Logistic Regression for Machine Learning

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Logistic regression is another technique borrowed by machine learning from the field of statistics.

It is the go-to method for binary classification problems (problems with two class values). In this post you will discover the logistic regression algorithm for machine learning.

After reading this post you will know:

  • The many names and terms used when describing logistic regression (like log odds and logit).
  • The representation used for a logistic regression model.
  • Techniques used to learn the coefficients of a logistic regression model from data.
  • How to actually make predictions using a learned logistic regression model.
  • Where to go for more information if you want to dig a little deeper.

This post was written for developers interested in applied machine learning, specifically predictive modeling. You do not need to have a background in linear algebra or statistics.

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Learning Algorithm for Logistic Regression

Learning Algorithm for Logistic Regression
Photo by Michael Vadon, some rights reserved.

Logistic Function

Logistic regression is named for the function used at the core of the method, the logistic function.

The logistic function, also called the sigmoid function was developed by statisticians to describe properties of population growth in ecology, rising quickly and maxing out at the carrying capacity of the environment. It’s an S-shaped curve that can take any real-valued number and map it into a value between 0 and 1, but never exactly at those limits.

1 / (1 + e^-value)

Where e is the base of the natural logarithms (Euler’s number or the EXP() function in your spreadsheet) and value is the actual numerical value that you want to transform. Below is a plot of the numbers between -5 and 5 transformed into the range 0 and 1 using the logistic function.

Logistic Function

Logistic Function

Now that we know what the logistic function is, let’s see how it is used in logistic regression.

Representation Used for Logistic Regression

Logistic regression uses an equation as the representation, very much like linear regression.

Input values (x) are combined linearly using weights or coefficient values (referred to as the Greek capital letter Beta) to predict an output value (y). A key difference from linear regression is that the output value being modeled is a binary values (0 or 1) rather than a numeric value.

Below is an example logistic regression equation:

y = e^(b0 + b1*x) / (1 + e^(b0 + b1*x))

Where y is the predicted output, b0 is the bias or intercept term and b1 is the coefficient for the single input value (x). Each column in your input data has an associated b coefficient (a constant real value) that must be learned from your training data.

The actual representation of the model that you would store in memory or in a file are the coefficients in the equation (the beta value or b’s).

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Logistic Regression Predicts Probabilities (Technical Interlude)

Logistic regression models the probability of the default class (e.g. the first class).

For example, if we are modeling people’s sex as male or female from their height, then the first class could be male and the logistic regression model could be written as the probability of male given a person’s height, or more formally:

P(sex=male|height)

Written another way, we are modeling the probability that an input (X) belongs to the default class (Y=1), we can write this formally as:

P(X) = P(Y=1|X)

We’re predicting probabilities? I thought logistic regression was a classification algorithm?

Note that the probability prediction must be transformed into a binary values (0 or 1) in order to actually make a probability prediction. More on this later when we talk about making predictions.

Logistic regression is a linear method, but the predictions are transformed using the logistic function. The impact of this is that we can no longer understand the predictions as a linear combination of the inputs as we can with linear regression, for example, continuing on from above, the model can be stated as:

p(X) = e^(b0 + b1*X) / (1 + e^(b0 + b1*X))

I don’t want to dive into the math too much, but we can turn around the above equation as follows (remember we can remove the e from one side by adding a natural logarithm (ln) to the other):

ln(p(X) / 1 – p(X)) = b0 + b1 * X

This is useful because we can see that the calculation of the output on the right is linear again (just like linear regression), and the input on the left is a log of the probability of the default class.

This ratio on the left is called the odds of the default class (it’s historical that we use odds, for example, odds are used in horse racing rather than probabilities). Odds are calculated as a ratio of the probability of the event divided by the probability of not the event, e.g. 0.8/(1-0.8) which has the odds of 4. So we could instead write:

ln(odds) = b0 + b1 * X

Because the odds are log transformed, we call this left hand side the log-odds or the probit. It is possible to use other types of functions for the transform (which is out of scope_, but as such it is common to refer to the transform that relates the linear regression equation to the probabilities as the link function, e.g. the probit link function.

We can move the exponent back to the right and write it as:

odds = e^(b0 + b1 * X)

All of this helps us understand that indeed the model is still a linear combination of the inputs, but that this linear combination relates to the log-odds of the default class.

Learning the Logistic Regression Model

The coefficients (Beta values b) of the logistic regression algorithm must be estimated from your training data. This is done using maximum-likelihood estimation.

Maximum-likelihood estimation is a common learning algorithm used by a variety of machine learning algorithms, although it does make assumptions about the distribution of your data (more on this when we talk about preparing your data).

The best coefficients would result in a model that would predict a value very close to 1 (e.g. male) for the default class and a value very close to 0 (e.g. female) for the other class. The intuition for maximum-likelihood for logistic regression is that a search procedure seeks values for the coefficients (Beta values) that minimize the error in the probabilities predicted by the model to those in the data (e.g. probability of 1 if the data is the primary class).

We are not going to go into the math of maximum likelihood. It is enough to say that a minimization algorithm is used to optimize the best values for the coefficients for your training data. This is often implemented in practice using efficient numerical optimization algorithm (like the Quasi-newton method).

When you are learning logistic, you can implement it yourself from scratch using the much simpler gradient descent algorithm.

Logistic Regression for Machine Learning

Logistic Regression for Machine Learning
Photo by woodleywonderworks, some rights reserved.

Making Predictions with Logistic Regression

Making predictions with a logistic regression model is as simple as plugging in numbers into the logistic regression equation and calculating a result.

Let’s make this concrete with a specific example.

Let’s say we have a model that can predict whether a person is male or female based on their height (completely fictitious). Given a height of 150cm is the person male or female.

We have learned the coefficients of b0 = -100 and b1 = 0.6. Using the equation above we can calculate the probability of male given a height of 150cm or more formally P(male|height=150). We will use EXP() for e, because that is what you can use if you type this example into your spreadsheet:

y = e^(b0 + b1*X) / (1 + e^(b0 + b1*X))

y = exp(-100 + 0.6*150) / (1 + EXP(-100 + 0.6*X))

y = 0.0000453978687

Or a probability of near zero that the person is a male.

In practice we can use the probabilities directly. Because this is classification and we want a crisp answer, we can snap the probabilities to a binary class value, for example:

0 if p(male) < 0.5

1 if p(male) >= 0.5

Now that we know how to make predictions using logistic regression, let’s look at how we can prepare our data to get the most from the technique.

Prepare Data for Logistic Regression

The assumptions made by logistic regression about the distribution and relationships in your data are much the same as the assumptions made in linear regression.

Much study has gone into defining these assumptions and precise probabilistic and statistical language is used. My advice is to use these as guidelines or rules of thumb and experiment with different data preparation schemes.

Ultimately in predictive modeling machine learning projects you are laser focused on making accurate predictions rather than interpreting the results. As such, you can break some assumptions as long as the model is robust and performs well.

  • Binary Output Variable: This might be obvious as we have already mentioned it, but logistic regression is intended for binary (two-class) classification problems. It will predict the probability of an instance belonging to the default class, which can be snapped into a 0 or 1 classification.
  • Remove Noise: Logistic regression assumes no error in the output variable (y), consider removing outliers and possibly misclassified instances from your training data.
  • Gaussian Distribution: Logistic regression is a linear algorithm (with a non-linear transform on output). It does assume a linear relationship between the input variables with the output. Data transforms of your input variables that better expose this linear relationship can result in a more accurate model. For example, you can use log, root, Box-Cox and other univariate transforms to better expose this relationship.
  • Remove Correlated Inputs: Like linear regression, the model can overfit if you have multiple highly-correlated inputs. Consider calculating the pairwise correlations between all inputs and removing highly correlated inputs.
  • Fail to Converge: It is possible for the expected likelihood estimation process that learns the coefficients to fail to converge. This can happen if there are many highly correlated inputs in your data or the data is very sparse (e.g. lots of zeros in your input data).

Further Reading

There is a lot of material available on logistic regression. It is a favorite in may disciplines such as life sciences and economics.

Logistic Regression Resources

Checkout some of the books below for more details on the logistic regression algorithm.

Logistic Regression in Machine Learning

For a machine learning focus (e.g. on making accurate predictions only), take a look at the coverage of logistic regression in some of the popular machine learning texts below:

If I were to pick one, I’d point to An Introduction to Statistical Learning. It’s an excellent book all round.

Summary

In this post you discovered the logistic regression algorithm for machine learning and predictive modeling. You covered a lot of ground and learned:

  • What the logistic function is and how it is used in logistic regression.
  • That the key representation in logistic regression are the coefficients, just like linear regression.
  • That the coefficients in logistic regression are estimated using a process called maximum-likelihood estimation.
  • That making predictions using logistic regression is so easy that you can do it in excel.
  • That the data preparation for logistic regression is much like linear regression.

Do you have any questions about logistic regression or about this post?
Leave a comment and ask, I will do my best to answer.


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80 Responses to Logistic Regression for Machine Learning

  1. Ahmaf kamran September 29, 2016 at 5:53 am #

    How to assign weights in logistic regression?

  2. Raghav November 16, 2016 at 12:26 am #

    how is e^(b0 + b1*X) / (1 + e^(b0 + b1*X)) a logistic function

    Isn’t the hypothesis function in logistic regression g(transpose(theta)x) where g = 1/1+e^-x

  3. Rishu February 7, 2017 at 5:39 am #

    Hi Jason, Thanks for such an informative post.

    I have a question which i am struggling with for some time now. Can you please help me with it.

    Let’s say i want to do customer attrition prediction. Now customer attrition can happen anytime during an year. There are 2 ways i can think of setting up the problem.

    1. Fix a reference data e.g. 1 Nov’16. Dependent variable (in observation period) calculated by considering customers who churned in next 3 months (Nov/Dec/Jan). Independent variables duration can be fixed between Nov’15-Oct’16 (1 yr) & variables such transaction in last 6 months can be created. (I think this is a better approach. Also makes more sense if i want to score the model and build campaigns)

    2. Consider year 2016. For customers who churned in July’16 (observation period) consider Jan-June’16 as the duration for creating independent variables, for customer churned in Aug’16 consider Feb-July’16 for independent variable creation along with an indicator whether the customer had churned in last month or not (auto regression blind of case). Append this data row-wise, take a random sample from it for training and rest for testing. (here i feel dependent variables will have seasonality as variable created would have considered different months)

    Can you please let me which of these is right (or if anyone is correct). This is will be helpful as i have not been able to figure this out.

    Regards,
    Rishabh

  4. abdul June 3, 2017 at 6:19 am #

    HI jason sir …i am working on hot weather effects human health ..like (skin diseases) ..i have two data sets i.e weather and patient data of skin diseases ,,after regressive study i found that ,as my data sets are small i plan to work Logistic regression algorithm with R..can u help to solve this i will b more graceful to u ..

  5. Dan August 1, 2017 at 3:56 pm #

    Hello! While studying for ML, I was just wondering how I can state differences between a normal logistic regression model and a deep learning logistic regression model which has two hidden layers. Let’s say five variables for x.

    • Jason Brownlee August 2, 2017 at 7:45 am #

      They are indeed very different. What do you mean “state the difference”? Apples and oranges?

      • Dan August 2, 2017 at 3:33 pm #

        Thank you for fast response. I was actually wondering formula for each. I know the normal logistic regression goes by, “ln(Y) = a + b1X1 + … +bnXn”. However, I was wondering a formula of a deep learning logistic regression model with two hidden layer (10 nodes each). I just want to know How I can express it as short version of formula.

        • Jason Brownlee August 3, 2017 at 6:45 am #

          Hi Dan, I would encourage you to switch to neural net terminology/topology when trying to describe hierarchical models. It is no longer a simple linear question.

          • Dan August 3, 2017 at 5:35 pm #

            Thanks again for your comment. How about a formula for a deeplearning model which has two hidden layers (10 nodes each) and five X variable and Y (the target value is binary). I know the difference between two models I mentioned earlier. I just want to express a deeplearning model in a mathematical way.

          • Jason Brownlee August 4, 2017 at 6:50 am #

            Normally the equations are described for a forward pass or back pass for a single node, not the whole network.

      • Daniel August 2, 2017 at 4:10 pm #

        Thank you for fast response. I was actually wondering formula for each. I know the normal logistic regression goes by, “ln(Y) = a + b1X1 + … +bnXn”. However, I was wondering a formula of a deep learning logistic regression model with two hidden layer (10 nodes each). I just want to know How I can express it as short version of formula.

  6. Anjali G August 27, 2017 at 10:59 am #

    Hi. Please let me know how we can proceed if the distribution of the data is skewed- right skew.

    Thank you.

    • Jason Brownlee August 28, 2017 at 6:46 am #

      Consider a power transform like a box-cox transform.

  7. Jobayer sheikh September 26, 2017 at 10:28 pm #

    Hello sir, can you please explain why p=exp(b0+b1*x)/(exp(b0+b1*x)+1) is probability

    • Jason Brownlee September 27, 2017 at 5:41 am #

      Hi Jobayer,

      I would recommend reading a textbook on the topic, such as “An Introduction to Statistical Learning” or “Elements of Statistical Learning”.

  8. Marjorie Escanan October 1, 2017 at 8:55 am #

    Hello Mr. Brownlee,

    thank you for a very informative this very informative piece..

    i am currently working on a paper in object detection algorithm…just wondering, how could i use logistics regression in my paper exactly?

  9. vinay kumar November 4, 2017 at 4:40 pm #

    A short video tutorial on Logistic Regression for beginners:
    https://quickkt.com/tutorials/artificial-intelligence/machine-learning/logistic-regression-theory/

  10. John Serpano December 1, 2017 at 1:39 am #

    In a binary classification problem, is there a good way to optimize the program to solve only for 1 (as opposed to optimizing for best predicting 1 and 0) – what I would like to do is predict as close as accurately as possible when 1 will be the case. (I do not care at all about 0 and if I miss a 1, that’s ok, but when it predicts a 1, I want it to be really confident – so I am trying to see if there is a good way to only solve for 1 (as opposed to 1 and 0)? Thanks.

    • Jason Brownlee December 1, 2017 at 7:36 am #

      Yes, in the literature we call this anomaly detection.

  11. Pararawendy December 15, 2017 at 9:13 am #

    This post is best! Crisp yet clear!

    Thank you, Sir!

  12. Harinee December 19, 2017 at 4:45 am #

    Wonderful post. I have been trying to read up a book and it just kept getting convoluted despite having done a project using LR. Thanks for the sheer simplicity with which you have covered this.

  13. Swati February 11, 2018 at 8:32 pm #

    Nice post..easy to understand..Thanks!

  14. Rick March 4, 2018 at 4:15 am #

    Thank you for your article!!!!!!!! A LOT OF HELP!!!

  15. Sam April 3, 2018 at 4:25 am #

    Hi Jason,

    Thanks so much for the article and blog in general. It’s all been tremendously helpful as I’ve been diving into machine learning.

    My question is on topic, but in a little different direction…

    I’ve got a trained and tested logistic regression. I trust it as a predictor, but now I’ve got a set of people that I need to apply it to. Let’s say this is a group of ten people, and for each of them, I’ve run a logistic regression that outputs a probability that they will buy a pack of gum. So now I have ten probability outputs [0.83, 0.71, 0.63, 0.23, 0.25, 0.41, 0.53, 0.95, 0.12, 0.66]. (From this point on, I’m a little less sure about each successive sentence). I can sum them together and see that my most likely outcome is that I’ll sell 5.32 packs of gum. But I also want to know what the probability is that I sell 6 packs of gum or 5, or 4, or 9. I’d like to plot some sort of probability distribution for the number of packs of gum that I expect to sell to this whole group of people. I’ve got an error measure, so I can calculate a standard deviation and plot some sort of normal distribution, with 5.32 at the center, to show the probability of different outcomes, right?

    I think all of that makes sense, but then it gets a little more complicated. I have some other people, with different features and a different classifier. I’m testing the same outcome (that they’ll buy a pack of gum), but these are people who are maybe already at the counter in my shop. I’ve got five of them and their probabilities are [0.93, 0.85, 0.75, 0.65, 0.97]. So, I’d expect the most likely outcome is that I would sell 4.15 packs of gum to this group of five. Great, but now I’ve got two different classifiers, with two different groups of people and two different error measures. I assume the most likely outcome is that I sell 9.47 packs of gum in total (5.32 from the first group, 4.15 from the second group). But how can I go about determining the likelihood that I sell 10 packs in total between the two groups? 12? 5? How can I come up with a normal distribution when I’ve got two different classifiers working on two different groups of people?

  16. Jallepalli Prerna April 29, 2018 at 5:41 pm #

    Hey Jason, your tutorials are amazing for beginners like me, thank you for explaining it systematically and in an easy manner.

  17. Mike Goldweber June 17, 2018 at 10:29 pm #

    Can you please tell me what the processing speed of logistic regression is? How does it compare to other predictive modeling types (like random forests or One-R)?

    Thank you!

    • Jason Brownlee June 18, 2018 at 6:42 am #

      Perhaps you can write code to compare the execution time?

  18. Gaurav July 27, 2018 at 4:41 pm #

    Hi Jason,

    I have a question regarding the example you took here, where prediction of sex is made based on height.

    With the logit function it is concluded that the p(male | height = 150cm) is close to 0. Using this information, what can I say about the p(female| height = 150cm) when I know that the output is classified as male or female?

  19. German August 29, 2018 at 8:50 pm #

    Amazing detailed and still clear content, as usually 😉

  20. Shailendra Acharya September 10, 2018 at 12:10 pm #

    Thank you so much it cleared many of my doubts

  21. Merve Bayram Durna October 2, 2018 at 11:52 am #

    Thank you for your article and the others! they are very helpfull for beginners like me

  22. Jay October 8, 2018 at 5:33 am #

    Thank you it is helpful for me

  23. CH December 18, 2018 at 7:44 pm #

    Jason, you are great! You can always explain very complex methodology in a layman way!
    Thank you!

  24. ML December 20, 2018 at 9:27 am #

    Thank you for the informative post. I have a questions on determining the value of input variables that optimize the response of a logistic regression (probability of a primary event).

    Assume the independent variables refers to treatment options, dependent variables refer to not-being-readmitted-to-hospital. The objective is to determine the most effective treatment options, more likely a combination of treatment options that maximize the probability of not-being-readmitted.

    How would you suggest me to determine which options or combinations are the most effective?

    Should I follow: 1) build a logistic regression model 2) with the coefficients figured out, assume maximizing prob, and then determine the value of independent variables? What’s a better way to find input values that optimize response variable?

    Thank you.

    • Jason Brownlee December 20, 2018 at 1:58 pm #

      Good question, perhaps treat it as an optimization problem with the fit model to seek the values that maximize the output.

  25. Maarten Over January 18, 2019 at 9:03 pm #

    Hi Jason,

    thank you vey much for sharing your knowledge in such an understandable way! This helps me a lot. However, I still struggle with a (maybe quite specific?) problem.

    In the case I’m studying, the Probability of success is expected not to reach 100%. In fact, realistic probabilities range between 0 – a%. While a is unknown.

    I believe in my case, I will need something like P(X) = a / (1 + e^(b + c*(X))
    Applying the logit and ML approach to this however causes problems..

    Do you maybe know how to solve this? What would be a good approach?
    Or maybe logistic regression is not the best option to tackle this problem? Would another approach like Naive Bayes be a better alternative?

    Thanks a lot in advance!
    Regards, Maarten

    • Jason Brownlee January 19, 2019 at 5:40 am #

      Great question!

      I would encourage you to re-post this question on math overflow, and get an answer from a real math person, I expect there is a way to constrain the model correctly for what you need and I don’t want to make something up and mislead you.

  26. mt February 21, 2019 at 10:19 am #

    Hi Jason,
    thanks for your helpful informations. those helped me a lot.
    I am struggling with one question that I can’t quite understand yet. I saw some specialists and teachers say that the logistic regression makes no assumption about the distribution of the independent variables and they do not have to be normally distributed, linearly related or of equal variance within each group. how does it fit with your explanation of logestic regression? I hope you can help me understand that. thanks
    here is a link that mentioned it:
    http://userwww.sfsu.edu/efc/classes/biol710/logistic/logisticreg.htm

    • Jason Brownlee February 21, 2019 at 2:04 pm #

      That does not match my understanding, perhaps talk to the owner/author of the material?

      • mt February 21, 2019 at 8:55 pm #

        I asked them and am waiting for their respond
        but meanwhile, here is another link
        https://www.quora.com/Does-logistic-regression-require-independent-variables-to-be-normal-distributed
        someone asked this question and some specialists answered that logistic regression doesn’t assum that your independent variable is normally distributed

        • Jason Brownlee February 22, 2019 at 6:17 am #

          Interesting, thanks.

          • mt February 22, 2019 at 5:14 pm #

            this is what I found out from their answers: logistic or linear regression algorithms do assum that there is a linear relationship between your indepndent and dependent variables but they have no assumption about independent variables having any particular distribution. just if I transform my continuous indepent variables distribution to a normal distribution form it exposes this linear relationship a lot better.
            what do you think? is it right?

          • Jason Brownlee February 23, 2019 at 6:29 am #

            Doesn’t match my understanding – at least as far as linear regression. There is no distribution when it comes to logistic regression, the target is binary.

  27. Jin Real April 6, 2019 at 1:04 pm #

    Hi Jason, should the page number of the referenced book “The Elements of Statistical Learning: Data Mining, Inference, and Prediction” be 119-128?

  28. Satish S K June 3, 2019 at 8:07 pm #

    Hi Jason,

    Thank you for this detailed explanation/tutorial on Logistic Regression.

    I have few queries related to Logistic Regression which I am not able to find answers over the internet or in books. It would be of great help if you could help me understand these uncleared questions.

    1. Has Logit function (i.e. Logit equation LN(P/1-P)) being derived from Logistic Regression equation or its the other way around?
    2. What is the purpose of Logit equation in logistic regression equation? How logit function is used in Logistic regression algorithm? Reason for asking this question will get clear after going through point no. 3 & 4.
    3. Upon building a logistic regression model, we get model coefficients. When we substitute these model coefficients and respective predictor values into the
    logistic regression equation, we get probability value of being default class (same as the values returned by predict()). Does this mean that estimated model coefficient values are determined based on the probability values (computed using logistic regression equation not logit equation) which will be inputed to the likelihood function to determine if it maximizes it or not? If this understanding is correct then, where the logit function is used in the entire process of model building.
    4. Neither logit function is used during model building not during predicting the values. If this is the case then why do we give importance to logit function which is used to map probability values to real number values (ranging between -Inf to +Inf). Where exactly the logit function is used in the entire logistic regression model buidling process? Is it while estimating the model coefficients?
    5. The model coefficient estimates that we see upon running summary(lr_model) are determined using linear form of logistic regression equation (logit equation) or the actual logistic regression equation?

    • Jason Brownlee June 4, 2019 at 7:49 am #

      Sorry, I don’t go into the derivation of the equations on this blog.

      Perhaps try posting your questions on mathoverlow?

  29. Ashwini Iral September 4, 2019 at 7:25 pm #

    Log odds is the logit and not probit right ?

  30. se_du September 5, 2019 at 1:07 am #

    Hi everyone,

    I have a question that I splitted my data as 80% train and 20% test. And I applied Gradient Boosting however, test score result is 1.0 . It comes to me a little bit strange. How could I infere this result?

    Thank you.

    • Jason Brownlee September 5, 2019 at 6:57 am #

      Perhaps the problem is too simple/trivial?

      • se_du September 5, 2019 at 6:51 pm #

        If so, should I rely on the result, although it is very simple?I mean, Should I trust the results if I believe that I have correctly identified the problem, even though I received the test result too high?

        If not, what is the way to get the problem out of too simple state? Increased number of columns and observations? Or something else?

        Thank you dude.

        • Jason Brownlee September 6, 2019 at 4:54 am #

          Your goal is to design a test harness that allows the results to be trusted.

          This means ensuring the training dataset is reliable, and using a technique such as k-fold cross validation:
          https://machinelearningmastery.com/k-fold-cross-validation/

          • se_du September 8, 2019 at 4:22 am #

            Hi Jason,

            I have tried k-fold and the test accuracy still is around %98. So, can I now trust the results and use this model ?

            (btw;
            # of observation : 3000,
            # of feature : 1131 ,
            n component used in PCA = 20
            )

            Logistic regression’s result according to above info is train accuracy=%99 , test accuracy=%98.3

            Thank you.

          • Jason Brownlee September 8, 2019 at 5:20 am #

            I believe so.

          • se_du September 8, 2019 at 4:25 am #

            Updated** :

            (btw;
            # of observation : 3000,
            # of feature : 1131 ,
            cross validation* : 20
            )

  31. se_du September 5, 2019 at 7:25 pm #

    Have another question: My target column (y) type is object and it includes values as “A”, “B” and “C”. Should I convert it from object to Categorical as below;

    y = pd.Categorical(y)

    Or I should leave it as object.

    Thank you.

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