{VERSION 5 0 "SUN SPARC SOLARIS" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 13 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 266 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 11 "" 1 "" {TEXT 256 13 "ACCESS - MATH" }}{PARA 257 "" 0 "" {TEXT -1 9 "July 2006" }}{PARA 256 "" 0 "" {TEXT -1 49 "Notes on \+ Body Mass Index and actual national data" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 28 "W hat is the Body Mass Index?" }}{PARA 0 "" 0 "" {TEXT -1 383 " If y ou read newspapers and magazines it is likely that once or twice a yea r you run across an article about the body mass index (B.M.I.) , and i ts use in determining health risk factors for overweight and underweig ht people. If you search the internet for \"body mass index\" you wil l find many sites which let you compute your B.M.I., and which tell yo u a little bit about it. " }}{PARA 0 "" 0 "" {TEXT -1 931 " A pe rson's B.M.I. is computed by dividing their weight by the square of th eir height, and then multiplying by a universal constant. If you meas ure weight in Newtons, i.e. mass in kilograms, and height in meters, t his constant is the fine number one. If you use pounds and inches inst ead, and are good at conversions you will work out that the constant i s about 703. Or you might just find a calculator on the internet whic h confirms this. Since your B.M.I. is supposed to indicate health ris ks, proponants of the B.M.I index are claiming that if for two diffe rent people, their weights are the same multiple (i.e. their common B. M.I.) of their heights-squared, then these people have comparable heal th risks. Thus, unless height itself is a health risk, the B.M.I. hyp othesis can be interpreted as a biometric scaling law that says that o n average, weight should scale proportionately to the square of height in people." }}{PARA 0 "" 0 "" {TEXT -1 907 " As we have discussed in class, if people were to scale equally in all directions (\"self-s imilar\") when they grew, volume and hence weight would scale as the c ube of height. That particular power law seems a little high, since \+ adults don't look like uniformly expanded versions of babies; we seem \+ to get relatively stretched out when we grow taller. One might expect the best predictive power for weight as a function of height to be so mewhere between 2 and 3, if one expected a power law at all. If there is a predictive power, and if it is much larger than 2, then one coul d argue that the body mass index might need to be modified to reflect \+ this fact. (In fact, when you find body mass index tables, they often \+ explain how and why you should modifiy the acceptable BMI values for c hildren. Is it possible that if a different power law had been used, n o such modification would have been needed?)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 561 " You women did a great job collecting height-weight data -we got well over 100 points. There wa s a shortage of people between 30 and 50 inches high, however, so I ad ded 23 data points in that height range - which I stole from the last \+ time I had one of my linear algebra classes do this project. Each gro up will use our common data, see if it is consistent with a power law relating weights to heights, and decide whether the B.M.I. power of 2 is a good choice. You may choose to use the data you alone collected, or its union with the 23 points I added." }}{PARA 0 "" 0 "" {TEXT -1 873 " Several years ago I found a national data base at the U.S. C enter for Disease Control web site. It contained a wide variety of bo dy measurements collected between 1976 and 1980, including national me dian heights and weights for boys and girls, age 2-19. By using only \+ the national medians a lot of the variance has been taken out of the d ata, compared to what yours will look like. The national data is very \+ consistent with a power law, with power = 2.6. When you do a literat ure and computer search into the history of the B.M.I. index and heigh t-weight scaling laws, as part of your project work for this week, you might discover that (by centuries) we're not the first people to stum ble upon a power other than 2, based on empirical data. As far as I k now their is no mathematical model to explain the correct empirical la w for height-weight scaling in people!" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 260 31 "How do you test for power laws?" }} {PARA 0 "" 0 "" {TEXT -1 159 " Yesterday Meagan explained how to \+ look for power law fits to data points: you look for a best line fit \+ to the corresponding ln-ln data points. Recall: " }}{PARA 0 "" 0 "" {TEXT -1 148 " Suppose we have a set of \"n\" data points, which \+ you can think of as your height-weight data, but which could really be any set of paired data:" }}{PARA 259 "" 0 "" {XPPEDIT 18 0 "[[x[1], y [1]], [x[2], y[2]], [x[3], y[3]], `...`, [x[n], y[n]]];" "6#7'7$&%\"xG 6#\"\"\"&%\"yGF'7$&F&6#\"\"#&F*F-7$&F&6#\"\"$&F*F2%$...G7$&F&6#%\"nG&F *F8" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 68 "We want to see if \+ there is a power m and a proportionality constant " }{XPPEDIT 18 0 "b; " "6#%\"bG" }{TEXT -1 21 " so that the formula" }}{PARA 260 "" 0 "" {XPPEDIT 18 0 "y = b*x^m;" "6#/%\"yG*&%\"bG\"\"\")%\"xG%\"mGF'" } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "effectively mirrors the r eal data. " }{TEXT 259 60 "Taking (natural) logarithms of the propose d power law yields" }}{PARA 261 "" 0 "" {XPPEDIT 18 0 "ln(y) = ln(b)+m *ln(x);" "6#/-%#lnG6#%\"yG,&-F%6#%\"bG\"\"\"*&%\"mGF,-F%6#%\"xGF,F," } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 16 "So, if we write " } {XPPEDIT 18 0 "Y = ln(y);" "6#/%\"YG-%#lnG6#%\"yG" }{TEXT -1 7 " and \+ " }{XPPEDIT 18 0 "X = ln(x);" "6#/%\"XG-%#lnG6#%\"xG" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "B = ln(b);" "6#/%\"BG-%#lnG6#%\"bG" }{TEXT -1 59 ", this becomes the equation of a line in the new variables " }{XPPEDIT 18 0 "X;" "6#%\"XG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Y;" "6#%\"YG" }{TEXT -1 1 ":" }}{PARA 262 "" 0 "" {XPPEDIT 18 0 "Y = mX+B;" "6#/%\"Y G,&%#mXG\"\"\"%\"BGF'" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Th us," }{TEXT 263 1 " " }{TEXT 264 349 "in order for there to be a power law for the original data, the ln-ln data should (approximately) sati sfy the equation of a line. Furthermore, this process is reversible; \+ if the ln-ln data approximately lies on a line with slope m and interc ept B, then the original data approximately satisfies a power law with power m and proportionality constant" }{TEXT 265 1 " " }{XPPEDIT 18 0 "b = exp(B);" "6#/%\"bG-%$expG6#%\"BG" }{TEXT 262 1 "." }{TEXT -1 43 " That's because of the rules of exponents:" }}{PARA 263 "" 0 "" {XPPEDIT 18 0 "Y = B+m*X;" "6#/%\"YG,&%\"BG\"\"\"*&%\"mGF'%\"XGF'F'" } {TEXT -1 0 "" }}{PARA 264 "" 0 "" {XPPEDIT 18 0 "exp(Y) = exp(B+m*X); " "6#/-%$expG6#%\"YG-F%6#,&%\"BG\"\"\"*&%\"mGF,%\"XGF,F," }{TEXT -1 0 "" }}{PARA 265 "" 0 "" {XPPEDIT 18 0 "exp(Y) = exp(B)*exp(X)^m;" "6#/- %$expG6#%\"YG*&-F%6#%\"BG\"\"\")-F%6#%\"XG%\"mGF," }{TEXT -1 0 "" }} {PARA 266 "" 0 "" {XPPEDIT 18 0 "y = b*x^m;" "6#/%\"yG*&%\"bG\"\"\")% \"xG%\"mGF'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 148 "In real e xperiments, it is not too hard to see if the ln-ln data is well approx imated by a line, so this trick with the logarithm is quite useful. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 261 24 "National data example: " }{TEXT -1 162 "I used colons a fter most of these commands to suppress the output. If you want to go back and see what each command is doing, replace the colons with semi colons." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 0 "" }{TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 334 "boyhw:=[[35.9,29.8],[38.9,34.1],[41.9,38.8],[ 44.3,42.8],\n [47.2,48.6],[49.6,54.8],[51.4,60.8],[53.6,66.5], \n [55.7,76.8],[57.3,82.3],[59.8,93.8],[62.8,106.8],\n [ 66.0,124.3],[67.3,132.6],[68.4,142.4],[68.9,145.1],\n [69.6,155 .3],[69.6,153.2]]:\n#boy heights (inches) weights (pounds): Ntl median s for ages 2-19" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 336 "girlhw: =[[35.4,28.0],[38.4,32.6],[41.1,36.8],[43.9,41.8],\n [46.6,47.0 ],[48.9,52.5],[51.4,60.8],[53.1,65.5],\n [55.7,76.1],[58.2,89.0 ],[61.0,100.1],[62.6,108.1],\n [63.3,117.1],[64.2,117.6],[64.3, 122.6],[64.2,128.8],\n [64.1,124.5],[64.5,126]]:\n#girl heights (inches) weights (pounds): Ntl medians for ages 2-19 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "boys:=pointplot(boyhw):\ngirls:=po intplot(girlhw):\ndisplay(\{boys,girls\},\n title=`plot of [height, weight], National medians ages 2-19`);\n" }}}{PARA 0 "" 0 "" {TEXT -1 29 "And now for the ln-ln data. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "with(linalg): #one of two linear algebra packages in Maple" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 237 "B:=evalm(boyhw): #\"evalm\" stands for evaluate as a matrix,\n #converts an array o f points\n #into a matrix, which will be easiser to manipulate in Ma ple\nG:=evalm(girlhw):\nBG:=stackmatrix(B,G): #stack the matrices on \+ top of eachother." }}}{EXCHG }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 240 "lnBGa:=map(ln,BG): #take ln of the boy and girl height-weights\n lnBG:=map(evalf,lnBGa): #Get decimal (floating point) values.\n #Thi s speeds up computations later in the\n #least squares fit - otherwis e Maple tries working\n #symbolically." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "lnlnplot:=pointplot(lnBG):\ndisplay(lnlnplot,title=`l n-ln data`);\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 266 46 "Findin g the best line fit to the ln-ln data: " }{TEXT -1 18 "As Erin explai ned:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(stats):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 270 "Xs:=convert(col(lnBG,1),lis t): #convert the first column of\n #the ln-ln data into a list of the \"x's\" The MAPLE least squares\n #command wants to have lists inpu t, not matrix columns, even\n #though it's hard for us to see any dif ference\nYs:=convert(col(lnBG,2),list):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "fit[leastsquare[[X,Y]]]([Xs,Ys]);#this is the bizarre syntax\n #Erin just introduced you to. \n" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 65 "We can paste in the equation of the line and see how well we did." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "line:=plo t(-6.038703303+2.593828004*X, X=3.4..4.5,Y=2.7..5.7,\n color=black): \ndisplay(\{line,lnlnplot\}, title=`least squares fit`);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "Finally, we can go back from the least sq uares line fit to a power law" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "m:=2.593828004; #power\nb:=exp(-6.038703303); #proportionality \+ constant\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 254 "powerplot:=p lot(b*h^m,h=0..80,w=0..200,color=black):\ndisplay(\{powerplot,boys,gir ls\},title=\n `power law approximation for national height-weight dat a`);\n#by calling the variables h and w, and giving their ranges, I\n# get Maple to label the axes as I want\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {MARK "10 0" 510 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }