最小二乘法可以將誤差方程轉化為有確定解的代數方程組(其方程式數目正好等於未知數的個數),從而可求解出這些未知參數。這個有確定解的代數方程組稱為最小二乘法估計的正規方程(或稱為法方程)。
基本介紹
- 中文名:正規方程
- 外文名:regular expression
- 套用學科:數學術語
- 範疇:數理科學
- 涉及:線性參數
- 屬於:最小二乘法
基本信息,線性參數,
基本信息
為了獲得更可靠的結果,測量次數
總要多於未知參數的數目
,即所得誤差方程式的數目總是要多於未知數的數目。因而直接用一般解代數方程的方法是無法求解這些未知參數的。最小二乘法則可以將誤差方程轉化為有確定解的代數方程組(其方程式數目正好等於未知數的個數),從而可求解出這些未知參數。這個有確定解的代數方程組稱為最小二乘法估計的正規方程(或稱為法方程)。
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線性參數的最小二乘法處理程式可歸結為:首先根據具體問題列出誤差方程式;再按最小二乘法原理,利用求極值的方法將誤差轉化為正規方程;然後求解正規方程,得到待求的估計量;最後給出精度估計。對於非線性參數,可先將其線性化,然後按上述線性參數的最小二乘法處理程式去處理。因此,建立正規方程是待求參數最小二乘法處理的基本環節。
線性參數
線性參數的誤差方程式為
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在精度測量中,應滿足最小二乘條件式,即:
最小。
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現求上式子的估計量
,
,
,
可利用求極值的方法來滿足上式的條件。為此,對殘餘誤差的平方和
求導數,並令其為零,有
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由此可知,上面各方程求得的極值是最小值,滿足最小二乘條件,因而也是所要求的估計量,最後把它寫成
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上式即為等精度測量的線性參數最小二乘法處理的正規方程。這是一個
元線性方程組,當其係數行列式不為零時,有唯一確定的解,由此可解得欲求的估計量。
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