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author Daniele Nicolodi <nicolodi@science.unitn.it>
date Mon, 05 Dec 2011 16:20:06 +0100
parents f0afece42f48
children
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Daniele Nicolodi <nicolodi@science.unitn.it>
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1 % Training session Topic 5 exercise 01
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Daniele Nicolodi <nicolodi@science.unitn.it>
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2 %
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Daniele Nicolodi <nicolodi@science.unitn.it>
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3 % System identification in z-domain 1
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Daniele Nicolodi <nicolodi@science.unitn.it>
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4 %
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Daniele Nicolodi <nicolodi@science.unitn.it>
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5 % 1) Generate white noise
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Daniele Nicolodi <nicolodi@science.unitn.it>
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6 % 2) Filter white noise with a miir filter
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Daniele Nicolodi <nicolodi@science.unitn.it>
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7 % 3) Extract transfer function from data
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Daniele Nicolodi <nicolodi@science.unitn.it>
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8 % 4) Fit TF with zDomainFit
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Daniele Nicolodi <nicolodi@science.unitn.it>
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9 % 5) Compare results
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Daniele Nicolodi <nicolodi@science.unitn.it>
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10 % 6) Comments
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Daniele Nicolodi <nicolodi@science.unitn.it>
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11 %
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Daniele Nicolodi <nicolodi@science.unitn.it>
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12 % L FERRAIOLI 22-02-09
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13 %
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Daniele Nicolodi <nicolodi@science.unitn.it>
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14 % $Id: TrainigSession_T5_Ex01.m,v 1.2 2009/02/25 18:18:45 luigi Exp $
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15 %
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16 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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17
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Daniele Nicolodi <nicolodi@science.unitn.it>
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18 %% 1) Generate White Noise
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19
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Daniele Nicolodi <nicolodi@science.unitn.it>
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20 % Build a gaussian random noise data series
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Daniele Nicolodi <nicolodi@science.unitn.it>
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21 a = ao(plist('tsfcn', 'randn(size(t))', 'fs', 1, 'nsecs', 10000,'yunits','m'));
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22
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Daniele Nicolodi <nicolodi@science.unitn.it>
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23 %% 2) Filtering White Noise
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Daniele Nicolodi <nicolodi@science.unitn.it>
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24
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Daniele Nicolodi <nicolodi@science.unitn.it>
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25 % set a proper plist for a miir filter converting meters to volt
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Daniele Nicolodi <nicolodi@science.unitn.it>
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26 plmiir = plist('a',[2 0.3 0.4],'b',[1 .05 .3 0.1],'fs',1,'iunits','m','ounits','V');
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27
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Daniele Nicolodi <nicolodi@science.unitn.it>
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28 % Building the filter
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Daniele Nicolodi <nicolodi@science.unitn.it>
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29 filt = miir(plmiir);
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30
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Daniele Nicolodi <nicolodi@science.unitn.it>
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31 % filtering white noise
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Daniele Nicolodi <nicolodi@science.unitn.it>
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32 ac = filter(a,filt);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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33 ac.simplifyYunits;
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Daniele Nicolodi <nicolodi@science.unitn.it>
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34
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Daniele Nicolodi <nicolodi@science.unitn.it>
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35 %% Checking
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Daniele Nicolodi <nicolodi@science.unitn.it>
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36 % set a plist for psd calculation
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Daniele Nicolodi <nicolodi@science.unitn.it>
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37 plpsd = plist('Nfft',1000);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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38
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Daniele Nicolodi <nicolodi@science.unitn.it>
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39 % calculate psd of white data
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Daniele Nicolodi <nicolodi@science.unitn.it>
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40 axx = psd(a,plpsd);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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41
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Daniele Nicolodi <nicolodi@science.unitn.it>
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42 % calculate psd of colored data
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Daniele Nicolodi <nicolodi@science.unitn.it>
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43 acxx = psd(ac,plpsd);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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44
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Daniele Nicolodi <nicolodi@science.unitn.it>
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45 % plotting psd
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Daniele Nicolodi <nicolodi@science.unitn.it>
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46 iplot(axx,acxx)
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Daniele Nicolodi <nicolodi@science.unitn.it>
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47
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48 %% 3) Extracting transfer function
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Daniele Nicolodi <nicolodi@science.unitn.it>
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49
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Daniele Nicolodi <nicolodi@science.unitn.it>
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50 % settung plist for TF estimation
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Daniele Nicolodi <nicolodi@science.unitn.it>
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51 pltf = plist('Nfft',1000);
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52
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Daniele Nicolodi <nicolodi@science.unitn.it>
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53 % TF estimation
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Daniele Nicolodi <nicolodi@science.unitn.it>
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54 tf = tfe(a,ac,pltf);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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55
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Daniele Nicolodi <nicolodi@science.unitn.it>
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56 % Removing first bins form tf
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Daniele Nicolodi <nicolodi@science.unitn.it>
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57 tf12 = split(tf(1,2),plist('split_type', 'frequencies', 'frequencies', [3e-3 5e-1]));
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Daniele Nicolodi <nicolodi@science.unitn.it>
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58
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Daniele Nicolodi <nicolodi@science.unitn.it>
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59 % plotting
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Daniele Nicolodi <nicolodi@science.unitn.it>
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60 iplot(tf(1,2),tf12)
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Daniele Nicolodi <nicolodi@science.unitn.it>
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61
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Daniele Nicolodi <nicolodi@science.unitn.it>
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62 %% 4.1) Fitting TF - checking residuals log difference
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Daniele Nicolodi <nicolodi@science.unitn.it>
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63
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Daniele Nicolodi <nicolodi@science.unitn.it>
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64 % Perform an automatic fit of TF data in order to identify a discrete
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Daniele Nicolodi <nicolodi@science.unitn.it>
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65 % model. The parameters controlling fit accuracy are
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Daniele Nicolodi <nicolodi@science.unitn.it>
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66 %
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Daniele Nicolodi <nicolodi@science.unitn.it>
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67 % 'ResLogDiff' check that the normalized difference between data and fit
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Daniele Nicolodi <nicolodi@science.unitn.it>
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68 % residuals in log scale is larger than the value specified.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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69 % if d is the input value for the parameter, than the algorithm check that
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Daniele Nicolodi <nicolodi@science.unitn.it>
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70 % log10(data) - log10(res) > d
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Daniele Nicolodi <nicolodi@science.unitn.it>
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71 % In other words if d = 1 an accuracy of 10% is guaranteed
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Daniele Nicolodi <nicolodi@science.unitn.it>
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72 % if d = an accuracy of 1% is guaranteed
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Daniele Nicolodi <nicolodi@science.unitn.it>
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73 % Note1 - fit residuals are calculated as res = model - data
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Daniele Nicolodi <nicolodi@science.unitn.it>
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74 % Note2 - In fitting noisy data, this option should be relaxed otherwise
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Daniele Nicolodi <nicolodi@science.unitn.it>
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75 % the function try to fit accurately the random oscillations due to the
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Daniele Nicolodi <nicolodi@science.unitn.it>
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76 % noise
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Daniele Nicolodi <nicolodi@science.unitn.it>
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77 %
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Daniele Nicolodi <nicolodi@science.unitn.it>
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78 % 'RMSE' if r is the parameter value, then the algorithm check that the
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Daniele Nicolodi <nicolodi@science.unitn.it>
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79 % step-by-step Root Mean Squared Error variation is less than 10^-r
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Daniele Nicolodi <nicolodi@science.unitn.it>
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80
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Daniele Nicolodi <nicolodi@science.unitn.it>
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81 plfit1 = plist('FS',1,... % Sampling frequency for the model filters
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Daniele Nicolodi <nicolodi@science.unitn.it>
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82 'AutoSearch','on',... % Automatically search for a good model
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Daniele Nicolodi <nicolodi@science.unitn.it>
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83 'StartPolesOpt','c1',... % Define the properties of the starting poles - complex distributed in the unitary circle
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Daniele Nicolodi <nicolodi@science.unitn.it>
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84 'maxiter',50,... % maximum number of iteration per model order
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Daniele Nicolodi <nicolodi@science.unitn.it>
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85 'minorder',2,... % minimum model order
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Daniele Nicolodi <nicolodi@science.unitn.it>
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86 'maxorder',9,... % maximum model order
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Daniele Nicolodi <nicolodi@science.unitn.it>
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87 'weightparam','abs',... % assign weights as 1./abs(data)
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Daniele Nicolodi <nicolodi@science.unitn.it>
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88 'ResLogDiff',2,... % Residuals log difference
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Daniele Nicolodi <nicolodi@science.unitn.it>
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89 'ResFlat',[],... % Rsiduals spectral flatness (no need to assign this parameter for the moment)
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Daniele Nicolodi <nicolodi@science.unitn.it>
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90 'RMSE',7,... % Root Mean Squared Error Variation
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Daniele Nicolodi <nicolodi@science.unitn.it>
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91 'Plot','on',... % set the plot on or off
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Daniele Nicolodi <nicolodi@science.unitn.it>
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92 'ForceStability','on',... % force to output a stable ploes model
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Daniele Nicolodi <nicolodi@science.unitn.it>
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93 'CheckProgress','off'); % display fitting progress on the command window
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Daniele Nicolodi <nicolodi@science.unitn.it>
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94
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Daniele Nicolodi <nicolodi@science.unitn.it>
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95 % Do the fit
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Daniele Nicolodi <nicolodi@science.unitn.it>
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96 fobj1 = zDomainFit(tf12,plfit1);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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97 % setting input and output units for fitted model
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Daniele Nicolodi <nicolodi@science.unitn.it>
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98 fobj1(:).setIunits('m');
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Daniele Nicolodi <nicolodi@science.unitn.it>
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99 fobj1(:).setOunits('V');
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Daniele Nicolodi <nicolodi@science.unitn.it>
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100
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Daniele Nicolodi <nicolodi@science.unitn.it>
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101 %% 4.2) Fitting TF - checking residuals spectral flatness
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Daniele Nicolodi <nicolodi@science.unitn.it>
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102
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Daniele Nicolodi <nicolodi@science.unitn.it>
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103 % Perform an automatic fit of TF data in order to identify a discrete
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Daniele Nicolodi <nicolodi@science.unitn.it>
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104 % model. The parameters controlling fit accuracy are
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Daniele Nicolodi <nicolodi@science.unitn.it>
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105 %
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Daniele Nicolodi <nicolodi@science.unitn.it>
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106 % 'ResFlat' Check that normalized fit residuals spectral flatness is larger
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Daniele Nicolodi <nicolodi@science.unitn.it>
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107 % than the parameter value. Admitted values for the parameter are 0<r<1.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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108 % Suggested values are 0.5<r<0.7.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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109 % Note - normalized fit residuals are calculated as
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Daniele Nicolodi <nicolodi@science.unitn.it>
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110 % res = (model - data)./data
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Daniele Nicolodi <nicolodi@science.unitn.it>
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111 % Note2 - This option could be useful with noisy data, when ResLogDiff
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Daniele Nicolodi <nicolodi@science.unitn.it>
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112 % option is not useful
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Daniele Nicolodi <nicolodi@science.unitn.it>
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113 %
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Daniele Nicolodi <nicolodi@science.unitn.it>
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114 % 'RMSE' if r is the parameter value, then the algorithm check that the
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Daniele Nicolodi <nicolodi@science.unitn.it>
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115 % step-by-step Root Mean Squared Error variation is less than 10^-r
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Daniele Nicolodi <nicolodi@science.unitn.it>
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116
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Daniele Nicolodi <nicolodi@science.unitn.it>
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117 plfit2 = plist('FS',1,... % Sampling frequency for the model filters
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Daniele Nicolodi <nicolodi@science.unitn.it>
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118 'AutoSearch','on',... % Automatically search for a good model
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Daniele Nicolodi <nicolodi@science.unitn.it>
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119 'StartPolesOpt','c1',... % Define the properties of the starting poles - complex distributed in the unitary circle
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Daniele Nicolodi <nicolodi@science.unitn.it>
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120 'maxiter',50,... % maximum number of iteration per model order
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Daniele Nicolodi <nicolodi@science.unitn.it>
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121 'minorder',2,... % minimum model order
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Daniele Nicolodi <nicolodi@science.unitn.it>
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122 'maxorder',9,... % maximum model order
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Daniele Nicolodi <nicolodi@science.unitn.it>
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123 'weightparam','abs',... % assign weights as 1./abs(data)
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Daniele Nicolodi <nicolodi@science.unitn.it>
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124 'ResLogDiff',[],... % Residuals log difference (no need to assign this parameter for the moment)
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Daniele Nicolodi <nicolodi@science.unitn.it>
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125 'ResFlat',0.65,... % Rsiduals spectral flatness
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Daniele Nicolodi <nicolodi@science.unitn.it>
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126 'RMSE',7,... % Root Mean Squared Error Variation
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Daniele Nicolodi <nicolodi@science.unitn.it>
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127 'Plot','on',... % set the plot on or off
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Daniele Nicolodi <nicolodi@science.unitn.it>
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128 'ForceStability','on',... % force to output a stable ploes model
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Daniele Nicolodi <nicolodi@science.unitn.it>
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129 'CheckProgress','off'); % display fitting progress on the command window
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Daniele Nicolodi <nicolodi@science.unitn.it>
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130
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Daniele Nicolodi <nicolodi@science.unitn.it>
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131 fobj2 = zDomainFit(tf12,plfit2);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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132 % setting input and output units for fitted model
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Daniele Nicolodi <nicolodi@science.unitn.it>
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133 fobj2(:).setIunits('m');
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Daniele Nicolodi <nicolodi@science.unitn.it>
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134 fobj2(:).setOunits('V');
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Daniele Nicolodi <nicolodi@science.unitn.it>
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135
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Daniele Nicolodi <nicolodi@science.unitn.it>
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136 %% 5) Evaluating accuracy
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Daniele Nicolodi <nicolodi@science.unitn.it>
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137
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Daniele Nicolodi <nicolodi@science.unitn.it>
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138 % fobj1 and fobj2 are parallel bank of miir filters. Therofore we can
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Daniele Nicolodi <nicolodi@science.unitn.it>
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139 % calculate filter response and compare with our starting model
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Daniele Nicolodi <nicolodi@science.unitn.it>
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140
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Daniele Nicolodi <nicolodi@science.unitn.it>
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141 % set plist for filter response
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Daniele Nicolodi <nicolodi@science.unitn.it>
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142 plrsp = plist('bank','parallel','f1',1e-5,'f2',0.5,'nf',100,'scale','log');
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Daniele Nicolodi <nicolodi@science.unitn.it>
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143
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Daniele Nicolodi <nicolodi@science.unitn.it>
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144 % calculate filters responses
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Daniele Nicolodi <nicolodi@science.unitn.it>
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145 rfilt = resp(filt,plrsp); % strating model response
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Daniele Nicolodi <nicolodi@science.unitn.it>
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146 rfobj1 = resp(fobj1,plrsp); % result of first fit
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Daniele Nicolodi <nicolodi@science.unitn.it>
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147 rfobj2 = resp(fobj2,plrsp); % result of second fit
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Daniele Nicolodi <nicolodi@science.unitn.it>
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148
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Daniele Nicolodi <nicolodi@science.unitn.it>
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149 % plotting filters
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Daniele Nicolodi <nicolodi@science.unitn.it>
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150 iplot(rfilt,rfobj1,rfobj2)
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Daniele Nicolodi <nicolodi@science.unitn.it>
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151
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Daniele Nicolodi <nicolodi@science.unitn.it>
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152 % plotting difference
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Daniele Nicolodi <nicolodi@science.unitn.it>
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153 iplot(abs(rfobj1-rfilt),abs(rfobj2-rfilt))
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Daniele Nicolodi <nicolodi@science.unitn.it>
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154
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Daniele Nicolodi <nicolodi@science.unitn.it>
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155 %% 6) Comments
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Daniele Nicolodi <nicolodi@science.unitn.it>
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156
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Daniele Nicolodi <nicolodi@science.unitn.it>
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157 % Decomposition in partial fractions of the starting filter provides
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Daniele Nicolodi <nicolodi@science.unitn.it>
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diff changeset
158 % residues =
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
159 %
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
160 % 0.531297296250259 - 0.28636710753776i
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
161 % 0.531297296250259 + 0.28636710753776i
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
162 % 0.937405407499481
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
163 %
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
164 %
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
165 % poles =
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
166 %
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
167 % 0.112984252455744 + 0.591265379634664i
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
168 % 0.112984252455744 - 0.591265379634664i
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
169 % -0.275968504911487
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
170 %
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
171 % If we look inside the fitted objects fobj1 and fobj2 the following
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
172 % results can be extracted
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
173 % rsidues =
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
174 % 0.531391341497363 - i*0.286395031447695
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
175 % 0.531391341497363 + i*0.286395031447695
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
176 % 0.937201773620951
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
177 %
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
178 % poles =
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
179 % 0.112971704620295 + i*0.591204610859687
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
180 % 0.112971704620295 - i*0.591204610859687
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
181 % -0.276032231799774
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
182 %
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
183 % Model order is correct and residues and poles values are accurate to the
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
184 % third decimal digit.
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
185 %
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
186 % With low order model the algorithm works fine. For higher order model the
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
187 % algorithm is capable to provide a model within the prescribed accuracy
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
188 % but the order could be not guaranteed. Lets see what happen with higher
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
189 % order models opening the second exercise script:
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
190 % TrainingSession_T5_Ex02.m
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
191
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
192