All it’s essential find out about Linear Regression is here (including an application in Python)

If you’re approaching Machine Learning, one in every of the primary models chances are you’ll encounter is Linear Regression. It’s probably the simplest model to know, but don’t underestimate it: there are quite a lot of things to know and master.

In the event you’re a beginner in Data Science or an aspiring Data Scientist, you’re probably facing some difficulties because there are quite a lot of resources on the market, but are fragmented. I understand how you’re feeling, and because of this I created this entire guide: I would like to offer you all of the knowledge you would like without trying to find the rest.

So, if you must have complete knowledge of Linear Regression this text is for you. You may study it deeply and re-read it every time you would like it essentially the most. Also, consider that, to cover this topic, we’ll need some knowledge generally related to regression evaluation: we’ll cover it in deep.

And…you’ll excuse me if I’ll link a resource you’ll need: prior to now, I’ve created an article on some topics related to Linear Regression so, to have a whole overview, I counsel you to read it (I’ll link later once we’ll need it).

Table of Contents:
What will we mean by "regression evaluation"?
Understanding correlation
The difference between correlation and regression
The Linear Regression model
Assumptions for the Linear Regression model
Finding the road that most closely fits the information
Graphical methods to validate your model
An example in Python

Here we’re studying Linear Regression, but what will we mean by “regression evaluation”? Paraphrasing from Wikipedia:

Regression evaluation is a mathematical technique used to search out a functional relationship between a dependent variable and a number of independent variable(s).

In other words, we all know that in mathematics we are able to define a function like so: y=f(x). Generally, y is known as the dependent variable and x the independent. So, we express y in relationship with x, using a certain function f. The aim of regression evaluation is, then, to search out the function f .

Now, this seems easy but shouldn’t be. And I do know you understand it. And the rationale why shouldn’t be easy is:

• We all know x and y. For instance, if we’re working with tabular data (with Pandas, for instance) x are the features and y is the label.
• Unfortunately, the information rarely follow a really clear path. So our job is to search out the perfect function f that approximates the connection between x and y.

So, let me summarize it: regression evaluation goals to search out an estimated relationship (a very good one!) between the dependent and the independent variable(s).

Now, let’s visualize why this process could also be difficult. Consider the next code and its final result:

import numpy as np
import matplotlib.pyplot as plt
# Create random linear data
a = 130
x = 6*np.random.rand(a,1)-3
y = 0.5*x+5+np.random.rand(a,1)
# Labels
plt.xlabel('x')
plt.ylabel('y')
# Plot a scatterplot
plt.scatter(x,y)

Now, tell me: can the connection between x and y be a line? So…can this data be approximated by a line? Like the next, for instance:

Stop reading for a moment and take into consideration that.

Well, it could. And the way in regards to the following one?

Well, even this might! So, what’s the perfect one? And why not one other one?

That is the aim of regression: to search out the best-estimated function that may approximate the given data. And it does so using some methodologies: we’ll cover them later in this text. We’ll apply them to the Linear Regression model but a few of them might be used with another regression technique. Don’t worry: I’ll be very specific so that you don’t get confused.

Quoting from Wikipedia:

In statistics, correlation is any statistical relationship, whether causal or not, between two random variables. Although within the broadest sense, “correlation” may indicate any variety of association, in statistics it often refers back to the degree to which a pair of variables are linearly related.

In other words, correlation is a statistical measure that expresses the linear relationship between variables.

We will say that two variables are correlated if each value of the primary variable corresponds to a price for the second variable, following a path. If two variables are highly correlated, the trail can be linear, since the correlation describes the linear relation between the variables.

The mathematics behind the correlation

This can be a comprehensive guide, as promised. So, I would like to cover the mathematics behind the correlation, but don’t worry: we’ll make it easy so which you can understand it even if you happen to’re not specialized in math.

We generally check with the correlation coefficient, also often known as the Pearson correlation coefficient. This offers an estimate of the correlation between two variables. Suppose we’ve two variables, a and b and so they can reach n values. We will calculate the correlation coefficient as follows:

Where we’ve:

• the mean value of a(however it applies to each variables, a and b):

So, putting all of it together:

As chances are you’ll know:

• the mean is the sum of all of the values of a variable divided by the variety of values. So, for instance, if our variable a has the values 1,3,7,13,25 the mean value of a shall be:

• the standard deviation is an index of statistical dispersion and is an estimate of the variability of a variable (or of a population, as we’d say in statistics). It’s one in every of the ways to specific the dispersion of information around an index; within the case of the correlation coefficient, the index around which we calculate the dispersion is the mean (see the above formula). The more the usual deviation is high, the more the dispersion across the mean is high: nearly all of the information points are distant from the mean value.

Numerically speaking, we’ve to do not forget that the worth of the correlation coefficient is constrained between 1 and -1; which means that:

• if r=1: the variables are highly positively correlated; it signifies that if one variable increases its value, the opposite does the identical, following a linear path.
• if r=-1: the variables are highly negatively correlated; it signifies that if one variable increases its value, the opposite one decreases its value, following a linear path.
• if r=0: there isn’t any correlation between the variables.

Finally, two variables are generally considered highly correlated if r>0.75.

Correlation shouldn’t be causation

We’d like to have very clear in our mind the indisputable fact that “correlation shouldn’t be causation”; we intend to make an example that is likely to be useful to recollect it.

It’s a hot summer; we don’t just like the high temperatures in our city, so we go to the mountain. Luckily, we get to the mountain top, measure the temperature and find it’s lower than in our city. We get a bit of suspicious, and we resolve to go to the next mountain, finding that the temperature is even lower than the one on the previous mountain.

We try mountains with different heights, measure the temperature, and plot a graph; we discover that with the peak of the mountain increasing, the temperature decreases, and we are able to see a linear trend.

What does it mean? It signifies that the temperature is said to the peak of the mountains, with a linear path: so there’s a correlation between the decrease in temperature and the peak (of the mountains). It doesn’t mean the peak of the mountain caused the decrease in temperature; in reality, if we get to the identical height, at the identical latitude, with a hot air balloon we’d measure the identical temperature.

The correlation matrix

So, how will we calculate the correlation coefficient in Python? Well, we generally calculate the correlation matrix. Suppose we’ve two variables, X and y; we store them in a knowledge frame called df and we are able to plot the correlation matrix using seaborn like so:

import pandas as pd
import numpy as np
import seaborn as sns
import matplotlib.pyplot as plt
# Create data
x = np.array([1, 1, 2, 3, 4, 4, 5, 6, 7, 7, 8, 9])
y = np.array([13, 14, 17, 12, 23, 24, 25, 25, 24, 28, 32, 33])
# Create the dataframe 
df = pd.DataFrame({'x':x, 'y':y})
# Plot heat map for correlation coefficient
sns.heatmap(df.corr(), annot=True, fmt="0.2")

If we’ve a 0 correlation coefficient, it signifies that the information points don’t are inclined to increase or decrease following a linear path, because we’ve no correlation.

Allow us to have a have a look at some plots of correlation coefficients with different values (image from Wikipedia here):

As we are able to see, when the correlation coefficient is the same as 1 or -1 the tendency of the information points is clearly to be along a line. But, because the correlation coefficient deviates from the 2 extreme values, the distribution of the information points deviates from a linear path. Finally, for the correlation coefficient of 0, the distribution of the information might be anything.

So, once we get a correlation coefficient of 0 we are able to’t say anything in regards to the distribution of the information, but we are able to investigate it (if needed) with a regression evaluation.

So, correlation and regression are linked but are different:

• Correlation analyzes the tendency of variables to be linearly distributed.
• Regression is the study of the connection between variables.

Now we have two sorts of Linear Regression models: the Easy and the Multiple ones. Let’s see them each.

The Easy Linear Regression model

The goal of the Easy Linear Regression is to model the connection between a single feature and a continuous label. That is the mathematical equation that describes this ML model:

y = wx + b

The parameter b (also called “bias”) represents the y-axis intercept (is the worth of ywhen X=0), and w is the load coefficient. Our goal is to learn the load w that describes the connection between x and y. This weight will later be used to predict the response for brand new values of x.

Let’s consider a practical example:

import numpy as np
import matplotlib.pyplot as plt
# Create data
x = np.array([1, 1, 2, 3, 4, 4, 5, 6, 7, 7, 8, 9])
y = np.array([13, 14, 17, 12, 23, 24, 25, 25, 24, 28, 32, 33])
# Show scatterplot
plt.scatter(x, y)

The query is: can this data distribution be approximated with a line? Well, we could create something like that:

import numpy as np
import matplotlib.pyplot as plt
# Create data
x = np.array([1, 1, 2, 3, 4, 4, 5, 6, 7, 7, 8, 9])
y = np.array([13, 14, 17, 12, 23, 24, 25, 25, 24, 28, 32, 33])
# Create basic scatterplot
plt.plot(x, y, 'o')
# Obtain m (slope) and b (intercept) of a line
m, b = np.polyfit(x, y, 1)
# Add linear regression line to scatterplot 
plt.plot(x, m*x+b)
# Labels
plt.xlabel('x variable')
plt.ylabel('y variable')

Well, as in the instance we’ve seen above, it might be a line however it might be a general curve.

And, in a moment we’ll see how we are able to say if the information distribution might be higher described by a line or by a general curve.

The Multiple Linear Regression model

Since reality is complex, the standard cases we’ll face are related to the Multiple Linear Regression case. We mean that the feature x shouldn’t be a single one: we’ll have multiple features. For instance, if we work with tabular data, a knowledge frame with 9 columns has 8 features and 1 label: which means that our problem is eight-dimensional.

As we are able to understand, this case may be very complicated to visualise and the equation of the road needs to be expressed with vectors and matrices, becoming:

So, the equation of the road becomes the sum of all of the weights (w) multiplied by the independent variable (x) and it may well even be written because the product of two matrices.

Now, to use the Linear Regression model, our data should respect some assumptions. These are:

1. Linearity: the connection between the dependent variable and independent variables must be linear. Which means that a change within the independent variable should end in a proportional change within the dependent variable, following a linear path.
2. Independence: the observations within the dataset must be independent of one another. Which means that the worth of 1 commentary mustn’t depend upon the worth of one other commentary.
3. Homoscedasticity: the variance of the residuals must be constant across all levels of the independent variable. In other words, the spread of the residuals must be roughly the identical across all levels of the independent variable.
4. Normality: the residuals must be normally distributed. In other words, the distribution of the residuals must be a standard (or bell-shaped) curve.
5. No multicollinearity: the independent variables mustn’t be highly correlated with one another. If two or more independent variables are highly correlated, it may well be difficult to differentiate the person effects of every variable on the dependent variable.

Unfortunately, testing all these hypotheses shouldn’t be at all times possible, especially within the case of the Multiple Linear Regression model. Anyway, there’s a method to test all of the hypotheses. It’s called the p-value test, and possibly you heard of that before. Anyway, we won’t cover this test here for 2 reasons:

1. It’s a general test, not specifically related to the Linear Regression model. So, it needs a selected treatment in a dedicated article.
2. I’m one in every of those (possibly one in every of the few) who believes that calculating the p-value shouldn’t be at all times a must when we’d like to research data. For that reason, I’ll create in the long run a dedicated article on this controversial topic. But only for the sake of curiosity, since I’m an engineer I actually have a really practical approach, and I like applied mathematics. I wrote an article on this topic here:

So, above we were reasoning which one in every of the next might be the perfect fit:

To know if the perfect model is the left one (the road) or the suitable one (a general curve) we proceed as follows:

• We split the information we’ve into the training and the test set.
• We validate each models on each sets, testing how well our models generalize their learning.

We won’t cover the polynomial model here (useful for general curves), but consider that there are two approaches to validate ML models:

• The analytical one.
• The graphical one.

Generally speaking, we’ll use each to get a greater understanding of the performance of the model. Anyway, generalizing signifies that our ML model learns from the training set and applies appropriately its learning to the test set. If it doesn’t, we try one other ML model. Here’s the method:

Which means that an ML model generalizes well when it has good performances on each the training and the test set.

I’ve discussed the analytical method to validate an ML model within the case of linear regression in the next article:

I counsel you to read it because we’ll use some metrics discussed there in the instance at the tip of this text.

After all, the metrics discussed might be applied to any ML model within the case of a regression problem. But you’re lucky: I’ve used the linear model for instance.

The graphical ways to validate an ML model within the case of a regression problem are discussed in the subsequent paragraph.

Let’s see three graphical ways to validate our ML models.

1. The residual evaluation plot

This method is particular to the Linear Regression model and consists in visualizing how the residuals are distributed. Here’s what we expect:

To plot this we are able to use the built-in function sns.residplot() in Seaborn (here’s the documentation).

A plot like that is sweet because we would like to see randomly distributed data points along the horizontal axis. One among the assumptions of the linear regression model, in reality, is that the residuals have to be normally distributed (assumption n°4 listed above). If the residuals are normally distributed, it signifies that the errors of the observed values from the expected ones are randomly distributed around zero, with no clear pattern or trend; and this is precisely the case in our plot. So, in these cases, our ML model could also be a very good one.

As a substitute, if there’s a specific pattern in our residual plot, our model shouldn’t be good for our ML problem. For instance, consider the next:

On this case, we are able to see that there’s a parabolic trend: which means that our model (the Linear model) shouldn’t be good to resolve our ML problem.

2. The actual vs. predicted values plot

One other plot we may use to validate our ML model is the actual vs. predicted plot. On this case, we plot a graph having the actual values on the horizontal axis and the expected values on the vertical axis. The goal is to search out the information points distributed as much as possible to a line, within the case of Linear Regression. We will even use the tactic within the case of a polynomial regression: on this case, we’d expect the information distributed as much as possible to a generic curve.

Suppose we’ve a result as follows:

The above graph shows that the expected data points are distributed along a line. It shouldn’t be an ideal linear distribution, so the linear model is probably not ideal.

If, for our specific problem, we’vey_train (the label on the training set) and we’ve calculated y_train_pred (the prediction on the training set), we are able to plot the next graph like so:

import matplotlib.pyplot as plt
# Scatterplot of y_train and y_train_pred
plt.scatter(y_train, y_train_pred)
plt.plot(y_test, y_test, color='r') # Plot the road
# Labels
plt.title('ACTUAL VS PREDICTED VALUES')
plt.xlabel('ACTUAL VALUES')
plt.ylabel('PREDICTED VALUES')

3. The Kernel Density Estimation (KDE) plot

The last graph we would like to speak about to validate our ML models is the Kernel Density Estimation (KDE) plot. This can be a general method and might be used to validate each regression and classification models.

The KDE is the applying of a kernel smoother for probability density estimation. A kernel smoother is a statistical method that’s used to estimate a function because the weighted average of the neighbor observed data. The kernel defines the load, giving the next weight to closer data points.

To know the usefulness of a smoother function, see the graph below:

It is useful to approximate our data points with a smoothing function if we would like to match two quantities. Within the case of an ML problem, in reality, we typically prefer to see the comparison between the actual labels and the labels predicted by our model, so we use the KDE to match two smoothed functions.

Let’s say we’ve predicted our labels using a linear regression model. We would like to match the KDE for our training set’s actual and predicted labels. We will accomplish that with Seaborn invoking the tactic sns.kdeplot() (here’s the documentation).

Suppose we’ve the next result:

As we are able to see, the comparison between the actual and the expected label is straightforward to do, since we’re comparing two smoothed functions; in a case like that, our model is sweet since the curves are very similar.

The truth is, what we expect from a “good” ML model are:

1. The curves are much like bell curves, as much as possible.
2. The 2 curves are similar between them, as much as possible.

Now, let’s apply all of the things we’ve learned to date here. We’ll use the famous “Ames Housing” dataset, which is ideal for our scopes.

This dataset has 80 features, but for simplicity, we’ll work with only a subset of them that are:

• Overall Qual: it’s the rating of the general material and finish of the home on a scale from 1 (bad) to 10 (excellent).
• Overall Cond: it’s the rating of the general condition of the home on a scale from 1 (bad) to 10 (excellent).
• Gr Liv Area: it’s the above-ground living area, measured in squared feet.
• Total Bsmt SF: it’s the overall basement area, measured in squared feet.
• SalePrice: it’s the sale price, in USD $.

We’ll consider our SalePrice column because the goal (label) variable, and the opposite columns because the features.

Exploratory Data Evaluation EDA

Let’s import our data, create a subset with the mentioned features, and display some statistics:

import pandas as pd
# Define the columns
columns = ['Overall Qual', 'Overall Cond', 'Gr Liv Area',
'Total Bsmt SF', 'SalePrice']
# Create dataframe
df = pd.read_csv('http://jse.amstat.org/v19n3/decock/AmesHousing.txt',
sep='t', usecols=columns)
# Show statistics
df.describe()

A vital commentary here is that the mean values for all labels have a special range (the Overall Qual mean value is 6.09 while Gr Liv Area mean value is 1499.69). This tells us a vital fact: we’ve to scale the features.

Data preparation

What does “features scaling” mean?

Scaling a feature implies that the feature range is scaled between 0 and 1 or between 1 and -1. There are two typical methods to scale the features:

• Mean normalization: Mean normalization is a technique of scaling numeric data in order that it has a minimum value of zero and a maximum value of every one the values are normalized across the mean value. Suppose c is a price reached by our feature; to scale across the mean (c′ is the brand new value of c after the normalization process):

Let’s see an example in Python:

import numpy as np
# Create a listing of numbers
data = [1, 2, 3, 4, 5]
# Find min and max values
data_min = min(data)
data_max = max(data)
# Normalize the information
data_normalized = [(x - data_min) / (data_max - data_min) for x in data]
# Print the normalized data
print(f'normalized data: {data_normalized}')
>>>
normalized data: [0.0, 0.25, 0.5, 0.75, 1.0]

• Standardization (or z-score normalization): This method transforms a variable in order that it has a mean of zero and a regular deviation of 1. The formula is the next (c′c’c′ is the brand new value of ccc after the normalization process):

Let’s see an example in Python:

import numpy as np
# Original data
data = [1, 2, 3, 4, 5]
# Calculate mean and standard deviation
mean = np.mean(data)
std = np.std(data)
# Standardize the information
data_standardized = [(x - mean) / std for x in data]
# Print the standardized data
print(f'standardized values: {data_standardized}')
print(f'mean of standardized values: {np.mean(data_standardized)}')
print(f'std. dev. of standardized values: {np.std(data_standardized): .2f}')
>>>
standardized values: [-1.414213562373095, -0.7071067811865475, 0.0, 0.7071067811865475, 1.414213562373095]
mean of standardized values: 0.0
std. dev. of standardized values:  1.00

As we are able to see, the normalized data have a mean of 0 and a regular deviation of 1, as we wanted. The excellent news is that we are able to use the library scikit-learn to standardize the features, and we will do it in a moment.

Features scaling is a vital thing to do when working on an ML problem, for an easy reason:

• If we perform exploratory data evaluation with features that will not be scaled, when calculating the mean values (for instance, through the calculation of the coefficient of correlation) we’ll get numbers which might be very different from one another. If we take a have a look at the statistics we’ve got above once we’ve invoked the df.describe() method, we are able to see that, for every column, we get a really different value of the mean. If we scale or normalize the features, as an alternative, we’ll get 0s, 1s, and -1s: and this may help us mathematically.

Now, this dataset has some NaN values. We won’t show it for brevity (try it on your individual), but we’ll remove them. Also, we’ll calculate the correlation matrix:

import seaborn as sns
import matplotlib.pyplot as plt
import numpy as np
# Drop NaNs from dataframe
df = df.dropna(axis=0)
# Apply mask
mask = np.triu(np.ones_like(df.corr()))
# Heat map for correlation coefficient
sns.heatmap(df.corr(), annot=True, fmt="0.1", mask=mask)

So, with np.triu(np.ones_like(df.corr())) we’ve created a mask that it’s useful to display a triangular correlation matrix, which is more readable (especially when we’ve rather more features than on this case).

So, there’s a moderate 0.6 correlation between Total Bsmt SF and SalePrice, quite a high 0.7 correlation between Gr Liv Area and SalePrice, and a high correlation 0.8 between Overall Qual and SalePrice; Also, there’s a moderate correlation between Overall Qual and Gr Liv Area 0.6 and 0.5 between Overall Qual and Total Bsmt SF.

Here there’s no multicollinearity, so no features are highly correlated with one another (so, our features satisfy the hypothesis n°5 listed above). If we’d found some highly correlated features, we could delete them because two highly correlated features have the identical effect on the label (this is applicable to each general ML model: if two features are highly correlated, we are able to drop one in every of the 2).

Finally, we subdivide the information frame dfinto X ( the features) and y(the label) and scale the features:

from sklearn.preprocessing import StandardScaler
# Define the features
X = df.iloc[:,:-1]
# Define the label
y = df.iloc[:,-1]
# Scale the features
scaler = StandardScaler() # Call the scaler
X = scaler.fit_transform(X) # Fit the features to scale them

Fitting the linear regression model

Now we’ve to separate the features X into the training and the test set and we’re fitting them with the Linear Regression model. Then, we calculate R² for each sets:

from sklearn.model_selection import train_test_split
from sklearn.linear_model import LinearRegression
from sklearn import metrics
# Split
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
# Fit the LR model
reg = LinearRegression().fit(X_train, y_train)
# Calculate R^2
coeff_det_train = reg.rating(X_train, y_train)
coeff_det_test = reg.rating(X_test, y_test)
# Print metrics
print(f" R^2 for training set: {coeff_det_train}")
print(f" R^2 for test set: {coeff_det_test}")
>>>
R^2 for training set:  0.77
R^2 for test set:  0.73

Notes:
1) your results might be barely different as a consequence of the stocastical
nature of the ML models.
2) here we are able to see generalization on motion: 
we fitted the Linear Regression model to the train set with
reg = LinearRegression().fit(X_train, y_train).
The, we have calculated R^2 on the training and test sets with:
coeff_det_train = reg.rating(X_train, y_train)
coeff_det_test = reg.rating(X_test, y_test
In other words: we do not fit the information to the test set.
We fit the information to the training set and we calculate the scores
and predictions (see next snippet of code with KDE) on each sets
to see the generalization of our modelon latest unseen data
(the information of the test set).

So we get R² of 0.77 on the training test and 0.73 on the test set that are quite good, suggesting the Linear model is a very good one to resolve this ML problem.

Let’s see the KDE plots for each sets:

# Calculate predictions
y_train_pred = reg.predict(X_train) # train set
y_test_pred = reg.predict(X_test) # test set
# KDE train set
ax = sns.kdeplot(y_train, color='r', label='Actual Values') #actual values
sns.kdeplot(y_train_pred, color='b', label='Predicted Values', ax=ax) #predicted values
# Show title
plt.title('Actual vs Predicted values')
# Show legend
plt.legend()

# KDE test set
ax = sns.kdeplot(y_test, color='r', label='Actual Values') #actual values
sns.kdeplot(y_test_pred, color='b', label='Predicted Values', ax=ax) #predicted values
# Show title
plt.title('Actual vs Predicted values')
# Show legend
plt.legend()

Whatever the indisputable fact that we’ve obtained an R² of 0.73 on the test set which is sweet (but remember: the upper, the higher), this plot shows us that the linear model is indeed a very good model to resolve this ML problem. This is the reason I like the KDE plot: is a really powerful tool, as we are able to see.

Also, this shows why shouldn’t depend on only one method to validate our ML model: a mix of 1 analytical method with one graphical one generally gives us the suitable insights to determine whether to vary our ML model or not. On this case, the Linear Regression model is ideal to make predictions.

I hope you’ll find useful this text. I understand it’s very long, but I wanted to offer you all of the knowledge you would like on this topic, so which you can return to it every time you would like it essentially the most.

A few of the things we’ve discussed listed below are general topics, while others are specific to the Linear Regression model. Let’s summarize them:

• The definition of regression is, after all, a general definition.
• Correlation is usually known as the Linear model. The truth is, as we said before, correlation is the tendency of two variables to be linearly dependent. Nevertheless, there are methods to define non-linear correlations, but we leave them for other articles (but, as knowledge for you: just consider that they exist).
• We’ve discussed the Easy and the Multiple Linear Regression models with their assumptions (the assumptions apply to each models).
• When talking about how you can find the road that most closely fits the information, we’ve referred to the article “Mastering the Art of Regression Evaluation: 5 Key Metrics Every Data Scientist Should Know”. Here, we discover all of the metrics to know to resolve a regression evaluation. So, it is a generical topic that applies to any regression model, including the Linear one, after all.
• We’ve shown three methods to validate our ML models: 1) The residual evaluation plot: which applies to Linear Regression models, 2) The actual vs. predicted values plot: which might be applied to Linear and Polynomial models, 3) the KDE plot: this might be applied to any ML model, even within the case of a classification problem

Finally, I would like to remind you that we’ve spent a few lines stressing the indisputable fact that we are able to avoid using p-values to check the hypotheses of our ML models. I’m writing an article on this topic very soon, but, as you possibly can see, the KDE has shown us that our Linear model is sweet to resolve this ML problem, and we haven’t validated our hypothesis with p-values.

Up to now in this text, we’ve used some plots. You may clone this repo I’ve created so which you can import the code and use it to simply plot the graphs. If you might have some difficulties, you discover examples of usages on my projects on GitHub. If you might have another difficulties, you possibly can contact me and I’ll enable you.

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