﻿ 多项式趋势曲线分析及其在白条猪价格指数中的应用 Analysis of Polynomial Trend Curve and Its Application in Chinese Pig Price Index

Pure Mathematics
Vol. 09  No. 08 ( 2019 ), Article ID: 32406 , 8 pages
10.12677/PM.2019.98111

Analysis of Polynomial Trend Curve and Its Application in Chinese Pig Price Index

Ling Lan, Kai Zuo*

School of Mathematics, Chengdu Normal University, Chengdu Sichuan

Received: Sep. 9th, 2019; accepted: Sep. 22nd, 2019; published: Sep. 29th, 2019

ABSTRACT

The polynomial trend curve is a very important model to describe the trend of time series, which is widely used in agricultural and industry, environmental and energy, and other fields. In this paper, the mathematical expressions of polynomial model parameters are given by using local summation method, and the time series analysis theory is used to analyze whether the model residual errors are white noise series to establish the corresponding autoregressive moving average model. On the basis of these discussions and with the help of MATLAB 2013b software and EViews 8.0software, taking the Chinese pig price index as an example, the steps of determining the parameters of the polynomial trend model, establishing the model, solving the model and error analysis are all shown in details.

Keywords:Polynomial Trend Curve, Local Summation Method, White Noise Test, Chinese Pig Price Index

1. 引言

2. 多项式趋势曲线

2.1. 模型参数的确定

$x\left(k\right)={a}_{0}+{a}_{1}k+{a}_{2}{k}^{2}+\cdots +{a}_{p-1}{k}^{p-1}+{a}_{p}{k}^{p}$ (1)

${S}_{i}=\underset{k=\left(i-1\right)m+1}{\overset{im}{\sum }}x\left(k\right),\text{\hspace{0.17em}}i=1,2,\cdots ,p+1$ . (2)

$\left\{\begin{array}{l}{S}_{1}=\underset{k=1}{\overset{m}{\sum }}x\left(k\right)={a}_{0}m+{a}_{1}\underset{k=1}{\overset{m}{\sum }}k+\cdots +{a}_{p-1}\underset{k=1}{\overset{m}{\sum }}{k}^{p-1}+{a}_{p}\underset{k=1}{\overset{m}{\sum }}{k}^{p},\\ {S}_{2}=\underset{k=m+1}{\overset{2m}{\sum }}x\left(k\right)={a}_{0}m+{a}_{1}\underset{k=m+1}{\overset{2m}{\sum }}k+\cdots +{a}_{p-1}\underset{k=m+1}{\overset{2m}{\sum }}{k}^{p-1}+{a}_{p}\underset{k=m+1}{\overset{2m}{\sum }}{k}^{p},\\ \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }⋮\\ {S}_{p+1}=\underset{k=pm+1}{\overset{\left(p+1\right)m}{\sum }}x\left(k\right)={a}_{0}m+{a}_{1}\underset{k=pm+1}{\overset{\left(p+1\right)m}{\sum }}k+\cdots +{a}_{p-1}\underset{k=pm+1}{\overset{\left(p+1\right)m}{\sum }}{k}^{p-1}+{a}_{p}\underset{k=pm+1}{\overset{\left(p+1\right)m}{\sum }}{k}^{p}.\end{array}$ (3)

$\left(\begin{array}{ccccccc}m& \underset{k=1}{\overset{m}{\sum }}k& \underset{k=1}{\overset{m}{\sum }}{k}^{2}& \cdots & \underset{k=1}{\overset{m}{\sum }}{k}^{p-2}& \underset{k=1}{\overset{m}{\sum }}{k}^{p-1}& \underset{k=1}{\overset{m}{\sum }}{k}^{p}\\ m& \underset{k=m+1}{\overset{2m}{\sum }}k& \underset{k=m+1}{\overset{2m}{\sum }}{k}^{2}& \cdots & \underset{k=m+1}{\overset{2m}{\sum }}{k}^{p-2}& \underset{k=m+1}{\overset{2m}{\sum }}{k}^{p-1}& \underset{k=m+1}{\overset{2m}{\sum }}{k}^{p}\\ m& \underset{k=2m+1}{\overset{3m}{\sum }}k& \underset{k=2m+1}{\overset{3m}{\sum }}{k}^{2}& \cdots & \underset{k=2m+1}{\overset{3m}{\sum }}{k}^{p-2}& \underset{k=2m+1}{\overset{3m}{\sum }}{k}^{p-1}& \underset{k=2m+1}{\overset{3m}{\sum }}{k}^{p}\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮\\ m& \underset{k=\left(p-2\right)m+1}{\overset{\left(p-1\right)m}{\sum }}k& \underset{k=\left(p-2\right)m+1}{\overset{\left(p-1\right)m}{\sum }}{k}^{2}& \cdots & \underset{k=\left(p-2\right)m+1}{\overset{\left(p-1\right)m}{\sum }}{k}^{p-2}& \underset{k=\left(p-2\right)m+1}{\overset{\left(p-1\right)m}{\sum }}{k}^{p-1}& \underset{k=\left(p-2\right)m+1}{\overset{\left(p-1\right)m}{\sum }}{k}^{p}\\ m& \underset{k=\left(p-1\right)m+1}{\overset{pm}{\sum }}k& \underset{k=\left(p-1\right)m+1}{\overset{pm}{\sum }}{k}^{2}& \cdots & \underset{k=\left(p-1\right)m+1}{\overset{pm}{\sum }}{k}^{p-2}& \underset{k=\left(p-1\right)m+1}{\overset{pm}{\sum }}{k}^{p-1}& \underset{k=\left(p-1\right)m+1}{\overset{pm}{\sum }}{k}^{p}\\ m& \underset{k=pm+1}{\overset{\left(p+1\right)m}{\sum }}k& \underset{k=pm+1}{\overset{\left(p+1\right)m}{\sum }}{k}^{2}& \cdots & \underset{k=pm+1}{\overset{\left(p+1\right)m}{\sum }}{k}^{p-2}& \underset{k=pm+1}{\overset{\left(p+1\right)m}{\sum }}{k}^{p-1}& \underset{k=pm+1}{\overset{\left(p+1\right)m}{\sum }}{k}^{p}\end{array}\right)\left(\begin{array}{c}{a}_{0}\\ {a}_{1}\\ {a}_{2}\\ ⋮\\ {a}_{p-2}\\ {a}_{p-1}\\ {a}_{p}\end{array}\right)=\left(\begin{array}{c}{S}_{1}\\ {S}_{2}\\ {S}_{3}\\ ⋮\\ {S}_{p-1}\\ {S}_{p}\\ {S}_{p+1}\end{array}\right)$ .(4)

2.2. 计算精度的评估

· 百分比误差(APE)

$\text{APE}\left(k\right)=\frac{x\left(k\right)-\stackrel{^}{x}\left(k\right)}{x\left(k\right)}×100%,k=1,2,\cdots ,h$ . (5)

· 平均绝对百分比误差(MAPE)

$\text{MAPE}=\frac{1}{r-l+1}\underset{k=l}{\overset{r}{\sum }}\frac{|x\left(k\right)-\stackrel{^}{x}\left(k\right)|}{x\left(k\right)}×100%$ , (6)

· 均方根误差(RMSPE)

$\text{RMSPE}=\sqrt{\frac{1}{r-l+1}\underset{k=l}{\overset{r}{\sum }}{\left(\frac{x\left(k\right)-\stackrel{^}{x}\left(k\right)}{x\left(k\right)}\right)}^{2}}×100%$ . (7)

2.3. 残差的白噪声检验及建模

P值法：对于残差序列APE，其样本自相关系数记为 ${\stackrel{^}{\rho }}_{k},k=1,2,\cdots ,r$，r为指定的延迟期数。构造LB统计量

$\text{LB}\left(h,r\right)=h\left(h+2\right)\underset{k=1}{\overset{r}{\sum }}\frac{{\stackrel{^}{\rho }}^{2}{}_{k}}{h-k}~{\chi }^{2}\left(r\right)$ , (8)

${H}_{0}:{\rho }_{1}={\rho }_{2}=\cdots ={\rho }_{r}=0,\forall r\ge 1$ ; ${H}_{1}:\text{ }1\le k\le r,{\rho }_{k}\ne 0,\forall r\ge 1$ . (9)

3. 多项式趋势曲线在白条猪价格指数中的应用

Table 1. White pig price index data (2019)

· p = 2时的表达式

$\stackrel{^}{x}\left(k\right)=132.2933+0.2485k+0.0846{k}^{2}$ . (10)

· p = 3时的表达式

$\stackrel{^}{x}\left(k\right)=134.3811-0.7934k+0.2035{k}^{2}-0.0037{k}^{3}$ . (11)

· p = 4时的表达式

$\stackrel{^}{x}\left(k\right)=127.4445+3.9015k-0.7084{k}^{2}+0.0634{k}^{3}-0.0016{k}^{4}$ . (12)

Table 2. Computation results of three polynomial trend curves (2019)

Figure 1. Sequence diagram of residual APE

Figure 2. APE correlation coefficient diagram

$\stackrel{^}{x}\left(k\right)=134.3811-0.7934k+0.2035{k}^{2}-0.0037{k}^{3}$ . (13)

Table 3. Prediction results of trend curves of third-order polynomials

4. 结束语

Analysis of Polynomial Trend Curve and Its Application in Chinese Pig Price Index[J]. 理论数学, 2019, 09(08): 849-856. https://doi.org/10.12677/PM.2019.98111

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11. NOTES

*通讯作者。