Fitting models to time series

A general approach to model fitting

Pick a model, including parameter values

Evaluate how likely the observed data are, given the model

Tweak the model to make the observations more likely

Is this model superior to other models?

Maximum likelihood (ML) inference

In ML, you have some set of data $D$ and a model for generating this data. This model has parameters $\theta$. The probability of observing data is $\mathrm{Pr}(D \, | \, \theta)$. The best parameter point estimate $\hat{\theta}$ is simply the value that maximizes $\mathrm{Pr}(D \, | \, \theta)$.

Maximum likelihood (ML) inference

For example, if we have data $D$ from a Bernoulli observation model representing $k$ successes in $n$ trials, then the probability of observing $k$ and $n$ given coin flip probability parameter $p$ is simply $$\mathrm{Pr}(k,n \, | \, p) = p^k \, (1-p)^{n-k}.$$

Maximum likelihood (ML) inference

For the Bernoulli model $\mathrm{Pr}(k,n \, | \, p) = p^k \, (1-p)^{n-k}$, we have $\hat{p} = k/n$. For example, with $k=8$ and $n=10$, $\hat{p}=0.8$ the likelihood curve follows

Likelihood in timeseries models

Observed trajectory $D=(t_0,...,t_n)$ depends on unknown parameter(s) $\theta$

Probability density function of $D$ is $f_{\theta}$

$$L_D(\theta)=f_{\theta}(D)$$

Problem: These data aren't independent.

Likelihood from non-independent data

Solution: Factor the joint density into conditional and marginal

e.g., $f(y_3,y_2,y_1)=f(y_3 \, | \, y_2,y_1) \cdot f(y_2,y_1)$

$$f(y_3,y_2,y_1)=f(y_3 \, | \, y_2,y_1) \cdot f(y_2 \, | \, y_1) \cdot f(y_1)$$

$$L(\theta)=\prod_{t=2}^{T}f(y_t|l_{t-1})\cdot f(y_1)$$

where $l_{t-1}$ is information through $t-1$ (i.e., $y_{t-1},...,y_1$), and $T$ is the time series length

What's $D$?

Case counts at different times

Sequences

Titers

or some composite of observations

Maximizing the likelihood

means maximizing the log-likelihood

or minimizing the negative log-likelihood

Finding the maximum likelihood

Can be analytically tractable

For our models, it's not

General approaches to likelihood maximization

Brute force

Derivative-based methods

Simplex

Simulated annealing

Sequential Monte Carlo

Many others... but few tried and true

Inference for time series

POMP (partially observed Markov Process)

pMCMC (particle Markov chain Monte Carlo)

TSIR, if conditions met

Likelihood profiles

Hold parameter(s) constant, fit the rest

Bayesian inference

Generally, it's difficult to make probability statements using frequentist statistics. You cannot directly say that model 1 is twice as likely as model 2. People misuse p values in this sort of fashion all the time.

Bayes' rule

Bayes' rule forms the basis of Bayesian inference, it states: $$\mathrm{Pr}(A \, | \, B) = \cfrac{ \mathrm{Pr}(B \, | \, A) \, \mathrm{Pr}(A) }{ \mathrm{Pr}(B) }$$

Bayesian inference

Bayesian inference applies Bayes' rule in a likelihood context, so that $$\mathrm{Pr}(\theta \, | \, D) = \cfrac{ \mathrm{Pr}(D \, | \, \theta) \, \mathrm{Pr}(\theta) }{ \mathrm{Pr}(D) },$$ where $D$ is data and $\theta$ are parameters. $\mathrm{Pr}(D)$ is constant with respect to $\theta$, so that $\mathrm{Pr}(\theta \, | \, D) \propto \mathrm{Pr}(D \, | \, \theta) \, \mathrm{Pr}(\theta)$. This relationship is often referred to as $\mathrm{posterior} \propto \mathrm{likelihood} \times \mathrm{prior}$.

Bayesian inference for Bernoulli model

Following our previous Bernoulli example, we've observed $k$ successes in $n$ trials, and so the likelihood $\mathrm{Pr}(k,n \, | \, p) = p^k \, (1-p)^{n-k}$. We'll assume a flat prior $\mathrm{Pr}(p) = 1$. In this case, the marginal likelihood follows $$\mathrm{Pr}(k,n) = \int_0^1 \mathrm{Pr}(k,n \, | \, p) \, \mathrm{Pr}(p) \, dp = \cfrac{k! \, (n-k)!}{(n+1)!}.$$ And the full posterior follows $$\mathrm{Pr}(p \, | \, k,n) = \cfrac{(n+1)! \, p^k \, (1-p)^{n-k}}{k! \, (n-k)!}.$$

Bayesian inference allows for probability statements

If $k=8$ and $n=10$, the mean posterior $\mathrm{E}[p] = 0.75$, while the 95% credible interval extends from $0.482$ to $0.896$, and the posterior distribution follows

Maximum likelihood (ML) inference

For the Bernoulli model $\mathrm{Pr}(k,n \, | \, p) = p^k \, (1-p)^{n-k}$, we have $\hat{p} = k/n$. For example, with $k=8$ and $n=10$, $\hat{p}=0.8$ the likelihood curve follows

Methods for Bayesian integration

Markov Chain Monte Carlo

Metropolis-Hastings MCMC

Particle MCMC

Hybrid/Hamiltonian Monte Carlo

Many others

Simulation-based inference

e.g., R-package pomp

Fit models to time series or (new) longitudinal data

A simple model of HPV

$\lambda_{i,j,t}=\lambda_{0_j}f(\overrightarrow{\theta_j} \boldsymbol{X_{it}}) + \text{I(prev. inf.)}d_{jc_i}e^{-w_j(t-t_\mathrm{clr})}$

Probes and arbitrary metrics

Approximate Bayesian Computation

Time-series probes

Challenges fitting multistrain models

Many parameters

Multiple minima

Noise, nonstationarity, etc.

How would we estimate how strongly two strains are competing?

Assume we have time series of cases of each.

One approach (1/3)

• Assume yesterday's model structure. (Potential modifications: allow coinfections, noise in rates.)
• Create a mapping from latent states (number infected, $X(t)$) to data (cases, $Y(t)$), e.g., $$Y(t) \sim \text{nbinom}(\text{mean} = \rho X(t), \text{size} = \psi )$$
• with $\rho$ representing the reporting rate. (N.B. Potential problems with overcounting here; better to use an accumulator variable.)

One approach (3/3)

Constrain/fix some parameters, fit (vary) $\alpha$, a, maybe $\rho$, etc., to find the parameter combinations that maximize the likelihood of the data, e.g., multiple iterated filtering using a particle filter (sequential Monte Carlo)

Model "validation"

Confirm convergence

AIC and WAIC

Leave-one-out cross-validation (LOO)

Out-of-sample prediction

Replicate on simulated data

Appendix 1: Inference via state-space reconstruction

A system with two state variables

$$H'=aH-bHL$$

$$L'=cHL-dL$$

$H$ hares, $L$ lynxes

hare birth rate $a$, predation rate $b$,

consumption rate $c$, death rate $d$

A more complex system

$$x'=\sigma(y-x)$$

$$y'=x(\rho-z)-y$$

$$z'=xy-\beta z$$

Implications of state-space reconstruction

We can detect underlying structure

We can detect and predict without understanding

New claim: We can infer causal interactions

Takens' theorem

Very roughly, the system's attractor is diffeomorphic to (can be mapped without loss of information) to the individual attractors of the state variables in some delay-embedding space.

Causal inference from "ecological" data?

Through their shadow manifolds, variables in the same dynamical system can predict each other.

If $X$ drives $Y$, increasing the number of observations of $Y$ should improve predictions of states of $X$.

Convergent cross-mapping

To infer if $X$ drives $Y$:

• Construct the shadow manifold of $Y$, $\pmb{M}_Y$ (for some $E$, $\tau$). (Each point in $\pmb{M}_Y$ is given by $\vec{y}(t) = \{y_t,y_{t-\tau},y_{t-2\tau},...,y_{t-(E-1)\tau}\}$.)
• For each $X(t)$, identify its analogues $\vec{x}(t)$ and $\vec{y}(t)$.
• Find the $E+1$ nearest neighbors of $\vec{y}(t)$ and weight them by their Euclidean distances to $\vec{y}(t)$.
• To make a prediction $\hat{X}(t)$, multiply these weights by the respective points in $\pmb{M}_X$. Let $\rho$ be the correlation between $\vec{x}(t)$ and $\hat{X}(t)$.
• First make predictions from $\pmb{M}_Y$ constructed with only a few points in the time series, $L_\text{min}$, and then with many, $L_\text{max}$.
• If $\rho$ increases with more information on $\pmb{M}_Y$, $X$ drives $Y$.

Appendix 2: Example of model validation

Predict something else

Exploit natural and unnatural disturbances

Streptococcus pneumoniae

Carried by 20-80% of young children

Transmitted mostly between healthy carriers

>90 serotypes

Some serotypes seem better at everything

Approach

For each value of serotype-specific immunity, $\sigma$

Fit the transmission rate to obtain 40% prevalence in kids

(Later, sensitivity analysis on fixed parameters)

Other matched patterns

Increase in serotype diversity with age

Stable rank order

Decrease in carriage duration with age

Frequency of co-colonizations

Epidemics of rarer serotypes