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

*To*: mathgroup at smc.vnet.net*Subject*: [mg46255] Estimating parameters p and q in y'' + p y' + q y = Tide(t)*From*: gilmar.rodriguez at nwfwmd.state.fl.us (Gilmar Rodr?guez Pierluissi)*Date*: Thu, 12 Feb 2004 22:46:15 -0500 (EST)*Sender*: owner-wri-mathgroup at wolfram.com

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 application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info at wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 21225, 425]*) (*NotebookOutlinePosition[ 21931, 449]*) (* CellTagsIndexPosition[ 21887, 445]*) (*WindowFrame->Normal*) 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", FontFamily->"Arial"], Cell[BoxData[ \(\(tide = {{1, 0}, {2, \(-0.015\)}, {3, 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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, \ \(-0.12503\)}, {a\_3, \(-0.03388\)}, {a\_4, \(-0.34959\)}, {b\_1, 8.093077}, {b\_2, 10.51904}, {b\_3, 2.014087}, {b\_4, 7.077294}, {c\_1, \(-0.05472\)}, {c\_2, \(-0.05678\)}, {c\_3, \ \(-0.1125\)}, {c\_4, \(-0.06463\)}, {d\_1, \(-5.01326\)}, {d\_2, \ \(-6.27323\)}, {d\_3, \(-1.67511\)}, {d\_4, \(-10.6115\)}, {v\_1, 0.126851}, {v\_2, 0.135713}, {v\_3, 0.089487}, {v\_4, 0.130158}, {w\_1, 0.049697}, {w\_2, 0.049183}, {w\_3, \(-0.00844\)}, {w\_4, 0.046353}};\)\)], "Input",\ FontFamily->"Arial"], Cell[BoxData[ \(\(model\ = \ a\_0 + Sum[a\_i*Sin[v\_i*t - b\_i], {i, 1, 4}] + Sum[c\_i*Sin[w\_i*t - d\_i], {i, 1, 4}];\)\)], "Input", FontFamily->"Arial"], Cell[BoxData[ \(<< Statistics`NonlinearFit`\)], "Input", FontFamily->"Arial"], Cell[BoxData[ RowBox[{\(TIDE[t_]\), "=", RowBox[{"Chop", "[", RowBox[{"NonlinearFit", "[", RowBox[{"tide", ",", "model", ",", "t", ",", "initparams", ",", FormBox[ FormBox[\(AccuracyGoal \[Rule] 2\), "TraditionalForm"], "TraditionalForm"]}], "]"}], "]"}]}]], "Input", FontFamily->"Arial"], 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", FontFamily->"Arial"], Cell[BoxData[ \( (*\(\(*\)\(\ \)\(Gravitational\)\(\ \)\(Constant\)\(\ \)\(G\)\)\ = \ 9.8\ m/sec\^2\ **) \)], "Input", FontFamily->"Arial"], Cell[BoxData[ \( (*\(*\)\(\ \)\(Tide\)\(\ \)\(acting\)\(\ \)\(as\)\(\ \)\(a\)\(\ \ \)\(Forcing\)\(\ \)\(Function\)\(\ \)\(term\)\(\ \)\(in\)\(\ \)\(the\)\(\ \ \)\(following\)\(\ \)\(Differential\)\(\ \)\(\(Equation\)\(:\)\)\ **) \)], \ "Input", FontFamily->"Arial"], Cell[BoxData[ RowBox[{"(*", RowBox[{\(\(\(*\)\(\ \)\(Using\)\(\ \)\("\<trial and error\>"\)\(\ \ \)\(we\)\(\ \)\(are\)\(\ \)\(using\)\(\ \)\(values : \ p\)\)\ = \ 9.8\ \((buoyancy\ factor)\)\), ",", " ", \(q = 1\), ",", " ", \(and\ two\ initial\ conditions; \ y[0] = 0\), ",", " ", RowBox[{ RowBox[{"and", " ", RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "[", "0", "]"}]}], "=", "1"}], ",", " ", \(to\ set\ up\ our\ differential\ \(\(equation\)\(:\)\)\)}], "**)"}]], "Input", FontFamily->"Arial"], Cell[BoxData[ RowBox[{ RowBox[{"solution", "=", RowBox[{"NDSolve", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["y", "\[DoublePrime]", MultilineFunction->None], "[", "t", "]"}], "+", RowBox[{\((9.8)\), "*", RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "[", "t", "]"}]}], "+", \(y[t]\)}], "\[Equal]", \(TIDE[t]\)}], ",", \(y[0] \[Equal] 0\), ",", RowBox[{ RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "[", "0", "]"}], "\[Equal]", "1"}]}], "}"}], ",", "y", ",", \({t, 0, L}\)}], 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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"], Cell[BoxData[ \(\(M = 50;\)\)], "Input", FontFamily->"Arial"], Cell[BoxData[ \( (*\(*\)\(\ \)\(Lag\)\(\ \)\(\(Factor\)\(:\)\)\ **) \)], "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\ \ 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