Modeling, simulation, and prediction of global energy indices: a differential approach

Abstract

Modeling, simulation, and prediction of global energy indices remain veritable tools for econometric, engineering, analysis, and prediction of energy indices. Thus, this paper differentially modeled, simulated, and non-differentially predicated the global energy indices. The state-of-the-art of the research includes normalization of energy indices, generation of differential rate terms, and regression of rate terms against energy indices to generate coefficients and unexplained terms. On imposition of initial conditions, the solution to the system of linear differential equations was realized in a Matlab environment. There was a strong agreement between the simulated and the field data. The exact solutions are ideal for interpolative prediction of historic data. Furthermore, the simulated data were upgraded for extrapolative prediction of energy indices by introducing an innovative model, which is the synergy of deflated and inflated prediction factors. The innovative model yielded a trendy prediction data for energy consumption, gross domestic product, carbon dioxide emission and human development index. However, the oil price was untrendy, which could be attributed to odd circumstances. Moreover, the sensitivity of the differential rate terms was instrumental in discovering the overwhelming effect of independent indices on the dependent index. Clearly, this paper has accomplished interpolative and extrapolative prediction of energy indices and equally recommends for further investigation of the untrendy nature of oil price.

Graphical abstract

Keywords

energy indices / differential model / normalization / simulation / inflation/deflation / predictive factor and prediction rate

1 Introduction

The nexus between energy indices, energy consumption (EC), gross domestic product (GDP), carbon dioxide emission (CDE), human development index (HDI) and oil price (OP), obviously reflects the intricate relationship or causality in different connections, which is the basis for formulating the system of qualitative and quantitative mathematical models, and describing the complex (or conglomeration of individual) behaviors immanent between the energy indices. Analogically, Nnamchi et al. [1], Bianco et al. [2], van den Brom et al. [3], Marion and Lawson [4], and Afgan et al. [5] advocated that the elements of complex energy systems should be modeled as a specific structure reflecting the different characteristics of the system, a sequel to the interaction between the system and its stimulating factors (like economic, social and environmental energy indices) which is consistent with Bianco [6] and Li et al. [7] index decomposition approach, which has the potential to establish a critical nexus between energy indices. Moreover, Sun et al. [8] investigated a life cycle assessment of pulp and paper production and established a nexus between greenhouse gas (GHG) emissions with a large energy consumption. Quantitatively, Sun et al. [8] affirmed that 1 ton of pulp and paper production emits about 950 kg of carbon dioxide (CO₂) equivalent (CO₂-eq) GHG emissions on average. Thus, a systematic integration of the individual characteristics of a system will certainly produce a workable mathematical model describing the macroscopic behavior of the energy system.

Therefore, modeling remains an art of representing the nature and natural phenomena in a mathematical notation [9–11], which could be either a deterministic differential or non-differential (Tobin) regression as employed by Wang et al. [12] in studying the energy conversion potential of the nonferrous metal industry in China with an emphasis on the population density, gross domestic product, and oil price or stochastic (or probabilistic model [13]).

Dominantly, the econometric models employed in the analysis of the causality between the energy indices are mainly deterministic, non-mechanistic linear models as implemented by Zeng et al. [14] in investigating the causal relationship between the price of carbon emission allowance with respect to economic development and the price of energy. Principally, econometric models relate the dependent variable (the responsive) to independent variables (stimulants) with inherently explained and unexplained terms [15,16]. Obviously, a wide application of mathematical models over other variants of models is justified by deductive or inferential mathematical argument (or proof) leading to a logical decision. These models are classified as time trendy and untrendy mathematical ones if the change in the dependent variable is not susceptible to fluctuations in time change and vice versa, respectively. Thus, time trendy and untrendy models econometrically (or statistically) designate stationary and non-stationary energy indices, respectively [17–19]. Thus, irrespective of the change in indices over time, stationary indices have a pattern of change over time and could be easily predicted, but non-stationary energy indices are void of a pattern of change and could not be easily predicted [20]. However, the invisible time variable associated with econometric model analysis is not physically represented in the model structure, but the differential model is expected to have derivative of time or consider time as a variable. This implies that time is an additional independent variable influencing the energy indices in differential models.

Thus, the current work strongly proposes the application of the deterministic differential (or mechanistic) model in modeling the intricate relationship between energy indices. The ideal differential model to achieve this adventure is first order linear differential system of equations. The differential terms clearly integrate the time variable and vividly represent the trendy and untrendy components of the system of differential equations.

Axiomatically, a steady-state equation portrays an equilibrium condition between energy indices, as the change in time is equal to zero. In this case, the mechanistic equation reduces to a non-mechanistic model or canonical econometric and statistical models (vector error correction model, VECM and variants of vector autoregression model, VAR). VECM has been extensively applied in the analysis of causality between energy indices in Refs. [21–30] whereas the VAR model has been applied in establishing the causality between energy indices [31–40]. VECM and VAR are not suitable for interpolation of energy index data due to multiple independent variables associated with ample error. However, the system of differential equations is very effective in interpolating the energy index data as a single independent variable (time) is associated with minimal error. Canonically, Tang et al. [41] developed a relationship between carbon emission and economic development via a decomposition model. In their further work, Tang et al. [41] developed a computational model to estimate carbon emission and energy consumption. Contrarily, Sun [42] established a causal relationship between carbon emission and economic growth. However, a significant value of the rate of change of energy indices, the system of energy indices is marked by a disequilibrium or instability (which portrays the tendency of non-stationarity), and the instability in energy indices could be ideally modeled and simulated by the system of differential equations [8,43,44]. The analytical solution to the system of differential equations will yield a closed (or exact) solution with a definite mathematical (or special) function, which precisely describes the intricate behavior of the energy system. Obviously, the instability (or disequilibrium) becomes remarkable when the rate term is either less or greater than zero, which warrants inflation (or increase) and deflation (or decrease) in energy indices, respectively [8,14].

For the purposes of having an amenable model solution, the system of differential equations is transformed from dimensional into normalized differential models without distorting their physical meaning. This could be achieved by normalizing all the variables involved in the formulation of energy indices and models [5]. Furthermore, the rate term is derived from the trending equation on the plot of normalized dependent variable (energy indices) against normalized time. Sufficiently, the trended equation with a high coefficient of regression (R²(1)) is differentiated with respect to normalized time to generate the energy index rate terms. The normalized rate terms are regressed against other normalized energy indices to obtain their dynamic coefficients (or elasticities) and unexplained normalized terms. The initial value problem formulated by assigning the initial or reference normalized variables as the initial condition is useful for generating the solution to normalized energy indices as a function of normalized time, which is characterized by exponential or power (or Taylor) series [43].

Subsequently, the normalized function is useful in translating normalized energy indices into dimensional ones [5,45]. The dimensional energy index function can be employed in the simulation and quality prediction (or interpolation) of energy indices under the auspices of historic energy index data. However, the solution to the differential energy index model just like VECM and VAR is not capable of extrapolating energy indices, as it lacks a stochastic characteristic. Emphatically, all the models, i.e., VECM, VAR, and system of linear differential equations (SLDE) cannot extrapolate energy indices. Consequently, the present work has basically developed innovative prediction models [46–48], whose probabilistic terms (the predictive factors are composed of the predictive rates) are to be regulated by the historic energy index data in accordance with Wang et al. [12] and Zeng et al. [14]. Essentially, the predicted results are the synergy (or average) of the deflated and inflated prediction indices.

Therefore, this work is pivoted on developing time dependent first order system of linear differential equation which describes energy indexes, solving the system of differential equations, simulating the solutions, and developing innovative (probabilistic) models to accomplish the extrapolative prediction (or forecasting) of energy indices.

2 Materials and method

For the purpose of this study, five energy indices, i.e., energy consumption (EC), gross domestic product (GDP), carbon dioxide emission (CDE), human development index (HDI), and oil price (OP) were selected. About 36 year historic data on the five selected indices (EC, GDP, CDE, HDI and OP) were acquired from regulatory bodies and institutions, such as Birol [49], Amadeo [50], Roser [51], McMahon [52] and Chair [53], respectively.

The time history of energy indices is to be modeled by dimensional and normalized first order differential system of linear equations. The normalized model is proposed for fairness in the analysis of energy indices (since every value bounds between zero and one) and for amenability of the solution. Subsequently, the normalized solution will be translated or transformed into dimensional form with the aid of normalized function. Error analysis of the simulated results (relative and absolute) is desired to show the fitness or agreement between the simulated and field data.

However, the simulation result obtained from the model solution is solely suitable for interpolative prediction of energy indices. To overcome this shortcoming, an innovative (or stochastic) model is developed by introducing paired prediction rates based on the historical data. The upper and lower limits of the predicted rate is used to generate random number of prediction rates across historic energy indices. Moreover, the prediction rates are useful for computing the prediction factor, which could be either deflated or inflated factor based on the sign (±) of the predicted rates. Notably, positive and negative prediction rates yield inflated and deflated prediction factors, respectively. The product of the prediction factor but not the historic energy index gives the prediction value (deflated or inflated). The arithmetic average of inflated and deflated prediction values yields the predicted (or forecasted) values (or results), which are ideal for extrapolative prediction of future energy indices.

2.1 Dimensional first order differential model

The rate of change of energy indices is expressed as the sum of the explained terms (products of energy indices and their elasticities) and the unexplained term (linear coefficient). Thus, the dimensional EC differential model is developed in Eq. (1) as

(1){\displaystyle \frac{\mathrm{d}\mathrm{EC}}{\mathrm{d}t}}=-{\alpha }_{\mathrm{ec}}\mathrm{EC}-{\alpha }_{\mathrm{gdp}}\mathrm{GDP}+{\alpha }_{\mathrm{cde}}\mathrm{CDE}-{\alpha }_{\mathrm{hdi}}\mathrm{HDI}\hspace{0.17em}+{\alpha }_{\mathrm{op}}\mathrm{OP}+g_{\mathrm{ec}}.

The dimensional GDP differential model is established in Eq. (2) as

(2){\displaystyle \frac{\mathrm{d}\mathrm{GDP}}{\mathrm{d}t}}={\beta }_{\mathrm{gdp}}\mathrm{GDP}+{\beta }_{\mathrm{ec}}\mathrm{EC}-{\beta }_{\mathrm{cde}}\mathrm{CDE}+{\beta }_{\mathrm{hdi}}\mathrm{HDI}\hspace{0.17em}-{\beta }_{\mathrm{op}}\mathrm{OP}-g_{\mathrm{gdp}}.

The dimensional CDE differential model is represented in Eq. (3) as

(3){\displaystyle \frac{\mathrm{dCDE}}{\mathrm{d}t}}=-{\delta }_{\mathrm{cde}}\mathrm{CDE}\hspace{0.17em}+{\delta }_{\mathrm{ec}}\mathrm{EC}+{\delta }_{\mathrm{gdp}}\mathrm{GDP}+{\delta }_{\mathrm{hdi}}\mathrm{HDI}-{\delta }_{\mathrm{op}}\mathrm{OP}-g_{\mathrm{cde}}.

The dimensional HDI differential model is written in Eq. (4) as

(4){\displaystyle \frac{\mathrm{d}\mathrm{HDI}}{\mathrm{d}t}}=-{\epsilon }_{\mathrm{hdi}}\mathrm{HDI}\hspace{0.17em}-{\epsilon }_{\mathrm{ec}}\mathrm{EC}-{\epsilon }_{\mathrm{gdp}}\mathrm{GDP}+\hspace{0.17em}{\epsilon }_{\mathrm{cde}}\mathrm{CDE}+\hspace{0.17em}{\epsilon }_{\mathrm{op}}\mathrm{OP}-g_{\mathrm{hdi}}.

The dimensional OP differential model is expressed in Eq. (5) as

(5){\displaystyle \frac{\mathrm{d}\mathrm{OP}}{\mathrm{d}t}}=-{\varphi }_{\mathrm{op}}\mathrm{OP}+{\varphi }_{\mathrm{ec}}\mathrm{EC}+{\varphi }_{\mathrm{gdp}}\mathrm{GDP}\hspace{0.17em}+\hspace{0.17em}{\varphi }_{\mathrm{cde}}\mathrm{CDE}-{\varphi }_{\mathrm{hdi}}\mathrm{HDI}-g_{\mathrm{op}},

where {\alpha }_i, {\beta }_i, {\delta }_i\hspace{0.17em}, {\epsilon }_i\hspace{0.17em} and {\varphi }_i are coefficients of dimensional first order differential model, i=\{\mathrm{1},\hspace{0.17em}\mathrm{2},\hspace{0.17em}\mathrm{3},\hspace{0.17em}\mathrm{4},\mathrm{5}\}\equiv \left\{\mathrm{ec,}\hspace{0.17em}\mathrm{gdp,}\hspace{0.17em}\mathrm{cde,}\hspace{0.17em}\mathrm{hdi,}\hspace{0.17em}\mathrm{op}\right\}, EC is energy consumption (million tons of oil equivalent (Mtoe)), GDP is gross domestic product (US$ Billion), CDE is carbon dioxide emission (million tons of oil equivalent of CO₂), HDI is human development index (%), and OP is oil price (US$ Billion/Mtoe). g_i,\hspace{0.17em}i=\left\{\mathrm{1},\hspace{0.17em}\mathrm{2},\hspace{0.17em}\mathrm{3},\hspace{0.17em}\mathrm{4},\hspace{0.17em}\mathrm{5}\right\}\equiv \{\mathrm{ec,}\hspace{0.17em}\mathrm{gdp,}\hspace{0.17em}\mathrm{cde,}\hspace{0.17em}\mathrm{hdi,}\hspace{0.17em}\mathrm{op}\} are unexplained dimensional constants associated with Eqs. (1)–(5), and t is time (year).

2.2 Normalized first order differential model

Conveniently, Eqs. (1)–(5) can be written in normalized form for quick realization of the solutions.

The normalized EC differential model is given in Eq. (6) as

(6){\displaystyle \frac{\mathrm{d}\mathrm{ec}}{\mathrm{d}\tau }}=-{{\alpha }^{\prime }}_{\mathrm{ec}}\mathrm{ec}-{{\alpha }^{\prime }}_{\mathrm{gdp}}\mathrm{gdp}+{{\alpha }^{\prime }}_{\mathrm{cde}}\mathrm{cde}-{{\alpha }^{\prime }}_{\mathrm{hdi}}\mathrm{hdi}\hspace{0.17em}+{{\alpha }^{\prime }}_{\mathrm{op}}\mathrm{op}+{g^{\prime }}_{\mathrm{ec}}.

The normalized GDP differential model is explicated in Eq. (7) as

(7){\displaystyle \frac{\mathrm{d}\mathrm{gdp}}{\mathrm{d}\tau }}={{\beta }^{\prime }}_{\mathrm{gdp}}\mathrm{gdp}+{{\beta }^{\prime }}_{\mathrm{ec}}\mathrm{ec}-{{\beta }^{\prime }}_{\mathrm{cde}}\mathrm{cde}+{{\beta }^{\prime }}_{\mathrm{hdi}}\mathrm{hdi}\hspace{0.17em}-{{\beta }^{\prime }}_{\mathrm{op}}\mathrm{op}-{g^{\prime }}_{\mathrm{gdp}}.

The normalized CDE differential model is stated in Eq. (8) as

(8){\displaystyle \frac{\mathrm{d}\mathrm{cde}}{\mathrm{d}\tau }}=-{{\delta }^{\prime }}_{\mathrm{cde}}\mathrm{cde}\hspace{0.17em}+{{\delta }^{\prime }}_{\mathrm{ec}}\mathrm{ec}+{{\delta }^{\prime }}_{\mathrm{gdp}}\mathrm{gdp}+{{\delta }^{\prime }}_{\mathrm{hdi}}\mathrm{hdi}-{{\delta }^{\prime }}_{\mathrm{op}}\mathrm{op}-{g^{\prime }}_{\mathrm{cde}}.

The normalized HDI differential model is written in Eq. (9) as

(9){\displaystyle \frac{\mathrm{d}\mathrm{hdi}}{\mathrm{d}\tau }}=-{{\epsilon }^{\prime }}_{\mathrm{hdi}}\mathrm{hdi}\hspace{0.17em}-{{\epsilon }^{\prime }}_{\mathrm{ec}}\mathrm{ec}-{{\epsilon }^{\prime }}_{\mathrm{gdp}}\mathrm{gdp}+\hspace{0.17em}{{\epsilon }^{\prime }}_{\mathrm{cde}}\mathrm{cde}+\hspace{0.17em}{{\epsilon }^{\prime }}_{\mathrm{op}}\mathrm{op}-{g^{\prime }}_{\mathrm{hdi}}.

The normalized OP differential model is expounded in Eq. (10) as

(10){\displaystyle \frac{\mathrm{d}\mathrm{op}}{\mathrm{d}\tau }}=-{{\varphi }^{\prime }}_{\mathrm{op}}\mathrm{op}+\hspace{0.17em}{{\varphi }^{\prime }}_{\mathrm{ec}}\mathrm{ec}+{{\varphi }^{\prime }}_{\mathrm{gdp}}\mathrm{gdp}\hspace{0.17em}+{{\varphi }^{\prime }}_{\mathrm{cde}}\mathrm{cde}-{{\varphi }^{\prime }}_{\mathrm{hdi}}\mathrm{hdi}-{g^{\prime }}_{\mathrm{op}},

where {{\alpha }^{\prime }}_i\hspace{0.17em}, {{\beta }^{\prime }}_i\hspace{0.17em}, {{\delta }^{\prime }}_i\hspace{0.17em}, {{\epsilon }^{\prime }}_i\hspace{0.17em} and {{\varphi }^{\prime }}_i,\hspace{0.17em}\hspace{0.17em}i=\left\{\mathrm{1},\hspace{0.17em}\mathrm{2},\hspace{0.17em}\mathrm{3},\hspace{0.17em}\mathrm{4},\hspace{0.17em}\mathrm{5}\right\}\equiv \left\{\mathrm{ec,}\hspace{0.17em}\mathrm{gdp,}\hspace{0.17em}\mathrm{cde,}\hspace{0.17em}\mathrm{hdi,}\hspace{0.17em}\mathrm{op}\right\}\hspace{0.17em} are normalized elasticity coefficients, ec is energy consumption (-), gdp is gross domestic product (-), cde is carbon dioxide emission (-), hdi is human development index (-), op is oil price (-). {g^{\prime }}_i,\hspace{0.17em}\hspace{0.17em}i=\left\{\mathrm{1},\hspace{0.17em}\mathrm{2},\hspace{0.17em}\mathrm{3},\hspace{0.17em}\mathrm{4},\hspace{0.17em}\mathrm{5}\right\}\equiv \left\{\mathrm{ec,}\hspace{0.17em}\mathrm{gdp,}\hspace{0.17em}\mathrm{cde,}\hspace{0.17em}\mathrm{hdi,}\hspace{0.17em}\mathrm{op}\right\}, are normalized unexplained coefficients associated with Eqs. (6)–(10), and t is normalized time.

The normalized ec (-) is represented in Eq. (11) as

(11)\mathrm{ec}={\displaystyle \frac{{\mathrm{EC}}_t-{\mathrm{EC}}_{\max }}{{\mathrm{EC}}_{\min }-{\mathrm{EC}}_{\max }}};\Rightarrow {\mathrm{EC}}_t=\mathrm{EC}\left(t\right)=\mathrm{ec}\left({\mathrm{EC}}_{\min }-{\mathrm{EC}}_{\max }\right)+{\mathrm{EC}}_{\max }.

The normalized gdp (-) is expressed in Eq. (12) as

(12)\mathrm{gdp}={\displaystyle \frac{{\mathrm{GDP}}_t-{\mathrm{GDP}}_{\max }}{{\mathrm{GDP}}_{\min }-{\mathrm{GDP}}_{\max }}};\Rightarrow {\mathrm{GDP}}_t=\mathrm{GDP}\left(t\right)=\mathrm{gdp}\left({\mathrm{GDP}}_{\min }-{\mathrm{GDP}}_{\max }\right)+{\mathrm{GDP}}_{\max }.

The normalized cde (-) is explained in Eq. (13) as

(13)\mathrm{cde}\hspace{0.17em}={\displaystyle \frac{{\mathrm{CDE}}_t-{\mathrm{CDE}}_{\max }}{{\mathrm{CDE}}_{\min }-{\mathrm{CDE}}_{\max }}};\Rightarrow {\mathrm{CDE}}_t=\mathrm{CDE}\left(t\right)=\mathrm{cde}\hspace{0.17em}\left({\mathrm{CDE}}_{\min }-{\mathrm{CDE}}_{\max }\right)+{\mathrm{CDE}}_{\max }.

The normalized hdi (-) is written in Eq. (14) as

(14)\mathrm{hdi}\hspace{0.17em}={\displaystyle \frac{{\mathrm{HDI}}_t-{\mathrm{HDI}}_{\max }}{{\mathrm{HDI}}_{\min }-{\mathrm{HDI}}_{\max }}};\Rightarrow {\mathrm{HDI}}_t=\mathrm{HDI}\left(t\right)=\mathrm{hdi}\hspace{0.17em}\left({\mathrm{HDI}}_{\min }-{\mathrm{HDI}}_{\max }\right)+{\mathrm{HDI}}_{\max }.

The normalized op (-) is developed in Eq. (15) as

(15)\mathrm{op}={\displaystyle \frac{{\mathrm{OP}}_t-{\mathrm{OP}}_{\max }}{{\mathrm{OP}}_{\min }-{\mathrm{OP}}_{\max }}};\Rightarrow {\mathrm{OP}}_t=\mathrm{OP}\left(t\right)=\mathrm{op}\left({\mathrm{OP}}_{\min }-{\mathrm{OP}}_{\max }\right)+{\mathrm{OP}}_{\max }.

The normalized time, t in Eq. (16) is represented as

(16)\tau ={\displaystyle \frac{t-t_{\mathrm{0}}}{t_{\mathrm{f}}-t_{\mathrm{0}}}};\Rightarrow t=\tau \left(t_{\mathrm{f}}-t_{\mathrm{0}}\right)+t_{\mathrm{0}},

where t is the dimensional time, t₀ is the initial (0) time, and t_f is the final (f) time.

Equations (6)–(10) could be represented in a matrix form in Eq. (17)

(17)\left[\begin{array}{c}\hfill {\displaystyle \frac{\mathrm{d}\mathrm{ec}}{\mathrm{d}\tau }}\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}\mathrm{gdp}}{\mathrm{d}\tau }}\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}\mathrm{cde}}{\mathrm{d}\tau }}\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}\mathrm{hdi}}{\mathrm{d}\tau }}\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}\mathrm{op}}{\mathrm{d}\tau }}\hfill \end{array}\right]=\left[\begin{array}{ccccc}\hfill \begin{array}{c}\hfill -{{\alpha }^{\prime }}_{\mathrm{ec}}\hfill \end{array}\hfill & \hfill \begin{array}{c}\hfill -{{\alpha }^{\prime }}_{\mathrm{gdp}}\hfill \end{array}\hfill & \hfill \begin{array}{c}\hfill {{\alpha }^{\prime }}_{\mathrm{cde}}\hfill \end{array}\hfill & \hfill \begin{array}{c}\hfill -{{\alpha }^{\prime }}_{\mathrm{hdi}}\hfill \end{array}\hfill & \hfill \begin{array}{c}\hfill {{\alpha }^{\prime }}_{\mathrm{op}}\hfill \end{array}\hfill \\ \hfill \begin{array}{c}\hfill {{\beta }^{\prime }}_{\mathrm{ec}}\hfill \end{array}\hfill & \hfill \begin{array}{c}\hfill {{\beta }^{\prime }}_{\mathrm{gdp}}\hfill \end{array}\hfill & \hfill \begin{array}{c}\hfill -{{\beta }^{\prime }}_{\mathrm{cde}}\hfill \end{array}\hfill & \hfill \begin{array}{c}\hfill {{\beta }^{\prime }}_{\mathrm{hdi}}\hfill \end{array}\hfill & \hfill \begin{array}{c}\hfill -{{\beta }^{\prime }}_{\mathrm{op}}\hfill \end{array}\hfill \\ \hfill \begin{array}{c}\hfill {{\delta }^{\prime }}_{\mathrm{ec}}\hfill \end{array}\hfill & \hfill \begin{array}{c}\hfill {{\delta }^{\prime }}_{\mathrm{gdp}}\hfill \end{array}\hfill & \hfill \begin{array}{c}\hfill -{{\delta }^{\prime }}_{\mathrm{cde}}\hfill \end{array}\hfill & \hfill \begin{array}{c}\hfill {{\delta }^{\prime }}_{\mathrm{hdi}}\hfill \end{array}\hfill & \hfill \begin{array}{c}\hfill -{{\delta }^{\prime }}_{\mathrm{op}}\hfill \end{array}\hfill \\ \hfill \begin{array}{c}\hfill -{{\epsilon }^{\prime }}_{\mathrm{ec}}\hfill \end{array}\hfill & \hfill \begin{array}{c}\hfill -{{\epsilon }^{\prime }}_{\mathrm{gdp}}\hfill \end{array}\hfill & \hfill \begin{array}{c}\hfill {{\epsilon }^{\prime }}_{\mathrm{cde}}\hfill \end{array}\hfill & \hfill \begin{array}{c}\hfill -{{\epsilon }^{\prime }}_{\mathrm{hdi}}\hfill \end{array}\hfill & \hfill \begin{array}{c}\hfill {{\epsilon }^{\prime }}_{\mathrm{op}}\hfill \end{array}\hfill \\ \hfill {{\varphi }^{\prime }}_{\mathrm{ec}}\hfill & \hfill {{\varphi }^{\prime }}_{\mathrm{gdp}}\hfill & \hfill {{\varphi }^{\prime }}_{\mathrm{cde}}\hfill & \hfill -{{\varphi }^{\prime }}_{\mathrm{hdi}}\hfill & \hfill -{{\varphi }^{\prime }}_{\mathrm{op}}\hfill \end{array}\right]\left[\begin{array}{c}\hfill \begin{array}{c}\hfill \mathrm{ec}\hfill \end{array}\hfill \\ \hfill \begin{array}{c}\hfill \mathrm{gdp}\hfill \end{array}\hfill \\ \hfill \begin{array}{c}\hfill \mathrm{cde}\hfill \end{array}\hfill \\ \hfill \begin{array}{c}\hfill \mathrm{hdi}\hfill \end{array}\hfill \\ \hfill \mathrm{op}\hfill \end{array}\right]+\left[\begin{array}{c}\hfill \begin{array}{c}\hfill {g^{\prime }}_{\mathrm{ec}}\hfill \end{array}\hfill \\ \hfill \begin{array}{c}\hfill -{g^{\prime }}_{\mathrm{gdp}}\hfill \end{array}\hfill \\ \hfill \begin{array}{c}\hfill -{g^{\prime }}_{\mathrm{cde}}\hfill \end{array}\hfill \\ \hfill \begin{array}{c}\hfill -{g^{\prime }}_{\mathrm{hdi}}\hfill \end{array}\hfill \\ \hfill -{g^{\prime }}_{\mathrm{op}}\hfill \end{array}\right],

where g_i′ is the unexplained coefficients in system of equations, and i= {1, 2, 3, 4, 5} ≡ {ec, gdp, cde, hdi, op}.

The initial condition corresponds to the value of normalized energy indices at the reference time.

(18)\mathrm{ec}\left(\mathrm{0}\right)={\mathrm{ec}}_{\mathrm{0}},\mathrm{gdp}\left(\mathrm{0}\right)={\mathrm{gdp}}_{\mathrm{0}},\mathrm{cde}\left(\mathrm{0}\right)={\mathrm{cde}}_{\mathrm{0}},\mathrm{hdi}\left(\mathrm{0}\right)={\mathrm{hdi}}_{\mathrm{0}}\mathrm{and\; op}\left(\mathrm{0}\right)={\mathrm{op}}_{\mathrm{0}},\mathrm{for}\tau =\mathrm{0.}

Equations (17)–(18) can be combined to obtain an explicit expression for normalized energy indices as a function of normalized time.

The general solution to Eqs. (17)–(18) given in Eq. (19) as

(19)\left[\begin{array}{c}\hfill {\mathrm{ec}}_{\tau }\hfill \\ \hfill {\mathrm{gdp}}_{\tau }\hfill \\ \hfill {\mathrm{cde}}_{\tau }\hfill \\ \hfill {\mathrm{hdi}}_{\tau }\hfill \\ \hfill {\mathrm{op}}_{\tau }\hfill \end{array}\right]={\mathrm{e}}^{A\tau }\hspace{0.17em}\left[\begin{array}{c}\hfill {\mathrm{ec}}_{\mathrm{0}}\hfill \\ \hfill {\mathrm{gdp}}_{\mathrm{0}}\hfill \\ \hfill {\mathrm{cde}}_{\mathrm{0}}\hfill \\ \hfill {\mathrm{hdi}}_{\mathrm{0}}\hfill \\ \hfill {\mathrm{op}}_{\mathrm{0}}\hfill \end{array}\right]+{\displaystyle {\int }_{\mathrm{0}}^{\tau }{\mathrm{e}}^{A\left(\tau -t\right)}}BU\left(t\right)\mathrm{d}t,

where

(20)A=\left[\begin{array}{ccccc}\hfill -{{\alpha }^{\prime }}_{\mathrm{ec}}\hfill & \hfill -{{\alpha }^{\prime }}_{\mathrm{gdp}}\hfill & \hfill {{\alpha }^{\prime }}_{\mathrm{cde}}\hfill & \hfill -{{\alpha }^{\prime }}_{\mathrm{hdi}}\hfill & \hfill {{\alpha }^{\prime }}_{\mathrm{op}}\hfill \\ \hfill {{\beta }^{\prime }}_{\mathrm{ec}}\hfill & \hfill {{\beta }^{\prime }}_{\mathrm{gdp}}\hfill & \hfill -{{\beta }^{\prime }}_{\mathrm{cde}}\hfill & \hfill {{\beta }^{\prime }}_{\mathrm{hdi}}\hfill & \hfill -{{\beta }^{\prime }}_{\mathrm{op}}\hfill \\ \hfill {{\delta }^{\prime }}_{\mathrm{ec}}\hfill & \hfill {{\delta }^{\prime }}_{\mathrm{gdp}}\hfill & \hfill -{{\delta }^{\prime }}_{\mathrm{cde}}\hfill & \hfill {{\delta }^{\prime }}_{\mathrm{hdi}}\hfill & \hfill -{{\delta }^{\prime }}_{\mathrm{op}}\hfill \\ \hfill -{{\epsilon }^{\prime }}_{\mathrm{ec}}\hfill & \hfill -{{\epsilon }^{\prime }}_{\mathrm{gdp}}\hfill & \hfill {{\epsilon }^{\prime }}_{\mathrm{cde}}\hfill & \hfill -{{\epsilon }^{\prime }}_{\mathrm{hdi}}\hfill & \hfill {{\epsilon }^{\prime }}_{\mathrm{op}}\hfill \\ \hfill {{\varphi }^{\prime }}_{\mathrm{ec}}\hfill & \hfill {{\varphi }^{\prime }}_{\mathrm{gdp}}\hfill & \hfill {{\varphi }^{\prime }}_{\mathrm{cde}}\hfill & \hfill -{{\varphi }^{\prime }}_{\mathrm{hdi}}\hfill & \hfill -{{\varphi }^{\prime }}_{\mathrm{op}}\hfill \end{array}\right],B=\left[\begin{array}{c}\hfill {g^{\prime }}_{\mathrm{ec}}\hfill \\ \hfill -{g^{\prime }}_{\mathrm{gdp}}\hfill \\ \hfill -{g^{\prime }}_{\mathrm{cde}}\hfill \\ \hfill -{g^{\prime }}_{\mathrm{hdi}}\hfill \\ \hfill -{g^{\prime }}_{\mathrm{op}}\hfill \end{array}\right],\mathrm{and}U=\left[\mathrm{1}\right].

The exponential matrix, {\mathrm{e}}^{A\tau }, is computed in Eq. (21) as

(21){\mathrm{e}}^{A\tau }=\mathrm{E}\mathrm{X}\mathrm{P}\left[A\tau \right]=I+A\tau +{\displaystyle \frac{A^{\mathrm{2}}{\tau }^{\mathrm{2}}}{\mathrm{2}!}}+{\displaystyle \frac{A^{\mathrm{3}}{\tau }^{\mathrm{3}}}{\mathrm{3}!}}+{\displaystyle \frac{A^{\mathrm{4}}{\tau }^{\mathrm{4}}}{\mathrm{4}!}}+{\displaystyle \frac{A^{\mathrm{5}}{\tau }^{\mathrm{5}}}{\mathrm{5}!}}+\cdots ={\displaystyle \sum _{k=\mathrm{0}}^{\infty }{\displaystyle \frac{A^k{\tau }^k}{k!}},}

where I is an identity matrix.

Translating normalized energy indices and time to their corresponding dimensional data, the simulated dimensional energy indices are defined in Eq. (22) as

(22)\left[\begin{array}{c}\hfill {\mathrm{EC}}_{t,\mathrm{simulated}}\hfill \\ \hfill {\mathrm{GDP}}_{t,\mathrm{simulated}}\hfill \\ \hfill {\mathrm{CDE}}_{t,\mathrm{simulated}}\hfill \\ \hfill {\mathrm{HDI}}_{t,\mathrm{simulated}}\hfill \\ \hfill {\mathrm{OP}}_{t,\mathrm{simulated}}\hfill \end{array}\right]=\left[\begin{array}{c}\hfill {\mathrm{ec}}_{\tau ,\mathrm{simulated}}\left({\mathrm{EC}}_{\min }-{\mathrm{EC}}_{\max }\right)+{\mathrm{EC}}_{\max }\hfill \\ \hfill {\mathrm{gdp}}_{\tau ,\mathrm{simulated}}\left({\mathrm{GDP}}_{\min }-{\mathrm{GDP}}_{\max }\right)+{\mathrm{GDP}}_{\max }\hfill \\ \hfill {\mathrm{cde}}_{\tau ,\mathrm{simulated}}\left({\mathrm{CDE}}_{\min }-{\mathrm{CDE}}_{\max }\right)+{\mathrm{CDE}}_{\max }\hfill \\ \hfill {\mathrm{hdi}}_{\tau ,\mathrm{simulated}}\left({\mathrm{HDI}}_{\min }-{\mathrm{HDI}}_{\max }\right)+{\mathrm{HDI}}_{\max }\hfill \\ \hfill {\mathrm{op}}_{\tau ,\mathrm{simulated}}\left({\mathrm{OP}}_{\min }-{\mathrm{OP}}_{\max }\right)+{\mathrm{OP}}_{\max }\hfill \end{array}\right].

The deflated dimensional indices are suggested as the product of randomized predictive factors (within the confines of maximum and minimum historic rate of indices as supported by Mojie et al. [7] and Zeng et al. [14] and the nth-simulated indices in Eq. (23)).

(23)\begin{array}{c}\hfill {\left[\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathrm{EC}}_{n_{\mathrm{year}}+k}\hfill \end{array}\hfill \\ \hfill \begin{array}{c}\hfill {\mathrm{GDP}}_{n_{\mathrm{year}}+k}\hfill \end{array}\hfill \\ \hfill \begin{array}{c}\hfill {\mathrm{CDE}}_{n_{\mathrm{year}}+k}\hfill \end{array}\hfill \\ \hfill \begin{array}{c}\hfill {\mathrm{HDI}}_{n_{\mathrm{year}}+k}\hfill \end{array}\hfill \\ \hfill {\mathrm{OP}}_{n_{\mathrm{year}}+k}\hfill \end{array}\right]}_{\mathrm{predicted}}=\left[\begin{array}{c}\hfill {\mathrm{EC}}_{n_{\mathrm{y}\mathrm{e}\mathrm{a}\mathrm{r}}}{\left(\mathrm{1}-\left(\mathrm{rand}()*\left({\left|{\displaystyle \frac{{\mathrm{EC}}_t-{\mathrm{EC}}_{t-\mathrm{1}}}{{\mathrm{EC}}_{t-\mathrm{1}}}}\right|}_{\mathrm{upper}}-{\left|{\displaystyle \frac{{\mathrm{EC}}_t-{\mathrm{EC}}_{t-\mathrm{1}}}{{\mathrm{EC}}_{t-\mathrm{1}}}}\right|}_{\mathrm{lower}}\right)+{\left|{\displaystyle \frac{{\mathrm{EC}}_t-{\mathrm{EC}}_{t-\mathrm{1}}}{{\mathrm{EC}}_{t-\mathrm{1}}}}\right|}_{\mathrm{lower}}\right)\right)}^k\hfill \\ \hfill {\mathrm{GDP}}_{n_{\mathrm{year}}}{\left(1-\left(\mathrm{rand()*}\left({\left|{\displaystyle \frac{{\mathrm{GDP}}_t-{\mathrm{GDP}}_{t\mathrm{-1}}}{{\mathrm{GDP}}_{t\mathrm{-1}}}}\right|}_{\mathrm{upper}}-{\left|{\displaystyle \frac{{\mathrm{GDP}}_t{-\mathrm{GDP}}_{t\mathrm{-1}}}{{\mathrm{GDP}}_{t\mathrm{-1}}}}\right|}_{\mathrm{lower}}\right)+{\left|{\displaystyle \frac{{\mathrm{GDP}}_t{-\mathrm{GDP}}_{t\mathrm{-1}}}{{\mathrm{GDP}}_{t\mathrm{-1}}}}\right|}_{\mathrm{lower}}\right)\right)}^k\hfill \\ \hfill {\mathrm{CDE}}_{n_{\mathrm{year}}}{\left(\mathrm{1}-\left(\mathrm{rand()*}\left({\left|{\displaystyle \frac{{\mathrm{CDE}}_t-{\mathrm{CDE}}_{t\mathrm{-1}}}{{\mathrm{CDE}}_{t\mathrm{-1}}}}\right|}_{\mathrm{upper}}-{\left|{\displaystyle \frac{{\mathrm{CDE}}_t-{\mathrm{CDE}}_{t\mathrm{-1}}}{{\mathrm{CDE}}_{t\mathrm{-1}}}}\right|}_{\mathrm{lower}}\right)+{\left|{\displaystyle \frac{{\mathrm{CDE}}_t{-\mathrm{CDE}}_{t\mathrm{-1}}}{{\mathrm{CDE}}_{t\mathrm{-1}}}}\right|}_{\mathrm{lower}}\right)\right)}^k\hfill \\ \hfill {\mathrm{HDI}}_{n_{\mathrm{year}}}{\left(1-\left(\mathrm{rand()*}\left({\left|{\displaystyle \frac{{\mathrm{HDI}}_t{-\mathrm{HDI}}_{t\mathrm{-1}}}{{\mathrm{HDI}}_{t\mathrm{-1}}}}\right|}_{\mathrm{upper}}-{\left|{\displaystyle \frac{{\mathrm{HDI}}_t{-\mathrm{HDI}}_{t\mathrm{-1}}}{{\mathrm{HDI}}_{t\mathrm{-1}}}}\right|}_{\mathrm{lower}}\right)+{\left|{\displaystyle \frac{{\mathrm{HDI}}_t{-\mathrm{HDI}}_{t\mathrm{-1}}}{{\mathrm{HDI}}_{t\mathrm{-1}}}}\right|}_{\mathrm{lower}}\right)\right)}^k\hfill \\ \hfill {\mathrm{OP}}_{n_{\mathrm{year}}}{\left(\mathrm{1}-\left(\mathrm{rand}()*\left({\left|{\displaystyle \frac{{\mathrm{OP}}_t-{\mathrm{OP}}_{t-\mathrm{1}}}{{\mathrm{OP}}_{t-\mathrm{1}}}}\right|}_{\mathrm{upper}}-{\left|{\displaystyle \frac{{\mathrm{OP}}_t-{\mathrm{OP}}_{t-\mathrm{1}}}{{\mathrm{OP}}_{t-\mathrm{1}}}}\right|}_{\mathrm{lower}}\right)+{\left|{\displaystyle \frac{{\mathrm{OP}}_t-{\mathrm{OP}}_{t-\mathrm{1}}}{{\mathrm{OP}}_{t-\mathrm{1}}}}\right|}_{\mathrm{lower}}\right)\right)}^k\hfill \end{array}\right],\hfill \\ k=\mathrm{1},\hspace{0.17em}\mathrm{2},\hspace{0.17em}...,\hspace{0.17em}{n}_{\mathrm{year}\hspace{0.17em}}\mathrm{;}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}t=\mathrm{1},\hspace{0.17em}\mathrm{2},\hspace{0.17em}...,\hspace{0.17em}{n}_{\mathrm{year}},\hspace{0.17em}\hfill \end{array}

where k is the number of years ahead of the last historic indices and rand ( ) is a random number between the lower and upper historic rate of the indices, which introduces the probability or randomness in the predictive factors in Eqs. (25)–(26), which designate the deflection and inflection of the predictive factors, respectively.

The inflated dimensional indices are proposed as the product of randomized predictive factors (bounded between maximum and minimum historic rate of indices as employed by Mojie et al. [7] and Zeng et al. [14] and the nth-simulated indices in Eq. (24)).

(24)\begin{array}{c}\hfill {\left[\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathrm{EC}}_{n_{\mathrm{year}}+k}\hfill \end{array}\hfill \\ \hfill \begin{array}{c}\hfill {\mathrm{GDP}}_{n_{\mathrm{year}}+k}\hfill \end{array}\hfill \\ \hfill \begin{array}{c}\hfill {\mathrm{CDE}}_{n_{\mathrm{year}}+k}\hfill \end{array}\hfill \\ \hfill \begin{array}{c}\hfill {\mathrm{HDI}}_{n_{\mathrm{year}}+k}\hfill \end{array}\hfill \\ \hfill {\mathrm{OP}}_{n_{\mathrm{year}}+k}\hfill \end{array}\right]}_{\mathrm{predicted}}\hfill \end{array}\begin{array}{c}\hfill ={\left[\begin{array}{c}\hfill {\mathrm{EC}}_{n_{\mathrm{year}}}{\left(\mathrm{1}+\left(\mathrm{rand}()*\left({\left|{\displaystyle \frac{{\mathrm{EC}}_t-{\mathrm{EC}}_{t-\mathrm{1}}}{{\mathrm{EC}}_{t-\mathrm{1}}}}\right|}_{\mathrm{upper}}-{\left|{\displaystyle \frac{\mathrm{E}{\mathrm{C}}_t-\mathrm{E}{\mathrm{C}}_{t-\mathrm{1}}}{\mathrm{E}{\mathrm{C}}_{t-\mathrm{1}}}}\right|}_{\mathrm{lower}}\right)+{\left|{\displaystyle \frac{{\mathrm{EC}}_t-\mathrm{E}{\mathrm{C}}_{t-\mathrm{1}}}{\mathrm{E}{\mathrm{C}}_{t-\mathrm{1}}}}\right|}_{\mathrm{lower}}\right)\right)}^k\hfill \\ \hfill {\mathrm{GDP}}_{n_{\mathrm{year}}}{\left(\mathrm{1}+\left(\mathrm{rand}()*\left({\left|{\displaystyle \frac{{\mathrm{GDP}}_t-{\mathrm{GDP}}_{t-\mathrm{1}}}{{\mathrm{GDP}}_{t-\mathrm{1}}}}\right|}_{\mathrm{upper}}-{\left|{\displaystyle \frac{{\mathrm{GDP}}_t-{\mathrm{GDP}}_{t-\mathrm{1}}}{{\mathrm{GDP}}_{t-\mathrm{1}}}}\right|}_{\mathrm{lower}}\right)+{\left|{\displaystyle \frac{{\mathrm{GDP}}_t-{\mathrm{GDP}}_{t-\mathrm{1}}}{{\mathrm{GDP}}_{t-\mathrm{1}}}}\right|}_{\mathrm{lower}}\right)\right)}^k\hfill \\ \hfill {\mathrm{CDE}}_{n_{\mathrm{year}}}{\left(\mathrm{1}+\left(\mathrm{rand}()*\left({\left|{\displaystyle \frac{{\mathrm{CDE}}_t-{\mathrm{CDE}}_{t-\mathrm{1}}}{{\mathrm{CDE}}_{t-\mathrm{1}}}}\right|}_{\mathrm{upper}}-{\left|{\displaystyle \frac{{\mathrm{CDE}}_t-{\mathrm{CDE}}_{t-\mathrm{1}}}{{\mathrm{CDE}}_{t-\mathrm{1}}}}\right|}_{\mathrm{lower}}\right)+{\left|{\displaystyle \frac{\mathrm{C}\mathrm{D}{\mathrm{E}}_t-\mathrm{C}\mathrm{D}{\mathrm{E}}_{t-1}}{\mathrm{C}\mathrm{D}{\mathrm{E}}_{t-1}}}\right|}_{\mathrm{lower}}\right)\right)}^k\hfill \\ \hfill {\mathrm{HDI}}_{n_{\mathrm{year}}}{\left(\mathrm{1}+\left(\mathrm{rand}()*\left({\left|{\displaystyle \frac{{\mathrm{HDI}}_t-{\mathrm{HDI}}_{t-\mathrm{1}}}{{\mathrm{HDI}}_{t-\mathrm{1}}}}\right|}_{\mathrm{upper}}-{\left|{\displaystyle \frac{{\mathrm{HDI}}_t-{\mathrm{HDI}}_{t-\mathrm{1}}}{{\mathrm{HDI}}_{t-\mathrm{1}}}}\right|}_{\mathrm{lower}}\right)+{\left|{\displaystyle \frac{{\mathrm{HDI}}_t-{\mathrm{HDI}}_{t-\mathrm{1}}}{{\mathrm{HDI}}_{t-\mathrm{1}}}}\right|}_{\mathrm{lower}}\right)\right)}^k\hfill \\ \hfill {\mathrm{OP}}_{n_{\mathrm{year}}}{\left(\mathrm{1}+\left(\mathrm{rand}()*\left({\left|{\displaystyle \frac{{\mathrm{OP}}_t-{\mathrm{OP}}_{t-\mathrm{1}}}{{\mathrm{OP}}_{t-\mathrm{1}}}}\right|}_{\mathrm{upper}}-{\left|{\displaystyle \frac{{\mathrm{OP}}_t-{\mathrm{OP}}_{t-\mathrm{1}}}{{\mathrm{OP}}_{t-\mathrm{1}}}}\right|}_{\mathrm{lower}}\right)+{\left|{\displaystyle \frac{{\mathrm{OP}}_t-{\mathrm{OP}}_{t-\mathrm{1}}}{{\mathrm{OP}}_{t-\mathrm{1}}}}\right|}_{\mathrm{lower}}\right)\right)}^k\hfill \end{array}\right]}_{\mathrm{simulated}},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hfill \\ \hfill \\ k=\mathrm{1},\hspace{0.17em}\mathrm{2},\hspace{0.17em}...,\hspace{0.17em}{n}_{\mathrm{year}\hspace{0.17em}}\mathrm{;}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}t=\mathrm{1},\hspace{0.17em}\mathrm{2},\hspace{0.17em}...,\hspace{0.17em}{n}_{\mathrm{year}}.\hspace{0.17em}\hfill \end{array}

Systematically, the predicted indices will be based on the mean value of the inflated and deflated indices.

Economically, the deflated prediction factors are defined in Eq. (25) as

(25)\begin{array}{c}\hfill \begin{array}{c}\hfill {\left(\mathrm{1}-\left(\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}()*\left({\left|{\displaystyle \frac{\mathrm{E}{\mathrm{C}}_t-\mathrm{E}{\mathrm{C}}_{t-\mathrm{1}}}{\mathrm{E}{\mathrm{C}}_{t-\mathrm{1}}}}\right|}_{\mathrm{u}\mathrm{p}\mathrm{p}\mathrm{e}\mathrm{r}}-{\left|{\displaystyle \frac{\mathrm{E}{\mathrm{C}}_t-\mathrm{E}{\mathrm{C}}_{t-\mathrm{1}}}{\mathrm{E}{\mathrm{C}}_{t-\mathrm{1}}}}\right|}_{\mathrm{l}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r}}\right)+{\left|{\displaystyle \frac{\mathrm{E}{\mathrm{C}}_t-\mathrm{E}{\mathrm{C}}_{t-\mathrm{1}}}{\mathrm{E}{\mathrm{C}}_{t-\mathrm{1}}}}\right|}_{\mathrm{l}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r}}\right)\right)}^k,\hfill \\ \hfill {\left(\mathrm{1}-\left(\mathrm{rand}()*\left({\left|{\displaystyle \frac{{\mathrm{GDP}}_t-{\mathrm{GDP}}_{t-\mathrm{1}}}{{\mathrm{GDP}}_{t-\mathrm{1}}}}\right|}_{\mathrm{upper}}-{\left|{\displaystyle \frac{{\mathrm{GDP}}_t-{\mathrm{GDP}}_{t-\mathrm{1}}}{{\mathrm{GDP}}_{t-\mathrm{1}}}}\right|}_{\mathrm{lower}}\right)+{\left|{\displaystyle \frac{{\mathrm{GDP}}_t{\mathrm{-GDP}}_{\mathrm{t-1}}}{{\mathrm{GDP}}_{\mathrm{t-1}}}}\right|}_{\mathrm{lower}}\right)\right)}^k,\hfill \\ \hfill {\left(\mathrm{1}-\left(\mathrm{rand}()*\left({\left|{\displaystyle \frac{{\mathrm{CDE}}_t-{\mathrm{CDE}}_{t-\mathrm{1}}}{{\mathrm{CDE}}_{t-\mathrm{1}}}}\right|}_{\mathrm{upper}}-{\left|{\displaystyle \frac{{\mathrm{CDE}}_t-{\mathrm{CDE}}_{t-\mathrm{1}}}{{\mathrm{CDE}}_{t-\mathrm{1}}}}\right|}_{\mathrm{lower}}\right)+{\left|{\displaystyle \frac{{\mathrm{CDE}}_t-{\mathrm{CDE}}_{t-\mathrm{1}}}{{\mathrm{CDE}}_{t-\mathrm{1}}}}\right|}_{\mathrm{lower}}\right)\right)}^k,\hfill \\ \hfill {\left(\mathrm{1}-\left(\mathrm{rand}()*\left({\left|{\displaystyle \frac{{\mathrm{HDI}}_t-{\mathrm{HDI}}_{t-\mathrm{1}}}{{\mathrm{HDI}}_{t-\mathrm{1}}}}\right|}_{\mathrm{upper}}-{\left|{\displaystyle \frac{{\mathrm{HDI}}_t-{\mathrm{HDI}}_{t-\mathrm{1}}}{{\mathrm{HDI}}_{t-\mathrm{1}}}}\right|}_{\mathrm{lower}}\right)+{\left|{\displaystyle \frac{{\mathrm{HDI}}_t-{\mathrm{HDI}}_{t-\mathrm{1}}}{{\mathrm{HDI}}_{t-\mathrm{1}}}}\right|}_{\mathrm{lower}}\right)\right)}^k,\hfill \\ \hfill {\left(\mathrm{1}-\left(\mathrm{rand}()*\left({\left|{\displaystyle \frac{{\mathrm{OP}}_t-{\mathrm{OP}}_{t-\mathrm{1}}}{{\mathrm{OP}}_{t-\mathrm{1}}}}\right|}_{\mathrm{upper}}-{\left|{\displaystyle \frac{{\mathrm{OP}}_t-{\mathrm{OP}}_{t-\mathrm{1}}}{{\mathrm{OP}}_{t-\mathrm{1}}}}\right|}_{\mathrm{lower}}\right)+{\left|{\displaystyle \frac{{\mathrm{OP}}_t-{\mathrm{OP}}_{t-\mathrm{1}}}{{\mathrm{OP}}_{t-\mathrm{1}}}}\right|}_{\mathrm{lower}}\right)\right)}^k.\hfill \end{array}\hfill \\ k=\mathrm{1},\hspace{0.17em}\mathrm{2},\hspace{0.17em}...,\hspace{0.17em}{n}_{\mathrm{year}\hspace{0.17em}}\mathrm{;}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}t=\mathrm{1},\hspace{0.17em}\mathrm{2},\hspace{0.17em}...,\hspace{0.17em}{n}_{\mathrm{year}}\hspace{0.17em}.\hfill \end{array}

Similarly, the inflated prediction factors are presented in Eq. (26) as

(26)\begin{array}{c}\hfill \begin{array}{c}\hfill {\left(\mathrm{1}+\left(\mathrm{rand}()*\left({\left|{\displaystyle \frac{{\mathrm{EC}}_t-{\mathrm{EC}}_{t-\mathrm{1}}}{{\mathrm{EC}}_{t-\mathrm{1}}}}\right|}_{\mathrm{upper}}-{\left|{\displaystyle \frac{{\mathrm{EC}}_t-{\mathrm{EC}}_{t-\mathrm{1}}}{{\mathrm{EC}}_{t-\mathrm{1}}}}\right|}_{\mathrm{lower}}\right)+{\left|{\displaystyle \frac{{\mathrm{EC}}_t-{\mathrm{EC}}_{t-\mathrm{1}}}{{\mathrm{EC}}_{t-\mathrm{1}}}}\right|}_{\mathrm{lower}}\right)\right)}^k,\hfill \\ \hfill {\left(\mathrm{1}+\left(\mathrm{rand}()*\left({\left|{\displaystyle \frac{{\mathrm{GDP}}_t-{\mathrm{GDP}}_{t-\mathrm{1}}}{{\mathrm{GDP}}_{t-\mathrm{1}}}}\right|}_{\mathrm{upper}}-{\left|{\displaystyle \frac{{\mathrm{GDP}}_t-{\mathrm{GDP}}_{t-\mathrm{1}}}{{\mathrm{GDP}}_{t-\mathrm{1}}}}\right|}_{\mathrm{lower}}\right)+{\left|{\displaystyle \frac{{\mathrm{GDP}}_t-{\mathrm{GDP}}_{t-\mathrm{1}}}{{\mathrm{GDP}}_{t-\mathrm{1}}}}\right|}_{\mathrm{lower}}\right)\right)}^k,\hfill \\ \hfill {\left(\mathrm{1}+\left(\mathrm{rand}()*\left({\left|{\displaystyle \frac{{\mathrm{CDE}}_t-{\mathrm{CDE}}_{t-\mathrm{1}}}{{\mathrm{CDE}}_{t-\mathrm{1}}}}\right|}_{\mathrm{upper}}-{\left|{\displaystyle \frac{{\mathrm{CDE}}_t-{\mathrm{CDE}}_{t-\mathrm{1}}}{{\mathrm{CDE}}_{t-\mathrm{1}}}}\right|}_{\mathrm{lower}}\right)+{\left|{\displaystyle \frac{{\mathrm{CDE}}_t-{\mathrm{CDE}}_{t-\mathrm{1}}}{{\mathrm{CDE}}_{t-\mathrm{1}}}}\right|}_{\mathrm{lower}}\right)\right)}^k,\hfill \\ \hfill {\left(\mathrm{1}+\left(\mathrm{rand}()*\left({\left|{\displaystyle \frac{{\mathrm{HDI}}_t-{\mathrm{HDI}}_{t-\mathrm{1}}}{{\mathrm{HDI}}_{t-\mathrm{1}}}}\right|}_{\mathrm{upper}}-{\left|{\displaystyle \frac{{\mathrm{HDI}}_t-{\mathrm{HDI}}_{t-\mathrm{1}}}{{\mathrm{HDI}}_{t-\mathrm{1}}}}\right|}_{\mathrm{lower}}\right)+{\left|{\displaystyle \frac{{\mathrm{HDI}}_t-{\mathrm{HDI}}_{t-\mathrm{1}}}{{\mathrm{HDI}}_{t-\mathrm{1}}}}\right|}_{\mathrm{lower}}\right)\right)}^k,\hfill \\ \hfill {\left(\mathrm{1}+\left(\mathrm{rand}()*\left({\left|{\displaystyle \frac{{\mathrm{OP}}_t-{\mathrm{OP}}_{t-\mathrm{1}}}{{\mathrm{OP}}_{t-\mathrm{1}}}}\right|}_{\mathrm{upper}}-{\left|{\displaystyle \frac{{\mathrm{OP}}_t-{\mathrm{OP}}_{t-\mathrm{1}}}{{\mathrm{OP}}_{t-\mathrm{1}}}}\right|}_{\mathrm{lower}}\right)+{\left|{\displaystyle \frac{{\mathrm{OP}}_t-{\mathrm{OP}}_{t-\mathrm{1}}}{{\mathrm{OP}}_{t-\mathrm{1}}}}\right|}_{\mathrm{lower}}\right)\right)}^k,\hfill \end{array}\hfill \\ k=\mathrm{1},\hspace{0.17em}\mathrm{2},\hspace{0.17em}...,\hspace{0.17em}{n}_{\mathrm{year}\hspace{0.17em}}\mathrm{;}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}t=\mathrm{1},\hspace{0.17em}\mathrm{2},\hspace{0.17em}...,\hspace{0.17em}{n}_{\mathrm{year}}.\hfill \end{array}

Subsequently, R_i is the predicted rates, where i= {1, 2, 3, 4, 5}≡ {EC, GDP, CDE, HDI, OP}, are defined in Eq. (27) as

(27)

2.3 Input data

The five major energy indicators selected for this work are EC, GDP, CDE, HDI, and OP in order to achieve the objectives of the present study. The data were retrieved from Birol [49], Amadeo [50], Roser [51], and jointly from Amadeo [50], McMahon [52], and Chair [53], respectively.

Notably, EC represents utility, GDP primarily denotes economic development, CDE indicates environmental development, HDI designates social development, and OP portrays economic crises or changes in oil price due to inflation and oil slumps. The raw data are represented in Fig. 1.

3 Results and discussion

3.1 Results

The numerical results are inclusively presented in Tables 1–6. Table 1 gives the differential of normalized indices with respect to normalized time (t). Table 2 presents the coefficients of the normalized system of linear Eqs. (6)–(10). Table 3 shows the constant terms in the normalized system of linear Eqs. (6)–(10).

Subsequently, the regression of rate functions (Table 1) with energy indices yielded the historic elastic constants in the models in Eqs. (6)–(10), and (20), and the unexplained terms in Table 3. Pertinently, the initial conditions (Table 4) were based on the reference of normalized energy indexes. The information contained in Tables 1–4 are combined in the Matlab environment to obtain the numerical solution to the mathematical models in Eq. (19).

Table 4 contains the initial conditions of normalized indices in Eqs. (6)–(10). Table 5 holds the simulated or interpolated results of dimensional energy indices. Table 6 presents the predicted rates (possible and effective) for the prediction of energy indices.

Graphically, more results are displayed in Figs. 2–11. Figure 2 describes the normalized history of energy indexes, Fig. 3 explains the rate of change of normalized energy indices, Fig. 4 portrays the simulation of normalized energy indices, Fig. 5 shows the global energy indices, Fig. 6 depicts the relative error in the simulation or interpolative prediction of energy indices, and Figs. 7–11 elucidate the innovative (or extrapolative) prediction of energy indices.

3.2 Discussion

Essentially, Fig. 2 yielded a historic information on the normalized energy indexes (quadratic function) upon which the time derivatives (or rates) were developed in Fig. 3, respectively. Considering the reciprocity between the dimensioned and normalized indexes, the normalized energy indices showed a decrease with normalized time, whereas dimensional energy indices progressed by normalized time. Thus, the former trend could be attributed to the normalization functions in Eqs. (11)–(15) whose stabilization term is based on the maximum indices which constrain the normalized indexes between 0 and 1, and equally maintains a maximum stability within the energy indices.

Differentially, Fig. 2 gave rise to Fig. 3, the normalized rates portray that the energy indices are time trending (or stationary), since their plots were straight lines, meaning that the indexes were less susceptible to fluctuation (or instability) in the normalized domain or exhibited a quality trend. Econometrically, this supports the fact that the normalized energy index data are quite stationary without embarking on the unit root tests [14]. In addition, based on the observed reciprocity, the positive slopes in Fig. 3 indicated that the dimensional counterparts produced a negative rate, whereas the negative slopes in Fig. 3 transformatively (or dimensionally) gave rise to a positive rate. By implication, the positive rate meant that the presence of the other indexes in the system of differential equations was catalytic in nature, whereas the negative rate impliy that the presence of the other indices is inhibiting.

Superficially, the presence of positive and negative rates is an evidence that there is an interaction between energy indices. However, the nexus of causalities between energy indices could be fathomed by conducting Granger tests, which is outside the scope of this work.

The visual display of the model solution is presented in Fig. 4, which is the amenable solution to the system of linear differential equations, functionally characterized by exponential or power (Taylor) series. Evidently, there was a good agreement between the field data and simulated results. Since all the data lie between zero and one for all normalized indices, substantiates the fact that there was fairness (or objectivity) in the analysis of the results.

Obviously, Fig. 4 was transformed (or translated) into its dimensional equivalence in Fig. 5 using Eqs. (11)–(15) or Eq. (22). The agreement between the dimensional field data and simulated results in the real time domain is stronger than those of the former (in the normalized time domain), due to the overriding effect of stabilization terms (the maximum indexes). Moreover, the fitness and agreement between the field data and simulated results is vividly portrayed in Fig. 6, the relative error curves for the energy indices, which qualitatively describe the prevailing agreement. Furthermore, the information in Fig. 6 is translated into absolute error in order to quantify the degree of fitness between energy indices. Notably, the indexes EC, GDP, CDE, HDI, and OP recorded an absolute error of 0.018934881, 0.104790607, 0.057333145, 0.011561588, and 0.414148282 (-), respectively. Deductively, EC, GDP, CDE, HDI EC, GDP, CDE, and HDI showed a strong agreement between the field data and simulated results. Exceptionally, OP recorded the highest absolute error in 2008–2017. Prior to 2008, there was a good agreement between the field data and simulated results in Figs. 5 and 6, since the absolute error (0.250591407) was much less than that of the historic data (1982–2017; 0.414148282).

Thus, the untrendy instability associated with OP lies between the crisis period of 2008–2017. Essentially, the simulated results in Fig. 5 and Table 5 is germane for interpolative prediction of the energy indices.

Therefore, the innovative (extrapolative) models were presented in Eqs. (22) and (23), whose results are featured in Figs. 7–11 for EC, GDP, CDE, HDI, and OP, respectively.

Historically, the normalized change in energy indices (the inflating/deflating factors) was computed in Eqs. (25) and (26), respectively. The upper (maximum) and lower (minimum) values were computed for the purposes of achieving extrapolative predictions of energy indexes. The feasible probability factors were based on a randomly restricted or regulated effective prediction rates as presented in Table 6. Remarkably, the probability factors built and dropped led to the inflation and deflation of energy indices and the arithmetic mean of the former and latter energy indices yielded extrapolative prediction values. A helicopter view on Figs. 7–11 shows that all the energy indices have a tendency to rise and is likely to remain trending with the exception of OP which appears to be untrended sequel to Organization of Petroleum Exporting Countries (OPEC) major influence on the oil price [55]. Thus, the incessant fluctuations in OP could be attributed to several factors like drop in production, slump in supply, high demand, inimical regional and national policies, vandalism (sabotage), political bureaucracy, sanctions, etc. Amidst all these odds, the futuristic control and regulation of oil price becomes difficult, leading to an untrendy instability.

Both Fig. (8) and (9) have a positive trend in the future, which concur with the findings of Bianco et al. [2] who attributed the increase in carbon emission to the increase in the economic growth, despite the fact that they applied decomposition model in his methodology. However, there is a good agreement between their result and that of the innovative model employed in the current wok.

Thus, the upcoming years will witness a tremendous increase in the GDP which will inevitably raise the amount of carbon emission and equally increase EC in the future [2,21,24,28]. Thus, appropriate legislation or regulatory policies [7,27,32] have to be enacted globally in order to checkmate the exclusive emission of carbon in the years of 2040 to 2050.

4 Conclusions

This work differently modeled, simulated, and non-deferentially predicted energy indices. The introduction of differential terms (rate of change of indices) made it possible to determine the trendiness and overall influence of independent energy indices on the dependent energy index, whether catalytic (having positive trend) or inhibitive (having negative trend) in nature.

Virtually, the model solutions or simulated results were in strong agreement with the field data, since minimal absolute errors (0.018934881, 0.104790607, 0.057333145, 0.011561588 and 0.414148282) were recorded for EC, GDP, CDE, HDI, and OP, respectively. Certainly, the simulated results could be employed in the interpolative prediction of energy indices. However, the simulated results lacked inferential quality and cannot be deployed in the extrapolative prediction of energy indices.

Thus, innovative (or extrapolative) prediction models were developed to cater for the extrapolative prediction of energy indices by introducing the prediction factor, which is controlled by paired prediction rates ({0.025431, 0.034663}, {0.075294, 0.082303}, {0.040113, 0.045870}, {0.002981, 0.030839} and {0.088102, 0.089833}) for EC, GDP, CDE, HDI, and OP, respectively. These factors introduced inferential quality into the simulated results. Moreover, the extrapolative prediction showed that there was an increase in EC, GDP, CDE, and HDI with the assurance of a positive time trend whereas OP was marked with a lot of fluctuations which is the evidence of untrendy instability in oil price. Pertinently, the present work has suggested several factors responsible for instability like drop in production, slump in supply, high demand, inimical regional and national policies, vandalism (sabotage), political bureaucracy, sanctions, etc.

Remarkably, the present work has modeled, simulated, and duly conducted interpolative and extrapolative prediction of energy indices and candidly recommends for the proper investigation untrendy instability observed on both historic and predicted oil prices.

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