5. Conclusions

Taking the approximated values as <sup>x</sup> <sup>¼</sup> ½ � 0 0 thres 0 0 <sup>T</sup> <sup>≈</sup> <sup>0</sup><sup>T</sup> and the same loop elements given above, the iteration numbers are handled as 138 (X), 178 (Y), 78 (Z), 16 (S) in 64Bit Python. For <sup>x</sup> <sup>¼</sup> ½ � 0 0 <sup>c</sup><sup>0</sup> 0 0 <sup>T</sup> they are 33 (X), 235 (Y), 96 (Z), 11 (S) in 64Bit Python. Different approximated value selections cause different iteration numbers (namely convergence rate).

Since the estimated standard deviations <sup>b</sup>σ<sup>0</sup> = {�56.552, �38.018, �51.901, �0.491} (for X, Y, Z, S in Table 1) of the CPT functions for the coordinates of the C06 satellite are not statically equal to their expected values (σ<sup>0</sup> ¼ �5 cm), the CPT model should be expanded. As an example,

The added terms represent an amplitude f <sup>ϕ</sup>, an initial phase g<sup>ϕ</sup> and a frequency h<sup>ϕ</sup> of a new

<sup>f</sup> <sup>0</sup> <sup>¼</sup> maxð Þ absð Þ <sup>b</sup><sup>ε</sup> <sup>g</sup><sup>0</sup> <sup>¼</sup> arcsin <sup>b</sup>ε1=<sup>f</sup> <sup>0</sup>

� � � arcsin <sup>b</sup>ε1=<sup>f</sup> <sup>0</sup> � � � � <sup>=</sup>f g <sup>t</sup><sup>2</sup> � <sup>t</sup><sup>1</sup>

� �<sup>T</sup> <sup>¼</sup> <sup>b</sup><sup>a</sup> <sup>b</sup><sup>b</sup> <sup>b</sup><sup>c</sup> <sup>b</sup><sup>d</sup> <sup>b</sup>e f <sup>0</sup> <sup>g</sup><sup>0</sup> <sup>h</sup><sup>0</sup>

The approximate values are substituted in the following linearized model as initial values for

� � � � h i

1 tj sin d<sup>0</sup> þ e<sup>0</sup> tj � �

c<sup>0</sup> cos d<sup>0</sup> þ e<sup>0</sup> tj � �

tj c<sup>0</sup> cos d<sup>0</sup> þ e<sup>0</sup> tj � �

cos g<sup>0</sup> þ h<sup>0</sup> tj � �

�f <sup>0</sup> sin g<sup>0</sup> þ h<sup>0</sup> tj � �

�tj f <sup>0</sup> sin g<sup>0</sup> þ h<sup>0</sup> tj

After the evaluation, the improved solution for the C06 satellite is represented in Table 2. We can readily see the improvements upon the downs of the standard deviations from <sup>b</sup>σ<sup>0</sup> <sup>=</sup> {�56.552, �38.018, �51.901, �0.491} (Table 1) into <sup>b</sup>σ<sup>0</sup> <sup>=</sup> {�0.178, �0.137, �0.191, �0.003}

� �

� �

wave carried by first (sine) wave. After the first estimation with respect to Eq. (12),

� � <sup>þ</sup> <sup>f</sup> <sup>ϕ</sup> cos <sup>g</sup><sup>ϕ</sup> <sup>þ</sup> <sup>h</sup><sup>ϕ</sup> tj

� �

, j∈f g 1; 2;…; n ¼ 96 . The approximate value

h i<sup>T</sup>

� � <sup>þ</sup> <sup>e</sup><sup>0</sup> cos <sup>h</sup><sup>0</sup> <sup>þ</sup> <sup>g</sup><sup>0</sup> tj

4.1. An expanded model example by an auxiliary cosine wave

three more unknowns are added to the model given in Eq. (12)

we can choose the approximate values of the new parameters as

<sup>h</sup><sup>0</sup> <sup>¼</sup> arcsin <sup>b</sup>ε2=<sup>f</sup> <sup>0</sup>

l<sup>j</sup> ¼ ϕ<sup>j</sup> � a<sup>0</sup> þ b<sup>0</sup> tj þ c<sup>0</sup> sin d<sup>0</sup> þ f <sup>0</sup> tj

AT <sup>j</sup> ¼

h i n o � �

x ¼ a<sup>0</sup> b<sup>0</sup> c<sup>0</sup> d<sup>0</sup> e<sup>0</sup> f <sup>0</sup> g<sup>0</sup> h<sup>0</sup>

from <sup>b</sup>ε<sup>j</sup> <sup>¼</sup> <sup>ϕ</sup><sup>j</sup> � <sup>b</sup><sup>a</sup> <sup>þ</sup> <sup>b</sup>b tj <sup>þ</sup>b<sup>c</sup> sin <sup>b</sup><sup>d</sup> <sup>þ</sup>be tj

the loop in LS estimation.

68 Optimization Algorithms - Examples

vector of the expanded model by a new wave is:

ϕ<sup>j</sup> ¼ a<sup>ϕ</sup> þ b<sup>ϕ</sup> tj þ c<sup>ϕ</sup> sin d<sup>ϕ</sup> þ e<sup>ϕ</sup> tj

In this chapter, the least squares (LS) estimations of the artificial satellite orbital movements by a combination of polynomial and trigonometric (CPT) functions have been given after a general overview has been made on the hard and soft computations. In practice, the orbital motions are modeled on Keplerian orbital elements. In contrary to this, the coordinate components have been selected for this chapter due to the nonlinear relations of the components and the unknowns which are the elements of CPT functions. The relations cause inconsistencies in the LS solutions. The inconsistencies result from the two injectivity defects, c-defects and i-defects. We can readily see the defects from the differences of the convergence rates (in other words the iteration numbers) in different computer platforms and architectures as shown in the chapter. The defects are not fully removed as long as not change the mathematic models. However, we can surpass the effects of those defects in part by means of the pseudoinverse based on the eigendecomposition or the singular value decomposition (SVD) as in here. The surjectivity defect (ds) of the CPT functions not including the datum defects (d-defects) was eliminated by the LS objective function.

Appendix

Epoch(j) tjt0 [h] Xj [km] Yj [km] Zj [km] tj [μsec] 0.00 17531.307506 21541.792054 31834.209680 691.175390 0.25 19992.147900 20678.913485 30924.464057 691.487182 0.50 22367.333395 19727.437365 29882.214843 691.798804 0.75 24646.589754 18691.353300 28711.797065 692.110527 1.00 26820.032098 17575.021170 27418.102912 692.422088 1.25 28878.208475 16383.154007 26006.563346 692.733923 1.50 30812.142166 15120.799311 24483.127129 693.045689 1.75 32613.372313 13793.318783 22854.237453 693.357416 2.00 34273.992670 12406.366506 21126.806228 693.668853 2.25 35786.688260 10965.865676 19308.186097 693.980453 2.50 37144.769753 9477.983947 17406.140275 694.291853 2.75 38342.205344 7949.107471 15428.810302 694.603337 3.00 39373.649964 6385.813755 13384.681845 694.914789 3.25 40234.471624 4794.843425 11282.548651 695.226157 3.50 40920.774721 3183.071022 9131.474825 695.537828 3.75 41429.420160 1557.474971 6940.755568 695.849062 4.00 41758.042128 74.893170 4719.876568 696.160359 4.25 41905.061383 1706.940006 2478.472204 696.471627 4.50 41869.694990 3331.562017 226.282782 696.782836 4.75 41651.962330 4941.677516 2026.889008 697.094239 5.00 41252.687387 6530.258698 4271.222183 697.405373 5.25 40673.497209 8090.363611 6496.921900 697.716659 5.50 39916.816531 9615.167884 8694.263981 698.028206 5.75 38985.858544 11097.996030 10853.639319 698.339619 6.00 37884.611845 12532.352178 12965.597898 698.651055 6.25 36617.823589 13911.950035 15020.892223 698.962566 6.50 35190.978917 15230.741942 17010.519896 699.273893 6.75 33610.276761 16482.946839 18925.765115 699.585363 7.00 31882.602129 17663.077000 20758.238873 699.896674 7.25 30015.495009 18765.963379 22499.917620 700.208082 7.50 28017.116067 19786.779434 24143.180195 700.519378 7.75 25896.209299 20721.063292 25680.842806 700.830889 8.00 23662.061860 21564.738140 27106.191889 701.142117 8.25 21324.461276 22314.130731 28413.014646 701.453707

On Non-Linearity and Convergence in Non-Linear Least Squares

http://dx.doi.org/10.5772/intechopen.76313

71

For the sake of simplicity for readers, a simple CPT function has been chosen at first. After the initial estimation of the function, the estimated errors vector has been found. We have seen that the errors have had a periodic characteristic in time. So, a new wave defining the error characteristic and been able to carry by the first wave has been planned for expanding the CPTs. It is shown that we can expand a CPT function until ensuring statically equivalency between a priory and a posteriori variances. For instance, one may secure the equivalency of the variances if one would expand more by a new wave in the last estimated model in the same manner.

The convergence rates (upon the iteration numbers) of the LS estimation have been inspected according to the threshold (thres = 5e-12) which is a good value for the estimation of the nonlinear CPT function. An algorithm compiled by different compilers and run in different architectures (with 32 Bit or 64 Bit) changes the convergence rate of the estimations in such as the inconsistent scientific problems. It is also observed that the iteration numbers change when the 64-bit Python software is run on Linux platform which has a different framework than Windows. But, the numbers have not been given in the example part of the chapter. Contrary to inconsistency model, namely in a consistent one, the iteration numbers can take equivalence values in all circumstances. Another way to determine the inconsistency of a model is to obtain its condition number which is computed from a rate of maximum and minimum eigenvalues or of singular values under LS. If the condition number is close to one, the projected model is accepted as a consistent model.

We can use the Soft Computing Methods (SCM) if not an exact mathematical relationship between the data and unknowns. The mathematical model is established by the trial-and-error method in training part of SCM by means of arbitrary weights and activation functions depending on SCM expert forecasts. For the solution of the SCM model during the training, we can use least absolute residuals (LAR) and minmax absolute residuals (MAR) objective functions by the linear programming or the LS estimation as in hard computing method (HCM). In the state, the inconsistency problem can erase whatever the solution method (LAR, MAR or LS) is. The inconstancy can be removed by means of experiences gained from HCMs.

Prior information is very important to select a suitable mathematical model for a scientific problem. For example, comparing a priori variance with a posteriori variance at the end of the estimation is a useful warning to the user to determine the correct mathematical model as seen from the expanded model in the example section of the chapter.

In numerical computation, there are two main phenomena which are the mathematical model (as a combination of functional and stochastic models) and objective function. The solution strategy is of no importance if the same mathematical model and objective function are preferred in the same problem of hard computing. All solution strategies always give same results, only their solution time spans can be distinct from each other (Table 3).
