## Pages

### Regression Example With ARDRegression in Python

Automatic Relevance Determination (ARD) is based on Bayesian inference method. Scikit-learn API provides ARDRegression class to fit the regression model by using ARD method. The ARDRegression considers the model weights as a Gaussian distributed and  estimates the lambda and alpha parameters through the iteration.

In this tutorial, we'll briefly learn how to fit and predict regression data by using ARDRegression class in Python. We'll apply the model for a randomly generated regression data and Boston housing price dataset to check the performance. The tutorial covers:

1. Preparing the data
2. Training the model
3. Predicting and accuracy check
4. Boston housing dataset prediction
5. Source code listing

```from sklearn.linear_model import ARDRegression
from sklearn.datasets import make_regression
from sklearn.metrics import mean_squared_error
from sklearn.model_selection import train_test_split
from sklearn.preprocessing import scale
import matplotlib.pyplot as plt
from sklearn import set_config ```
```
```

Preparing the data

First, we'll generate random regression data with make_regression() function. The dataset contains 10 features and 5000 samples.

```x, y = make_regression(n_samples=5000, n_features=10)
print(x[0:2])
print(y[0:2])```
` `
`[[ 1.773  2.534  0.693 -1.11   1.492  0.631 -0.577  0.085 -1.308  1.024]`
`[ 1.953 -1.362  1.294  1.025  0.463 -0.485 -1.849  1.858  0.483 -0.52 ]][120.105 262.69 ] `

To improve the model accuracy we'll scale both x and y data then, split them into train and test parts. Here, we'll extract 10 percent of the samples as test data.

```x = scale(x)
y = scale(y)xtrain, xtest, ytrain, ytest=train_test_split(x, y, test_size=0.10)
```

Training the model

Next, we'll define the regressor model by using the ARDRegression class. Here, we can use default parameters of the ARDRegression class. The default values can be seen in below.

`set_config(print_changed_only=False) `

```ardr = ARDRegression()
print(ardr)
```
`ARDRegression(alpha_1=1e-06, alpha_2=1e-06, compute_score=False, copy_X=True,              fit_intercept=True, lambda_1=1e-06, lambda_2=1e-06, n_iter=300,              normalize=False, threshold_lambda=10000.0, tol=0.001,              verbose=False) `
` `

Then, we'll fit the model on train data and check the model accuracy score.

`dtr.fit(xtrain, ytrain) `
` `
```ardr.fit(xtrain, ytrain)

score = ardr.score(xtrain, ytrain)
print("R-squared:", score)```
`  `
`R-squared: 1.0 `
` `

Predicting and accuracy check

Now, we can predict the test data by using the trained model. We can check the accuracy of predicted data by using MSE and RMSE metrics.

```ypred = ardr.predict(xtest)

mse = mean_squared_error(ytest, ypred)
print("MSE: ", mse)
print("RMSE: ", mse*(1/2.0)) ```
` `
`MSE:  1.0459020366671401e-22RMSE:  5.2295101833357005e-23 `
` `

Finally, we'll visualize the original and predicted data in a plot.

```x_ax = range(len(ytest))
plt.plot(x_ax, ytest, linewidth=1, label="original")
plt.plot(x_ax, ypred, linewidth=1.1, label="predicted")
plt.title("y-test and y-predicted data")
plt.xlabel('X-axis')
plt.ylabel('Y-axis')
plt.grid(True)
plt.show() ```

Running the above code provides a plot that shows the the original and predicted test data.

Boston housing dataset prediction

We'll apply the same method we've learned above to the Boston housing price regression dataset. We'll load it by using load_boston() function, scale and split into the train and test parts. Then, we'll define model by changing some of the parameter values, check training accuracy, and predict test data.

```print("Boston housing dataset prediction.")
x, y = boston.data, boston.target

x = scale(x)
y = scale(y)
xtrain, xtest, ytrain, ytest = train_test_split(x, y, test_size=.15)

ardr = ARDRegression()
ardr.fit(xtrain, ytrain)

score = ardr.score(xtrain, ytrain)
print("R-squared:", score)

ypred = ardr.predict(xtest)

mse = mean_squared_error(ytest, ypred)
print("MSE: ", mse)
print("RMSE: ", mse*(1/2.0))

x_ax = range(len(ytest))
plt.plot(x_ax, ytest, label="original")
plt.plot(x_ax, ypred, label="predicted")
plt.title("Boston test and predicted data")
plt.xlabel('X-axis')
plt.ylabel('Y-axis')
plt.grid(True)
plt.show()  ```

` `
`Boston housing dataset prediction.R-squared: 0.730951555514822MSE:  0.1362112343271604RMSE:  0.0681056171635802  `
` `

In this tutorial, we've briefly learned how to fit and predict regression data by using Scikit-learn API's ARDRegression class in Python. The full source code is listed below.

Source code listing

` `
```from sklearn.linear_model import ARDRegression
from sklearn.datasets import make_regression
from sklearn.metrics import mean_squared_error
from sklearn.model_selection import train_test_split
from sklearn.preprocessing import scale
import matplotlib.pyplot as plt
from sklearn import set_config

x, y = make_regression(n_samples=5000, n_features=10)
print(x[0:2])
print(y[0:2])

x = scale(x)
y = scale(y)
xtrain, xtest, ytrain, ytest = train_test_split(x, y, test_size=.10)

set_config(print_changed_only=False)
ardr = ARDRegression()
print(ardr)

ardr.fit(xtrain, ytrain)

score = ardr.score(xtrain, ytrain)
print("R-squared:", score)

ypred = ardr.predict(xtest)

mse = mean_squared_error(ytest, ypred)
print("MSE: ", mse)
print("RMSE: ", mse*(1/2.0))

x_ax = range(len(ytest))
plt.plot(x_ax, ytest, linewidth=1, label="original")
plt.plot(x_ax, ypred, linewidth=1.1, label="predicted")
plt.title("y-test and y-predicted data")
plt.xlabel('X-axis')
plt.ylabel('Y-axis')
plt.grid(True)
plt.show()

print("Boston housing dataset prediction.")
x, y = boston.data, boston.target

x = scale(x)
y = scale(y)
xtrain, xtest, ytrain, ytest = train_test_split(x, y, test_size=.15)

ardr = ARDRegression()
ardr.fit(xtrain, ytrain)

score = ardr.score(xtrain, ytrain)
print("R-squared:", score)

ypred = ardr.predict(xtest)

mse = mean_squared_error(ytest, ypred)
print("MSE: ", mse)
print("RMSE: ", mse*(1/2.0))

x_ax = range(len(ytest))
plt.plot(x_ax, ytest, label="original")
plt.plot(x_ax, ypred, label="predicted")
plt.title("Boston test and predicted data")
plt.xlabel('X-axis')
plt.ylabel('Y-axis')
` `