# GLM: Model Selection¶

In [1]:

%matplotlib inline
import pymc3 as pm
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns
from collections import OrderedDict
from ipywidgets import interactive, fixed

plt.style.use('seaborn-darkgrid')
print('Running on PyMC3 v{}'.format(pm.__version__))
rndst = np.random.RandomState(0)

Running on PyMC3 v3.3


A fairly minimal reproducable example of Model Selection using WAIC, and LOO as currently implemented in PyMC3.

This example creates two toy datasets under linear and quadratic models, and then tests the fit of a range of polynomial linear models upon those datasets by using Widely Applicable Information Criterion (WAIC), and leave-one-out (LOO) cross-validation using Pareto-smoothed importance sampling (PSIS).

The example was inspired by Jake Vanderplas’ blogpost on model selection, although Cross-Validation and Bayes Factor comparison are not implemented. The datasets are tiny and generated within this Notebook. They contain errors in the measured value (y) only.

## Local Functions¶

In [2]:

def generate_data(n=20, p=0, a=1, b=1, c=0, latent_sigma_y=20):
'''
Create a toy dataset based on a very simple model that we might
imagine is a noisy physical process:
1. random x values within a range
2. latent error aka inherent noise in y
3. optionally create labelled outliers with larger noise

Model form: y ~ a + bx + cx^2 + e

NOTE: latent_sigma_y is used to create a normally distributed,
'latent error' aka 'inherent noise' in the 'physical' generating
process, rather than experimental measurement error.
Please don't use the returned latent_error values in inferential
models, it's returned in the dataframe for interest only.
'''

df = pd.DataFrame({'x':rndst.choice(np.arange(100), n, replace=False)})

## create linear or quadratic model
df['y'] = a + b*(df['x']) + c*(df['x'])**2

## create latent noise and marked outliers
df['latent_error'] = rndst.normal(0, latent_sigma_y, n)
df['outlier_error'] = rndst.normal(0, latent_sigma_y*10, n)
df['outlier'] = rndst.binomial(1, p, n)

## add noise, with extreme noise for marked outliers
df['y'] += ((1-df['outlier']) * df['latent_error'])
df['y'] += (df['outlier'] * df['outlier_error'])

## round
for col in ['y','latent_error','outlier_error','x']:
df[col] = np.round(df[col],3)

df['source'] = 'linear' if c == 0 else 'quadratic'

## create simple linspace for plotting true model
plotx = np.linspace(df['x'].min() - np.ptp(df['x'])*.1,
df['x'].max() + np.ptp(df['x'])*.1, 100)

ploty = a + b * plotx + c * plotx ** 2
dfp = pd.DataFrame({'x':plotx, 'y':ploty})

return df, dfp

def interact_dataset(n=20, p=0, a=-30, b=5, c=0, latent_sigma_y=20):
'''
Convenience function:
Interactively generate dataset and plot
'''

df, dfp = generate_data(n, p, a, b, c, latent_sigma_y)

g = sns.FacetGrid(df, size=8, hue='outlier', hue_order=[True,False]
,palette=sns.color_palette('Set1'), legend_out=False)

g.map(plt.errorbar, 'x', 'y', 'latent_error', marker="o",

plt.plot(dfp['x'], dfp['y'], '--', alpha=0.8)

g.fig.suptitle('Sketch of Data Generation ({})'.format(df['source'][0]), fontsize=16)

'''
Convenience function:
Plot the two generated datasets in facets with generative model
'''

g = sns.FacetGrid(col='source', hue='source', data=df, size=6,
sharey=False, legend_out=False)

g.map(plt.scatter, 'x', 'y', alpha=0.7, s=100, lw=2, edgecolor='w')

g.axes[0][0].plot(dfp_lin['x'], dfp_lin['y'], '--', alpha=0.6)

def plot_traces(traces, retain=1000):
'''
Convenience function:
Plot traces with overlaid means and values
'''

ax = pm.traceplot(traces[-retain:], figsize=(12,len(traces.varnames)*1.5),
lines={k: v['mean'] for k, v in pm.summary(traces[-retain:]).iterrows()})

for i, mn in enumerate(pm.summary(traces[-retain:])['mean']):
ax[i,0].annotate('{:.2f}'.format(mn), xy=(mn,0), xycoords='data',
xytext=(5,10), textcoords='offset points', rotation=90,
va='bottom', fontsize='large', color='#AA0022')

def create_poly_modelspec(k=1):
'''
Convenience function:
Create a polynomial modelspec string for patsy
'''
return ('y ~ 1 + x ' + ' '.join(['+ np.power(x,{})'.format(j)
for j in range(2, k+1)])).strip()

def run_models(df, upper_order=5):
'''
Convenience function:
Fit a range of pymc3 models of increasing polynomial complexity.
Suggest limit to max order 5 since calculation time is exponential.
'''

models, traces = OrderedDict(), OrderedDict()

for k in range(1, upper_order+1):

nm = 'k{}'.format(k)
fml = create_poly_modelspec(k)

with pm.Model() as models[nm]:

print('\nRunning: {}'.format(nm))
pm.glm.GLM.from_formula(fml, df,
priors={'Intercept':pm.Normal.dist(mu=0, sd=100)},
family=pm.glm.families.Normal())

traces[nm] = pm.sample(2000)

return models, traces

def plot_posterior_cr(models, traces, rawdata, xlims,
datamodelnm='linear', modelnm='k1'):
'''
Convenience function:
Plot posterior predictions with credible regions shown as filled areas.
'''

## Get traces and calc posterior prediction for npoints in x
npoints = 100
mdl = models[modelnm]
trc = pm.trace_to_dataframe(traces[modelnm][-1000:])
trc = trc[[str(v) for v in mdl.cont_vars[:-1]]]

ordr = int(modelnm[-1:])
x = np.linspace(xlims[0], xlims[1], npoints).reshape((npoints,1))
pwrs = np.ones((npoints,ordr+1)) * np.arange(ordr+1)
X = x ** pwrs
cr = np.dot(X, trc.T)

## Calculate credible regions and plot over the datapoints
dfp = pd.DataFrame(np.percentile(cr,[2.5, 25, 50, 75, 97.5], axis=1).T,
columns=['025','250','500','750','975'])
dfp['x'] = x

pal = sns.color_palette('Greens')
f, ax1d = plt.subplots(1,1, figsize=(7,7))
f.suptitle('Posterior Predictive Fit -- Data: {} -- Model: {}'.format(datamodelnm,
modelnm), fontsize=16)

ax1d.fill_between(dfp['x'], dfp['025'], dfp['975'], alpha=0.5,
color=pal[1], label='CR 95%')
ax1d.fill_between(dfp['x'], dfp['250'], dfp['750'], alpha=0.5,
color=pal[4], label='CR 50%')
ax1d.plot(dfp['x'], dfp['500'], alpha=0.6, color=pal[5], label='Median')

plt.legend()
ax1d.set_xlim(xlims)
sns.regplot(x='x', y='y', data=rawdata, fit_reg=False,
scatter_kws={'alpha':0.7,'s':100, 'lw':2,'edgecolor':'w'}, ax=ax1d)


## Interactively draft data¶

Throughout the rest of the Notebook, we’ll use two toy datasets created by a linear and a quadratic model respectively, so that we can better evaluate the fit of the model selection.

Right now, lets use an interactive session to play around with the data generation function in this Notebook, and get a feel for the possibilities of data we could generate.

$y_{i} = a + bx_{i} + cx_{i}^{2} + \epsilon_{i}$
where:
$$i \in n$$ datapoints

$$\epsilon \sim \mathcal{N}(0,latent\_sigma\_y)$$

NOTE on outliers:

• We can use value p to set the (approximate) proportion of ‘outliers’ under a bernoulli distribution.
• These outliers have a 10x larger latent_sigma_y
• These outliers are labelled in the returned datasets and may be useful for other modelling, see another example Notebook GLM-robust-with-outlier-detection.ipynb
In [3]:

interactive(interact_dataset, n=[5,50,5], p=[0,.5,.05], a=[-50,50],
b=[-10,10], c=[-3,3], latent_sigma_y=[0,1000,50])


Observe:

• I’ve shown the latent_error in errorbars, but this is for interest only, since this shows the inherent noise in whatever ‘physical process’ we imagine created the data.
• There is no measurement error.
• Datapoints created as outliers are shown in red, again for interest only.

## Create datasets for modelling¶

We can use the above interactive plot to get a feel for the effect of the params. Now we’ll create 2 fixed datasets to use for the remainder of the Notebook.

1. For a start, we’ll create a linear model with small noise. Keep it simple.
2. Secondly, a quadratic model with small noise
In [4]:

n = 12
df_lin, dfp_lin = generate_data(n=n, p=0, a=-30, b=5, c=0, latent_sigma_y=40)


Scatterplot against model line

In [5]:

plot_datasets(df_lin, df_quad, dfp_lin, dfp_quad)


Observe:

• We now have two datasets df_lin and df_quad created by a linear model and quadratic model respectively.
• You can see this raw data, the ideal model fit and the effect of the latent noise in the scatterplots above
• In the folowing plots in this Notebook, the linear-generated data will be shown in Blue and the quadratic in Green.

## Standardize¶

In [6]:

dfs_lin = df_lin.copy()
dfs_lin['x'] = (df_lin['x'] - df_lin['x'].mean()) / df_lin['x'].std()



Create ranges for later ylim xim

In [7]:

dfs_lin_xlims = (dfs_lin['x'].min() - np.ptp(dfs_lin['x'])/10,
dfs_lin['x'].max() + np.ptp(dfs_lin['x'])/10)

dfs_lin_ylims = (dfs_lin['y'].min() - np.ptp(dfs_lin['y'])/10,
dfs_lin['y'].max() + np.ptp(dfs_lin['y'])/10)



### Demonstrate simple linear model¶

This linear model is really simple and conventional, an OLS with L2 constraints (Ridge Regression):

$y = a + bx + \epsilon$

## Define model using explicit PyMC3 method¶

In [8]:

with pm.Model() as mdl_ols:
## define Normal priors to give Ridge regression
b0 = pm.Normal('b0', mu=0, sd=100)
b1 = pm.Normal('b1', mu=0, sd=100)

## define Linear model
yest = b0 + b1 * df_lin['x']

## define Normal likelihood with HalfCauchy noise (fat tails, equiv to HalfT 1DoF)
sigma_y = pm.HalfCauchy('sigma_y', beta=10)
likelihood = pm.Normal('likelihood', mu=yest, sd=sigma_y, observed=df_lin['y'])

traces_ols = pm.sample(2000)

Auto-assigning NUTS sampler...
Multiprocess sampling (2 chains in 2 jobs)
NUTS: [sigma_y_log__, b1, b0]
100%|██████████| 2500/2500 [00:03<00:00, 826.32it/s]

In [9]:

plot_traces(traces_ols, retain=1000)


Observe:

• This simple OLS manages to make fairly good guesses on the model parameters - the data has been generated fairly simply after all - but it does appear to have been fooled slightly by the inherent noise.

## Define model using PyMC3 GLM method¶

PyMC3 has a module - glm - for defining models using a patsy-style formula syntax. This seems really useful, especially for defining simple regression models in fewer lines of code.

Here’s the same OLS model as above, defined using glm.

In [10]:

with pm.Model() as mdl_ols_glm:
# setup model with Normal likelihood (which uses HalfCauchy for error prior)
pm.glm.GLM.from_formula('y ~ 1 + x', df_lin, family=pm.glm.families.Normal())

traces_ols_glm = pm.sample(2000)

Auto-assigning NUTS sampler...
Multiprocess sampling (2 chains in 2 jobs)
NUTS: [sd_log__, x, Intercept]
100%|██████████| 2500/2500 [00:03<00:00, 788.53it/s]
There were 1 divergences after tuning. Increase target_accept or reparameterize.

In [11]:

plot_traces(traces_ols_glm, retain=1000)


Observe:

• The output parameters are of course named differently to the custom naming before. Now we have:

b0 == Intercept
b1 == x
sigma_y == sd
• However, naming aside, this glm-defined model appears to behave in a very similar way, and finds the same parameter values as the conventionally-defined model - any differences are due to the random nature of the sampling.

• We can quite happily use the glm syntax for further models below, since it allows us to create a small model factory very easily.

### Create higher-order linear models¶

Back to the real purpose of this Notebook, to demonstrate model selection.

First, let’s create and run a set of polynomial models on each of our toy datasets. By default this is for models of order 1 to 5.

## Create and run polynomial models¶

Please see run_models() above for details. Generally, we’re creating 5 polynomial models and fitting each to the chosen dataset

In [12]:

models_lin, traces_lin = run_models(dfs_lin, 5)

Auto-assigning NUTS sampler...


Running: k1

Multiprocess sampling (2 chains in 2 jobs)
NUTS: [sd_log__, x, Intercept]
100%|██████████| 2500/2500 [00:01<00:00, 1251.87it/s]


Running: k2

Auto-assigning NUTS sampler...
Multiprocess sampling (2 chains in 2 jobs)
NUTS: [sd_log__, np.power(x, 2), x, Intercept]
100%|██████████| 2500/2500 [00:02<00:00, 1036.36it/s]


Running: k3

Auto-assigning NUTS sampler...
Multiprocess sampling (2 chains in 2 jobs)
NUTS: [sd_log__, np.power(x, 3), np.power(x, 2), x, Intercept]
100%|██████████| 2500/2500 [00:04<00:00, 530.45it/s]


Running: k4

Auto-assigning NUTS sampler...
Multiprocess sampling (2 chains in 2 jobs)
NUTS: [sd_log__, np.power(x, 4), np.power(x, 3), np.power(x, 2), x, Intercept]
100%|██████████| 2500/2500 [00:07<00:00, 347.64it/s]
There were 2 divergences after tuning. Increase target_accept or reparameterize.
There were 3 divergences after tuning. Increase target_accept or reparameterize.
The number of effective samples is smaller than 25% for some parameters.


Running: k5

Auto-assigning NUTS sampler...
Multiprocess sampling (2 chains in 2 jobs)
NUTS: [sd_log__, np.power(x, 5), np.power(x, 4), np.power(x, 3), np.power(x, 2), x, Intercept]
100%|██████████| 2500/2500 [00:24<00:00, 101.41it/s]
There were 33 divergences after tuning. Increase target_accept or reparameterize.
There were 28 divergences after tuning. Increase target_accept or reparameterize.
The number of effective samples is smaller than 25% for some parameters.

In [13]:

models_quad, traces_quad = run_models(dfs_quad, 5)


Running: k1

Auto-assigning NUTS sampler...
Multiprocess sampling (2 chains in 2 jobs)
NUTS: [sd_log__, x, Intercept]
100%|██████████| 2500/2500 [00:02<00:00, 875.37it/s]
The acceptance probability does not match the target. It is 0.993112884126, but should be close to 0.8. Try to increase the number of tuning steps.
Auto-assigning NUTS sampler...


Running: k2

Multiprocess sampling (2 chains in 2 jobs)
NUTS: [sd_log__, np.power(x, 2), x, Intercept]
100%|██████████| 2500/2500 [00:04<00:00, 578.24it/s]
The acceptance probability does not match the target. It is 0.883398371777, but should be close to 0.8. Try to increase the number of tuning steps.
The acceptance probability does not match the target. It is 0.999997671953, but should be close to 0.8. Try to increase the number of tuning steps.
The chain reached the maximum tree depth. Increase max_treedepth, increase target_accept or reparameterize.
The gelman-rubin statistic is larger than 1.2 for some parameters.
The estimated number of effective samples is smaller than 200 for some parameters.


Running: k3

Auto-assigning NUTS sampler...
Multiprocess sampling (2 chains in 2 jobs)
NUTS: [sd_log__, np.power(x, 3), np.power(x, 2), x, Intercept]
100%|██████████| 2500/2500 [03:07<00:00, 13.32it/s]
The acceptance probability does not match the target. It is 0.999381198882, but should be close to 0.8. Try to increase the number of tuning steps.
The acceptance probability does not match the target. It is 0.899455530613, but should be close to 0.8. Try to increase the number of tuning steps.


Running: k4

Auto-assigning NUTS sampler...
Multiprocess sampling (2 chains in 2 jobs)
NUTS: [sd_log__, np.power(x, 4), np.power(x, 3), np.power(x, 2), x, Intercept]
100%|██████████| 2500/2500 [00:05<00:00, 430.11it/s]
The acceptance probability does not match the target. It is 0.920287453662, but should be close to 0.8. Try to increase the number of tuning steps.
The acceptance probability does not match the target. It is 0.993538143406, but should be close to 0.8. Try to increase the number of tuning steps.


Running: k5

Auto-assigning NUTS sampler...
Multiprocess sampling (2 chains in 2 jobs)
NUTS: [sd_log__, np.power(x, 5), np.power(x, 4), np.power(x, 3), np.power(x, 2), x, Intercept]
100%|██████████| 2500/2500 [00:06<00:00, 407.53it/s]
The acceptance probability does not match the target. It is 0.924822538129, but should be close to 0.8. Try to increase the number of tuning steps.
The acceptance probability does not match the target. It is 0.986333585677, but should be close to 0.8. Try to increase the number of tuning steps.


### A really bad method for model selection: compare likelihoods¶

Evaluate log likelihoods straight from model.logp

In [14]:

dfll = pd.DataFrame(index=['k1','k2','k3','k4','k5'], columns=['lin','quad'])
dfll.index.name = 'model'

for nm in dfll.index:
dfll.loc[nm,'lin'] = - models_lin[nm].logp(pm.summary(traces_lin[nm],
traces_lin[nm].varnames)['mean'].to_dict())

dfll = pd.melt(dfll.reset_index(), id_vars=['model'],
var_name='poly', value_name='log_likelihood')


Plot log-likelihoods

In [15]:

g = sns.factorplot(x='model', y='log_likelihood', col='poly',
hue='poly', data=dfll, size=6)


Observe:

• Again we’re showing the linear-generated data at left (Blue) and the quadratic-generated data on the right (Green)
• For both datasets, as the models get more complex, the likelhood increases monotonically
• This is expected, since the models are more flexible and thus able to (over)fit more easily.
• This overfitting makes it a terrible idea to simply use the likelihood to evaluate the model fits.

## View posterior predictive fit¶

Just for the linear, generated data, lets take an interactive look at the posterior predictive fit for the models k1 through k5.

As indicated by the likelhood plots above, the higher-order polynomial models exhibit some quite wild swings in the function in order to (over)fit the data

In [16]:

interactive(plot_posterior_cr, models=fixed(models_lin), traces=fixed(traces_lin),
rawdata=fixed(dfs_lin), xlims=fixed(dfs_lin_xlims), datamodelnm=fixed('linear'),
modelnm = ['k1','k2','k3','k4','k5'])


### Compare models using WAIC¶

The Widely Applicable Information Criterion (WAIC) can be used to calculate the goodness-of-fit of a model using numerical techniques. See (Watanabe 2013) for details.

Observe:

• We get three different measurements:
• waic: widely available information criterion
• waic_se: standard error of waic
• p_waic: effective number parameters

In this case we are interested in the WAIC score. We also plot error bars for the standard error of the estimated scores. This gives us a more accurate view of how much they might differ.

Now loop through all the models and calculate the WAIC

In [18]:

dfwaic = pd.DataFrame(index=['k1','k2','k3','k4','k5'], columns=['lin','quad'])
dfwaic.index.name = 'model'

for nm in dfwaic.index:
dfwaic.loc[nm, 'lin'] = pm.waic(traces_lin[nm], models_lin[nm])[:2]
dfwaic = pd.melt(dfwaic.reset_index(), id_vars=['model'], var_name='poly', value_name='waic_')
dfwaic[['waic', 'waic_se']] = dfwaic['waic_'].apply(pd.Series)

# Define a wrapper function for plt.errorbar
def errorbar(x, y, se, order, color, **kws):
xnum = [order.index(x_i) for x_i in x]
plt.errorbar(xnum, y, yerr=se, color='k', ls='None')

g = sns.factorplot(x='model', y='waic', col='poly', hue='poly', data=dfwaic, size=6)
order = sns.utils.categorical_order(dfwaic['model'])
g.map(errorbar,'model', 'waic', 'waic_se', order=order);

/home/osvaldo/proyectos/00_PyMC3/pymc3/pymc3/stats.py:194: UserWarning: For one or more samples the posterior variance of the
log predictive densities exceeds 0.4. This could be indication of
WAIC starting to fail see http://arxiv.org/abs/1507.04544 for details

""")


Observe

• We should prefer the model(s) with lower WAIC
• Linear-generated data (lhs):
• The WAIC seems quite flat across models
• The WAIC seems best (lowest) for simpler models.
• The WAIC is also quite flat across the models
• The lowest WAIC is model k4, but k3 - k5 are more or less the same.

### Compare leave-one-out Cross-Validation [LOO]¶

Leave-One-Out Cross-Validation or K-fold Cross-Validation is another quite universal approach for model selection. However, to implement K-fold cross-validation we need to paritition the data repeatly and fit the model on every partition. It can be very time consumming (computation time increase roughly as a factor of K). Here we are applying the numerical approach using the posterier trace as suggested in Vehtari et al 2015.

In [19]:

dfloo = pd.DataFrame(index=['k1','k2','k3','k4','k5'], columns=['lin','quad'])
dfloo.index.name = 'model'

for nm in dfloo.index:
dfloo.loc[nm, 'lin'] = pm.loo(traces_lin[nm], models_lin[nm])[:2]
dfloo = pd.melt(dfloo.reset_index(), id_vars=['model'], var_name='poly', value_name='loo_')
dfloo[['loo', 'loo_se']] = dfloo['loo_'].apply(pd.Series)

g = sns.factorplot(x='model', y='loo', col='poly', hue='poly', data=dfloo, size=6)
order = sns.utils.categorical_order(dfloo['model'])
g.map(errorbar,'model', 'loo', 'loo_se', order=order);

/home/osvaldo/proyectos/00_PyMC3/pymc3/pymc3/stats.py:270: UserWarning: Estimated shape parameter of Pareto distribution is
greater than 0.7 for one or more samples.
You should consider using a more robust model, this is because
importance sampling is less likely to work well if the marginal
posterior and LOO posterior are very different. This is more likely to
happen with a non-robust model and highly influential observations.
happen with a non-robust model and highly influential observations.""")


Observe

• We should prefer the model(s) with lower LOO. You can see that LOO is nearly identical with WAIC. That’s because WAIC is asymptotically equal to LOO. However, PSIS-LOO is supposedly more robust than WAIC in the finite case (under weak priors or influential observation).
• Linear-generated data (lhs):
• The LOO is also quite flat across models
• The LOO is also seems best (lowest) for simpler models.
• The same pattern as the WAIC

### Final remarks and tips¶

It is important to keep in mind that, with more data points, the real underlying model (one that we used to generate the data) should outperform other models.

There is some agreement that PSIS-LOO offers the best indication of a model’s quality. To quote from avehtari’s comment: “I also recommend using PSIS-LOO instead of WAIC, because it’s more reliable and has better diagnostics as discussed in http://link.springer.com/article/10.1007/s11222-016-9696-4 (preprint https://arxiv.org/abs/1507.04544), but if you insist to have one information criterion then leave WAIC”.

Alternatively, Watanabe says “WAIC is a better approximator of the generalization error than the pareto smoothing importance sampling cross validation. The Pareto smoothing cross validation may be the better approximator of the cross validation than WAIC, however, it is not of the generalization error”.