introduction paul j. hurtado mathematical biosciences institute (mbi), the ohio state university 19...
DESCRIPTION
Why do statistics? Scientific vs. Mathematical Inference Estimation & Uncertainty Quantification Statistics with dynamic models? Challenges of statistics with ODEs?TRANSCRIPT
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IntroductionPaul J. Hurtado
http://www.pauljhurtado.com/Mathematical Biosciences Institute
(MBI),The Ohio State University
19 May 2014 (Monday a.m.)
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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
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Why do statistics?•Scientific vs. Mathematical
Inference•Estimation & Uncertainty
Quantification
Statistics with dynamic models?•Challenges of statistics with ODEs?
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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!
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Quick Review
•Linear Models•Probability•Parameter Space Bifurcations•Visualization
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Linear Equations
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X
Y
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X
Y
Y = m X + b
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X
Y
Y = m X + b
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X
Y
Y = m X + b
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X
Y
Y = m X + b + ε
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X
Y
Y = m X + b
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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
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Matrices & Vectors…… useful notation. For example, y =
Ax vs
… essential tools for math/computing.
or
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Computers :: Matrix
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Matrix ApplicationsTwo common ways matrices are used:
1. Storage variables: data, etc.* Easier, faster computations!
2. Maps/Transformations
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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?
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Matrix transformations
Example:
y1’ = A11y1+A12y2+…+A1nyn
y2’ = A21y1+A22y2+…+A2nyn
...
yn’ = An1y1+An2y2+…+Annyn
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Matrix transformations
Example:
y1’ A11 A12 … A1n yn
y2’ A21 A22 … A2n yn
...
yn’ An1 An2 … Ann yn
=
A
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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
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Matrix transformations
Example:
y1’ λ1 0 … 0 yn
y2’ 0 λ2 … 0 yn
...
yn’ 0 0 … λn yn
= Q-1Q Q-1Q-1
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Matrix transformations
Example:
Y1’ λ1 0 … 0 Y1
Y2’ 0 λ2 … 0 Y2
...
Yn’ 0 0 … λn Yn
=
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Matrix transformations
Example:
Y1’ = λ1 Y1
Y2’ = λ2 Y2
...
Yn’ = λn Yn
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Matrix transformations
Example:
Y1(t) = Y1(0)exp(λ1t)
Y2(t) = Y2(0)exp(λ2t)
...
Yn(t) = Yn(0)exp(λnt)
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Matrix transformations
Example:
Y1(t) Y1(0)exp(λ1t)
Y2(t) Y2(0)exp(λ2t)
...
Yn(t) Yn(0)exp(λnt)
=
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Matrix transformations
Example:
y1(t) Y1(0)exp(λ1t)
y2(t) Y2(0)exp(λ2t)
...
yn(t) Yn(0)exp(λnt)
= Q
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Matrix transformations
Example:
y1(t) Y1(0)exp(λ1t)
y2(t) Y2(0)exp(λ2t)
...
yn(t) Yn(0)exp(λnt)
= q1 … qn
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Matrix transformations
Example:
y1(t) y2(t)
…
yn(t)
= Y1(0)exp(λ1t) q1 + … + Yn(0)exp(λnt) qn
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Matrix transformations
Summary #1: Eigenpairs tells us about the geometry of matrix transformations
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Matrices & ModelsLinear Model in matrix form:
Yi = β0 + β1 Xi + εi where εi ~ N(0,σ2)
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Matrices & ModelsLinear Model in matrix form:
Y1 = β0 + β1 X1 + ε1
Y2 = β0 + β1 X2 + ε2
…Yn = β0 + β1 Xn + εn
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Matrices & ModelsLinear Model in matrix form:
Y1 β0 + β1 X1 ε1
Y2 β0 + β1 X2 ε2
…Yn β0 + β1 Xn εn
= +
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Matrices & ModelsLinear Model in matrix form:
Y1 1 X1 ε1
Y2 1 X2 ε2
…Yn 1 Xn εn
= +β0
β1
Unknown!
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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!
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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!
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X
Y
Y = m X + b + ε
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Probability Basics
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Distributions Density CDF
Continuous Random Variables: Ex: Normal, Gamma, etc.
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Distributions Mass CDF
Discrete Random Variables: Ex: Poisson, Binomial, etc.
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Distributions Mass+Density CDF
20%
80%
20%
Mixed Distributions: Zero-inflated Normal,
etc.
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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
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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.
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Multivariate If Yi all independent, identically
distributedYi ~ f(y|θ)
then their joint distribution is the product
Y = (Y1, …,Yn) ~ f(yi|θ).
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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)).
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Parameter Space Bifurcations
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Consumption Rate (a)
Satu
ratio
n Pa
ram
eter
(k)
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Optimization
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Visualization
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GDP
Life
Exp
ecta
ncy
R2 = … p = …
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Questions?