Data Analysis

or why I like to draw straight lines

Engineers like Lines Two parameters for a line

m slope of the line b the y intercept

-5

0

5

10

15

-4 -2 0 2 4

b = 5

m = (-5/2.5) = -2

y = -2x +5

How Do We Make Trend Lines?

4

6

0 0.25 0.5 0.75 1

How Do We Make Trend Lines?

4

6

0 0.25 0.5 0.75 1

e1

e2

e3

e4

e5

e6

i i ie y mx b 2 2

0 0i i

i i

e e

m b

How do we evaluate lines?

y = 1.0054x + 0.0461

R2 = 0.6975

-0.4

0

0.4

0.8

1.2

0 0.25 0.5 0.75 1 1.25

y = 1.0326x - 0.0779

R2 = 0.8984

0

0.4

0.8

1.2

0 0.25 0.5 0.75 1 1.25

y = 1.0201x + 0.1187

R2 = 0.9532

0

0.4

0.8

1.2

0 0.25 0.5 0.75 1 1.25

One of these things is not like the other, one of these things does not belong

Plot ei vs xi

4

6

0 0.25 0.5 0.75 1

e1

e2

e3

e4

e5

e6

Good lines have random, uncorrelated errors

Residual Plots

-0.2

-0.1

0

0.1

0.2

0 0.25 0.5 0.75 1 1.25

-0.2

-0.1

0

0.1

0.2

0 0.2 0.4 0.6 0.8 1 1.2

-0.3

-0.15

0

0.15

0.3

0 0.2 0.4 0.6 0.8 1 1.2

Why do we plot lines?

0

0.5

1

1.5

2

0 0.25 0.5 0.75 1

Why do we plot lines?

0

0.5

1

1.5

2

0 0.25 0.5 0.75 1

y = mx + b

Why do we plot lines?

0

0.5

1

1.5

2

0 0.25 0.5 0.75 1

y = Aebx

Why do we plot lines?

0

0.5

1

1.5

2

0 0.25 0.5 0.75 1

y = Ax2 + Bx + C

Why do we plot lines?

Lines are simple to comprehend and draw We are familiar with slope and intercept as

parameters We can linearize many functions and plot

them as lines Many functions can be expressed as Taylor

Series

Taylor Series

2 3( ) ( )( )

2! 3!

f a f af x f a f a x a x a x a

2 311

1x x x

x

3 5 7

sin3! 5! 7!

x x xx x

2 4 6

cos 12! 4! 6!

x x xx

2 3 4

12! 3! 4!

x x x xe x

Linearizing Equations

We have non linear function v = f(u) v = u3

v = 2eu+5u v=u/(u-4)

We want to transform the equation into y=mx+b

Linearizing Data continued33y x

0

30

60

90

0 0.5 1 1.5 2 2.5 3 3.5x

y

Linearizing Data continued

-8

-6

-4

-2

0

2

4

6

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5

ln( ) ln(3) 3ln( )y x

ln(x)

ln(y)

Linearizing Data continued

2 3 5xy

-10

0

10

20

30

40

50

0 0.5 1 1.5 2 2.5 3

x

y

Linearizing Data

-2

0

2

4

-3 -2 -1 0 1 2 3 4

ln( 5) ln 3 ln(2)y x

x

ln(y+5)

Enzyme Kinetics

0

2

4

6

8

10

0 5 10 15 20 25

Substrate Concentration

Ra

te o

f P

rod

uc

t F

orm

ati

on

Enzyme Production

max[ ]

[ ]m

V Sv

K S

Vmax

½*Vmax

Km

Vmax = 10

Km = 1

Michaelis - Menten

Linearization of Enzyme Kinetics

0

0.1

0.2

0.3

0.4

0.5

0 1 2 3 4

1/[S]

1/v

max[ ]

[ ]m

V Sv

K S

max max max

[ ]1 1

[ ] [ ]m mK S K

v V S V S V

Engineers often use logarithms to solve problems

What is a Log?

logab = x b = ax

Logarithms are the inverse functions of exponential functions

Most important log bases

log10 = log We like to count in powers of 10

loge = ln Nature likes to count in powers of e

And maybe …

log2

Computers count in bits

What are the important properties of logs? log(a*b) = log(a) +log(b)

log(ab) = b*log(a)

Why do we care about logs?

Nature likes power law relationships

y = k*uavbwc

For some reason a,b,c are usually either integers, or nice fractions

log(y) = log(k)+a*log(u)+b*log(v)+c*log(w) Pretty close to linear - we can use linear regression

Buckling in the Materials Lab

From studying the problem we expect that buckling load (P) is a power law function of Radius R, and Length L

How would you design an experiment for the pendulum?

L

M

g

Keep Mass constant – vary L

Keep Length constant – vary M

Keep mass and length constant – vary g

Where do log-log plots break down? Two or more power laws

y=k1*uavb + k2ucvd

s(t)=-g/2*t2+v0t+s0

E=mgh+1/2*mv2

y = 1.0074x - 0.0434

0

0.5

1

1.5

0 0.2 0.4 0.6 0.8 1

Extra Stuff on Lines

Extra Stuff on Lines

y = 0.935x + 0.014

0

0.5

1

1.5

0 0.2 0.4 0.6 0.8 1

More Extra Stuff on Lines

y = 1.2853x - 0.2709

0

0.5

1

1.5

0 0.2 0.4 0.6 0.8 1