Functional Time Series Analysis of Land Surface Temperature

Parametric modeling imposes rigid assumption on abstraction of physical characteristics of a phenomenon, which in case of model misspecification could give erroneous results. To address the drawbacks, efforts have been channeled on semiparametric and nonparametric modeling and inference. This study focuses on constructing an estimator and consequently modeling a meteorological temperature time series first by constructing a penalized spline estimator based on cubic splines. The penalized spline estimator proposed, which are known to impose very minimal restrictions on estimation process, provides good fits to observed data with very attractive properties namely consistent as observed in values of the Mean Squared Error from the analysis. The results of our simulations compared favorably with the empirical analysis on average monthly meteorological temperature data obtained from Climate Knowledge Portal World Bank Organization on Ghana for periods 1901-2016.


Introduction
A major fundamental problem of all types of meteorological time series variables is their nonlinear features see e.g. (Wang et al., 2013;Bradley and Kantz, 2015;Hyrkkänen et al., 2016;Krishnamurthy, 2019;Benrhmach et al., 2020;Md. Karimuzzaman & Md.Hossain, 2020). Approaching from the data generating processes and its inherent physical conditions. Nonlinearity can cause severe methodological inaccuracies in modeling meteorological times series variables if not taken into account. As a corrective or remedy, the numerical technique of Spline functions (B-spline interpolation) has been modified to the field of statistical time series modeling,see e.g. (Eilers and Marx, 2010;Grimstad and Sandnes, 2016;Jauch et al., 2017;Singh et al., 2020;Yeh et al., 2020;Michel and Zidna, 2020). An unlimited advantage of this method is the ability to correct nonlinearity in the meteorological time series variable. As cited in (Tamsir et al., 2016;Bluemm et al., 2010;Remontet et al., 2019) with a simple extension of this technique, the elimination of systematic influences, such as the effect of nonstationarity coming from the seasonality and cyclical, is it possible to model temperature variable to obtain a high degree efficient temperature model. The study proposed to derive a penalized spline estimator and study its mean squared error measure in relations to selection of smoothing parameter with illustration to real meteorological temperature variable.
climate. However, its effects are felt locally, but the global distribution of climate response to many global climate changes is reasonably congruent in climate models, suggesting that the global metric measure is useful, (Hansen et al., 2006). By IPCC (2007) the Intergovernmental Panel on Climate Change (IPCC), has forecasted a temperature rise of 2.5 to 10 degrees Fahrenheit over the next century.
According to Godfrey et al. (2012) climate change is one of the extreme significant problems on the worldwide political and economic agendas. The complexities, confusing, and at times contested scientific information resulted in a slow public and political response to the climate crisis.
In the African context, climate change is far from intangible, it is presently defining the course of people's lives. Africa experienced extreme weather events and more irregularity in weather patterns, leading to serious outcomes for the people, who depend on land and some water bodies to survive. As a result, Africa's engagement with the issue is evolving rapidly, presenting an opportunity to leapfrog the slow evolution of western public opinion and political action, (Godfrey et al., 2012).
According to Adedeji et al. (2014), climate change is one of the major challenges of our time and adds considerable stress to our societies and to the environment. This is seen from shifting weather patterns that threaten the survival of the likelihood of mankind from food production, rising sea levels increasing the risk of catastrophic flooding, this impacts of climate change are global in scope and unprecedented in scale. This requires a drastic action today, adapting to these impacts in the present and future will be more difficult and costly.
A report by DeGhetto et al. (2016) on African Agenda 2063 of the Africa we want, point out that, whilst Africa at present contributes less than 5% of global carbon emissions, it bears the brunt of the impact of climate change. Africa shall address the global challenge of climate change by prioritizing adaptation in all our actions, drawing upon skills of diverse disciplines to ensure implementation of actions for the survival of the most vulnerable populations, including islands states, and for sustainable development and shared prosperity.
Africa will participate in global efforts for climate change mitigation that support and broaden the policy space for sustainable development on the continent. Africa shall continue to speak with one voice and unity of purpose in advancing its position and interests on climate change, (DeGhetto et al., 2016).
Another study by, Amegah et al. (2016) on sub-Saharan Africa found that, sub-Saharan Africa augments very little to global climate change and nonetheless it is projected to endure the utmost problem of climate change, with 34% of the global Disability-Adjusted Life Years (DALYs) attributable to the effects of climate change found in sub-Saharan Africa.
As a result, Ye et al. (2013); Mudelsee (2018) modeled the absolute temperature using the traditional parametric methods to model the relationships and using the "trial and error" method to determine the order of the polynomial function for estimation process. These concerns are to be address by using appropriate spline functions.
The study sought to achieved the following objectives: to derive a penalized spline estimator and to study its mean squared error measure in relations to selection of smoothing parameter with illustration to real meteorological temperature variable;to study the gains of the models obtained here over other existing models;to assess the accuracy as well as the consequence of the results achieved.

Spline Models
A spline model is a piece-wise well-defined function with the separable pieces joined together employing continuity and smoothness constraints. They are worthwhile in explaining the relationship between a response variable and one or more independent variables when the relationship involves a curve or flexible model.The segments of a spline function are usually low order polynomials of up to third degree and the polynomial segments connect at a set of finite points known as knots.There are numerous forms of spline functions and in order to offer a definition, the ensuing representation is presented. LetY and X be jointly distributed random variables such that where Y is the response variable and X is the explanatory variable. The goal is to estimate the function l based on observed data,(y i , x i ), where i = 1, . . . , n. Given that the observed data exhibits a relationship such that linear and quadratic functions would not be a good fit to the data and using a high-order polynomial does not essentially offer an improved fit to the data and pose challenges in understanding the coefficients of the large number of polynomial terms. By way of an alternative methodology, one can subdivide the domain [a, b] of the function l into k + 1 equally spaced segment.
Definition 2.1. Let a < t 1 < · · · < t k < b be fixed points called knots. Let t 0 = aand t k+1 = b. Generally, splines functions International Journal of Statistics and Probability Vol. 9, No. 5; are piecewise polynomials joined together smoothly at the knots. Formally,a spline function of order m, is a real-valued function on the closed interval [a, b], such that (i) l is piecewise polynomial of order m on the [t i , t i+1 ), i = 0, 1, · · · , k (ii) l has m − 2 continuous derivatives and the (m − 1)st derivative is a step function with jumps at the knots. For orders represented by m = 2r,the function l is a natural spline function of order 2r if, in addition to (i) and (ii), it satisfies the natural boundary conditions The natural boundary conditions ensure that l is spline function of order r on the two outside intervals [a, t 1 ] and [t k , b]. Represent the function space of the natural spline function of order 2r with knots t 1 , · · · , t k as NS 2r (t 1 , · · · , t k ).
To obtain a good approximation of the natural spline function,we define smoothing spline approach where the number and location of knots is worth considering. We begin, by defining a well-defined model space for the function and introduce a penalty to account for overfitting.
Suppose l is "smooth". Specifically, assume that l ∈ W r 2 [a, b] where the Sobolev space Let L(t) be a natural cubic interpolation spline. On each interval,[t i , t i+1 ], i = 1, · · · , m − 1, where, l(t) is given by a different cubic polynomial, l i , define as since L(t) is natural cubic spline, the following must hold, Using Truncated Power Basis to model natural cubic spline function as given in equation (2.3) is where η k is the knots and K is number of knots of the data-set The truncated power basis has r +l−1 basis function. Hence the truncated power basis function is indeed a basis of the vector space of the splines function. In general the truncated power basis of a function with degree r and K knots can be written as where K and η k are the number of knots and knots position of the data set. The data represents average monthly meteorological temperature data collected from (https://climateknowledgeportal.worldbank. org/download-data) for the period between January,1901-December,2016 on Ghana.
Based on the data pairs (Y t , y t−1 ) , y t−1 ∈ [a, b],t = 1, . . . , n with a true relationship we aim to estimate the unknown smooth function l (.) ∈ C p+1 ([a, b]), a p + 1 times continuously differentiable function, with penalized splines. The residuals ε i are assumed to be uncorrelated with zero mean and variance σ 2 > 0.
Suppose that, n ≥ r = 2 and a < t 1 , < t 2 < · · · < t n ≤ b. Then, for fixed 0 < λ < ∞ (2.9) has a unique minimizerl andl ∈ NS 2 (t 1 , · · · , t n ), Eubank (1999). By this result indicates that even though we begun with the infinite dimensional space W r 2 [a, b] as the model space for l, the solution is the minimizer of the equation (2.9) belonging to a finite dimensional space. Specifically,the solution is a natural spline function with knots at distinct design points. One approach to computing the polynomial spline function estimate is to denotel as a linear combination of a basis of NS 2 (t 1 , · · · , t n ).

The Penalized Least Squares Estimation Method
The function l (t) is to be estimated using penalized cubic spline least squares method where generalized cross-validation (GCV) will be used to select smoothing parameter. Given any twice-differentiable function l(t) defined on [a, b], and a smoothing parameter λ > 0, define the penalized sum of squares as where l(t i ) is defined in equation (2.4). The addition of the roughness penalty term λ l (t) 2 in equation (2.9) ensures that the cost S 2 ( t ) of a particular curve is determined not only by its goodness-of-fit to the data as quantified by the residual sum of squares n i=1 {Y(t i ) − l(t i )} 2 but also by its roughness l (t) 2 . The smoothing parameter λ represents the "rate of exchange" between residual error and local variation and gives the amount in terms of summed square residual error that corresponds to one unit of integrated squared second derivative.
In general, we want the resulting function to exhibit some degree of smoothness. The general approach to a formal generalization of this is to introduce the roughness measure, which clearly measure the total curvature of the smoothing function. A fundamental results in spline theory is that the natural cubic spline (the cubic spline imposing, l (t i ) = l (t m ) = 0 ensures the smoothest fit by minimizing equation (2.10) among all C 2 .
Theorem 2.2. With l denoting the natural cubic interpolation spline, we have for any interpolation function f ∈ C 2 such that P( f ) ≥ P(l), where P(l) is defined in equation (2.10), with equality if and only if f = l.
using partial integration, the continuity of the derivatives up to and including second order,and (2.11). Recall the definition of τ, making the second term zero. Remaining with, We note that by letting, l (t 1 ) = l (t m ) = 0 (2.14) This immediately allows us to conclude, with equality if and only if l = f In other words, the function l is expressed as a linear combination of the basis functions,l(t) = n j=1 N j (t)θ j , where N 1 , . . . , N n are the natural spline basis functions: dt, the minimization problem over l now becomes a minimization problem over θ = (θ 1 , . . . , θ n ) T ∈ R n : We can solve for θ, thus, Since l (t) = n j=1 N l (t) θ, from equation (2.17) From equation (2.18), we represent the smoother matrix S λ = N N T N + λΩ N −1 N T Hencel (λ) = S λ y. By the Reinsch (1967) form, for a smoothing spline is by manipulating the singular value decomposition (SVD) of N = UZV T , we can rewrite S λ as

Selecting the Smoothing Parameter and Model Diagnostics
In this section, we discuss various data driven criteria for selecting λ in fitting criterion (2) conditional on the value of K: Akaike's information criterion corrected (AICc), Bayesian information criterion (BIC), Cross-Validation(CV) and Generalized Cross-Validation (GCV) criterion. Each of these data driven criteria provides an approach to select the value of λ conditional on the number of knots, and each criterion is a function of λ. For the AICc, BIC, CV, and GCV criteria, the value of λ that gives the minimum value of the criteria is taken to be a good value for the smoothing parameter conditional on the value of K. Each of these methods is dependent on the sum of squares error, Ruppert et al. (2003) The procedure explained by Ruppert et al. (2003)using a Demmler-Reinsch Orthogonalization to compute our fitted values for each smoothing parameter value is adopted in this study. This is accomplished using the identity S S E(λ) = y T y − 2y Tl λ +l Tl λ (2.21) The Demmler-Reinsch Orthogonalization is explained below Ruppert et al. (2003). Assumed that • The Procedure 1. Obtain matrix R using the Cholesky decomposition: N T N = R T R.

Use Singular
The CV method is also known as the leave-one-out method. The value of λ that gives the minimum CV score is taken to be a good choice for the smoothing parameter. If a smoother matrix exists,then the CV formula Silverman (1985) may be expressed as Where S ii denotes the i th diagonal element of S λ , defined as andl(t i )is the spline smoother with λ. Akaike's information criterion corrected, AICc, was introduced by Hurvich et al.
(1998) because the commonly used Akaike's information criterion Eubank (1999) may have a tendency to over-fit the curve estimate for small samples. The AICc criterion may be expressed as The BIC criterion Schwarz et al. (1978) may be defined as, where d f (λ)=tr(S λ ) which is similar to AICc but penalizes a model fit with a larger d f (λ) more strongly than the AICc for large n.
Developed by Craven and Wahba (1979),Generalized cross-validation (GCV) may be defined by To compare the proposed model, the mean squared error,predictive squared error, Mallow's C p measures will be employed. Given that our model is defined as y = µ + (2.29) we assume that E (y) = µ. The vector µ = (µ 1 , . . . , µ n ) is the spline function and evaluated at the design model µ i = l (t i ) and the components of the error vector are normally with mean zero and σ 2 < ∞. Sincel (λ) = S λ y, then its expectation is E l (λ) = E (S λ y) = S λ E (y) = S λ µ (2.30)

Bias of the Estimator
Let b λ be bias of the estimator, then

Variance of the Estimator
The variance of y is the same as , that is, Note that the variance of the residual vector is given as

The Variance Estimate,σ 2
To estimate the variance, σ 2 ,in case of regression, we divide the residual sum of squares ||y −l (λ) || 2 = n i=1 y −l (λ) i by the degrees of freedom for the error to obtained an unbiased estimator of σ 2 . We need the expectation of the residual sum of squares. For any random variable, W, The sum of the diagonal elements of a matrix is called its trace. A trace of a matrix A = tr (A) The estimate of σ 2 is to divide the MS E by the degrees of freedom given s To obtain a uniform criterion of performance, we averaged the MS E and PS E over the observed observations as is a sensible estimate of the predictive squared error.

Asymptotic Properties of Penalized Spline Estimator
In studying the theoretical properties of the penalized spline estimator as the minimizer of the (2.9) , we considered the average mean squared error (AMSE) and the asymptotic bias and variance of the model. In addition, we also deliberate on the optimum choice of the smoothing parameter. Asymptotic properties of the penalized spline estimator are discussed under subsection 3.4, subsection 3.5 Average mean squared error of the estimator and Asymptotic of the variance and the bias in subsection 3.6.
See Zhou et al. (1998) for asymptotic properties of the regression spline estimator and we adopt the following assumptions.
Assumption 2. For deterministic designs points y t ∈ [a, b], t = 1, . . . n, assume that there exists a distribution function Q with corresponding positive continuous design density ρ such that, with Q n the empirical distribution of y 1 , . . . , y n sup y∈[a,b] |Q n (y) − Q(y)| = o(K −1 ).
Assumption 3. The number of knots K = o (n) International Journal of Statistics and Probability Vol. 9, No. 5; The asymptotic bias and variance for the regression splines are obtainable from Zhou et al. (1998) as given that B p+1 (.) is the (p + 1) th Bernoulli polynomial see Abramowitz and Stegun (1972)

Average Mean Squared Error of the Estimator
Agreeing to Demmler and Reinsch (1975), it is possible to express the average bias and variance in terms of the eigenvalues having been obtained from the singular value decomposition.
where U is the matrix of vectors and s is the vector of eigenvalues s j .
We obtain the AMS E as where AA t l = E(l reg ), l = {l(t 1 ) . . . l(t n )} t ,b = A t l with components b j and AA t an Idempotent matrix, obtaining From AMS E equation (3.10) the first term is the average variance while the second term is the average squared bias (shrinkage) as a result of penalization and the third term is the average squared approximation bias.
Let define K q = (K + p + 1 − q)(αc 1 ) 1/(2q) n −1/(2q) (3.11) Theorem 3.1. Under Assumptions 1-3, we consider two asymptotic cases: (i) if K q < 1 and l(.) ∈ C p+1 [a, b], then where K ∼ C 1 n 1/(2p+3) , with C 1 a constant, α = O(n γ ) with γ (p + 2 − q)/(2p + 3), the penalized spine estimator attains the optimal rate of convergence for l ∈ C p+1 [a, b] with AMSE(l) = O(n −(2p+2)/(2p+3) ). (ii)if K q 1 and l ∈ W q [a, b] and for α = O(n 1/(2q+1) ), such that αn 2q−1 → ∞ and K ∼ C 2 n v with v 1/(2q + 1) and C 2 a constant, the penalized spline estimator attains the optimal rate of convergence for l ∈ W q [a, b] with AMSE(l) = O(n −2q/(1+2q) ) Considering case (i) with K q < 1 the result we obtained is similar to the regression splines. In this case, the AMSE is determined by the squared approximation bias and the mean asymptotic variance. Thus the smaller the smoothing parameter, α, the insignificant the shrinkage bias is, that is γ < (p + 2 − q)/(2p + 3) with the asymptotically optimal number of knots having same order as the regression splines, that is K ∼ C 1 n 1/(2p+3) . We also notice in the case(ii) with K q 1 that this result obtained is similar to the smoothing spline. Where AMSE in this case is dominated by the squared shrinkage bias and average asymptotic variance. The mean squared approximation bias has the same order as that of the shrinkage bias for K q = 1 and is negligible order for K q > 1. The results suggest that convergence rate for the penalized spline estimator is much faster in the case (i) where K q < 1 relying on the assumption that q p.

Asymptotic of the Variance and the Bias
Theorem 3.2. Under Assumptions 1-3, we derive the Asymptotic of the Variance and the Bias under following two cases: The shrinkage bias b α is defined as b α (t) = −αn −1 N(t)(G + αD q /n) −1 D q β, where G and β are define as in Zhou et al. (1998).

Simulation Studies
A climate data (indicator) was generated using simulation method to measure strength of our proposed modeling adequacy. We anticipate the climate or whether signal may have composed of different components, namely, trend, periodicity as well as stochastic patterns. In Figure 2a, using a spline function to estimate the function for varying order k with a fixed number of m = 8 inner equidistant knot. For k = 1 the function consists of horizontal straight lines with jumps at the knots and for k > 2 indicates a smooth function but the orders cannot well be distinguished visually anymore. In contrast, Figure 2b shows the curves for polynomials of degreep = 3, · · · , 12. The curves are characterized with degrees of fluctuations and for higher values of p, the polynomial curves exhibit an oscillatory form. Also, in Figure 2c, the curve is estimated using Penalized splines with m = 20 equidistant inner knots and varying smoothing parameter values. Figure 2d shows the plot of selected smoothing parameter for simulated data for the penalized spline function. While Figure 2e represents the smoothing with k = 4 and varying values of λ. Figure 2f shows the plot of selected smoothing parameter for simulated data for the smoothing spline function. http://ijsp.ccsenet.org International Journal of Statistics and Probability Vol. 9, No. 5; 2020 (a) Spline functions with 8 equidistant inner knots and different orders k = 1, · · · , 6 (b) A polynomial function fitting the simulated temperature with different orders p = 3, · · · , 12.  The Figure3 below comprises panels 3a-3h representing different spline function of different orders, number of knots and smoothing parameter λ. From Figure 3a shows the time series plots of the average monthly temperature exhibiting irregular oscillations and increasing trend. It is also, observed in Figure 3b that, using a spline function to estimate the function for varying order k with a fixed number of m = 8 inner knots in an equidistant knot sequence. For k = 1 the function consists of horizontal straight lines with jumps at the knots and for k = 2 the curve is constructed from straight lines with non-zero slopes which are continuously connected. For k = 3 and k = 4 the quadratic and cubic pieces can be observed but for k > 5 the orders cannot well be distinguished visually anymore. In contrast, Figure 3c shows the curves for polynomials of degreep = 3, · · · , 12. The curves is characterized with degree of fluctuations and for higher values of p, the polynomial curves exhibit an undesirable behavior by oscillating more and more especially at the end point. Also, in Figure 3d, the curve is estimated using Penalized splines with m = 20 equidistant inner knots and varying smoothing parameter values. Figure 3e shows the plot of selected smoothing parameter for data under considerations. With the selected smoothing parameters in Figure 3g, the B-spline, the penalized spline and smoothing splines were used to fit data in Figure 3h.
The performance criteria of fitted models are presented below based on MSE,PSE, and ASR taken into account the influence of the smoothing parameter, position and the number of knots as in the theoretical studies shown in section 3 http://ijsp.ccsenet.org International Journal of Statistics and Probability Vol. 9, No. 5; 2020 (a) A Time Series Plot of the Average Monthly Temperature (b) Spline functions with 8 equidistant inner knots and different orders k = 1, · · · , 6 (c) A polynomial function fitting the average monthly temperature data with different orders p = 3, · · · , 12.  From Table(1), the model performance measures for B-spline, Penalized spline and the Smoothing spline functions fitted to temperature data. The three spline functions perform fairly comparable, particularly the B-spline function and Penalized spline curves with suitably selected values for λ using the CV and GCV criteria. Largely, the specifications for the three curves seem all well chosen to represent the average monthly temperature for Ghana.

Conclusion
Considering the average monthly temperature data characterized by nonlinear the trend, seasonal and periodic components, the study proposed a penalized spline model. The study compared the B-spline, Penalized spline and the smoothing spline which in all cases with the appropriate determination of the smoothing parameter adequately fit the average monthly temperature for Ghana.