﻿ 基于灰色分数阶马尔科夫模型的房价预测 House Price Forecast Based on Grey Fractional Markov Model

Computer Science and Application
Vol. 09  No. 04 ( 2019 ), Article ID: 29951 , 9 pages
10.12677/CSA.2019.94091

House Price Forecast Based on Grey Fractional Markov Model

Yixuan Yang1, Lang Yu1, Hai Li2

1School of Science, Southwest University of Science and Technology, Mianyang Sichuan

2School of Civil Engineering and Architecture, Southwest University of Science and Technology, Mianyang Sichuan

Received: Apr. 10th, 2019; accepted: Apr. 19th, 2019; published: Apr. 26th, 2019

ABSTRACT

According to the nonlinearity and complexity of the house price data, this paper combines the fractional-order cumulative generation operator and the Markov correction model to establish the MKFGM(1,1) model, and forecasts and analyzes the house prices of Chengdu from April 2017 to May 1818. Firstly, the particle swarm optimization algorithm is used to search for the optimal order of the cumulative generation operator in the real number domain; combined with the GM(1,1) model, the FGM(1,1) model after changing the order can be obtained; and a preliminary prediction of Chengdu city’s house price is made. Secondly, the relative error between the predicted value and the actual value is divided depending on the states, then the Markov model is used for error correction, and finally the MKFGM(1,1) model is established. Comparing and analyzing the fitting and prediction results of GM(1,1), FGM(1,1) and MKFGM(1,1) models, it can be seen that the MKFGM(1,1) model has a higher accuracy in housing price forecasting.

Keywords:House Price Forecast, Fractional Gray Model, Markov, Prediction Accuracy, PSO

1西南科技大学理学院，四川 绵阳

2西南科技大学土木工程与建筑学院，四川 绵阳

1. 引言

2. 灰色分数阶马尔科夫模型

2.1. 分数阶FGM(1,1)模型

${x}^{\left(r\right)}\left(k\right)=\sum _{i=1}^{k}\frac{\Gamma \left(r+k-i\right)}{\Gamma \left(k-i+1\right)\Gamma \left(r\right)}{x}^{\left(0\right)}\left(i\right),k=1,2,\cdots ,n$ (1)

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

FGM(1,1)模型 ${x}^{\left(r\right)}\left(k\right)-{x}^{\left(r\right)}\left(k-1\right)+a{z}^{\left(r\right)}\left(k\right)=b$$\stackrel{^}{u}={\left[a,b\right]}^{\text{T}}$ 可以运用最小二乘法估计：

$\stackrel{^}{u}={\left({B}^{\text{T}}B\right)}^{-1}{B}^{\text{T}}Y$ (3)

$Y=\left[\begin{array}{c}{x}^{\left(r-1\right)}\left(2\right)\\ {x}^{\left(r-1\right)}\left(3\right)\\ ⋮\\ {x}^{\left(r-1\right)}\left(n\right)\end{array}\right],B=\left[\begin{array}{cc}-{z}^{\left(r\right)}\left(2\right)& 1\\ -{z}^{\left(r\right)}\left(3\right)& 1\\ ⋮& ⋮\\ -{z}^{\left(r\right)}\left(n\right)& 1\end{array}\right]$ (4)

$\frac{\text{d}{x}^{\left(r\right)}\left(t\right)}{\text{d}t}+a{x}^{\left(r\right)}\left(t\right)=b$ (5)

${\stackrel{^}{x}}^{\left(r\right)}\left(t\right)=\left[{x}^{\left(0\right)}\left(1\right)-\frac{b}{a}\right]{\text{e}}^{-a\left(t-1\right)}+\frac{b}{a},t=1,2,\cdots ,n$ (6)

${\stackrel{^}{x}}^{\left(r\right)}\left(k\right)=\left[{x}^{\left(0\right)}\left(1\right)-\frac{b}{a}\right]{\text{e}}^{-a\left(k-1\right)}+\frac{b}{a},k=1,2,\cdots ,n$ (7)

${\stackrel{^}{x}}^{\left(0\right)}\left(k\right)={\left({\stackrel{^}{x}}^{\left(r\right)}\right)}^{\left(-r\right)}\left(k\right)=\sum _{i=0}^{k-1}{\left(-1\right)}^{i}\frac{\Gamma \left(r+1\right)}{\Gamma \left(i+1\right)\Gamma \left(r-i+1\right)}{\stackrel{^}{x}}^{\left(r\right)}\left(k-i\right),k=2,3,\cdots ,n$ (8)

2.2. 马尔科夫模型

${C}_{h+1}={C}_{h}\ast P$ (9)

2.2.1. 划分预测状态

${E}_{i}=\left[{Q}_{i1},{Q}_{i2}\right],\left[i=1,2,\cdots ,u\right]$ (10)

2.2.2. 构造状态转移概率矩阵

(11)

2.2.3. 确定预测值

$y\left(k\right)=\frac{{\stackrel{^}{x}}^{\left(r\right)}\left(k\right)}{1+\left({Q}_{i1}+{Q}_{i2}\right)}$ (12)

2.3. 模型误差检验准则

$\text{MAPEF}=\frac{1\text{00}}{m}\sum _{i=1}^{m}|\frac{{\stackrel{^}{x}}^{\left(0\right)}\left(k\right)-{x}^{\left(0\right)}\left(k\right)}{{x}^{\left(0\right)}\left(k\right)}|$ (13)

$\text{MAPEP}=\frac{1\text{00}}{n-m}\sum _{i=m+1}^{n}|\frac{{\stackrel{^}{x}}^{\left(0\right)}\left(k\right)-{x}^{\left(0\right)}\left(k\right)}{{x}^{\left(0\right)}\left(k\right)}|$ (14)

$\text{MAPET}=\frac{1\text{00}}{n}\sum _{i=1}^{n}|\frac{{\stackrel{^}{x}}^{\left(0\right)}\left(k\right)-{x}^{\left(0\right)}\left(k\right)}{{x}^{\left(0\right)}\left(k\right)}|$ (15)

2.4. 最优阶数的确定

$\begin{array}{l}\mathrm{min}\text{MAPET}=\frac{1\text{00}}{n}\sum _{i=1}^{n}|\frac{{\stackrel{^}{x}}^{\left(0\right)}\left(k\right)-{x}^{\left(0\right)}\left(k\right)}{{x}^{\left(0\right)}\left(k\right)}|\\ s.t.\left\{\begin{array}{l}r\in R,\\ k=2,3,\cdots ,n\end{array}\end{array}$ (16)

1) 初始化群体粒子个数，粒子维数等。

2) 初始化种群粒子位置和速度，计算每个粒子的适应度值，分别与个体极值和全局极值的最优位置及最优值比较，择优选取。

3) 迭代更新粒子的速度和位置，进行边界条件处理。

4) 判断是否满足终止条件；若不满足，则继续搜索；若满足，则将该粒子作为粒子群算法的最优计算结果。

3. 成都市房价预测分析

3.1. 样本数据

3.2. FGM(1,1)模型预测

${\stackrel{^}{x}}^{\left(-0.37\right)}\left(k\right)=193884.364{\text{e}}^{0.1513\left(k-1\right)}-184481.364,k=1,2,\cdots ,12$ (17)

Table 1. Comparison data between GM(1,1) and FGM(1,1) models.

3.3. 马尔科夫模型修正

3.3.1. 对每个月进行状态区间划分

Table 2. State partition

3.3.2. 构建状态转移矩阵

$P=\left[\begin{array}{cccc}\text{0}\text{.6667}& \text{0}& \text{0}\text{.3333}& \text{0}\\ \text{0}& \text{0}& \text{0}& \text{1}\text{.0000}\\ \text{0}\text{.2500}& \text{0}& \text{0}\text{.5000}& \text{0}\text{.2500}\\ \text{0}& \text{0}& \text{0}\text{.6667}& \text{0}\text{.3333}\end{array}\right]$ , ${P}^{\left(2\right)}=\left[\begin{array}{cccc}0.5278& 0& 0.3889& 0.0833\\ 0& 0& 0.6667& 0.3333\\ 0.2917& 0& 0.5000& 0.2083\\ 0.1667& 0& 0.5556& 0.2778\end{array}\right]$

${P}^{\left(3\right)}=\left[\begin{array}{cccc}\text{0}\text{.4491}& 0& \text{0}\text{.4259}& \text{0}\text{.1250}\\ \text{0}\text{.1667}& 0& \text{0}\text{.5556}& \text{0}\text{.2778}\\ \text{0}\text{.3194}& 0& \text{0}\text{.4861}& \text{0}\text{.1944}\\ \text{0}\text{.2500}& 0& \text{0}\text{.5185}& \text{0}\text{.2315}\end{array}\right]$ , ${P}^{\left(4\right)}=\left[\begin{array}{cccc}\text{0}\text{.4059}& 0& \text{0}\text{.4460}& \text{0}\text{.1481}\\ \text{0}\text{.2500}& 0& \text{0}\text{.5185}& \text{0}\text{.2315}\\ \text{0}\text{.3345}& 0& \text{0}\text{.4792}& \text{0}\text{.1863}\\ \text{0}\text{.2963}& 0& \text{0}\text{.4969}& \text{0}\text{.2068}\end{array}\right]$

3.3.3. 计算预测值

Table 3. State transition matrix predicting the state of April 2018

Table 4. Comparison data between FGM(1,1) and MKFGM(1,1) models

Table 5. The average absolute percentage error of the fitting, prediction and total data for the three models

Figure 1. Three models fitting prediction comparison chart

4. 结论

1) 三个模型的MAPEF、MAPEP、MAPET均在5%以下，说明这三个模型应用于房价预测都是切实有效的。

2) 本文建立的MKFGM(1,1)模型的预测精度最高，MAPET仅为0.748%，远低于GM(1,1)模型和FGM(1,1)模型，改进效果显著。

3) MKFGM(1,1)模型作为一种优秀且稳定的预测模型，可考虑将其进一步应用在金融、能源、交通运输等领域。

House Price Forecast Based on Grey Fractional Markov Model[J]. 计算机科学与应用, 2019, 09(04): 802-810. https://doi.org/10.12677/CSA.2019.94091

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