线性预测模型

线性预测的数学模型在很多领域都有应用, 包括时间序列分析, 数字信号处理, 最优化方法等等.

考虑一个序列 $x[n]$, 最简单的线性预测方法是假设每个序号的值和前一个序号的值有线性关系, 线性系数设为 $a$. 也就是

定义误差函数 $e[n] equiv x[n] - hat{x}[n]$, 使用最小均方差误差准则控制误差达到最小值:

最大值通常是极大值, 我们对 $a$ 求导, 令导函数为0来获得极值点. 故令

因为 $e[n] = x[n] + ax[n-1]$

所以,

代回上式得到

定义离散自相关系数(参见后面)后, 有

带入整理得,

将表达式带回到最小值公式, 便可得到求解最小值的公式为:

这样的一阶线性预测滤波器如下图所示:

大专栏  线性预测模型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" alt="一阶线性预测滤波器"/>

p 阶的情况

对于前p个元素来进行线性预测的情况,

同样有残差的定义:

我们令 $a_0 = 1$, 得到求和符号表示的残差公式

为了得到矩阵表达式, 我们记

可以得到线性预测模型的矩阵表达式:

残差平方和

对参数 $boldsymbol{a}$ 求偏导,

得到

AR 线性预测模型

在 AR 模型的系统中, 每个输出 $x(n)$ 是过去 p 个输出的线性组合.

其中 $p$ 为 AR 模型阶数, $w(n)$ 为均值为 0, 方差为 $sigma_w^2$ 的高斯白噪声.

由给定的时间序列求解 AR 模型的参数 $a_i$ 和阶数 $p$ 是进行线性预测的两个关键问题.

我们给出估计值:

以及误差:

求:

令:

其中, $lin{ 1,2,cdots,p }$

按照这个条件推导出来的极值点其实是边界点:

用到的一些基本概念梳理

自相关和互相关

互相关用来衡量两个时间序列在两个不同时刻之间取值的相似程度. 或者叫两个随机序列的相关性. 设连个不同时刻之间的差距为 $tau$. $int f^*(t)g(t+tau)dt$ 表示互相关.

f(t)与g(t)做相关性相当于 f*(-t) 与 g(t) 做卷积.

自相关表示序列自身不同时刻间的相似程度. 定义如下:

对于离散的情况,

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转载自www.cnblogs.com/liuzhongrong/p/12390595.html
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