In [ ]:
%matplotlib inline


# Machine learning with missing values¶

Here we use simulated data to understanding the fundamentals of statistical learning with missing values.

This notebook reveals why a HistGradientBoostingRegressor ( :class:sklearn.ensemble.HistGradientBoostingRegressor ) is a good choice to predict with missing values.

We use simulations to control the missing-value mechanism, and inspect it's impact on predictive models. In particular, standard imputation procedures can reconstruct missing values without distortion only if the data is missing at random.

A good introduction to the mathematics behind this notebook can be found in https://arxiv.org/abs/1902.06931

.. topic:: Missing values in categorical data

If a categorical column has missing values, the simplest approach is
to create a specific category "missing" and assign missing values to
this new category, to represent missingness in the classifier.
Indeed, as we will see, imputation is not crucial for prediction.
In the following we focus on continuous columns, where the discrete
nature of a missing value poses more problems.

## The fully-observed data: a toy regression problem¶

We consider a simple regression problem where X (the data) is bivariate gaussian, and y (the prediction target) is a linear function of the first coordinate, with noise.

### The data-generating mechanism¶

In [ ]:
import numpy as np

def generate_without_missing_values(n_samples, rng=42):
mean = [0, 0]
cov = [[1, 0.5], [0.5, 1]]
if not isinstance(rng, np.random.RandomState):
rng = np.random.RandomState(rng)
X = rng.multivariate_normal(mean, cov, size=n_samples)

epsilon = 0.1 * rng.randn(n_samples)
y = X[:, 0] + epsilon

return X, y


A quick plot reveals what the data looks like

In [ ]:
import matplotlib.pyplot as plt
plt.rcParams['figure.figsize'] = (5, 4) # Smaller default figure size

plt.figure()
X_full, y_full = generate_without_missing_values(1000)
plt.scatter(X_full[:, 0], X_full[:, 1], c=y_full)
plt.colorbar(label='y')


## Missing completely at random settings¶

We now consider missing completely at random settings (a special case of missing at random): the missingness is completely independent from the values.

### The missing-values mechanism¶

In [ ]:
def generate_mcar(n_samples, missing_rate=.5, rng=42):
X, y = generate_without_missing_values(n_samples, rng=rng)
if not isinstance(rng, np.random.RandomState):
rng = np.random.RandomState(rng)

M = rng.binomial(1, missing_rate, (n_samples, 2))

return X, y


A quick plot to look at the data

In [ ]:
X, y = generate_mcar(500)

plt.figure()
plt.scatter(X_full[:, 0], X_full[:, 1], color='.8', ec='.5', label='All data')
plt.colorbar(label='y')
plt.scatter(X[:, 0], X[:, 1], c=y, label='Fully observed')
plt.legend()


We can see that the distribution of the fully-observed data is the same than that of the original data

### Conditional Imputation with the IterativeImputer¶

As the data is MAR (missing at random), an imputer can use the conditional dependencies between the observed and the missing values to impute the missing values.

We'll use the IterativeImputer, a good imputer, but it needs to be enabled

In [ ]:
from sklearn.experimental import enable_iterative_imputer
from sklearn import impute
iterative_imputer = impute.IterativeImputer()


Let us try the imputer on the small data used to visualize

The imputation is learned by fitting the imputer on the data with missing values

In [ ]:
iterative_imputer.fit(X)


The data are imputed with the transform method

In [ ]:
X_imputed = iterative_imputer.transform(X)


We can display the imputed data as our previous visualization

In [ ]:
plt.figure()
plt.scatter(X_full[:, 0], X_full[:, 1], color='.8', ec='.5',
label='All data', alpha=.5)
plt.scatter(X_imputed[:, 0], X_imputed[:, 1], c=y, marker='X',
label='Imputed')
plt.colorbar(label='y')
plt.legend()


We can see that the imputer did a fairly good job of recovering the data distribution

### Supervised learning: imputation and a linear model¶

Given that the relationship between the fully-observed X and y is a linear relationship, it seems natural to use a linear model for prediction. It must be adapted to missing values using imputation.

To use it in supervised setting, we will pipeline it with a linear model, using a ridge, which is a good default linear model

In [ ]:
from sklearn.pipeline import make_pipeline
from sklearn.linear_model import RidgeCV

iterative_and_ridge = make_pipeline(impute.IterativeImputer(), RidgeCV())


We can evaluate the model performance in a cross-validation loop (for better evaluation accuracy, we increase slightly the number of folds to 10)

In [ ]:
from sklearn import model_selection
scores_iterative_and_ridge = model_selection.cross_val_score(
iterative_and_ridge, X, y, cv=10)

scores_iterative_and_ridge


Computational cost: One drawback of the IterativeImputer to keep in mind is that its computational cost can become prohibitive of large datasets (it has a bad computation scalability).

### Mean imputation: SimpleImputer¶

We can try a simple imputer: imputation by the mean

In [ ]:
mean_imputer = impute.SimpleImputer()


A quick visualization reveals a larger disortion of the distribution

In [ ]:
X_imputed = mean_imputer.fit_transform(X)
plt.figure()
plt.scatter(X_full[:, 0], X_full[:, 1], color='.8', ec='.5',
label='All data', alpha=.5)
plt.scatter(X_imputed[:, 0], X_imputed[:, 1], c=y, marker='X',
label='Imputed')
plt.colorbar(label='y')


Evaluating in prediction pipeline

In [ ]:
mean_and_ridge = make_pipeline(impute.SimpleImputer(), RidgeCV())
scores_mean_and_ridge = model_selection.cross_val_score(
mean_and_ridge, X, y, cv=10)

scores_mean_and_ridge


### Supervised learning without imputation¶

The HistGradientBoosting models are based on trees, which can be adapted to model directly missing values

In [ ]:
from sklearn.experimental import enable_hist_gradient_boosting



### Recap: which pipeline predicts well on our small data?¶

Let's plot the scores to see things better

In [ ]:
import pandas as pd
import seaborn as sns

scores = pd.DataFrame({'Mean imputation + Ridge': scores_mean_and_ridge,
'IterativeImputer + Ridge': scores_iterative_and_ridge,
})

sns.boxplot(data=scores, orient='h')
plt.title('Prediction accuracy\n linear and small data\n'
'Missing Completely at Random')
plt.tight_layout()


Not much difference with the more sophisticated imputer. A more thorough analysis would be necessary, with more cross-validation runs.

### Prediction performance with large datasets¶

Let us compare models in regimes where there is plenty of data

In [ ]:
X, y = generate_mcar(n_samples=20000)


Iterative imputation and linear model

In [ ]:
scores_iterative_and_ridge= model_selection.cross_val_score(
iterative_and_ridge, X, y, cv=10)


Mean imputation and linear model

In [ ]:
scores_mean_and_ridge = model_selection.cross_val_score(
mean_and_ridge, X, y, cv=10)


And now the HistGradientBoostingRegressor, which does not need imputation

In [ ]:
score_hist_gradient_boosting = model_selection.cross_val_score(


We plot the results

In [ ]:
scores = pd.DataFrame({'Mean imputation + Ridge': scores_mean_and_ridge,
'IterativeImputer + Ridge': scores_iterative_and_ridge,
})

sns.boxplot(data=scores, orient='h')
plt.title('Prediction accuracy\n linear and large data\n'
'Missing Completely at Random')
plt.tight_layout()


When there is a reasonnable amout of data, the HistGradientBoostingRegressor is the best strategy even for a linear data-generating mechanism, in MAR settings, which are settings favorable to imputation + linear model [#]_.

.. [#] Even in the case of a linear data-generating mechanism, the optimal prediction one data imputed by a constant is a piecewise affine function with 2^d regions ( http://proceedings.mlr.press/v108/morvan20a.html ). The larger the dimensionality (number of features), the more a imperfect imputation is hard to approximate with a simple model.

|

## Missing not at random: censoring¶

We now consider missing not at random settings, in particular self-masking or censoring, where large values are more likely to be missing.

### The missing-values mechanism¶

In [ ]:
def generate_censored(n_samples, missing_rate=.4, rng=42):
X, y = generate_without_missing_values(n_samples, rng=rng)
if not isinstance(rng, np.random.RandomState):
rng = np.random.RandomState(rng)

B = rng.binomial(1, 2 * missing_rate, (n_samples, 2))
M = (X > 0.5) * B

return X, y


A quick plot to look at the data

In [ ]:
X, y = generate_censored(500, missing_rate=.4)

plt.figure()
plt.scatter(X_full[:, 0], X_full[:, 1], color='.8', ec='.5',
label='All data')
plt.colorbar(label='y')
plt.scatter(X[:, 0], X[:, 1], c=y, label='Fully observed')
plt.legend()


Here the full-observed data does not reflect well at all the distribution of all the data

### Imputation fails to recover the distribution¶

With MNAR data, off-the-shelf imputation methods do not recover the initial distribution:

In [ ]:
iterative_imputer = impute.IterativeImputer()
X_imputed = iterative_imputer.fit_transform(X)

plt.figure()
plt.scatter(X_full[:, 0], X_full[:, 1], color='.8', ec='.5',
label='All data', alpha=.5)
plt.scatter(X_imputed[:, 0], X_imputed[:, 1], c=y, marker='X',
label='Imputed')
plt.colorbar(label='y')
plt.legend()


Recovering the initial data distribution would need much more mass on the right and the top of the figure. The imputed data is shifted to lower values than the original data.

Note also that as imputed values typically have lower X values than their full-observed counterparts, the association between X and y is also distorted. This is visible as the imputed values appear as lighter diagonal lines.

An important consequence is that the link between imputed X and y is no longer linear, although the original data-generating mechanism is linear [#]_. For this reason, it is often a good idea to use non-linear learners in the presence of missing values.

.. [#] As mentionned above, even in the case of a linear data-generating mechanism, imperfect imputation leads to complex functions to link to y ( http://proceedings.mlr.press/v108/morvan20a.html )

### Predictive pipelines¶

Let us now evaluate predictive pipelines

In [ ]:
scores = dict()

# Iterative imputation and linear model
scores['IterativeImputer + Ridge'] = model_selection.cross_val_score(
iterative_and_ridge, X, y, cv=10)

# Mean imputation and linear model
scores['Mean imputation + Ridge'] = model_selection.cross_val_score(
mean_and_ridge, X, y, cv=10)

# IterativeImputer and non-linear model
iterative_and_gb = make_pipeline(impute.SimpleImputer(),
iterative_and_gb, X, y, cv=10)

# Mean imputation and non-linear model
mean_and_gb = make_pipeline(impute.SimpleImputer(),
mean_and_gb, X, y, cv=10)

# And now the HistGradientBoostingRegressor, whithout imputation

# We plot the results
sns.boxplot(data=pd.DataFrame(scores), orient='h')
plt.title('Prediction accuracy\n linear and small data\n'
'Missing not at Random')
plt.tight_layout()


We can see that the imputation is not the most important step of the pipeline [#]_, rather what is important is to use a powerful model. Here there is information in missingness (if a value is missing, it is large), information that a model can use to predict better.

.. [#] Note that there are less missing values in the example here compared to the section above on MCAR, hence the absolute prediction accuracies are not comparable.

.. topic:: Prediction with missing values

The data above are very simple: linear data-generating mechanism, Gaussian, and low dimensional. Yet, they show the importance of using non-linear models, in particular the HistGradientBoostingRegressor which natively deals with missing values.

## Using a predictor for the fully-observed case¶

Let us go back to the "easy" case of the missing completely at random settings with plenty of data

In [ ]:
n_samples = 20000

X, y = generate_mcar(n_samples, missing_rate=.5)


Suppose we have been able to train a predictive model that works on fully-observed data:

In [ ]:
X_full, y_full = generate_without_missing_values(n_samples)
full_data_predictor.fit(X_full, y_full)

model_selection.cross_val_score(full_data_predictor, X_full, y_full)


The cross validation reveals that the predictor achieves an excellent explained variance; it is a near-perfect predictor on fully observed data

Now we turn to data with missing values. Given that our data is MAR (missing at random), we will use imputation to build a completed data that looks like the full-observed data

In [ ]:
iterative_imputer = impute.IterativeImputer()
X_imputed = iterative_imputer.fit_transform(X)


The full data predictor can be used on the imputed data

In [ ]:
from sklearn import metrics
metrics.r2_score(y, full_data_predictor.predict(X_imputed))


This prediction is less good than on the full data, but this is expected, as missing values lead to a loss of information. We can compare it to a model trained to predict on data with missing values

In [ ]:
X_train, y_train = generate_mcar(n_samples, missing_rate=.5)
na_predictor.fit(X_train, y_train)

metrics.r2_score(y, na_predictor.predict(X))


Applying a model valid on the full data to imputed data work almost as well as a model trained for missing values. The small loss in performance is because the imputation is imperfect.

### When the data-generation is non linear¶

We now modify a bit the example above to consider the situation where y is a non-linear function of X

In [ ]:
X, y = generate_mcar(n_samples, missing_rate=.5)
y = y ** 2

# Train a predictive model that works on fully-observed data:
X_full, y_full = generate_without_missing_values(n_samples)
y_full = y_full ** 2
full_data_predictor.fit(X_full, y_full)

model_selection.cross_val_score(full_data_predictor, X_full, y_full)


Once again, we have a near-perfect predictor on fully-observed data

On data with missing values:

In [ ]:
iterative_imputer = impute.IterativeImputer()
X_imputed = iterative_imputer.fit_transform(X)

from sklearn import metrics
metrics.r2_score(y, full_data_predictor.predict(X_imputed))


The full-data predictor works much less well

Now we use a model trained to predict on data with missing values

In [ ]:
X_train, y_train = generate_mcar(n_samples, missing_rate=.5)
y_train = y_train ** 2