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W2D01 Piscine AI - Data Science

The goal of this day is to understand practical Linear regression and supervised learning.

Author:

Introduction

The word "regression" was introduced by Sir Francis Galton (a cousin of C. Darwin) when he studied the size of individuals within a progeny. He was trying to understand why large individuals in a population appeared to have smaller children, more close to the average population size; hence the introduction of the term "regression".

Today we will learn a basic algorithm used in supervised learning : The Linear Regression. We will be using Scikit-learn which is a machine learning library. It is designed to to interoperate with the Python libraries NumPy and Pandas. We will also learn progressively the Machine Learning methodology for supervised learning - today we will focus on evalutatig a machine learning model by splitting the data set in a train set and a test set.

'0.22.1'

Rules

Ressources

To start with Scikit-learn:

https://scikit-learn.org/stable/modules/linear_model.html

Machine learning methodology and algorithms:

This course provides a broad introduction to machine learning, datamining, and statistical pattern recognition. Andrew Ng is a star in the Machine Learning community. I recommend to spend some time during the projects to focus on some algorithms. However, Python is not the langage used for the course. https://www.coursera.org/learn/machine-learning
https://docs.microsoft.com/en-us/azure/machine-learning/algorithm-cheat-sheet

https://scikit-learn.org/stable/tutorial/index.html

Linear Regression

Train test split

Exercise 1 Scikit-learn estimator

The goal of this exercise is to learn to fit a Scikit-learn estimator and use it to predict.


X, y = [[1],[2.1],[3]], [[1],[2],[3]]

Fit a LinearRegression from Scikit-learn with X the features and y the target.
Predict for x_pred = [[4]]
Print the coefficients (coefs_) and the intercept (intercept_), the score (score)of the regression of X and y.

Correction

This question is validated if the ouput of the fitted model is:


LinearRegression(copy_X=True, fit_intercept=[[1], [2.1], [3]], n_jobs=None,
                normalize=[[1], [2], [3]])

This question is validated if the ouput is:
```
array([[3.96013289]])
```

This question is validated if the ouptut is:

Coefficients:  [[0.99667774]]
Intercept:  [-0.02657807]
Score:  0.9966777408637874

Exercise 2 Linear regression in 1D

The goal of this exercise is to understand how the linear regression works in one dimension. To do so, we will generate a data in one dimension. Using make regression from Scikit-learn, generate a data set with 100 observations:

```
X, y, coef = make_regression(n_samples=100,
                         n_features=1,
                         n_informative=1,
                         noise=10,
                         coef=True,
                         random_state=0,
                         bias=100.0)
```

Plot the data using matplotlib. The plot should look like this:

Fit a LinearRegression from Scikit-learn on the generated data and give the equation of the fitted line. The expected output is: y = coef * x + intercept
Add the fitted line to the plot. the plot should look like this:

Predict on X
Create a function that computes the Mean Squared Error (MSE) and compute the MSE on the data set. The MSE is frequently used as well as other regression metrics that will be studied later this week.
```
def compute_mse(y_true, y_pred):
    #TODO
    return mse
```

Change the noise parameter of make_regression to 50

Repeat question 2, 4 and compute the MSE on the new data.

https://scikit-learn.org/stable/modules/generated/sklearn.metrics.mean_squared_error.html

Correction

This question is validated if the plot looks like:

This question is validated if the equation of the fitted line is: y = 42.619430291366946 * x + 99.18581817296929
This question is validated if the plot looks like:

This question is validated if the outputted prediction for the first 10 values are:

array([ 83.86186727, 140.80961751, 116.3333897 ,  64.52998689,
        61.34889539, 118.10301628,  57.5347917 , 117.44107847,
       108.06237908,  85.90762675])

This question is validated if the MSE returned is 114.17148616819485
This question is validated if the MSE returned is 2854.2871542048706

Exercise 3: Train test split

The goal of this exercise is to learn to split a data set. It is important to understand why we split the data in two sets. To put it in a nutshell: the Machine Learning algorithms learns on the training data and is evaluated on the that it hasn't seen before: the testing data.

This video gives a basic and nice explanation: https://www.youtube.com/watch?v=_vdMKioCXqQ

This article explains the conditions to split the data and how to split it: https://machinelearningmastery.com/train-test-split-for-evaluating-machine-learning-algorithms/

```
X = np.arange(1,21).reshape(10,-1)
y = np.arange(1,11)
```

Split the data using train_test_split with shuffle=False. The test set represents 20% of the total size of the data set. Print X_train, y_train, X_test, y_test.

https://scikit-learn.org/stable/modules/generated/sklearn.model_selection.train_test_split.html

Correction

This question is validated if X_train, y_train, X_test, y_test match this output:

X_train:  
 [[ 1  2]
 [ 3  4]
 [ 5  6]
 [ 7  8]
 [ 9 10]
 [11 12]
 [13 14]
 [15 16]]


y_train:  
 [1 2 3 4 5 6 7 8]


X_test:  
 [[17 18]
 [19 20]]


y_test:  
 [ 9 10]

Exercise 4 Forecast diabetes progression

The goal of this exercise is to use Linear Regression to forecast the progression of diabetes. It will not always be precised, you should ALWAYS start doing an exploratory data analysis in order to have a good understanding of the data you model. As a reminder here an introduction to EDA: https://towardsdatascience.com/exploratory-data-analysis-eda-a-practical-guide-and-template-for-structured-data-abfbf3ee3bd9

The data set used is described in https://scikit-learn.org/stable/modules/generated/sklearn.datasets.load_diabetes.

from sklearn.datasets import load_diabetes
diabetes = load_diabetes()
X, y = diabetes.data, diabetes.target

Using train_test_split, split the data set in a train set and test set (20%). Use random_state=43 for results reproducibility.
Fit the Linear Regression on all the variables. Give the coefficients and the intercept of the Linear Regression. What is then the equation ?
Predict on the test set. Predicting on the test set is like having new patients for who, as a physician, need to forecast the disease progression in one year given the 10 baseline variables.
Compute the MSE on the train set and test set. Later this week we will learn about the R2 which will help us to evaluate the performance of this fitted Linear Regression. The MSE returns an arbitrary value depending on the range of error.

WARNING: This will be explained later this week. But here, we are doing something "dangerous". As you may have read in the data documentation the data is scaled using the whole dataset whereas we should first scale the data on the training set and then use this scaling on the test set. This is a toy example, so let's ingore this detail for now.

https://scikit-learn.org/stable/datasets/toy_dataset.html#diabetes-dataset

Correction

This question is validated if the output of y_train.values[:10] and y_test.values[:10]are:

y_train.values[:10]: 
[[202.]
[ 55.]
[202.]
[ 42.]
[214.]
[173.]
[118.]
[ 90.]
[129.]
[151.]]

y_test.values[:10]: 
[[ 71.]
[ 72.]
[235.]
[277.]
[109.]
[ 61.]
[109.]
[ 78.]
[ 66.]
[192.]]

This question is validated if the coefficients and the intercept are:

[('age', -60.40163046086952),
('sex', -226.08740652083418),
('bmi', 529.383623302316),
('bp', 259.96307686274605),
('s1', -859.121931974365),
('s2', 504.70960058378813),
('s3', 157.42034928335502),
('s4', 226.29533600601638),
('s5', 840.7938070846119),
('s6', 34.712225788519554),
('intercept', 152.05314895029233)]

This question is validated if the output of predictions_on_test[:10] is:

array([[111.74351759],
    [ 98.41335251],
    [168.36373195],
    [255.05882934],
    [168.43764643],
    [117.60982186],
    [198.86966323],
    [126.28961941],
    [117.73121787],
    [224.83346984]])

This question is validated if the mse on the train set is 2888.326888 and the mse on the test set is 2858.255153.

Exercise 5 Gradient Descent

The goal of this exercise is to understand how the Linear Regression algorithm finds the optimal coefficients.

The goal is to fit a Linear Regression on a one dimensional features data without using Scikit-learn. Let's use the data set we generated for the exercise 1:

```
X, y, coef = make_regression(n_samples=100,
                         n_features=1,
                         n_informative=1,
                         noise=10,
                         coef=True,
                         random_state=0,
                         bias=100.0)
```

Warning: The shape of X is not the same as the shape of y. You may need (for some questions) to reshape X using: X.reshape(1,-1)[0].

Plot the data using matplotlib:

As a reminder, fitting a Linear Regression on this data means finding (a,b) that fits well the data points.

- y_pred = a*x +b

Mathematically, it means finding (a,b) that minimizes the MSE, which is the loss used in Linear Regression. If we consider 3 data points:

- Loss(a,b) = MSE(a,b) = 
1/3 *((y_pred1 - y_true1)**2 + (y_pred2 - y_true2)**2) + (y_pred3 - y_true3)**2) 

and we know: 
    y_pred1 = a*x1 + b
    y_pred2 = a*x2 + b 
    y_pred3 = a*x3 + b

Greedy approach

Create a function compute_mse. Compute mse for a = 1 and b = 2. Warning: X.shape is (100, 1) and y.shape is (100, ). Make sure that y_preds and y have the same shape before to compute y_preds-y.

def compute_mse(coefs, X, y):
    '''
    coefs is a list that contains a and b: [a,b]
    X is the features set 
    y is the target

    Returns a float which is the MSE 
    '''

    #TODO

    y_preds = 
    mse = 

    return mse

Create a grid of 640000 points that combines a and b with. Check that the grid contains 640000 points.
- a between -200 and 200, step= 0.5
- b between -200 and 200, step= 0.5
This is how to compute the grid with the combination of a and b:
```
aa, bb = np.mgrid[-200:200:0.5, -200:200:0.5]
grid = np.c_[aa.ravel(), bb.ravel()]
```
Compute the MSE for all points in the grid. If possible, parallelize the computations. It may be needed to use functools.partial to parallelize a function with many parameters on a list. Put the result in a variable named losses.

Use this chunk of code to plot the MSE in 2D:

aa, bb = np.mgrid[-200:200:.5, -200:200:.5]
grid = np.c_[aa.ravel(), bb.ravel()]
losses_reshaped = np.array(losses).reshape(aa.shape)

f, ax = plt.subplots(figsize=(8, 6))
contour = ax.contourf(aa,
                    bb,
                    losses_reshaped,
                    100,
                    cmap="RdBu",
                    vmin=0,
                    vmax=160000)
ax_c = f.colorbar(contour)
ax_c.set_label("MSE")

ax.set(aspect="equal",
    xlim=(-200, 200),
    ylim=(-200, 200),
    xlabel="$a$",
    ylabel="$b$")

The expected output is:

From the losses list, find the optimal value of a and b and plot the line in the scatter point of question 1.

In this example we computed 160 000 times the MSE. It is frequent to deal with 50 features, which requires 51 parameters to fit the Linear Regression. If we try this approach with 50 features we would need to compute 5.07e+132 MSE. Even if we reduce the scope and try only 5 values per coefficients we would have to compute the MSE 4.4409e+35 times. This approach is not scalable and that is why is not used to find optimal coefficients for Linear Regression.

Gradient Descent

In a nutshel, Gradient descent is an optimization algorithm used to minimize some function by iteratively moving in the direction of steepest descent as defined by the negative of the gradient. In machine learning, we use gradient descent to update the parameters (a and b) of our model. Parameters refer to the coefficients used in Linear Regression. Before to start implementing the questions, take the time to read the article. https://jairiidriss.medium.com/gradient-descent-algorithm-from-scratch-using-python-2b36c1548917. It explains the gradient descent and how to implement it. The "tricky" part is the computation of the derivative of the mse. You can admit the formulas of the derivatives to implement the gradient descent (d_theta_0 and d_theta_1 in the article).

Implement the gradient descent to find optimal a and b with learning rate = 0.1 and nbr_iterations=100.
Save the a and b through the iterations in a two dimensional numpy array. Add them to the plot of the previous part and observe a and b that converge towards the minimum. The plot should look like this:

9. Use Linear Regression from Scikit-learn. Compare the results.

Correction

This question is validated if the outputted plot looks like:

This question is validated if the output is: 11808.867339751561
This question is validated if grid.shape is (640000,2).

This question is validated if the 10 first values of losses are:

array([158315.41493175, 158001.96852692, 157689.02212209, 157376.57571726,
    157064.62931244, 156753.18290761, 156442.23650278, 156131.79009795,
    155821.84369312, 155512.39728829])

This question is validated if the outputted plot looks like

This question is validated if the point returned is array([42.5, 99. ]). It means that a= 42.5 and b=99.

This question is validated if the coefficients returned are

Coefficients (a): 42.61943031121358
Intercept (b): 99.18581814447936

This question is validated if the outputted plot is

This question is validated if the coefficients and intercept returned are:
```
Coefficients:  [42.61943029]
Intercept:  99.18581817296929
```

16 KiB Raw Blame History

W2D01 Piscine AI - Data Science

Table of Contents:

Introduction

Rules

Ressources

To start with Scikit-learn:

Machine learning methodology and algorithms:

Linear Regression

Train test split

Exercise 1 Scikit-learn estimator

Correction

Exercise 2 Linear regression in 1D

Correction

Exercise 3: Train test split

Correction

Exercise 4 Forecast diabetes progression

Correction

Exercise 5 Gradient Descent

Greedy approach

Gradient Descent

Correction

16 KiB

Raw Blame History