# Linear regression without scikit-learn¶

In this notebook, we introduce linear regression. Before presenting the available scikit-learn classes, we will provide some insights with a simple example. We will use a dataset that contains measurements taken on penguins.

Note

If you want a deeper overview regarding this dataset, you can refer to the Appendix - Datasets description section at the end of this MOOC.

```import pandas as pd

```
Flipper Length (mm) Body Mass (g)
0 181.0 3750.0
1 186.0 3800.0
2 195.0 3250.0
3 193.0 3450.0
4 190.0 3650.0

We will formulate the following problem: using the flipper length of a penguin, we would like to infer its mass.

```import seaborn as sns

feature_names = "Flipper Length (mm)"
target_name = "Body Mass (g)"
data, target = penguins[[feature_names]], penguins[target_name]

ax = sns.scatterplot(data=penguins, x=feature_names, y=target_name,
color="black", alpha=0.5)
ax.set_title("Flipper length in function of the body mass")
```
```Text(0.5, 1.0, 'Flipper length in function of the body mass')
``` Tip

The function `scatterplot` from searborn take as input the full dataframe and the parameter `x` and `y` allows to specify the name of the columns to be plotted. Note that this function returns a matplotlib axis (named `ax` in the example above) that can be further used to add elements on the same matplotlib axis (such as a title).

In this problem, penguin mass is our target. It is a continuous variable that roughly varies between 2700 g and 6300 g. Thus, this is a regression problem (in contrast to classification). We also see that there is almost a linear relationship between the body mass of the penguin and its flipper length. The longer the flipper, the heavier the penguin.

Thus, we could come up with a simple formula, where given a flipper length we could compute the body mass of a penguin using a linear relationship of the form `y = a * x + b` where `a` and `b` are the 2 parameters of our model.

```def linear_model_flipper_mass(flipper_length, weight_flipper_length,
intercept_body_mass):
"""Linear model of the form y = a * x + b"""
body_mass = weight_flipper_length * flipper_length + intercept_body_mass
return body_mass
```

Using the model we defined above, we can check the body mass values predicted for a range of flipper lengths. We will set `weight_flipper_length` to be 45 and `intercept_body_mass` to be -5000.

```import numpy as np

weight_flipper_length = 45
intercept_body_mass = -5000

flipper_length_range = np.linspace(data.min(), data.max(), num=300)
predicted_body_mass = linear_model_flipper_mass(
flipper_length_range, weight_flipper_length, intercept_body_mass)
```

We can now plot all samples and the linear model prediction.

```label = "{0:.2f} (g / mm) * flipper length + {1:.2f} (g)"

ax = sns.scatterplot(data=penguins, x=feature_names, y=target_name,
color="black", alpha=0.5)
ax.plot(flipper_length_range, predicted_body_mass)
_ = ax.set_title(label.format(weight_flipper_length, intercept_body_mass))
``` The variable `weight_flipper_length` is a weight applied to the feature `flipper_length` in order to make the inference. When this coefficient is positive, it means that penguins with longer flipper lengths will have larger body masses. If the coefficient is negative, it means that penguins with shorter flipper lengths have larger body masses. Graphically, this coefficient is represented by the slope of the curve in the plot. Below we show what the curve would look like when the `weight_flipper_length` coefficient is negative.

```weight_flipper_length = -40
intercept_body_mass = 13000

predicted_body_mass = linear_model_flipper_mass(
flipper_length_range, weight_flipper_length, intercept_body_mass)
```

We can now plot all samples and the linear model prediction.

```ax = sns.scatterplot(data=penguins, x=feature_names, y=target_name,
color="black", alpha=0.5)
ax.plot(flipper_length_range, predicted_body_mass)
_ = ax.set_title(label.format(weight_flipper_length, intercept_body_mass))
``` In our case, this coefficient has a meaningful unit: g/mm. For instance, a coefficient of 40 g/mm, means that for each additional millimeter in flipper length, the body weight predicted will increase by 40 g.

```body_mass_180 = linear_model_flipper_mass(
flipper_length=180, weight_flipper_length=40, intercept_body_mass=0)
body_mass_181 = linear_model_flipper_mass(
flipper_length=181, weight_flipper_length=40, intercept_body_mass=0)

print(f"The body mass for a flipper length of 180 mm "
f"is {body_mass_180} g and {body_mass_181} g "
f"for a flipper length of 181 mm")
```
```The body mass for a flipper length of 180 mm is 7200 g and 7240 g for a flipper length of 181 mm
```

We can also see that we have a parameter `intercept_body_mass` in our model. This parameter corresponds to the value on the y-axis if `flipper_length=0` (which in our case is only a mathematical consideration, as in our data, the value of `flipper_length` only goes from 170mm to 230mm). This y-value when x=0 is called the y-intercept. If `intercept_body_mass` is 0, the curve will pass through the origin:

```weight_flipper_length = 25
intercept_body_mass = 0

# redefined the flipper length to start at 0 to plot the intercept value
flipper_length_range = np.linspace(0, data.max(), num=300)
predicted_body_mass = linear_model_flipper_mass(
flipper_length_range, weight_flipper_length, intercept_body_mass)
```
```ax = sns.scatterplot(data=penguins, x=feature_names, y=target_name,
color="black", alpha=0.5)
ax.plot(flipper_length_range, predicted_body_mass)
_ = ax.set_title(label.format(weight_flipper_length, intercept_body_mass))
``` Otherwise, it will pass through the `intercept_body_mass` value:

```weight_flipper_length = 45
intercept_body_mass = -5000

predicted_body_mass = linear_model_flipper_mass(
flipper_length_range, weight_flipper_length, intercept_body_mass)
```
```ax = sns.scatterplot(data=penguins, x=feature_names, y=target_name,
color="black", alpha=0.5)
ax.plot(flipper_length_range, predicted_body_mass)
_ = ax.set_title(label.format(weight_flipper_length, intercept_body_mass))
``` In this notebook, we have seen the parametrization of a linear regression model and more precisely meaning of the terms weights and intercepts.