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class: center, middle, inverse, title-slide

# Stan @ ISEC 2020
### The Crew
### 20 June 2020

---

class: inverse, center, middle





# Building a Logistic Regression Model


---
## Cabo Pulmo, Baja California Sur, México

.center[![cabo pulmo](photos/cabopulmosearch.png)]

---
## Pelagios Kakunjá -- Bull Shark Data

Frida Lara -- http://migramar.org/hi/en/lara/

James Ketchum -- https://www.pelagioskakunja.org/james-ketchum

.center[![pelagios](photos/pelagioskakunja.png)]

https://www.pelagioskakunja.org/


---
## Presence/Absence Data from an Acoustic Array 

.pull-left[
`\(Y\)` can takes on the value of `1` when the shark is detected, and `0` when it is not. 

Observation Process: 


$$
Y \sim Bernoulli(p)
$$

Prior: 

$$
p \sim unif(0, 1)
$$
$$
p \sim beta(1, 1)
$$
]

.pull-right[
Photo by Miguel Grau Gómez: 

&lt;img src="photos/bullshark_acousticarray.jpg" width="70%" /&gt;
]

---
## Stan Code
  

```stan

data{
  int&lt;lower=1&gt; TT;
  int y[TT];
}

parameters{
  real&lt;lower=0, upper=1&gt; p;
}


model{

  p ~ uniform(0, 1);
  //p ~ beta(1,1);

  y ~ bernoulli(p);
}

```

.footnote[Order doesn't matter in the model code.]


---
class: center
### Priors over the interval (0,1): 

Combinations: 

( `\(\nu_1 = 1, \nu_2=1\)` ), ( `\(\nu_1=2, \nu_2=1\)` ) , ( `\(\alpha=1, \beta=2\)` )

( `\(\alpha=10, \beta=1\)` ), ( `\(\alpha=1, \beta=10\)` ), ( `\(\alpha=10, \beta=10\)` )

![](index_files/figure-html/unnamed-chunk-3-1.png)&lt;!-- --&gt;


---
### Is the probability constant over time? 

`\(Y\)` can takes on the value of `1` when the shark is detected, and `0` when it is not. 


$$
Y_t \sim Bernoulli(p_t)
$$

$$
logit(p_t) = f(t)
$$
 Commonly, `\(f(t)\)` is expressed as a linear function of parameters and covariate values. 
 
$$
f(t) = \beta_0 + \beta_1 x_t
$$

Priors: 
 
`$$\beta_0 \sim N(m_{\beta_0}, sd_{\beta_0})  \qquad \beta_1 \sim N(0, sd_{\beta_1})$$`

.footnote[ `\(f(t)\)` can also be a smooth non-linear function! -- `brms`, `rstanarm`] 
 

---
### Stan Code
  
Changes: allowing for time-varying covariates  
  
  

```stan

data{
  int&lt;lower=1&gt; TT;
  int y[TT];

  int&lt;lower=1&gt; ncov;
  matrix[TT, ncov + 1] x;
}

parameters{
  vector[ncov + 1] beta;
}


model{
 
 beta ~ normal(0, 0.5);

 y ~ bernoulli_logit(x*beta);
}

```


---
## Accounting for individual size/sex differences

We may expect and want to account for differences across size and sex. 

`$$f(t) = \beta_0 (sex, size) + \beta_1 * x_t$$`




`$$\beta_0 (sex, size) = \alpha_0 + \alpha_1 *male + \alpha_2 *(size - baselinesize)$$`

Where we have: 



$$
male = 1 \quad \text{if shark is male (has claspers); zero o.w.}
$$

$$
baselinesize: \text{ e.g. size at which the sharks reach maturity, here approx. 220 cm }
$$

---
## Stan Code 

Changes: removing intercept term `\(\beta_0\)` and allowing it to reflect baseline differences across sharks. Change `x` to no longer contain a column of ones. 
  
.pull-left[  

```stan

data{
  ...
  int&lt;lower=1&gt; ncov;
  matrix[TT, ncov] x;
  
  vector[TT] bsize;
  int sex[TT];
}

parameters{
  vector[ncov] beta;
  vector[3] alpha;
}
```
]


.pull-right[

```stan

model{
 alpha[1] ~ normal(0, 0.5);
 alpha[2] ~ normal(0, 0.5);
 //values range from -40 to 40
 alpha[3] ~ normal(0, 0.1);
 beta ~ normal(0, 0.5);
 y ~ bernoulli_logit(alpha[1] + 
                    alpha[2]*sex + 
                    alpha[3]*bsize +
                    x*beta);
}

```
]

---
class: inverse, middle
# Missing Discrete-Valued Covariates
## (or categorical covariates)

---
class: middle, center
### Marginalizing over discrete-valued covariates/categorical covariates


`Law of Total Probability`

`$$P(Y) = \sum_{n=1}^N P(Y|X = x_n)P(X=x_n)$$`

---
### Marginalizing in the shark example
Sex `\(\in\)` {female, male}  `\(\rightarrow\)` {0, 1}


We go from this: 

$$
Y_t \sim Bernoulli(p_t)
$$


To: 

$$
Y_t \sim \pi Bernoulli(p_t(female)) + (1-\pi)Bernoulli(p_t(male))
$$
`\(\pi \in (0, 1)\)`

`$$p_t(female) = \beta_0(female, size) + \beta_1x_t$$`

`$$p_t(male) = \beta_0(male, size) + \beta_1x_t$$`


Where `\(\pi\)` represents the probability that the shark is female. 

---
## Stan Code
Changes: need to identify the observations from the shark that is missing a value for sex.


```stan
data{
  int&lt;lower=1&gt; TT;
  int y[TT];
  
  //vector with 1 if sex is missing
  int sexmissing[TT];
  int&lt;lower=1&gt; ncov;
  matrix[TT, ncov] x;
  
  vector bsize[TT];
  vector sex[TT];

}

parameters{
  vector[ncov] beta;
  vector[3] alpha;

  real&lt;lower=0, upper=1&gt; pi;
}
```


---
## Stan Code
Changes: need to identify the observations from the shark that is missing a value for sex.


```stan
model{
 ...
 for(t in 1:TT){
  if(sexmissing[t] == 1){
    target += log_mix(pi, 
                bernoulli_logit_lpmf(y[t] | alpha[1] + 
                    alpha[3]*bsize +
                    x[t]*beta), 
                bernoulli_logit_lpmf(y[t] | alpha[1] + 
                    alpha[2] + 
                    alpha[3]*bsize[t] +
                    x[t]*beta));
  } else {
    y[t] ~ bernoulli_logit(alpha[1] +
                    alpha[2]*sex[t] + 
                    alpha[3]*bsize[t] +
                    x[t]*beta);
  }
 }
}
```


---
class: inverse

## A very not profound overview of a hierarchical model but definitely an example of one


---
## More differences across individuals

A simple hierarchical model: 

`$$Y_{jt} \sim Bernoulli(p_{jt})$$`



`$$logit(p_{jt}) = \beta_0 (sex, size) + \beta_{j1} * x_t$$`


`$$\beta_{j1} \sim N(\mu, \sigma)$$`
`$$\mu \sim N(m_{\mu}, s_{\mu}) \quad \sigma \sim N^+(m_{\sigma}, s_{\sigma})$$`
Why might this be important to include? Aside from the differences across individuals related to their sex and size (related to maturity), the sharks may not all have the same temporal patterns. Some may arrive before others, leave a bit later, and understanding individual variation plays an important role in their conservation. 

---
## Stan Code

Changes: have an index for individual shark and `\(\beta_1\)` is now a hierarchical term in the model



```stan
data{
   ...
  int nsharks;
  vector[TT] sharkid;
}

parameters{
  ...
  vector[nsharks] beta;
  real betamu;
  real&lt;lower=0&gt; betasig;
}
```

---
## Stan Code

Changes: have an index for individual shark and `\(\beta_1\)` is now a hierarchical term in the model



```stan
model{
 alpha[1] ~ normal(0, 0.5);
 alpha[2] ~ normal(0, 0.5);
 //values range from -40 to 40
 alpha[3] ~ normal(0, 0.1);
 
 betamu ~ normal(0, 0.1);
 betasig ~ normal(0, 0.1);
 
 beta ~ normal(betamu, betasig);

 y[t] ~ bernoulli_logit(alpha[1] + alpha[2]*sex[t] +
                       alpha[3]*bsize[t] + x[t]*beta[sharkid]);

}

```

---
class: inverse
# Model Assessment: 

# To Describe, Explain or Predict? 




---
### Describe(/Explain): 

- Use full data set to fit model
- Evaluate posterior predictive checks
- For logistic regression, `binned residuals for discrete data`


Bayesian Data Analysis: http://www.stat.columbia.edu/~gelman/book/ (pg 157-158)

Galit Shmueli's: To Explain or Predict? https://www.stat.berkeley.edu/~aldous/157/Papers/shmueli.pdf

---
class: inverse 
## Predict: `projpred`

### Projection predictive variable selection



```r
library(rstanarm)
library(projpred)
library(ggplot2)
library(bayesplot)
theme_set(theme_classic())
```

https://mc-stan.org/projpred/articles/quickstart.html

---
### projpred

Back to the fixed effects for a minute: 


```r
n &lt;- 7300
D &lt;- 5
p0 &lt;- 2 # prior guess for the number of relevant variables
tau0 &lt;- p0/(D-p0) * 1/sqrt(n) 
# regularized horseshoe prior
prior_coeff &lt;- hs(global_scale = tau0, slab_scale = 1) 


logregcov.horseshoe &lt;- stan_glm(formula = Presence~., 
                      family = binomial(link="logit"), 
                      data = sharkfun, 
                      prior = hs(global_scale = 0.007802743, 
                                 slab_scale = 1))

logregcov.defaultpriors &lt;- stan_glm(formula = Presence~., 
                      family = binomial(link="logit"), 
                      data = sharkfun)

logregcov.n01 &lt;- stan_glm(formula = Presence~., 
                      family = binomial(link="logit"), 
                      data = sharkfun, 
                      prior = normal(0,1))
```

---
### projpred (cont.)

K-fold Cross-Validation: 


```r
cvs.horseshoe &lt;- cv_varsel(logregcov.horseshoe, 
                           method='forward', 
                           cv_method='kfold', K=5)

cvplot.horseshoe &lt;- varsel_plot(cvs.horseshoe, 
                                stats=c('elpd, acc'))  + 
  theme_minimal() + theme(text=element_text(size=15), 
                          legend.position = "none") + 
  ggtitle("Classification Accuracy (5-fold CV)")
```

---
### What you can get from projpred: 

-- Out-of-sample and In-sample accuracies


```r
## out of sample predictive accuracies
cvplot.horseshoe
```


-- Suggested size of the models


```r
## suggested size
suggest_size(cvs)
```


-- Order of the variables

```r
vs &lt;- varsel(logregcov.horseshoe, method='forward')
vs$vind
varsel_plot(vs, stats=c('elpd', 'acc'), deltas=F)

#----------

cvs.horseshoe$vind
```

---
### Other: Occupancy Models


Observation process: 

`$$Y \sim Bernoulli(Zp)$$`
Latent process: 

`$$Z \sim Bernoulli(\psi)$$`


`\(Z\)` -- True occupancy

`\(p\)` -- Detection probability

Not logistic regression as usually thought of, but similar structure and can easily put into Stan as well. Can extend to Binomial/Poisson structures. 


---
class: inverse, center
### Last Slide for Now

Check out: https://stanecology.github.io

![](https://media.giphy.com/media/IoP0PvbbSWGAM/giphy.gif)
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