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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 21 "Multidimensional Data" }}{PARA
257 "" 0 "" {TEXT -1 9 "Jim Herod" }}{PARA 258 "" 0 "" {TEXT -1 22 "Sc
hool of Mathematics " }}{PARA 259 "" 0 "" {TEXT -1 12 "Georgia Tech" }
}{PARA 260 "" 0 "" {TEXT -1 21 "herod@math.gatech.edu" }}{PARA 0 "" 0
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 176 " In this workshee
t, we explore a collection of data which could arise in a physical sit
uation. The tools used are ones which arise in studies of multidimensi
onal functions." }}{PARA 0 "" 0 "" {TEXT -1 311 " In this first pr
oblem, temperature is collected at a set of twenty-five points spread \+
over a rectangular region. We want to have a picture for the location \+
of isotherms, an average for the temperature over the plate, and the m
aximum for the temperature spread over a ring placed in the center of \+
the plate." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 ""
{TEXT -1 26 "Section 1: Collecting Data" }}{PARA 0 "" 0 "" {TEXT -1 0
"" }}{PARA 0 "" 0 "" {TEXT -1 101 " We simulate the collection of \+
data. Data points are located at the following twenty-five points." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "pts:=[seq(seq([p/4,q/4],p=0.
.4),q=0..4)];" }}}{PARA 0 "" 0 "" {TEXT -1 57 "We plot these points so
their location can be visualized." }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 22 "plot(pts,style=POINT);" }}}{PARA 0 "" 0 "" {TEXT -1
34 "Here are the temperature readings." }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 180 "T:=(x,y)->sinh(Pi*(1-x))/sinh(Pi)*sin(Pi*y)+\n \+
sinh(2*Pi*x)/sinh(2*Pi)*sin(2*Pi*y):\nDigits:=3:\nreadings:=[seq(
seq([p/4,q/4,evalf(T(p/4,q/4))],p=0..4),q=0..4)];\nDigits:=10:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT
-1 252 " Of course, in a physical situation, the data might be fou
nd by reading a thermometer and the data typed into a computer. Here w
e have simulated the determination of the data and have saved the stud
ent the task of typing in twenty-five data points." }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 37 "Section 2: Plot
ting and Fitting Data" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 ""
0 "" {TEXT -1 74 "We plot this collection of data. To do this, we read
in the plots package." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "wi
th(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "pointplot3d(r
eadings,axes=NORMAL,orientation=[-25,75],symbol=circle);" }}}{PARA 0 "
" 0 "" {TEXT -1 179 " This collection of data does not appear to ha
ve structure. The next collection of ideas is designed to give underst
anding to the data. We use the techniques of least squares. " }}{PARA
0 "" 0 "" {TEXT -1 321 " For this collection of data, we find a tw
o-dimensional polynomial, cubic in x and y. Other type fits could be u
sed, This one gives indication of the wide varieties available. Choice
s for the coefficients are made by requiring the resulting data should
fit the data as close as possible in the sense of least squares." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(stats);" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "xcord:=[seq(readings[p][1],p=1..25
)];\nycord:=[seq(readings[p][2],p=1..25)];\nzcord:=[seq(readings[p][3]
,p=1..25)];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "result:=fit
[leastsquare[[x,y,z],\nz=a*x^3+b*x^2+c*x+d*y^3+e*y^2+f*y+g*x^2*y+h*x*y
^2+i*x*y+j]]\n([xcord,ycord,zcord]);" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 35 "approx:=unapply(rhs(result),(x,y));" }}}{EXCHG {PARA
0 "> " 0 "" {MPLTEXT 1 0 46 "plot3d(approx(x,y),x=0..1,y=0..1,axes=NOR
MAL);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0
"" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 34 "Section 3: Dra
wing Isotherm Curves" }}{PARA 0 "" 0 "" {TEXT -1 146 " In this sec
tion, we try to understand the data by plotting the fitting surface wi
th contour lines. These plots correspond to isotherm curves." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "contourplot(approx(x,y),x=0.
.1,y=0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "plot3d(appro
x(x,y),x=0..1,y=0..1,style=contour,axes=NORMAL,\n orientation = [2
5,35]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 ""
0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3
"" 0 "" {TEXT -1 43 "Section 4: Average Temperature on the Plate" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 381 " We \+
want to know the average temperature on the plate. One way to think of
determing this is by finding the average of all the data points. We c
ould use the Statistics Package in Maple to find the average. We save \+
this package for examination and exploration in another course at anot
her time. For now, we compute the average of this collection of number
s in a standard manner." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "s
um(zcord[i],i=1..25)/25;" }}}{PARA 0 "" 0 "" {TEXT -1 300 " It see
ms appropriate to recall here, that the average of a function was dete
rmined in one-dimensional calculus by computing an integral. The same \+
idea is appropriate here. In the general setting, if z(x,y) is defined
over an interval [a,b]x[c,d], the average of z can be computed by the
quotient" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1
23 " " }{XPPEDIT 18 0 "int(int(z(x,y),x=a..b),y=
c..d)/((b-a)*(d-c))" "6#*&-%$intG6$-F%6$-%\"zG6$%\"xG%\"yG/F,;%\"aG%\"
bG/F-;%\"cG%\"dG\"\"\"*&,&F1F6F0!\"\"F6,&F5F6F4F9F6F9" }{TEXT -1 1 ".
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 154 " \+
Here is this computation of the average for this example. While the n
ext computed number will not be exactly the same as the previous, reme
mber that " }{TEXT 256 6 "approx" }{TEXT -1 23 " is a fit for the data
." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "int(int(approx(x,y),x=0
..1),y=0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 57 "S
ection 5: Maximum Temperature along a Ring on the Plate." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 155 " We next plac
e a ring on the center of the plate. The ring has diameter 1/2 -- radi
us 1/4. We draw a picture to illustration the placement of the ring."
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "implicitplot((x-1/2)^2+(y-
1/2)^2=1/16,x=-1..1,y=-1..1,view=[0..1,0..1]);" }}}{PARA 0 "" 0 ""
{TEXT -1 64 " The task is to find the the maximum temperature on th
e ring." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "approx(cos(t)/4+1
/2,sin(t)/4+1/2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "normal
(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "T:=unapply(%,t);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot(T(t),t=0..2*Pi);" }}}
{PARA 0 "" 0 "" {TEXT -1 110 " From the graph, it seems that the m
aximum temperature occurs between t = 3 and t = 4. We seek this value.
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "diff(T(t),t);\nMaxAt:=fs
olve(%=0,t=3..4);\nT(MaxAt);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 24 "Exercise for the studen
t" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 307 "The
Health and Performance Science Department at Georgia Tech gathered da
ta on males taking PE. Three parameters for the students were measured
: thigh circumference, leg press ability, and height of a leap. Here i
s the data. Circumference is measured in centimeters, leg press in pou
nds, and leap in inches." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "
thigh:=[52,54.2,64.5,52.3,54.5,58.0,48.0,58.4,58.5,60.0,49.2,55.5];" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "press:=[140,110,150,120,13
0,120,95,180,125,125,95,115];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 52 "leap:=[13,8.5,13,13,13,13,8.5,19,14,18.5,16.5,10.5];" }}}{PARA
0 "" 0 "" {TEXT -1 124 "Fit this data to formulate the height of a lea
p as a function of thigh circumference and pounds achieved with the le
g press." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(stats):" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "result:=fit[leastsquare[[x
,y,z], z=a*x^2+b*x*y+c*y^2+d*x+e*y+f,\n \{a,b,c,d,e,f\}]]([thigh,p
ress,leap]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "jump:=unapp
ly(rhs(result),(x,y));" }}}{PARA 0 "" 0 "" {TEXT -1 169 "(1) Plot the
function which predicts the ability to leap as a function of thigh ci
rcumference between 45 and 60 and of pounds lifted in a leg press betw
een 100 and 150." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "plot3d(j
ump(x,y),x=45..60,y=100..150,axes=NORMAL);" }}}{PARA 0 "" 0 "" {TEXT
-1 62 "Suppose a person has a thigh circumference of 53 centimeters?
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "plot(jump(53,y),y=100..1
50);" }}}{PARA 0 "" 0 "" {TEXT -1 28 " Answer these two questions." }}
{PARA 0 "" 0 "" {TEXT -1 111 " (2)At what level of leg press does th
is result predict the person would be able to leap the smallest amount
?" }}{PARA 0 "" 0 "" {TEXT -1 148 " (3) What is the average height t
hat people can leap if their thigh circumference is 53 centimeters and
they can press between 100 and 150 pounds?" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{MARK "0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }