Re: Estimating parameters p and q in y'' + p y' + q y = Tide(t)

• To: mathgroup at smc.vnet.net
• Subject: [mg46284] Re: Estimating parameters p and q in y'' + p y' + q y = Tide(t)
• From: drbob at bigfoot.com (Bobby R. Treat)
• Date: Fri, 13 Feb 2004 21:56:48 -0500 (EST)
• References: <c0hhrh\$lgd\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```There's no need to go so 'round about to get that into a notebook.
Just copy and paste it into Mathematica. The front end asks if you
want it to interpret the text; you say yes, and it's done.

Bobby

gilmar.rodriguez at nwfwmd.state.fl.us (Gilmar Rodr?guez Pierluissi) wrote in message news:<c0hhrh\$lgd\$1 at smc.vnet.net>...
> Dear Math User friends:
> I have two data sets; the first one corresponds to tide data, and the
> second one corresponds to
> water elevation data obtained from a groundwater monitoring well.  The
> tide affects the water level
> inside the well.  If we let the variable y(t) represent the height of
> the water column inside the pipe,
> and Tide(t) be a least square fit representation of our tide record,
> with t representing time,
> then we can form a Differential Equation:  y'' + p y' + q y = Tide(t),
> where Tide(t) acts as a forcing
> function.  Since I have a water elevation record; what I'm seeking is
> to find a way to estimate
> two values p and q, such that the solution y(t) to the above DE,
> becomes a model that fits my
> water elevation data; i.e. a model in the least square sense, showing
> a correlation (of say) 0.95,
> or above.  The following is an unevaluated Mathematica notebook to
> elaborate this question
> with the aid of a specific example. Please copy the following text and
> paste it into Wordpad, or
> Notepad and save it as DE.txt  Then change the name of this file to
> DE.nb, (ignore the "are you
> sure that you want to  change extention name" message) and open the
> new notebook using
> Mathematica (version 5.0, or version above 5.0) as usual.  Thank you
> for your help!
>
> Start copying here:
> (************** Content-type: application/mathematica **************
>                      CreatedBy='Mathematica 5.0'
>
>                     Mathematica-Compatible Notebook
>
> This notebook can be used with any Mathematica-compatible
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> for the notebook starts with the line containing stars above.
>
> To get the notebook into a Mathematica-compatible application, do
> one of the following:
>
> * Save the data starting with the line of stars above into a file
>   with a name ending in .nb, then open the file inside the
>   application;
>
> * Copy the data starting with the line of stars above to the
>   clipboard, then use the Paste menu command inside the application.
>
> Data for notebooks contains only printable 7-bit ASCII and can be
> sent directly in email or through ftp in text mode.  Newlines can be
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>
> NOTE: If you modify the data for this notebook not in a Mathematica-
> compatible application, you must delete the line below containing
> the word CacheID, otherwise Mathematica-compatible applications may
> try to use invalid cache data.
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> Notebook[{
> Cell[BoxData[
>     StyleBox[\( (*\(\(*\)\(\ \)\(I\)\(\ \)\(have\)\(\ \)\(two\)\(\
> \)\(data\)\
> \(\ \)\(sets\)\); \ the\ first\ one\ corresponds\ to\ tide\ data, \
>         and\ the\ second\ one\ corresponds\ to\n
>           water\ elevation\ data\ obtained\ from\ a\ groundwater\
> monitoring\ \
> well . \ \ The\ tide\ affects\ the\ water\ level\n
>           inside\ the\ well . \ \ If\ we\ let\ the\ variable\ y
> \((t)\)\ \
> represent\ the\ height\ of\ the\ water\ column\ inside\ the\ pipe, \n
>         and\ Tide \((t)\)\ be\ a\ least\ square\ fit\ representation\
> of\ our\
> \ tide\ record, \ with\ t\ reperesenting\ time, \n
>         then\ we\ can\ form\ a\ Differential\ \(Equation : \ \ y''\  +
> \
>                 p\ y'\  + \ q\ y\)\  = \ Tide \((t)\), \
>         where\ Tide \((t)\)\ acts\ as\ a\ forcing\n
>           function . \ \ Since\ I\ have\ a\ water\ elevation\ record;
> \
>         what\ I' m\ seeking\ is\ to\ find\ a\ way\ to\ estimate\n
>           two\ values\ p\ and\ q, \
>         such\ that\ the\ solution\ y \((t)\)\ to\ the\ above\ DE, \
>         becomes\ a\ model\ that\ fits\ my\nwater\ elevation\ data; \
>         i . e . \ a\ model\ in\ the\ least\ square\ sense, \
>         showing\ a\ correlation\ \((of\ say)\)\ 0.95, \n
>         or\ above . \ \ The\ following\ example\ is\ an\ attempt\ to\
> clarify\
> \ my\ \(question : \n\(\(Here\ is\ the\ tide\ record\ \((the\ values\
> are\ \
> measured\ in\ decimal\ meters)\)\)\(:\)\)\)\ **) \),
>       FormatType->StandardForm,
>       FontFamily->"Arial"]], "Input",
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>   FontFamily->"Arial"],
>
> Cell[BoxData[
>     \(L = Length[tide]\)], "Input",
>   FontFamily->"Arial"],
>
> Cell[BoxData[
>     \(plt1 =
>       ListPlot[tide, PlotJoined \[Rule] True,
>         PlotStyle \[Rule] RGBColor[1, 0, 0]]\)], "Input",
>   FontFamily->"Arial"],
>
> Cell[BoxData[
>     \(\(initparams = {{a\_0, \(-0.31278\)}, {a\_1, \(-0.21078\)},
> {a\_2, \
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>             0.130158}, {w\_1, 0.049697}, {w\_2,
>             0.049183}, {w\_3, \(-0.00844\)}, {w\_4, 0.046353}};\)\)],
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>
>   FontFamily->"Arial"],
>
> Cell[BoxData[
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>         a\_0 + Sum[a\_i*Sin[v\_i*t - b\_i], {i, 1, 4}] +
>           Sum[c\_i*Sin[w\_i*t - d\_i], {i, 1, 4}];\)\)], "Input",
>   FontFamily->"Arial"],
>
> Cell[BoxData[
>     \(<< Statistics`NonlinearFit`\)], "Input",
>   FontFamily->"Arial"],
>
> Cell[BoxData[
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>       RowBox[{"Chop", "[",
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>               "TraditionalForm"]}], "]"}], "]"}]}]], "Input",
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>
> Cell[BoxData[
>     \(plt2 =
>       Plot[TIDE[t], {t, 1, 264}, PlotStyle -> RGBColor[0, 1, 0]]\)],
> "Input",
>   FontFamily->"Arial"],
>
> Cell[BoxData[
>     \(Show[{plt1, plt2}, ImageSize\  \[Rule] \ 540]\)], "Input",
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>
> Cell[BoxData[
>     \( (*\(\(*\)\(\ \)\(Gravitational\)\(\ \)\(Constant\)\(\
> \)\(G\)\)\  = \
>         9.8\ m/sec\^2\ **) \)], "Input",
>   FontFamily->"Arial"],
>
> Cell[BoxData[
>     \( (*\(*\)\(\ \)\(Tide\)\(\ \)\(acting\)\(\ \)\(as\)\(\ \)\(a\)\(\
> \
> \)\(Forcing\)\(\ \)\(Function\)\(\ \)\(term\)\(\ \)\(in\)\(\
> \)\(the\)\(\ \
> \)\(following\)\(\ \)\(Differential\)\(\ \)\(\(Equation\)\(:\)\)\ **)
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>       RowBox[{\(\(\(*\)\(\ \)\(Using\)\(\ \)\("\<trial and
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> \)\(we\)\(\ \)\(are\)\(\ \)\(using\)\(\ \)\(values : \ p\)\)\  = \
>           9.8\ \((buoyancy\ factor)\)\), ",", " ", \(q = 1\), ",",
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> Cell[BoxData[
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> \)\(data\)\); \ \(\(i . e . \
>             the\ water\ elevation\ data\ obtained\ from\ a\
> groundwater\ \
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> {105,
>               0.004}, \n\ \ \ \ \ \ \ \ {106, 0.004}, {107, 0.004},
> {108,
>               0.004}, {109, 0.005}, {110, 0.005}, {111, 0.004}, {112,
>               0.003}, {113, 0.002}, {114,
>               0.001}, \n\ \ \ \ \ \ \ \ {115, \(-0.001\)}, {116,
> \(-0.002\)}, \
> {117, \(-0.003\)}, {118, \(-0.004\)}, {119, \(-0.004\)}, {120,
> \(-0.004\)}, \
> {121, \(-0.005\)}, {122, \(-0.007\)}, \n\ \ \ \ \ \ \ \ {123,
> \(-0.008\)}, \
> {124, \(-0.01\)}, {125, \(-0.011\)}, {126, \(-0.011\)}, {127,
> \(-0.012\)}, \
> {128, \(-0.013\)}, {129, \(-0.014\)}, {130, \(-0.014\)}, \n\ \ \ \ \ \
> \ \ \
> {131, \(-0.013\)}, {132, \(-0.014\)}, {133, \(-0.014\)}, {134,
> \(-0.014\)}, \
> {135, \(-0.015\)}, {136, \(-0.014\)}, {137, \(-0.014\)}, {138,
> \(-0.015\)}, \n\
> \ \ \ \ \ \ \ \ {139, \(-0.014\)}, {140, \(-0.014\)}, {141,
> \(-0.012\)}, \
> {142, \(-0.011\)}, {143, \(-0.011\)}, {144, \(-0.009\)}, {145,
> \(-0.009\)}, \
> {146, \(-0.007\)}, \n\ \ \ \ \ \ \ \ {147, \(-0.007\)}, {148,
> \(-0.006\)}, \
> {149, \(-0.004\)}, {150, \(-0.004\)}, {151, \(-0.004\)}, {152,
> \(-0.004\)}, \
> {153, \(-0.003\)}, {154, \(-0.003\)}, \n\ \ \ \ \ \ \ \ {155,
> \(-0.003\)}, \
> {156, \(-0.004\)}, {157, \(-0.004\)}, {158, \(-0.004\)}, {159,
> \(-0.004\)}, \
> {160, \(-0.004\)}, {161, \(-0.004\)}, {162, \(-0.005\)}, \n\ \ \ \ \ \
> \ \ \
> {163, \(-0.006\)}, {164, \(-0.007\)}, {165, \(-0.007\)}, {166,
> \(-0.008\)}, \
> {167, \(-0.01\)}, {168, \(-0.011\)}, {169, \(-0.011\)}, {170,
> \(-0.013\)}, \n\
> \ \ \ \ \ \ \ \ {171, \(-0.015\)}, {172, \(-0.017\)}, {173,
> \(-0.017\)}, \
> {174, \(-0.018\)}, {175, \(-0.018\)}, {176, \(-0.018\)}, {177,
> \(-0.018\)}, \
> {178, \(-0.019\)}, \n\ \ \ \ \ \ \ \ {179, \(-0.02\)}, {180,
> \(-0.021\)}, \
> {181, \(-0.021\)}, {182, \(-0.021\)}, {183, \(-0.021\)}, {184,
> \(-0.022\)}, \
> {185, \(-0.019\)}, {186, \(-0.021\)}, \n\ \ \ \ \ \ \ \ {187,
> \(-0.02\)}, \
> {188, \(-0.018\)}, {189, \(-0.017\)}, {190, \(-0.017\)}, {191,
> \(-0.011\)}, \
> {192, \(-0.014\)}, {193, \(-0.011\)}, {194, \(-0.011\)}, \n\ \ \ \ \ \
> \ \ \
> {195, \(-0.01\)}, {196, \(-0.01\)}, {197, \(-0.005\)}, {198,
> \(-0.008\)}, \
> {199, \(-0.008\)}, {200, \(-0.006\)}, {201, \(-0.004\)}, {202,
> \(-0.004\)}, \n\
> \ \ \ \ \ \ \ \ {203, \(-0.006\)}, {204, \(-0.005\)}, {205,
> \(-0.003\)}, \
> {206, \(-0.004\)}, {207, \(-0.003\)}, {208, \(-0.003\)}, {209,
> \(-0.003\)}, \
> {210, \(-0.004\)}, \n\ \ \ \ \ \ \ \ {211, \(-0.004\)}, {212,
> \(-0.004\)}, \
> {213, \(-0.005\)}, {214, \(-0.005\)}, {215, \(-0.006\)}, {216,
> \(-0.008\)}, \
> {217, \(-0.008\)}, {218, \(-0.01\)}, \n\ \ \ \ \ \ \ \ {219,
> \(-0.011\)}, \
> {220, \(-0.013\)}, {221, \(-0.013\)}, {222, \(-0.015\)}, {223,
> \(-0.017\)}, \
> {224, \(-0.018\)}, {225, \(-0.018\)}, {226, \(-0.018\)}, \n\ \ \ \ \ \
> \ \ \
> {227, \(-0.018\)}, {228, \(-0.018\)}, {229, \(-0.02\)}, {230,
> \(-0.02\)}, \
> {231, \(-0.019\)}, {232, \(-0.023\)}, {233, \(-0.02\)}, {234,
> \(-0.021\)}, \n\
> \ \ \ \ \ \ \ \ {235, \(-0.02\)}, {236, \(-0.019\)}, {237,
> \(-0.018\)}, {238, \
> \(-0.016\)}, {239, \(-0.016\)}, {240, \(-0.014\)}, {241, \(-0.013\)},
> {242, \
> \(-0.013\)}, \n\ \ \ \ \ \ \ \ {243, \(-0.012\)}, {244, \(-0.011\)},
> {245, \
> \(-0.01\)}, {246, \(-0.009\)}, {247, \(-0.009\)}, {248, \(-0.007\)},
> {249, \
> \(-0.005\)}, {250, \(-0.004\)}, \n\ \ \ \ \ \ \ \ {251, \(-0.004\)},
> {252, \
> \(-0.004\)}, {253, \(-0.003\)}, {254, \(-0.004\)}, {255, \(-0.005\)},
> {256, \
> \(-0.004\)}, {257, \(-0.005\)}, {258, \(-0.006\)}, \n\ \ \ \ \ \ \ \
> {259, \
> \(-0.007\)}, {260, \(-0.007\)}, {261, \(-0.008\)}, {262, \(-0.009\)},
> {263, \
> \(-0.01\)}, {264, \(-0.011\)}, {265, \(-0.011\)}, {266, \(-0.012\)},
> \n\ \ \ \
> \ \ \ \ \ {267, \(-0.013\)}, {268, \(-0.017\)}, {269, \(-0.017\)}};\),
>       FormatType->StandardForm,
>       FontFamily->"Arial",
>       FontSize->12]], "Input",
>   FontFamily->"Arial"],
>
> Cell[BoxData[
>     \(Length[Elev]\)], "Input",
>   FontFamily->"Arial"],
>
> Cell[BoxData[
>     \( (*\(\(*\)\(\ \)\(To\)\(\ \)\(compare\)\(\ \)\(our\)\(\ \
> \)\(solution\)\(\ \)\(model\)\(\ \)\(to\)\(\ \)\(the\)\(\
> \)\(elevation\)\(\ \
> \)\(record\)\); \
>       it\ is\ necessary\ to\ magnify\ the\ elevation\ record\ by\ a\
> factor\ \
> of\ 50. \ \ There\ is\ also\ a\ time\ lag\ between\ the\ solution\
> model\ and\
> \ the\ elevation\ record\ of\ 6\ time\ \(\(units\)\(:\)\)\ **) \)],
> "Input",
>   FontFamily->"Arial"],
>
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>
> Cell[BoxData[
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> "Input",
>   FontFamily->"Arial"],
>
> Cell[BoxData[
>     \(\(Lag = 6;\)\)], "Input",
>   FontFamily->"Arial"],
>
> Cell[BoxData[
>     \( (*\(*\)\(\ \)\(The\)\(\ \)\(abbreviation\)\(\
> \)\("\<rsElev\>"\)\(\ \)\
> \(stands\)\(\ \)\(for\)\(\ \)\("\<re-scaled Elevation\>"\)\ **) \)],
> "Input"],
>
> Cell[BoxData[
>     \(\(rsElev =
>         Table[{i - Lag, M*\(Elev[\([i]\)]\)[\([2]\)]}, {i, 1,
>             L}];\)\)], "Input",
>   FontFamily->"Arial"],
>
> Cell[BoxData[
>     \(plt4 =
>       ListPlot[rsElev, PlotJoined \[Rule] True,
>         PlotStyle \[Rule] RGBColor[0, 0, 1]]\)], "Input",
>   FontFamily->"Arial"],
>
> Cell[BoxData[
>     \( (*\(\(*\)\(\ \)\(Here\)\(\ \)\(we\)\(\ \)\(compare\)\(\
> \)\(our\)\(\ \
> \)\(solution\)\(\ \)\(model\)\(\ \)\(with\)\(\ \)\(our\)\(\
> \)\(well\)\(\ \
> \)\(elevation\)\(\ \)\(data . \ \ "\<Not close, and no cigar\>"\)\); \
>       because\ we\ are\ attempting\ two\ arbitrary\ values\ for\ p\
> and\ \
> \(\(q\)\(:\)\)\ \ **) \)], "Input"],
>
> Cell[BoxData[
>     \(Show[{plt3, plt4}]\)], "Input",
>   FontFamily->"Arial"],
>
> Cell[BoxData[
>     \( (*\(\(*\)\(\ \)\(Remark : \ \ One\ can\ see\ a\ downward\
> trend\ in\ \
> the\ elevation\ record . \ \ I\ don'
>             t\ know\ what\ physical\ factor\ "\<out there\>"\ causes\
> this\ \
> trend . \ \ If\ we\ could\ identify\ it\)\), \
>       and\ incorporate\ it\ into\ the\ Differential\ Equation;
>       perhaps\ our\ "\<Elevation DE Model\>"\ could\ be\ more\
> accurate\ in\ \
> accounting\ for\ "\<the behavior\>"\ of\ the\ elevation\
> \(\(record\)\(.\)\)\ \
> **) \)], "Input",
>   FontFamily->"Arial"],
>
> Cell[BoxData[
>     StyleBox[\(\( (*\(\(*\)\(\ \)\(Again\)\); \
>         what\ I' m\ looking\ for\ is\ to\ find\ a\ way\ to\ estimate\n
>           the\ values\ p\ and\ q, \
>         such\ that\ the\ solution\ y \((t)\)\ to\ the\ above\ DE, \
> \(becomes\
> \ a\ model\ that\ fits\ my\)\n\(water\ elevation\ data; \
>           i . e . \ a\ model\ in\ the\ least\ square\ sense\), \
>         showing\ a\ correlation\ \((of\ say)\)\ 0.95, \nor\ above, \
>         without\ having\ to\ engage\ into\ trial\ and\
> \(\(error\)\(.\)\)\ \ \
> **) \)\(\ \)\),
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> },
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> PrintingPageRange->{Automatic, Automatic}
> ]
>
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> you save this file from within Mathematica.
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> *******************************************************************)

```

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