# 具有振荡项的指数曲线及其在一次能源消费中的应用An Exponential Curve with Oscillating Term and Its Application in China’s Primary En-ergy Consumption

Abstract: Aiming at modelling and prediction for oscillation sequences which exist widely in the real world, this paper proposes a new type of exponential curve model with oscillation term based on the classical exponential curve model. With the characteristics of the model and the least squares es-timation method, an optimization problem is developed to evaluate the system parameters. Based on this problem, the Matlab software is used to solve the optimization problem to obtain the system parameters. Further, the model is applied to forecast the China’s primary energy consumption. The computational results are compared with the classical exponential curve and the modified exponential curve, and it shows the oscillating exponential curve has higher accuracy in China’s primary energy consumption.

1. 引言

2. 指数曲线和修正的指数曲线

2.1. 指数曲线

${Y}_{t}=a{b}^{t}$(1)

$\mathrm{ln}{Y}_{t}=\mathrm{ln}a+t\mathrm{ln}b$(2)

$\left\{\begin{array}{l}\sum \mathrm{ln}Y=n\mathrm{ln}a+\left(\sum t\right)\mathrm{ln}b\\ \sum t\mathrm{ln}Y=\left(\sum t\right)\mathrm{ln}a+\left(\sum {t}^{2}\right)\mathrm{ln}b\end{array}$(3)

2.2. 修正指数曲线

${Y}_{t}=a{b}^{t}+K$(4)

${S}_{1}=\underset{t=0}{\overset{m-1}{\sum }}{Y}_{t}$${S}_{2}=\underset{t=m}{\overset{2m-1}{\sum }}{Y}_{t}$${S}_{2}=\underset{t=2m}{\overset{3m-1}{\sum }}{Y}_{t}$(5)

$\left\{\begin{array}{l}{S}_{1}=mK+a+ab+a{b}^{2}+\cdots +a{b}^{m-1}\\ {S}_{2}=mK+a{b}^{m}+a{b}^{m+1}+\cdots +a{b}^{2m-1}\\ {S}_{3}=mK+a{b}^{2m}+a{b}^{2m+1}+\cdots +a{b}^{3m-1}\end{array}$(6)

$\left\{\begin{array}{l}b={\left(\frac{{S}_{3}-{S}_{2}}{{S}_{2}-{S}_{1}}\right)}^{\frac{1}{m}}\\ a=\left({S}_{2}-{S}_{1}\right)\frac{b-1}{{\left({b}^{m}-1\right)}^{2}}\\ K=\frac{1}{m}\left({S}_{1}-\frac{a\left({b}^{m}-1\right)}{b-1}\right)\end{array}$(7)

3. 振荡型的指数曲线

${Y}_{t}=a{b}^{t}+ct+d+r\mathrm{sin}\left(t\right)$(8)

$\left\{\begin{array}{c}{Y}_{1}=\stackrel{^}{a}\stackrel{^}{b}+\stackrel{^}{c}+\stackrel{^}{d}+\stackrel{^}{r}\mathrm{sin}\left(1\right),\text{\hspace{0.17em}}\text{ }\\ {Y}_{2}=\stackrel{^}{a}{\stackrel{^}{b}}^{2}+2\stackrel{^}{c}+\stackrel{^}{d}+\stackrel{^}{r}\mathrm{sin}\left(2\right),\text{\hspace{0.17em}}\\ ⋮\\ {Y}_{m}=\stackrel{^}{a}{\stackrel{^}{b}}^{m}+m\stackrel{^}{c}+\stackrel{^}{d}+\stackrel{^}{r}\mathrm{sin}\left(m\right).\end{array}$ (9)

$S=\underset{a,b,c,d,r}{\mathrm{min}}\underset{i=1}{\overset{m}{\sum }}{\left\{x\left(1\right)-\left(a{b}^{i}+ci+d+r\mathrm{sin}\left(i\right)\right)\right\}}^{2}$(10)

$\left\{\begin{array}{c}\frac{\partial S}{\partial a}=\underset{i=1}{\overset{m}{\sum }}\left\{x\left(1\right)-\left(a{b}^{i}+ci+d+r\mathrm{sin}\left(i\right)\right)\right\}{b}^{i}=0,\text{ }\text{\hspace{0.17em}}\\ \frac{\partial S}{\partial b}=\underset{i=1}{\overset{m}{\sum }}\left\{x\left(1\right)-\left(a{b}^{i}+ci+d+r\mathrm{sin}\left(i\right)\right)\right\}ai{b}^{i-1}=0,\\ \frac{\partial S}{\partial c}=\underset{i=1}{\overset{m}{\sum }}\left\{x\left(1\right)-\left(a{b}^{i}+ci+d+r\mathrm{sin}\left(i\right)\right)\right\}i=0,\text{ }\text{ }\\ \begin{array}{l}\frac{\partial S}{\partial d}=\underset{i=1}{\overset{m}{\sum }}\left\{x\left(1\right)-\left(a{b}^{i}+ci+d+r\mathrm{sin}\left(i\right)\right)\right\}=0,\\ \frac{\partial S}{\partial r}=\underset{i=1}{\overset{m}{\sum }}\left\{x\left(1\right)-\left(a{b}^{i}+ci+d+r\mathrm{sin}\left(i\right)\right)\right\}\mathrm{sin}\left(i\right)=0.\end{array}\end{array}$ (11)

4. 应用实例

Table 1. The statistical data of the primary energy consumption of China (mtoe)

$\text{APE}=|\frac{{Y}_{t}-{\stackrel{^}{Y}}_{t}}{{Y}_{t}}|×100%,t=1,2,\cdots ,n$(12)

$\text{MAPE}=\frac{1}{v-l+1}\underset{t=l}{\overset{v}{\sum }}|\frac{{Y}_{t}-{\stackrel{^}{Y}}_{t}}{{Y}_{t}}|×100%,v\le n$(13)

Figure 1. Modelling and forecasting the primary energy consumption by the exponential curve, the modified exponential curve and the oscillation exponential curve models

Table 2. The results of China’s primary energy consumption by the exponential curve, the modified exponential curve and the oscillation exponential curve models

Figure 2. The absolute percentage errors of the primary energy consumption by the exponential curve, the modified exponential curve and the oscillation exponential curve models

Figure 3. The forecasting error and the total error of the primary energy consumption by the exponential curve, the modified exponential curve and the oscillation exponential curve models

5. 结论

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