,calculus ,a1 ,lab #7

    ,9troduc;n4
  ,? lab is ab curve fitt+ & l9e>
regres.n4 ,x repres5ts a basic applic,n
( calculus 6/ati/ics & data model+4
  ,a -mon pro#m 9 appli$ ma? is 6f9d a
l9e or o!r curve pass+ "? a set ( data
po9ts4 ,typic,y ! data po9ts -e f f
exp]i;tal process2 s !y >rive 3ta9+
approxim,ns & ]rors4 ,!y don't exactly
lie on ! l9e or smoo? curve "o is seek+4
,! purpose ( ? lab is 6see h[ we c use
:at we "k ab derivatives & optimiz,n
6f9d ! /rai<t l9e or o!r curve t be/
fits a cl\d ( data4

    ,:at 6m9imize8
  ,figure #1 %[s a collec;n ( data po9ts
&a /rai<t l9e t mi<t approximate !m4 ,!
f/ "q we h 6ask is h[ we wd m1sure !
gd;s ( fit ( ? l9e 6! data4
  ,! f/ answ] 6? "q t wd -e 6my m9d wd 2
6draw p]p5dicul> l9e seg;ts f ea* (!
                                      #j
data po9ts 6! l9e4 ,! sum (! l5g?s ( ^!
seg;ts is a m1sure (! gd;s ( fit (! l9e4
,we wd !n try 6m9imize ! sum ( ^! l5g?s1
: >e %[n 9 ,figure #2_4
  ,"! >e #2 di6iculties ) ? *oice (a
quant;y 6m9imize4 ,! f/ is %[n 9 ,pro#m
#1

  #1_4 ,-pute ! p]p5dicul> 4t.e f ! po9t
(4, 5) 6! l9e y .k mx+b_4 ,i wd d ?
0rememb]+ t any l9e p]p5dicul> 6! l9e
y .k mx+b has slope -?1_/m#_4 ,? wd let
me write ! equ,n (! l9e "? (4, 5)
p]p5dicul> to y .k mx+b_4 ,i cd !n f9d
": ^? two l9es met1 & -pute ! 4t.e 2t !
po9t ( 9t]sec;n &! po9t (4, 5)_4
  ,wd y feel l -put+ ? 4t.e =a bun* (
po9ts1 summ+ ! results1 & !n m9imiz+ !
sum8

  ,! o!r pro#m ) m9imiz+ ? expres.n is t
(t5 ! quant;y on ! ;x-axis c 2 m1sur$
exactly or q a3urately1 &! ]ror 9 !
exp]i;t is 9 ! det]m9,n ( ;y_4 ,if ? is
                                      #a
s1 !n x wd seem 6make m s5se 6use z a
m1sure (! ]ror 9 ! l9e n ! sum (!
p]p5dicul> 4t.es 6! l9e1 b ! sum (!
l5g?s ( v]tical seg;ts dropp$ f ! po9ts
6! l9e1 z %[n 9 ,figure #3_4

  #2_4 ,write ! sum (! l5g?s (! v]tical
seg;ts jo9+ ! po9ts (1, 1), (2, 1),
(3, 2), (4, 5), & (5, 6) 6! l9e
y .k mx+b_4 ,yr answ] w obvi\sly dep5d
on ;m & ;b_4 ,x %d 3ta9 a bun* (
absolute values4

  ,! expres.n y got 9 ,pro#m #2 %d look
a :ole lot m tracta# ?an ! "o f ,pro#m
#1, b "!'s / a di6iculty4 ,! tr\# ) us+
calculus 6m9imize ! expres.n f ,pro#m #2
is t ! absolute value func;n doesn't alw
h a derivative4 ,"! isn't a simple
algebraic expres.n =! derivative ( \x\_4
,"o has 6say t ! derivative is #1 if
x .1 #0, t x is -#1 if x "k #0, & t x is
undef9$ at x .k #0_4 ,un.s y re,y lov$
"w+ ) absolute values 9 s*ool1 x's "!=e
                                      #b
agd idea 6avoid !m "h if we c4
  ,s let's try a ?ird approa*1 less
9tuitive ?an ! o!r two4 ,let's use z \r
m1sure =! gd;s ( fit (! l9e n ! sum (!
absolute values (! v]tical 4t.es 2t !
po9ts &! l9e1 b ! sum (! squ>es (!
v]tical 4t.es 2t ! po9ts &! l9e4 ,!
squ>es1 l ! absolute values1 >e gu>ante$
6be non-negative2 b unlike \x\, x^2 has
a simple derivative4 ,we c "!=e use
calculus 6m9imize ! sum (! squ>es (!
4t.es1 &! l9e we get %d 2 a pretty gd
fit 6! data4 (,there >e "s fanci]
ju/ific,ns = ? *oice (a func;n 6m9imize1
b ,i / ?9k t at bottom ! r1son =! *oice
is t x results 9 a r1sona# fit & we c d
! ma?_4) ,? process ( f9d+ a l9e t
m9imizes ! sum (! squ>es (! v]tical
4t.es 2t ! po9ts &! l9e is call$ l1/
squ>es l9e> regres.n4
  #3_4 ,write ! sum ;,s (! squ>es (!
l5g?s (! v]tical seg;ts jo9+ ! po9ts
(1, 1), (2, 1), (3, 2), (4, 5), & (5, 6)
6! l9e y .k mx+b_4 ,don't bo!r 6simplify
                                      #c
yr answ]1 : w dep5d on ;m & b, & : w
3ta9 a bun* ( squ>es4

  ,! goal n[ is 6f9d values = ;m & ;b t
make ! sum f ,pro#m #3 z small z possi#4
,"! >e a v>iety ( ways 6d ?1 b "h's a
naive "o t "ws4 ,?9k ( bo? ;m & ;b z
v>ia#s1 & ?9k (! sum ;,s ( squ>es (
v]tical 4t.es z a func;n ( ;m & ;b_4 ,if
"! >e values ( ;m & ;b t m9imize ,s, !n
at ! m9imum1 we c reg>d ;b z a 3/ant &
;m z a v>ia#2 & we h 6be at a m9imum (
;,s z a func;n ( ;m_4 ,alt]natively1 we
c reg>d ;m z a 3/ant & ;b z a v>ia#2 &
we h 6be at a m9imum ( ;,s z a func;n (
;b_4 ,s =! be/-fit l9e1 we expect t
  ?d,s_/dm# .k ?d,s_/db# .k #0_4
  #4_4 ,reg>d ;m z a 3/ant & ;b z a
v>ia#1 set ?d,s_/db# .k #0, & solve =
;b_4

  ,! equ,n y get = ;b 9 t]ms ( ;m has a
simple geometric 9t]pret,n4 ,! c5t] (
mass (! po9ts we >e "w+ ) is ! po9t
                                      #d
(3, 3) ^: ;x coord9ate is ! av]age (! ;x
coord9ates (! #5 po9ts & ^: ;y coord9ate
is ! av]age ( _! ;y coord9ates4 ,y %d 2
a# 69t]pret ! equ,n 9 ,pro#m #4 z say+ t
! l9e y .k mx+b m/ pass "? ! c5t] ( mass
(! po9ts4

  #5_4 ,n[ t y "k ! m9imum value = b, y
c plug ? value 96yr =mula = ,s, -pute
?d,s_/dm#, & set x equal to #0 z well4
,solve = ;m & ;b_4 ,plot ! l9e y .k mx+b
tgr )! #5 data po9ts4 ,! l9e "\ 6look z
if x fits ! po9ts at l1/ z well z any
o!r /rai<t l9e wd4 ,if x doesn't1 !n g
back & look = yr mistake4

  ,an alt]native approa* wd h be5 6set
bo? ?d,s_/db# & ?d,s_/dm# equal to #0
simultane\sly1 & 6solve4 ,? wdn't h giv5
u ! geometric 9t]pret,n 9 t]ms (! c5t] (
mass1 b x mi<t h 9volv$ less ?9k+4

  ,n[ let's try a fanci] pro#m4 ,suppose
we don't want 6run a /rai<t l9e "? \r
                                      #e
data po9ts1 b a p>abola y .k ax^2"+bx+c
_4 ,we mi<t d ? if we expect$ =
!oretical r1sons t \r data po9ts %d
foll[ a curve ( ? =m4 ,s9ce m9imiz+ !
sums (! squ>es (! l5g?s ( v]tical seg;ts
jo9+ ! data po9ts 6! curve "w$ 2f1 :y n
try ! same idea ag8

  #6_4 ,write ! sum ;,s (! squ>es (!
l5g?s (! v]tical seg;ts jo9+ ! po9ts
(1, 1), (2, 1), (3, 2), (4, 5), & (5, 6)
6! p>abola y .k ax^2"+bx+c_4 ,don't bo!r
6simplify yr answ]1 : w dep5d on ;m & b,
& : w 3ta9 a bun* ( squ>es4

  ,! goal n[ is 6f9d values = a, b, & ;c
t make ! sum f ,pro#m #6 z small z
possi#4 ,proce$ j z 2f3 ?9k ( a, b, & ;c
z v>ia#s1 &! sum ;,s z a func;n ( a, b,
& ;c_4 ,=! be/-fit p>abola1 we expect
  ?d,s_/da# .k ?d,s_/db# .k ?d,s_/dc#
    .k #0_4


                                      #f
  #7_4 ,-pute ?d,s_/da#, ?d,s_/db#, &
?d,s_/dc#_4 ,set !m all equal to #0, &
solve = a, b, & ;c_4 ,i'd use ,maple
6solve !m tgr1 b x is al j a hi< s*ool
algebra pro#m4 ,!n plot ! p>abola
y .k ax^2"+bx+c tgr )! #5 data po9ts4 ,!
p>abola "\ 6look z if x fits ! po9ts at
l1/ z well z any o!r p>abola wd4 ,if x
doesn't1 !n g back & look = yr mistake4

  ,at ? po9t1 x %d seem hi<ly likely t
giv5 a collec;n ( po9ts1 we cd fit a
curve ( any give degree 6t set ( po9ts
0us+ ! same apprao* we us$ = l9es (
curves ( degree #1) & p>abolas (curves (
degree #2)_4
  ,"! >e re,y only two bits left 6f9i%+
up a -pletely g5]al !ory ( l9e>
regres.n4 ,!ye >e a bit m not,n,y dem&+
?an :at we've d"o s f>1 b if y /ay calm1
y %d 2 f9e4 ,y're / j "w+ ) l9e> equ,ns2
x's j t ! coe6ici5ts w 9volve sli<tly
messy symbolic expres.ns4

                                      #g
  ,suppose y want 6fit a /rai<t l9e 6!
?ree data po9ts (x1, y1), (x2, y2), &
(x3, y3)_4 ,suppose al t
  "x<:] .k ?x1+x2+x3_/3#
is ! av]age ( x1, x2, x3, & t "y<:] is !
av]age ( y1, y2, y3_4 ,! sum (! squ>es
(! v]tical 4t.es f ! ?ree data po9ts 6!
l9e y .k mx+b is !n
  ,s .k (mx1+b-y1)^2"+(mx2+b-y2)^2"+(mx3
    +b-y3)^2_4
  #8_4 (a) ,reg>d ;m z a 3/ant1 & set
?d,s_/db# .k #0_4 ,%[ t at ! 5d ( all !
algebra1 y h
      ,'2g9.(equ,n.),'
b .k "y<:]-m"x<:]_4
      ,'5d.(equ,n.),'
  (;b) ,plug ? value = ;b 9to ;,s & y
get
  ,s .k @(m(x1-"x<:])-(y1-"y<:])@)^2"
    +@(m(x2-"x<:])-(y2-"y<:])@)^2"
    +@(m(x3-"x<:])-(y3-"y<:])@)^2_4
,n[ set ?d,s_/dm# .k #0 & solve = ;m_4
,y %d get
      ,'2g9.(equ,n.),'
                                      #h
m .k ?(x1-"x<:])(y1-"y<:])+(x2-"x<:])(y2
-"y<:])+(x3-"x<:])(y3-"y<:])_/(x1-"x<:]
)^2"+(x2-"x<:])^2"+(x3-"x<:])^2"#_4
      ,'5d.(equ,n.),'
,=mulas (1) & (2) = ;b & m, : g5]alize
immly 6m ?an #3 data po9ts1 %[ t "o c
get be/ fit l9es )\t do+ ! calculus anew
= ea* set ( data4 ,y j -pute ! av]ages &
plug 96! =mulas4 ,at ! 5d ( all !
calculus1 we "!=e h simple proc$ures =
fitt+ l9es 6data1 : c easily 2 imple;t$
9 a calculator or simple /ati/ics
program4

  #9_4 ,f9,y1 an op5-5d$ "q 6?9k ab4 ,c
y design a way 6m1sure h[ close a set (
po9ts -es 6ly+ on a /rai<t l9e8 ,ide,y1
y wd l a m1sure t doesn't *ange if y
%ift all ! po9ts a 3/ant 4t.e horizont,y
or v]tic,y1 & : doesn't *ange :5 y
multiply e ;x coord9ate 0a fix$ 3/ant1
or :5 y multiply e ;y coord9ate 0a fix$
3/ant4 ,s* a m1sure wd let y say
num]ic,y h[ gd yr be/ fit l9e is z an
                                      #i
approxim,n 6! data po9ts4























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