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Abstract
ARIMA (p, d, q) models were fitted to areal annual rainfall of two homogeneous regions in East Africa with rainfall records extending between the period 1922–80. The areal estimates of the regional rainfall were derived from the time series of the first eigenvector, which was significantly dominant at each of the two regions. The first eigenvector accounted for about 80% of the total rainfall variance in each region. The class of ARIMA (p, d, q) models which best fitted the areal indices of relative wetness/dryness were the A R M A (3, 1) models. Tests of forecasting skill however indicated low skill in the forecasts given by these models. In all cases the models accounted for less than 50% of the total variance. Spectral analysis of the indices time series indicated dominant quasi-periodic fluctuations around 2.2–2.8 years, 3–3.7 years, 5–6 years and 10–13 years. These spectral bands however accounted for very low proportion of the total rainfall variance.
... There are also vast studies on modelling and forecasting of the frequency of rainfall outside Nigeria. In Bangladesh [22], Iran [23], Jordan [24], Golastan Province [25], East Africa [26] and Turkey [27]. ...
... [25] analyzed the precipitation forecast using SARIMA model and found the seasonality measure in SARIMA to be highly useful in measuring precipitation. [26] used time series methods based on ARIMA and Spectral analysis of a real annual rainfall of two homogenous region in East Africa and recommended ARMA (3,1) as the best fit for areal indices of relative wetness\dryness and dominant quasi-periodic fluctuation around 2.2-2.8 years, 3-3.7 years, 5-6 years and 10-13 years. ...
The need to have a quantitative means of modelling and predicting rainfall is very important for the purposes of Airplane movements, Agriculture, planning and policy formulation. The aim of this study is to propose a model of Seasonal Autoregressive Integrated Moving Average (SARIMA) model to model and forecast the monthly frequency of the rainfall in Abuja, Nigeria for the period 1996 to 2018. The data was obtained from Nigerian Meteorological Agency (NiMet) Abuja, Nigeria and the analysis was based on probability Seasonal time series modelling approach. The Plot of the original data shows that the time series is stationary and the Augmented Dickey-Fuller test did not suggest otherwise. The graph further displays evidence of seasonality and it was removed by seasonal differencing. The plots of the ACF and PACF show spikes at seasonal lags, the minimum Akaike information criterion (AIC) was 3618.5, Bayesian Information Criterion (BIC) was 3624.3, maximum Coefficient of Determination (R 2) was 0.799 and all the parameters of the proposed SARIMA model were statistically significant at p < 0.05, suggesting SARIMA (0, 0, 2)(0, 1, 2) 12 as the best for modeling and predicting the monthly rainfall in Abuja, Nigeria. The fitted model was used to make forecast for the next four proceeding years, the average predicted rainfall for the peak periods i. and 2022 were; 335.5mm, 337.5mm, 339.3mm and 341.2mm respectively, this implies; a 3.5%, increase in rainfall from 2018 to 2019, 4.% increase in rainfall from 2018 to 2020, 4.7%, increase in rainfall from 2018 to 2021 and a 5.2%, increase in rainfall from 2018 to 2022. The results are useful for predicting the expected rainfall in Abuja in the next four years and also provide information that would be helpful for decision makers in formulating policies, planning and mitigating the problems of flooding in Abuja. Thus, the study recommends among others that Federal Government and minister of the federal capital territory, Abuja should commence the process of avoiding flood by building dikes, levees and provision of adequate drainage systems in the city.
... A very common quasi-periodic oscillation is the quasi-biennial oscillation (QBO), in which the climatic events recur every 2 to 2.5 years. Laban [7] uses time series methods based on ARIMA and Spectral Analysis of areal annual rainfall of two homogenous region in East Africa and recommended ARMA (3,1) as the best fit for areal indices of relative wetness\dryness and dominant quasi-periodic fluctuation around 2.2-2.8 years, 3-3.7 years, 5-6 years and 10-13 years. ...
Objective: Nigerian agriculture is mainly rain-fed and highly dependent on weather especially rainfall. Therefore modeling of monthly rainfall in some selected stations in Nigeria was undertaken in this study. Methodology: Data (rainfall) spanning a period of 30 years (1981-2010) for fourteen stations which were collected from the Nigerian Meteorological Agency (NIMET) were utilized. Autoregressive Integrated Moving Average (ARIMA) and Seasonal Autoregressive Integrated Moving Average (SARIMA) models were used. The accuracy and trend of time series was analyzed to give the monthly rainfall prediction for the succeeding year. The results showed that the model fitted into the data well and the stochastic seasonal fluctuation was successfully modeled. Rainfall was minimal in January, February, March and December over the selected stations in northern, Nigeria but increased progressively in strength and amount in the months of June, August and September over the stations in South west, and June and September over the stations in South -south, Nigeria. The highest rainfall of 230 mm was recorded in September over Warri and the lowest rainfall of 52 mm was recorded in August over Maiduguri. The rainfall recorded over the selected stations in South-south stations was visibly higher than what was recorded over the stations in the northern and the South-west stations. In northern Nigeria, the peak monthly mean rainfall amount of 91 mm was observed in August and rainfall amount was very low in January (0.0 mm), February (0.0 mm), March (0.0 mm) and December (0.0 mm). Over South-west, the Peak monthly mean rainfall amount of 215 mm was observed in June and September and rainfall amount was very low in January (0.0 mm) and December (0.0 mm). Over the stations in South-south, the Peak monthly mean rainfall amount of 325 mm was experienced on September and rainfall amount is very low in December (0.0 mm). Conclusion: The study concluded that Seasonal Autoregressive Integrated Moving Average (SARIMA) model was a proper method for modeling and predicting the monthly rainfall. The results are useful for forecasting the pattern of rainfall in the study area and provide information that would be helpful for decision makers in formulating policies to mitigate the problems of water resources management, soil erosion, flooding and drought.
... AC for the residuals from the global model was strong, perhaps, due to their direct relationship with rainfall. Some studies of rainfall in the Greater Horn of Africa have observed serial autocorrelation in as much as up to AC(3) or AC(4) (Ogallo, 1986;Eklundh, 1998;Mistry and Conway, 2003;Mebrhatu and Walker, 2004). Therefore, it was possible that the autocorrelation observed in the residuals from the global model were due to salient influence of climatic characteristics in their time-series pattern. ...
... AC for the residuals from the global model was strong, perhaps, due to their direct relationship with rainfall. Some studies of rainfall in the Greater Horn of Africa have observed serial autocorrelation in as much as up to AC(3) or AC(4) ( Ogallo, 1986;Eklundh, 1998;Mistry and Conway, 2003;Mebrhatu and Walker, 2004). Therefore, it was possible that the autocorrelation observed in the residuals from the global model were due to salient influence of climatic characteristics in their time-series pattern. ...
The frequency of famines and rainfall deficiency is described for the last quarter of the nineteenth century. Historical records suggest that there were 14 famines between the 11th and 17th centuries and another 12 famines in 9O.year period 1769 and 1958, with a sharp increase in number between 1860 and 1908. But, famines did not always coincide with years of drought. Past evidence suggests that many famines were caused by lack of 223 adequate communications. In recent years research has been directed towards development of drought indices. Using an index Bhalme and Mooley (1981) found droughts in 1951, 1965-66,1972 and 1.974. Two years of consecutive drought in 1.965 and 1966 was a rare event. In the second part of this paper we describe .a stochastic prediction technique based on autoregresson for monsoon rainfall. This is a variation of ARIMA, which has been used in econometrics but has an interesting application in Meteorology.
An attempt has been made to develop a Stochastic dynamic model that could be used for forecasting monsoon rainfall, June to September, in the larger sub-division of India, viz., Peninsula. For building such a model the atmosphere bas been considered as a linear dynamic system that converts various inputs into the output, say the rainfall. In this study the 500 mb mean April sub-tropical ridge position along Long. 75° E has been used as input to the atmosphere. The input~ output data for the recent 38 years (1939-1976) have been utilised for developing the model which utilises the dynamics of the atmosphere and also that of the ARIMA process to forecast the rainfall. The performance of the model has been found good during the sample and the test (1977 to 1980) periods. Even in rank drought and excess rainfall years the closeness of the predicted and realised values stands out well. In terms of seven categories currently being used by the India Meteorological Department for describing its long range forecasts, the skill score of the model forecast for the test period has been found equal to one which is the highest that a forecast formula can have. This suggests that the Stochastic Dynamic Model developed here can therefore, be used for issuing more accurate long range monsoon rainfall forecasts about a month ahead of the season for the Peninsula. This would provide enough time for planning adequate strategies for mitigating the disastrous effects that are produced due to the large vagaries of monsoon.
Using an exponential smoothing procedure and an autoregressive-moving average process; forecasts for the monthly Palmer Drought Severity Index were calculated. The autocorrelation and partial autocorrelation functions of severity index values were used as a starting point for the autoregressive-moving average model selection process. Of the many possible autoregressive-moving average models, the one that was selected provided the best forecasts based on the mean square error. Monthly data for the period 1929–1969 were utilized in a nonlinear least-squares computer routine to arrive at estimated parameter values for the autoregressive-moving average model. Monthly forecasts with a lead time of one month were generated using the exponential smoothing and autoregressive-moving average procedures for the period 1970–1972. These forecasts were compared with the myopic (persistence) forecasts, Xt+1=Xt. The mean square errors of the forecasts were 0.63 for the autoregressive-moving average model, 0.65 for the myopic model, and 0.79 for the exponential smoothing model. From the mean-square-error calculations, it appears that there is no statistically significant difference between the forecasts given by the Box-Jenkins and myopic models; however, the 95% confidence intervals for these two models overlap only slightly during the first part of the forecast period indicating that there may be some advantage to using the Box-Jenkins model instead of the myopic model. Both of these models are superior to the exponential smoothing model. These results demonstrate the usefulness of the relatively new autoregressive-moving average time series analysis procedures.
Using an exponential smoothing procedure and an autoregressive-moving average process; forecasts for the monthly Palmer Drought Severity Index were calculated. The autocorrelation and partial autocorrelation functions of severity index values were used as a starting point for the autoregressive-moving average model selection process. Of the many possible autoregressive-moving average models, the one that was selected provided the best forecasts based on the mean square error. Monthly data for the period 1929–1969 were utilized in a nonlinear least-squares computer routine to arrive at estimated parameter values for the autoregressive-moving average model. Monthly forecasts with a lead time of one month were generated using the exponential smoothing and autoregressive-moving average procedures for the period 1970–1972. These forecasts were compared with the myopic (persistence) forecasts, Xt+1=Xt. The mean square errors of the forecasts were 0.63 for the autoregressive-moving average model, 0.65 for the myopic model, and 0.79 for the exponential smoothing model. From the mean-square-error calculations, it appears that there is no statistically significant difference between the forecasts given by the Box-Jenkins and myopic models; however, the 95% confidence intervals for these two models overlap only slightly during the first part of the forecast period indicating that there may be some advantage to using the Box-Jenkins model instead of the myopic model. Both of these models are superior to the exponential smoothing model. These results demonstrate the usefulness of the relatively new autoregressive-moving average time series analysis procedures.
The Integrated Autoregressive Moving Average (ARIMA) model was applied
to the average monthly rainfall time series over 15 basins located in
Indiana, Illinois and Kentucky, with areas varying between 240 and 4000
mi2 approximately. The record length varied from 492 to 684
months. The first-order, mixed, autoregressive, moving average model
emerged as the most suitable one for forecasting and generation of
cyclicly standardized monthly rainfall square roots series. The model
passed the goodness-of-fit test in all cases studied. The seasonally
differenced, multiplicative model applied to monthly rainfall square
roots also passed the goodness-of-fit test in all cases. This model has
the advantage of requiring fewer parameters than the previous one.
However, the use of the differenced models is limited to forecasting of
monthly rainfall series and cannot be used for the generation of
synthetic rainfall time series, as it does not preserve the monthly
standard deviations. Seasonal differencing is effective in removing the
periodicities but distorts the spectral structure of the original
rainfall series, whereas cyclic standardization only introduces a
negligible distortion in the random component while effectively removing
the circularly stationary part.
Principal time and space variations of snowfall (precipitation) over the Japan Sea coastal region are described in terms of empirical orthogonal function (EOF). Daily precipitation data at -40 stations in 15 winters (30 months) are used.
The first two functions account for -65% of the total variance of precipitation. The spatical functions (B1 and B2) are related to the topography of the analyzed area. Basing on the magnitude of time variation functions (A1 and A2), we classify precipitation pattern of each day into three types; "mountain-", "normal-" and "plain-snowfall type". Inspecting time series of time variation functions for 30 months, we find that a certain snowfall type appears frequently in a particular winter. Especially P-type is dominant in the extraordinarily heavy snowfall winter.
In the latter part of this paper, we focus our attention on the heavy snowfall days (areaaveraged precipitation_??_20mm/day). The heavy snowfalls continue generally for a few days as a "heavy snowfall period", and a certain snowfall type tends to continue through a heavy snowfall period. It is inferred that the heavy snowfalls of a certain type occur under a certain large-scale situation. This will be studied in Part II of this paper.
In order to use the information present in past observations and simultaneously to take advantage of the benefits of stochastic dynamic prediction we formulate the lagged average forecast (LAF) method. In a LAF, just as in a Monte Carlo forecast (MCF), sample statistics are calculated from an ensemble of forecasts. Each LAF ensemble member is an ordinary dynamical forecast (ODF) started from the initial conditions observed at a time lagging the start of the forecast period by a different amount. These forecasts are averaged at their proper verification times to obtain an LAF. The LAF method is operationally feasible since the LAF ensemble members are produced during the normal operational cycle. To test the LAF method, we use a two-layer, f-plane, highly truncated spectral model, forced by asymmetric Newtonian heating of the lower layer. In the experiments, a long run is generated by the primitive equation version of the model which is taken to represent nature, while forecasts are made by the quasigeostrophic version of the model. On the basis of forecast skill, the LAF and MCF are superior to the ODF; this occurs principally because ensemble averaging hedges the LAF and MCF toward the climate mean. The LAF, MCF and ODF are all improved when tempered by a simple regression filter; this procedure yields different weights for the different members of the LAF ensemble. The tempered LAF is the most skillful of the forecast methods tested. The LAF and MCF can provide a priori estimates of forecast skill because there is a strong correlation between the dispersion of the ensemble and the loss of predictability. In this way the time at which individual forecasts lose their skill can be predicted. The application of the LAF method to more realistic models and to monthly or seasonally averaged forecasts is briefly discussed. DOI: 10.1111/j.1600-0870.1983.tb00189.x
Delleur and Kavaas (1978) have fitted ARIMA models to the average monthly rainfall time series over 15 basins. The writers feel that Akaike's information criterion as developed by Ozaki (1977) should be used for choosing or comparing the models instead of the Portmanteau test.- from Authors
In this paper the difficulty in deciding the order of an ARIMA (autoregressive integrated moving average) model is discussed. The possibility of removing this difficulty by using the MAICE (minimum AIC estimation) procedure, which selects a model by using Akaike's Information Criterion (AIC), is checked with the numerical examples treated in the book by Box and Jenkins.
Some do's and don'ts in univariate Box-Jenkins analysis are discussed. An emphasis is given to looking at the raw data and keeping an open (but informed) mind when interpreting the correlation structure. The importance of the analyst not forgetting that he usually only forms a very small part of the overall policy-making machinery must be emphasized; and this cog has to remember that the onus is very much on him to mesh smoothly with the other organizational components if he wishes his work to be accepted and himself not rejected as a misfit by management. Practitioners have the need of avoiding being too proud (or shortsighted) to compromise; for statistical truth is neither absolute nor a fully convertible currency, and there are other gods that deserve (and will exact) recognition.
Empirical orthogonal functions for the 500 mb height and 850 mb temperature fields over a limited domain are obtained. These functions are then used as predictors in a regression model to forecast the height field 24 h in the future.Several regression models are formulated using predictors from the current 850 mb temperature field, 6 and 12 h barotropic forecast fields, as well as the current 500 mb height field. The nonlinear predictors derived from the barotropic forecast fields predominate as selected predictors in those models where they were available for selection and contribute considerable skill to the forecasts. The predictors of the 12 h forecast field are also selected more frequently than those of the 6 h forecast field.The regression models were developed from seven winters of data and then tested against an independent three-winter sample.The error fields of these regression forecasts are compared to those of climatology, persistence and the NMC primitive equation model. The NMC model is found to be considerably better in baroclinic and data-dense regions than any of the regression models. In data-spare regions the distribution of error is similar.
Statistical problems that may be encountered in fitting autoregressive-moving average (ARMA) processes to meteorological time series are described. Techniques that lead to an increased likelihood of choosing the most appropriate ARMA process to model the data at hand are emphasized. 1 specific meteorological application of ARMA processes, the modeling of Palmer Drought Index time series for climatic divisions of the US is considered in detail. -from Authors
Empirical orthogonal function analysis was applied to outgoing longwave radiation (OLR) data obtained from NOAA polar orbiting satellites during the winter months in 1974–77, over the monsoon region extending from 50°E to the date-line and from 30°N to 20°S. Spectral analyses of the amplitude functions for the ten largest eigenvectors exhibit marked peaks in a period range of about 15 to 30 days. Standard deviations of 15°30 day filtered OLR data am extremely large over the Arafura-Indonesian Seas region with a maximum at 10°S and 130°E.
A compositing technique was applied to 15–30 day filtered OLR data to investigate the relationships between long-period changes at a reference point (10°S, 130°E) and those over other regions. Composite maps, which were constructed by considering only the first ten eigenvectors, indicate distinct southward propagation of OLR perturbations along 100–115°E from about 25°N to 20°S. These longitudes appear to correspond to a region of maximum penetration of Northern Hemisphere midlatitude effects and, perhaps, interhemispheric interactions in 15–30 day filtered OLR fields. Intensification of negative OLR perturbations over the South China Sea-Bay of Bengal region, which is presumably associated with active winter monsoonal surges, tends to occur about 5–7 days prior to a decrease in OLR values (increase in convection and/or rainfall) over the Arafura-Indonesian Seas region.
Summer rainfall in Arizona exhibits tremendous spatial variability due
to topographic irregularities and the fact that convection is the
dominant precipitation generating mechanism. As such it has proven
difficult to document changes over time, changes in either total amounts
or in spatial distribution. In this study, eigenvector analysis is
utilized to generate an effective index of this climatic parameter. From
an original data matrix consisting of summer rainfall totals at 101
climatological stations for 50 years, the first time-series eigenvector
is extracted and accounts for 70% of the variance. Geographic mapping of
the multipliers (coefficients) reveals that the first eigenvector does
adhere to physical reality, reflecting the dominant pattern of rainfall
over the state. Thus, its validity as an effective index is established.
Several investigators have pointed out the changes in microclimate
brought about by large urban areas. In particular, the precipitation and
precipitation-related activity have been shown to be larger in and
around urban areas than in the surrounding countryside. The seasonality
of the increased precipitation is another question which has been
investigated.The present paper deals with testing the significance of
observed changes in the mean and in the trends of annual precipitation
series in and around urban areas. The results of analysis of data from
St. Louis, Missouri and LaPorte, Indiana and other stations in the
vicinity of these stations are reported.The annual precipitation data
may not be assumed to be independent random variables because they may
be correlated. For example, the annual rainfall series at LaPorte and
St. Louis were found to be correlated. Consequently, the traditional
univariate and multivariate tests which are based on the assumption that
the variates are independent random variables may not be applicable to
testing changes in the characteristics of annual precipitation data. As
a result, Integrated Moving Average (IMA) models of the annual
precipitation data are used in the present study.The posterior
distributions of the parameters of the IMA models have been obtained by
assuming locally uniform and independent prior distributions of the
parameters. Significance tests based on these posterior distributions
are used to determine the statistical significance of the observed
changes in precipitation which are attributed to the effects of
urbanization. Simpler tests based on the standard errors of the
parameter estimates are also discussed. The importance of model
validation tests and their relationship to the inference are
demonstrated by considering the IMA models of different orders.The
statistical significance of the observed changes in the urban
precipitation data are discussed and a set of conclusions are presented.
The problem of selecting one of a number of models of different dimensions is treated by finding its Bayes solution, and evaluating the leading terms of its asymptotic expansion. These terms are a valid large-sample criterion beyond the Bayesian context, since they do not depend on the a priori distribution.
- Anderson O. D.
Rainfall-Run-off relation in Ruvu Basin of Tanzania
- P Kato