7.-Forecasting surface .FULL

International Journal of Mathematics and Computer

Applications Research (IJMCAR)

ISSN 2249-6955

Vol. 3, Issue 2, Jun 2013, 65-78

© TJPRC Pvt. Ltd.

FORECASTING SURFACE AIR TEMPERATURE USING NEURAL NETWORKS

K. ANITHA KUMARI1, NAVEEN KUMAR BOIROJU

2, T. GANESH

3 & P. RAJASHEKARA REDDY

4

1,3,4Department of Statistics, Sri Venkateswara University, Tirupati, Andhra Pradesh, India

2Department of Statistics, Osmania University, Hyderabad, Andhra Pradesh, India

ABSTRACT

In this paper, forecasting of monthly mean of minimum surface air temperature of India using seasonal

autoregressive integrated moving average (SARIMA) model, feed forward neural networks (FFNN) and higher order

neural networks (HONN) is discussed. The prediction ability of the models also tested using sign test, Diebold-Mariano

test and Bootstrap test procedure for absolute errors. FFNN and HONN models outperforming than that of SARIMA

model in out-of-sample forecasts.

KEYWORDS: Box-Jenkins Methodology, Neural Networks, Prediction Accuracy Tests

INTRODUCTION

Surface air temperature (SAT) is a measurement of the average kinetic energy of the air near the surface of the

earth and it also measures the radiation. SAT is very important in all fields of natural sciences, including physics, geology,

chemistry, atmospheric sciences and biology. SAT plays a vital role in environmental and agricultural issues, water cycle,

energy cycle, weather forecasts and global climate change. Atmospheric stability is determined by SAT. Hypothermia and

Frostbite are due to the changes in SAT. Air temperature prediction is of a concern in environment, industry, agriculture,

and health management. Tosadduq et.al. (2005) discussed on application of neural networks for the prediction of hourly

mean surface temperatures. Afzali et.al. (2011) used artificial neural networks to predict the ambient air temperature.

Shrivastva et.al. (2012) presented a review on applications of neural networks in weather forecasting. Smith (2006)

discussed on air temperature prediction using neural networks. Stein and Lloret (2001) used Box-Jenkins methodology for

the forecasting of air and water temperatures. Some of the authors, Brunetti et.al. (2000), Anisimov (2001), Bodri (2003),

Alfaro (2004), Lee and Sohn(2007), Kulkarni et.al. (2008), FAN Ke (2009), Hejase and Assi (2012), Kemajou (2012)

discussed on modelling of air temperatures.

Monthly mean of minimum surface air temperature in degrees Celsius data of all India is collected from Indian

Institute of Tropical Meteorology (IITM), Pune, India. This data consists of 1284 monthly observations during 1901 to

2007, in which 100 years of data (1212 monthly observations) during1901-2001 are used for model fitting and remaining 6

years of data (72 monthly observations) during 2002-2007 are used as out-of-sample set to measure the predictability of the

selected models using mean absolute error, mean absolute percentage error and root mean squared error.

The following section presents the Box-Jenkins methodology and neural networks methodologies. Section 3

presents the forecasting models using seasonal autoregressive integrated moving average (SARIMA) model, feed forward

neural networks (FFNN) and higher order neural networks (HONN). Comparison of models reported in the Section 4 and

final conclusion presented in Section 5 followed by references.

METHODOLOGY

This section presents the forecasting model building procedures using Box-Jenkins methodology and neural

networks methodologies.

66 K. Anitha Kumari, Naveen Kumar Boiroju, T. Ganesh & P. Rajashekara Reddy

Box-Jenkins Methodology

Box Jenkins methodology provides us a class of univariate time series models. SARIMA model is one of the

models in Box Jenkins methodology. SARIMA model sometimes consider as a benchmark model for comparison of the

models applied to same data set. Box-Jenkins methodology is an iterative procedure to build an adequate model for the

given time series. The basic class of SARIMA model is denoted by S

QDPXqdpSARIMA ,,,, and the model is

given by

t

s

Qqt

D

s

ds

Pp aBBZBB

where tZ is the time series value at time t and and ,, are polynomials of order of p, P, q and Q

respectively. B is the backward shift operator, stt

s ZZB and B 1 . Order of seasonality is represented by s.

Non-seasonal and seasonal difference order is denoted by d and D respectively. White noise process is denoted by ta . (Box

et.al. 1994).

The Box-Jenkins procedure consists of the following four steps: (1) model identification, where the orders d, D, p,

P, q and Q are determined by observing the behaviour of the corresponding autocorrelation function (ACF) and partial

autocorrelation function (PACF); (2) estimation, where the parameters of the model are estimated by the maximum

likelihood method; (3) diagnostic checking by the “Portmanteau test”, where the adequacy of the fitted model is checked

by the Ljung-Box statistic applied to the residuals of the model; (4) forecasts are obtained from an adequate model using

minimum mean squared error method. If the model is judged to be inadequate, steps 1-3 are repeated with different values

of d, D, p, P, q and Q until an adequate model is obtained. The detailed procedure of SARIMA Model building is explained

in the Section 3.

Feed Forward Neural Networks

An artificial neural networks, usually called neural networks, is a mathematical model or computational model

that is inspired by the structure and/or functional aspects of biological neural networks. A neural network consists of an

interconnected group of artificial neurons, and it processes information using a connectionist approach to computation. In a

feed forward neural network (FFNN) structure, the only appropriate connections are between the outputs of each layer and

the inputs of the next layer. Therefore, no connections exist between the outputs of a layer and the inputs of either the same

layer or previous layers. In this topology, the inputs of each neuron are the weighted sum of the outputs from the previous

layer. There are weighted connections between the outputs of each layer and the inputs of the next layer. If the weight of a

branch is assigned a zero, it is equivalent to no connection between correspondence nodes. The inputs are connected to

each neuron in hidden layer via their correspondence weights. Outputs of the last layer are considered the outputs of the

network. Selecting the best number of hidden neurons involves experimentation. The forward selection method involves

adding hidden neurons until network performance starts deteriorating. A neural network is required to go through training

before it is actually being applied. Training involves feeding the network with data so that it would be able to learn the

knowledge among inputs through its learning rule. Backpropogation algorithm is used in supervised learning of the

network. The main idea of the backpropagation algorithm is to minimize the error, which is the difference between the

expected value and the output of the model. Weights between neurons are adjusted until the error reaches an acceptable

value. In order to train the network successfully, the output of the network is made to approach the desired output by

continually reducing the error between the network's output and the desired output. This is achieved by adjusting the

Forecasting Surface Air Temperature Using Neural Networks 67

weights between layers by calculating the approximation error and backpropagating this error from the final layer to the

first layer. The weights are then adjusted in such a way to reduce the approximation error. The approximation error is

minimized using the gradient descent optimization technique (Rojas 1996).

Faraway and Chatfield (1998) compared FFNN models with a SARIMA model on their accuracy for forecasting

airline data. In their paper, they discovered that FFNN model also reduces the mean square errors (MSEs) of out-of-sample

prediction. Rao (2011), Naveen Kumar Boiroju (2012) and Zhang et.al. (1998) provide a comprehensive review of the

current status of research in this area. Forecasting of minimum SAT using FFNN model is explained in Section 3.

Higher Order Neural Networks

HONNs have only started recently to be used in time series modeling. Higher order neural network is a feed

forward neural network trained with the higher order inputs. HONNs use joint activation functions; this technique reduces

the need to establish the relationships between inputs when training. Furthermore this reduces the number of free weights

and means that HONNS are faster to train than even MLPs.

However because the number of inputs can be very large for higher order architectures, orders of 4 and over are

rarely used. Another advantage of the reduction of free weights means that the problems of over fitting and local optima

affecting the results of neural networks can be largely avoided. For a detailed description of HONNs see Knowles et al.

(2005) and Naveen Kumar Boiroju (2012). Forecasting of minimum SAT using HONN is explained in the section 3.

Measures of Errors

The mean absolute error (MAE) measures forecast accuracy by averaging the magnitudes of the forecast errors

(i.e. absolute values of each error).

N

t

t

N

t

tt eN

ZZN

MAE11

1ˆ1

The mean squared error (MSE) is another method for evaluating a forecasting technique.

N

t

tt

N

t

t eN

ZZN

MSE1

22

1

1)(

1

And the root mean squared error (RMSE) is given as

N

t

tt

N

t

t eN

ZZN

RMSE1

22

1

1)(

1

Mean absolute percentage error is given by 1001

1

N

t t

t

Z

e

NMAPE .

These error measures used to compare the accuracy of the different techniques applied on a time series data.

FORECASTING MODELS

This section presents some forecasting models for forecasting monthly mean of minimum SAT using Box-Jenkins

methodology and neural networks.


Building SARIMA Model

In this section, monthly mean of minimum SAT is modeled using Box-Jenkins methodology. The development of

SARIMA model for any variable involves following four steps: Identification, Estimation, Diagnostic checking and

Forecasting.

Time plot of the minimum SAT reveals that the data is seasonal and non stationary.

Figure 1: Time Plot of Monthly Mean of Minimum Surface Air Temperature in India

Sample autocorrelation function (ACF) also computed to check whether the time series is non-stationary and

seasonal or not.

Figure 2: Autocorrelation Function for Minimum SAT


From the above ACF, it is observed that the ACF is dies out slowly for higher lags which indicates non-

stationarity of the series and the significant spikes at seasonal lags shows that the series is a seasonal time series. Seasonal

difference of order one (D=1) is sufficient to achieve stationarity of the series. Autocorrelation function and partial

autocorrelation function (PACF) is computed for the differenced series and are presented below.

Figure 3: Sample ACF for Minimum SAT with Seasonal Difference D=1

Figure 4: Sample PACF for Minimum SAT with Seasonal Difference D=1

From the above ACF and PACF it is observed that the order of p is at most 1, P is at most 4, q is at most 2 and Q

is at most 1. All the tentative models are considered with the above specifications and observed that the most suitable

model is SARIMA (1, 0, 2) X (0, 1, 1)12.

Model parameters (without constant term in the model) are estimated using SPSS for selected model. Estimates of

parameters are given below.


Table 1: SARIMA Model Parameters

Transformation Parameter Estimate SE T Sig.

No Transformation

AR Lag 1 .895 .040 22.551 .000

MA Lag 1 .647 .050 12.949 .000

Lag 2 .113 .035 3.240 .001

Seasonal Difference 1

MA, Seasonal Lag 1 .951 .012 80.540 .000

with the above parameters the fitted model is

tt aBBBZB )951.01)(113.0647.01(~

)895.01( 1221

12

Adequacy of the model is tested using Portmanteau test. For this purpose, the various autocorrelations of residuals

for 25 lags are computed and the same along with their significance which is tested by Box-Ljung Q- test statistic. Let the

hypothesis on the model is

Ho: The selected model is adequate.

H1: The selected model is inadequate.

Table 2: Portmanteau Test

Ljung-Box Q-Test

Statistic DF Sig.

9.421 14 0.803

Since the probability corresponding to Ljung-Box Q-statistic is greater than 0.05, therefore, we accept Ho and we

may conclude that the selected SARIMA model is an adequate model for the given time series on Mean of Minimum

Surface Air temperature.

One can forecast the future mean of minimum SAT using the equation (fitted model)

12

1 2 12(1 0.895 ) (1 0.647 0.113 )(1 0.951 )t tB Z B B B a by minimum mean square error method.

The above model is used to forecast the future values of monthly mean of minimum SAT and the forecasts for

out-of-sample are presented in the Table 5. Prediction performance of the model is measured using the error measures and

the results are presented in the Table 6.

Building FFNN Model

In this section, building of forecasting model for minimum SAT using FFNN is discussed. The in-sample data set

is partitioned into two sets namely training set and testing set. For model building 70% of the in-sample data is taken as

training set and 30% of the data is taken as testing set. The feed forward neural network (FFNN) consists of input layer,

hidden layer and output layer. Input layer consists of 14 units representing the month (numbers from 1 to 12), 1tZ and

12tZ values. Output layer consists of only one neuron and represents the forecast value ( tZ ) of the series. Number of

hidden neurons in the hidden layer is determined using forward selection method. The optimum number of hidden neurons

is four. Hyperbolic tangent function is used as an activation function and scaled conjugate gradient algorithm is used to


train the network. The network is trained until the number of epochs is equivalent to 10,000. SPSS software is used to train

the network. With the above specifications the following synaptic weights are obtained.

Table 3: Synoptic Weights of FFNN Model

Predictor

Predicted

Hidden Layer 1 Output Layer

H(1:1) H(1:2) H(1:3) H(1:4) Min_SAT

Input

Layer

(Bias) -0.118 0.151 -0.298 0.209

[MONTH=1] -0.310 -0.361 0.708 0.144

[MONTH=2] -0.256 -0.043 0.278 0.280

[MONTH=3] -0.360 0.969 -0.278 0.341

[MONTH=4] 0.779 0.193 -0.129 -0.384

[MONTH=5] 0.715 0.564 -0.031 -0.381

[MONTH=6] 0.504 0.207 -0.312 -0.821

[MONTH=7] 0.303 0.154 -0.381 -0.567

[MONTH=8] 0.288 0.273 -0.422 -0.241

[MONTH=9] 0.409 -0.118 -0.371 -0.399

[MONTH=10] -0.253 0.232 -0.286 0.160

[MONTH=11] -0.113 -0.514 0.065 0.151

[MONTH=12] -0.747 -0.457 0.599 0.070

1tZS 0.080 0.121 0.260 -0.672

12tZS 0.265 -0.187 -0.368 0.103

Hidden

Layer 1

(Bias)

-0.233

H(1:1)

0.601

H(1:2)

0.708

H(1:3)

-0.411

H(1:4)

-0.697

FFNN forecasting model can be constructed using above synoptic weights as follows

st ZZ ˆˆ Where and are the mean and standard deviation of the in-sample data set and

)4:1(697.0)3:1(411.0)2:1(708.01:1601.0233.0ˆ HHHHZs

wher


M= month, )(/)( 1111 tttt ZZZZS , 12121212 / tttt ZZZZS and AI is an

indicator function.


out-of-sample are presented in the table 5. Prediction performance of the model is measured using the error measures and


Building HONN Model

In this section, building of forecasting model for minimum SAT using HONN is discussed. The in-sample data set

is partitioned into two sets namely training set and testing set. For model building 70% of the in-sample data is taken as

training set and 30% of the data is taken as testing set. The HONN model is similar to FFNN model and it consists of input

layer, hidden layer and output layer. Input layer consists of 14 units representing the month (numbers from 1 to 12), 1tZ ,

12tZ , 2

1tZ , 2

12tZ and 121 tt ZZ values. Output layer consists of only one neuron and represents the forecast value (tZ )

of the series. Number of hidden neurons in the hidden layer is determined using forward selection method. The optimum

number of hidden neurons is four. Hyperbolic tangent function is used as an activation function and scaled conjugate

gradient algorithm is used to train the network. The network is trained until the number of epochs is equivalent to 1,000.

SPSS software is used to train the network. With the above specifications the following synaptic weights are obtained.

Table 4: Synoptic Weights for HONN Model

Predictor

Predicted

Hidden Layer 1 Output Layer

H(1:1) H(1:2) H(1:3) H(1:4) Min_SAT

Input Layer

(Bias) .232 -.069 -.470 -.355

[MONTH=1] -.232 .227 .274 .377

[MONTH=2] .241 -.220 .155 -.070

[MONTH=3] -.179 .504 -.533 -.549

[MONTH=4] -.072 -.821 -.757 .092

[MONTH=5] -.315 -.064 -.187 -.518

[MONTH=6] -.268 -.240 .127 -.394

[MONTH=7] -.198 -.312 .174 -.020

[MONTH=8] .032 .413 .465 -.334

[MONTH=9] -.059 -.364 .230 .240

[MONTH=10] -.078 -.155 .007 .893

[MONTH=11] .156 .676 .653 -.065

[MONTH=12] -.409 .169 .548 .343

1tZS .343 -.394 .020 .101


Table – 4 Contd.,

12tZS -.111 -.706 -.139 .177

2

1tZS .544 -.275 -.165 -.469

2

12tZS -.158 -.249 -.099 -.005

121 tt ZZS .495 .094 -.161 .086

Hidden Layer 1

(Bias)

-.433

H(1:1)

.132

H(1:2)

-.585

H(1:3)

-.589

H(1:4)

-.709

From the above weight matrix the forecasting model can be constructed as

st ZZ ˆˆ Where and are the mean and standard deviation of the in-sample data set

and )4:1(709.0)3:1(589.0)2:1(585.0)1:1(32.1433.0ˆ HHHHZs

M= month, ZS is the standardized values of Z and AI is an indicator function.


out-of-sample are presented in the table 5. Prediction performance of the model is measured using the error measures and



Table 5: Out-of-Sample Forecasts from SARIMA, FFNN and HONN Models

COMPARISON OF FORECASTING MODELS

This section presents the error measures and the significance of equal prediction ability of the forecasting models.

Table 6: Error Measures for SARIMA, FFNN and HONN Models

Sample Error SARIMA FFNN HONN

In-Sample

MAE 0.431 0.425 0.419

MAPE 2.809 2.793 2.753

RMSE 0.573 0.566 0.557

Out-of-Sample

MAE 0.519 0.440 0.426

MAPE 3.186 2.719 2.584

RMSE 0.653 0.581 0.558

From the above table, it is observed that the HONN model has minimum error measures in both the in-sample and

out-of-sample sets compared to SARIMA and FFNN models. FFNN model has minimum error measures than that of

SARIMA model. From the above study it is observed that, HONN model is good at forecasting of minimum SAT.

Testing Equal Forecasting Accuracy with Respect to Absolute Errors

This section presents the results of Sign test, Diebold-Mariano (DM) test and Bootstrap test to test the equal

prediction accuracy of the forecasting models with respect to the out-of-sample absolute errors. The detailed procedure of


these tests presented in the paper of Naveen Kumar Boiroju et.al. (2011). the following table presents the test statistic

values for testing the null hypothesis of equal prediction accuracy of the models.

Table 7: Equal Prediction Accuracy Tests

Models Test Statistics Bootstrap Test

Sign test DM test LDL UDL

SARIMA vs FFNN 2.593 2.228 0.0188 0.1399

SARIMA vs HONN 2.828 2.742 0.0284 0.1565

FFNN vs HONN 1.179 1.037 -0.0109 0.0385

Sign test statistic value (2.593) is greater than the critical value (1.96) at 5% level of significance for comparison

of SARIMA and FFNN models. Therefore null hypothesis is rejected and we may conclude that there is a significant

difference in forecasting ability between SARIMA and FFNN models.


of SARIMA and HONN models. Therefore null hypothesis is rejected and we may conclude that there is a significant

difference in forecasting ability between SARIMA and HONN models.


of FFNN and HONN models. Therefore null hypothesis is accepted and we may conclude that there is no significant

difference in forecasting ability between FFNN and HONN models.

DM test statistic value (2.228) which is greater than the critical value (1.96) at 5% level of significance for

comparison of SARIMA and FFNN models. Therefore the null hypothesis is rejected and we may conclude that there is a

significant difference in prediction ability between SARIMA and FFNN models.


comparison of SARIMA and HONN models. Therefore the null hypothesis is rejected and we may conclude that there is a

significant difference in prediction ability between SARIMA and HONN models.


comparison of FFNN and HONN models. Therefore the null hypothesis is accepted and we may conclude that there is no

significant difference in prediction ability between FFNN and HONN models.

Bootstrap test procedure for absolute errors is also applied to test the equal prediction accuracy of the models. The

decision limits (0.0284, 0.1565) are obtained using the bootstrap test procedure for comparison of SARIMA and FFNN

models. Since the hypothetical difference zero does not belongs to the interval of decision limits that is, 0(0.0284,

0.1565). Therefore null hypothesis is rejected and we may conclude that there is a significant difference in prediction

ability between the SARIMA and HONN models.

The decision limits (0.0188, 0.1399) are obtained using the bootstrap test procedure for comparison of SARIMA

and HONN models. Since the hypothetical difference zero does not belongs to the interval of decision limits that is,

0(0.0188, 0.1399). Therefore null hypothesis is rejected and we may conclude that there is a significant difference in

prediction ability between the SARIMA and HONN models.

The decision limits (-0.0109, 0.0385) are obtained using the bootstrap test procedure for comparison of FFNN and

HONN models. Since the hypothetical difference zero belongs to the interval of decision limits that is, 0 (-0.0188,


0.0385). Therefore null hypothesis is accepted and we may conclude that there is no significant difference in prediction

ability between the FFNN and HONN models.

From the above tests, it is observed that, the forecasting accuracy of the models is not same and empirical

evidence shows that FFNN model is good at forecasting than that of SARIMA model. FFNN and HONN models are

equally efficient at forecasting of minimum SAT.

CONCLUSIONS

Three different forecasting models, SARIMA, FFNN and HONN models have been developed in this study to

predict the mean of minimum surface air temperature in India. Error measures showed small values that demonstrate the

developed models are suitable and adequate for the forecasting of SAT. Forecasting errors of neural network models is less

compared to SARIMA model. Equal prediction accuracy tests reports that the neural networks models are significantly

different from SARIMA model. FFNN and HONN models are equally efficient at forecasting of minimum SAT. Hence,

neural networks are robust models for forecasting the mean of minimum surface air temperature.

ACKNOWLEDGEMENTS

Author is very thankful to Department of Science and Technology (DST), India, for providing Inspire fellowship

to carry out this research work.

REFERENCES

1. Afzali, M., Afzali, A. & Zahedi, G. (2011). Ambient Air Temperature Forecasting Using Artificial Neural

Network Approach, 2011 International Conference on Environmental and Computer Science , IPCBEE Vol.19,

IACSIT Press, Singapore.

2. Alfaro, E. (2004). A Method for Prediction of California Summer Air Surface Temperature, Eos, Vol. 85, No. 51,

21 December 2004.

3. Anisimov O.A., (2001). Predicting Patterns of Near-Surface Air Temperature Using Empirical Data, Climatic

Change, Vol. 50, No. 3, 297-315.

4. Bodri, L.,Čermák, V. (2003). Prediction of Surface Air Temperatures by Neural Network, Example Based on

Three-Year Temperature Monitoring at Spořilov Station, Studia Geophysica et Geodaetica, Vol. 47, 1, 173-184.

5. Box, G. E. P., Jenkins, G. M. & Reinsel, G. C. (1994). Time Series Analysis Forecasting and Control, 3rd ed.,

Englewood Cliffs, N.J. Prentice Hall.

6. Brunetti, M., Buffoni,L., Maugeri, M. & Nanni, T., (2000). Trends of minimum and maximum daily temperatures

in Italy from 1865 to 1996. Theor. Appl. Climatol., 66, 49–60.

7. FAN Ke, (2009). Predicting Winter Surface Air Temperature in Northeast China, Atmospheric and Oceanic

Science Letters, Vol. 2, NO. 1, 14−17.

8. Faraway, J. & Chatfield, C. (1998). Time series forecasting with neural networks: a comparative study using the

air line data, Journal of the Royal Statistical Society, Series C, Vol. 47, 2, 231-250.

9. Hejase, H.A.N. & Assi, A.H. (2012). Time-Series Regression Model for Prediction of Mean Daily Global Solar

Radiation in Al-Ain, UAE, ISRN Renewable Energy, Vol. 2012, Article ID 412471, 11 pages.

http://link.springer.com/search?facet-author=%22L.+Bodri%22

http://link.springer.com/search?facet-author=%22V.+%C4%8Cerm%C3%A1k%22

http://link.springer.com/journal/11200


10. Kémajou, A., Mba, L. & Meukam, P. (2012). Application of Artificial Neural Network for Predicting the Indoor

Air Temperature in Modern Building in Humid Region, British Journal of Applied Science & Technology 2(1),

23-34.

11. Knowles, A., Hussein, A., Deredy, W., Lisboa, P. & Dunis, C. L. (2005). Higher-Order Neural Networks with

Bayesian Confidence Measure for Prediction of EUR/USD Exchange Rate, CIBEF Working Papers. Available at

www.cibef.com.

12. Kulkarni,M.A., Patil, S., Rama, G.V. & Sen, P.N. (2008). Wind speed prediction using statistical regression and

neural network, J. Earth Syst. Sci. 117, No. 4, 457–463.

13. Lee, J.H., Sohn, K. (2007). Prediction of monthly mean surface air temperature in a region of China, Advances in

Atmospheric Sciences, Vol. 24, 3, 503-508.

14. Naveen Kumar Boiroju (2012). Forecasting Foreign Exchange Rates using Neural Networks, Unpublished Ph.D.

Thesis, Department of Statistics, Osmania University.

15. Naveen Kumar Boiroju, Ramu, Y., Venugopala Rao, M. & Krishna Reddy, M. (2011). A bootstrap test for

equality of mean absolute errors, ARPN Journal of engineering and applied sciences, 6(5), 9-11.

16. Rao, S.S. (2011). Forecasting of Monthly Rainfall in Andhra Pradesh using Neural Networks, Unpublished Ph.D.

Thesis, Department of Statistics, Osmania University.

17. Rojas, R. (1996). Neural Networks: A systematic introduction, Springer-Verlag.

18. Shrivastava, G., Karmakar, S., Kowar, M.K., Guhathakurta, P. (2012). Application of Artificial Neural Networks

in Weather Forecasting: A Comprehensive Literature Review, International Journal of Computer Applications,

Vol. 51, No.18, August 2012.

19. Smith, B.A. (2006). Air temperature prediction using artificial neural networks, M.Sc. (Thesis), University of

Gerogia.

20. Stein, M. & Lloret, J. (2001). Forecasting of Air and Water Temperatures for Fishery Purposes with Selected

Examples from Northwest Atlantic, J. Northw. Atl. Fish. Sci., Vol. 29, 23-30.

21. Tasadduq. I., Rehman, S., Bubshait, K. (2005). Application of neural networks for the prediction of hourly mean

surface temperatures in Saudi Arabia. Renewable Energy, 25, 545-554.

22. Zhang, G., Patuwo, B.E. & Hu, M.Y. (1998). Forecasting with Artificial Neural Networks: The State of the Art,

International Journal of Forecasting, 14, 35-62.

7.-Forecasting surface .FULL

Documents

Transcript of 7.-Forecasting surface .FULL