# How much data for learning model (not) similar to the test sample?

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Consider one of the scenarios in which your machine learning model may be useless.

There is a saying:

In real-life cases, this situation is ubiquitous. When developing machine learning models, we are faced with a situation where our model works well with the training sample, but the quality of the model drops sharply on test data.

And this is not about retraining. Suppose that we have built a model that gives an excellent result for cross-validation, but shows a poor result on the test. So in the test sample there is information that we do not take into account.

Imagine a situation in which we predict customer behavior in the store. If the training and test sample looks like the image below, this is a clear problem:

In this text we will talk about "naive" approaches that allow to identify such phenomena and try to eliminate them.

We give a more accurate definition of this concept.

In real problems with a large number of variables, the covariant shift is difficult to detect. The article discusses the method of identifying and also taking into account the covariant shift in the data.

Let's understand why this is so. Let us return to the example with customers, where age was a “shifted” sign of a training and test sample. If you take a classifier (for example, based on a random forest) and try to divide the mixed sample into training and test, then age will be a very important feature for such a classification.

Let's try to apply the described idea to a real dataset. Use dataset from Kaggle competition.

First of all, let's perform a series of standard steps: clean, fill in the blanks, perform label encoding for categorical features. For the dataset in question, the step was not required, so let's skip its description.

To both parts of the dataset - training and test - it is necessary to add a new indicator indicator. For the training sample with the value "1", for the test, respectively, "0".

Now it is necessary to combine two datasets. Because the training dataset contains a column of target values 'target', which is not in the test dataset, this column must be deleted.

For classification purposes, we will use the Random Forest Classifier, which will be configured to predict the labels of the data source in the combined dataset. You can use any other classifier.

We use a stratified randomized split into 4 folds. This way we will keep the ratio of the 'is_train' tags in each fold as in the original merged sample. For each splitting, we will train the classifier on most of the splitting and predict the class label for the smaller deferred part.

We calculate the value of the ROC AUC metric for our classifier. Based on this value, we conclude how well our classifier detects a covariant shift in the data.

Find the ROC AUC value:

The resulting value is close to 0.5. This means that our quality classifier is the same as the random predictor of tags. There is no evidence of a covariant shift in the data.

Since it’s taken from Kaggle, the result is pretty predictable. As in other machine learning competitions, the data are carefully checked to ensure that there are no shifts.

But such an approach can be applied in other data science problems to check for the presence of a covariant shift just before the start of the solution.

So, either we are seeing a covariant shift, or not. What to do to improve the quality of the model on the test?

Let's look at the predictions we received in the previous section. For each object, the prediction contains the probability that this object belongs to the training set for our classifier.

For example, for the first object, our Random Forest Classifier considers that it belongs to the training set with a probability of 0.397. Let's call this value$$. Or you can say that the probability of belonging to the test data is 0.603. Similarly, let's call the probability$$.

Now a little trick: for each object of the training dataset, we calculate the coefficient$$.

Coefficient$$tells us how close the object from the training set is to the test data. Basic thought:

These weights can be calculated using the code:

The resulting coefficients can be transferred to the model, for example, as follows:

A couple of words about the resulting histogram:

We hope that this post will help you in identifying the "covariant shift" in the data and in dealing with it.

[1] Shimodaira, H. (2000). The log-likelihood function. Journal of Statistical Planning and Inference, 90, 227–244.

[2] Bickel, S. et al. (2009). Discriminative Learning Under Covariate Shift. Journal of Machine Learning Research, 10, 2137–2155

[3] github.com/erlendd/covariate-shift-adaption

[4] Link to dataset used

PS Notebook with the code from the article can be viewed here .

There is a saying:

*"Do not compare apples with oranges*.*"*But what if you need to compare one set of apples with oranges with another, but the distribution of fruits in two sets is different? Can you work with the data? And how will you do it?In real-life cases, this situation is ubiquitous. When developing machine learning models, we are faced with a situation where our model works well with the training sample, but the quality of the model drops sharply on test data.

And this is not about retraining. Suppose that we have built a model that gives an excellent result for cross-validation, but shows a poor result on the test. So in the test sample there is information that we do not take into account.

Imagine a situation in which we predict customer behavior in the store. If the training and test sample looks like the image below, this is a clear problem:

*In this example, the model is trained on the data with the average value of the sign “customer age” lower than the average value of the same sign on the test. In the process of learning, the model has never “seen” the larger values of the sign “age”. If age is an important feature for a model, then one should not expect good results on the test sample.*In this text we will talk about "naive" approaches that allow to identify such phenomena and try to eliminate them.

### Covariant shift

We give a more accurate definition of this concept.

**Covariance**refers to the values of features, and by**covariant shift**is meant the situation when the distributions of feature values in the training and test samples have different characteristics (parameters).In real problems with a large number of variables, the covariant shift is difficult to detect. The article discusses the method of identifying and also taking into account the covariant shift in the data.

### main idea

*If there is a shift in the data, then when mixing two samples, we will be able to build a classifier that can determine whether an object belongs to a training or test sample.*Let's understand why this is so. Let us return to the example with customers, where age was a “shifted” sign of a training and test sample. If you take a classifier (for example, based on a random forest) and try to divide the mixed sample into training and test, then age will be a very important feature for such a classification.

### Implementation

Let's try to apply the described idea to a real dataset. Use dataset from Kaggle competition.

#### Step 1: Data Preparation

First of all, let's perform a series of standard steps: clean, fill in the blanks, perform label encoding for categorical features. For the dataset in question, the step was not required, so let's skip its description.

```
import pandas as pd
#загрузка датасетов train и test
train = pd.read_csv('train.csv',low_memory=True)
test = pd.read_csv('test.csv',low_memory=True)
```

#### Step 2: Add Data Source Indicator

To both parts of the dataset - training and test - it is necessary to add a new indicator indicator. For the training sample with the value "1", for the test, respectively, "0".

```
#добавляем новый столбец к данным, нулевой к test, единичный к train
test['is_train'] = 0
train['is_train'] = 1
```

#### Step 3: Combine Training and Test Samples

Now it is necessary to combine two datasets. Because the training dataset contains a column of target values 'target', which is not in the test dataset, this column must be deleted.

```
#объединение выборок train, test
df_combine = pd.concat([train, test], axis=0, ignore_index=True)
#удаление метки target
df_combine = df_combine.drop('target', axis =1)
y = df_combine['is_train'].values #индикатор источника
x = df_combine.drop('is_train', axis=1).values #объединенный датасет
tst, trn = test.values, train.values
```

#### Step 4: building and testing the classifier

For classification purposes, we will use the Random Forest Classifier, which will be configured to predict the labels of the data source in the combined dataset. You can use any other classifier.

```
from sklearn.ensemble import RandomForestClassifier
import numpy as np
rfc = RandomForestClassifier(n_jobs=-1, max_depth=5, min_samples_leaf = 5)
predictions = np.zeros(y.shape) #пустой массив для вероятностей классов
```

We use a stratified randomized split into 4 folds. This way we will keep the ratio of the 'is_train' tags in each fold as in the original merged sample. For each splitting, we will train the classifier on most of the splitting and predict the class label for the smaller deferred part.

```
from sklearn.model_selection import StratifiedKFold, cross_val_score
skf = StratifiedKFold(n_splits=4, shuffle=True, random_state=100)
for fold, (train_idx, test_idx) in enumerate(skf.split(x, y)):
X_train, X_test = x[train_idx], x[test_idx]
y_train, y_test = y[train_idx], y[test_idx]
rfc.fit(X_train, y_train)
probs = rfc.predict_proba(X_test)[:, 1] #предсказание вероятностей классов
predictions[test_idx] = probs
```

#### Step 5: Interpret Results

We calculate the value of the ROC AUC metric for our classifier. Based on this value, we conclude how well our classifier detects a covariant shift in the data.

*If the classifier with well separates objects into training and test datasets, then the value of the ROC AUC metric should be significantly greater than 0.5, ideally close to 1. This picture indicates a strong covariant shift in the data.*Find the ROC AUC value:

```
from sklearn.metrics import roc_auc_score
print('ROC-AUC:', roc_auc_score(y_true=y, y_score=predictions))
# ROC-AUC: 0.49974692698385287
```

The resulting value is close to 0.5. This means that our quality classifier is the same as the random predictor of tags. There is no evidence of a covariant shift in the data.

Since it’s taken from Kaggle, the result is pretty predictable. As in other machine learning competitions, the data are carefully checked to ensure that there are no shifts.

But such an approach can be applied in other data science problems to check for the presence of a covariant shift just before the start of the solution.

## Further steps

So, either we are seeing a covariant shift, or not. What to do to improve the quality of the model on the test?

- Remove offset features
- Use object importance weights based on density coefficient estimate

### Removing displaced features:

**Note: the**method is applicable if there is a covariant shift in the data.- Extract the importance of signs from the Random Forest Classifier classifier, which we have built and trained earlier.
- The most important features are those that are shifted and cause a shift in the data.
- Starting with the most important ones, delete by one feature, build the target model and look at its quality. Collect all signs for which the quality of the model is not reduced.
- Discard the collected signs from the data and build the final model.

*This algorithm will remove the signs from the red basket on the chart.*### Using object importance weights based on density coefficient estimation

**Note: the**method is applicable regardless of whether there is a covariant shift in the data.Let's look at the predictions we received in the previous section. For each object, the prediction contains the probability that this object belongs to the training set for our classifier.

```
predictions[:10]
#array([0.39743827 ...
```

For example, for the first object, our Random Forest Classifier considers that it belongs to the training set with a probability of 0.397. Let's call this value$$. Or you can say that the probability of belonging to the test data is 0.603. Similarly, let's call the probability$$.

Now a little trick: for each object of the training dataset, we calculate the coefficient$$.

Coefficient$$tells us how close the object from the training set is to the test data. Basic thought:

*We can use$$as weights in any of the models, in order to increase the weight of those observations that look similar to the test sample. Intuitively, this makes sense, since our model will be more data-oriented as in the test set.*These weights can be calculated using the code:

```
import seaborn as sns
import matplotlib.pyplot as plt
plt.figure(figsize=(20,10))
predictions_train = predictions[:len(trn)]
weights = (1./predictions_train) - 1.
weights /= np.mean(weights) #Нормализация весов
plt.xlabel('Значение весов w')
plt.ylabel('Число сэмплов')
sns.distplot(weights, kde=False)
```

The resulting coefficients can be transferred to the model, for example, as follows:

```
rfc = RandomForestClassifier(n_jobs=-1,max_depth=5)
m.fit(X_train, y_train, sample_weight=weights)
```

A couple of words about the resulting histogram:

- Larger weights correspond to observations more similar to the test sample.
- Almost 70% of the objects from the training sample have a weight close to 1, and, therefore, are in a subspace that is equally similar to the training and the test sample. This corresponds to the AUC value that we calculated earlier.

## Conclusion

We hope that this post will help you in identifying the "covariant shift" in the data and in dealing with it.

## Links

[1] Shimodaira, H. (2000). The log-likelihood function. Journal of Statistical Planning and Inference, 90, 227–244.

[2] Bickel, S. et al. (2009). Discriminative Learning Under Covariate Shift. Journal of Machine Learning Research, 10, 2137–2155

[3] github.com/erlendd/covariate-shift-adaption

[4] Link to dataset used

PS Notebook with the code from the article can be viewed here .