(original) (raw)

(*^ ::[ Information = "This is a Mathematica Notebook file. It contains ASCII text, and can be transferred by email, ftp, or other text-file transfer utility. It should be read or edited using a copy of Mathematica or MathReader. If you received this as email, use your mail application or copy/paste to save everything from the line containing (*^ down to the line containing ^*) into a plain text file. On some systems you may have to give the file a name ending with ".ma" to allow Mathematica to recognize it as a Notebook. 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In particular, it considers some of the techniques employed by Tim Sauer and described in his article in \cite{WeigendandGershenfeld94}. This notebook makes no attempt to completely implement Sauer's prediction algorithm (Mathematica probably isn't the right language for a practical implementation), but it should give you enough information so that you can understand it better and implement it yourself if you are ambitious. This notebook is not entirely self contained; in order to understand it you should read the introduction to delay coordinate embedding and to the Fourier transform in this tutorial, and you should have read Sauer's article. To begin with choose a simple function to use as an example of a continuous dynamical system. For a somewhat more exotic dynamical system, you might experiment with a chaotic dynamical system defined by the Lorenz equations. See Page 702 in [Wolfram, 1991] for details or check out the notebook Dynamics`LorenzEquations`. Here is plot of the function we use for illustration. *) :[font = input; preserveAspect; startGroup] Plot[Sin[x],{x,0,8 Pi}] ; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; endGroup] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.037894 0.309017 0.294302 [ [(5)] .21328 .30902 0 2 Msboxa [(10)] .40275 .30902 0 2 Msboxa [(15)] .59222 .30902 0 2 Msboxa [(20)] .78169 .30902 0 2 Msboxa [(25)] .97116 .30902 0 2 Msboxa [(-1)] .01131 .01472 1 0 Msboxa [(-0.5)] .01131 .16187 1 0 Msboxa [(0.5)] .01131 .45617 1 0 Msboxa [(1)] .01131 .60332 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .21328 .30902 m .21328 .31527 L s P [(5)] .21328 .30902 0 2 Mshowa p .002 w .40275 .30902 m .40275 .31527 L s P 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(Gaussian) distribution with zero mean and 0.02 standard deviation. *) :[font = input; preserveAspect] gaussian = NormalDistribution[0.0,0.02] ; :[font = input; preserveAspect] (* Generate a time series using the Sin function and our noise source. *) :[font = input; preserveAspect] timeSeries = Table[N[Sin[x]+Random[gaussian]],{x,0,8 Pi, Pi/50}] ; :[font = input; preserveAspect] (* Here's what the time series looks like. *) :[font = input; preserveAspect; startGroup] ListPlot[timeSeries] ; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; endGroup] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.00237501 0.308594 0.285168 [ [(100)] .26131 .30859 0 2 Msboxa [(200)] .49881 .30859 0 2 Msboxa [(300)] .73631 .30859 0 2 Msboxa [(400)] .97382 .30859 0 2 Msboxa [(-1)] .01131 .02343 1 0 Msboxa 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.18762 Mdot .72444 .19851 Mdot .72681 .22388 Mdot .72919 .24515 Mdot .73156 .25877 Mdot .73394 .27248 Mdot .73631 .28842 Mdot .73869 .31163 Mdot .74106 .33428 Mdot .74344 .33162 Mdot .74581 .36143 Mdot .74819 .37354 Mdot .75056 .40455 Mdot .75294 .41129 Mdot .75531 .438 Mdot .75769 .44416 Mdot .76006 .45543 Mdot .76244 .47215 Mdot .76481 .48596 Mdot .76719 .49451 Mdot .76956 .51418 Mdot .77194 .52728 Mdot .77431 .53628 Mdot .77669 .538 Mdot .77906 .55944 Mdot .78144 .56701 Mdot .78381 .5755 Mdot .78619 .57331 Mdot .78856 .5789 Mdot .79094 .58904 Mdot .79331 .5919 Mdot .79569 .59171 Mdot .79806 .59076 Mdot .80044 .59635 Mdot .80281 .58567 Mdot .80519 .59258 Mdot .80756 .58548 Mdot .80994 .59125 Mdot .81231 .58804 Mdot .81469 .55852 Mdot .81706 .56255 Mdot .81944 .5363 Mdot .82181 .53975 Mdot .82419 .52754 Mdot .82656 .52439 Mdot .82894 .50041 Mdot .83131 .49506 Mdot .83369 .47198 Mdot .83606 .47267 Mdot .83844 .45365 Mdot .84081 .42566 Mdot .84319 .40927 Mdot .84556 .39594 Mdot .84794 .38614 Mdot .85031 .3713 Mdot .85269 .33554 Mdot .85506 .33091 Mdot .85744 .30899 Mdot .85981 .28859 Mdot .86219 .26858 Mdot .86456 .25644 Mdot .86694 .2332 Mdot .86931 .2249 Mdot .87169 .1957 Mdot .87406 .19708 Mdot .87644 .17563 Mdot .87881 .15239 Mdot .88119 .14116 Mdot .88356 .12248 Mdot .88594 .11362 Mdot .88831 .09699 Mdot .89069 .08866 Mdot .89306 .08314 Mdot .89544 .06495 Mdot .89781 .05463 Mdot .90019 .05016 Mdot .90257 .04463 Mdot .90494 .01558 Mdot .90732 .01659 Mdot .90969 .02474 Mdot .91207 .01652 Mdot .91444 .0269 Mdot .91682 .0245 Mdot .91919 .02519 Mdot .92157 .02586 Mdot .92394 .03102 Mdot .92632 .03398 Mdot .92869 .04733 Mdot .93107 .04184 Mdot .93344 .04715 Mdot .93582 .06123 Mdot .93819 .06648 Mdot .94057 .06187 Mdot .94294 .08934 Mdot .94532 .10881 Mdot .94769 .11436 Mdot .95007 .13063 Mdot .95244 .14399 Mdot .95482 .15033 Mdot .95719 .17564 Mdot .95957 .19112 Mdot .96194 .20208 Mdot .96432 .21988 Mdot .96669 .23041 Mdot .96907 .25943 Mdot .97144 .28341 Mdot .97382 .28637 Mdot .97619 .32207 Mdot P % End of Graphics MathPictureEnd :[font = input; preserveAspect] (* Delay coordinate embedding relies on using sequences of observations called delay coordinates as proxies for the states of the dynamical system. The space determined by the delay coordinates is called the embedding space. The following procedure computes delay vectors of a specified width from a given time series. *) :[font = input; preserveAspect] ComputeDelayVectors[series_,width_] := Table[Table[series[[i+j]], {j,0,width-1}], {i,1,Length[series]-width}] ; :[font = input; preserveAspect] (* Compute delay vectors of width-2 from the time series data. *) :[font = input; preserveAspect] shortDelayVectors = ComputeDelayVectors[timeSeries,2] ; :[font = input; preserveAspect] (* Consider the delay vector phase-space plot. *) :[font = input; preserveAspect; startGroup] ListPlot[shortDelayVectors, PlotJoined -> True] ; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; endGroup] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.499316 0.461411 0.308594 0.285168 [ [(-1)] .0379 .30859 0 2 Msboxa [(-0.5)] .26861 .30859 0 2 Msboxa [(0.5)] .73002 .30859 0 2 Msboxa [(1)] .96073 .30859 0 2 Msboxa [(-1)] .48682 .02343 1 0 Msboxa [(-0.5)] .48682 .16601 1 0 Msboxa [(0.5)] .48682 .45118 1 0 Msboxa [(1)] .48682 .59376 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .0379 .30859 m .0379 .31484 L s P [(-1)] .0379 .30859 0 2 Mshowa p .002 w .26861 .30859 m .26861 .31484 L s P [(-0.5)] .26861 .30859 0 2 Mshowa p .002 w .73002 .30859 m .73002 .31484 L s P [(0.5)] .73002 .30859 0 2 Mshowa p .002 w .96073 .30859 m .96073 .31484 L s P [(1)] .96073 .30859 0 2 Mshowa p .001 w .08405 .30859 m .08405 .31234 L s P p .001 w .13019 .30859 m .13019 .31234 L s P p .001 w .17633 .30859 m .17633 .31234 L s P p .001 w .22247 .30859 m .22247 .31234 L s P p .001 w .31475 .30859 m .31475 .31234 L s P p .001 w .36089 .30859 m .36089 .31234 L s P p .001 w .40703 .30859 m .40703 .31234 L s P p .001 w .45317 .30859 m .45317 .31234 L s P p .001 w .54546 .30859 m .54546 .31234 L s P p .001 w .5916 .30859 m .5916 .31234 L s P p .001 w .63774 .30859 m .63774 .31234 L s P p .001 w .68388 .30859 m .68388 .31234 L s P p .001 w .77616 .30859 m .77616 .31234 L s P p .001 w .8223 .30859 m .8223 .31234 L s P p .001 w .86844 .30859 m .86844 .31234 L s P p .001 w .91459 .30859 m .91459 .31234 L s P p .002 w 0 .30859 m 1 .30859 L s P p .002 w .49932 .02343 m .50557 .02343 L s P [(-1)] .48682 .02343 1 0 Mshowa p .002 w .49932 .16601 m .50557 .16601 L s P [(-0.5)] .48682 .16601 1 0 Mshowa p .002 w .49932 .45118 m .50557 .45118 L s P [(0.5)] .48682 .45118 1 0 Mshowa p .002 w .49932 .59376 m .50557 .59376 L s P [(1)] .48682 .59376 1 0 Mshowa p .001 w .49932 .05194 m .50307 .05194 L s P p .001 w .49932 .08046 m .50307 .08046 L s P p .001 w .49932 .10898 m .50307 .10898 L s P p .001 w .49932 .13749 m .50307 .13749 L s P p .001 w .49932 .19453 m .50307 .19453 L s P p .001 w .49932 .22304 m .50307 .22304 L s P p .001 w .49932 .25156 m .50307 .25156 L s P p .001 w .49932 .28008 m .50307 .28008 L s P p .001 w .49932 .33711 m .50307 .33711 L s P p .001 w .49932 .36563 m .50307 .36563 L s P p .001 w .49932 .39414 m .50307 .39414 L s P p .001 w .49932 .42266 m .50307 .42266 L s P p .001 w .49932 .47969 m .50307 .47969 L s P p .001 w .49932 .50821 m .50307 .50821 L s P p .001 w .49932 .53673 m .50307 .53673 L s P p .001 w .49932 .56524 m .50307 .56524 L s P p .002 w .49932 0 m .49932 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .004 w .48672 .32934 m .53288 .35372 L .57233 .36824 L .59582 .37289 L .60335 .39255 L .63516 .425 L .68766 .42761 L .69189 .43224 L .69938 .46154 L .74679 .46514 L .75262 .4945 L .80011 .50896 L .82352 .51917 L .84003 .52767 L .85379 .53427 L .86447 .55241 L .89382 .54577 L .88308 .57228 L .92597 .57478 L .93002 .59089 L .95608 .57691 L .93346 .58339 L .94394 .59164 L .95729 .60332 L .97619 .59148 L .95703 .59956 L .9701 .59334 L .96005 .58585 L .94792 .58441 L .94559 .57158 L .92484 .57538 L .93099 .56704 L .91748 .55451 L .89721 .55337 L .89537 .53235 L .86136 .52746 L .85344 .50127 L .81106 .49812 L .80598 .49159 L .79542 .47059 L .76142 .46373 L .75033 .45737 L .74005 .42084 L .68094 .41313 L .66845 .39649 L .64153 .37872 L .61278 .3527 L .57068 .35084 L .56768 .33327 L .53925 .30499 L Mistroke .49349 .29193 L .47235 .2736 L .44269 .2553 L .41308 .23403 L .37866 .22242 L .35988 .20142 L .32591 .19814 L .32059 .16464 L .26639 .15831 L .25614 .14506 L .23472 .13591 L .2199 .11967 L .19362 .09921 L .16053 .08245 L .13341 .07182 L .11621 .06852 L .11087 .05492 L .08887 .04564 L .07385 .04147 L .0671 .04138 L .06695 .02758 L .04462 .02998 L .04851 .02025 L .03276 .02044 L .03307 .01574 L .02547 .02015 L .03261 .01472 L .02381 .02102 L .03401 .03157 L .05108 .04325 L .06999 .03786 L .06125 .05176 L .08375 .06324 L .10233 .06703 L .10845 .0815 L .13186 .09402 L .15212 .10542 L .17057 .11666 L .18876 .12223 L .19778 .14394 L .2329 .15246 L .24668 .17736 L .28698 .18073 L .29243 .20287 L .32826 .22805 L .36899 .2379 L .38493 .25511 L .41278 .27416 L .4436 .28572 L .4623 .30325 L Mistroke .49067 .32756 L .53 .35099 L .56792 .36493 L .59047 .37871 L .61276 .39915 L .64584 .40655 L .65782 .4321 L .69915 .45678 L .73908 .46789 L .75705 .47023 L .76085 .50092 L .81051 .50716 L .8206 .5077 L .82148 .52849 L .85511 .54169 L .87647 .55249 L .89394 .55763 L .90227 .57941 L .9375 .57653 L .93285 .5793 L .93733 .58004 L .93852 .59486 L .9625 .59557 L .96366 .59524 L .96312 .58308 L .94345 .59649 L .96514 .59641 L .96501 .57717 L .93387 .58299 L .94329 .58694 L .94969 .56802 L .91908 .56276 L .91056 .57151 L .92472 .54563 L .88285 .54729 L .88553 .52946 L .85668 .51361 L .83103 .5057 L .81824 .47887 L .77483 .477 L .77181 .45547 L .73696 .44789 L .7247 .44326 L .71722 .41211 L .6668 .39516 L .63938 .37971 L .61438 .35885 L .58062 .34692 L .56133 .32346 L .52337 .30811 L Mistroke .49853 .30381 L .49157 .25779 L .41712 .25852 L .4183 .23948 L .38748 .21747 L .35188 .20659 L .33427 .18325 L .29651 .16782 L .27154 .15021 L .24305 .14137 L .22874 .12408 L .20076 .11272 L .18238 .09992 L .16167 .0923 L .14934 .07215 L .11674 .06132 L .09921 .05617 L .09088 .05037 L .0815 .04071 L .06588 .0433 L .07007 .03455 L .0559 .0319 L .05162 .02242 L .03628 .02174 L .03518 .02466 L .0399 .01969 L .03185 .0296 L .04789 .03577 L .05788 .03243 L .05248 .03932 L .06362 .04878 L .07893 .05135 L .08309 .06875 L .11124 .05846 L .0946 .07238 L .11711 .09194 L .14876 .10054 L .16268 .1051 L .17006 .12345 L .19974 .15236 L .24652 .15317 L .24783 .18238 L .2951 .18541 L .30001 .1992 L .3223 .2292 L .37085 .23533 L .38077 .25811 L .41763 .27465 L .44439 .29376 L .47531 .30746 L Mistroke .49748 .32224 L .52139 .34564 L .55926 .36131 L .58462 .38139 L .61711 .39828 L .64443 .41712 L .67491 .44071 L .71308 .44467 L .71949 .45408 L .73471 .48224 L .78028 .49507 L .80104 .50938 L .8242 .51528 L .83375 .53112 L .85936 .54736 L .88565 .55118 L .89183 .56591 L .91566 .55428 L .89684 .56309 L .9111 .58918 L .95332 .58703 L .94984 .58262 L .9427 .58618 L .94846 .59612 L .96455 .59838 L .96821 .59277 L .95912 .59685 L .96573 .59218 L .95817 .5766 L .93295 .5844 L .94559 .56915 L .9209 .57531 L .93087 .54718 L .88536 .55073 L .89109 .52966 L .85701 .52793 L .85421 .52399 L .84784 .50597 L .81868 .48802 L .78964 .47525 L .76897 .45958 L .74362 .44318 L .71708 .4308 L .69706 .41475 L .67108 .38777 L .62743 .37352 L .60437 .35541 L .57507 .34143 L .55244 .329 L .53234 .29985 L Mistroke .48516 .29202 L .4725 .2818 L .45597 .26109 L .42246 .24351 L .394 .23262 L .37638 .20404 L .33014 .18431 L .29822 .16477 L .26661 .15019 L .24301 .13153 L .21283 .13314 L .21542 .11357 L .18376 .08932 L .14452 .08772 L .14193 .06709 L .10855 .06714 L .10863 .05549 L .08978 .04867 L .07875 .04074 L .06592 .04385 L .07096 .0333 L .05387 .02453 L .03969 .02298 L .03718 .01755 L .02839 .02156 L .03488 .02685 L .04344 .02709 L .04384 .04116 L .0666 .03239 L .05241 .03827 L .06193 .04177 L .06759 .04735 L .07662 .05832 L .09437 .06836 L .11061 .07341 L .11877 .08244 L .13339 .09498 L .15368 .10576 L .17113 .12259 L .19836 .14371 L .23254 .14485 L .23437 .16502 L .26701 .18762 L .30357 .19851 L .3212 .22388 L .36225 .24515 L .39667 .25877 L .4187 .27248 L .44088 .28842 L .46667 .31163 L Mistroke .50423 .33428 L .54087 .33162 L .53658 .36143 L .58481 .37354 L .6044 .40455 L .65458 .41129 L .66549 .438 L .7087 .44416 L .71866 .45543 L .7369 .47215 L .76396 .48596 L .78631 .49451 L .80013 .51418 L .83197 .52728 L .85316 .53628 L .86772 .538 L .8705 .55944 L .90519 .56701 L .91745 .5755 L .93118 .57331 L .92764 .5789 L .93669 .58904 L .95309 .5919 L .95771 .59171 L .95741 .59076 L .95587 .59635 L .96491 .58567 L .94764 .59258 L .95881 .58548 L .94732 .59125 L .95666 .58804 L .95146 .55852 L .9037 .56255 L .91023 .5363 L .86775 .53975 L .87333 .52754 L .85357 .52439 L .84848 .50041 L .80968 .49506 L .80103 .47198 L .76367 .47267 L .7648 .45365 L .73402 .42566 L .68874 .40927 L .66221 .39594 L .64065 .38614 L .62479 .3713 L .60077 .33554 L .54292 .33091 L .53542 .30899 L Mistroke .49995 .28859 L .46695 .26858 L .43457 .25644 L .41493 .2332 L .37732 .2249 L .3639 .1957 L .31664 .19708 L .31889 .17563 L .28418 .15239 L .24658 .14116 L .22839 .12248 L .19818 .11362 L .18385 .09699 L .15693 .08866 L .14346 .08314 L .13453 .06495 L .10509 .05463 L .08839 .05016 L .08116 .04463 L .07222 .01558 L .02521 .01659 L .02685 .02474 L .04004 .01652 L .02673 .0269 L .04352 .0245 L .03965 .02519 L .04076 .02586 L .04185 .03102 L .05019 .03398 L .05498 .04733 L .07657 .04184 L .0677 .04715 L .07629 .06123 L .09908 .06648 L .10757 .06187 L .1001 .08934 L .14455 .10881 L .17606 .11436 L .18503 .13063 L .21137 .14399 L .23298 .15033 L .24324 .17564 L .28419 .19112 L .30924 .20208 L .32697 .21988 L .35577 .23041 L .37281 .25943 L .41976 .28341 L .45856 .28637 L Mfstroke % End of Graphics MathPictureEnd :[font = input; preserveAspect] (* Sauer suggests computing somewhat longer delay vectors of width w and then applying a transformation to obtain delay vectors of width m (m < w) to use for the actual embedding. The transformation consists of applying a filter that eliminates all high frequency components of the window, preserving only the lowest m/2 frequency contributions, and keeping only m points of the filtered w-vector. The effect of the transformation is equivalent to taking the discrete Fourier transform of order w of each width-w delay vector, set to zero all but the lowest m/2 frequency contributions, and then take the inverse Fourier transform of order m using the remaining frequencies, to obtain the filtered width-m delay vector. In the following implementation, we take the absolute value to eliminate the imaginary parts in the resulting filtered delay vectors. *) :[font = input; preserveAspect] Filter[v_,m_] := Abs[InverseFourier[Take[N[Fourier[v]],m]]] ; :[font = input; preserveAspect] (* Consider some longer (w = 4) delay vectors. *) :[font = input; preserveAspect] longDelayVectors = ComputeDelayVectors[timeSeries,4] ; :[font = input; preserveAspect] (* Filter these vectors to obtain the (m = 2) filtered delay vectors. *) :[font = input; preserveAspect] filteredDelayVectors = Map[(Filter[#,2])&,longDelayVectors] ; :[font = input; preserveAspect] (* Consider the filtered delay vector phase-space plot. *) :[font = input; preserveAspect; startGroup] ListPlot[filteredDelayVectors, PlotJoined -> True] ; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; endGroup] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.656501 0.0147151 0.407942 [ [(0.2)] .15511 .01472 0 2 Msboxa [(0.4)] .28641 .01472 0 2 Msboxa [(0.6)] .41771 .01472 0 2 Msboxa [(0.8)] .54901 .01472 0 2 Msboxa [(1)] .68031 .01472 0 2 Msboxa [(1.2)] .81161 .01472 0 2 Msboxa [(1.4)] .94291 .01472 0 2 Msboxa [(0.2)] .01131 .0963 1 0 Msboxa [(0.4)] .01131 .17789 1 0 Msboxa [(0.6)] .01131 .25948 1 0 Msboxa [(0.8)] .01131 .34107 1 0 Msboxa [(1)] .01131 .42266 1 0 Msboxa [(1.2)] .01131 .50425 1 0 Msboxa [(1.4)] .01131 .58583 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .15511 .01472 m .15511 .02097 L s P [(0.2)] .15511 .01472 0 2 Mshowa p .002 w .28641 .01472 m .28641 .02097 L s P [(0.4)] .28641 .01472 0 2 Mshowa p .002 w .41771 .01472 m .41771 .02097 L s P [(0.6)] .41771 .01472 0 2 Mshowa p .002 w .54901 .01472 m .54901 .02097 L s P 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from a linear transformation and scaling, the result doesn't look a lot different, but you should try varying the amount of noise and experiment with different values for the parameters m and w. Sauer mentions that this method constitutes performing a low-pass filter on the length w delay vectors. He writes that "In the absence of noise, there is no advantage to this method over deciminating the series by a factor of w/m (choosing every w/m point). In the presence of high-frequency noise (though digitization error, for example), however, increased representation accuracy is achieved by this type of down sampling." The w parameter should depend on the sampling frequency and be independent of m. Note that if w is a power of 2, this makes using the fast Fourier transform more efficient. Note also that there all sorts of ways stochasticity can complicate time series analysis. In the above generated data, the underlying dynamical system is deterministic but the observations are noisy. Consider a contrived case in which the system is stochastic. Suppose a system that proceeds in discrete steps. From step i to step j the system is deterministic. But at the end of each step the state is perturbed (instantaneously) by a Gaussian disturbance. In this contrived example, suppose that the observations are noiseless. How does the behavior of this system differ from the behavior we would expect of a completely deterministic with Gaussian observation noise. *) :[font = input; preserveAspect] (* Now consider another (even more contrived) time series. *) :[font = input; preserveAspect] timeSeries = Join[Table[N[Sin[t]], {t, 0, 2 Pi, Pi/100}], Table[N[Sin[2 t - 4 Pi/1000]], {t, 2 Pi, 4 Pi, Pi/100}], Table[N[Sin[4 t - 12 Pi/1000]], {t, 4 Pi, 6 Pi, Pi/100}]] ; :[font = input; preserveAspect] (* In this time series the frequency doubles every 2 Pi and there are small delays introduced at the first and second frequency changes. These delays are a kludge to keep the amount of data required for the experiment to a minimum. 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.30968 .13603 Mdot .31126 .14359 Mdot .31284 .15132 Mdot .31442 .15921 Mdot .316 .16724 Mdot .31758 .17541 Mdot .31916 .18371 Mdot .32074 .19214 Mdot .32232 .20068 Mdot .3239 .20933 Mdot .32548 .21807 Mdot .32706 .22691 Mdot .32863 .23583 Mdot .33021 .24482 Mdot .33179 .25387 Mdot .33337 .26298 Mdot .33495 .27213 Mdot .33653 .28132 Mdot .33811 .29054 Mdot .33969 .29977 Mdot .34127 .30902 Mdot .34285 .30532 Mdot .34443 .3238 Mdot .34601 .34223 Mdot .34759 .36053 Mdot .34917 .37862 Mdot .35075 .39644 Mdot .35233 .41391 Mdot .35391 .43097 Mdot .35548 .44755 Mdot .35706 .46358 Mdot .35864 .479 Mdot .36022 .49375 Mdot .3618 .50777 Mdot .36338 .52101 Mdot .36496 .53341 Mdot .36654 .54492 Mdot .36812 .5555 Mdot .3697 .56511 Mdot .37128 .57371 Mdot .37286 .58127 Mdot .37444 .58775 Mdot .37602 .59313 Mdot .3776 .59739 Mdot .37918 .60051 Mdot .38075 .60248 Mdot .38233 .6033 Mdot .38391 .60295 Mdot .38549 .60144 Mdot .38707 .59878 Mdot .38865 .59497 Mdot .39023 .59004 Mdot .39181 .58399 Mdot .39339 .57686 Mdot .39497 .56868 Mdot .39655 .55947 Mdot .39813 .54927 Mdot .39971 .53812 Mdot .40129 .52607 Mdot .40287 .51316 Mdot .40445 .49945 Mdot .40603 .48498 Mdot .4076 .46982 Mdot .40918 .45403 Mdot .41076 .43766 Mdot .41234 .42079 Mdot .41392 .40347 Mdot .4155 .38578 Mdot .41708 .36779 Mdot .41866 .34957 Mdot .42024 .33119 Mdot .42182 .31272 Mdot .4234 .29423 Mdot .42498 .2758 Mdot .42656 .25751 Mdot .42814 .23941 Mdot .42972 .2216 Mdot .4313 .20412 Mdot .43288 .18707 Mdot .43445 .17049 Mdot .43603 .15446 Mdot .43761 .13904 Mdot .43919 .12429 Mdot .44077 .11027 Mdot .44235 .09703 Mdot .44393 .08463 Mdot .44551 .07311 Mdot .44709 .06253 Mdot .44867 .05292 Mdot .45025 .04432 Mdot .45183 .03677 Mdot .45341 .03028 Mdot .45499 .0249 Mdot .45657 .02064 Mdot .45815 .01752 Mdot .45973 .01555 Mdot .4613 .01474 Mdot .46288 .01509 Mdot .46446 .0166 Mdot .46604 .01926 Mdot .46762 .02306 Mdot .4692 .028 Mdot .47078 .03404 Mdot .47236 .04117 Mdot .47394 .04936 Mdot .47552 .05857 Mdot .4771 .06877 Mdot .47868 .07991 Mdot .48026 .09197 Mdot .48184 .10487 Mdot .48342 .11859 Mdot .485 .13305 Mdot .48658 .14821 Mdot .48815 .16401 Mdot .48973 .18037 Mdot .49131 .19725 Mdot .49289 .21456 Mdot .49447 .23225 Mdot .49605 .25024 Mdot .49763 .26847 Mdot .49921 .28685 Mdot .50079 .30532 Mdot .50237 .3238 Mdot .50395 .34223 Mdot .50553 .36053 Mdot .50711 .37862 Mdot .50869 .39644 Mdot .51027 .41391 Mdot .51185 .43097 Mdot .51342 .44755 Mdot .515 .46358 Mdot .51658 .479 Mdot .51816 .49375 Mdot .51974 .50777 Mdot .52132 .52101 Mdot .5229 .53341 Mdot .52448 .54492 Mdot .52606 .5555 Mdot .52764 .56511 Mdot .52922 .57371 Mdot .5308 .58127 Mdot .53238 .58775 Mdot .53396 .59313 Mdot .53554 .59739 Mdot .53712 .60051 Mdot .5387 .60248 Mdot .54027 .6033 Mdot .54185 .60295 Mdot .54343 .60144 Mdot .54501 .59878 Mdot .54659 .59497 Mdot .54817 .59004 Mdot .54975 .58399 Mdot .55133 .57686 Mdot .55291 .56868 Mdot .55449 .55947 Mdot .55607 .54927 Mdot .55765 .53812 Mdot .55923 .52607 Mdot .56081 .51316 Mdot .56239 .49945 Mdot .56397 .48498 Mdot .56555 .46982 Mdot .56712 .45403 Mdot .5687 .43766 Mdot .57028 .42079 Mdot .57186 .40347 Mdot .57344 .38578 Mdot .57502 .36779 Mdot .5766 .34957 Mdot .57818 .33119 Mdot .57976 .31272 Mdot .58134 .29423 Mdot .58292 .2758 Mdot .5845 .25751 Mdot .58608 .23941 Mdot .58766 .2216 Mdot .58924 .20412 Mdot .59082 .18707 Mdot .5924 .17049 Mdot .59397 .15446 Mdot .59555 .13904 Mdot .59713 .12429 Mdot .59871 .11027 Mdot .60029 .09703 Mdot .60187 .08463 Mdot .60345 .07311 Mdot .60503 .06253 Mdot .60661 .05292 Mdot .60819 .04432 Mdot .60977 .03677 Mdot .61135 .03028 Mdot .61293 .0249 Mdot .61451 .02064 Mdot .61609 .01752 Mdot .61767 .01555 Mdot .61925 .01474 Mdot .62082 .01509 Mdot .6224 .0166 Mdot .62398 .01926 Mdot .62556 .02306 Mdot .62714 .028 Mdot .62872 .03404 Mdot .6303 .04117 Mdot .63188 .04936 Mdot .63346 .05857 Mdot .63504 .06877 Mdot .63662 .07991 Mdot .6382 .09197 Mdot .63978 .10487 Mdot .64136 .11859 Mdot .64294 .13305 Mdot .64452 .14821 Mdot .64609 .16401 Mdot .64767 .18037 Mdot .64925 .19725 Mdot .65083 .21456 Mdot .65241 .23225 Mdot .65399 .25024 Mdot .65557 .26847 Mdot .65715 .28685 Mdot .65873 .30532 Mdot .66031 .29792 Mdot .66189 .33487 Mdot .66347 .37141 Mdot .66505 .40697 Mdot .66663 .44098 Mdot .66821 .47291 Mdot .66979 .50225 Mdot .67137 .52855 Mdot .67294 .55138 Mdot .67452 .5704 Mdot .6761 .58529 Mdot .67768 .59582 Mdot .67926 .60183 Mdot .68084 .60323 Mdot .68242 .59998 Mdot .684 .59214 Mdot .68558 .57984 Mdot .68716 .56327 Mdot .68874 .54269 Mdot .69032 .51842 Mdot .6919 .49085 Mdot .69348 .46042 Mdot .69506 .42759 Mdot .69664 .3929 Mdot .69822 .35688 Mdot .69979 .32011 Mdot .70137 .28316 Mdot .70295 .24662 Mdot .70453 .21107 Mdot .70611 .17706 Mdot .70769 .14513 Mdot .70927 .11578 Mdot .71085 .08949 Mdot .71243 .06665 Mdot .71401 .04764 Mdot .71559 .03275 Mdot .71717 .02221 Mdot .71875 .0162 Mdot .72033 .01481 Mdot .72191 .01805 Mdot .72349 .02589 Mdot .72507 .03819 Mdot .72664 .05476 Mdot .72822 .07534 Mdot .7298 .09961 Mdot .73138 .12718 Mdot .73296 .15762 Mdot .73454 .19044 Mdot .73612 .22514 Mdot .7377 .26115 Mdot .73928 .29792 Mdot .74086 .33487 Mdot .74244 .37141 Mdot .74402 .40697 Mdot .7456 .44098 Mdot .74718 .47291 Mdot .74876 .50225 Mdot .75034 .52855 Mdot .75192 .55138 Mdot .75349 .5704 Mdot .75507 .58529 Mdot .75665 .59582 Mdot .75823 .60183 Mdot .75981 .60323 Mdot .76139 .59998 Mdot .76297 .59214 Mdot .76455 .57984 Mdot .76613 .56327 Mdot .76771 .54269 Mdot .76929 .51842 Mdot .77087 .49085 Mdot .77245 .46042 Mdot .77403 .42759 Mdot .77561 .3929 Mdot .77719 .35688 Mdot .77876 .32011 Mdot .78034 .28316 Mdot .78192 .24662 Mdot .7835 .21107 Mdot .78508 .17706 Mdot .78666 .14513 Mdot .78824 .11578 Mdot .78982 .08949 Mdot .7914 .06665 Mdot .79298 .04764 Mdot .79456 .03275 Mdot .79614 .02221 Mdot .79772 .0162 Mdot .7993 .01481 Mdot .80088 .01805 Mdot .80246 .02589 Mdot .80404 .03819 Mdot .80561 .05476 Mdot .80719 .07534 Mdot .80877 .09961 Mdot .81035 .12718 Mdot .81193 .15762 Mdot .81351 .19044 Mdot .81509 .22514 Mdot .81667 .26115 Mdot .81825 .29792 Mdot .81983 .33487 Mdot .82141 .37141 Mdot .82299 .40697 Mdot .82457 .44098 Mdot .82615 .47291 Mdot .82773 .50225 Mdot .82931 .52855 Mdot .83089 .55138 Mdot .83246 .5704 Mdot .83404 .58529 Mdot .83562 .59582 Mdot .8372 .60183 Mdot .83878 .60323 Mdot .84036 .59998 Mdot .84194 .59214 Mdot .84352 .57984 Mdot .8451 .56327 Mdot .84668 .54269 Mdot .84826 .51842 Mdot .84984 .49085 Mdot .85142 .46042 Mdot .853 .42759 Mdot .85458 .3929 Mdot .85616 .35688 Mdot .85774 .32011 Mdot .85931 .28316 Mdot .86089 .24662 Mdot .86247 .21107 Mdot .86405 .17706 Mdot .86563 .14513 Mdot .86721 .11578 Mdot .86879 .08949 Mdot .87037 .06665 Mdot .87195 .04764 Mdot .87353 .03275 Mdot .87511 .02221 Mdot .87669 .0162 Mdot .87827 .01481 Mdot .87985 .01805 Mdot .88143 .02589 Mdot .88301 .03819 Mdot .88459 .05476 Mdot .88616 .07534 Mdot .88774 .09961 Mdot .88932 .12718 Mdot .8909 .15762 Mdot .89248 .19044 Mdot .89406 .22514 Mdot .89564 .26115 Mdot .89722 .29792 Mdot .8988 .33487 Mdot .90038 .37141 Mdot .90196 .40697 Mdot .90354 .44098 Mdot .90512 .47291 Mdot .9067 .50225 Mdot .90828 .52855 Mdot .90986 .55138 Mdot .91143 .5704 Mdot .91301 .58529 Mdot .91459 .59582 Mdot .91617 .60183 Mdot .91775 .60323 Mdot .91933 .59998 Mdot .92091 .59214 Mdot .92249 .57984 Mdot .92407 .56327 Mdot .92565 .54269 Mdot .92723 .51842 Mdot .92881 .49085 Mdot .93039 .46042 Mdot .93197 .42759 Mdot .93355 .3929 Mdot .93513 .35688 Mdot .93671 .32011 Mdot .93828 .28316 Mdot .93986 .24662 Mdot .94144 .21107 Mdot .94302 .17706 Mdot .9446 .14513 Mdot .94618 .11578 Mdot .94776 .08949 Mdot .94934 .06665 Mdot .95092 .04764 Mdot .9525 .03275 Mdot .95408 .02221 Mdot .95566 .0162 Mdot .95724 .01481 Mdot .95882 .01805 Mdot .9604 .02589 Mdot .96198 .03819 Mdot .96356 .05476 Mdot .96513 .07534 Mdot .96671 .09961 Mdot .96829 .12718 Mdot .96987 .15762 Mdot .97145 .19044 Mdot .97303 .22514 Mdot .97461 .26115 Mdot .97619 .29792 Mdot P % End of Graphics MathPictureEnd :[font = input; preserveAspect] (* Compute the width-2 delay vectors for this time series. *) :[font = input; preserveAspect] delayVectors = ComputeDelayVectors[timeSeries,2] ; :[font = input; preserveAspect] (* Notice that the width-2 delay vectors trace out several ellipses in phase space. Locally these ellipses appear as distinct trajectories. *) :[font = input; preserveAspect; startGroup] ListPlot[delayVectors, PlotJoined -> True] ; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; endGroup] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.47619 0.309017 0.294302 [ [(-1)] .02381 .30902 0 2 Msboxa [(-0.5)] .2619 .30902 0 2 Msboxa [(0.5)] .7381 .30902 0 2 Msboxa [(1)] .97619 .30902 0 2 Msboxa [(-1)] .4875 .01472 1 0 Msboxa [(-0.5)] .4875 .16187 1 0 Msboxa [(0.5)] .4875 .45617 1 0 Msboxa [(1)] .4875 .60332 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .02381 .30902 m .02381 .31527 L s P [(-1)] .02381 .30902 0 2 Mshowa p .002 w .2619 .30902 m .2619 .31527 L s P [(-0.5)] .2619 .30902 0 2 Mshowa p .002 w .7381 .30902 m .7381 .31527 L s P [(0.5)] .7381 .30902 0 2 Mshowa p .002 w .97619 .30902 m .97619 .31527 L s P [(1)] .97619 .30902 0 2 Mshowa p .001 w .07143 .30902 m .07143 .31277 L s P p .001 w .11905 .30902 m .11905 .31277 L s P p .001 w .16667 .30902 m .16667 .31277 L s P p .001 w .21429 .30902 m .21429 .31277 L s P p .001 w .30952 .30902 m .30952 .31277 L s P p .001 w .35714 .30902 m .35714 .31277 L s P p .001 w .40476 .30902 m .40476 .31277 L s P p .001 w .45238 .30902 m .45238 .31277 L s P p .001 w .54762 .30902 m .54762 .31277 L s P p .001 w .59524 .30902 m .59524 .31277 L s P p .001 w .64286 .30902 m .64286 .31277 L s P p .001 w .69048 .30902 m .69048 .31277 L s P p .001 w .78571 .30902 m .78571 .31277 L s P p .001 w .83333 .30902 m .83333 .31277 L s P p .001 w .88095 .30902 m .88095 .31277 L s P p .001 w .92857 .30902 m .92857 .31277 L s P p .002 w 0 .30902 m 1 .30902 L s P p .002 w .5 .01472 m .50625 .01472 L s P [(-1)] .4875 .01472 1 0 Mshowa p .002 w .5 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.54481 .3459 L .55968 .35506 L .57449 .36416 L .58923 .37322 L .60388 .38221 L .61842 .39112 L .63285 .39996 L .64715 .40871 L .6613 .41736 L .6753 .4259 L .68912 .43432 L .70275 .44263 L .71619 .4508 L .72941 .45883 L .7424 .46671 L .75516 .47444 L .76766 .482 L .7799 .4894 L .79186 .49661 L .80354 .50364 L .81491 .51048 L .82597 .51712 L .83672 .52355 L .84713 .52978 L .8572 .53578 L .86691 .54156 L .87626 .54711 L .88525 .55243 L .89385 .5575 L .90206 .56234 L .90988 .56692 L .91729 .57124 L .92429 .57531 L .93087 .57911 L .93703 .58265 L .94275 .58592 L .94804 .58891 L .95288 .59163 L .95728 .59407 L .96123 .59623 L .96472 .59811 L .96776 .5997 L .97033 .601 L .97244 .60201 L .97408 .60274 L .97525 .60317 L .97596 .60332 L Mistroke .97619 .60317 L .97596 .60274 L .97525 .60201 L .97408 .601 L .97244 .5997 L .97033 .59811 L .96776 .59623 L .96472 .59407 L .96123 .59163 L .95728 .58891 L .95288 .58592 L .94804 .58265 L .94275 .57911 L .93703 .57531 L .93087 .57124 L .92429 .56692 L 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Sauer's method of prediction (see below) relies on the data being densely sampled. To compensate for sparse data, he "upsamples" the supplied data to artificially increase the sampling rate. Upsampling is accomplished as follows. For each pair of consecutive points in the time series, choose some number of points on either side of the pair for a total of s points. Fit an interpolation function to these points and use the function to fill in "samples" between the pair. Sauer uses a Fourier polynomial with s = 16 allowing him combine upsampling and filtering, but other interpolation functions are also reasonable. You might try Mathematica's Interpolation procedure which returns an InterpolationFunction object (see Page 676 in [Wolfram, 1991]). One of the reasons that Sauer has to sample frequently is that his method of continuing a time series from a point b relies on finding the nearest neighbors b_1,...,b_k of b such that for each b_i the ray from b_i to b should be close to perpendicular to the (local) portion of the trajectory on which b_i is found. As you will see shortly, we contrived the time series (remember those little delays we added) so that we can easily pick a b such that its nearest neighbors in the embedding space will satisfy Sauer's condition. Now that we know how to compute delay vectors, consider how to make predictions. Suppose that we want to predict beyond the end of our sampled data. Suppose that b is the final delay vector extracted from the time series, and suppose that we want to predict what we will observe (the price of a stock for example) h time units in the future. Let X denote that observation. If the dynamical system is deterministic, the observations noise free, and we sample "often enough" and "long enough", then prediction is trivial. Since the delay vectors are proxies for their corresponding states, all we have to do is find a previous instance of b, call it b, in the time series and then march forward h time units from b' to find X' which should be an accurate prediction for X. Observation noise complicates the picture somewhat. Sauer recommends the following method. Find the k nearest neighbors to b, denoted, b_1,...,b_k, such that each vector is representative of a distinct portion (trajectory) of the time series. Compute their corresponding observations h time units in the future and denote these X_1,...,X_k. Fit a function to data defined by {{b_1,X_1},...,{b_k,X_k}} and use this function to compute an estimate for X. The actual implementation is more complicated, but worth exploring as it provides a nice application of singular value decomposition. You might want to have an introductory linear algebra text handy if you are rusty on vector spaces, basis vectors, linear transformations and the like. *) :[font = input; preserveAspect] (* Suppose that b is the following delay vector. *) :[font = input; preserveAspect] b = {0.11,0.18} ; :[font = input; preserveAspect] (* Load a package that will allow us to compute nearest neighbors. *) :[font = input; preserveAspect] Needs["Dynamics`Packages`NearestNeighbors`"] ; :[font = input; preserveAspect] (* Compute the indices of the k (k = 6) nearest neighbors of b. Given the indices we can recover the actual vectors from the list of all the vectors. We will need these indices in order to compute the X_i for the b_i. *) :[font = input; preserveAspect] nearestIndices = NearestNeighborsOnDistinctTrajectories[b,delayVectors,6,8] ; :[font = input; preserveAspect] (* Extract the actual neighbors from the list of delay vectors. *) :[font = input; preserveAspect] nearestVectors = Map[(delayVectors[[#]])&,nearestIndices] ; :[font = input; preserveAspect] (* We are going to create a new coordinate system in which to search for a function with which to estimate X. We choose the centroid of the delay vectors corresponding to the nearest neighbors of b to be the origin of this new coordinate system. *) :[font = input; preserveAspect] (* Find the centroid of the list of nearest delay vectors. *) :[font = input; preserveAspect] c = Apply[Plus,nearestVectors]/Length[nearestVectors] ; :[font = input; preserveAspect] (* Look at the relative positions of b, c, and the nearest neighbors of b. Notice that the lines from b to its nearest neighbors are roughly perpendicular to the trajectories on which those lines appear. Remember we added those little delays to ensure that this would indeed by the case. 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2.5722 1.51453 Mdot 2.01453 .9251 Mdot 1.4251 .31319 Mdot .81319 -0.31155 Mdot .18845 -0.93926 Mdot -0.43926 -1.56004 Mdot -1.06004 -2.1641 Mdot -1.6641 -2.74192 Mdot -2.24192 -3.28438 Mdot -2.78438 -3.78293 Mdot -3.28293 -4.22971 Mdot -3.72971 -4.61766 Mdot -4.11766 -4.94068 Mdot -4.44068 -5.19367 Mdot -4.69367 -5.37263 Mdot -4.87263 -5.47476 Mdot -4.97476 -5.49842 Mdot -4.99842 -5.44326 Mdot -4.94326 -5.31014 Mdot -4.81014 -5.10116 Mdot -4.60116 -4.81962 Mdot -4.31962 -4.46995 Mdot -3.96995 -4.05768 Mdot -3.55768 -3.5893 Mdot -3.0893 -3.0722 Mdot -2.5722 -2.51453 Mdot -2.01453 -1.9251 Mdot -1.4251 -1.31319 Mdot P % End of Graphics MathPictureEnd :[font = input; preserveAspect] (* Transform the delay vectors into the new coordinate system. The list of these transformed vectors constitutes a matrix and the vectors span a subspace (perhaps the entire space) of the embedding space containing all of the delay vectors. The rows of the resulting matrix consist of the vectors b_1-c,...,b_2-c. *) :[font = input; preserveAspect] newVectors = Map[(#-c)&,nearestVectors] ; :[font = input; preserveAspect] (* Now we use the built-in Mathematica function that computes the singular value decomposition of this matrix. SingularValues[m] gives the singular value decomposition (SVD) for the numerical matrix m. The result is a list {u,w,v} where u and v are orthogonal matrices, w is the list of nonzero singular values, and m can be written as Conjugate[Transpose[u]].DiagonalMatrix[w].v (taking the conjugate isn't necessary if m contains no entries with imaginary parts). The singular vectors v constitute a basis (set of spanning vectors) for the vector space spanned by the vectors b_1-c,...,b_2-c. However, the vectors in v correspond 1-1 with the singular values in w which are in decreasing magnitude. Intuitively, if eliminate basis vectors in v with small singular values, then we will end up with a new vector space of lower dimension which provides a pretty good representation of original transformed delay vectors. Technically, we want to find an l-dimensional (l < m) subspace of the space spanned by the m-dimensional delay vectors that minimizes the sum of the squared distances to the nearest neighbors of b. The first l rows of v span the desired subspace. *) :[font = input; preserveAspect] singularValueDecomposition = SingularValues[newVectors] ; :[font = input; preserveAspect] (* Extract the singular vectors. *) :[font = input; preserveAspect] singularVectors = singularValueDecomposition[[3]] ; :[font = input; preserveAspect] (* Normalize the vectors, i.e., make them unit-length vectors. *) :[font = input; preserveAspect] orthonormalBasis = Map[(#/Sqrt[#.#])&,singularVectors] ; :[font = input; preserveAspect] (* Retain only the dominant normalized vectors. Since m = 2 in this example we choose l = 1 and the resulting subspace is a line. *) :[font = input; preserveAspect] reducedOrthonormalBasis = Take[orthonormalBasis,1] ; (* Given an arbitrary delay vector u, the following transformation projects u onto the l-dimensional subspace. *) :[font = input; preserveAspect] Projection[u_] := u.Transpose[reducedOrthonormalBasis] ; :[font = input; preserveAspect] (* Fit the data {{Projection[b_i-c],X_i}} with an affine (constant plus linear) model. This works for the case in which l = 1. We generate a different transformation for each choice of h. *) :[font = input; preserveAspect] AffineTransformation[h_] := Function[x, f] /. f -> Fit[Table[Flatten[{Projection[newVectors[[i]]-c], delayVectors[[nearestIndices[[i]]+h]][[1]]}], {i,Length[newVectors]}], {x,1}, x] :[font = input; preserveAspect] (* Now we can make predictions for h and h + 1 and use them to construct a new delay vector d. *) :[font = input; preserveAspect] ContinueSeries[h_] := Module[{f1 = AffineTransformation[h], f2 = AffineTransformation[h+1],d}, (* Combine the predictions to construct a delay vector. *) d = Join[f1[Projection[b-c]],f2[Projection[b-c]]] ; (* Display b and d along with the delay vectors. *) ListPlot[delayVectors, AspectRatio -> 1, Prolog -> {Text["b",b],Text["d",d]}]] :[font = input; preserveAspect] (* Here we compute a length-4 continuation. If you have a graphical front end for Mathematica, select the four resulting graphics cells and invoke the Animate Selected Graphics selection under the Graph menu. *) :[font = input; preserveAspect; startGroup] Do[ContinueSeries[h],{h,1,4}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 282; startGroup] %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.47619 0.5 0.47619 [ [(-1)] .02381 .5 0 2 Msboxa [(-0.5)] .2619 .5 0 2 Msboxa [(0.5)] .7381 .5 0 2 Msboxa [(1)] .97619 .5 0 2 Msboxa [(-1)] .4875 .02381 1 0 Msboxa [(-0.5)] .4875 .2619 1 0 Msboxa [(0.5)] .4875 .7381 1 0 Msboxa [(1)] .4875 .97619 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 1.001 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .02381 .5 m .02381 .50625 L s P [(-1)] .02381 .5 0 2 Mshowa p .002 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.17841 Mdot .17841 .157 Mdot .157 .13693 Mdot .13693 .1183 Mdot .1183 .10118 Mdot .10118 .08563 Mdot .08563 .07171 Mdot .07171 .05949 Mdot .05949 .049 Mdot .049 .04029 Mdot .04029 .0334 Mdot .0334 .02835 Mdot .02835 .02516 Mdot .02516 .02385 Mdot .02385 .02441 Mdot .02441 .02685 Mdot .02685 .03116 Mdot .03116 .03732 Mdot .03732 .0453 Mdot .0453 .05508 Mdot .05508 .06662 Mdot .06662 .07986 Mdot .07986 .09476 Mdot .09476 .11127 Mdot .11127 .1293 Mdot .1293 .1488 Mdot .1488 .16969 Mdot .16969 .19188 Mdot .19188 .21528 Mdot .21528 .23981 Mdot .23981 .26537 Mdot .26537 .29185 Mdot .29185 .31915 Mdot .31915 .34717 Mdot .34717 .37579 Mdot .37579 .4049 Mdot .4049 .43439 Mdot .43439 .46413 Mdot .46413 .49402 Mdot .49402 .52393 Mdot .52393 .55374 Mdot .55374 .58334 Mdot .58334 .61262 Mdot .61262 .64145 Mdot .64145 .66972 Mdot .66972 .69732 Mdot .69732 .72414 Mdot .72414 .75008 Mdot .75008 .77503 Mdot .77503 .7989 Mdot .7989 .82159 Mdot .82159 .843 Mdot .843 .86307 Mdot .86307 .8817 Mdot .8817 .89882 Mdot .89882 .91437 Mdot .91437 .92829 Mdot .92829 .94051 Mdot .94051 .951 Mdot .951 .95971 Mdot .95971 .9666 Mdot .9666 .97165 Mdot .97165 .97484 Mdot .97484 .97615 Mdot .97615 .97559 Mdot .97559 .97315 Mdot .97315 .96884 Mdot .96884 .96268 Mdot .96268 .9547 Mdot .9547 .94492 Mdot .94492 .93338 Mdot .93338 .92014 Mdot .92014 .90524 Mdot .90524 .88873 Mdot .88873 .8707 Mdot .8707 .8512 Mdot .8512 .83031 Mdot .83031 .80812 Mdot .80812 .78472 Mdot .78472 .76019 Mdot .76019 .73463 Mdot .73463 .70815 Mdot .70815 .68085 Mdot .68085 .65283 Mdot .65283 .62421 Mdot .62421 .5951 Mdot .5951 .56561 Mdot .56561 .53587 Mdot .53587 .50598 Mdot .50598 .47607 Mdot .47607 .44626 Mdot .44626 .41666 Mdot .41666 .38738 Mdot .38738 .35855 Mdot .35855 .33028 Mdot .33028 .30268 Mdot .30268 .27586 Mdot .27586 .24992 Mdot .24992 .22497 Mdot .22497 .2011 Mdot .2011 .17841 Mdot .17841 .157 Mdot .157 .13693 Mdot .13693 .1183 Mdot .1183 .10118 Mdot .10118 .08563 Mdot .08563 .07171 Mdot .07171 .05949 Mdot .05949 .049 Mdot .049 .04029 Mdot .04029 .0334 Mdot .0334 .02835 Mdot .02835 .02516 Mdot .02516 .02385 Mdot .02385 .02441 Mdot .02441 .02685 Mdot .02685 .03116 Mdot .03116 .03732 Mdot .03732 .0453 Mdot .0453 .05508 Mdot .05508 .06662 Mdot .06662 .07986 Mdot .07986 .09476 Mdot .09476 .11127 Mdot .11127 .1293 Mdot .1293 .1488 Mdot .1488 .16969 Mdot .16969 .19188 Mdot .19188 .21528 Mdot .21528 .23981 Mdot .23981 .26537 Mdot .26537 .29185 Mdot .29185 .31915 Mdot .31915 .34717 Mdot .34717 .37579 Mdot .37579 .4049 Mdot .4049 .43439 Mdot .43439 .46413 Mdot .46413 .49402 Mdot .49402 .48205 Mdot .48205 .54183 Mdot .54183 .60096 Mdot .60096 .65849 Mdot .65849 .71352 Mdot .71352 .76518 Mdot .76518 .81266 Mdot .81266 .85521 Mdot .85521 .89216 Mdot .89216 .92292 Mdot .92292 .94702 Mdot .94702 .96406 Mdot .96406 .97379 Mdot .97379 .97604 Mdot .97604 .97079 Mdot .97079 .95811 Mdot .95811 .93821 Mdot .93821 .91139 Mdot .91139 .87809 Mdot .87809 .83883 Mdot .83883 .79422 Mdot .79422 .74497 Mdot .74497 .69186 Mdot .69186 .63572 Mdot .63572 .57745 Mdot .57745 .51795 Mdot .51795 .45817 Mdot .45817 .39904 Mdot .39904 .34151 Mdot .34151 .28648 Mdot .28648 .23482 Mdot .23482 .18734 Mdot .18734 .14479 Mdot .14479 .10784 Mdot .10784 .07708 Mdot .07708 .05298 Mdot .05298 .03594 Mdot .03594 .02621 Mdot .02621 .02396 Mdot .02396 .02921 Mdot .02921 .04189 Mdot .04189 .06179 Mdot .06179 .08861 Mdot .08861 .12191 Mdot .12191 .16117 Mdot .16117 .20578 Mdot .20578 .25503 Mdot .25503 .30814 Mdot .30814 .36428 Mdot .36428 .42255 Mdot .42255 .48205 Mdot .48205 .54183 Mdot .54183 .60096 Mdot .60096 .65849 Mdot .65849 .71352 Mdot .71352 .76518 Mdot .76518 .81266 Mdot .81266 .85521 Mdot .85521 .89216 Mdot .89216 .92292 Mdot .92292 .94702 Mdot .94702 .96406 Mdot .96406 .97379 Mdot .97379 .97604 Mdot .97604 .97079 Mdot .97079 .95811 Mdot .95811 .93821 Mdot .93821 .91139 Mdot .91139 .87809 Mdot .87809 .83883 Mdot .83883 .79422 Mdot .79422 .74497 Mdot .74497 .69186 Mdot .69186 .63572 Mdot .63572 .57745 Mdot .57745 .51795 Mdot .51795 .45817 Mdot .45817 .39904 Mdot .39904 .34151 Mdot .34151 .28648 Mdot .28648 .23482 Mdot .23482 .18734 Mdot .18734 .14479 Mdot .14479 .10784 Mdot .10784 .07708 Mdot .07708 .05298 Mdot .05298 .03594 Mdot .03594 .02621 Mdot .02621 .02396 Mdot .02396 .02921 Mdot .02921 .04189 Mdot .04189 .06179 Mdot .06179 .08861 Mdot .08861 .12191 Mdot .12191 .16117 Mdot .16117 .20578 Mdot .20578 .25503 Mdot .25503 .30814 Mdot .30814 .36428 Mdot .36428 .42255 Mdot .42255 .48205 Mdot .48205 .54183 Mdot .54183 .60096 Mdot .60096 .65849 Mdot .65849 .71352 Mdot .71352 .76518 Mdot .76518 .81266 Mdot .81266 .85521 Mdot .85521 .89216 Mdot .89216 .92292 Mdot .92292 .94702 Mdot .94702 .96406 Mdot .96406 .97379 Mdot .97379 .97604 Mdot .97604 .97079 Mdot .97079 .95811 Mdot .95811 .93821 Mdot .93821 .91139 Mdot .91139 .87809 Mdot .87809 .83883 Mdot .83883 .79422 Mdot .79422 .74497 Mdot .74497 .69186 Mdot .69186 .63572 Mdot .63572 .57745 Mdot .57745 .51795 Mdot .51795 .45817 Mdot .45817 .39904 Mdot .39904 .34151 Mdot .34151 .28648 Mdot .28648 .23482 Mdot .23482 .18734 Mdot .18734 .14479 Mdot .14479 .10784 Mdot .10784 .07708 Mdot .07708 .05298 Mdot .05298 .03594 Mdot .03594 .02621 Mdot .02621 .02396 Mdot .02396 .02921 Mdot .02921 .04189 Mdot .04189 .06179 Mdot .06179 .08861 Mdot .08861 .12191 Mdot .12191 .16117 Mdot .16117 .20578 Mdot .20578 .25503 Mdot .25503 .30814 Mdot .30814 .36428 Mdot .36428 .42255 Mdot .42255 .48205 Mdot .48205 .54183 Mdot .54183 .60096 Mdot .60096 .65849 Mdot .65849 .71352 Mdot .71352 .76518 Mdot .76518 .81266 Mdot .81266 .85521 Mdot .85521 .89216 Mdot .89216 .92292 Mdot .92292 .94702 Mdot .94702 .96406 Mdot .96406 .97379 Mdot .97379 .97604 Mdot .97604 .97079 Mdot .97079 .95811 Mdot .95811 .93821 Mdot .93821 .91139 Mdot .91139 .87809 Mdot .87809 .83883 Mdot .83883 .79422 Mdot .79422 .74497 Mdot .74497 .69186 Mdot .69186 .63572 Mdot .63572 .57745 Mdot .57745 .51795 Mdot .51795 .45817 Mdot .45817 .39904 Mdot .39904 .34151 Mdot .34151 .28648 Mdot .28648 .23482 Mdot .23482 .18734 Mdot .18734 .14479 Mdot .14479 .10784 Mdot .10784 .07708 Mdot .07708 .05298 Mdot .05298 .03594 Mdot .03594 .02621 Mdot .02621 .02396 Mdot .02396 .02921 Mdot .02921 .04189 Mdot .04189 .06179 Mdot .06179 .08861 Mdot .08861 .12191 Mdot .12191 .16117 Mdot .16117 .20578 Mdot .20578 .25503 Mdot .25503 .30814 Mdot .30814 .36428 Mdot .36428 .42255 Mdot P % End of Graphics MathPictureEnd ^*)