple elements operating in parallel. Theseelements are inspired by biological nervous systems. As in nature, the networkfunction is determined largely by the connections between elements. A neuralnetwork can be trained to perform a particular function by adjusting the values of theconnections (weights) between the elements. Commonly neural networks areadjusted, or trained, so that a particular input leads to a specific target output. Such asituation is shown in Fig. 1. The network is adjusted, based on a comparison of theoutput and the target, until the network output matches the target. Typically manysuch input/target output pairs are used to train a network. Entire set (batch) trainingof a network proceeds by making weight and bias changes based on an batch of inputvectors. Incremental training changes the weights and biases of a network as neededafter presentation of each individual input vector. Incremental training is sometimesreferred to as “on line” or “adaptive” training. [Lin C.T., Lee C.S.G. , (1996), Stull,R. B., (1989), Fu L.M., (1994), Tsoukalas L.H., Uhrig R.E., (1997)].


Fig. 1. Basic Principles of Artificial Neural Networks


There are different learning algorithm s that can be applied to train a neuralnetwork. The m ost popular of them is the back-propagation algorithm ,which has different variants. Standard back-propagation is a gradientdescent algorithm . It is very difficult to know which training algorithm willbe the fastest for a given problem and the best one is usually chosen bytrial and error.

ANN with back-propagation algorithm learns by changing the connectionweights and these changes are stored as knowledge. Som e statisticalm ethods, such as the Root-M ean-Squared (RM S), the coefficient ofm ultiple determ ination (R 2) and the coefficient of variation (cov) m ay beused to com pare predicted and actual values. During learning the error isestim ated by RM S defined as [Bechtler H ., Browne M .W ., Bansal P.K .,Kecm an V ., (2001)]:

( )n

tyRMS

n

1m

2m,mm,p∑

=

−= (1)

In addition, the coefficient of m ultiple determ ination (R 2) and coefficientof variation (cov) in percent are defined as follows [Bechtler H ., BrowneM .W ., Bansal P.K ., Kecm an V ., (2001)]:


( )

( )∑

∑

=

=

−

−−= n

1m

2m,mm,m

2n

1mm,pm,m

2

tt

yt1R (2)

100tRMScov

m,m

= (3)

where n is the number of data patterns, yp,m indicates thepredicted, tm,m is the measured value of one data point m, and m,mt

is the mean value of all measure data points.


2.2. Study Area and Data

Flight data measured in and near vicinity of Isparta (Latitude:37 º18’and 38º30’ N, Longitude: 30º02’ and 31º33’E, Altitude:1000-1050m (msl) and Lake Eğirdir. This area is called as theArea of Lakes. This region is under the effect of central andsouth-western Anatolian climatological conditions. It is underthe combined effects of Mediterranean and terrestrial climateconditions with hot and dry summers and cold and wet winters.Annual rainfall rate is 600 mm in and near vicinity of Isparta.Complex topography generally causes orographic or convectiverain formation in the study area.Flight measurements are recorded on August 31th. 1998, (Fig 3).


Fig. 3. Study area (Eğridir Lake/Isparta)


3. Determination of sed Values by Using ANN Many of the thermodynamic variables that have been used in theliterature to describe the state of the cloudy boundary layer.Static energies are based on assumption that any kinetic energyis locally dissipated into heat. Dry static energy (also known as the Montgomery stream function), [Stull (1989)]. These valuesare indicator for soaring conditions. sed = CpT + gz (4) where; T: Air temperature, Cp: specific heat at constants pressure for moist air, g: acceleration due to gravity, z: height, relative to local sea-level horizontal surface.


The inputs of the network are pressure, temperature, humidity, surface temperature,whereas output is the sed values. For this purpose the back-propagation learningalgorithm has been used in a feed-forward, single hidden layer neural network. Thealgorithm used in the study is Levenberg-Marquardt (LM). Inputs and outputs arenormalised in the (0, 1) range. Logistic sigmoid (logsig) transfer function has been usedfor both the hidden layer and the output layer. The transfer function used is given by:

zezF −+=

11)(

(5)where z is the weighted sum of the input.

The computer program was performed under MATLAB environment using the neuralnetwork toolbox. In the training, a variable number of neurons are used in the hiddenlayer to define the output accurately. The data set for the sed values available included133 data patterns. From these 107 data patterns were used for the training of the networkand the remaining 26 patterns were randomly selected and used as test data set.


Different network structures, sizes and learning parameters havebeen tried. The best network that was ultimately selected has onehidden layer and seven neurons. Statistical values such as RMS,R2 and cov are given in Table 1 for sed values.

Table 1. Statistical values for sed valuesAlgorithmneurons

RMS cov R2

LM-7 0,1467 4,758E-06 0,9999

The decrease of the mean square error (MSE) during the trainingprocess of this topology is shown in Fig 3.

OSTIV2005, 16-18 September, Istanbul/TürkiyeFig. 3. Variation of mean square error with training epochs


4. Results and Discussion

In order to calculate the sed values, mathematical formulations are derived from theresulting weights and the activation functions used in the ANN. As the statisticalresults obtained from both the training and testing of the ANNs were extremely goodin both cases it is believed that the results thus obtained would be accurate.

Mathematical formulations derived from the ANN model are presented here. In thefollowing formulas the coefficients of the input parameters are used to evaluate the Ei(summation function of neuron i) and Fi (activation function of neuron i). Thesecoefficients represent the weight values of the summation function of each neuronbelonging to the hidden layer of the trained network. For this purpose, and for the caseof ANN model used for sed values prediction, seven pairs of equations are required asthe neural network model has 7 hidden neurons. The activation function chosen is thelog-sigmoid as shown from the Fi function. In the ANN model, the inputs of thenetwork are the pressure (p), temperature (t), humidity (rh), surface temperature (tr)and the output is sed value. In order to calculate the sed values, the following equationsare derived:


P=p/931 (6)

T=t/299 (7)

RH=rh/53 (8)

TR=tr/301 (9)

E1=P*(59.3432)+T*(-229.3963)+RH*(130.2788)+ TR*(2133.5085)- 2103.5068 (10)

1E1 e11F −+

= (11)

E2=P*(-13.6595)+T*(48.565)+RH*(0.01995)+ TR*(0.29086)- 35.1055 (12)

2E2 e11F −+

= (13)

E3=P*(-263.677)+T*(548.7833)+RH*(112.0004)+ TR*(660.7908) -957.4588 (14)

3E3 e11F −+

= (15)

E4=P*(160.7355)+T*(1009.67)+RH*(35.0175)+ TR*(-582.4141)- 575.1013 (16)


4E4 e11F −+

= (17)

E5=P*(-82.5772)+T*(553.222)+RH*(16.0673)+ TR*(204.9591)- 688.5121 (18)

5E5 e11F −+

= (19)

E6=P*(-187.0695)+T*(-382.3831)+RH*(-125.7718)+ TR*(1310.1903)- 654.195 (20)

6E6 e11F −+

= (21)

E7=P*( -541.8007)+T*(247.9551)+RH*(7.1307)+ TR*(-501.7395)+747.8141 (22)

7E7 e11F −+

= (23)


In th e a b o v e e q u a t io n s fo r E i th e f ir s t fo u r v a lu e s a re th em u lt ip l ic a t io n o f th e in p u t p a ra m e te r s P , T , R H , T R w ith th e irw e ig h ts a t lo c a tio n i a n d th e la s t c o n s ta n t v a lu e re p re s e n t th eb ia s te rm . T h u s th e te rm s E 1 to E 7 a n d F 1 to F 7 re p re s e n ts u m m a tio n a n d a c tiv a tio n fu n c tio n s o f e a c h n e u ro n o f th e h id d e nla y e r , r e s p e c tiv e ly . E q s . (5 -8 ) a re u s e d to c o n v e r t th e a c tu a l in p u td a ta o f te m p e ra tu re v a lu e s to n o rm a lis e d v a lu e s in th e ra n g e [0 -1 ] . T h e s e d v a lu e s d e p e n d in g o n p re s s u re , te m p e ra tu re ,h u m id ity , s u r fa c e te m p e ra tu re v a lu e s c a n b e c o m p u te d a sb e lo w :

E 8 = F 1 * ( -8 .4 7 0 5 )+ F 2 * (0 .0 9 0 6 6 6 )+ F 3 * (0 .2 4 3 2 3 )+ F 4 * (0 .7 9 9 0 4 )+ F 5 * ( -0 .0 0 0 1 4 8 8 6 )

+ F 6 * ( -3 .8 5 8 8 )+ F 7 * ( -9 .1 4 6 3 ) -3 .2 8 9 4 (2 4 )

309710.e11sed

8E ⎟⎠⎞

⎜⎝⎛

+= − (2 5 )

T h e c o e f f ic ie n t s h o w n in E q . (2 4 ) is u s e d to c o n v e r t th en o rm a lis e d o u tp u t to a c tu a l o u tp u t ( s e d ) .


Table 2. Comparison between actual sed values and sed values

obtained with equations derived from ANN

In Table 2 a comparison is presented between the actual sed values and sed values predicted with the equations derived from ANN.


5. Wavelet Analysis and Comparison with ANN Model:

Figure 4(a-b) illustrate 1-D wavelet packets of sed values. Lastpart of the period corresponds higher value of dry static energy.More favourable soaring conditions are available in this area.Figure 4(c) illustrates 1-D continuous wavelet analysis of valuesof dry static energy values. Small scale effects have beenobserved all period based on db wavelet level 7.


Fig. 4(a) 1D- wavelet packet, sed values, haar wavelet, level 4, August 31, 1998.


Fig. 4(b) 1D- wavelet packet, sed values, db2 wavelet, level 7, August 31, 1998.


Fig. 4(c) 1D- continuous wavelet, sed values, db wavelet, level 7, August 31, 1998.


6. Conclusion In this paper an ANN is successfully applied to determine of sedvalues. The R2-value for sed value is 0.9999 which can be considered as very satisfactory. In order to calculate the sedvalues, mathematical formulations were derived by using ANN model. Mathematical formulations have been obtained fromformulations of the summation and activation functions used inthe ANN model and weights of neurons. The new methodologyprovides faster and simpler solutions instead of complexequations. Also the use of the derived equations, which can be employed with any programming language or spreadsheetprogram for the estimation of the sed values, as described in thispaper, may make the use of dedicated ANN softwareunnecessary. 1D- db wavelets and wavelet packets show small scalefluctuations over the study area.


Acknowledgements Authors would like to thank for his support to Mr. CarstenLindemann from Friee University.


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