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Basic Information About GFLS:
Outline of GFLS Program Logic:
Variables:
x_t = n x 1 column vector of state variables
y_t = m x 1 column vector of observations
Dynamic and Measurement Equation Matrices and Forcing Terms:
F(t) = n x n dynamic equation matrix, t = 1,...,T-1
a(t) = n x 1 dynamic equation forcing term, t = 1,...,T-1
H(t) = m x n measurement equation matrix, t = 1,...,T
b(t) = m x 1 measurement equation forcing term, t = 1,...,T
Prior Dynamic Specification (Approximately Linear Dynamics):
x_{t+1} \approx= F(t)x_t + a(t) , t = 1,...,T-1
Prior Measurement Specification (Approximately Linear Measurements):
y_t \approx= H(t)x_t + b(t) , t = 1,...,T
Penalty Weight Specifications for the Incompatibility Cost Function:
D(t) = symmetric positive definite nxn matrix, t = 1,...,T-1
M(t) = symmetric positive definite mxm matrix, t = 1,...,T
Q_0 = symmetric positive definite nxn matrix
p_0 = nx1 column vector
r_0 = scalar
INCOMPATIBILITY COST FUNCTION SPECIFICATIONS:
Let x* = (x*_1,...,x*_T) denote any possible choice for a sequence of
state vector estimates.
Dynamic Cost Associated with the Choice of x*:
T-1
C_D(x*;T) = SUM [x*_{t+1}- F(t)x*_t - a(t)]'D(t)[x*_{t+1}-F(t)x*_t - a(t)]
t=1
Measurement Cost Associated with the Choice of x*:
T
C_M(x*;T) = SUM [y_t - H(t)x*_t - b(t)]'M(t)[y_t - H(t)x*_t - b(t)]
t=1
Initialization Cost Associated with the Choice of x*_1:
C_I(x*,T) = [x*_1'Q_0x*_1] - [2x*_1'p_0] + r_0
Total Incompatibility Cost Associated with the Choice of x* for a Given
Nonnegative Trade-off Value \mu:
C(x*;\mu,T) = \mu[C_D(x*,T)] + C_M(x*,T) + C_I(x*,T) .
GFLS PROBLEM FOR GIVEN \mu AND T:
Determine the trajectory x* which minimizes the total cost function
C(x*;\mu,T). This trajectory is called the "general flexible least
squares (GFLS) solution conditional on \mu and T."
COMPUTER IMPLEMENTATION:
The Fortran program GFLS generates the general flexible least
squares solution in a sequential manner. Comment statements are
interspersed throughout the program GFLS explaining the sequential
updating equations. The order of derivation is as follows:
Time-T Updating Equations, T \geq 1:
In storage at time T \geq 1 are the exogenously given matrices
and vectors F(T), a(T), H(T), b(T), D(T), and M(T) and the
previously calculated (if T exceeds 1) or exogenously given (if T = 1)
matrix, vector, and scalar Q_{T-1}(\mu), p_{T-1}(\mu), and
r_{T-1}(\mu). A new observation y_T is obtained. Determine in
order:
U_T(\mu) = H(T)'M(T)H(T) + Q_{T-1}(\mu)
C(T) = F(T)'D(T)
W_T(\mu) = \mu [C(T)F(T)] + U_T(\mu)
V_T(\mu) = [W_T(\mu)]^{-1}
e_T = y_T - b(T)
z_T(\mu) = H(T)'M(T)e_T + p_{T-1}(\mu)
G_T(\mu) = \mu [V_T(\mu)C(T)]
Q_T(\mu) = \mu D(T)[I - F(T)G_T(\mu)]
p_T(\mu) = G_T(\mu)'z_T(\mu) + Q_T(\mu)'a(T)
s_T(\mu) = V_T(\mu)[z_T(\mu) - \mu C(T)a(T)]
r_T(\mu) = r_{T-1}(\mu) + e_T'M(T)e_T + \mu [a(T)'D(T)a(T)]
- s_T(\mu)'W_T(\mu)s_T(\mu)
GFLS Filtered State Estimate Obtained at time T:
x_T^{FLS}(\mu,T) = [U_T(\mu)]^{-1}z_T(\mu)
GFLS Smoothed State Estimates Obtained at time T:
x_t^{FLS}(\mu,T) = s_t(\mu) + G_t(\mu)x_{t+1}^{FLS}(\mu,T),
for 0 less than or equal to t, and for t less than T.