考虑一组
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{\displaystyle (x_{1},y_{1}),(x_{2},y_{2}),\dots ,(x_{m},y_{m})}
m
{\displaystyle m}
y
^
=
f
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β
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{\displaystyle {\hat {y}}=f(x,{\boldsymbol {\beta }})}
x
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β
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n
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{\displaystyle {\boldsymbol {\beta }}=(\beta _{1},\beta _{2},\dots ,\beta _{n})}
n
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n
{\displaystyle m\geq n}
β
{\displaystyle {\boldsymbol {\beta }}}
S
=
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m
r
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2
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{\displaystyle S=\sum _{i=1}^{m}r_{i}^{2},}
其中残差 ri
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{\displaystyle r_{i}=y_{i}-f(x_{i},{\boldsymbol {\beta }}),\qquad (i=1,2,\dots ,m).}
S 最小值 时的梯度 为零。由于模型包含n n
∂
S
∂
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2
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r
i
∂
r
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0
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{\displaystyle {\frac {\partial S}{\partial \beta _{j}}}=2\sum _{i}r_{i}{\frac {\partial r_{i}}{\partial \beta _{j}}}=0\quad (j=1,\ldots ,n).}
在非线性系统中，偏导数
∂
r
i
∂
β
j
{\textstyle {\frac {\partial r_{i}}{\partial \beta _{j}}}}
x
β
{\displaystyle {\boldsymbol {\beta }}}
β
j
≈
β
j
k
+
1
=
β
j
k
+
Δ
β
j
.
{\displaystyle \beta _{j}\approx \beta _{j}^{k+1}=\beta _{j}^{k}+\Delta \beta _{j}.}
其中，k
Δ
β
{\displaystyle \Delta {\boldsymbol {\beta }}}
β
k
{\displaystyle {\boldsymbol {\beta }}^{k}}
泰勒级数 展开以线性化模型：
f
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≈
f
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+
∑
j
∂
f
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∂
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−
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=
f
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+
∑
j
J
i
j
Δ
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j
.
{\displaystyle f(x_{i},{\boldsymbol {\beta }})\approx f(x_{i},{\boldsymbol {\beta }}^{k})+\sum _{j}{\frac {\partial f(x_{i},{\boldsymbol {\beta }}^{k})}{\partial \beta _{j}}}\left(\beta _{j}-\beta _{j}^{k}\right)=f(x_{i},{\boldsymbol {\beta }}^{k})+\sum _{j}J_{ij}\,\Delta \beta _{j}.}
雅可比矩阵 J J
∂
r
i
∂
β
j
=
−
J
i
j
,
{\displaystyle {\frac {\partial r_{i}}{\partial \beta _{j}}}=-J_{ij},}
残差的表达式则为
Δ
y
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=
y
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−
f
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β
k
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{\displaystyle \Delta y_{i}=y_{i}-f(x_{i},{\boldsymbol {\beta }}^{k}),}
r
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−
f
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=
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f
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f
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≈
Δ
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∑
s
=
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n
J
i
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Δ
β
s
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{\displaystyle r_{i}=y_{i}-f(x_{i},{\boldsymbol {\beta }})=\left(y_{i}-f(x_{i},{\boldsymbol {\beta }}^{k})\right)+\left(f(x_{i},{\boldsymbol {\beta }}^{k})-f(x_{i},{\boldsymbol {\beta }})\right)\approx \Delta y_{i}-\sum _{s=1}^{n}J_{is}\Delta \beta _{s}.}
将上述表达式代入梯度方程，可以得到
−
2
∑
i
=
1
m
J
i
j
(
Δ
y
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−
∑
s
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1
n
J
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s
Δ
β
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=
0
,
{\displaystyle -2\sum _{i=1}^{m}J_{ij}\left(\Delta y_{i}-\sum _{s=1}^{n}J_{is}\ \Delta \beta _{s}\right)=0,}
以上方程可化简为n 正规方程 （normal equations）：
∑
i
=
1
m
∑
s
=
1
n
J
i
j
J
i
s
Δ
β
s
=
∑
i
=
1
m
J
i
j
Δ
y
i
(
j
=
1
,
…
,
n
)
.
{\displaystyle \sum _{i=1}^{m}\sum _{s=1}^{n}J_{ij}J_{is}\ \Delta \beta _{s}=\sum _{i=1}^{m}J_{ij}\ \Delta y_{i}\qquad (j=1,\dots ,n).}
正规方程可用矩阵表示法写成
(
J
T
J
)
Δ
β
=
J
T
Δ
y
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{\displaystyle \left(\mathbf {J} ^{\mathsf {T}}\mathbf {J} \right)\Delta {\boldsymbol {\beta }}=\mathbf {J} ^{\mathsf {T}}\ \Delta \mathbf {y} .}
上述方程是使用高斯-牛顿算法
需要注意的是雅可比矩阵定义中导数的符号约定。某些文献中的J
不同数据点（观测结果）的可靠性并不一定相同，此时可使用加权平方和
S
=
∑
i
=
1
m
W
i
i
r
i
2
.
{\displaystyle S=\sum _{i=1}^{m}W_{ii}r_{i}^{2}.}
权重矩阵W 对角矩阵 ，理想情况下每个权重系数应等于观测误差方差 的倒数。[ 1]
(
J
T
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J
)
Δ
β
=
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Δ
y
.
{\displaystyle \left(\mathbf {J} ^{\mathsf {T}}\mathbf {WJ} \right)\Delta {\boldsymbol {\beta }}=\mathbf {J} ^{\mathsf {T}}\mathbf {W} \ \Delta \mathbf {y} .}
