Utilitas Mathematica Journal has become a fully open-access journal. This journal publishes mainly in areas of pure and applied mathematics, Statistics. Our journal is an official publication of the Utilitas mathematical journal’s original research articles and aspects of both pure and applied mathematics.
Statistical analysis of an orographic rainfall for eight north-east region of...IJICTJOURNAL
Autoregressive integrated moving average (ARIMA) models are used to predict the rain rate for orographic rainfall over a long period of time, from 1980 to 1918. As the orographic rainfall may cause landslides and other natural disaster issues, So, this study is very important for the analysis of rainfall prediction. In this research, statistical calculations have been done based on the rainfall data for twelve regions of India (Cherrapunji, Darjeling, Dawki, Ghum, Itanagar, Kamchenjunga, Mizoram, Nagaland, Pakyong, Saser Kangri, Slot Kangri, and Tripura) from the eight states, i.e., Sikkim, Meghalaya, West Bengal, Ladakh (Union Territory of India), Arunachal Pradesh, Mizoram, Tripura, and Nagaland) with varying altitude. The model's output is assessed using several error calculations. The model's performance is represented by the fit value, which is reliable for the northeast region of India with increasing altitude. The statistical dependability of the rainfall prediction is shown by the parameters. The lowest value of root mean square error (RMSE) indicates better prediction for orographic rainfall.
Assessment of two Methods to study Precipitation PredictionAI Publications
Presipitation analysis plays an important role in hydrological studies. In this study, using 50 years of rainfall data and ARIMA model, critical areas of Iran were determined. For this purpose, annual rainfall data of 112 different synoptic stations in Iran were gathered. To summarize, it could be concluded that: ARIMA model was an appropriate tool to forecast annual rainfall. According to obtained results from relative error, five stations were in critical condition. At 45 stations accrued rainfalls with amounts of less than half of average in the 50-year period. Therefore, in these 45 areas, chance of drought is more than other areas of Iran.
Time Series Data Analysis for Forecasting – A Literature ReviewIJMER
In today's world there is ample opportunity to clout the numerous sources of time series data
available for decision making. This time ordered data can be used to improve decision making if the data
is converted to information and then into knowledge which is called knowledge discovery. Data Mining
(DM) methods are being increasingly used in prediction with time series data, in addition to traditional
statistical approaches. This paper presents a literature review of the use of DM and statistical approaches
with time series data, focusing on weather prediction. This is an area that has been attracting a great deal
of attention from researchers in the field.
Statistical analysis of an orographic rainfall for eight north-east region of...IJICTJOURNAL
Autoregressive integrated moving average (ARIMA) models are used to predict the rain rate for orographic rainfall over a long period of time, from 1980 to 1918. As the orographic rainfall may cause landslides and other natural disaster issues, So, this study is very important for the analysis of rainfall prediction. In this research, statistical calculations have been done based on the rainfall data for twelve regions of India (Cherrapunji, Darjeling, Dawki, Ghum, Itanagar, Kamchenjunga, Mizoram, Nagaland, Pakyong, Saser Kangri, Slot Kangri, and Tripura) from the eight states, i.e., Sikkim, Meghalaya, West Bengal, Ladakh (Union Territory of India), Arunachal Pradesh, Mizoram, Tripura, and Nagaland) with varying altitude. The model's output is assessed using several error calculations. The model's performance is represented by the fit value, which is reliable for the northeast region of India with increasing altitude. The statistical dependability of the rainfall prediction is shown by the parameters. The lowest value of root mean square error (RMSE) indicates better prediction for orographic rainfall.
Assessment of two Methods to study Precipitation PredictionAI Publications
Presipitation analysis plays an important role in hydrological studies. In this study, using 50 years of rainfall data and ARIMA model, critical areas of Iran were determined. For this purpose, annual rainfall data of 112 different synoptic stations in Iran were gathered. To summarize, it could be concluded that: ARIMA model was an appropriate tool to forecast annual rainfall. According to obtained results from relative error, five stations were in critical condition. At 45 stations accrued rainfalls with amounts of less than half of average in the 50-year period. Therefore, in these 45 areas, chance of drought is more than other areas of Iran.
Time Series Data Analysis for Forecasting – A Literature ReviewIJMER
In today's world there is ample opportunity to clout the numerous sources of time series data
available for decision making. This time ordered data can be used to improve decision making if the data
is converted to information and then into knowledge which is called knowledge discovery. Data Mining
(DM) methods are being increasingly used in prediction with time series data, in addition to traditional
statistical approaches. This paper presents a literature review of the use of DM and statistical approaches
with time series data, focusing on weather prediction. This is an area that has been attracting a great deal
of attention from researchers in the field.
Application of mathematical modelling in rainfall forcast a csae study in...eSAT Journals
Abstract Malaysia receives rainfall from 2000 mm to 4000 mm annually where it is greatly influenced by two monsoon periods in November to March and May to September. The state of Sarawak is well known for its long and wide rivers. Numerous activities such as commercial, industrial and residential can always be found in the vicinity of the rivers. The activities have started since decades ago and still continue to grow and spatially expanding through times providing incomes ranging from small farmers to the largest corporations. Unfortunately, these areas are expected to experience frequent flood events as well as possible receding water level in rivers based on the findings of previous studies. If the projections are accurate, the productivity of these activities will be reduced, hence, in a longer term may affect the economy of the state as whole as well. Therefore, there is an urgent need for existing knowledge on rainfall behavior to be revised as effects of climate change with the intention that the state can fully utilize the favorable conditions and make scientific based decisions in the future. Recent study reveals that the Fourier series (FS), has the ability to simulate long-term rainfall up to 300 years is viewed as an important finding in the study of rainfall forecast. Long-term rainfall forecasting is viewed to be beneficial to the state of Sarawak in its future planning in various sectors such as water supply, flood mitigation, river transportation as well as agriculture. The main goal of the study is to apply a mathematical modeling in rainfall forecasting for the Sungai Sarawak basin. Data from eight rain gauge stations was analyzed and prepared for missing data, consistency check and adequacy of number of stations. Simple statistical analysis was conducted on the data such as maximum, minimum, mean and standard deviation. 27 years of annual rainfall data were simulated with the Fourier Series equation using spreadsheet. Hence, the result was compared with the Fitting N-term Harmonic Series. The model result reveals that the Fourier Series has the ability to simulate the observed data by being able to describe the rainfall pattern and there is a reasonable relationship between the simulation and observed data with p-value of 0.93. Keywords: Fourier series, Mathematical
ANNUAL PRECIPITATION IN SOUTHERN OF MADAGASCAR: MODELING USING HIGH ORDER FUZ...ijfls
The objective of this research is to find the best conventional high order fuzzy time series model for annual precipitation series in southern Madagascar. This work consists on finding the hyper parameters (number of partition of the universe of discourse and model order) to obtain the best conventional high
order fuzzy time series model for our experimental data. In previous works, entitled spatial and temporal variability of precipitation in southern Madagascar, we subdivided the study area between 22 ° S to 30 ° S latitude and 43 ° Eto 48 ° E longitude into four zones of homogeneous precipitation. In this article, we seek to model annual precipitation data representative of one of these four areas. These data were taken between 1979 and 2017. Our approach consists on subdividing the data: data obtained from 1979 to 2001 (60%) for the training and data from 2002 to 2017 (40%) to test the model. To determine the number of partitions and model order, we fix first the number of partitions to 10 and then to 15, 20, 25,30, 35, 40, 45 and 50.For each of these values, we vary the model order from 1 to 10.Thenwe locate the model order which corresponds to the minimum of the average curve between the Mean Absolute Errors (MAE) between the training data and the test data. Thus, the orders of the candidate model are 2, 3, 5, and 6.The next step is to fix the model order with the previous values and vary the number of partitions from 3 to 50.For each couple of hyper parameter of the model (number of partitions, model order), we locate the value of number of partitions corresponding to the minimum of the average curve between the absolute mean of the errors or MAE (Mean Absolute Error) between the train and test data. We obtain the hyper-parameter pairs (37, 2), (20, 3), (35, 5) and (35, 6).The first pair gives the lowest Mean Absolute Error. As a final result, we obtain the best high order fuzzy time series model with hyperparameters umber of partition equals thirty seven and of order equals two for annual precipitation in Southern of Madagascar.
Comparison of Tropical Thunderstorm Estimation between Multiple Linear Regres...journalBEEI
Thunderstorms are dangerous and it has increased due to highly precipitation and cloud cover density in the Mesoscale Convective System area. Climate change is one of the causes to increasing the thunderstorm activity. The present studies aimed to estimate the thunderstorm activity at the Tawau area of Sabah, Malaysia based on the Multiple Linear Regression (MLR), Dvorak technique, and Adaptive Neuro-Fuzzy Inference System (ANFIS). A combination of up to six inputs of meteorological data such as Pressure (P), Temperature (T), Relative Humidity (H), Cloud (C), Precipitable Water Vapor (PWV), and Precipitation (Pr) on a daily basis in 2012 were examined in the training process to find the best configuration system. By using Jacobi algorithm, H and PWV were identified to be correlated well with thunderstorms. Based on the two inputs that have been identified, the Sugeno method was applied to develop a Fuzzy Inference System. The model demonstrated that the thunderstorm activities during intermonsoon are detected higher than the other seasons. This model is comparable to the thunderstorm data that was collected manually with percent error below 50%.
Best Fit and Selection of Probability Distribution Models for Frequency Analy...IJERD Editor
Frequency analysis of extreme low mean annual rainfall events is important to water resource planners at catchment level because mean annual rainfall is an important parameter in determining mean annual runoff. Mean annual runoff is an important input in determining surface water available for water resource infrastructure development. In order to carry out frequency analysis of extreme low mean annual rainfall events, it is necessary to identify the best fit probability distribution models (PDMs) for the frequency analysis. The primary objective of the study was to develop two model identification criteria. The first criterion was developed to identify candidate probability distribution models from which the best fit probability distribution models were identified. The second criterion was applied to select the best fit probability distribution models from the candidate models. The secondary objectives were:
Determination of homogenous regions in the Tensift basin (Morocco).IJERA Editor
The aim of this study is to determine homogenous region in the Tensift basin within which the hydrological behavior is similar. In order to do this we used two methods: The Principal components analysis on the monthly precipitation registered at the 23 rainfall stations. This resulted in setting apart 4 groups of stations. The second method is analysis of land use map, geological map, pedagogical map, vegetation map and slope map of the studied area. This method allowed us to delineate 4 homogenous areas. The two methods yielded complementary results and the superposition of groups and regions obtained allowed us to retain 4 homogenous regions corresponding to 3 groups of stations.
Explore the cutting-edge methods and technologies utilized in rain forecasting, from traditional meteorological models to machine learning algorithms. Discover how these predictive tools enable accurate anticipation of rainfall patterns, aiding in disaster preparedness, agriculture planning, and urban infrastructure management. To learn in detail about analysis and prediction visit: https://bostoninstituteofanalytics.org/data-science-and-artificial-intelligence/
Linear Regressions of Predicting Rainfall over Kalay Regionijtsrd
Regression analysis is a statistical technique for investigating the relationship between variables. In this paper, rainfall and water level prediction models are discussed with the use of empirical statistical technique, Simple Linear Regression and analyzed the development of the predictive power of Linear Regression model to forecast the predicting rainfall and water level over Kalay in Sagaing Region for 10 years 2008 2017 .The data of the monthly rainfall and water level used in this study were obtained from Meteorology and Hydrology Department of Kalay, Myanmar. In July 2015, Kalay was affected by the floods. So the rainfall and water level are predicted for next five years in this paper. Ohnmar Myint "Linear Regressions of Predicting Rainfall over Kalay Region" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-5 , August 2019, URL: https://www.ijtsrd.com/papers/ijtsrd26711.pdf Paper URL: https://www.ijtsrd.com/mathemetics/computational-science/26711/linear-regressions-of-predicting-rainfall-over-kalay-region/ohnmar-myint
On the performance analysis of rainfall prediction using mutual information...IJECEIAES
Monsoon rainfall prediction over a small geographic region is indeed a challenging task. This paper uses monthly means of climate variables, namely air temperature (AT), sea surface temperature (SST), and sea level pressure (SLP) over the globe, to predict monthly and seasonal summer monsoon rainfall over the state of Maharashtra, India. Mutual information correlates the temperature and pressure from a grid of 10 longitude X 10 latitude with Maharashtra’s monthly rainfall time series. Based on the correlations, selected features over the respective latitude and longitudes are given as inputs to an artificial neural network. It was observed that AT and SLP could predict monthly monsoon rainfall with excellent accuracy. The performance of the test dataset was evaluated through mean absolute error; root mean square error, correlation coefficient, Nash Sutcliffe model efficiency coefficient, and maximum rainfall prediction capability of the network. The individual climate variable model for AT performed better in all evaluation parameters except maximum rainfall capability, where the combined model 2 with AT, SLP and SST as predictors outperformed. The SLP-only model’s performance was comparable to the AT-only model. The combined model 1 with AT and SLP as predictors was found better than the combined model 2.
International Journal of Computational Engineering Research(IJCER)ijceronline
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology
Development of an Equation to Estimate the Monthly Rainfall A Case Study for ...ijtsrd
This study aimed to derived an equation to estimate the monthly rainfall for Catarman, Northern Samar.The observed monthly rainfall data for Catarman N. Samar, Catbalogan Samar, Legazpi City and Masbate were obtained from the Philippine Atmospheric Geographical Astronomical Services Administration PAGASA . The monthly rainfall records of the three 3 neighboring stations Catbalogan, Legazpi, Masbate were used to identify which of the existing rainfall prediction methods, namely, Normal Ratio Method, Distance Power Method and Multi Linear Regression Method is the basis in the development of a new equation. The accuracy by which the existing methods predict the observed monthly rainfall in Catarman was evaluated using T test for correlated samples and the Pearson’s Correlation Coefficient. Since none of the methods produced estimates nearest to the observed monthly rainfall in Catarman, an equation has been derived Celeste A. De Asis | Benjamin D. Varela "Development of an Equation to Estimate the Monthly Rainfall: A Case Study for Catarman, Northern Samar, Philippines" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-5 | Issue-1 , December 2020, URL: https://www.ijtsrd.com/papers/ijtsrd35875.pdf Paper URL : https://www.ijtsrd.com/engineering/civil-engineering/35875/development-of-an-equation-to-estimate-the-monthly-rainfall-a-case-study-for-catarman-northern-samar-philippines/celeste-a-de-asis
WATERSHED MODELING USING ARTIFICIAL NEURAL NETWORKS IAEME Publication
Artificial Neural Networks analysis was used for modeling rainfall-runoff relationship. A new Instantaneous ANN watershed model was built and tried herein using Walnut Gulch watershed (catchment) area. For modeling the instantaneous response of a catchment to a rainfall event an ANN model was built shown herein. The built model can represent the actual response using descritized
rainfall-runoff values, over a selected time interval (∆t). As this time interval decreases the actual response is more accurately modeled. This model was applied to one of the sub-catchment of Walnut Gulch watershed (sub-catchment No.9 (flume 11)). The model was found successful to represent the lag-time and time of runoff related to the hyetograph properties
ijerst offers a fast publication schedule whilst maintaining rigorous peer review; the use of recommended electronic formats for article delivery expedites the process. International Journal of Engineering Research and Science & Technology (IJERST) is an international online journal in English published Quarterly. All submitted research articles are subjected to immediate rapid screening by the editors, in consultation with the Editorial Board or others working in the field as appropriate, to ensure they are likely to be of the level of interest and importance appropriate for the journal.
Alinteri Journal of Agriculture Sciences The journal is an open access, international, double-blind peer-reviewed journal publishing research articles, Invited reviews, short communications, and letters to the Editor in the field of agriculture, fisheries, veterinary, biology, and closely related disciplines. We adopt the policy of providing open access to readers who may be interested in recent developments. Is being published online biannually as of 2007.
Application of mathematical modelling in rainfall forcast a csae study in...eSAT Journals
Abstract Malaysia receives rainfall from 2000 mm to 4000 mm annually where it is greatly influenced by two monsoon periods in November to March and May to September. The state of Sarawak is well known for its long and wide rivers. Numerous activities such as commercial, industrial and residential can always be found in the vicinity of the rivers. The activities have started since decades ago and still continue to grow and spatially expanding through times providing incomes ranging from small farmers to the largest corporations. Unfortunately, these areas are expected to experience frequent flood events as well as possible receding water level in rivers based on the findings of previous studies. If the projections are accurate, the productivity of these activities will be reduced, hence, in a longer term may affect the economy of the state as whole as well. Therefore, there is an urgent need for existing knowledge on rainfall behavior to be revised as effects of climate change with the intention that the state can fully utilize the favorable conditions and make scientific based decisions in the future. Recent study reveals that the Fourier series (FS), has the ability to simulate long-term rainfall up to 300 years is viewed as an important finding in the study of rainfall forecast. Long-term rainfall forecasting is viewed to be beneficial to the state of Sarawak in its future planning in various sectors such as water supply, flood mitigation, river transportation as well as agriculture. The main goal of the study is to apply a mathematical modeling in rainfall forecasting for the Sungai Sarawak basin. Data from eight rain gauge stations was analyzed and prepared for missing data, consistency check and adequacy of number of stations. Simple statistical analysis was conducted on the data such as maximum, minimum, mean and standard deviation. 27 years of annual rainfall data were simulated with the Fourier Series equation using spreadsheet. Hence, the result was compared with the Fitting N-term Harmonic Series. The model result reveals that the Fourier Series has the ability to simulate the observed data by being able to describe the rainfall pattern and there is a reasonable relationship between the simulation and observed data with p-value of 0.93. Keywords: Fourier series, Mathematical
ANNUAL PRECIPITATION IN SOUTHERN OF MADAGASCAR: MODELING USING HIGH ORDER FUZ...ijfls
The objective of this research is to find the best conventional high order fuzzy time series model for annual precipitation series in southern Madagascar. This work consists on finding the hyper parameters (number of partition of the universe of discourse and model order) to obtain the best conventional high
order fuzzy time series model for our experimental data. In previous works, entitled spatial and temporal variability of precipitation in southern Madagascar, we subdivided the study area between 22 ° S to 30 ° S latitude and 43 ° Eto 48 ° E longitude into four zones of homogeneous precipitation. In this article, we seek to model annual precipitation data representative of one of these four areas. These data were taken between 1979 and 2017. Our approach consists on subdividing the data: data obtained from 1979 to 2001 (60%) for the training and data from 2002 to 2017 (40%) to test the model. To determine the number of partitions and model order, we fix first the number of partitions to 10 and then to 15, 20, 25,30, 35, 40, 45 and 50.For each of these values, we vary the model order from 1 to 10.Thenwe locate the model order which corresponds to the minimum of the average curve between the Mean Absolute Errors (MAE) between the training data and the test data. Thus, the orders of the candidate model are 2, 3, 5, and 6.The next step is to fix the model order with the previous values and vary the number of partitions from 3 to 50.For each couple of hyper parameter of the model (number of partitions, model order), we locate the value of number of partitions corresponding to the minimum of the average curve between the absolute mean of the errors or MAE (Mean Absolute Error) between the train and test data. We obtain the hyper-parameter pairs (37, 2), (20, 3), (35, 5) and (35, 6).The first pair gives the lowest Mean Absolute Error. As a final result, we obtain the best high order fuzzy time series model with hyperparameters umber of partition equals thirty seven and of order equals two for annual precipitation in Southern of Madagascar.
Comparison of Tropical Thunderstorm Estimation between Multiple Linear Regres...journalBEEI
Thunderstorms are dangerous and it has increased due to highly precipitation and cloud cover density in the Mesoscale Convective System area. Climate change is one of the causes to increasing the thunderstorm activity. The present studies aimed to estimate the thunderstorm activity at the Tawau area of Sabah, Malaysia based on the Multiple Linear Regression (MLR), Dvorak technique, and Adaptive Neuro-Fuzzy Inference System (ANFIS). A combination of up to six inputs of meteorological data such as Pressure (P), Temperature (T), Relative Humidity (H), Cloud (C), Precipitable Water Vapor (PWV), and Precipitation (Pr) on a daily basis in 2012 were examined in the training process to find the best configuration system. By using Jacobi algorithm, H and PWV were identified to be correlated well with thunderstorms. Based on the two inputs that have been identified, the Sugeno method was applied to develop a Fuzzy Inference System. The model demonstrated that the thunderstorm activities during intermonsoon are detected higher than the other seasons. This model is comparable to the thunderstorm data that was collected manually with percent error below 50%.
Best Fit and Selection of Probability Distribution Models for Frequency Analy...IJERD Editor
Frequency analysis of extreme low mean annual rainfall events is important to water resource planners at catchment level because mean annual rainfall is an important parameter in determining mean annual runoff. Mean annual runoff is an important input in determining surface water available for water resource infrastructure development. In order to carry out frequency analysis of extreme low mean annual rainfall events, it is necessary to identify the best fit probability distribution models (PDMs) for the frequency analysis. The primary objective of the study was to develop two model identification criteria. The first criterion was developed to identify candidate probability distribution models from which the best fit probability distribution models were identified. The second criterion was applied to select the best fit probability distribution models from the candidate models. The secondary objectives were:
Determination of homogenous regions in the Tensift basin (Morocco).IJERA Editor
The aim of this study is to determine homogenous region in the Tensift basin within which the hydrological behavior is similar. In order to do this we used two methods: The Principal components analysis on the monthly precipitation registered at the 23 rainfall stations. This resulted in setting apart 4 groups of stations. The second method is analysis of land use map, geological map, pedagogical map, vegetation map and slope map of the studied area. This method allowed us to delineate 4 homogenous areas. The two methods yielded complementary results and the superposition of groups and regions obtained allowed us to retain 4 homogenous regions corresponding to 3 groups of stations.
Explore the cutting-edge methods and technologies utilized in rain forecasting, from traditional meteorological models to machine learning algorithms. Discover how these predictive tools enable accurate anticipation of rainfall patterns, aiding in disaster preparedness, agriculture planning, and urban infrastructure management. To learn in detail about analysis and prediction visit: https://bostoninstituteofanalytics.org/data-science-and-artificial-intelligence/
Linear Regressions of Predicting Rainfall over Kalay Regionijtsrd
Regression analysis is a statistical technique for investigating the relationship between variables. In this paper, rainfall and water level prediction models are discussed with the use of empirical statistical technique, Simple Linear Regression and analyzed the development of the predictive power of Linear Regression model to forecast the predicting rainfall and water level over Kalay in Sagaing Region for 10 years 2008 2017 .The data of the monthly rainfall and water level used in this study were obtained from Meteorology and Hydrology Department of Kalay, Myanmar. In July 2015, Kalay was affected by the floods. So the rainfall and water level are predicted for next five years in this paper. Ohnmar Myint "Linear Regressions of Predicting Rainfall over Kalay Region" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-5 , August 2019, URL: https://www.ijtsrd.com/papers/ijtsrd26711.pdf Paper URL: https://www.ijtsrd.com/mathemetics/computational-science/26711/linear-regressions-of-predicting-rainfall-over-kalay-region/ohnmar-myint
On the performance analysis of rainfall prediction using mutual information...IJECEIAES
Monsoon rainfall prediction over a small geographic region is indeed a challenging task. This paper uses monthly means of climate variables, namely air temperature (AT), sea surface temperature (SST), and sea level pressure (SLP) over the globe, to predict monthly and seasonal summer monsoon rainfall over the state of Maharashtra, India. Mutual information correlates the temperature and pressure from a grid of 10 longitude X 10 latitude with Maharashtra’s monthly rainfall time series. Based on the correlations, selected features over the respective latitude and longitudes are given as inputs to an artificial neural network. It was observed that AT and SLP could predict monthly monsoon rainfall with excellent accuracy. The performance of the test dataset was evaluated through mean absolute error; root mean square error, correlation coefficient, Nash Sutcliffe model efficiency coefficient, and maximum rainfall prediction capability of the network. The individual climate variable model for AT performed better in all evaluation parameters except maximum rainfall capability, where the combined model 2 with AT, SLP and SST as predictors outperformed. The SLP-only model’s performance was comparable to the AT-only model. The combined model 1 with AT and SLP as predictors was found better than the combined model 2.
International Journal of Computational Engineering Research(IJCER)ijceronline
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology
Development of an Equation to Estimate the Monthly Rainfall A Case Study for ...ijtsrd
This study aimed to derived an equation to estimate the monthly rainfall for Catarman, Northern Samar.The observed monthly rainfall data for Catarman N. Samar, Catbalogan Samar, Legazpi City and Masbate were obtained from the Philippine Atmospheric Geographical Astronomical Services Administration PAGASA . The monthly rainfall records of the three 3 neighboring stations Catbalogan, Legazpi, Masbate were used to identify which of the existing rainfall prediction methods, namely, Normal Ratio Method, Distance Power Method and Multi Linear Regression Method is the basis in the development of a new equation. The accuracy by which the existing methods predict the observed monthly rainfall in Catarman was evaluated using T test for correlated samples and the Pearson’s Correlation Coefficient. Since none of the methods produced estimates nearest to the observed monthly rainfall in Catarman, an equation has been derived Celeste A. De Asis | Benjamin D. Varela "Development of an Equation to Estimate the Monthly Rainfall: A Case Study for Catarman, Northern Samar, Philippines" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-5 | Issue-1 , December 2020, URL: https://www.ijtsrd.com/papers/ijtsrd35875.pdf Paper URL : https://www.ijtsrd.com/engineering/civil-engineering/35875/development-of-an-equation-to-estimate-the-monthly-rainfall-a-case-study-for-catarman-northern-samar-philippines/celeste-a-de-asis
WATERSHED MODELING USING ARTIFICIAL NEURAL NETWORKS IAEME Publication
Artificial Neural Networks analysis was used for modeling rainfall-runoff relationship. A new Instantaneous ANN watershed model was built and tried herein using Walnut Gulch watershed (catchment) area. For modeling the instantaneous response of a catchment to a rainfall event an ANN model was built shown herein. The built model can represent the actual response using descritized
rainfall-runoff values, over a selected time interval (∆t). As this time interval decreases the actual response is more accurately modeled. This model was applied to one of the sub-catchment of Walnut Gulch watershed (sub-catchment No.9 (flume 11)). The model was found successful to represent the lag-time and time of runoff related to the hyetograph properties
ijerst offers a fast publication schedule whilst maintaining rigorous peer review; the use of recommended electronic formats for article delivery expedites the process. International Journal of Engineering Research and Science & Technology (IJERST) is an international online journal in English published Quarterly. All submitted research articles are subjected to immediate rapid screening by the editors, in consultation with the Editorial Board or others working in the field as appropriate, to ensure they are likely to be of the level of interest and importance appropriate for the journal.
Alinteri Journal of Agriculture Sciences The journal is an open access, international, double-blind peer-reviewed journal publishing research articles, Invited reviews, short communications, and letters to the Editor in the field of agriculture, fisheries, veterinary, biology, and closely related disciplines. We adopt the policy of providing open access to readers who may be interested in recent developments. Is being published online biannually as of 2007.
Authors are invited to submit theoretical or empirical papers in all aspects of management, including strategy, human resources, marketing, operations, technology, information systems, finance and accounting, business economics, and public sector management. IJMRR is an international forum for research that advances the theory and practice of management. The journal publishes original works with practical significance and academic value.
ijerst offers a fast publication schedule whilst maintaining rigorous peer review; the use of recommended electronic formats for article delivery expedites the process. International Journal of Engineering Research and Science & Technology (IJERST) is an international online journal in English published Quarterly. All submitted research articles are subjected to immediate rapid screening by the editors, in consultation with the Editorial Board or others working in the field as appropriate, to ensure they are likely to be of the level of interest and importance appropriate for the journal.
IJMRR is an international forum for research that advances the theory and practice of management. All papers submitted to IJMRR are subject to a double-blind peer review process. Authors are invited to submit theoretical or empirical papers in all aspects of management, including strategy, human resources, marketing, operations, technology, information systems, finance and accounting, business economics, and public sector management.
Alinteri Journal of Agriculture Sciences aims to create an environment for researchers to introduce, share, read, and discuss recent scientific progress. We adopt the policy of providing open access to readers who may be interested in recent developments.
Alinteri Journal of Agriculture Sciences is being published online biannually as of 2007. The journal is an open access, international, double-blind peer-reviewed journal publishing research articles, Invited reviews, short communications, and letters to the Editor in the fields of agriculture, fisheries, veterinary, biology, and closely related disciplines. Alinteri Journal of Agriculture Sciences aims to create an environment for researchers to introduce, share, read, and discuss recent scientific progress. We adopt the policy of providing open access to readers who may be interested in recent developments.
Alinteri Journal of Agriculture Sciences aims to create an environment for researchers to introduce, share, read, and discuss recent scientific progress. We adopt the policy of providing open access to readers who may be interested in recent developments. Alinteri Journal of Agriculture Sciences is being published online biannually as of 2007. The journal is an open access, international, double-blind peer-reviewed journal publishing research articles, Invited reviews, short communications, and letters to the Editor in the fields of agriculture, fisheries, veterinary, biology, and closely related disciplines.
IJERST offers a fast publication schedule whilst maintaining rigorous peer review; the use of recommended electronic formats for article delivery expedites the process. International Journal of Engineering Research and Science & Technology (IJERST) is an international online journal in English published Quarterly. All submitted research articles are subjected to immediate rapid screening by the editors, in consultation with the Editorial Board or others working in the field as appropriate, to ensure they are likely to be of the level of interest and importance appropriate for the journal.
ijerst offers a fast publication schedule whilst maintaining rigorous peer review; the use of recommended electronic formats for article delivery expedites the process. Editorial Board or others working in the field as appropriate, to ensure they are likely to be of the level of interest and importance appropriate for the journal. International Journal of Engineering Research and Science & Technology (IJERST) is an international online journal in English published Quarterly. All submitted research articles are subjected to immediate rapid screening by the editors.
The journal publishes original works with practical significance and academic value. Authors are invited to submit theoretical or empirical papers in all aspects of management, including strategy, human resources, marketing, operations, technology, information systems, finance and accounting, business economics, and public sector management.
All papers submitted to IJMRR are subject to a double-blind peer review process. IJMRR is an international forum for research that advances the theory and practice of management. The journal publishes original works with practical significance and academic value. Authors are invited to submit theoretical or empirical papers in all aspects of management, including strategy, human resources, marketing, operations, technology, information systems, finance and accounting, business economics, and public sector management.
ijerst offers a fast publication schedule whilst maintaining rigorous peer review; the use of recommended electronic formats for article delivery expedites the process. Editorial Board or others working in the field as appropriate, to ensure they are likely to be of the level of interest and importance appropriate for the journal. International Journal of Engineering Research and Science & Technology (IJERST) is an international online journal in English published Quarterly. All submitted research articles are subjected to immediate rapid screening by the editors.
The journal publishes original works with practical significance and academic value. All papers submitted to IJMRR are subject to a double-blind peer review process. Authors are invited to submit theoretical or empirical papers in all aspects of management, including strategy, human resources, marketing, operations, technology, information systems, finance and accounting, business economics, and public sector management.
Alinteri Journal of Agriculture Sciences is being published online biannually as of 2007. Alinteri Journal of Agriculture Sciences aims to create an environment for researchers to introduce, share, read, and discuss recent scientific progress. The journal is an open access, international, double-blind peer-reviewed journal publishing research articles, Invited reviews, short communications, and letters to the Editor in the fields of agriculture, fisheries, veterinary, biology, and closely related disciplines. We adopt the policy of providing open access to readers who may be interested in recent developments.
All submitted research articles are subjected to immediate rapid screening by the editors, in consultation with the Editorial Board or others working in the field as appropriate, to ensure they are likely to be of the level of interest and importance appropriate for the journal. ijerst offers a fast publication schedule whilst maintaining rigorous peer review; the use of recommended electronic formats for article delivery expedites the process.
The journal publishes original works with practical significance and academic value. All papers submitted to IJMRR are subject to a double-blind Authors are invited to submit theoretical or empirical papers in all aspects of management, including strategy, human resources, marketing, operations, technology, information systems, finance and accounting, business economics, public sector management. peer review process. IJMRR is an international forum for research that advances the theory and practice of management.
All papers submitted to IJMRR are subject to a double-blind peer review process. The journal publishes original works with practical significance and academic value. Authors are invited to submit theoretical or empirical papers in all aspects of management, including strategy, human resources, marketing, operations, technology, information systems, finance and accounting, business economics, and public sector management. IJMRR is an international forum for research that advances the theory and practice of management.
The journal publishes original works with practical significance and academic value. Authors are invited to submit theoretical or empirical papers in all aspects of management, including strategy, human resources, marketing, operations, technology, information systems, finance and accounting, business economics, and public sector management. IJMRR is an international forum for research that advances the theory and practice of management. All papers submitted to IJMRR are subject to a double-blind peer review process.
Alinteri Journal of Agriculture Sciences aims to create an environment for researchers to introduce, share, read, and discuss recent scientific progress. We adopt the policy of providing open access to readers who may be interested in recent developments. The journal is an open access, international, double-blind peer-reviewed journal publishing research articles, Invited reviews, short communications, and letters to the Editor in the field of agriculture, fisheries, veterinary, biology, and closely related disciplines. Alinteri Journal of Agriculture Sciences is being published online biannually as of 2007.
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
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A Strategic Approach: GenAI in EducationPeter Windle
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The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
Honest Reviews of Tim Han LMA Course Program.pptxtimhan337
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journal publication
1. UtilitasMathematica
ISSN 0315-3681 Volume 118, 2021
58
A SARIMA and Adjusted SARIMA Models in a Seasonal Nonstationary Time
Series; Evidence of Enugu Monthly Rainfall
Amaefula Chibuzo Gabriel
Federal University Otuoke, Nigeria, wordwithflmae@gmail.com
Abstract
The paper compares SARIMA and adjusted SARIMA(ASARIMA) in a regular stationary series
where the underlying variable is seasonally nonstationary. Adopting empirical rainfall data and
Box-Jenkins iterative algorithm that calculates least squares estimates, Out of 11 sub-classes of
SARIMA and 7 sub-classes of ASARIMA models, AIC chose ASARIMA(2,1,1)12 over all
sub-classes of SARIMA(p,0,q)x(P,1,Q)12 identified. Diagnostic test indicates absence of
autocorrelation up to the 48th lag. The forecast values generated by the fitted model are closely
related to the actual values. Hence, ASARIMA can be recommended for regular stationary time
series with seasonal characteristics and where parameter redundancy and large sum of square
errors are penalized.
Keywords: AIC, ASARIMA model, rainfall, seasonal nonstationary time series.
I. Introduction
The use of seasonal autoregressive integrated moving average (SARIMA) terms for monthly or
quarterly data with systematic seasonal movements was recommended by[7]. Technical details
can be obtained from the aforementioned citation. Situation could arise when the underlying
variable of interest is regularly stationary but it is characterized by cyclical pattern that is
seasonally nonstationary and needs seasonal differencing. Time series variables with such
characteristics can be better modelled with Adjusted SARIMA(P,D,Q)s rather than
SARIMA(p,d,q)x(P,D,Q)s model.However, for such time series, SARIMA(p,d,q)x(P,D,Q)s
increases the sum of square residuals due to some redundant parameters and the autocorrelation
of the model residuals may be strong in higher lag orders. These are the advantages of Adjusted
SARIMA over SARIMA model. Adjusted SARIMA models are frugal in parameter
representation. Rainfall is one of the most important natural factors that determine the
agricultural production in and across the globe, particularly in Nigeria. The variability of rainfall
and the pattern of extreme high or low precipitation are very important for agriculture as well as
the economy of the state. Even the global climatic change has increased the quest for more
research on the subject matter due to high flood risk disaster at the peak of rainy season.
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Enugu State is one of the states in the eastern part of Nigeria located at the foot of the Udi
Plateau, a tropical rain forest zone with a derived savannah. The state shares borders with Abia
State and Imo State to the south, Ebonyi State to the east, Benue State to the northeast, Kogi
State to the northwest and Anambra State to the west. Enugu has good soil-land and climatic
conditions all year round, sitting at about 223 metres (732 ft) above sea level, and the soil is well
drained during its rainy seasons.Enugu is in the tropical rain forest zone with a derived savannah,
with humidity highest between March and November [13]. For the whole of Enugu State
the mean daily temperature is 26.7 °C (80.1
°F). The mean temperature in Enugu State in the hottest month of February is about 87.16 °F
(30.64 °C), while the lowest temperatures occur in the month of November, reaching 60.54 °F
(15.86 °C). The lowest rainfall of about
0.16 cubic centimetres (0.0098 cubic inch) is normal in February, while the highest is about 35.7
cubic centimetres (2.18 cu in) in July. Enugu State had a population of 3,267,837 people at the
census held in 2006 (estimated at over 3.8 million in 2012).
A lot of researchers have paid considerable attention towards modelling and forecasting the
amount of rainfall pattern in various places. For instance, [14] fitted a SARIMA(0, 1, 1)x(0, 1,
1)12 monthly rainfall in Tamilnadu, India. [16] fitted the SARIMA models of orders (1, 1, 2)x(1,
1, 1)12 and (4, 0, 2)x(1, 0, 1)12 respectively for monthly rainfall in Malaaca and Kuantan in
Malaysia. [1] examined the SARIMA model suitable for rainfall prediction in the Brong Ahafo
(BA) Region of Ghana using a data from 1975 to 2009. The results revealed that the region
experience much rainfall in the months of September and October, and least amount of rainfall in
the months of January, December and February. They fitted SARIMA (0,0,0)×(1,1,1)12, model
for predicting monthly average rainfall figures for the Brong Ahafo Region of Ghana.
[12] modelled monthly rainfall in Port Harcourt, Nigeria,
using seasonal SARIMA (5, 1, 0)x(0, 1, 1)12 model. The time- plot shows no noticeable trend.
The known and expected seasonality is clear from the plot. Seasonal (i.e. 12-point) differencing
of the data is done, then a nonseasonal differencing is done of the seasonal differences. The
correlogram of the resultant series reveals the expected 12- monthly seasonality, and the
involvement of a seasonal moving average component in the first place and a nonseasonal
autoregressive component of order 5. Hence the model mentioned above. The adequacy of the
modelled has been established. [15] modelled quarterly rainfall in Port Harcourt, Nigeria, as a
SARIMA(0, 0, 0)x(2, 1, 0)4 model.
[3] examined the time series analysis on rainfall in Oshogbo Osun State, Nigeria, using monthly
data of rainfall between 2004-2015. The time plot reveals that the rainfall data show high level of
volatility characterized by seasonal and irregular variations. And the logistic model applied
showed to be better and then used to forecast the rainfall for the next 2 years. [5] examined the
modelling of mean annual rainfall pattern in Port Harcourt, Nigeria using ARMA(p,q) model..
The data on rainfall used covered the period of 1981 to 2016. Sum of squares deviation forecast
criteria (SSDFC) was adopted to select the best performing sub-classes of ARMA(p, q) that fits
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the data. Among ARMA(1, 1), ARMA(1, 2) ARMA(2, 1) and ARMA(2, 2) models
estimated, SSDFC chose ARMA(1, 2) as the best performing model. The selected model were
supported by AIC and BIC respectively. And concluded that ARMA(1, 2) can be used to predict
long term quality of water foragriculture and hydrological purpose and to create long term
awareness against flood and control strategy for Port Harcourt.
[6] modelled monthly rainfall pattern in Imo state using seasonal autoregressive integrated
moving average (SARIMA) model with univariate monthly rainfall data spanning from 1981M1-
2017M12. Sum of square deviation forecast criterion (SSDFC) was used to compare nine (9)
different sub-classes of SARIMA( p,d,q)x(P,D,Q)12 models identified. And the result indicated
that SARIMA(0,0,0)x(1,1,1)12 is more appropriate in predicting monthly rainfall in the state.
The modelling of monthly rainfall in any state is essential in understanding the temporal and
spatial variability which is very important in flood risk management, irrigation and surface water
management and so on. Moreover, the need to diversify the economy towards agricultural base in
Nigeria has made it necessary to model seasonal pattern of rainfall in the state for agricultural
planning. Hence, this study examines the best fitted model between seasonal ARIMA (SARIMA)
model and adjusted seasonal ARIMA (ASARIMA) model for rainfall forecast in Enugu State.
The study presents a simple analytical model adjusted from SARIMA process. The remaining
part of the paper is arranged as follows; section two presents the materials and methods, section
three presents data analysis and results and section four deals with conclusion.
II. MATERIALS AND METHODS
This section highlights the methods and sources of data collection, variable measurement,
method of unit root test, model specification, and model identification, method of data analysis,
model comparison techniques and diagnostic checks.
A. Source of Data and Variable Measurement
The monthly rainfall data was obtained from central bank of Nigeria (CBN) (2018) statistical
bulletin. The univariate time series data collected covered the period of 1981M1- 2017M12 (432
observations of monthly rainfall data). Rainfall is usually measured in millimetre using rain
gauge.
B. SARIMA Model Specification
If the time series Xt is nonstationarity due to the presence of one or several of five
conditions: outliers, random walk, drift, trend, or changing variance, it is conventional that first
or second differencing (d) is necessary to achieve stationarity. Hence, the original series is said to
follow an autoregressive integrated moving average model or orders p, d and q denoted by
ARIMA(p, d, q) of the form
If the series Xt exhibits seasonal patterns of nonstationarity, this may be detected
using time plot, correlograms or even unit root test. And according to [7] Seasonal ARIMA
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models sometimes called SARIMA models has the general form SARIMA( p, d, q) (P, D,Q)S
and it is given as
where A(L) is the autoregressive (AR) operator, given by and B(L) is the
moving average (MA)operator, given by B(L) =1− 1L − − qL. For L denotes the backshift
operator. (Ls ) and (Ls ) are lagged seasonal AR and MA operators of order P and Q
respectively. The operator d
denotes the difference operator defined by d
= 1− L and d 2. The
represents the seasonal difference operator defined by = 1− Ls
and D is the seasonal
differencing order. The seasonal differencing (1− Ls )is called the simplifying operator, which
renders the residual series stationary and amenable to further analysis.
C. Adjusted SARIMA Model
The SARIMA model in (2) is the combination of nonseasonal AR and MA operators of order p
and q and seasonal AR and MA operators of order P and Q. If a univariate time series is
stationary in non-seasonal component (where d=0) and exhibits a purely seasonal pattern that is
nonstationary (where D=1). It could be parsimoniously better to only fit the seasonal AR and MA
operators of order P and Q. In such cases, it is appropriate to assume that A(L) 1, B(L) 1and
d=0 so that (2) can be of the form;
where (Ls ) is the seasonal autoregressive (SAR) operator, given by
and (L ) is the seasonal moving average (SMA) operator, given by .
Generally, the Adjusted SARIMA(P,D,Q)s model which hereafter is known as
ASARIMA(P,D,Q)s model with the inbuilt constant term is specifically of the form;
where is the constant parameter and s is the seasonal index. ASARIMA(P,D,Q)s model is
special case of SARIMA( p, d, q) (P, D,Q)S model.
D. Model Identification
The ACF of an MA(q) model cuts off after lag q whereas that of an AR(p) model is a
combination of sinusoidals dying off slowly. On the other hand, the PACF of an MA(q) model
dies off slowly whereas that of an AR(p) model cuts off after lag p. The AR and MA models are
known to exhibit some duality relationships. Parametric parsimony consideration in model
building entails the use of the mixed ARMA fit in preference to either the pure AR or the pure
MA fit.
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Note that SARIMA can be fitted irrespective of whether the underlying variable is seasonally
stationary or not.The differencing operators d 0 for stationary series and for nonstationary
series d could be 1 or 2 depending on the order of integration of the variable under study. The
seasonal difference D may be chosen to be at most equal to 1. The nonseasonal and seasonal AR
orders p and P are fitted by the nonseasonal and the seasonal cut-off lags of the partial
autocorrelation function (PACF) respectively. Similarly the nonseasonal and the seasonal MA
orders q and Q are fitted respectively by the nonseasonal and seasonal cut-off points of the ACF.
E. Conditions for ASARIMA(P,D,Q)s Model
The following conditions should lead to the adoption of ASARIMA(P,D,Q)s model;
1) The underlying univariate time series must be non- seasonally stationary (d=0) and
exhibits cognizable seasonal pattern. Note, seasonal differencing(D) may be 0 or 1.
2) The ACF must reveal seasonal oscillation with significant spikes at every kth lag, here k =
s i and i =1,2 , K .
3) The PACF tends to cut-off at every kth lag and cut-in.
4) If the spikes in (iii) tails off at every kth lag consider fitting ASARIMA(P,D,0)s
5) If the spikes in (iii) do not tails off at every kth lag consider fitting ASARIMA(0,D,Q)s
6) If the spikes indicate mixture of (iv) and (v) consider fitting ASARIMA(P,D,Q)s
7) Use some information criteria such SSDFC, AIC BIC, SC etc to select the best fitted
model.
F. ADF Unit Root Test
ADF unit root test helps to check the order of integration of the variables under study. The unit
root test here, is based on Augmented Dickey Fuller (ADF) test and is of the form
where k is the number of lag variables. In (5) there is intercept term, the drift term and the
deterministic trend. The non deterministic trend term removes the trend term in (5). And it can be
carried out with the choice of removing both the constant and deterministic trend term in the
above regression.
ADF unit root test null hypothesis H0 : = 0 and alternative Ha : 0. According to [7], if the
ADF test statistic is greater than 1%, 5% and 10% critical values, the null hypothesis of a unit
root test is accepted. ERS unit root test will used to consolidate the result provided by ADF test.
See the technical details in [11].
G. Model Comparison
There are several model selection criteria in literature such as; Bayesian information
criterion(BIC),Aikaike information criterion(AIC), residual sum of squares and so on. If n is the
sample size and RSS is the residual sum of squares, then, BIC and AIC are given as follows;
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where, n is the sample size, k is the number of estimated parameters (for the case of regression, k
is the number of regressors) and RSS is the residual sum of squares based on the estimated
model. However, it is good to note that both BIC and AIC are affected by the number of
parameters included to be estimated in a model. For the case of BIC, it penalizes free parameters
while AIC becomes smaller as the number of free parameters to be estimated increases. But for
this study, model selection will be based on AIC. The sum of squares deviation forecast criterion
introduced by [4] will be used to check models output performance for 150 forecast lead time .
And it is of the form;
Where l is the lead time, m is the number of forecast values to be deviated from the actual values
(m should be reasonably large), yt (l ,i) is the actual values of the time series corresponding to the
ith position of the forecast values and yˆt ,(l ,i) is the forecast values corresponding to the ith
position of the actual values. In comparison, the model with the smallest value of SSDFC is the
best output performing model that can describe, to the closest precision, the behavior of the
underlying fitted model.
H. Model Estimation
The coefficients are estimated using an iterative algorithm that calculates least squares estimates.
At each point of iteration, the back forecasts are computed and sum of squares error (SSE) is
calculated. For more details, see [8].
III. DATA ANALYSIS AND RESULTS
This section presents the time series plot of Enugu monthly rainfall data, results of ADF unit root
test, plots of ACF and PACF and estimates of SARIMA(p,d,q)×(P,D,Q)s model.
Fig.1. Time plot of EMR (1981M1 – 2016M12)
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The plot of monthly rainfall in Figure1 exhibits seasonal nonstationary pattern. It is also
observable that the time series plot lacks trend with the highest precipitation of 508.3 Millimeters
in July 1990 and lowest precipitation of 0.5 Millimeters in January and February the same year.
Fig.2. Time plot of Seasonally differenced EMR (1981M1 – 2016M12)
The seasonally differenced EMR data in Fig.2 is seasonally stationary with most of the data
concentrated around zero.
TABLE I. ANALYSIS OF ORDER OF INTEGRATION OF EMR
Note that DT represents ‘deterministic term’.
The results of ADF and ERS unit root tests in Table I above generally indicate that EMR variable
is integrated order zero I(0), significant at 5% level. Hence, the monthly rainfall under
investigation is stationary. Having the EMR variable exhibiting stationarity, then, it will be
modeled using seasonal autoregressive moving average SARIMA (p, 0, q)×(P,D,Q)s model.
A. Correlogram
The correlogram presents the plots of autocorrelation function (ACF) and the partial
autocorrelation function (PACF) for model identification as presented in Figure3 and Figure4
below.
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Fig.3. Plots of ACF EMR (with 5% significance limits for the correlogram)
Fig.4. Plots of PACF for EMR (with 5% significance limits for the correlogram)
The plot of autocorrelation function in Figure3 exhibits presence of seasonal effect. The cyclical
correlogram with a seasonal frequency suggests fitting a seasonal ARMA model to the rainfall
data. The result indicates the need for seasonal differencing in the model. The time plot revealed
seasonality in the rainfall variable. But where this is not too clear via time plot, the
autocorrelation function (ACF) could reveal the value of s, as the significant lag of the ACF.
There appear to be annual or 12-month spikes in the ACF and PACF as shown in Figure3 and
Figure4. The ACF clearly exhibits this prima facie evidence of seasonal nonstationarity. The
PACF reveals the seasonal spikes at lags 12, 24 36 and
48. Slow attenuation of the seasonal peaks in the Figure4 ACF signifies seasonal nonstationarity.
The 12-month PACF periodicity can be seen in the periodic peaks at every 12th lag up to 48th
lag evocatiing of seasonal differencing at lag 12.
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B. Model Comparison
This section presents a comparison of 27 possible models using SSDFC as presented in Table II
below;
TABLE II. MODEL SELECTION USING AIC
The 18 models in Table II above showed at least no serial correlation in the model residuals up to
12th lag using Modified Box-Pierce statistic. Model comparison using AIC indicates that
ASARIMA(2,1,1)12 is preferred to the other sub-classes of SARIMA(p,d,q)×(P,D,Q)12 and
ASARIMA (P,D,Q)12 models since it has the smallest value of AIC. Though the chosen
information criterion is AIC, BIC also preferred ASARIMA to SARIMA. However, based on
output performance such as forecast (for 150 lead time), SSDFC prefers and
SARIMA(1,0,0)×(1,1,1)12 followed by ASARIMA(1,1,1)12 .
TABLE III. FINAL ESTIMATES OF ASARIMA(2,1,1)12 PARAMETERS
Differencing: 0 regular, 1 seasonal of order 12, Number of observations: Original series 432,
after differencing 420, Residuals: SS = 1571863 (backforecasts excluded),
MS = 3779 DF = 416
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The model result in Table III shows that the parameters SMA lag 12 and SAR at lag 24 are
significant under 5% and 10% respectively. The ASARIMA(2,1,1)12 model is of the form;
TABLE IV. MODIFIED BOX-PIERCE (LJUNG-BOX) CHI-SQUARE STATISTIC
The result of Table IV shows that the probability of Modified Box-Pierce (Ljung-Box) Chi-
Square statistic are all greater than 5% significant level, this indicates that the residuals of the
ASARIMA(2,1,1)12 are not correlated up to 48th lag. Hence the model is adequate.
Fig.5. Plot of ACF of Residuals (with 5% significance limits for autocorrelations)
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Fig.6. Plot of PACF of Residuals (with 5% significance limits for autocorrelations)
The ACF and PACF of residuals in Figure5 and Figure6 respectively for the Enugu rainfall data
showed no significant spikes (the spikes are within the confidence limits) indicating that the
residuals are uncorrelated. Therefore, the ASARIMA(2,1,1)12 model appears to fit well and can
be used to make forecasts for Enugu monthly rainfall.
Fig.7. Time plot of forecast and actual values
The generated forecast values in Figure7 above showed close relation with the actual values.
Hence, it can be said that the fitted model has performed pretty good.
C. Discussion of Results
In a regular stationary time series variable with seasonal nonstationary behaviour, such as that of
Enugu monthly rainfall (EMR) pattern, the comparison study using AIC reveals that
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ASARIMA(2,1,1)12 performed better than all the sub-classes of SARIMA(p,d,q)×(P,D,Q)s
model. Modified Box-Pierce (Ljung-Box) Chi-Square statistic indicates that the residuals of the
ASARIMA(2,1,1)12 are not correlated up to 48th lag. Again, the ACF and PACF of the model
residuals are uncorrelated too and the forecast values are very close, indicating the adequacy of
the fitted model.
However, unlike past studies by researchers have clustered on the application of SARIMA model
introduced by Box and Jenkins(1979), the Adjusted SARIMA introduced here, has spiced up a
new dimension in the modeling of seasonal behaviour of variables that are adjudged to be
regularly stationary (where d = 0) and seasonally nonstationary(where D =1).
IV. CONCLUSION
The paper compared SARIMA and Adjusted SARIMA models in a regular stationary time series
with seasonal nonstationary behaviour such as Enugu monthly rainfall data, and the finding
indicates that ASARIMA(2,1,1)12 subclass is better than all SARIMA sub-classes as reported by
AIC.
Therefore, it can be recommended that for such pattern of time series, ASARIMA is preferred
due to its ability to reduce parameter redundancy and sum of square errors in the model.
References
[1] Afrifa-Yamoah E , I. I. Bashiru Saeed , A. Karim ( 2016) Sarima Modelling and Forecasting
of Monthly Rainfall in the Brong Ahafo Region of Ghana, World Environment, Vol. 6 No.
1, 2, pp. 1-9.
[2] Akpanta A. C, I. E. Okorie1 and N. N. Okoye (2015). SARIMA Modelling of the Frequency
of Monthly Rainfall in Umuahia, Abia State of Nigeria. American Journal of Mathematics
and Statistics; 5(2): 82-88.
[3] Alawaye A.I and A.N.Alao(2017). Time Series Analysis on Rainfall in Oshogbo Osun State,
Nigeria, International Journal of Engineering and Applied Sciences (IJEAS); 4(7): Pp.35-37.
[4] Amaefula, C. G..(2011). Optimal identification of subclass of autoregressive integrated
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