# Ordinal logistic regression¶

The ordinal logistic regression model is a statistical model that models the
relation between one or more features and an ordinal response variable.
That is, the response can take on more than two categories, but there exists
an ordering between these categories. For instance, this could be a ranking,
consisting of `bad`

, `acceptable`

and `good`

as its categories. For more
information about ordinal logistic regression, see Wikipedia.

## Setup¶

Before moving on with the guide, it is necessary to import a few modules and functions from crandas first:

```
import crandas as cd
from crandas.crlearn.logistic_regression import LogisticRegression
from crandas.crlearn.metrics import classification_accuracy
from crandas.crlearn.utils import min_max_normalize
from crandas.crlearn.metrics import confusion_matrix
```

## Reading the data¶

In this example, the dataset is read from a local CSV:

```
tab = cd.read_csv("../../test/logreg_test_data/white_wine_quality_scaled.csv")
```

This imports the *White wine quality* dataset, which contains records of white
wines with features such as the `pH`

or the `alcohol`

content, along with
the `quality`

. The `quality`

column is an ordinal variable with a value
from 0 to 4, indicating the graded `quality`

of the wine. This guide
demonstrates how ordinal logistic regression can be used to predict the quality
of a wine based on these features.

The dataset looks as following:

```
>>> print(tab.open().head())
class fixed acidity volatile acidity citric acid residual sugar chlorides free sulfur dioxide total sulfur dioxide density pH sulphates alcohol
0 3 0.428572 0.150 0.303030 0.029126 0.131673 0.214022 0.473283 0.253345 0.662921 0.351352 0.303572
1 2 0.515873 0.070 0.242424 0.288026 0.391459 0.391144 0.641221 0.373327 0.359550 0.256757 0.321428
2 2 0.412699 0.170 0.191919 0.190939 0.128114 0.701107 0.629771 0.280967 0.370787 0.121622 0.303572
3 2 0.476191 0.140 0.353536 0.009708 0.153025 0.236162 0.519084 0.253345 0.696630 0.256757 0.285714
4 2 0.317460 0.170 0.272727 0.268608 0.149467 0.295203 0.480916 0.300820 0.471910 0.378378 0.410714
```

Note that the features are all numerical.

Note

See Kaggle for more information about this dataset.

The Wine quality dataset provided here is a slightly modified variant of the original dataset. Specifically, the features have already been normalized, some classes that are sparsely represented have been excluded, and the dataset has been sampled for a more balanced class distribution.

## Preparing the data¶

### Getting rid of null values¶

The ordinal regression can only be executed on a `CDataFrame`

without null values (specifically, without nullable columns).
If the dataset contains any missing values, one can get rid of all rows with null values using `CDataFrame.dropna()`

.

```
tab = tab.dropna()
```

An alternative to deleting the rows with null values is performing data imputation using `CSeries.fillna()`

. However, this might introduce bias and is not recommended in the general case.

### Normalizing¶

If the dataset contains any numerical values (e.g. `fixed acidity`

in this example), these
first need to be normalized to values between 0 and 1. The way in which this is
commonly done is through Min-Max-Normalization.

```
tab_normalized = min_max_normalize(tab, columns=['fixed acidity', 'volatile acidity', 'citric acid', 'residual sugar', 'chlorides', 'free sulfur dioxide', 'total sulfur dioxide', 'density', 'pH', 'sulphates', 'alcohol'])
```

Here, `columns`

can be used to specify which columns need to be normalized. The
remaining columns will remain untouched.

Attention

It is **essential** to normalize your numerical features to within [0, 1]
**before** you fit the model. Otherwise, fitting will **not** work correctly,
and will return erroneous results.

Negative values are also **not** allowed.

### Splitting into predictors, response¶

First, split the predictor variables from the response variable:

```
X = tab_normalized[['fixed acidity', 'volatile acidity', 'citric acid', 'residual sugar', 'chlorides', 'free sulfur dioxide', 'total sulfur dioxide', 'density', 'pH', 'sulphates', 'alcohol']]
y = tab_normalized[['quality']]
```

## Creating the model¶

The logistic regression functionality in crandas is made accessible through
the `LogisticRegression`

class, which can be used to fit the model and
make predictions. The model can be created using:

```
model = LogisticRegression(solver='lbfgs', multi_class="ordinal", n_classes=5)
```

Here, the `multi_class`

argument specifies the type of regression to be
performed, in this case `ordinal`

. The `n_classes`

argument specifies
the number of classes in the dataset.

Note

The `solver`

argument indicates which numerical solver the model should
use to fit the model. Currently, the available options are:

`lbfgs`

(which stands for Limited-memory BFGS)`gd`

(which stands for Gradient Descent)

The `lbfgs`

solver gives better results and fits the model faster. As such,
there is normally no reason to deviate from it.

Attention

It is **required** to specify the number of classes in the dataset. Unlike
in scikit-learn, the crandas model does not detect this automatically due to the
dataset being secret-shared.

## Fitting the model¶

Now that the data has been prepared and the model has been created, the model can be fitted to the training set:

```
model.fit(X, y, max_iter=20)
```

Here, the `max_iter`

argument specifies how many iterations the numerical
solver should perform to fit the model. The default of 10 is sufficient in some
cases but sometimes it is necessary to increase this number in order for the
model to fully converge. In this case, we need 20 iterations for the wine
quality dataset.

Note

Fitting a logistic regression model in crandas can take quite some time, depending on the number of records in the dataset, the number of features, and the number of iterations that you specify.

The fitted model parameters can now be accessed as following:

```
beta = model.get_beta()
```

The first `n_classes - 1`

columns represent the threshold values
that distinguish each of the classes. The remaining columns correspond to each
of the features.

## Predicting¶

Now that the model has been fitted, it can be used to make predictions. We distinguish two different types in crandas:

probabilities: the model can predict the probability of each class being associated with the record

classes: the model can predict the class with the highest likelihood

### Probabilities¶

First, to predict the probabilities corresponding to each record of the test dataset:

```
y_pred_probabilities = model.predict_proba(X)
```

This returns a table with five columns (one for each class), containing the point probability for each class. These sum up to one.

### Classes¶

Alternatively, if you are interested in making actual class predictions rather than the probabilities, you can directly predict the classes through:

```
y_pred_classes = model.predict(X)
```

## Assessing prediction quality¶

After fitting the model, it is important to assess the quality of the model and its predictions. crandas provides a couple of methods for doing this, namely:

Classification Accuracy

### Accuracy¶

To compute the accuracy of the (class) predictions, you can use:

```
accuracy = classification_accuracy(y, y_pred_classes, n_classes=5)
print("Classification Accuracy:", accuracy.open())
```

Attention

It is **required** to specify the number of classes in the dataset. The
function does not detect this automatically, due to the dataset being
secret-shared.

### Confusion Matrix¶

The confusion matrix visualizes the relation between the predicted classes and the actual classes. The Y-axis represents the true class, while the X-axis represents the class predicted by the model. To compute the confusion matrix obliviously, you can use:

```
matrix = confusion_matrix(y, y_pred_classes, n_classes=3)
matrix.open()
[[85, 15, 60, 0, 3],
[51, 21, 94, 0, 5],
[11, 13, 154, 0, 47],
[ 0, 4, 56, 0, 44],
[ 2, 2, 77, 0, 94]]
```