(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 4.0, MathReader 4.0, or any compatible application. The data 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 CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@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[ 230924, 7385]*) (*NotebookOutlinePosition[ 231883, 7416]*) (* CellTagsIndexPosition[ 231839, 7412]*) (*WindowFrame->Normal*) Notebook[{ Cell["Least Squares as a tool for regression", "Title"], Cell[TextData[{ "This notebook has been written in ", StyleBox["Mathematica ", FontSlant->"Italic"], "by \n\n", StyleBox["Mark J. McCready\nProfessor and Chair of Chemical Engineering\n\ University of Notre Dame\nNotre Dame IN 46556\nUSA", FontSize->14], "\n\nMark.J.McCready.1@nd.edu\n", ButtonBox["http://www.nd.edu/~mjm/", ButtonData:>{ URL[ "http://www.nd.edu/~mjm/"], None}, ButtonStyle->"Hyperlink"], "\n\n\nIt is copyrighted to the extent allowed by whatever laws pertain to \ the World Wide Web and the Internet.\n\nI would hope that as a professional \ courtesy, this notice remain visible to other users. \nThere is no charge for \ copying and dissemination \n\nVersion: 9/98\nMore recent versions of this \ notebook should be available at the web site:\n", Cell[BoxData[ FormBox[ ButtonBox[\(\(\(http\)\(:\)\) // \(www . nd . edu/\(\(~\)\(mjm\)\)\)/ linear . regression . nb\), ButtonData:>{ URL[ "http://www.nd.edu/~mjm/linear.regression.nb"], None}, ButtonStyle->"Hyperlink"], TraditionalForm]]] }], "Text"], Cell[TextData[{ "Suppose that we have a set of experimental data that we would like to fit \ to a curve. Generally, the idea would be to choose a functional form for the \ curve based on some known theory and then adjust the coefficients to minimize \ the error. We might be doing a transport experiment where dimensional \ analysis predicts a power-law relation, f ~ 1/Re or we could be searching for \ the value of a universal constant, say ", StyleBox["g. ", FontSlant->"Italic"], " " }], "Text"], Cell[CellGroupData[{ Cell["\<\ How about for the sake of an initial example in this course, we \ choose a known function to generate our data, do the fitting, and then check \ the result. \ \>", "Subsection"], Cell["Here are some data. 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t[i]^2 . \n\nThe \"goodness of \ fit\" needs some quantification. Why not use something that looks \ reasonable, the ", Cell[BoxData[ \(TraditionalForm\`\(\(\[Chi]\^2\ function, \)\ \)\)]], Cell[BoxData[ \(TraditionalForm\`\[Chi]\^2\)]], " = Sum[r[i]^2,{i,1,n}].\n\nThe idea is to choose {a0, a1, a2} so that ", Cell[BoxData[ \(TraditionalForm\`\[Chi]\^2\)]], " has the smallest possible value.\n\nHow do we do this? We need to \ minimize, ", Cell[BoxData[ \(TraditionalForm\`\[Chi]\^2\)]], ", with respect to the 3 coefficients. \n\nTake the partial derivatives \ and set the three equations to 0." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(chisquar = Sum[\((\((\(-xx[i]\)\ + \ a0\ + a1\ tt[i] + \ a2\ tt[i]^2\ )\))\)^2, { i, 0, 2, .5}]\)], "Input", CellLabel->"In[239]:="], Cell[BoxData[ \(TraditionalForm \`\((a2\ \(tt(0)\)\^2 + a1\ \(tt(0)\) + a0 - xx(0))\)\^2 + \((a2\ \(tt(0.5`)\)\^2 + a1\ \(tt(0.5`)\) + a0 - xx(0.5`))\)\^2 + \((a2\ \(tt(1.`)\)\^2 + a1\ \(tt(1.`)\) + a0 - xx(1.`))\)\^2 + \((a2\ \(tt(1.5`)\)\^2 + a1\ \(tt(1.5`)\) + a0 - xx(1.5`))\)\^2 + \((a2\ \(tt(2.`)\)\^2 + a1\ \(tt(2.`)\) + a0 - xx(2.`))\)\^2\)], "Output", CellLabel->"Out[239]="] }, Open ]], Cell["Here we take the derivative with respect to a0.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(d0 = FullSimplify[D[chisquar, a0]]\)], "Input", CellLabel->"In[240]:="], Cell[BoxData[ \(TraditionalForm \`2\ \((5\ a0 + a1\ \((tt(0) + tt(0.5`) + tt(1.`) + tt(1.5`) + tt(2.`))\) + a2\ \((\(tt(0)\)\^2 + \(tt(0.5`)\)\^2 + \(tt(1.`)\)\^2 + \(tt(1.5`)\)\^2 + \(tt(2.`)\)\^2)\) - 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xx(0))\)\ \(tt(0)\)\^2 + \(tt(0.5`)\)\^2\ \((a0 + \(tt(0.5`)\)\ \((a1 + a2\ \(tt(0.5`)\))\) - xx(0.5`))\) + \(tt(1.`)\)\^2\ \((a0 + \(tt(1.`)\)\ \((a1 + a2\ \(tt(1.`)\))\) - xx(1.`))\) + \(tt(1.5`)\)\^2\ \((a0 + \(tt(1.5`)\)\ \((a1 + a2\ \(tt(1.5`)\))\) - xx(1.5`))\) + \(tt(2.`)\)\^2\ \((a0 + \(tt(2.`)\)\ \((a1 + a2\ \(tt(2.`)\))\) - xx(2.`))\)) \)\)], "Output", CellLabel->"Out[242]="] }, Open ]], Cell["\<\ However, we need to see what is going on here to make some sense,\ \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Coefficient[ExpandAll[d0], a0]\)], "Input", CellLabel->"In[243]:="], Cell[BoxData[ \(TraditionalForm\`10\)], "Output", CellLabel->"Out[243]="], Cell[CellGroupData[{ Cell[BoxData[ \(Coefficient[ExpandAll[d0], a1]\)], "Input", CellLabel->"In[244]:="], Cell[BoxData[ \(TraditionalForm \`2\ \(tt(0)\) + 2\ \(tt(0.5`)\) + 2\ \(tt(1.`)\) + 2\ \(tt(1.5`)\) + 2\ \(tt(2.`)\)\)], "Output", CellLabel->"Out[244]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Coefficient[ExpandAll[d0], a2]\)], "Input", CellLabel->"In[245]:="], Cell[BoxData[ \(TraditionalForm \`2\ \(tt(0)\)\^2 + 2\ \(tt(0.5`)\)\^2 + 2\ \(tt(1.`)\)\^2 + 2\ \(tt(1.5`)\)\^2 + 2\ \(tt(2.`)\)\^2\)], "Output", CellLabel->"Out[245]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Coefficient[ExpandAll[d1], a0]\)], "Input", CellLabel->"In[246]:="], Cell[BoxData[ \(TraditionalForm \`2\ \(tt(0)\) + 2\ \(tt(0.5`)\) + 2\ \(tt(1.`)\) + 2\ \(tt(1.5`)\) + 2\ \(tt(2.`)\)\)], "Output", CellLabel->"Out[246]="], Cell[CellGroupData[{ Cell[BoxData[ \(Coefficient[ExpandAll[d1], a1]\)], "Input", CellLabel->"In[247]:="], Cell[BoxData[ \(TraditionalForm \`2\ \(tt(0)\)\^2 + 2\ \(tt(0.5`)\)\^2 + 2\ \(tt(1.`)\)\^2 + 2\ \(tt(1.5`)\)\^2 + 2\ \(tt(2.`)\)\^2\)], "Output", CellLabel->"Out[247]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Coefficient[ExpandAll[d1], a2]\)], "Input", CellLabel->"In[248]:="], Cell[BoxData[ \(TraditionalForm \`2\ \(tt(0)\)\^3 + 2\ \(tt(0.5`)\)\^3 + 2\ \(tt(1.`)\)\^3 + 2\ \(tt(1.5`)\)\^3 + 2\ \(tt(2.`)\)\^3\)], "Output", CellLabel->"Out[248]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Coefficient[ExpandAll[d2], a0]\)], "Input", CellLabel->"In[249]:="], Cell[BoxData[ \(TraditionalForm \`2\ \(tt(0)\)\^2 + 2\ \(tt(0.5`)\)\^2 + 2\ \(tt(1.`)\)\^2 + 2\ \(tt(1.5`)\)\^2 + 2\ \(tt(2.`)\)\^2\)], "Output", CellLabel->"Out[249]="], Cell[CellGroupData[{ Cell[BoxData[ \(Coefficient[ExpandAll[d2], a1]\)], "Input", CellLabel->"In[250]:="], Cell[BoxData[ \(TraditionalForm \`2\ \(tt(0)\)\^3 + 2\ \(tt(0.5`)\)\^3 + 2\ \(tt(1.`)\)\^3 + 2\ \(tt(1.5`)\)\^3 + 2\ \(tt(2.`)\)\^3\)], "Output", CellLabel->"Out[250]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Coefficient[ExpandAll[d2], a2]\)], "Input", CellLabel->"In[251]:="], Cell[BoxData[ \(TraditionalForm \`2\ \(tt(0)\)\^4 + 2\ \(tt(0.5`)\)\^4 + 2\ \(tt(1.`)\)\^4 + 2\ \(tt(1.5`)\)\^4 + 2\ \(tt(2.`)\)\^4\)], "Output", CellLabel->"Out[251]="] }, Open ]] }, Open ]], Cell[TextData[{ "It is fairly clear that there is a pattern. Note also that the xx's all \ appear as a linear sum. Someone has figured out the pattern, you could do \ this if you knew there was one and had enough time.\n\nHere is the problem \ that needs to be solved to get the coefficients\n\nWe start with the data, \ x[t[i]], t[i].\nWe ant to fite the coefficients, {a0,a1,a2} from the \ relation,\n\nf[t] = a0 +a1 t(i)+ a2 t(i)^2 \n\nWe construct a ", StyleBox["T", FontSlant->"Italic"], " matrix which is the table of the values of the independent variable. The \ functional relation for the independent variable can be nonlinear. The \ original data, x[t[i], t[i] are then related by the residual relations. \n\n\ ", StyleBox["T", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" . ", FontSlant->"Italic"], StyleBox["a", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" = ", FontSlant->"Italic"], StyleBox["x\n\n", FontWeight->"Bold", FontSlant->"Italic"], "If the number of data points (i.e., rows) was equal to the number of \ columns of ", StyleBox["T", FontWeight->"Bold", FontSlant->"Italic"], ", then there would be a unique solution and no minimization would be \ necessary. However, we have mroe data and thus want to mimimize the ", Cell[BoxData[ \(TraditionalForm\`\[Chi]\^2\)]], " function. \n\nSo starting with ", StyleBox["T", FontWeight->"Bold", FontSlant->"Italic"], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(T = Table[{1, tt[i], tt[i]^2}, {i, 0, 2, .5}]\)], "Input", CellLabel->"In[252]:="], Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ {"1", \(tt(0)\), \(\(tt(0)\)\^2\)}, {"1", \(tt(0.5`)\), \(\(tt(0.5`)\)\^2\)}, {"1", \(tt(1.`)\), \(\(tt(1.`)\)\^2\)}, {"1", \(tt(1.5`)\), \(\(tt(1.5`)\)\^2\)}, {"1", \(tt(2.`)\), \(\(tt(2.`)\)\^2\)} }], ")"}], TraditionalForm]], "Output", CellLabel->"Out[252]="] }, Open ]], Cell[TextData[{ "It turns out that to reproduce the pattern generated from taking the \ derivative above we need the following operation.\n\n", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["T", FontWeight->"Bold"], "T"], TraditionalForm]]], ".", StyleBox[" T", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" . ", FontSlant->"Italic"], StyleBox["a", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" = ", FontSlant->"Italic"], Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["T", FontWeight->"Bold"], "T"], TraditionalForm]]], StyleBox[" ", FontSlant->"Italic"], StyleBox["x\n", FontWeight->"Bold", FontSlant->"Italic"], "\nwhich gives \n\n", StyleBox[" ", FontSlant->"Italic"], StyleBox["a", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" = ", FontSlant->"Italic"], Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", RowBox[{ FormBox[ SuperscriptBox[ StyleBox["T", FontWeight->"Bold"], "T"], "TraditionalForm"], ".", " ", StyleBox["T", FontWeight->"Bold"], " ", "."}], ")"}], \(-1\)], TraditionalForm]]], StyleBox[" ", FontSlant->"Italic"], Cell[BoxData[ FormBox[ RowBox[{" ", SuperscriptBox[ StyleBox["T", FontWeight->"Bold"], "T"]}], TraditionalForm]]], StyleBox[" ", FontSlant->"Italic"], StyleBox["x\n \n ", FontWeight->"Bold", FontSlant->"Italic"], "to get ", StyleBox["a. \n \n ", FontWeight->"Bold", FontSlant->"Italic"], "Here are some checks," }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(test1 = \((Transpose[T] . T)\) . {a0, a1, a2}\)], "Input", CellLabel->"In[253]:="], Cell[BoxData[ \(TraditionalForm \`{5\ a0 + a1\ \((tt(0) + tt(0.5`) + tt(1.`) + tt(1.5`) + tt(2.`))\) + a2\ \((\(tt(0)\)\^2 + \(tt(0.5`)\)\^2 + \(tt(1.`)\)\^2 + \(tt(1.5`)\)\^2 + \(tt(2.`)\)\^2)\), a0\ \((tt(0) + tt(0.5`) + tt(1.`) + tt(1.5`) + tt(2.`))\) + a1\ \((\(tt(0)\)\^2 + \(tt(0.5`)\)\^2 + \(tt(1.`)\)\^2 + \(tt(1.5`)\)\^2 + \(tt(2.`)\)\^2)\) + a2\ \((\(tt(0)\)\^3 + \(tt(0.5`)\)\^3 + \(tt(1.`)\)\^3 + \(tt(1.5`)\)\^3 + \(tt(2.`)\)\^3)\), a0\ \((\(tt(0)\)\^2 + \(tt(0.5`)\)\^2 + \(tt(1.`)\)\^2 + \(tt(1.5`)\)\^2 + \(tt(2.`)\)\^2)\) + a1\ \((\(tt(0)\)\^3 + \(tt(0.5`)\)\^3 + \(tt(1.`)\)\^3 + \(tt(1.5`)\)\^3 + \(tt(2.`)\)\^3)\) + a2\ \((\(tt(0)\)\^4 + \(tt(0.5`)\)\^4 + \(tt(1.`)\)\^4 + \(tt(1.5`)\)\^4 + \(tt(2.`)\)\^4)\)}\)], "Output", CellLabel->"Out[253]="] }, Open ]], Cell["We can do some checks,", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Coefficient[test1[\([1]\)], a0]\)], "Input", CellLabel->"In[254]:="], Cell[BoxData[ \(TraditionalForm\`5\)], "Output", CellLabel->"Out[254]="], Cell[CellGroupData[{ Cell[BoxData[ \(Coefficient[test1[\([1]\)], a1]\)], "Input", CellLabel->"In[255]:="], Cell[BoxData[ \(TraditionalForm\`tt(0) + tt(0.5`) + tt(1.`) + tt(1.5`) + tt(2.`)\)], "Output", CellLabel->"Out[255]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Coefficient[test1[\([1]\)], a2]\)], "Input", CellLabel->"In[256]:="], Cell[BoxData[ \(TraditionalForm \`\(tt(0)\)\^2 + \(tt(0.5`)\)\^2 + \(tt(1.`)\)\^2 + \(tt(1.5`)\)\^2 + \(tt(2.`)\)\^2\)], "Output", CellLabel->"Out[256]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Coefficient[test1[\([2]\)], a0]\)], "Input", CellLabel->"In[257]:="], Cell[BoxData[ \(TraditionalForm\`tt(0) + tt(0.5`) + tt(1.`) + tt(1.5`) + tt(2.`)\)], "Output", CellLabel->"Out[257]="], Cell[CellGroupData[{ Cell[BoxData[ \(Coefficient[test1[\([2]\)], a1]\)], "Input", CellLabel->"In[258]:="], Cell[BoxData[ \(TraditionalForm \`\(tt(0)\)\^2 + \(tt(0.5`)\)\^2 + \(tt(1.`)\)\^2 + \(tt(1.5`)\)\^2 + \(tt(2.`)\)\^2\)], "Output", CellLabel->"Out[258]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Coefficient[test1[\([2]\)], a2]\)], "Input", CellLabel->"In[259]:="], Cell[BoxData[ \(TraditionalForm \`\(tt(0)\)\^3 + \(tt(0.5`)\)\^3 + \(tt(1.`)\)\^3 + \(tt(1.5`)\)\^3 + \(tt(2.`)\)\^3\)], "Output", CellLabel->"Out[259]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Coefficient[test1[\([3]\)], a0]\)], "Input", CellLabel->"In[260]:="], Cell[BoxData[ \(TraditionalForm \`\(tt(0)\)\^2 + \(tt(0.5`)\)\^2 + \(tt(1.`)\)\^2 + \(tt(1.5`)\)\^2 + \(tt(2.`)\)\^2\)], "Output", CellLabel->"Out[260]="], Cell[CellGroupData[{ Cell[BoxData[ \(Coefficient[test1[\([3]\)], a1]\)], "Input", CellLabel->"In[261]:="], Cell[BoxData[ \(TraditionalForm \`\(tt(0)\)\^3 + \(tt(0.5`)\)\^3 + \(tt(1.`)\)\^3 + \(tt(1.5`)\)\^3 + \(tt(2.`)\)\^3\)], "Output", CellLabel->"Out[261]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Coefficient[test1[\([3]\)], a2]\)], "Input", CellLabel->"In[262]:="], Cell[BoxData[ \(TraditionalForm \`\(tt(0)\)\^4 + \(tt(0.5`)\)\^4 + \(tt(1.`)\)\^4 + \(tt(1.5`)\)\^4 + \(tt(2.`)\)\^4\)], "Output", CellLabel->"Out[262]="] }, Open ]] }, Open ]], Cell["\<\ So we see that that sans a factor of 2, we have got it. Check the \ xx's\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(test2 = Transpose[aa] . {xx[0], xx[ .5], xx[1], xx[1.5], xx[2]}\)], "Input", CellLabel->"In[263]:="], Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{ RowBox[{\(xx(0)\), "+", RowBox[{"xx", "(", StyleBox["0.5`", StyleBoxAutoDelete->True, PrintPrecision->1], ")"}], "+", \(xx(1)\), "+", RowBox[{"xx", "(", StyleBox["1.5`", StyleBoxAutoDelete->True, PrintPrecision->2], ")"}], "+", \(xx(2)\)}], ",", RowBox[{\(\(tt(0)\)\ \(xx(0)\)\), "+", RowBox[{\(tt(0.5`)\), " ", RowBox[{"xx", "(", StyleBox["0.5`", StyleBoxAutoDelete->True, PrintPrecision->1], ")"}]}], "+", \(\(tt(1.`)\)\ \(xx(1)\)\), "+", RowBox[{\(tt(1.5`)\), " ", RowBox[{"xx", "(", StyleBox["1.5`", StyleBoxAutoDelete->True, PrintPrecision->2], ")"}]}], "+", \(\(tt(2.`)\)\ \(xx(2)\)\)}], ",", RowBox[{\(\(xx(0)\)\ \(tt(0)\)\^2\), "+", RowBox[{\(\(tt(0.5`)\)\^2\), " ", RowBox[{"xx", "(", StyleBox["0.5`", StyleBoxAutoDelete->True, PrintPrecision->1], ")"}]}], "+", \(\(tt(1.`)\)\^2\ \(xx(1)\)\), "+", RowBox[{\(\(tt(1.5`)\)\^2\), " ", RowBox[{"xx", "(", StyleBox["1.5`", StyleBoxAutoDelete->True, PrintPrecision->2], ")"}]}], "+", \(\(tt(2.`)\)\^2\ \(xx(2)\)\)}]}], "}"}], TraditionalForm]], "Output", CellLabel->"Out[263]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Coefficient[d0, xx[ .5]]\)], "Input", CellLabel->"In[264]:="], Cell[BoxData[ \(TraditionalForm\`\(-2\)\)], "Output", CellLabel->"Out[264]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(test2[\([1]\)]\)], "Input", CellLabel->"In[265]:="], Cell[BoxData[ FormBox[ RowBox[{\(xx(0)\), "+", RowBox[{"xx", "(", StyleBox["0.5`", StyleBoxAutoDelete->True, PrintPrecision->1], ")"}], "+", \(xx(1)\), "+", RowBox[{"xx", "(", StyleBox["1.5`", StyleBoxAutoDelete->True, PrintPrecision->2], ")"}], "+", \(xx(2)\)}], TraditionalForm]], "Output", CellLabel->"Out[265]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Coefficient[d1, xx[1. ]]\)], "Input", CellLabel->"In[266]:="], Cell[BoxData[ \(TraditionalForm\`\(-2\)\ \(tt(1.`)\)\)], "Output", CellLabel->"Out[266]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(test2[\([2]\)]\)], "Input", CellLabel->"In[267]:="], Cell[BoxData[ FormBox[ RowBox[{\(\(tt(0)\)\ \(xx(0)\)\), "+", RowBox[{\(tt(0.5`)\), " ", RowBox[{"xx", "(", StyleBox["0.5`", StyleBoxAutoDelete->True, PrintPrecision->1], ")"}]}], "+", \(\(tt(1.`)\)\ \(xx(1)\)\), "+", RowBox[{\(tt(1.5`)\), " ", RowBox[{"xx", "(", StyleBox["1.5`", StyleBoxAutoDelete->True, PrintPrecision->2], ")"}]}], "+", \(\(tt(2.`)\)\ \(xx(2)\)\)}], TraditionalForm]], "Output", CellLabel->"Out[267]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Coefficient[d2, xx[1. ]]\)], "Input", CellLabel->"In[268]:="], Cell[BoxData[ \(TraditionalForm\`\(-2\)\ \(tt(1.`)\)\^2\)], "Output", CellLabel->"Out[268]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(test2[\([3]\)]\)], "Input", CellLabel->"In[269]:="], Cell[BoxData[ FormBox[ RowBox[{\(\(xx(0)\)\ \(tt(0)\)\^2\), "+", RowBox[{\(\(tt(0.5`)\)\^2\), " ", RowBox[{"xx", "(", StyleBox["0.5`", StyleBoxAutoDelete->True, PrintPrecision->1], ")"}]}], "+", \(\(tt(1.`)\)\^2\ \(xx(1)\)\), "+", RowBox[{\(\(tt(1.5`)\)\^2\), " ", RowBox[{"xx", "(", StyleBox["1.5`", StyleBoxAutoDelete->True, PrintPrecision->2], ")"}]}], "+", \(\(tt(2.`)\)\^2\ \(xx(2)\)\)}], TraditionalForm]], "Output", CellLabel->"Out[269]="] }, Open ]], Cell["\<\ Thus the factor of 2 is still there but the set of equations to be \ solved is\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(eqs = \((Transpose[T] . T)\) . {a0, a1, a2}\ - Transpose[T] . {xx[0], xx[ .5], xx[1. ], xx[1.5], xx[2. ]}\)], "Input",\ CellLabel->"In[270]:="], Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{ RowBox[{ \(5\ a0\), "+", \(a1\ \((tt(0) + tt(0.5`) + tt(1.`) + tt(1.5`) + tt(2.`))\)\), "+", \(a2\ \((\(tt(0)\)\^2 + \(tt(0.5`)\)\^2 + \(tt(1.`)\)\^2 + \(tt(1.5`)\)\^2 + \(tt(2.`)\)\^2)\)\), "-", \(xx(0)\), "-", RowBox[{"xx", "(", StyleBox["0.5`", StyleBoxAutoDelete->True, PrintPrecision->1], ")"}], "-", RowBox[{"xx", "(", StyleBox["1.`", StyleBoxAutoDelete->True, PrintPrecision->1], ")"}], "-", RowBox[{"xx", "(", StyleBox["1.5`", StyleBoxAutoDelete->True, PrintPrecision->2], ")"}], "-", RowBox[{"xx", "(", StyleBox["2.`", StyleBoxAutoDelete->True, PrintPrecision->1], ")"}]}], ",", RowBox[{ \(a0\ \((tt(0) + tt(0.5`) + tt(1.`) + tt(1.5`) + tt(2.`))\)\), "+", \(a1\ \(( \(tt(0)\)\^2 + \(tt(0.5`)\)\^2 + \(tt(1.`)\)\^2 + \(tt(1.5`)\)\^2 + \(tt(2.`)\)\^2)\)\), "+", \(a2\ \(( \(tt(0)\)\^3 + \(tt(0.5`)\)\^3 + \(tt(1.`)\)\^3 + \(tt(1.5`)\)\^3 + \(tt(2.`)\)\^3)\)\), "-", \(\(tt(0)\)\ \(xx(0)\)\), "-", RowBox[{\(tt(0.5`)\), " ", RowBox[{"xx", "(", StyleBox["0.5`", StyleBoxAutoDelete->True, PrintPrecision->1], ")"}]}], "-", RowBox[{\(tt(1.`)\), " ", RowBox[{"xx", "(", StyleBox["1.`", StyleBoxAutoDelete->True, PrintPrecision->1], ")"}]}], "-", RowBox[{\(tt(1.5`)\), " ", RowBox[{"xx", "(", StyleBox["1.5`", StyleBoxAutoDelete->True, PrintPrecision->2], ")"}]}], "-", RowBox[{\(tt(2.`)\), " ", RowBox[{"xx", "(", StyleBox["2.`", StyleBoxAutoDelete->True, PrintPrecision->1], ")"}]}]}], ",", RowBox[{ \(\(-\(xx(0)\)\)\ \(tt(0)\)\^2\), "+", \(a0\ \(( \(tt(0)\)\^2 + \(tt(0.5`)\)\^2 + \(tt(1.`)\)\^2 + \(tt(1.5`)\)\^2 + \(tt(2.`)\)\^2)\)\), "+", \(a1\ \(( \(tt(0)\)\^3 + \(tt(0.5`)\)\^3 + \(tt(1.`)\)\^3 + \(tt(1.5`)\)\^3 + \(tt(2.`)\)\^3)\)\), "+", \(a2\ \(( \(tt(0)\)\^4 + \(tt(0.5`)\)\^4 + \(tt(1.`)\)\^4 + \(tt(1.5`)\)\^4 + \(tt(2.`)\)\^4)\)\), "-", RowBox[{\(\(tt(0.5`)\)\^2\), " ", RowBox[{"xx", "(", StyleBox["0.5`", StyleBoxAutoDelete->True, PrintPrecision->1], ")"}]}], "-", RowBox[{\(\(tt(1.`)\)\^2\), " ", RowBox[{"xx", "(", StyleBox["1.`", StyleBoxAutoDelete->True, PrintPrecision->1], ")"}]}], "-", RowBox[{\(\(tt(1.5`)\)\^2\), " ", RowBox[{"xx", "(", StyleBox["1.5`", StyleBoxAutoDelete->True, PrintPrecision->2], ")"}]}], "-", RowBox[{\(\(tt(2.`)\)\^2\), " ", RowBox[{"xx", "(", StyleBox["2.`", StyleBoxAutoDelete->True, PrintPrecision->1], ")"}]}]}]}], "}"}], TraditionalForm]], "Output", CellLabel->"Out[270]="] }, Open ]], Cell["Here we check this, just to be sure!!", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(FullSimplify[d0 - 2 Flatten[eqs[\([1]\)]]]\)], "Input", CellLabel->"In[271]:="], Cell[BoxData[ \(TraditionalForm\`0\)], "Output", CellLabel->"Out[271]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(FullSimplify[d1 - 2 Flatten[eqs[\([2]\)]]]\)], "Input", CellLabel->"In[272]:="], Cell[BoxData[ \(TraditionalForm\`0\)], "Output", CellLabel->"Out[272]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(FullSimplify[d2 - 2 Flatten[eqs[\([3]\)]]]\)], "Input", CellLabel->"In[273]:="], Cell[BoxData[ \(TraditionalForm\`0\)], "Output", CellLabel->"Out[273]="] }, Open ]], Cell["\<\ Now let's solve the first problem. In terms of the original data \ we have\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(t*t\)], "Input", CellLabel->"In[274]:="], Cell[BoxData[ \(TraditionalForm\`{0, 0.25`, 1.`, 2.25`, 4.`}\)], "Output", CellLabel->"Out[274]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(t\)], "Input", CellLabel->"In[275]:="], Cell[BoxData[ \(TraditionalForm\`{0, 0.5`, 1.`, 1.5`, 2.`}\)], "Output", CellLabel->"Out[275]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(x\)], "Input", CellLabel->"In[276]:="], Cell[BoxData[ \(TraditionalForm \`{0.971397136778580971`, 1.38058078289795528`, 1.40375308028514673`, 1.1295345023808776`, 0.397553874854338786`}\)], "Output", CellLabel->"Out[276]="] }, Open ]], Cell[TextData[{ "Here is the ", StyleBox["a", FontSlant->"Italic"], " matrix which is the matrix of the t values for each of the data points. \ This is one place to use * for a vector operation!!\n\nNote that this matrix \ has a number columns equal to the number of coefficients to fit. However, \ the number of rows is equal to the number of data points. We better have \ more data points than coefficients!!" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Tx = Transpose[{Table[1, {i, 0, 2, .5}], \ t, t*t}]\)], "Input", CellLabel->"In[277]:="], Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ {"1", "0", "0"}, {"1", "0.5`", "0.25`"}, {"1", "1.`", "1.`"}, {"1", "1.5`", "2.25`"}, {"1", "2.`", "4.`"} }, ColumnAlignments->{Decimal}], ")"}], TraditionalForm]], "Output", CellLabel->"Out[277]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(?? 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*) (*********************************************************************** End of Mathematica Notebook file. ***********************************************************************)