or why I like to draw straight lines 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 b = 5 m = (-5/2.5) = -2 y = -2x +5
How Do We Make Trend Lines?
How Do We Make Trend Lines?
How do we evaluate lines? One of these things is not like the other, one of these things does not belong
Plot ei vs xi e6 e5 e4 e3 e2 e1 Good lines have random, uncorrelated errors
Residual Plots
Why do we plot lines?
Why do we plot lines? y = mx + b
Why do we plot lines? y = Aebx
Why do we plot lines? 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
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 continued y x
Linearizing Data continued ln(y) ln(x)
Linearizing Data continued y x
Linearizing Data ln(y+5) x
Enzyme Kinetics
Enzyme Production Michaelis - Menten Vmax Vmax = 10 Km = 1 ½*Vmax Km
Linearization of Enzyme Kinetics
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? 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
Extra Stuff on Lines
Extra Stuff on Lines
More Extra Stuff on Lines