1. Experimental research in noise influence on estimation precision for polyharmonic model frequencies Natalia Visotska

2. Problem statement (1)Polyharmonic model - discrete time - harmonic frequency,- amplitudes for sin and cos parts of harmonic - number of harmonics - measured values - theoretical model values - noise The main idea for model parameter identification is to minimize the criterion

3. Three-stage method for parameter estimation First stage: “balancing coefficients” are evaluated from relation Second stage: calculated “balancing coefficients” are used in the next equation (2) (3) Equation (3) is transformed to “frequency polynomial” (4) from which m different frequencies can be found (4) Third stage: amplitudes are defined for all harmonic items from the best approximation of (5) with already evaluated frequencies (5)

4. Three-stage method for parameter estimation Noise-free datathree-stage method gives absolutely precise estimation of all parameters of polyharmonic model Noise-corrupted data displacement in parameter evaluations appears (First stage) displacement in “balancing coefficients” (Second stage) frequencies are not precisely evaluated (Third stage) harmonic amplitudes also have displacement Estimated parametersare not equal to

5. Orthogonal regression method “Balancing relation” (2) can be interpreted as a regression equation Output values (left part) and arguments (values in right part) are influenced by noise. Theoretically Ordinary linear regression leads to displaced estimation of parameters Orthogonal regression gives not displaced estimations (in the meaning of mathematical expectation)

6. Integrating difference equations on “sliding interval” Discrete dynamic model or can be represented in integral form Interpreting “balancing relation” (2) as discrete dynamic model it’s integral form can be analogically built. - length of the sliding interval

7. Numerical experiments Time-series for experiments were generated by model (1) with independent uniformly distributed noise. Noise level varied from 0 to 10% of pure polyharmonic signal level. Sliding interval length is 4. Real frequency Estimate by LS Estimate by orthogonal regression FrequencyErrorFrequencyError 0.05 0.0520091.1126030.0503560.313319 0.10.1012100.9103040.1002580.341675 0.20.2017321.2265600.2013350.958326 0.40.4000570.3043040.3998500.3130646 0.60.6005820.5010840.6004290.412734 0.90.269463 0.8999630.273468 1.21.2000350.2945611.2000050.293712 1.61.5999750.3003371.5999770.300108 Typical dependency of estimation error on frequency (noise level 5%, time series length 100)

8. Numerical experiments Typical dependency of estimation error on noise level, time series length 100). ( Noise level Estimation by LSEstimation by “sliding interval” Frequencies ErrorFrequenciesError 0.5%0.7001265800 0.4027570768 0.624082865380.7000227122 0.4001571654 0.05188996321 0.7003121596 0.4014532905 0.388363925040.7001085069 0.4002160055 0.08978823565 0.7003775879 0.4032954795 0.787341470970.7001268259 0.4003112239 0.11716300393 2.5%0.7058363527 0.4551633137 8.682080630790.70074644174 0.40248215615 0.77954468429 0.7088738959 0.4754817748 9.760121662550.70009745386 0.40270307734 0.62241961481 0.7059137619 0.4543798392 8.686191987440.70013033082 0.40143134067 0.34432603714 5%0.7489917282 0.5821464992 23.18231653620.70326640318 0.41053965170 3.220677802312 0.7618563492 0.5964025406 23.50848226370.70254548642 0.41298267142 3.348360503642 0.7568981144 0.5977817311 23.28795738840.70176638686 0.41161431985 2.831692160287

9. Numerical experiments Average square error for estimated frequency Noise level 2% Noise level 20%

10. Conclusions - orthogonal regression method has some advantages in accuracy (in comparison with ordinary LS-method) for “slow” harmonics, for long time- series and in the case of high noise level; - in all cases the method, which is based on integrating on “sliding interval” demonstrated significantly better accuracy then other two methods; - it seems that computational complexity of orthogonal regression method neglects its some advantages, especially minding that much more simple method “sliding interval” brings us to better noise-immunity.

11. Thank You for Your Attention