IntroductionPaul J. Hurtado

http://www.pauljhurtado.com/Mathematical Biosciences Institute

(MBI),The Ohio State University

19 May 2014 (Monday a.m.)

Workshop Overview• Why do we do statistics?• Estimation vs Uncertainty Quantification• ODEs vs “Classical” Models• Other useful topics…

I. Fundamental Concepts: Review/Overview• Linear models and ex

• Parameter Space & Bifurcations• Probability & Statistics• Optimization• Visualization

II. Computer Lab• Resources: (URL)• Scripts vs. console (R vs Matlab)• Simulating ODE Solutions• Graphics/Plotting• Random numbers• Manipulating Objects• …

III. Summary

Why do statistics?•Scientific vs. Mathematical

Inference•Estimation & Uncertainty

Quantification

Statistics with dynamic models?•Challenges of statistics with ODEs?

Additional Topics?•Markov Chain Monte Carlo (MCMC)•Bayesian Methods•Filtering (Kalman, Particle, etc)•Functional Data Analysis•SDEs, PDEs, SPDEs…•Decision Trees, Neural Networks,

etc.•etc!

Quick Review

•Linear Models•Probability•Parameter Space Bifurcations•Visualization

Linear Equations

X

Y

X

Y

Y = m X + b

X

Y

Y = m X + b

X

Y

Y = m X + b

X

Y

Y = m X + b + ε

X

Y

Y = m X + b

Why linear algebra?

• Curves: intuition based on lines.

• Models are rarely 1-dimensional! y1 = ax1 – bx3

y = m x vs y2 = – cx1 – dx2 + bx3

y3 = – bx3 + ax1

X

Y

Matrices & Vectors…… useful notation. For example, y =

Ax vs

… essential tools for math/computing.

or

Computers :: Matrix

Matrix ApplicationsTwo common ways matrices are used:

1. Storage variables: data, etc.* Easier, faster computations!

2. Maps/Transformations

Matrix transformations

Pick a random* matrix A. It can be written:

A = QDQ-1

where D=diag(λ1, …, λn) are eigenvalues, & the columns of Q are their eigenvectors.

y = A xQ: How does A convert x to y?

Matrix transformations

Example:

y1’ = A11y1+A12y2+…+A1nyn

y2’ = A21y1+A22y2+…+A2nyn

...

yn’ = An1y1+An2y2+…+Annyn

Matrix transformations

Example:

y1’ A11 A12 … A1n yn

y2’ A21 A22 … A2n yn

...

yn’ An1 An2 … Ann yn

=

A

Matrix transformations

Example:

y1’ λ1 0 … 0 yn

y2’ 0 λ2 … 0 yn

...

yn’ 0 0 … λn yn

= Q Q-1

A = Q D Q-1

Matrix transformations

Example:

y1’ λ1 0 … 0 yn

y2’ 0 λ2 … 0 yn

...

yn’ 0 0 … λn yn

= Q-1Q Q-1Q-1

Matrix transformations

Example:

Y1’ λ1 0 … 0 Y1

Y2’ 0 λ2 … 0 Y2

...

Yn’ 0 0 … λn Yn

=

Matrix transformations

Example:

Y1’ = λ1 Y1

Y2’ = λ2 Y2

...

Yn’ = λn Yn

Matrix transformations

Example:

Y1(t) = Y1(0)exp(λ1t)

Y2(t) = Y2(0)exp(λ2t)

...

Yn(t) = Yn(0)exp(λnt)

Matrix transformations

Example:

Y1(t) Y1(0)exp(λ1t)

Y2(t) Y2(0)exp(λ2t)

...

Yn(t) Yn(0)exp(λnt)

=

Matrix transformations

Example:

y1(t) Y1(0)exp(λ1t)

y2(t) Y2(0)exp(λ2t)

...

yn(t) Yn(0)exp(λnt)

= Q

Matrix transformations

Example:

y1(t) Y1(0)exp(λ1t)

y2(t) Y2(0)exp(λ2t)

...

yn(t) Yn(0)exp(λnt)

= q1 … qn

Matrix transformations

Example:

y1(t) y2(t)

…

yn(t)

= Y1(0)exp(λ1t) q1 + … + Yn(0)exp(λnt) qn

Matrix transformations

Summary #1: Eigenpairs tells us about the geometry of matrix transformations

Matrices & ModelsLinear Model in matrix form:

Yi = β0 + β1 Xi + εi where εi ~ N(0,σ2)

Matrices & ModelsLinear Model in matrix form:

Y1 = β0 + β1 X1 + ε1

Y2 = β0 + β1 X2 + ε2

…Yn = β0 + β1 Xn + εn

Matrices & ModelsLinear Model in matrix form:

Y1 β0 + β1 X1 ε1

Y2 β0 + β1 X2 ε2

…Yn β0 + β1 Xn εn

= +

Matrices & ModelsLinear Model in matrix form:

Y1 1 X1 ε1

Y2 1 X2 ε2

…Yn 1 Xn εn

= +β0

β1

Unknown!

Matrices & ModelsLinear Model in matrix form:

Goal: Minimize ε’ε = (Y-Xβ)’(Y-Xβ).

This is the same as solving (X’Y) = (X’X)β.

Y = X β + εUnknown!

Summary•Matrices are pervasive in scientific computing, statistics. - Computing with

vectors/matrices is faster, simpler than iteration/loops.

- Intuition improves use, interpretation.

•Linear algebra is a cornerstone of stats!

X

Y

Y = m X + b + ε

Probability Basics

Distributions Density CDF

Continuous Random Variables: Ex: Normal, Gamma, etc.

Distributions Mass CDF

Discrete Random Variables: Ex: Poisson, Binomial, etc.

Distributions Mass+Density CDF

20%

80%

20%

Mixed Distributions: Zero-inflated Normal,

etc.

Sampling CDFsLet r~Unif(0,1), CDF F(x) with inverse

F-1. Then F-1(r) ~ F(x). Ex: .67 5.1 .12 0.0 .85 5.9

Distributions in RR has many built-in densities and

CDFs!Density CDF Quantile Sample

dnorm pnorm qnorm rnorm

dpois ppois qpois rpois

… beta, binomial, Cauchy, χ2, exponential, F, gamma, geometric, hypergeometric, log-normal, multinomial, negative binomial, Student's t, uniform distribution, Weibull, etc.

Multivariate If Yi all independent, identically

distributedYi ~ f(y|θ)

then their joint distribution is the product

Y = (Y1, …,Yn) ~ f(yi|θ).

LiklihoodThe likelihood of data X=(X1,…,Xn)

under parameter θ is given byLik(θ|X) = f(Xi|θ).

The log-likelihood of data X=(X1,…,Xn) under parameter θ is given by

LL(θ|X) = log(f(Xi|θ)).

Parameter Space Bifurcations

Consumption Rate (a)

Satu

ratio

n Pa

ram

eter

(k)

Optimization

Visualization

GDP

Life

Exp

ecta

ncy

R2 = … p = …

Questions?