Modelling Seasonal Behaviour of Rainfall in Northeast Nigeria. A State Space Approach

Transcription

1 Inernaional Journal of Saisics and Applicaions 216, 6(4): DOI: /j.saisics Modelling Seasonal Behaviour of Rainfall in Norheas Nigeria. A Sae Space Approach O. J. Asemoa 1,*, M. A. Bamanga 2, O. J. Alaribe 1 1 Deparmen of Saisics, Universiy of Abuja, Abuja 2 Saisics Deparmen, Cenral Bank of Nigeria, Nigeria Absrac The modelling of seasonal behaviour of rainfall has become very imporan in he wake of he climaic change. Due o he agriculure base of he Norheas Nigeria, i is imperaive o explicily model seasonal behaviour of rainfall in he region for agriculural planning. Hence, his sudy examines he seasonal behaviour of monhly rainfall daa from 1981 o 213 in Maiduguri and Damauru areas of Borno and Yobe Saes respecively, using he sae space approach. We employed he local level model wih sochasic seasonal and he local level model wih deerminisic seasonal o modelling of he dynamic feaures in he daa. Our resuls clearly indicae ha he local level model wih deerminisic seasonal is he parsimonious model beween he wo sae space models considered in his sudy. This implies ha he seasonal paerns of rainfall in he wo areas have no significanly changed despie he challenges of global warming and climae change. In addiion, he CUSUM es indicaes he presence of srucural breaks in 1998 and 199 for Maiduguri and Damauru respecively. This implies ha here was an abrup change in he rainfall level in 1998 for Maiduguri and in 199 for Damauru. We, herefore, recommend ha seasonaliy should be explicily included in he modelling of rainfall series as he paern of seasonaliy could be useful for imporan decision making. In addiion, measures should be pu in place o curb human-made aciviies ha are derimenal o he climae since he region is highly vulnerable o he impacs of climae change. Keywords Rainfall, Climae Change, Kalman filer, Kalman Smoohing, Seasonaliy, Sae space model 1. Inroducion Agriculure in Norh-Easern region of Nigeria is predominanly dependen on rainfall. Hence, he oupu from he secor each year is largely deermined by he rainfall seasonal paern, rend and cycles. Weaher as we know is dynamic. Similarly, climae flucuaes or changes over ime and space. According o Nieuwol (1982), he non-saic behaviour in rainfall paerns are in various magniudes ranging from variabiliy hrough flucuaions, rends, and abrup o gradual changes. Some characerisics of rainfall which include is seasonal and diurnal disribuion, inensiy, duraion, onse, cessaion and frequency of rain days all show imporan variaions in respec o ime and places. I is herefore imporan o examine climaic daa o ascerain possible rends and changes in he daa generaing process. Trend could reflec eiher an increase or decrease in he observed phenomenon and may serve as a good indicaor for predicing fuure occurrence. A fair knowledge of he weaher is very necessary, in view of he fac ha farmers, * Corresponding auhor: asemoaomos@yahoo.com (O. J. Asemoa) Published online a hp://journal.sapub.org/saisics Copyrigh 216 Scienific & Academic Publishing. All Righs Reserved power generaion (hydroelecric power generaion), meeorological saions, miliary operaions, safey and emergency agencies such as Naional Emergency Managemen Agency (NEMA) heavily rely on i for heir operaions, among oher socioeconomic aciviies. The high amouns of rainfall which are paricularly received in many elevaed area of he ropics, consiue a reliable basis for he consrucion of many hydro-elecric power saions. There has always been a good deal of ineres in he possibiliy of seasonaliy in rainfall daa. Harvey (1987) poined ou ha he analysis of climaic ime series is essenial for building saisical models o generae synheic hydrologic records, o forecas hydrologic evens, o deec inrinsic sochasic or deerminisic characerisics of hydrologic variables. Assessing he seasonal behaviour of rainfall characerisics and rend is a vial for agriculural pracice in he Norh-Eas saes of Nigeria. In addiion, he ime series analysis of climaic daa provides a basis for asceraining climae change or variabiliy. The Box-Jenkins SARIMA model has been exensively used o model rainfall paerns in Nigeria, see for example, Gulumbe (213), Euk, Moffa and Chims (213) and Marins e al. (214). However, he Box-Jenkins approach is based on he eliminaion of rend and seasonaliy applying seasonal differencing o he series before fiing an

2 24 O. J. Asemoa e al.: Modelling Seasonal Behaviour of Rainfall in Norheas Nigeria. A Sae Space Approach appropriae ARMA model o he daa. Durbin and Koopman (21) noed ha eliminaion of rend and seasonaliy by differencing may pose a problem in applicaions where knowledge of he seasonal paern is needed. Forunaely, modern conrol and sysem heory has shown ha he behaviour of dynamic sysems can be convenienly and succincly described by using he sae space models. Sae space mehod is more general and is based on he modelling of all he observed feaures of he daa. The differen feaures inheren in a series such as rend, seasonal, cycle, explanaory variables and inervenions can be modelled separaely before being pu ogeher in he sae space model. The assumpion of saionariy is no needed in sae space model. Alhough he lieraure on sae space has increased recenly, here is however, no empirical applicaion o modelling rainfall paern in Nigeria. In addiion, using he sae space approach, he naure of seasonaliy presen in he daa will be explicily deermined. I is agains hese backdrops ha his sudy is being proposed. The rainfall received a he synopic saions a Maiduguri and Damauru for hiry-hree years period ( ) are analyzed o evaluae he seasonal behaviour and rend in he monhly daa rainfall. The research is useful in assessing he seasonal behaviour of rainfall characerisics and he rend in he rainfall daa which is a vial requiremen for agriculural pracice in he Norh-Eas saes of Nigeria. I is also inended o provide a basis for asceraining climae change or variabiliy. The res of his paper is organized as follows. In secion 2, sae space models are briefly inroduced, in secion 3, we discuss he modelling mehodology employed in his work. The daa and he empirical resuls are presened and discussed in chaper 4. The final secion concludes he paper. 2. The Model 2.1. Sae Space Model A sae space model consiss of wo equaions: he sae equaion (also called ransiion or sysem equaion) and he observaion equaion (also called measuremen equaion). The observaion equaion expresses he observed variables (daa) as a linear funcion of he sae variable(s) plus noise, while he ransiion equaion describes he evoluion of he sae variables. The ransiion equaion has he form of a firs-order difference equaion. The selecion of componens o include in he sae space model is based on he feaures inheren in he observed ime series. For a series ha have seasonal paerns, he basic srucural model is used, for a srongly rending non-seasonal series; he local rend model is employed. However, for a non-rending series, he local level model is used. Le y denoe an (n 1) vecors of variables observed a ime and be (r 1) unobserved sae vecor. The general sae space represenaion of he dynamics of y is given by: y = AX + H + (1) +1 = F +η (2) The (n x 1) vecor and he (k x 1) vecor η are whie noise: E( ) = R, for =, and oherwise (3) E(η η ) = Q, for =, and oherwise (4) The disurbances are assumed o be uncorrelaed a all lags, ha is; E( η ) = a all and (5) where is k x 1 vecor of unobserved sae variables, H is an n x k marix ha links he observed vecor y o he unobserved, X is an r x 1 vecor of exogenous or predeermined observed variables, A is a marix ha maps he exogenous variables ino he measuremen domain, F is he sae ransiion marix which applies he effec of each sae parameer a ime on he sysem sae a ime +1, R is (n x n) and Q is (k x k) marices of he measuremen equaion variance and ransiion equaion variance respecively. The R variance marices play he same role as in he classical regression model, while he Q variance marices allow he parameers in he sae equaions o evolve over ime. Equaion (1) is known as he observaion equaion and (2) is known as he sae equaion. The sysem of (1) hrough (5) is called sae space models. The essenial difference beween he sae space model and he convenional ARIMA model represenaion is ha in he former, he sae of naure analogous o he regression coefficiens of he laer is no assumed o be consan bu may change wih ime. 3. Modelling Mehodologies 3.1. Sae Space Approach: Local Level Model wih Seasonaliy The local level model or random walk-plus-noise model is a simple form of a linear Gaussian sae space model for modelling series wih no visible rend. The model conains only he level and irregular componens; he single sae (level) variable follows he random walk: y = μ + ε ε ~ NID(, σ 2 ε) μ +1 = μ + ~ NID(, σ 2 ) (6) where ε and are muually uncorrelaed whie-noise processes wih variance σ 2 ε and σ 2. The inerpreaion of his model is ha μ is an (unobservable) local level or mean for he process. The observable y is he underlying process mean conaminaed wih he measuremen error ε. Alhough i has a simple form, i provides he basis for he analysis of imporan real problems in pracical ime series analysis. I exhibis he characerisics srucure of sae space models in which here is a series of unobserved values μ 1 μ n which represens he developmen over ime of he sysem under sudy, ogeher wih a se of observaions y 1,, y n. The aim of he analysis is o sudy he developmen of he sae over ime

3 Inernaional Journal of Saisics and Applicaions 216, 6(4): using he observed values y 1,, y n. However, when a ime series consiss of daily, monhly, or quarerly observaions, he presence of seasonal effecs should be aken ino consideraion. In he sae space framework, seasonaliy can be handled by building he seasonal effecs direcly ino he model. Hence, adding seasonal componens o equaion above yields, y = µ + +, N(, 2 ) µ +1 = µ +, N(, 2 ) +1 = - +1-j + N(, 2 ) (7) where = s-1,..., n. When he seasonal effec is no allowed o change over ime, we require = for all = s -1,..., n. This is done by seing 2 = and (7) is called he local level wih deerminisic seasonal model. When he seasonal effec is allowed o vary over ime, ha is 2 >, he resuling model is called he local level wih sochasic seasonal model. Since he rainfall daa consiss of monhly observaions, he periodiciy of he seasonal is s =12. The sochasic formulaion of he seasonal effec in (7) follows from he sandard dummy variable mehods of modelling seasonal paern. An alernaive way of modelling seasonal effec is by using rigonomeric erms a he seasonal frequencies. Sae space models are esimaed using he Kalman filer. The Kalman filer is a saisical algorihm ha enables cerain compuaions o be carried ou for a model cas in sae space form. However, o obain a more accurae esimae of he sae vecor, he smoohing algorihm is performed. Kalman smoohing provides us wih a more accurae inference on, since i uses more informaion han he filering. Le Y 1 denoe he se of pas observaions y and assuming he condiional 1,, y 1 Y is N( u, p ) where disribuion of given 1 u and p are o be deermined. Assuming ha u and p have been deermined, he celebraed Kalman filer equaions for updaing he above local level model from ime o 1 are given by: u u k p p k k p f 2 1 = +, 1 = (1 - ) +, = /,, (8) 2 = y u-, f p = + for = 1, 2,, n, where denoes he Kalman filer residual or predicion errors, f is is variance and k is he Kalman gain. A random walk like has no naural level and o handle he iniial condiions ( u1, p 1) for he non-saionary model, we employed he exac iniial Kalman filer, (Durbin and Koopman, 21). The Kalman smoohed sae ( ˆ ) and smoohed sae variance ( V ) can be calculaed by he following backward recursions: 1 2 ˆ = u + pr 1, r 1 = f + lr, l = 1 k / f, n,,1 N =, for,,1 V p p N, N f l N, n,,1 (9) wih r n and n n. The unknown variance parameers in he sae space model are esimaed by he maximum likelihood esimaion via he Kalman filer predicion error decomposiion iniialized wih he exac iniial Kalman filer. Harvey and Peers (199) suggesed concenraion of he log likelihood when he variance parameers display difficul esimaion problems, as his helps o improve he behavior of difficul esimaion. Diagnosic checking in he sae space models are based on he hree assumpions concerning he residuals of he analysis. The residuals should saisfy hese hree properies, in order of imporance; independence, homoscedasiciy and normaliy. These assumpions are checked using he following es saisic; The assumpion of independence of he residuals can be checked wih he Ljung-Box saisic defined as; Q (k) = n(n + 2) k r2 l=1 n l 2 (k-w+1) (1) For lags l = 1,...,k. The number of diffuse iniial sae values which need o be esimaed for he level and seasonal componens in (7) corresponds o k, r l denoes he residual auocorrelaion for lag l and w is he number of hyperparameers (i.e. disurbance variances). The second mos imporan assumpion is he homoscedasiciy of he residuals. This is checked using he following es saisic; H(h) = n 2 =n h +1 ε =d+h ε2 =d+1 F (h, h) (11) where d is he number of diffuse iniial elemens, and h is he neares ineger o (n- d) /3. The leas imporan assumpion is ha he residuals are normally disribued. This assumpion can be checked wih he following es saisic; N = n s 6 + (k 3) df (12)

4 26 O. J. Asemoa e al.: Modelling Seasonal Behaviour of Rainfall in Norheas Nigeria. A Sae Space Approach where s denoes he skewness of he residuals, and k is he kurosis. In he sae space models, he sandardized smoohed disurbances are useful for deecion of possible ouliers and srucural breaks. The deecion of srucural break using he cumulaive sum (CUSUM) es is considered in his work. 4. Daa and Empirical Resuls 4.1. The Daa The monhly rainfall daa for wo Norh-Easern areas (Maiduguri and Yobe, see Annexure I) from January, 1981 o December 213 are examined in his sudy. The monhly rainfall daa measured in millimeer is obained from he Nigeria Meeorological Agency (NIMET), Lagos sae office. The plos of he rainfall series are presened in Figure 1 and Figure 2 for Maiduguri and Damauru respecively. 3 Maiduguri Rainfall ( ) Maiduguri Rainfall wih Time Lines for Years Figure 1. Time Series Plo of Maiduguri Rainfall 3 Yobe Rainfall ( ) Yobe Rainfall wih Time Lines for Years Figure 2. Time Series Plo of Damauru Rainfall

5 Inernaional Journal of Saisics and Applicaions 216, 6(4): From figure 1 he rainfall series flucuaes over he years bu had he highes spike in 1999 and he lowes in 1983 and figure 2 also indicaes ha he rainfall had is peak in 212 and he lowes in The descripive saisics of he subsamples of boh series are presened in able 1 and 2. Table 1 and 2 indicae ha he subsample means and variance are no he same over ime, which is an indicaion ha he subsample means and variance are no consan. For example, he subsample mean for Damauru beween 1981:1 o 1991:1 is 28.1 while for 1992:1 o 21:1 and 22: 1 o 213:12 are and respecively. Hence, he means varies across he samples. Thus, he sae space model ha accommodaes non-saionary feaures by represening he level of he series as a random walk process, since random walk process is non-saionary, would be appropriae for modelling he series. Table 1. Descripive Saisics of Maiduguri Rainfall Table 2. Descripive Saisics of Damauru Rainfall

6 28 O. J. Asemoa e al.: Modelling Seasonal Behaviour of Rainfall in Norheas Nigeria. A Sae Space Approach 4.2. Esimaion Resuls and Discussion The selecion of componens o be included in a sae space model is based on he characerisics of he observed series. Since seasonaliy is normally presen in a monhly ime series daa, we employed he local level wih sochasic seasonal and local level wih deerminisic seasonal models described in secion 3 o he Maiduguri and Damauru original series. Firs, using he daa, we esimae he unknown variance parameers (hyperparameers) of he models using maximum likelihood mehod. This is maximized using he BFGS (Broyden-Flecher-Goldfarb-Shannon) opimizaion mehod. The esimaion resuls for he local level model wih sochasic seasonal is presened below. Table 3. Esimaion Variance for Local Level Model wih Sochasic Seasonal for Maiduguri Rainfall series (1981:1 213:12) Noe: SIGSQEPS, SIGSQETA and SIGSQOMEGA denoe esimaes 2, 2 and 2 respecively. Table 4. Esimaion Variance for Local Level Model wih Sochasic Seasonal for Damauru Rainfall series (1981:1 213:12) Noe: SIGSQEPS, SIGSQETA and SIGSQOMEGA denoe esimaes 2, 2 and 2 respecively. The esimaion resuls in able 3 and 4 indicaes ha he hyperparameers (disurbance variances) of he measuremen and seasonal componen are highly saisically significan; however, he esimaed hyperparameers for he level equaions is no saisically significan. This indicaes ha any seasonal paern in he observed ime series changes over he years. Also, here are 396 observaions in boh series; however, he esimaion is done using only he final 384 observaions. This is because 12 diffuse iniial sae values are esimaed (11 for he seasonal componens and 1 for he local level componens). We proceed ino performing he Kalman filering and kalman smoohing using he esimaed hyperparameers. The resuls of he Kalman filer esimaes for Maiduguri rainfall series is presened in figure 3 and 4 while figure 5 and 6 presen ha of Damauru rainfall series respecively.

7 Inernaional Journal of Saisics and Applicaions 216, 6(4): Filered Lev el Filered Seasonal Fied Values Kalm an Filer Oupu for Local level + Sochasic Seasonal Figure 3. Kalman Filer Oupu for local level wih sochasic seasonal for Maiduguri series 9 6 Filered Level Variance 3 Filered Seasonal Variance Predicion Errors Kalman Filer Variances for Local level + Sochasic Seasonal Figure 4. Kalman Filer Oupu for local level wih sochasic seasonal for Maiduguri series

8 21 O. J. Asemoa e al.: Modelling Seasonal Behaviour of Rainfall in Norheas Nigeria. A Sae Space Approach 42.5 Filered Level Filered Seasonal Fied Values Kalman Filer Oupu f or Local lev el + Sochasic Seasonal Figure 5. Kalman Filer Oupu for local level wih sochasic seasonal for Damauru series Filered Level Variance Filered Seasonal Variance Predicion Errors Kalman Filer Variances for Local level + Sochasic Seasonal Figure 6. Kalman Filer Oupu for local level wih sochasic seasonal for Damauru series

9 Inernaional Journal of Saisics and Applicaions 216, 6(4): Smoohed Level Smoohed Seasonal 1-2 Smoohed Fied Values 1 - Kalman Smoohed Oupu for Local level + Sochasic Seasonal Figure 7. Kalman Smoohed Oupu for local level wih sochasic seasonal for Maiduguri series Smoohed Level Variance Smoohed Seasonal Variance Smoohed Predicion Errors Kalman Smoohed Variances f or Local lev el + Sochasic Seasonal Figure 8. Kalman Smoohed Oupu for local level wih sochasic seasonal for Maiduguri series

10 212 O. J. Asemoa e al.: Modelling Seasonal Behaviour of Rainfall in Norheas Nigeria. A Sae Space Approach Smoohed Level Smoohed Seasonal Smoohed Fied Values Kalman Smoohed Oupu f or Local level + Sochasic Seasonal Figure 9. Kalman Smoohed Oupu for local level wih sochasic seasonal for Damauru series Smoohed Level Variance Smoohed Seasonal Variance Smoohed Predicion Errors Kalman Smoohed Variances for Local level + Sochasic Seasonal Figure 1. Kalman Smoohed Oupu for local level wih sochasic seasonal for Damauru series

11 Inernaional Journal of Saisics and Applicaions 216, 6(4): Figure 3 o 6 presen graphically he dynamic evoluion of he level, seasonal componens and he fied values obained by he Kalman filer ogeher wih heir variances and he predicion errors (residuals). The evoluion of he wo Norh-Easern regions under sudy is refleced by he esimaed level componen and is presened in he upper graph of figure 3 and 5. The plos indicae ha he period of highes level of rainfall in Maiduguri occurred in he 26 and he lowes level of rainfall occurred in he From figure 5, he plo also indicaes ha he period of he highes level of rainfall in Damauru occurred in he 2, while he lowes level of rainfall occurred beween 1988 and One obvious feaures of figure 4 is ha, he Kalman filered sae variances converges o zero value, which implies ha here is no variabiliy, while in figure 6, he Kalman filered sae variance converges o a consan value as he sample size increases which empirically confirms ha he fied local level model has a seady sae soluion. Figure 7 o 1 presen he resuls of he Kalman smoohing recursion esimaes of he level, seasonal and he fied values of he series ogeher wih he sae variances of he level, seasonal componen and he smoohed predicion errors respecively. On comparing he graphs of he Kalman filered level and he smoohed level of he wo Norheas saes in figure 3 o 5 and 7 o 9 respecively, i is obvious ha figure 7 o 9 is smooher han ha of figure 3 o 5. In addiion, figure 7 o 9 reveal ha he seasonal paern in he Maiduguri and Damauru series have been relaively consan over he years. In order o deec any sysemaic and haphazard srucural breaks, we employed he cumulaive sum (CUSUM) and cumulaive sum squares (CUSUMSQ) inroduced by Brown e al. (19) o he residuals of he fied model. The resuls of he CUSUM es are displayed in figure 11 and 12. Figure 11 indicaes ha here was a srucural break around Similarly, figure indicaes he possibiliy of a srucural break in he year Sig Level CUSUM es CUSUMSQ es CUSUM Tess Figure 11. The CUSUM and CUSUMSQ Srucural Break es for Maiduguri Rainfall.

12 214 O. J. Asemoa e al.: Modelling Seasonal Behaviour of Rainfall in Norheas Nigeria. A Sae Space Approach 3 Sig Level CUSUM es CUSUMSQ es CUSUM Tess Figure 12. The CUSUM and CUSUMSQ Srucural Break es for Damauru Rainfall Table 5. Diagnosics Resuls for Local Level Model wih Sochasic Seasonal for Maiduguri Series Table 6. Diagnosics Resuls for Local Level Model wih Sochasic Seasonal for Damauru Series

13 Inernaional Journal of Saisics and Applicaions 216, 6(4): Table 5 indicaes ha he diagnosic ess for independence, homoscedasiciy, and normaliy of he residuals for he fied Maiduguri model are all saisfacory. These ess indicae ha he residuals saisfy all of he assumpions of he models since he p-value associaed wih all he ess are insignifican a he convenional.5 significance level. Also, from able 6, he diagnosic ess for independence and homoscedasiciy of he residuals are saisfacory for Damauru fied model, while ha of normaliy is no saisfacory, which suggess ha he residuals saisfy he wo mos imporan assumpions of he models as discussed in secion 3. However, he residuals are higher a he beginning and end of he sample, as heoreically expeced, since he smoohed saes are calculaed by a backwards recursions. Hence, he uncerainy in he esimaion is higher a he beginning of he backward recursions, unlike he Kalman filering ha uses a forward recursion. We now presen he resuls of analysis of Maiduguri and Damauru rainfall ime series wih a local level model wih a deerminisic seasonal. Table 7. Esimaion Resuls for Local Level Model wih Deerminisic Seasonal for Maiduguri Rainfall Series (1981:1-213:12) Noe: SIGSQEPS and SIGSQETA denoe esimaes 2 and 2 respecively, 2 is fixed a zero. Table 8. Esimaion Resuls for Local Level Model wih Deerminisic Seasonal for Damauru Rainfall Series (1981:1-213:12) Noe: SIGSQEPS and SIGSQETA denoe esimaes 2 and 2 respecively, 2 is fixed a zero. The esimaion resuls display above shows ha he esimaed hyperparameers (disurbance variances) of he measuremen equaion is significan while ha of he level equaion is no significan. Also, here are 396 observaions in our series; however, he esimaion is done using only he final 384 observaions. This is because 12 diffuse iniial sae values are esimaed (11 for he seasonal componens and 1 for he local level componens). We perform he Kalman filering and smoohing based on hese wo esimaes. We presen he resuls of he Kalman filer esimaes of he local level model wih deerminisic seasonal for Maiduguri and Damauru in figure 13 o 16 and Kalman smoohed esimaes of he local level model wih deerminisic seasonal in figure 16 and 19. Figures 13 and 16 presen he oupu of he Kalman filering of he local level model wih deerminisic seasonaliy for Maiduguri and Damauru respecively. The oupu is very similar o he oupu of he local level model wih sochasic seasonal (his is because he variance is small and insignifican), he evoluion of Maiduguri and Damauru rainfall is refleced by he esimaed level componen and is presened in he upper graph of figure 13 and 14. The plos also indicaes ha he period of highes level of rainfall in Maiduguri and Damauru occurred in 26 and 2 respecively, while he lowes level occurred in 1986 and respecively. Figure 14 also reveals ha he Kalman filered sae variances converges o zero, while he filered seasonal variance converges o a consan value. From figure 16, he filered level variance converges o consan values as he sample size increases which empirically confirms ha he local level model has a seady sae soluion.

14 216 O. J. Asemoa e al.: Modelling Seasonal Behaviour of Rainfall in Norheas Nigeria. A Sae Space Approach Filered Lev el Filered Seasonal Fied Values Kalman Filer Oupu for Local level + Deerminisic Seasonal Figure 13. Kalman Filer Oupu for local level wih deerminisic seasonal for Maiduguri series Filered Lev el Variance Filered Seasonal Variance Predicion Errors Kalman Filer Oupu for Local level + Deerminisic Seasonal Figure 14. Kalman Filer Oupu for local level wih deerminisic seasonal for Maiduguri series

15 Inernaional Journal of Saisics and Applicaions 216, 6(4): Filered Level Filered Seasonal 1-1 Fied Values 1 - Kalman Filer Oupu for Local level + Deerminisic Seasonal Figure 15. Kalman Filer Oupu for local level wih deerminisic seasonal for Damauru series 6 Filered Lev el Variance Filered Seasonal Variance Predicion Errors Kalman Filer Oupu f or Local lev el + Deerminisic Seasonal Figure 16. Kalman Filer Oupu for local level wih deerminisic seasonal for Damauru series

16 218 O. J. Asemoa e al.: Modelling Seasonal Behaviour of Rainfall in Norheas Nigeria. A Sae Space Approach Figures 17 o 2 presen he oupu of he Kalman smoohing recursion esimaes of he level, seasonal and he fied values of he series ogeher wih he sae variances of he level and seasonal componen and he smoohed predicion errors for Maiduguri and Damauru series. Comparing he graphs of he Kalman filered level and he smoohed level in figure 13 and 15 and figure 17and 19, we see ha he graph in figure 17 and 19 is smooher han ha of figure 13 and15 for boh saes. However, he consan seasonal paern in he series is clearly apparen in figure 18 and Smoohed Level Smoohed Seasonal 1 - Smoohed Fied Values Kalman Smoohed Oupu f or Local lev el + Deerminisic Seasonal Figure 17. Kalman Smoohed Oupu for local level wih deerminisic seasonal for Maiduguri series Table 9. Diagnosics Resuls for Local Level Model wih Deerminisic Seasonal for Maiduguri Rainfall Series Table 1. Diagnosics Resuls for Local Level Model wih Deerminisic Seasonal for Damauru Rainfall Series

17 Inernaional Journal of Saisics and Applicaions 216, 6(4): Smoohed Level Variance Smoohed Seasonal Variance Smoohed Predicion Errors Kalman Smoohed Variances for Local level + Deerminisic Seasonal Figure 18. Kalman Smoohed Oupu for local level wih deerminisic seasonal for Maiduguri series 4 Smoohed Lev el Smoohed Seasonal Smoohed Fied Values Kalman Smoohed Oupu for Local level + Deerminisic Seasonal Figure 19. Kalman Smoohed Oupu for local level wih deerminisic seasonal for Damauru series

18 22 O. J. Asemoa e al.: Modelling Seasonal Behaviour of Rainfall in Norheas Nigeria. A Sae Space Approach 14 Smoohed Level Variance Smoohed Seasonal Variance Smoohed Predicion Errors Kalman Smoohed Variances for Local level + Deerminisic Seasonal Figure 2. Kalman Smoohed Oupu for local level wih deerminisic seasonal for Damauru series From able 8 and 1, he resuls of he diagnosic ess for he residuals of he local level model wih deerminisic seasonal are again saisfacory excep for he normaliy of Damauru. From able 9, he assumpion of independence, homoscedasiciy and normaliy are saisfied a he convenional level while able 1 indicaes ha he assumpion of normaliy is no saisfied a convenional significance level. From he wo compeing model ha is; local level sochasic and local level deerminisic model, he Akaike Informaion Crierion (AIC) and he Bayesian Informaion Crierion (SIC) of he wo models are examined o obain he parsimonious model. Using he informaion crieria approach, models ha yield smaller values for he crierion are preferred, and regarded as bes fiing model. From able 5 and 6, he AIC and BIC for he local level model wih sochasic seasonal are 9.89, 9.21 and 9.92, 9.24, for Maiduguri and Damauru respecively, while he AIC and BIC for he local level model wih deerminisic seasonal displayed in able 9 and 1 are 9.88, 9.21 and 9.9, 9.23 for Maiduguri and Damauru respecively. Hence, he local level model wih deerminisic seasonal is slighly beer han he model wih sochasic seasonal componen. In addiion, he log-likelihood values of he wo models for he wo saes are almos idenical , and , for he local level model wih sochasic seasonal and he local level model wih deerminisic seasonal respecively. Hence, he improved fi of he local level model wih deerminisic model can compleely be aribued o is greaer parsimony. Commandeur and Koopman (27) poined ou ha, in sae space modelling, a small and insignifican sae disurbance variance indicaes ha he corresponding sae componen may as well be reaed as a deerminisic effec, resuling in a more parsimonious model. Therefore, he local level model wih deerminisic seasonal is able o model he dynamic feaures

19 Inernaional Journal of Saisics and Applicaions 216, 6(4): in he Maiduguri and Damauru rainfall ime series. 5. Conclusions The variabiliy in elemens of climae such as rainfall and emperaure exposes he counry o he negaive impac of climae change such as erraic rainfall, rise in emperaure, deserificaion, low agriculural yield, drying up of waer bodies, and sea level rise. This paper analysed he seasonal paern of rainfall in Maiduguri and Damauru in he sae space framework. We employ he local level model wih deerminisic and sochasic seasonal o modelling he monhly rainfall of he wo saes. The AIC and BIC of he wo sae space models sugges ha he local level model wih deerminisic seasonal provide a beer fi o he daa han he model wih sochasic seasonal. The deecion of deerminisic seasonal in he wo saes implies ha he rainfall fall paerns have lile variabiliy or indicaion of climae change as regards is rainfall elemen. In addiion, he CUSUM es indicaes he presence of srucural breaks in 1998 and 199 for Maiduguri and Damauru respecively. This implies ha here was abrup change in he rainfall level in 1998 for Maiduguri area and in 199 for Damauru area. We, herefore, recommend ha seasonaliy should be explicily included in he modelling of seasonal ime series daa as he paern of seasonaliy could be useful for imporan decision making. These echniques can be adoped o he analysis of ime series drawn from oher domains. In addiion, measures should be pu in place o curb human-made aciviies ha are derimenal o he climae since he region is highly vulnerable o he impacs of climae change. All compuaions performed in his sudy are done using he RATS economeric ime series sofware. Annexure I. Geographical Map of he Sudy Area

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