Identification Method - Climate Change: Identification and Projections - page 71

ˆ is that which minimizes the criteria

The OE estimate

θ

2

. To carry out this minimization, optimization algorithms
work through iteration, performing as many simulations as necessary.
Around the optimum level, the first order development of the function

J ( θ ) = y − f ( θ )

f ( θ )

is written as:

ˆ ˆ

f ()~ () θ f θ + F

θ

( θ − θ )

where F is the Jacobian matrix of partial derivatives:

θ

⎡ ⎤

∂ f ∂ f

"

1 1

⎢ ⎥

∂ θ ∂ θ

⎢ ⎥

1 n

⎢ ⎥

∂ f

[ ]

Ftj (, ) = ⎢ #

t

# ⎥

θ

∂ θ

⎢ ⎥

j

⎢ ⎥

∂ f ∂ f

⎢ ⎥

N

"

N

⎢ ∂ θ ∂ θ ⎥

⎣ ⎦

1 n

At optimum, the gradient of in relation to is zero. It develops as:

J ( θ ) θ

∂ J

⎡ ⎤
⎢ ⎥

∂ ⎢ ⎥
⎢ ⎥

θ

1

ˆ ˆ

T T

# = 2( Fyf − ( )2( θ = Fvf + () ( )0 θ − f θ =

θ θ

⎢ ⎥

∂ J

⎢ ⎥
⎢ ⎥

∂ θ

⎣ ⎦

n

by its development gives:

Replacing

f ( θ )

ˆ

2(

T FvF
θ

+

θ θθ

( − )~0

+
θ

+ T − 1 T

Define

F

as the pseudo-inverse of

F : F = ( F F ) F

. Pre-

θ θ θ θ θ

F T

− 1

multiplying the equation above by , gives the error estimation:

( F )

θ

θ

ˆ

θθ − ~ Fv

θ

+

,