Linear Correlation between Position Components
We collected position data for a fix point at 17 days and saved them to 17 files. Based on the equations (7) to (9), linear correlation between x,y and y,z and z,x for 17 data files were calculated. The obtained results of correlation coefficient were recorded in Table 2.
Where,r
xy, r
xz and r
yz are linear correlation between x,y and y,z and z,x , respectively. The results of these statistical studying show that the average correlation coefficients are: r
xy=0.58,r
xz=0.42 and z
i=0.51 Hence, the result shows linear correlation between x,y and y,z and x,z.
Table 2. x,y,z correlation coefficients for 17 days
Model Description
ARMA (Auto-Regressive Moving Average) is a famous model, which is used a discrete-time stochastic process. A time varying ARMA model of order (p,q) which is conventionally noted as ARMA(p,q), is mathematically described by [8]:
Where x(k-j) and y(k-i) are the input and output of the model and e(k) is the noise value at time k for k=1,2,... .a
i(k) for i=1,2,...,p and for b
j(k)=1,2,...,q are the sets of parameters which describes the model structure. Without loss of generality, it is always assumed that a
o(k) 1[9].
Proposed Time Varying ARMA Model
Because of linear correlation between x,y and y,z and also x,z , the ARMA model should be of such kinds that their input variables include patterns of positioning errors conjointly. Structure of this time varying ARMA model is as:
For simplicity, we assume in this research P1=P2=P3=P, q1=q2=q3=q and r1=r2=r3=r.
The Calculation Process
We start from dx component. Equation (11) can restate as:
If we rewrite equation (14) in matrix format, We get:
The parameter matrix
q may be estimated using the Least-Squares (LS) method. The LS estimation of the parameter matrix
q can be obtained by [9]:
Once the model parameters a
1i(n),a
1j(n) and a
1k(n) are known, the calculation of dx(n) for an arbitrary n can be accomplished by [9]:
Modeling for dy and dz components are similar to equations (14) to (23).
