Math

Question 1

inverse of matrix

Answer 1

AX = B

A^-1AX = A^-1B

IX = A^-1B

X = A^-1B

Question 2

outer product

Answer 2

Question 3

matrix operations

Answer 3

• Associativity of Addition: A + ( B + C ) = ( A + B ) + C
• Associativity of Scalar Multiplication: (cd) A = c (dA)
• Distributive: c(A + B) = cA + cB
• Distributive: (c + d) A = cA + dA
• Associativity of Multiplication: A(BC) = (AB)C
• Left Distributive: A(B + C) = AB + AC
• Right Distributive: (A + B )C = AC + BC
• Scalar Associativity / Commutativityc (AB) = (cA) B = A (cB) = (AB) c
• Multiplicative IdentityIA = AI = A

Question 4

add large number

Answer 4

take the log of the number

Question 5

statistical power

Answer 5

• is the probability that the test rejects the null hypothesis H₀
• power=(1-β)=Pr(reject H₀| H₁ is True)
• high power = low probability of type II error
• the alpha significance criterion (α)
• statistical power, or the chosen or implied beta (β)

Question 6

statistical power calculation

Answer 6

• no difference
  
  • test statistic: T_n
    
    • µ₀=0
    • find T_n where alpha > 0.05
  • find where alpha is met using quantile fn
• B(Ø)=Pr(T_n>value|µ_D=Ø)
  
  • greater than flipped with 1 -
• B(Ø)=1-φ(T_n-Ø/(σ_D/√n)), for 1 sample t-test

Question 7

one sample t-test (paired t-test)

Answer 7

Question 8

two sample unpooled t-test

Answer 8

Question 9

confidence

Answer 9

• confidence = 1 - alpha, where alpha = significance
  
  • .95 = 1 - alpha = ∑N(x, σ²/N)

Question 10

Gaussian confidence interval

Answer 10

• CI = µ±z_left(σ/√N)
• for CI = 0.95, z = ± 1.96
• z-score = (x-µ)/(σ/√N)

Question 11

bernouilli confidence interval

Answer 11

• binary case
• p = successes/N
• CI = p ± z√(p(p-1)/N)

Question 12

wilson interval

Answer 12

• p = successes/N
• (p+z²/2N)/(1+z²/N) ± z/(1+z²/N) √(p(1-p)/N + z²/4N²)

Question 13

conjugate priors

Answer 13

• pairs of likelihood and and priors such that the posterior has the same distribution as the prior
• precision: λ^-1=σ²
• likelihood: X~N(µ, τ^-1)
• conj prior for Gaussian likelihood is Gaussian
  
  • prior: µ~N(m₀, λ₀^-1)
• proportionality for posterior

Question 14

lift

Answer 14

• p(A, B)/p(A)p(B) = p(A | B)/p(A) = p(B | A)/p(B)
• if A and B are independent, lift = 1

Question 15

bayesian approach

Answer 15

1. start with likelihood: p(X|ø) = Πp(X_i|ø)

2. choose prior: p(ø)
3. calculate posterior: P(ø|X)=p(X|ø)p(ø)
   
   • posterior update given normal likelihood & prior
     
     • likelihood: N~(µ,τ^-1)
     • prior: N ~(m,λ^-1)
     • posterior mean: m’ = (λm+τ)∑x/(λ+Nτ)
     • update: λ’ = λ + Nτ

• Bernoulli has p, Gaussian has µ, σ
  
  • p(x|µ, σ) = exp((x-µ)²/2σ²)/√(2πσ²)

Question 16

sensitivity/recall

Answer 16

• true positive rate: TP / (TP + FN)
• cost of false negative high
• is a missile not coming?

Question 17

specificity

Answer 17

• true negative rage: TN / (TN + FP)
• cost of false positive high?

Question 18

precision

Answer 18

• TP / (TP + FP)
• how often its correct when it predicts True
• cost of false positive high
• skin cancer tests?

Question 19

accuracy

Answer 19

• TP+TN/(TP+TN+FP+FN)
• tells performance
• does not tell imbalance

Question 20

ROC

Answer 20

• logistic regression uses 0.5 threshold (P=50%)
• if you use different ones you get different true positive false positive rate

Question 21

f1 score

Answer 21

• 2*(precision*recall)/(precision+recall)
• good = low FP and low FN
• identifying threats but not disturbed

Question 22

4th order runge-kutta

Answer 22

• k₁= h f (t_n, y_n)
• k₂= h f ( t_n + h/2, y_n + k₁/2)
• k₃= h f ( t_n + h/2, y_n + k₂/2)
• k₄= h f ( t_n + h, y_n + k₃)