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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "LeastSquares915.mws\nThu S
ep 15 08:34:32 MDT 2005\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 
"restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 94 "These \"with\" commands make items available from the Map
le packages \"plots\" and \"CurveFitting\":" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "with(plots, displ
ay);  " }{TEXT -1 46 "To make \"display\" available to this worksheet.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "with(CurveFitting, Leas
tSquares); " }{TEXT -1 24 "Ditto for \"LeastSquares\"" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 223 "Maple ha
s recently overhauled/redesigned their Least-Squares.  The \"with(stat
s)\" environment described in chapter 3 is still available.  Here we l
ook at the new ways Maple has come up with to do least-squares curve f
itting." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
76 "We begin with a couple of famous list-processing functions, \"map
\" and \"zip\":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 136 "Here are data cribbed from page 42 of our text.  Th
e square brackets delimit Maple lists (as oppsed to Maple sets, delimi
ted by \{ and \}:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 28 "Ex2_xRow := [1,2,3,4,5,6,7];" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "Ex2_yRow := [2.2, 3.7, 5.5, 5.9, 5.
6, 4.2, 3.1];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 120 "To make a scatter plot of these data, we need to matc
h them up in points.  The Maple command \"zip\" does this as follows.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 217 "Firs
t we set up the function \"makepoint\" which takes an ordered pair of \+
values and resturns a Maple list of two items.  Such a list is how Map
le expresses points in the two-dimensional Cartesian plane.   We follo
w the" }}{PARA 0 "" 0 "" {TEXT -1 113 "\"makepoint\" definition with a
 test invocation just to make sure \"makepoint\" is \"there\" and doin
g what we expect. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 29 "makepoint := (x,y) -> [x,y]; " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "makepoint(3,Pi);" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "Nopw we \"zip\"
 our \"makepoint\" function through the pair of lists.  The Maple \"zi
p\" command  feeds" }}{PARA 0 "" 0 "" {TEXT -1 220 "\"makepoint\" its \+
x-value from the first list, and the y-value from the second list.  In
 this case, \"zip\" makes us a list of Maple points.    You might want
 to check what happens if we change \"makepoint\" to add x and y...." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 43 "PTS := zip( makepoint, Ex2_xRow, Ex2_yRow);" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "Next we go for \+
a scatterplot of our data points:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "plot(PTS);" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 "We'd rather a p
lot of the discrete points, so we add on plot option \"style=POINT\", \+
and then" }}{PARA 0 "" 0 "" {TEXT -1 60 "plot option \"symbolsize=20\"
  to make the data points bigger." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 23 "plot(PTS, style=POINT);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 38 "plot(PTS, style=POINT, symbolsize=20);" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 243 "Later on, we'll want t
o superimpose plots, so we assign this scatter-plot structure to a Map
le variable we have named \"Scatter\" (note the \":\" at the end of th
is command).    We have to use the \"display\" command to see what \"S
catter \"looks like" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 49 "Scatter := plot(PTS, style=POINT, symbols
ize=20):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "display(Scatter
);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 348 "Here we have Maple do us the least-squares LINE for our data. \+
 We've told Maple to return its result as an expression in variable x.
    This is DANGEROUS because x might already be in play.   Try puttin
g an assignment statement \"x := -13;\" in ahead of the following -- w
atch the ensuing carnage.   We should probably use some other variable
 name..." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "LSQ_line := LeastSquares(PTS, x);" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "Now we plot the l
east-squares line and store its plot in the variable \"PA\":" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "P
A := plot(LSQ_line, x):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 113 "Now we use the paste-together aspect of \+
 \"display\" to show the least-squares line in relation to the data po
ints:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "display([Scatter, PA]);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 174 "The above graph should d
isappoint us as to how well the least-squares line fits the data.  So \+
we try for a least-squares parabola by applying the \"curve=A*x^2+B*x+
C\" option: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 55 "LSQ_parabola := LeastSquares(PTS,x, curve=A*x^2+
B*x+C);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 247 "We don't have absolute freedom in what we can tell \"cur
ve\" to do.   In the following command, we supply a \"curve\" which we
 know to be a parabola.  But parameter \"h\" will be squared, not \"li
near\", so \"LeastSquares\" yarks at us, and refuses the job:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 39 "LeastSquares(PTS,x, curve=A*(x-h)^2+k);" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 110 "Here we plot the para
bola and the data points as we did for the least-squares line.  The pa
rabola fits better:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 45 "P_LSQ_parabola := plot(LSQ_parabola, x=1.
.7):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "display([P_LSQ_para
bola, Scatter]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 157 "We can interpolate in our table of values using o
ur parabola.    Our x-values are all integers.   But, what if we neede
d an idea about a y-value for x = 2.4?" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "Ex2_xRow;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "f := unapply(LSQ_parabola, x);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Interpolated_Value := f(2.4)
;" }}}{EXCHG }{EXCHG }{EXCHG }{EXCHG }{EXCHG }{EXCHG }}{MARK "45" 0 }
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