Cross Validation and Model Selection

class: center, middle

# Cross-Validation and AIC<br>

![:scale 55%](images/cv/princess_bride_cv.jpg)

---

# HOT ROACHES!
<br><br>
.pull-left[
- Do American Cockroaches sense surrounding temperatures and respond?

- Survey of activity by the prothoracic ganglion in different temperature regimes

- Murphy and Heath 1983 J. Exp. Biol

]

.pull-right[
![](./images/cv/roach_nipcam_com.jpg)
]

---

# Which is best?

<img src="crossvalidation_files/figure-html/roachplot_int-1.png" style="display: block; margin: auto;" />
  
---

# Which is best?

---

# Applying Different Styles of Inference

- **Null Hypothesis Testing**: What's the probability that things are not influencing our data?
      - Deductive

- **Cross-Validation**: How good are you at predicting new data?
      - Deductive
      
- **Model Comparison**: Comparison of alternate hypotheses
      - Deductive or Inductive

- **Probabilistic Inference**: What's our degree of belief in a data?
      - Inductive
      
---

# Applying Different Styles of Inference

.grey[
- **Null Hypothesis Testing**: What's the probability that things are not influencing our data?
      - Deductive
]

- **Cross-Validation**: How good are you at predicting new data?
      - Deductive
      
- **Model Comparison**: Comparison of alternate hypotheses
      - Deductive or Inductive

.grey[
- **Probabilistic Inference**: What's our degree of belief in a data?
      - Inductive
]

---

# Validating Across Models

1. Out of Sample Prediction

2. Train-Test Random Cross Validation

3. K-Fold Cross Validation

4. LOOCV

5. AIC and Model Comparison

---

# What is Out of Sample Prediction

1. We fit a model

2. We have a new sample(s) with a full set of predictors

3. We apply the model to our new prediction

4. We see how badly we deviate from our fit model

---

# Common Out of Sample Metrics

`$MSE = \frac{1}{n}\sum{(Y_i - \hat{Y})^2}$`
     - In units of sums of squares

`$RMSE = \sqrt{\frac{1}{n}\sum{(Y_i - \hat{Y})^2}}$`
     - In units of response variable!
     - Estimate of SD of out of sample predictions
     
`$Deviance = -2 \sum log(\space p(Y_i | \hat{Y}, \theta)\space)$`
     - Probability-based
     - Encompasses a wide number of probability distributions
     - Just MSE for gaussian linear models!
     
---
# Evaluating and Out of Sample Point: What if we had left out the 1st row of Roach data?

---
# We Can See This Model is Worse at Prediction

---
# Evaluating Predictive Ability of Statistical Models: Cross-Validation

1. Fit a model on a **training** data set

2. Evaluate a Model on a **TEST** data set

3. Compare predictive ability of competing models with MSE, Deviance, etc.

---

# But.... what data do I use for training and testing?
## Random Cross-Validation

- Cross-validating on only one point could be biased by choice of poing

- So, choose a random subset of your data!

- Typically use a 60:40 split, or 70:30 for lower sample size

- Calculate fit metrics for alternate models

---
# A Random Cross-Validation Sample

---
# The Fits

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# Cross-Validation: the Fits

---

# Results of Random CV

RMSE Temperature Model:

```
[1] 27.31889
```

RMSE Intercept Only Model:

```
[1] 32.00178
```

This is the estimate of the SD of the training set - which is acceptable for your predictions?

---

# Validating Across Models

1. Out of Sample Prediction

2. Train-Test Random Cross Validation

3. .red[K-Fold Cross Validation]

4. LOOCV

5. AIC and Model Comparison

---
class: center, middle

# But, wait, what if I choose a bad section of the data by chance?

---

# K-Fold Cross Validation

- Data is split into K sets of training and testing folds

- Performance is averaged over all sets

.center[
![](./images/cv/kfold_performance.png)
]

---
# Our Models

---

# Five-Fold CV

|id    |      mse|     rmse|
|:-----|--------:|--------:|
|Fold1 | 566.6716| 23.80487|
|Fold2 | 482.6707| 21.96977|
|Fold3 | 493.9591| 22.22519|
|Fold4 | 586.6068| 24.21997|
|Fold5 | 515.8802| 22.71300|

Temperature Model Score: 529.1576829

Intercept Only Score: 788.2974

---

# But what if there is some feature in your data that splits it into groups? 
## Stratified K-Fold CV!
.center[.middle[
![](images/cv/stratified_kfold.png)
]]

---

# Validating Across Models

1. Out of Sample Prediction

2. Train-Test Random Cross Validation

3. K-Fold Cross Validation

4. .red[LOOCV]

5. AIC and Model Comparison

---
# Problem: How Many Folds?

- What is a good number?
     - More folds = smaller test dataset
     - Bias-Variance tradeoff if also looking at coefficients

- 5 and 10 are fairly standard, depending on data set size

- More folds = closer to average out of sample error

- But, more folds can be computationally intensive

- Still, Leave One Out Cross Validation is very powerful

---
# LOOCV (Leave One Out Cross-Validation)

.center[.middle[
![](./images/cv/LOOCV.gif)

]]

---
# LOOCV (Leave One Out Cross-Validation)
.center[.middle[
![](./images/cv/loocv.png)

]]

---
# LOOCV Comparison of Out of Sample Deviance (MSE)

Rate Model:569.58

Intercept Only Model: 818.68

---
#What abour our polynomial models?

---

# Using LOOCV to Compare Polynomial Fits

---

# Validating Across Models

1. Out of Sample Prediction

2. Train-Test Random Cross Validation

3. K-Fold Cross Validation

4. LOOCV

5. .red[AIC and Model Comparison]

---
class: center, middle

# This is all well and very computationally intensive, but, what about when we have n = 1e5, or many different models?

---

# Can our Training Deviance Provide an Approximation of our Test Deviance?

---

# Does the Gap Increase with Number of Parameters?

.large[.center[Slope = 1.89 ≈ 2]]

---
# Enter the AIC
.pull-left[
![](./images/mmi/hirotugu_akaike.jpg)
]

.pull-right[
- `$E[D_{test}] = D_{train} + 2K$`  
  
- This is Akaike's Information Criteria (AIC)

]

`$$\Large AIC = Deviance + 2K$$`

---
# AIC and Prediction

- AIC optimized for forecasting (out of sample deviance)

- Approximates average out of sample deviance from test data

- Assumes large N relative to K  
    - AICc for a correction

---
# But Sample Size Can Influence Fit...
<br><br>
`$$\large AIC = -2log(L(\theta | x)) + 2K$$`

<br><br><br>
`$$\large AICc = AIC + \frac{2K(K+1)}{n-K-1}K$$`

---
# How can we Use AIC Values?

`$$\Delta AIC = AIC_{i} - min(AIC)$$`

Rules of Thumb from Burnham and Anderson(2002):

- `$\Delta$` AIC `$<$` 2 implies that two models are similar in their fit to the data

- `$\Delta$` AIC between 3 and 7 indicate moderate, but less, support for retaining a model

- `$\Delta$` AIC `$>$` 10 indicates that the model is very unlikely

---
# A Quantitative Measure of Relative Support

`$$w_{i} = \frac{e^{-\Delta_{i}/2 }}{\displaystyle \sum^R_{r=1} e^{-\Delta_{i}/2 }}$$`

Where `$w_{i}$` is the *relative support* for model i
compared to other models in the set being considered.

Model weights summed together = 1

---
# Let's Apply AIC to our Polynomials

---
# Model Information for Polynomial Fits

<table>
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:left;"> Modnames </th>
   <th style="text-align:right;"> K </th>
   <th style="text-align:right;"> AIC </th>
   <th style="text-align:right;"> Delta_AIC </th>
   <th style="text-align:right;"> ModelLik </th>
   <th style="text-align:right;"> AICWt </th>
   <th style="text-align:right;"> LL </th>
   <th style="text-align:right;"> Cum.Wt </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> 3 </td>
   <td style="text-align:left;"> Order 3 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 522.36 </td>
   <td style="text-align:right;"> 0.00 </td>
   <td style="text-align:right;"> 1.00 </td>
   <td style="text-align:right;"> 0.44 </td>
   <td style="text-align:right;"> -256.18 </td>
   <td style="text-align:right;"> 0.44 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 4 </td>
   <td style="text-align:left;"> Order 4 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 523.38 </td>
   <td style="text-align:right;"> 1.02 </td>
   <td style="text-align:right;"> 0.60 </td>
   <td style="text-align:right;"> 0.26 </td>
   <td style="text-align:right;"> -255.69 </td>
   <td style="text-align:right;"> 0.70 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 5 </td>
   <td style="text-align:left;"> Order 5 </td>
   <td style="text-align:right;"> 7 </td>
   <td style="text-align:right;"> 525.34 </td>
   <td style="text-align:right;"> 2.98 </td>
   <td style="text-align:right;"> 0.23 </td>
   <td style="text-align:right;"> 0.10 </td>
   <td style="text-align:right;"> -255.67 </td>
   <td style="text-align:right;"> 0.79 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 1 </td>
   <td style="text-align:left;"> Order 1 </td>
   <td style="text-align:right;"> 3 </td>
   <td style="text-align:right;"> 525.92 </td>
   <td style="text-align:right;"> 3.55 </td>
   <td style="text-align:right;"> 0.17 </td>
   <td style="text-align:right;"> 0.07 </td>
   <td style="text-align:right;"> -259.96 </td>
   <td style="text-align:right;"> 0.87 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 6 </td>
   <td style="text-align:left;"> Order 6 </td>
   <td style="text-align:right;"> 8 </td>
   <td style="text-align:right;"> 526.67 </td>
   <td style="text-align:right;"> 4.31 </td>
   <td style="text-align:right;"> 0.12 </td>
   <td style="text-align:right;"> 0.05 </td>
   <td style="text-align:right;"> -255.33 </td>
   <td style="text-align:right;"> 0.92 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 2 </td>
   <td style="text-align:left;"> Order 2 </td>
   <td style="text-align:right;"> 4 </td>
   <td style="text-align:right;"> 526.89 </td>
   <td style="text-align:right;"> 4.53 </td>
   <td style="text-align:right;"> 0.10 </td>
   <td style="text-align:right;"> 0.05 </td>
   <td style="text-align:right;"> -259.44 </td>
   <td style="text-align:right;"> 0.96 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 7 </td>
   <td style="text-align:left;"> Order 7 </td>
   <td style="text-align:right;"> 9 </td>
   <td style="text-align:right;"> 528.27 </td>
   <td style="text-align:right;"> 5.91 </td>
   <td style="text-align:right;"> 0.05 </td>
   <td style="text-align:right;"> 0.02 </td>
   <td style="text-align:right;"> -255.14 </td>
   <td style="text-align:right;"> 0.99 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 8 </td>
   <td style="text-align:left;"> Order 8 </td>
   <td style="text-align:right;"> 10 </td>
   <td style="text-align:right;"> 530.27 </td>
   <td style="text-align:right;"> 7.91 </td>
   <td style="text-align:right;"> 0.02 </td>
   <td style="text-align:right;"> 0.01 </td>
   <td style="text-align:right;"> -255.14 </td>
   <td style="text-align:right;"> 1.00 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 9 </td>
   <td style="text-align:left;"> Order 9 </td>
   <td style="text-align:right;"> 11 </td>
   <td style="text-align:right;"> 532.24 </td>
   <td style="text-align:right;"> 9.87 </td>
   <td style="text-align:right;"> 0.01 </td>
   <td style="text-align:right;"> 0.00 </td>
   <td style="text-align:right;"> -255.12 </td>
   <td style="text-align:right;"> 1.00 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 10 </td>
   <td style="text-align:left;"> Order 10 </td>
   <td style="text-align:right;"> 12 </td>
   <td style="text-align:right;"> 533.75 </td>
   <td style="text-align:right;"> 11.39 </td>
   <td style="text-align:right;"> 0.00 </td>
   <td style="text-align:right;"> 0.00 </td>
   <td style="text-align:right;"> -254.88 </td>
   <td style="text-align:right;"> 1.00 </td>
  </tr>
</tbody>
</table>

.center[So... how much better are the top models? What about Parsimody? What would you say?]

---
# Inferential Frameworks of Today

- **Cross-Validation**: How good are you at predicting new data?
      - Deductive
      - Chose model with best predictive ability!
      
- **Model Comparison**: Comparison of alternate hypotheses
      - Deductive or Inductive
      - Either compare models to elminiate poor fits
      - Or... compare multiple models to determine how a system works