ARMA模型建模过程中存在的问题分析.htm

11672

收藏 2005-05-17

<DIV class=Section1 style="LAYOUT-GRID-LINE: 15.6pt">
ARMA模型建模过程中存在的问题分析
毛友全，匡奕球
(西南交通大学经济管理学院成都 610031)
摘 要: 本文通过一个实例，简要介绍了随机时间序列分析中最常用的自回归移动平均模型（ARMA模型）的建模过程，分析讨论了应用ARMA模型建模过程中容易遇到的问题，针对相应的问题提出了改进建议，从而改善ARMA模型在实际应用中的预测效果。
关键词: 随机时间序列 ARMA模型 模型改进
 
随着改革开放的进一步深入，市场竞争愈加激烈，要求各经济部门提高经营决策的科学性、准确性，这就需要对各种经济指标进行必要的分析预测。目前流行的经济分析及预测技术包括回归分析、时间序列分析、马尔柯夫预测法、灰色预测法、神经网络预测法、投入产出技术等。由于各种经济指标受到的干扰因素十分复杂，加上管理机制的缺陷造成的统计数据缺失，有时难以找到分析指标的主要影响因素，或者虽能找到主要影响因素，但寻找的成本太高，因此很难利用变量之间的因果关系模型来描述预测对象，这时常常宜用已有的历史数据进行时间序列分析。与回归分析不同，时间序列分析中的模型不是描述因变量与其他自变量的因果关系，而是描述因变量自身变化的统计规律。用时间序列模型进行预测，就是从变量自身的历史资料中推测出其未来时刻的可能取值，这种预测仅仅依赖过去的变量值和过去的预测误差。从表面上看，时间序列分析撇开了系统变量之间存在因果关系和结构关系的影响，但将时间作为一个明确的自变量引入模型，实际上代表了决定变量变化的诸因素的联合影响[1]。
时间序列分析分为确定型和随机型两大类型。本文讨论随机型，为了叙述和阅读的方便，首先简单回顾了随机时间序列分析最常用的模型——自回归移动平均模型ARMA（p，q）及其建模过程，着重以某铁路局货物运输平均一日装车数（表2）为建模对象，重点阐述了建立经济数据ARMA（p，q）模型过程中容易遇到的一些问题，并针对性地提出了一些实用的改进建议。
1 ARMA模型概述
自回归移动平均模型ARMA是一种刻划平稳随机时间序列的线性模型，是有限个自身过去值的线性组合加上有限个过去白噪声误差值的线性组合。模型方程为：<v:shapetype> <v:stroke joinstyle="miter"></v:stroke><v:formulas><v:f eqn="if lineDrawn pixelLineWidth 0 "></v:f><v:f eqn="sum @0 1 0 "></v:f><v:f eqn="sum 0 0 @1 "></v:f><v:f eqn="prod @2 1 2 "></v:f><v:f eqn="prod @3 21600 pixelWidth "></v:f><v:f eqn="prod @3 21600 pixelHeight "></v:f><v:f eqn="sum @0 0 1 "></v:f><v:f eqn="prod @6 1 2 "></v:f><v:f eqn="prod @7 21600 pixelWidth "></v:f><v:f eqn="sum @8 21600 0 "></v:f><v:f eqn="prod @7 21600 pixelHeight "></v:f><v:f eqn="sum @10 21600 0 "></v:f></v:formulas><v:path connecttype="rect" gradientshapeok="t" extrusionok="f"></v:path><lock aspectratio="t" v:ext="edit"></lock></v:shapetype><v:shape><v:imagedata src="./jgqy5-1.files/image001.wmz" title=""></v:imagedata></v:shape>[1] （1）
其中<v:shape> <v:imagedata src="./jgqy5-1.files/image003.wmz" title=""></v:imagedata></v:shape>——提前p期的观测值，<v:shape> <v:imagedata src="./jgqy5-1.files/image005.wmz" title=""></v:imagedata></v:shape>——提前q期的白噪声误差值；
 <v:shape><v:imagedata src="./jgqy5-1.files/image007.wmz" title=""></v:imagedata></v:shape>为模型待定参数。
假设条件：①线性假设；②{<v:shape> <v:imagedata src="./jgqy5-1.files/image009.wmz" title=""></v:imagedata></v:shape>}为正态分布；③误差序列{<v:shape> <v:imagedata src="./jgqy5-1.files/image011.wmz" title=""></v:imagedata></v:shape>}是白噪声（最根本的假设）。 
适用条件：适用于一个平稳、零均值时间序列的短期预测。
2 ARMA模型建模过程及存在的问题分析
2.1 ARMA模型建模过程
 ARMA模型建模就是对采集到的时间数据序列{<v:shape> <v:imagedata src="./jgqy5-1.files/image009.wmz" title=""></v:imagedata></v:shape>}拟合出适用的ARMA（p，q）模型，其中关键是确定p和q的估计值，现给出基本建模过程如下：
（1）基本分析与预处理。首先对数据序列进行直观判断，观察序列的趋势特征，找出离群点，再通过定性分析初步确认离群点是否为异常值，最后通过定量的描述性特征分析（均值、方差、离散度、偏度、峰度等），可以发现数据序列中的趋势特征、周期特征或离散特征等，再检验时间序列是否符合正态分布，是否是平稳序列，以进一步确认是否有异常值存在。
几乎所有的经济数据记录都不是零均值序列，需要进行零化处理。根据基本分析，对数据序列进行趋势提取处理（变换：如转换成比率、差分等）和异常值处理，使处理后的序列满足零均值和方差的数学平稳性要求，以提高模型建模和预测估计的精度。
（2）模型识别。实现模型识别的基本手段是考察数据序列的自相关系数和偏自相关系数的截尾性（表1）。
 表1：模型识别表[5]
<DIV align=center>
<TABLE border=1 cellPadding=0 cellSpacing=0 style="BORDER-BOTTOM: medium none; BORDER-COLLAPSE: collapse; BORDER-LEFT: medium none; BORDER-RIGHT: medium none; BORDER-TOP: medium none; mso-padding-alt: 0cm 5.4pt 0cm 5.4pt; mso-border-alt: solid windowtext .5pt">

<TR >
<TD width=151>
类别 模型</TD>
<TD width=121>
自回归AR（p）</TD>
<TD width=136>
移动平均MA（q）</TD>
<TD width=136>
ARMA（p，q）</TD></TR>
<TR >
<TD width=151>
自相关系数</TD>
<TD width=121>
拖尾</TD>
<TD width=136>
q步截尾</TD>
<TD width=136>
拖尾</TD></TR>
<TR >
<TD width=151>
偏自相关系数</TD>
<TD width=121>
P步截尾</TD>
<TD width=136>
拖尾</TD>
<TD width=136>
拖尾</TD></TR></TABLE></DIV>
（3）模型参数估计。首先利用序列的相关特性，或者依据信息准则（AIC、BIC、FDE等），结合经验初步判定模型的阶次。然后根据选择的模型和确定的阶次运用有效的算法估计模型参数（往往需借助计算机才能完成）。
（4）模型检验及预测
由于ARAR模型成立的根本条件是误差序列{εt}为白噪声，因此，模型适用性的最根本的检验是检验{εt}是否为白噪声。
2.2 存在的问题分析
尽管ARMA模型有较成熟的理论支撑，但由于现实经济生活中各种复杂因素的影响，建模和预测结果常常难以令人满意。下面我们以某铁路局货物运输平均一日装车数（表2）为研究对象（选取1995年~1999年的60个数据作为样本数据，2000年的12个数据为检验数据），分析ARMA模型传统建模过程存在的问题。
表2 平均一日装车数据单位：车/日
<DIV align=center>
<TABLE border=0 cellPadding=0 cellSpacing=0 style="BORDER-COLLAPSE: collapse; WIDTH: 15cm; mso-padding-alt: 0cm 0cm 0cm 0cm" width=567>

<TR >
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: windowtext 0.5pt solid; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext 0.5pt solid; HEIGHT: 14.25pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm" vAlign=bottom width=38>
 月
年</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext 0.5pt solid; HEIGHT: 14.25pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm" width=38>
一月</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext 0.5pt solid; HEIGHT: 14.25pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm" width=38>
二月</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext 0.5pt solid; HEIGHT: 14.25pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm" width=38>
三月</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext 0.5pt solid; HEIGHT: 14.25pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm" width=38>
四月</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext 0.5pt solid; HEIGHT: 14.25pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm" width=38>
五月</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext 0.5pt solid; HEIGHT: 14.25pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm" width=38>
六月</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext 0.5pt solid; HEIGHT: 14.25pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm" width=38>
七月</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext 0.5pt solid; HEIGHT: 14.25pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm" width=38>
八月</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext 0.5pt solid; HEIGHT: 14.25pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm" width=38>
九月</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext 0.5pt solid; HEIGHT: 14.25pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm" width=38>
十月</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext 0.5pt solid; HEIGHT: 14.25pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm" width=38>
十一月</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext 0.5pt solid; HEIGHT: 14.25pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm" width=38>
十二月</TD></TR>
<TR >
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: windowtext 0.5pt solid; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
1995</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
7491</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
7803</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
7703</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
7680</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
7992</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
7832</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
7055</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
7206</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
7607</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
7672</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
7780</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
7317</TD></TR>
<TR >
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: windowtext 0.5pt solid; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
1996</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
7706</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
7751</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
8093</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
7720</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
7531</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
7442</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
7488</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
7372</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
7644</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
7413</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
7404</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
7642</TD></TR>
<TR >
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: windowtext 0.5pt solid; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
1997</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
7216</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
8483</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
7785</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
7652</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
7677</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
8214</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
8444</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
8232</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
8143</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
7871</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
7983</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
7724</TD></TR>
<TR >
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: windowtext 0.5pt solid; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
1998</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
7406</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
7033</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
6803</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
7117</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
6430</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
6219</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
6505</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
5645</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
5832</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
7238</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
7701</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
7120</TD></TR>
<TR >
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: windowtext 0.5pt solid; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
1999</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
7254</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
6782</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
6468</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
6815</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
7049</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
7042</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
6932</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
7137</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
7512</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
7747</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
6740</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
6569</TD></TR>
<TR >
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: windowtext 0.5pt solid; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
2000</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
6862</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
6893</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
7132</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
6786</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
6913</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
7168</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
6650</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
6801</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
7383</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
7179</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
6984</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 1cm; mso-border-top-alt: solid windowtext .5pt" vAlign=bottom width=38>
7045</TD></TR></TABLE></DIV>
（1）平稳性分析。从装车数据折线图看，序列有略微下降的趋势，但经过定量分析，数据满足平稳性要求。其中1998年8、9月的装车数异常，经过定性分析可知，由于当年货运价格上调、洪涝灾害等原因，可以初步判定其为异常值，但按3σ原则进行定量分析异常值假设未得到确认。从实际分析结果看，把1998年8、9月的装车数作为异常值处理，模型的预测效果更好些。
（2）数据预处理。对序列进行差分分析，发现差分后的序列标准差（426.65）小于原序列的标准差（585.70），平稳性提高，但增加了两个异常值，实际预测效果反而降低。对于这样的情况，如何判定其趋势达到什么程度才进行趋势项的提取方能有效，是一个值得进一步研究的问题。
（3）模型识别。模型识别先得计算自相关系数，其公式为[2]：
<v:shape><v:imagedata src="./jgqy5-1.files/image014.wmz" title=""></v:imagedata></v:shape> （2）
其中<v:shape> <v:imagedata src="./jgqy5-1.files/image016.wmz" title=""></v:imagedata></v:shape>，<v:shape> <v:imagedata src="./jgqy5-1.files/image018.wmz" title=""></v:imagedata></v:shape>
为了理论上与实践中的便利，通常以<v:shape> <v:imagedata src="./jgqy5-1.files/image020.wmz" title=""></v:imagedata></v:shape>代替<v:shape> <v:imagedata src="./jgqy5-1.files/image022.wmz" title=""></v:imagedata></v:shape>与<v:shape> <v:imagedata src="./jgqy5-1.files/image024.wmz" title=""></v:imagedata></v:shape>，再以整个序列的平方和代替部分平方和。于是得到下列最常用的公式[2]：
<v:shape><v:imagedata src="./jgqy5-1.files/image026.wmz" title=""></v:imagedata></v:shape> （3）
以上替代行为要求观察值的个数n必须远大于k，否则自相关系数的计算值将改变较大，而n大于k的定量程度需要根据实际情况而定。用SPSS统计软件分析得到平均一日装车数据的自相关系数和偏自相关系数都是截尾的，这时常规判别准则失效，只能依靠经验选择多个模型进行试验。经多次试验，移动平均模型MA（8）满足建模过程的所有要求，但该模型不仅计算复杂，而且实际预测结果也很不理想，而我们用高阶的自回归模型AR（16）却能得到较满意的效果（表3）。
表3：AR（16）模型预测结果
<DIV align=center>
<TABLE border=1 cellPadding=0 cellSpacing=0 style="BORDER-BOTTOM: medium none; BORDER-COLLAPSE: collapse; BORDER-LEFT: medium none; BORDER-RIGHT: medium none; BORDER-TOP: medium none; WIDTH: 15cm; mso-padding-alt: 0cm 0cm 0cm 0cm; mso-border-alt: solid windowtext .5pt" width=567>

<TR >
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: windowtext 0.5pt solid; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext 0.5pt solid; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 31.2pt" width=42>
月份</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext 0.5pt solid; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 31.2pt; mso-border-left-alt: solid windowtext .5pt" width=42>
1</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext 0.5pt solid; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 31.2pt; mso-border-left-alt: solid windowtext .5pt" width=42>
2</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext 0.5pt solid; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 31.2pt; mso-border-left-alt: solid windowtext .5pt" width=42>
3</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext 0.5pt solid; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 31.2pt; mso-border-left-alt: solid windowtext .5pt" width=42>
4</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext 0.5pt solid; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 31.2pt; mso-border-left-alt: solid windowtext .5pt" width=42>
5</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext 0.5pt solid; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 31.2pt; mso-border-left-alt: solid windowtext .5pt" width=42>
6</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext 0.5pt solid; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 31.2pt; mso-border-left-alt: solid windowtext .5pt" width=42>
7</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext 0.5pt solid; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 31.2pt; mso-border-left-alt: solid windowtext .5pt" width=42>
8</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext 0.5pt solid; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 31.2pt; mso-border-left-alt: solid windowtext .5pt" width=42>
9</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext 0.5pt solid; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 31.2pt; mso-border-left-alt: solid windowtext .5pt" width=42>
10</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext 0.5pt solid; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 31.2pt; mso-border-left-alt: solid windowtext .5pt" width=42>
11</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext 0.5pt solid; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 31.2pt; mso-border-left-alt: solid windowtext .5pt" width=42>
12</TD></TR>
<TR >
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: windowtext 0.5pt solid; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 31.2pt; mso-border-top-alt: solid windowtext .5pt" width=42>
预测值</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 31.2pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt" width=42>
6567.0</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 31.2pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt" width=42>
6828.6</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 31.2pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt" width=42>
7112.0</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 31.2pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt" width=42>
7169.3</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 31.2pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt" width=42>
6791.0</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 31.2pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt" width=42>
6837.9</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 31.2pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt" width=42>
7200.9</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 31.2pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt" width=42>
6787.7</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 31.2pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt" width=42>
6881.0</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 31.2pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt" width=42>
6937.8</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 31.2pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt" width=42>
6983.7</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 31.2pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt" width=42>
7246.8</TD></TR>
<TR >
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: windowtext 0.5pt solid; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 45.35pt; mso-border-top-alt: solid windowtext .5pt" width=60>
误差比例</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 31.2pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt" width=42>
-0.043</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 31.2pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt" width=42>
-0.009</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 31.2pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt" width=42>
-0.003</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 31.2pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt" width=42>
0.056</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 31.2pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt" width=42>
-0.018</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 31.2pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt" width=42>
-0.046</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 31.2pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt" width=42>
0.083</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 31.2pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt" width=42>
-0.002</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 31.2pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt" width=42>
-0.068</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 31.2pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt" width=42>
-0.034</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 31.2pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt" width=42>
0.000</TD>
<TD noWrap style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: medium none; BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: medium none; HEIGHT: 15.75pt; PADDING-BOTTOM: 0cm; PADDING-LEFT: 0cm; PADDING-RIGHT: 0cm; PADDING-TOP: 0cm; WIDTH: 31.2pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt" width=42>
0.029</TD></TR></TABLE></DIV>
 (4)模型参数的估计。目前许多统计软件都能有效地计算模型参数的估计值，但它都是基于零均值序列设计的，而我们估计AR（16）模型的参数时，是直接对原始序列进行分析。
3 建模过程的改进
 根据对哈尔滨铁路局多个货运指标的预测建模得出的经验，提出以下改进建议：
 （1）人员配备。对经济数据的预测建模，若要求较高，应该成立专门的工作小组，既要有专门的统计分析员，也要配备适量的对企业发展过程有充分了解的管理人员以及有关专家，这样才能做到定性与定量相结合，不脱离客观实际，提高所建模型的精度和适用性。
 （2）基本分析。由于ARMA建模一般要求数据序列不小于50[1]，是具有统计学意义的大样本，可以认为近似服从正态分布，所以一般情况下不考虑某种检验方法的条件限制。基本分析应该注重对数据序列的背景分析，因为单纯的统计分析无法准确识别序列中随机因素干扰下产生的异常情况，必须对数据序列的宏观背景有一个准确的把握，才能保证微观分析的正确性。
（3）预处理。数据序列是否进行预处理，既要看基本分析的结果，更应该注重对处理前后数据特征的分析。因为一般的统计处理方法只有在完全满足正态分布的情况下，才能获得满意的效果。比如差分处理既需要有大样本的特性，又要有明显的趋势特征，否则很难取得好的效果。因为差分处理将减少数据个数，而且倾向于增加差分变异，从而增加分析的难度，减弱估计与预测的精确性。对于预处理，目前尚没完善的理论和有效的方法，因此不能忽视定性分析的作用。总之，对数据序列的预处理要有针对性，才能给出合理的解释。
<v:rect><v:textbox style="mso-next-textbox: #_x0000_s1040" inset="0,0,0,0"></v:textbox></v:rect><v:rect><v:textbox style="mso-next-textbox: #_x0000_s1039" inset="0,0,0,0">
<TABLE cellPadding=0 cellSpacing=0 width="100%">

<TR>
<TD>
<DIV>
异常值</DIV></TD></TR></TABLE></v:textbox></v:rect><v:line></v:line><v:line></v:line><v:line><v:stroke endarrowwidth="narrow" endarrow="block"></v:stroke></v:line><v:rect><v:textbox style="mso-next-textbox: #_x0000_s1044" inset="0,0,0,0">
<TABLE cellPadding=0 cellSpacing=0 width="100%">

<TR>
<TD>
<DIV>
非异常值</DIV></TD></TR></TABLE></v:textbox></v:rect>
<TABLE cellPadding=0 cellSpacing=0 width="100%">

<TR>
<TD>
<DIV class=shape v:shape="_x0000_s1040">
异常值</DIV></TD></TR></TABLE> 
 
<v:rect><v:textbox style="mso-next-textbox: #_x0000_s1036" inset="0,0,0,0"></v:textbox></v:rect>
<TABLE cellPadding=0 cellSpacing=0 width="100%">

<TR>
<TD>
<DIV class=shape v:shape="_x0000_s1036">
下一步分析</DIV></TD></TR></TABLE><v:line><v:stroke endarrowwidth="narrow" endarrow="block"></v:stroke></v:line><v:rect><v:textbox style="mso-next-textbox: #_x0000_s1035" inset="0,0,0,0"></v:textbox></v:rect>
<TABLE cellPadding=0 cellSpacing=0 width="100%">

<TR>
<TD>
<DIV class=shape v:shape="_x0000_s1035">
预处理</DIV></TD></TR></TABLE><v:line><v:stroke endarrowwidth="narrow" endarrow="block"></v:stroke></v:line><v:rect><v:textbox style="mso-next-textbox: #_x0000_s1034" inset="0,0,0,0"></v:textbox></v:rect>
<TABLE cellPadding=0 cellSpacing=0 width="100%">

<TR>
<TD>
<DIV class=shape v:shape="_x0000_s1034">
定量分析</DIV></TD></TR></TABLE><v:line><v:stroke endarrowwidth="narrow" endarrow="block"></v:stroke></v:line><v:rect><v:textbox style="mso-next-textbox: #_x0000_s1033" inset="0,0,0,0"></v:textbox></v:rect>
<TABLE cellPadding=0 cellSpacing=0 width="100%">

<TR>
<TD>
<DIV class=shape v:shape="_x0000_s1033">
定性分析</DIV></TD></TR></TABLE><v:line><v:stroke endarrowwidth="narrow" endarrow="block"></v:stroke></v:line><v:rect><v:textbox style="mso-next-textbox: #_x0000_s1032" inset="0,0,0,0"></v:textbox></v:rect>
<TABLE cellPadding=0 cellSpacing=0 width="100%">

<TR>
<TD>
<DIV class=shape v:shape="_x0000_s1032">
直观判断</DIV></TD></TR></TABLE> 
<v:rect><v:textbox style="mso-next-textbox: #_x0000_s1037" inset="0,0,0,0"></v:textbox></v:rect>
<TABLE cellPadding=0 cellSpacing=0 width="100%">

<TR>
<TD>
<DIV class=shape v:shape="_x0000_s1037">
非异常值</DIV></TD></TR></TABLE><v:line><v:stroke endarrowwidth="narrow" endarrow="block"></v:stroke></v:line><v:line></v:line> 
<v:line></v:line> 
图1 异常值分析流程图
（4）模型识别。由于只有观测值的个数n远大于自相关系数滞后步距k时，替代计算公式才有效。至于n比k大到什么程度才可以按替代公式计算，目前还没有一个定量依据，因此建议将原始公式和替代公式的计算结果一并加以考察。另外，由于只能得到样本的自相关系数是一个参数估计值，而样本自相关系数和偏自相关系数与数据序列一样具有随机性，其精确度也存在问题。因此，在对一个实际的经济数据序列模型识别时，有时可能很难识别或得到几种不同的识别结果。建议尽量采用高阶自回归模型去拟合原始序列，一则模型参数估计算法简单，二则不需要对数据序列进行零化处理，能保证拟合模型是原始序列特征的客观描述，况且有严格的理论依据｛ARMA（p,q）=AR(∞)[5]｝。
（5）参数估计。对随机时间序列来说，模型阶次判定和参数估计是密不可分的，两者往往交织在一起。实际运用中，主要根据自相关系数截尾点和异常点初步定阶，然后通过试算，比较各阶次模型的预测结果（误差平方和最小）进行取舍。当然也可以考虑采用信息准则、统计特性等定阶，但这些方法计算量比较大，有时会出现不收敛的异常现象。对于高阶自回归模型，一般采用经验原则定阶和U-C算法[5]估计参数。因为采用U-C算法，可以不做零均值处理，而且使用正置与逆置时序共同估计参数，这样，正逆两次利用了观测时间序列所含的信息，提高了数据的利用率，是AR模型参数估计精度最高的估计方法。
（6）模型检验及预测。用检验模型预测结果的残差是否属于白噪声序列来判别模型的适用性，这只是统计中的一个标准处理，事实上，任何一个模型都可能产生随机的残差序列。另外，预测方法使得该序列的预测值具有自相关（Yule—Slutsky效应[2]），则误差也一定存在相关。因此，在实际建模时，并不一定需要用白噪声序列来判别，只需比较预测误差平方和的大小即可。
4 结束语
 一个经济数据序列蕴含了许多因素的共同作用，要把它分析准确，除了运用大量的统计分析方法之外，还必须充分依靠经济背景分析，有时专家、技术人员的经验技巧也很重要。随机时间序列分析作为一种经济适用的分析工具，有很大的应用前景。但也有它的局限性，在实际应用过程中，必须重视其固有的问题，作出针对性的分析研究，才能避免预测分析的失真。如果预测分析本身的要求不高，完全可以对模型的许多条件进行弱化，既节约分析成本，又能获得令人满意的效果。以上思想在对哈尔滨铁路局的货物运输预测分析中获得了较好效果。 
 
 
参考文献
[1] 李学伟，关忠良，陈景艳．经济数据分析预测学．中国铁道出版社，1998．7．
[2] 施锡铨，范正绮．数据分析方法．上海财经大学出版社，1997．10．
[3] Robert S．pindyck，Daniel l．Rubinfeld．计量经济模型与经济预测．钱小军等译．机械工业出版社，1999．11．
[4] 施仁杰，卢科学．时间序列分析引论．西安电子科技大学出版社，1988．6．
[5] 杨叔子，吴雅．时间序列分析的工程应用（上）．华中理工大学出版社，1991．5．</DIV>