~v200 200 ~w57 4 482 768 0 680 0 18 ~f? 14 12 10
? 2 0 1 1 ? ? ? "Arial" ? ? ? 1 ? 0 1
"Times" 12 ? ? 5 0 c n 106 1 0 0 k 468
i"?n             page ?p?a"
-2 1 26177 26178 26115 26178 1 1 1 1 0 0
8405120 0 -1 0 1 -1 -1 -1 -1 -1 0 1 ? ? 
~Q
]|Expr|[#b @`bb#_b#_b#_}).# b'4" *|: ;bP:&c0!)Total Least Squares|
  | vs,N Ordinary Least Squares}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1
0 ~V?m0 (X)~p0
1 ~V?m0 (Y)~p0
1 ~V?m0 (T)~p0
1 ~V?f0 (Mean)~p0
1 ~V?f144 (RowsOf)~p0
1 ~V?v0 (YY)~p0
1 ~V?v0 (XX)~p0
1 ~V?v0 (XY)~p0
1 ~V?v0 (C)~p0
1 ~V?v0 (k)~p0
1 ~V?v0 (m)~p0
1 ~V?v0 (b)~p0
1 ~V?v0 (y)~p0
1 ~V?v0 (x)~p0
1 ~V?c1 ('p)~p0
1 ~Q
]|Expr|[#b @`bb#_b#_b#_})!# b'4["! ) # b'4|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW})!# b'4" *|: ;bP8&c0!*Introduction|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}}}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0
0 ~Q
]|Expr|[#b @`bb#_b#_b#_})!# b$7[%! ) # b$7|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW})b!*# b$7" *|: ;bP8&c0!*When the|
  | X data are considered error,Mfree,L a `fb#@}vertical`f } projection|
  | is used to define the residual,N  This is the familiar linear|
  | regression model and an example of ordinary least squares ,H|
  |OLS,I curve fitting,N}& b!( b"0 b#8 b$@ b%H b&P!WW}) # b$7|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW})b W# b$7When the Y data are considered|
  | error,Mfree,L a `f0}horizontal`f } projection is used to define|
  | the residual,N  This also is an example of OLS curve fitting|
  |,N}& b!( b"0 b#8 b$@ b%H b&P!WW}) # b$7|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}}}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0
1 ~T~Q
]|Expr|[#b @`bb#_b#_b#_})!# b#/["! )$# b#/" *|: ;bP8&c0!*Point|
  | projections,Z}% b!( b"0 b$@ b%H b&P!WW})+# b#/`fb#@}vertical|
  |`f }    `f0}horizontal`f }    `f#}orthogonal|
  |}% b!( b"0 b$@ b%H b&P!WW}}}% b!( b"0 b$@ b%H b&P!WW}]|[~p0
0 ~G1 1 152 230
0 2 0 1 10
(-3.8000000000000003...4.8000000000000007):(~
 -3...2.7000000000000002):(?=0...2*'p):('p/5):(10)~Q
]|Expr|[#b @`bb#_b#_b#_})!# b!C" *|: ;bP8&c0!*Declarations|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1
0 ~V?c64 (left)~p0
1 ~V?c65 (right)~p0
1 ~V?c66 (bottom)~p0
1 ~V?c67 (top)~p0
1 ~A(T=(0,0;2,-2;-2,-2;2,-2;2,2))~p0
0 ~d~L1 11184810 ? (x,x):(x=left...right)~p0
0 ~gc1 2 ? 0 -4096 -5246 0
65535 4096 7114 0
~L2 255 ? (T_k):(k=~
 1...2)~p0
0 ~gc0 2 ? 0 -476 216 0
65535 1429 -2659 0
~L2 32768 ? (T_k):(k=~
 2...3)~p0
0 ~gc0 2 ? 0 1429 -2659 0
65535 -2381 -2659 0
~L2 16711680 ? (T_~
 k):(k=4...5)~p0
0 ~gc0 2 ? 0 1429 -2659 0
65535 1429 3090 0
~S17 ? 16744448 ? ? (~
 T_2):(?):(8)~p0
0 ~gc-1 1 ? 0 1429 -2659 0
~S17 ? 255 ? ? (T_1):(?):(8)~p0
0 ~gc-1 1 ? 0 -476 216 0
~S17 ? 32768 ? ? (T_3):(?):(8)~p0
0 ~gc-1 1 ? 0 -2381 -2659 0
~S17 ? 16711680 ? ? (T_5):(?):(8)~p0
0 ~gc-1 1 ? 0 1429 3090 0
~t~p0
0 ~t~p0
255 ~Q
]|Expr|[#b @`bb#_b#_b#_}`fb#L})!# b'4" *|: ;bP8&c0!*Data|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0
0 ~A(X=(-1.75;-1.1799999999999999;-0.88;-0.65000000000000002;-0.29999999999999999;~
 0.34000000000000002;0.5;0.78000000000000003;1.0800000000000001;~
 1.3999999999999999))~p0
1 ~d~A(Y=(-1.8200000000000001;-0.91000000000000003;-0.97999999999999998;~
 -0.42999999999999999;-0.32000000000000001;0.37;0.83999999999999997;~
 1.05;1.3400000000000001;2.5800000000000001))~p0
255 ~d~Q
]|Expr|[#b @`bb#_b#_b#_})b!K# b$@" *|: ;bP8&c0!*When both the|
  | X and Y data are subject to errors that are independently and|
  | identically distributed with zero mean and common variance,L|
  | an `f#}orthogonal`f } projection is used to define the residual|
  |,N  This technique is variously known as orthogonal distance |
  |regression,L errors,Min,Mvariables modeling,L and total least|
  | squares ,HTLS,I,N}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0
255 ~Q
]|Expr|[#b @`bb#_b#_b#_})!# b'4["! ) # b'4|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW})!# b'4" *|: ;bP8&c0!*Statistics|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}}}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1
0 ~A(Mean(?u)=1/(RowsOf(?u))*Summation(k):(1):(RowsOf(?u))*?u_k)~p0
1 ~d~Ht(C*(?u,?v)):(1/(RowsOf(?u)-1)*Summation(k):(1):(RowsOf(?u))*~
 (?u_k-Mean(?u))*(?v_k-Mean(?v)))~p0
1 ~A(C_XY=C*(X,Y))~p1
1 ~A(C_XY=1.3371355555555555)~p0
255 ~d~sb/_!!   }
$&!  c#T"!c"H$_c/__c/__}
^ _~A(C_XY=C*((-1.75;-~
 1.1799999999999999;-0.88;-0.65000000000000002;-0.29999999999999999;~
 0.34000000000000002;0.5;0.78000000000000003;1.0800000000000001;~
 1.3999999999999999),(-1.8200000000000001;-0.91000000000000003;~
 -0.97999999999999998;-0.42999999999999999;-0.32000000000000001;~
 0.37;0.83999999999999997;1.05;1.3400000000000001;2.5800000000000001)))~p0
2 ~sb/_!!   }
$&!  c#T"!c#L"_c/__c/__}
^ _~A(C_XY=1/9*Summation(~
 k):(1):(10)*((-1.75;-1.1799999999999999;-0.88;-0.65000000000000002;~
 -0.29999999999999999;0.34000000000000002;0.5;0.78000000000000003;~
 1.0800000000000001;1.3999999999999999)_k-Mean(-1.75;-1.1799999999999999;~
 -0.88;-0.65000000000000002;-0.29999999999999999;0.34000000000000002;~
 0.5;0.78000000000000003;1.0800000000000001;1.3999999999999999))*~
 ((-1.8200000000000001;-0.91000000000000003;-0.97999999999999998;~
 -0.42999999999999999;-0.32000000000000001;0.37;0.83999999999999997;~
 1.05;1.3400000000000001;2.5800000000000001)_k-Mean(-1.8200000000000001;~
 -0.91000000000000003;-0.97999999999999998;-0.42999999999999999;~
 -0.32000000000000001;0.37;0.83999999999999997;1.05;1.3400000000000001;~
 2.5800000000000001)))~p0
3 ~sb/_!!   }
#'!  c#T"!c#L"_c/__c/__!}
^ _~A(C_XY=0.1111111111111111*~
 Summation(k):(1):(10)*((-1.75;-1.1799999999999999;-0.88;-0.65000000000000002;~
 -0.29999999999999999;0.34000000000000002;0.5;0.78000000000000003;~
 1.0800000000000001;1.3999999999999999)_k+0.065999999999999975)*~
 ((-1.8200000000000001;-0.91000000000000003;-0.97999999999999998;~
 -0.42999999999999999;-0.32000000000000001;0.37;0.83999999999999997;~
 1.05;1.3400000000000001;2.5800000000000001)_k-0.17200000000000001))~p0
4 ~sb/_!!   }
$&!  c#T"!c"H$_c/__c/__}
^ _~A(C_XX=C*(X,X))~p1
1 ~A(C_XX=1.0887377777777778)~p0
255 ~d~sb/_!!   }
$&!  c#T"!c"H#_c/__c/__}
^ _~A(C_XX=C*((-1.75;-~
 1.1799999999999999;-0.88;-0.65000000000000002;-0.29999999999999999;~
 0.34000000000000002;0.5;0.78000000000000003;1.0800000000000001;~
 1.3999999999999999),(-1.75;-1.1799999999999999;-0.88;-0.65000000000000002;~
 -0.29999999999999999;0.34000000000000002;0.5;0.78000000000000003;~
 1.0800000000000001;1.3999999999999999)))~p0
2 ~sb/_!!   }
$&!  c#T"!c#L"_c/__c/__}
^ _~A(C_XX=1/9*Summation(~
 k):(1):(10)*((-1.75;-1.1799999999999999;-0.88;-0.65000000000000002;~
 -0.29999999999999999;0.34000000000000002;0.5;0.78000000000000003;~
 1.0800000000000001;1.3999999999999999)_k-Mean(-1.75;-1.1799999999999999;~
 -0.88;-0.65000000000000002;-0.29999999999999999;0.34000000000000002;~
 0.5;0.78000000000000003;1.0800000000000001;1.3999999999999999))^~
 2)~p0
3 ~sb/_!!   }
#'!  c#T"!c#L"_c/__c/__!}
^ _~A(C_YY=C*(Y,Y))~p1
1 ~A(C_YY=1.7210399999999999)~p0
255 ~d~sb/_!!   }
$&!  c#T"!c"H#_c/__c/__}
^ _~A(C_YY=C*((-1.8200000000000001;~
 -0.91000000000000003;-0.97999999999999998;-0.42999999999999999;~
 -0.32000000000000001;0.37;0.83999999999999997;1.05;1.3400000000000001;~
 2.5800000000000001),(-1.8200000000000001;-0.91000000000000003;~
 -0.97999999999999998;-0.42999999999999999;-0.32000000000000001;~
 0.37;0.83999999999999997;1.05;1.3400000000000001;2.5800000000000001)))~p0
2 ~sb/_!!   }
$&!  c#T"!c#L"_c/__c/__}
^ _~A(C_YY=1/9*Summation(~
 k):(1):(10)*((-1.8200000000000001;-0.91000000000000003;-0.97999999999999998;~
 -0.42999999999999999;-0.32000000000000001;0.37;0.83999999999999997;~
 1.05;1.3400000000000001;2.5800000000000001)_k-Mean(-1.8200000000000001;~
 -0.91000000000000003;-0.97999999999999998;-0.42999999999999999;~
 -0.32000000000000001;0.37;0.83999999999999997;1.05;1.3400000000000001;~
 2.5800000000000001))^2)~p0
3 ~sb/_!!   }
#'!  c#T"!c#L"_c/__c/__!}
^ _~Q
]|Expr|[#b @`bb#_b#_b#_}`fb#@})!# b'4`f }["! `fb#@}) # b'4|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}),# b'4" *|: ;bP8&c0!*Ordinary |
  |least squares,L X independent fit}& b!( b"0 b#8 b$@ b%H b&P!WW}}|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0
0 ~A(y_1=m_1*x+b_1)~p1
1 ~d~A(m_1=1.2281520700831952)~p0
255 ~d~sb/_!!!  }
$&!! c#T"!c"\"_c/__c/__}
^ _~A(b_1=0.25305803662549087)~p0
255 ~d~sb/_!!!  }
$&!! c#T"!c"L"_c/__c/__}
^ _~A(m_1=C_XY/C_XX)~p0
2 ~A(b_1=Mean(Y)-m_1*Mean(X))~p0
2 ~Q
]|Expr|[#b @`bb#_b#_b#_}`f0})!# b'4`f }["! `f0}) # b'4|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}),# b'4" *|: ;bP8&c0!*Ordinary |
  |least squares,L Y independent fit}& b!( b"0 b#8 b$@ b%H b&P!WW}}|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0
0 ~A(x_2=m_2*y+b_2)~p1
1 ~d~A(m_2=0.77693461834446353)~p0
255 ~d~sb/_!!!  }
$&!! c#T"!c"\"_c/__c/__}
^ _~A(b_2=-0.19963275435524772)~p0
255 ~d~sb/_!!!  }
$&!! c#T"!c"L"_c/__c/__}
^ _~A(m_2=C_XY/C_YY)~p0
2 ~A(b_2=Mean(X)-m_2*Mean(Y))~p0
2 ~Q
]|Expr|[#b @`bb#_b#_b#_}`f#})!# b'4`f }["! `f#}) # b'4|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW})'# b'4" *|: ;bP8&c0!*Total least|
  | squares fit}& b!( b"0 b#8 b$@ b%H b&P!WW}}|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0
0 ~A(y=m*x+b)~p1
1 ~d~A(m=1.2640107117810266)~p0
255 ~d~sb/_!!!  }
$&!! c#T"!c'L!_c/__c/__}
^ _~A(b=0.25542470697754771)~p0
255 ~d~sb/_!!!  }
$&!! c#T"!c"L"_c/__c/__}
^ _~A(m^2+m*(C_XX-C_YY)/~
 C_XY-1=0)~p1
2 ~T~A(m=1/2*(sqrt(((C_XX-C_YY)/C_XY)^2+4)-(C_XX-C_YY)/C_XY))~p0
0 ~sb/_!!!  }
)9!! c#T" c"L# c%X" c'8 _c/__c/__"c#T" c"L#!c"H" c'8 _c/__c/__ " !}
^ _~
 ~A(m=1.2640107117810266)~p0
1 ~sb/_!!!  }
$&!! c#T"!c"H"_c/__c/__}
^ _~t~p0
3 ~T~A(m=1/2*(-sqrt(((C_XX-C_YY)/C_XY)^2+4)-(C_XX-C_YY)/C_XY))~p0
0 ~sb/_!!!  }
)9!! c#T" c"L# c%X" c'8 _c/__c/__"c#T" c"L#!c"H" c'8 _c/__c/__!" !}
^ _~
 ~A(m=-0.79113253604549916)~p0
1 ~sb/_!!!  }
$&!! c#T"!c"H"_c/__c/__}
^ _~t~p0
3 ~A(m=Conditional(1/2*(-sqrt(((C_XX-C_YY)/C_XY)^2+4)-(C_XX-C_YY)/~
 C_XY),(C_XY<0);1/2*(sqrt(((C_XX-C_YY)/C_XY)^2+4)-(C_XX-C_YY)/~
 C_XY),(1>=0)))~p0
2 ~A(b=Mean(Y)-m*Mean(X))~p0
2 ~G1 1 268 496
0 1 0 4 10
(-2...1.5):(-2.5...2.5):(?=0...2*'p):(~
 'p/5):(10)~Q
]|Expr|[#b @`bb#_b#_b#_})!# b#@" *|: ;bP8&c0!*Declarations|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1
0 ~R8405120 ? (x,y):(y=bottom...top):(x=left...right):(0)~p0
1 ~gc1 2 ? 0 -1755 -4096 0
65535 -1755 4096 0
~gc1 2 ? 0 585 -4096 0
65535 585 4096 0
~gc1 2 ? 0 2926 -4096 0
65535 2926 4096 0
~
 ~R8405120 ? (x,y):(x=left...right):(y=bottom...top):(0)~p0
1 ~gc1 2 ? 0 -4096 -3277 0
65535 4096 -3277 0
~gc1 2 ? 0 -4096 -1638 0
65535 4096 -1638 0
~gc1 2 ? 0 -4096 0 0
65535 4096 0 0
~gc1 2 ? 0 -4096 1638 0
65535 4096 1638 0
~gc1 2 ? 0 -4096 3277 0
65535 4096 3277 0
~
 ~X2 8405120 (left,y):(y=bottom...top):(y)~p0
1 ~gc1 2 ? 0 -4096 -4096 0
65535 -4096 4096 0
~X1 8405120 (x,bottom):(~
 x=left...right):(x)~p0
1 ~gc1 2 ? 0 -4096 -4096 0
65535 4096 -4096 0
~V?c64 (left)~p0
1 ~V?c65 (right)~p0
1 ~V?c66 (bottom)~p0
1 ~V?c67 (top)~p0
1 ~L1 255 ? (x,y):(x=left...right)~p0
0 ~gc1 2 ? 0 -4096 -3723 0
65535 4096 3525 0
~L1 16711680 ? (x,~
 y_1):(x=left...right)~p0
0 ~gc1 2 ? 0 -4096 -3610 0
65535 4096 3433 0
~L1 32768 ? (x_2,y):(~
 y=top...bottom)~p0
0 ~gc1 2 ? 0 4664 4096 0
65535 -4428 -4096 0
~S17 ? 16744448 ? ? (~
 X_k,Y_k):(k=1...RowsOf(X)):(6)~p0
0 ~gc-1 10 ? 0 -3511 -2982 0
8191 -2177 -1491 0
14563 -1475 -1606 0
21845 -936 -705 0
28216 -117 -524 0
36408 1381 606 0
43690 1755 1376 0
51881 2411 1720 0
57343 3113 2195 0
65535 3862 4227 0
~
 ~S17 ? 255 ? ? (Mean(X),Mean(Y)):(?):(6)~p0
0 ~gc-1 1 ? 0 431 282 0
~t~p0
0 ~c1
27 -1 30 -1 ~c7
28 -1 26 -1 20 -1 1 -1 21 -1 2 -1 24 -1 4 -1 ~c2
29 -1 28 -1 25 -1 ~c3
30 -1 29 -1 24 -1 4 -1 ~c3
32 -1 34 -1 24 -1 4 -1 ~c5
33 -1 31 -1 20 -1 1 -1 24 -1 4 -1 ~c2
34 -1 33 -1 25 -1 ~c3
36 -1 38 -1 24 -1 4 -1 ~c5
37 -1 35 -1 21 -1 2 -1 24 -1 4 -1 ~c2
38 -1 37 -1 25 -1 ~c4
41 -1 43 -1 27 -1 9 -1 32 -1 ~c9
42 -1 44 -1 21 -1 2 -1 24 -1 4 -1 20 -1 1 -1 41 -1 11 -1 ~c4
47 -1 49 -1 27 -1 9 -1 36 -1 ~c9
48 -1 50 -1 21 -1 2 -1 24 -1 4 -1 20 -1 1 -1 47 -1 11 -1 ~c5
53 -1 58 -1 27 -1 9 -1 32 -1 36 -1 ~c9
54 -1 59 -1 21 -1 2 -1 24 -1 4 -1 20 -1 1 -1 53 -1 11 -1 ~c1
0 56 -1 55 -1 ~c5
1 56 -1 0 56 -1 32 -1 9 -1 36 -1 27 -1 ~c1
0 57 -1 55 -1 ~c5
1 57 -1 0 57 -1 32 -1 9 -1 36 -1 27 -1 ~e