Module 3 - Data, Learning, Systematic Relationships

Question 1

2 approaches to finding a systematic relationship

Answer 1

1. Graphical
2. Quantitative

Question 2

Use engineering judgement to ask:

Answer 2

Should there be a relationship?

Question 3

Scatterplot plotting process

Answer 3

1. Plot values of one var against another
2. Loo for trend in data (nature of trend = lin? exp? quad?; degree of scatter = indicates that variance isn’t cst over range)

Question 4

In scatterplots, look for…

Answer 4

• Trend betw. “independent” variables and dep variables
• Trend betw supposedly independent variables (indicates these quants may be correlated - codependencies, imp when mult x variables)
• Correlation can produce poor model estimation results

Question 5

T/F: If scatterplot data arbitrarily placed on graph, the experiment was designed and thought out

Answer 5

F: trends and patterns indicate designed experiment

Question 6

T/F: Use square graphs to put model on comparable basis

Answer 6

T

Question 7

Covariance

Answer 7

Expected value of joint distribution of X and Y:

Cov(X,Y) = E{(X-mux)(Y-muy)}

Question 8

Sign of covariance

Answer 8

Indicates sign of slope of systematic LINEAR relationship

Question 9

T/F: Correlation and covariance are non-lin’r relationships

Answer 9

F: they are lin’r

Question 10

Correlation

Answer 10

Dimensionless covariance:

Corr(X,Y) = p(X,Y) = Cov(X,Y)/(sigmaX*sigmaY)

Question 11

Properties of correlation

Answer 11

• Dimensionless (no units)
• Range (-1 <= p(X,Y) ,= 1)
• Close to -1 = strong lin’r with -ve slope
• Close to 1 = strong lin’r with +ve slope

Question 12

T/F: Correlation gives NO info abt actual numerical value of slope

Answer 12

T

Question 13

If we have N pairs of observations of X and Y values… (covariance & correlation)

Answer 13

Sample covariance:

R = 1/(N-1) SUM (Xi - Xbar)(Yi-Ybar)

Sample correlation:

r = R/sXsY

Question 14

Sample correlation and covariance characteristics

Answer 14

• Random fluctuations in data will produce random flucts in computed values
• Random variables
• Estimates of true covariance and correlation
• Work with values as guides without computing conf intervals

Question 15

Don’t assume ________ equals _________.

Answer 15

Correlation

Causality

Question 16

Rule of thumb for standard normal random variable

Answer 16

95% of values of Normal histogram occur within +/- 2 st. devs. of mean

Question 17

Another name for bivariate distribution

Answer 17

Joint distribution (betw 2 variables)

Question 18

T/F: Joint distributions will have covariance matrix, and diagonals are equal to variance of X (top left) and Y (bottom right)

Answer 18

T

Question 19

What happens to bivariate distribution as correlation increases?

Answer 19

distribution is stretched along X = Y line, contours more elliptical

Question 20

T/F: If X and Y are not strongly correlated, the distribution will be stretched

Answer 20

F: more circular, less of a trend

Question 21

The multivariate Normal distr describes frequency with which vectors of values X1, Y1, X2, Y2,… Xn, Yn occur

Answer 21

F: X1, X2, X3… Xn

Question 22

Bivariate UNIFORM probability distribution

Answer 22

Take non-0 values over certain interval: change of getting value in interval is same everywhere (contour is a single square)

Question 23

Linear model

Answer 23

Linear in parameter(s)

Question 24

Distinguish lin’r from nonlin’r regression models

Answer 24

Take first derivative wrt parameters - does derivative depend on parameters?

Question 25

Fundamental framework

Answer 25

There is always: - Fundamental behaviour (deterministic) - Little bit of random noise

Question 26

For the linear model, the observations vector/matrix/table form rows and columns are:

Answer 26

ROWS: # runs COLUMNS (in matrix): variables by which we're evaluating response (ie. T, V, P)

Question 27

Least Squares Estimation

Answer 27

Minimize sum of squared prediction errors (or min square lengths betw model prediction (line) and observed value)

Question 28

Residual eqn

Answer 28

epsilon = y - y(hat)

Question 29

T/F: For lin'r model, values of slope and intercept estimate will NEVER depend on each other

Answer 29

F: they will often

Question 30

Assumptions of LSE

Answer 30

1. Values of explanatory variables (x's) are known exactly 2. Model eqn form provides an adequate representation of the data ("model is correct") 3. Noise variance is cst over range of data collected 4. Noise in each obs is statistically ind from noise in other obs 5. Typically assume noise is Normally distr