New to machine learning? Dive into logistic regression in machine learning with us, a foundational technique in predictive modeling that bridges the gap between simple linear models and complex neural networks in deep learning. In this article, we will demystify logistic regression using Python, explore its role as a linear model, discuss its application alongside neural networks, and understand how regularization techniques enhance its predictive power. Whether you’re a beginner or looking to deepen your understanding, join us as we explore the intersection of logistic regression with Python, deep learning, linear models, neural networks, and regularization.
In this article, you will get understanding about the logistic regression, what is logistic regression and logistic regression model, logistic regression machine learning you get to know in this article. At the end of this article you will clear your concepts about logistic regression.
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Logistic regression is the appropriate regression analysis to conduct when the dependent variable is dichotomous (binary). Like all regression analyses, logistic regression is a predictive analysis. It is used to describe data and to explain the relationship between one dependent binary variable and one or more nominal, ordinal, interval or ratio-level independent variables.
I found this definition on google and now we’ll try to understand it. Logistic Regression is another statistical analysis method borrowed by Machine Learning. It is used when our dependent variable is dichotomous or binary. It just means a variable that has only 2 outputs, for example, A person will survive this accident or not, The student will pass this exam or not. The outcome can either be yes or no (2 outputs). This regression technique is similar to linear regression and can be used to predict the Probabilities for classification problems.
Let us explore types of logistic regression.
Binary logistic regression is used to predict the probability of a binary outcome, such as yes or no, true or false, or 0 or 1. For example, it could be used to predict whether a customer will churn or not, whether a patient has a disease or not, or whether a loan will be repaid or not.
Multinomial logistic regression is used to predict the probability of one of three or more possible outcomes, such as the type of product a customer will buy, the rating a customer will give a product, or the political party a person will vote for.
It is used to predict the probability of an outcome that falls into a predetermined order, such as the level of customer satisfaction, the severity of a disease, or the stage of cancer.
If you have this doubt, then you’re in the right place, my friend. After reading the definition of logistic regression we now know that it is only used when our dependent variable is binary and in linear regression this dependent variable is continuous.
The second problem is that if we add an outlier in our dataset, the best fit line in linear regression shifts to fit that point.
Now, if we use linear regression to find the best fit line which aims at minimizing the distance between the predicted value and actual value, the line will be like this:
Here the threshold value is 0.5, which means if the value of h(x) is greater than 0.5 then we predict malignant tumor (1) and if it is less than 0.5 then we predict benign tumor (0). Everything seems okay here but now let’s change it a bit, we add some outliers in our dataset, now this best fit line will shift to that point. Hence the line will be somewhat like this:
Do you see any problem here? The blue line represents the old threshold, and the yellow line represents the new threshold, which is maybe 0.2. To keep our predictions right, we had to lower our threshold value. Hence, we can say that linear regression is prone to outliers. Now, if h(x)h(x)h(x) is greater than 0.2, only this regression will give correct outputs. Another problem with linear regression is that the predicted values may be out of range. We know that probability can be between 0 and 1, but if we use linear regression, this probability may exceed 1 or go below 0. To overcome these problems, we use Logistic Regression, which converts this straight best-fit line in linear regression to an S-curve using the sigmoid function, which will always give values between 0 and 1. How this works and the math behind it will be covered in a later section.
If you want to know the difference between logistic regression and linear regression then you refer to this article.
Logistic regression is a statistical method commonly used to analyze data with binary outcomes (yes/no, 1/0) and identify the relationship between those outcomes and independent variables. Here are some key assumptions for logistic regression:
You must be wondering how logistic regression squeezes the output of linear regression between 0 and 1. If you haven’t read my article on Linear Regression then please have a look at it for a better understanding.
Well, there’s a little bit of math included behind this and it is pretty interesting trust me.
Let’s start by mentioning the formula of logistic function:
How similar it is too linear regression? If you haven’t read my article on Linear Regression, then please have a look at it for a better understanding.
We all know the equation of the best fit line in linear regression is:
Let’s say instead of y we are taking probabilities (P). But there is an issue here, the value of (P) will exceed 1 or go below 0 and we know that range of Probability is (0-1). To overcome this issue we take “odds” of P:
Do you think we are done here? No, we are not. We know that odds can always be positive which means the range will always be (0,+∞ ). Odds are nothing but the ratio of the probability of success and probability of failure. Now the question comes out of so many other options to transform this why did we only take ‘odds’? Because odds are probably the easiest way to do this, that’s it.
The problem here is that the range is restricted and we don’t want a restricted range because if we do so then our correlation will decrease. By restricting the range we are actually decreasing the number of data points and of course, if we decrease our data points, our correlation will decrease. It is difficult to model a variable that has a restricted range. To control this we take the log of odds which has a range from (- ∞,+ ∞ ).
If you understood what I did here then you have done 80% of the maths. Now we just want a function of P because we want to predict probability right? not log of odds. To do so we will multiply by exponent on both sides and then solve for P.
Now we have our logistic function, also called a sigmoid function. The graph of a sigmoid function is as shown below. It squeezes a straight line into an S-curve.
Linear regression and logistic regression, while both workhorses in machine learning, serve distinct purposes. The core difference lies in their target predictions. Linear regression excels at predicting continuous values along a spectrum. Imagine predicting house prices based on size and location – the resulting output would be a specific dollar amount, a continuous value on the price scale.
Logistic regression, on the other hand, deals with categories. It doesn’t predict a specific value but rather the likelihood of something belonging to a particular class. For instance, classifying emails as spam (category 1) or not spam (category 0). The output here would be a probability between 0 (not likely spam) and 1 (very likely spam). This probability is then used to assign an email to a definitive category (spam or not spam) based on a chosen threshold.
In simpler terms, linear regression answers “how much” questions, providing a specific value on a continuous scale. Logistic regression tackles “yes or no” scenarios, giving the probability of something belonging to a certain category.
In linear regression, we use the Mean squared error which was the difference between y_predicted and y_actual and this is derived from the maximum likelihood estimator. The graph of the cost function in linear regression is like this:
In logistic regression Yi is a non-linear function (Ŷ=1/1+ e -z ). If we use this in the above MSE equation then it will give a non-convex graph with many local minima as shown.
The problem here is that this cost function will give results with local minima, which is a big problem because then we’ll miss out on our global minima and our error will increase.
In order to solve this problem, we derive a different cost function for logistic regression called log loss which is also derived from the maximum likelihood estimation method.
In the next section, we’ll talk a little bit about the maximum likelihood estimator and what it is used for. We’ll also try to see the math behind this log loss function.
Logistic regression predicts yes/no outcomes (like email open). It analyzes data (age, email history) to estimate the chance (0-1) of an event. A sigmoid function turns this into a probability. We can then set a threshold (e.g. 0.5) to classify (open/not open). It’s useful because it’s easy to understand and interpret.sharemore_vert
The primary objective of Maximum Likelihood Estimation (MLE) in machine learning, particularly in the context of logistic regression, is to identify parameter values that maximize the likelihood function. This function represents the joint probability density function (pdf) of our sample observations. In essence, it involves multiplying the conditional probabilities for observing each example given the distribution parameters. In the realm of logistic regression in machine learning, this process aims to discover parameter values such that, when plugged into the model for P(x), it produces a value close to one for individuals with a malignant tumor and close to zero for those with a benign tumor.
Let’s start by defining our likelihood function. We now know that the labels are binary which means they can be either yes/no or pass/fail etc. We can also say we have two outcomes success and failure. This means we can interpret each label as Bernoulli random variable.
A random experiment whose outcomes are of two types, success S and failure F, occurring with probabilities p and q respectively is called a Bernoulli trial. If for this experiment a random variable X is defined such that it takes value 1 when S occurs and 0 if F occurs, then X follows a Bernoulli Distribution.
Where P is our sigmoid function
where σ( θ^T*x^i ) is the sigmoid function. Now for n observations,
We need a value for theta which will maximize this likelihood function. To make our calculations easier we multiply the log on both sides. The function we get is also called the log-likelihood function or sum of the log conditional probability
In machine learning, it is conventional to minimize a loss(error) function via gradient descent, rather than maximize an objective function via gradient ascent. If we maximize this above function then we’ll have to deal with gradient ascent to avoid this we take negative of this log so that we use gradient descent. We’ll talk more about gradient descent in a later section and then you’ll have more clarity. Also, remember,
The negative of this function is our cost function and what do we want with our cost function? That it should have a minimum value. It is common practice to minimize a cost function for optimization problems; therefore, we can invert the function so that we minimize the negative log-likelihood (NLL). So in logistic regression, our cost function is:
Here y represents the actual class and log(σ(θ^T*x^i) ) is the probability of that class.
If we combine both the graphs, we will get a convex graph with only 1 local minimum and now it’ll be easy to use gradient descent here:
The red line here represents the 1 class (y=1), the right term of cost function will vanish. Now if the predicted probability is close to 1 then our loss will be less and when probability approaches 0, our loss function reaches infinity.
The black line represents 0 class (y=0), the left term will vanish in our cost function and if the predicted probability is close to 0 then our loss function will be less but if our probability approaches 1 then our loss function reaches infinity.
This cost function is also called log loss. It also ensures that as the probability of the correct answer is maximized, the probability of the incorrect answer is minimized. Lower the value of this cost function higher will be the accuracy.
In this section, we will try to understand how we can utilize Gradient Descent to compute the minimum cost.
Gradient descent changes the value of our weights in such a way that it always converges to minimum point or we can also say that, it aims at finding the optimal weights which minimize the loss function of our model. It is an iterative method that finds the minimum of a function by figuring out the slope at a random point and then moving in the opposite direction.
The intuition is that if you are hiking in a canyon and trying to descend most quickly down to the river at the bottom, you might look around yourself 360 degrees, find the direction where the ground is sloping the steepest, and walk downhill in that direction.
At first gradient descent takes a random value of our parameters from our function. Now we need an algorithm that will tell us whether at the next iteration we should move left or right to reach the minimum point. The gradient descent algorithm finds the slope of the loss function at that particular point and then in the next iteration, it moves in the opposite direction to reach the minima. Since we have a convex graph now we don’t need to worry about local minima. A convex curve will always have only 1 minima.
We can summarize the gradient descent algorithm as:
Here alpha is known as the learning rate. It determines the step size at each iteration while moving towards the minimum point. Usually, a lower value of “alpha” is preferred, because if the learning rate is a big number then we may miss the minimum point and keep on oscillating in the convex curve.
Now the question is what is this derivative of cost function? How do we do this? Don’t worry, In the next section we’ll see how we can derive this cost function w.r.t our parameters.
Before we derive our cost function we’ll first find a derivative for our sigmoid function because it will be used in derivating the cost function.
Now, we will derive the cost function with the help of the chain rule as it allows us to calculate complex partial derivatives by breaking them down.
Hence the derivative of our cost function is:
Now since we have our derivative of the cost function, we can write our gradient descent algorithm as:
If the slope is negative (downward slope) then our gradient descent will add some value to our new value of the parameter directing it towards the minimum point of the convex curve. Whereas if the slope is positive (upward slope) the gradient descent will minus some value to direct it towards the minimum point.
This tutorial has provided a comprehensive overview of logistic regression, focusing on its application and implementation using scikit-learn ( sklearn ). Logistic regression is a powerful tool for handling categorical variables and predicting binary outcomes. Unlike decision trees, which create non-linear decision boundaries, logistic regression uses a linear relationship transformed by the exponential function through an activation function. We emphasized the importance of splitting data into a training set and a test set for proper model validation. By following this tutorial, you should now have a solid understanding of how to implement logistic regression in Python using scikit-learn and how it compares to other machine learning models like decision trees.
Hope you like the article, and getting clear on topics of logistic regression model, logistic regression machine learning, what is logistic regression and logistic regression model and how we have tell full about the logistic regression explained it fully.
A. Logistic regression, also known as logistics regression, is used for categorical outcomes, while linear regression is used for continuous outcomes.
Q2. What is Better than Logistics Regression?A. There are a number of machine learning algorithms that can outperform logistic regression on certain tasks. For example, random forests and gradient-boosting machines can often achieve higher accuracy on classification tasks. However, logistic regression is still a very popular algorithm due to its simplicity, interpretability, and efficiency.
Q3. What are the three types of logistic regression?A. Binary: Two outcomes (yes/no, spam/not spam).
Multinomial: Three or more unordered outcomes (cat/dog/bird).
Ordinal: Three or more ordered outcomes (low/medium/high).
A. Logistic regression is supervised.
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Anshul 28 Aug, 2024I have recently graduated with a Bachelor's degree in Statistics and am passionate about pursuing a career in the field of data science, machine learning, and artificial intelligence. Throughout my academic journey, I thoroughly enjoyed exploring data to uncover valuable insights and trends. I am eager to continue learning and expanding my knowledge in the field of data science. I am particularly interested in exploring deep learning and natural language processing, and I am constantly seeking out new challenges to improve my skills. My ultimate goal is to use my expertise to help businesses and organizations make data-driven decisions and drive growth and success.
