# 基于改进GM模型的房价预测模型The Housing Price Forecasting Model Based on Improved GM Model

DOI: 10.12677/CSA.2018.812200, PDF, HTML, XML, 下载: 586  浏览: 1,068

Abstract: In view of the dynamic changes characteristics of real estate price, this article takes the real estate price of The West Coast New Area of Qingdao as an example, combines the improved grey GM (1, 1) forecast model with Markov model, and puts forward a new time series basing forecast model to forecast the real estate price of The West Coast New Area of Qingdao. The grey GM (1, 1) model is suitable for short-term data forecasting with fewer data and fewer fluctuations, while the Markov model is suitable for prediction processes with large data fluctuations. By combining the advantages of the improved GM (1, 1) model with the Markov model, a new forecast model is established for real estate price forecasting. The experimental results show that the accuracy of this model is high, and it is a feasible model for house price forecasting.

1. 引言

2. 灰色–马尔可夫模型基本原理

2.1. 灰色GM (1, 1)模型

2.1.1. GM (1, 1)模型基本形式

${X}^{\left(0\right)}=\left({X}^{{}^{\left(0\right)}}\left(1\right)\text{},{X}^{{}^{\left(0\right)}}\left(2\right)\text{},{X}^{{}^{\left(0\right)}}\left(3\right)\cdots {X}^{\left(0\right)}\left(n\right)\right)$

${X}^{\left(1\right)}=\left({X}^{\left(1\right)}\left(1\right)\text{},{X}^{\left(1\right)}\left(2\right)\text{},{X}^{{}^{\left(1\right)}}\left(3\right)\cdots {X}^{\left(1\right)}\left(n\right)\right)$

${X}^{\left(1\right)}\left(i\right)=\sum {}_{\left(k=1\right)}^{i}{X}^{\left(0\right)}\left(k\right),i=1,2,3,\cdots ,n$

${Z}^{\left(1\right)}=\left({Z}^{\left(1\right)}\left(2\right),{Z}^{\left(1\right)}\left(3\right)\cdots {Z}^{\left(1\right)}\left(n\right)\right)$

${Z}^{\left(1\right)}\left(k\right)=\frac{{X}^{\left(1\right)}\left(k\right)+{X}^{\left(1\right)}\left(k-1\right)}{2},k=2,3,\cdots ,n$

${X}^{\left(0\right)}\left(k\right)+\alpha \cdot {Z}^{\left(1\right)}\left(k\right)=b$

2.1.2. GM (1, 1)模型作用机理

$\frac{\text{d}{x}^{\left(1\right)}\left(k\right)}{\text{d}t}+\alpha {x}^{\left(1\right)}\left(k\right)=\mu$ (1)

$\frac{\text{d}{x}^{\left(1\right)}\left(k\right)}{\text{d}t}+\alpha {z}^{\left(1\right)}\left(k\right)=\text{μ}$

$\frac{\text{d}{x}^{\left(1\right)}\left(k\right)}{\text{d}t}=\frac{{X}^{\left(1\right)}\left(k\right)-{X}^{\left(1\right)}\left(k-1\right)}{1},k=2,3,\cdots ,n$

${X}^{\left(1\right)}\left(k\right)-{X}^{\left(1\right)}\left(k-1\right)+\alpha {z}^{\left(1\right)}\left(k\right)=\text{μ},k=2,3,\cdots ,n$ (2)

$\left[\begin{array}{c}\alpha \\ \text{μ}\end{array}\right]={\left({B}^{T}B\right)}^{-1}{B}^{T}Y$

$B=\left[\begin{array}{cc}-{z}^{\left(1\right)}\left(2\right)& 1\\ \begin{array}{c}-{z}^{\left(1\right)}\left(3\right)\\ ⋮\\ -{z}^{\left(1\right)}\left(n\right)\end{array}& \begin{array}{c}1\\ ⋮\\ 1\end{array}\end{array}\right]Y=\left[\begin{array}{c}{x}^{\left(0\right)}\left(2\right)\\ \begin{array}{c}{x}^{\left(0\right)}\left(3\right)\\ ⋮\end{array}\\ {x}^{\left(0\right)}\left(n\right)\end{array}\right]$

${\stackrel{^}{X}}^{\left(1\right)}\text{=}c{e}^{-\alpha t}+\frac{\text{μ}}{\alpha }$ (3)

${\stackrel{^}{X}}^{\left(1\right)}\left(k+1\right)=c{e}^{-\alpha k}+\frac{\text{μ}}{\alpha },k=0,1,2,3,\cdots n-1$ (4)

${\stackrel{^}{X}}^{\left(1\right)}\left(1\right)=c+\frac{\text{μ}}{\alpha }={X}^{\left(0\right)}\left(1\right)$

${\stackrel{^}{X}}^{\left(1\right)}\left(k+1\right)=\left({X}^{\left(0\right)}\left(1\right)-\frac{\text{μ}}{\alpha }\right){e}^{-\alpha k}+\frac{\text{μ}}{\alpha }k=0,1,2,3，\cdots ，n-1$ (5)

${\stackrel{^}{X}}^{\left(0\right)}\left(k+1\right)={\stackrel{^}{X}}^{\left(1\right)}\left(k+1\right)-{\stackrel{^}{X}}^{\left(1\right)}\left(k\right),k=1,2,3,\cdots ,n-1$ (6)

${\stackrel{^}{X}}^{\left(0\right)}\left(k+1\right)=\left({X}^{\left(0\right)}\left(1\right)-\frac{\text{μ}}{\alpha }\right){e}^{-\alpha k}\cdot \left(1-{e}^{\alpha }\right)$ (7)

2.2. 马尔可夫模型

$P\left\{{x}_{n+1}={i}_{n+1}|{x}_{0}={i}_{0},{x}_{1}={i}_{1},\cdots {x}_{n}={i}_{n}\right\}=P\left\{{x}_{n+1}={i}_{n+1}|{x}_{n}={i}_{n}\right\}$

2.2.1. 状态区域划分

2.2.2. 状态转移概率矩阵

${P}_{ij}^{\left(k\right)}=\frac{{m}_{ij}^{\left(k\right)}}{{M}_{i}}$

${m}_{ij}^{\left(k\right)}$ 表示状态Ei经过k步转移到状态Ej的次数，Mi为状态Ei出现的总次数。

${P}^{\left(k\right)}\text{=}|\begin{array}{cc}\begin{array}{cc}{P}_{11}^{\left(k\right)}& {P}_{12}^{\left(k\right)}\\ {P}_{21}^{\left(k\right)}& {P}_{22}^{\left(k\right)}\end{array}& \begin{array}{cc}\cdots & {P}_{1n}^{\left(k\right)}\\ \cdots & {P}_{2n}^{\left(k\right)}\end{array}\\ \begin{array}{cc}⋮& ⋮\\ {P}_{n1}^{\left(k\right)}& {P}_{n2}^{\left(k\right)}\end{array}& \begin{array}{cc}\dots & ⋮\\ \dots & {P}_{nn}^{\left(k\right)}\end{array}\end{array}|$

2.2.3. 计算预测值

3. 改进的灰色模型

1) 传统GM (1, 1)算法：

${X}^{\left(0\right)}\left(t+1\right)=\left(1-{e}^{-0.02092}\right)\cdot \left(10.437-\frac{11.4556}{-0.02092}\right){e}^{0.02092t}$

2) 利用二次多项式拟合改进：

$Y\left(x\right)=0.12538{X}^{2}+11.428837X-0.707217$

3) 利用三次多项式拟合改进：

$Y\left(x\right)=0.010056{X}^{3}-0.115960{X}^{2}+13.023698X-3.16889$

4) 利用四次多项式拟合改进：

$Y\left(x\right)=-0.000504{X}^{4}+0.0182631{X}^{3}-0.2263819{X}^{2}+1.383664X+9.4540461$

5) 利用指数多项式拟合改进：

$Y\left(x\right)=14.5871851128\cdot {e}^{-\frac{{}^{0.420828817852}}{x}}$

6) 利用对数多项式拟合改进：

$Y\left(x\right)=1.7297367+\mathrm{log}\left(x\right)+10.13768756$

4. 应用实例

4.1. 建立改进的GM (1, 1)模型

Table 1. Comparison of GM (1, 1) model and modified GM (1, 1) model

X(0) = (10.437, 11.670, 12.311, 12.855, 13.067, 13.192, 12.824, 12.693, 12.885, 13.357, 13.856, 14.420, 14.981, 15.590, 16.049 )

GM (1, 1)模型与改进的GM (1, 1)模型，对原始实际数据的拟合效果如图1所示：

Figure 1. Comparison of the fitting effect between GM (1, 1) model and modified GM (1, 1) model

4.2. 利用马尔可夫模型修正

4.2.1. 状态区间划分

4.2.2. 构建状态转移概率矩阵

${P}^{\left(1\right)}=\left[\begin{array}{cc}\begin{array}{cc}\frac{2}{3}& \frac{1}{3}\\ \frac{1}{4}& \frac{1}{4}\end{array}& \begin{array}{cc}0& 0\\ \frac{1}{4}& \frac{1}{4}\end{array}\\ \begin{array}{cc}0& 0\\ 0& \frac{1}{6}\end{array}& \begin{array}{cc}0& 1\\ 0& \frac{5}{6}\end{array}\end{array}\right]\text{}{P}^{\left(2\right)}=\left[\begin{array}{cc}\begin{array}{cc}\frac{1}{3}& \frac{2}{3}\\ \frac{1}{4}& 0\end{array}& \begin{array}{cc}0& 0\\ \frac{1}{4}& \frac{2}{4}\end{array}\\ \begin{array}{cc}0& 0\\ \frac{1}{5}& \frac{1}{5}\end{array}& \begin{array}{cc}0& 1\\ 0& \frac{3}{5}\end{array}\end{array}\right]\text{}{P}^{\left(3\right)}=\left[\begin{array}{cc}\begin{array}{cc}0& \frac{2}{3}\\ 0& 0\end{array}& \begin{array}{cc}\frac{1}{3}& 0\\ 0& 1\end{array}\\ \begin{array}{cc}0& 0\\ \frac{2}{5}& \frac{1}{5}\end{array}& \begin{array}{cc}0& 1\\ 0& \frac{2}{5}\end{array}\end{array}\right]$

Table 2. Results of state interval division

Table 3. Results of state transition probability

$\frac{15.439}{1-2.78%}=15.880$

5. 结束语

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