Logistic Regression Course Overview

    What is Logistic Regression?

    If you are new to data analytics and machine learning, you may attempt to learn different techniques and tools related to these fields. One specific kind of analysis that data analysts widely use is logistic regression. However, before using it, you must first understand the logistic regression meaning.

    Logistic regression is a controlled classification algorithm. In a logistic regression classification problem, the output, i.e., target variable (y), can accept only discrete values for a given set of features, i.e., inputs (x).

    It is a logistic regression model that predicts the likelihood that a given data entry fits the category labeled as “1”. Similar to how the linear regression mandates the data to obey a linear function, the logistic regression models the data through the logistic regression sigmoid function.

    The corresponding binary outcome features two possible circumstances - either the event happens (shows 1) or doesn’t happen (shows 0). Note that independent variables can affect the outcome.

    The logistic regression analysis is helpful when you work with binary data. In binary logistic regression, you deal with binary data when the dependent variable or output is categorical. It means logistic reasoning outfits into the two categories - ‘yes’ or ‘no’, ‘pass’ or ‘fail’, ‘true’ or ‘false’, etc.

    Logistic regression turns out to be a classification technique if a decision threshold comes into the picture. The significant aspect of the logistic regression is to set the threshold value. The decision related to the threshold value primarily depends on recall and precision values. Although we need both recall and precision values to be 1, this may not always be the case.

    Binary classification problems, for example, if an email is spam or not, can be solved through logistic regression. Moreover, we can solve multiclass logistic regression problems like classifying random fruits using logistic regression. Its ability to solve such problems becomes more pronounced when you dive deep into the logistic regression in Python.

    When to use logistic regression?

    After understanding the logistic regression meaning and the logistic regression from scratch, the next vital thing is to know when to use it. Logistic regression helps predict the categorical dependent variable. It is useful when the prediction is definite, like true or false, yes or no, 0 or 1. The value of the predicted probability or outcome of logistic regression can be any one - no middle value is possible.

    Regarding the predictor variables, logistic regression can fall into any of the below categories: 

    i. Continuous:

    Continuous data is classified as ratio data or interval data. You can measure it on an infinite scale. It can accept any value from the two numbers. The example includes temperature weight in grams or temperature in degrees Celsius.

    ii. Discrete (ordinal):

    The ordinal logistic regression shows data suitable for inclusion into a specific type of order on a scale. The logistic example for this category can be the eyes’ color: black, brown, or blue. You can understand it with another example, like describing how satisfied you are with a service or product on a scale of 1-5.

    Logistic regression analysis is significant for forecasting the probability of an event. It lets you decide the probabilities among any two classes. You can only anticipate probability and classification outcomes via logistic regression.

    A logistic regression model can assist you to categorize data for extract, transform, and load (ETL) operations. Remember logistic regression must not be used if the number of observations is lower than the number of features. If used, it leads to overfitting.

    How does Logistic Regression work?

    The working of logistic regression involves forecasting the “log odds” in form of a linear equation, identical to linear regression. After you forecast this, you use a sigmoid function for this prediction to measure the probability. The outcomes are probability values, so they fall in the range from 0 to 1. The typical cutoff value is 0.5. Those values below 0.5 fall into one class and those values above 0.5 belong to another class.

    The logistic regression sigmoid function (also known as the logistic regression model) maps the forecasted predictions to probabilities. Here, the Sigmoid function depicts an S-shaped curve whenever its plotting takes place on a map. The corresponding graphs plot the forecasted values in the range of 0 to 1. These values are subsequently plotted near the margins at the upper and lower part of the Y-axis, along with the labels 0 and 1. As per these values, the classification of the target variable takes place in any one of the classes.

     

    Here is the formula of the Sigmoid function:

     

    y=1/(1+e^x)

     

    (here, e^x shows the exponential constant and its value is 2.718)

     

    The above equation provides the value of y near 0 if x is a substantial negative value. Identically, if x’s value is a large positive number, the value of y is forecasted close to 1.

    When a decision boundary is set, it helps to predict the particular class for the belonging data. As per the set value, the estimated values categorizes into classes.

    Let’s understand this with an example of categorizing emails to be spam or not. If the predicted value is below 0.5, the email is known as spam and vice versa.

    Sigmoid function and decision boundary

    In logistic regression, the sigmoid function works as an advanced regression method for solving diverse classification problems. It is a classification model, and so its alternate name is ‘regression’. Another reason behind this name is the fundamental techniques are identical to linear regression.

    You can understand the sigmoid function to be a mathematical function with a typical “S” — shaped curve. This curve converts the values in the range of 0 to 1 - it asymptotes both values. Other names for this function are the logistic function and the sigmoidal curve. This function is useful in terms of non-linear activation functions, adding to its popularity.


    The logit model in r or sigmoid helps to predict the probabilities of a binary result. Furthermore, the sigmoid function transforms a regression line into a decision boundary for the logistic regression binary classification. Its working is like the logit model.

     

    Let’s understand the Sigmoid function with an example. Suppose we assume a standard regression problem like - 

    z = βtx

    and it passes through a sigmoid function

    σ(z) = σ(βtx))

     

    When you implement the Sigmoid function, you get an S curve instead of a straight line.   This shape shows the growth increases till it attains climax and declines afterward. This makes it easy for the binary classification with 0 and 1 as possible output values. When the value of the linear regression model shows 2.5, 5, or 10, the sigmoid function arranges it into classes related to 1.

     

    σ(z) < 0.5    for z<0

     

    σ(z)≥0.5      for z≥0

     

    The sigmoid function helps in credit card fraud detection using logistic regression. Let’s understand it with an example. Suppose a sigmoid function intends to classify credit card transactions as fraudulent or genuine. When the value of the function shows a 70% probability, the transaction is s fraudulent. To represent this, you will write - 

    hβ(x) = 0.7


    Decision Boundary:


    The primary application of logistic regression helps determine a decision boundary for use in binary logistic regression classification problems. The baseline helps to recognize a binary decision boundary. So, this approach is useful for scenarios featuring logistic regression for multiclass classification.

    In the multiple logistic regression, the decision boundary shows a linear line separating class A and class B. Certain points from class A fall into the area of class B. The reason is in the linear model, it's hard to obtain the precise boundary line discriminating the two classes.

    When logistic regression training works on a classifier on a dataset with the help of a precise classification algorithm, it is necessary to state a set of hyper-planes, known as ‘Decision Boundary’. It discriminates the data points into explicit classes wherein the algorithm transits from one class to another.

    One side of a decision boundary shows data points, probably called ‘Class A’. The other side of the boundary is probably Class B.

    Logistic regression aims to come up with an approach to divide the data points to get a precise prediction of any particular observation’s class. For this, the information available in the features is helpful.

    Now let’s take the logistic regression example to understand this.Suppose we specify a line describing a decision boundary. Every point on one side of this boundary will feature all data points belonging to Class A. All those points present on another side of the boundary will feature all the data points belonging to class B.

    For the use of logistic regression, the following formula is useful: 

    S(z)=1/(1+e^-z)


    Here, S(z) = output in range of 0 to 1 (probability estimate)


    z = Input to the function (z= mx + b)


    e = Base of natural log


    The prediction function in the use shows a probability score in the range of 0 to 1. To map the same to a discrete class (A/B), you should choose a threshold value, or you can say a tipping point. Any value above this threshold value or a tipping point will be categorized into class A. Any value below this point will classify into class B.


    p >= 0.5       for class=A


    p <= 0.5       for class=B


    In case the threshold value is 0.5 and the prediction function returns 0.7, this observation classifies into class A. If the prediction value was 0.2, the observation classifies into class B. Hence, the line with 0.5 is the decision boundary.


    Assumptions of logistic regression

    In a logistic regression model, certain assumptions help enhance its performance. Let’s go through the details of each of the assumptions:

     logistic regression assumptions

    Assumption-1: The nature of the response variable is binary


    Logistic regression mandates the response variable to take only one of the two probable outcomes. The corresponding examples are:

    • Yes or No

    • Pass or Fail

    • Male or Female

    • Malignant or Benign

    • Drafted or Not Drafted

    Assumption-2: The observations are independent


    Logistic regression presumes the observations in the dataset to be independent of each other. The observations must not derive from recurrent quantities of the same individual or be associated with each other in any form.

     

    Assumption-3: No Multicollinearity between explanatory variables


    No multicollinearity exists between the explanatory variables as per the assumption of logistic regression.

    Multicollinearity takes place whenever two or multiple explanatory variables are highly linked to each other, in a way they don’t offer exceptional or independent info in the regression model. If the amount of correlation is large among the variables, it leads to issues when interpreting logistic regression and fitting its model.

    The typical method to identify multicollinearity is to use the variance inflation factor (VIF). This factor calculates the strength and correlation among the predictor variables existing in a regression model. 


    Assumption-4: No extreme outliers


    In the logistic regression concept, the assumption is there are zero influential observations or extreme outliers in the dataset.

    Now, the question is - how can you test this assumption? 

    The typical approach to check for influential observations and extreme outliers in a dataset is to measure the object’s distance for every observation. In case there exist indeed outliers, you can select to (i) discard them, (ii) substitute them with a value like the median or mean, or (3) save them in the model and make a note of this when recording the regression results.

    Assumption-5: Existence of logit of the response variable and linear relationship between explanatory variables


    Logistic regression presumes the existence of a linear relationship between every explanatory variable and the
    logit model of the response variable.


    The logit function is defined as below:


    Logit(p)  = log(p / (1-p)) (here p shows the probability of a positive outcome)

    The simplest way to check this assumption is to use a Box-Tidwell test. 


    Assumption-6: The sample size is significantly big


    Another assumption in the Logistic regression is the sample size of the dataset is sufficiently big to derive accurate conclusions from a particular fitted logistic regression model.


    To test this assumption, you must use at least 5 cases with the minimum frequent outcome for every explanatory variable. Suppose you work on two explanatory variables and suppose the predictable likelihood of the least frequent result is 0.10. In this case, the sample size must be minimum (5*2) / 0.10 = 100.


    Linear vs. Logistic

    The main difference between linear and logistic regression is the relationship between the variables. Another major difference between linear regression and logistic regression is in terms of the predictive methods they use. Linear regression and logistic regression also differ in predicting the next weight value. The difference between linear and logistic regression in machine learning is also noticeable. Let’s go through the details of these differences below:


    Logistic Regression

    Linear Regression

    Logistic Regression shows an S-shaped curve when plotting happens on a map. The relevant graphs plot the predicted values from 0 to 1.

    Linear regression represents a straight line, and lets analysts prepare graphs and charts for tracking the movement of linear relationships.

     

    Independent variables have no correlations. The reason is that all of them are independent without any dependent variables.

     

    Linear regression testing is applicable to recognize correlations among variables. A correlation exists between the independent and dependent variables in a typical linear regression.

     

    Multiple linear regressions can detect one or multiple possible correlations between variables, like the case with the cause-and-effect relationships.

    Logistic regression gives only two outcomes. Whenever analysts reach either outcome, the outcome is 0 or 1.

     

    Linear regression makes use of positive and negative whole numbers to forecast value. Due to the infinite nature of the numerical possibilities over a straight line, linear regression can present you with a range of values as results.

    Logistic regression can either use the least-square estimation method or maximum likelihood estimation.

    Linear regression uses the ‘root-mean-square error’ as the standard deviation for measuring the extent of data points over the line shown by linear regression. Linear regression utilizes only one estimation method to measure the unidentified values of a system's features, functions, or other parameters.

     

    Data analysts or architects must program logistic models to trigger when the system or AI network fulfills certain parameters. It is based on the logistic regression in ai.

     

    For linear regression, the activation function is not mandatory. But it’s useful if you aim to transform a linear regression model into logistic regression equivalence.

    In addition to being familiar with its application as logistic regression in data science, it is useful in many industries like database management, credit scoring, customer behavior tracking, booking accommodation, and text editing.

     

     

    The applications of linear regression include information technology, data and computer science, business and finance, and accounting.


    Mathematical Formulation for Logistic Regression


    The standard mathematical equation for calculating logistic regression:


    y = 1/(1+e^-(a+b1x1+b2x2+b3x3+...))


    The description of parameters is as below:

    y: response variable.

    x: predictor variable.

    a and b: coefficients, working as numeric constants


    Advantages and Disadvantages of Logistic Regression

    From the application in the form of logistic regression machine learning to logistic regression machine learning in Python, there are tons of applications of this approach. But it comes with both advantages and disadvantages. Before you learn logistic regression, you should know its advantages and disadvantages discussed below:

    Advantages of Logistic regression:

    logistic regression advantages


    1. Easy to implement:

    Logistic regression is easier to execute than other methods, specifically from the perspective of machine learning. A typical machine learning model can feature a mathematical representation of a real-world procedure. This procedure to set up a machine learning model entails tasks like training and testing the model.

    The logistic regression training process finds patterns in the input data. Hence, the corresponding model can plot a specific input to a certain type of output, for example, a label. Compared to other similar methods, the logistic regression is simpler to train and implement.

     2. Works well for a linearly separable dataset:

    Logistic regression functions flawlessly for cases featuring the linearly separable dataset. The ability to plot a straight line separating the two classes of data from one another makes sure the dataset is linearly separable. Logistic regression is useful when a ‘y variable’ can accept only two values. Being linearly separable, the data proves to be more competent to categorise into two distinct classes.

    3. Measures correlation of independent variables:

    Logistic regression provides a measure of the correlation of the independent variable. It accordingly measures the coefficient size. In addition, it informs you regarding the relationship’s direction - positive or negative.

    Any two variables will possess a positive correlation when a rise in the value of one variable also raises the value of another variable. To understand better, for instance, the more hours you dedicate to training, your efficiency will increase for the particular sport.

    The correlation doesn’t always depict causation. The logistic regression might indicate a positive correlation between sales and outdoor temperature. However, this doesn’t essentially imply that sales are increasing due to the temperature.

    Disadvantages of logistic regression:

    logistic regression disadvantages

    1. Cannot predict a continuous outcome:

    Logistic regression is incapable to forecast a continuous outcome. Here's  an example to depict this better. In medical applications, logistic regression can't be used to forecast how much a patient's body temperature will rise. The reason is the measurement scale is continuous. The logistic regression works only when the dependent variable is dichotomous.

    2. Not accurate for small sample size:

    Logistic regression might not provide enough accuracy for a small sample size. When the sample size is small, the logistic regression model generated depends on a smaller number of observations, leading to overfitting.

    In logistic regression statistics, the overfitting represents modeling error when the model is a very close fit to a finite set of data due to inadequate training data. Alternatively, the particular model has insufficient input data to search for patterns inside. For this case, the model can’t precisely predict the results of a new or forthcoming dataset.

     3. Assumes linearity between the dependent variable:

    Logistic regression takes into account the linearity between the dependent variable and the independent variables. The linearity arises from the fact that it’s quite unlikely the observations are linearly discrete.

    Suppose you want to categorize the Iris plant into any of the two families like versicolor or sentosa. To differentiate between these two categories, the factors like sepal size and petal size prove useful. Creating an algorithm helps to categorize this plant. But there is no perceptible distinction between these parameters. Hence, the linearly separable data is an assumption for the logistic regression. However, it’s not always possible in the real world.

    Classification problem and its solution using logistic regression

    The logistic regression algorithm is nothing but a standard classification algorithm that works based on the categorical dependent variables.  Typical examples of logistic regression algorithms can include classifying an email into spam or not spam and predicting whether or not a patient has symptoms of cancer.

    The classification problem is a controlled problem wherein the target variable is definite. For this purpose, the logistic regression function and the name are used. This algorithm works as a regression problem, although it performs classification. The reason is rather than offering the class, logistic regression informs us about the likelihood of a data point relating to every class. Essentially, logistic regression functions as a simple logistic regression classification algorithm.

    Let’s take a detailed example to understand the classification problem better. Suppose the cancer is benign or malignant is a classification problem. This example classifies outcomes into various classes. So, the result falls into either of the two classes - benign or malignant.

    Another example is whether a customer will default on their loan or not can be a classification problem. This problem is of high interest to those companies capitalizing on finance-related issues. Different algorithms for unsupervised learning are Hierarchical clustering, k means clustering, and Neural Network (logistic regression with a neural network mindset).


    Logistic Regression


    Logistic Regression is a popular classification method with application in machine learning. The logistic regression algorithm in machine learning adopts a logistic function to exemplify the dependent variable.


    The logistic regression definition in machine learning states that the dependent variable has a dichotomous nature. So, there can be only two probable classes. In the classification problem discussed above, the classes can be benign or malignant. So, this basic logistic regression technique is beneficial when dealing with binary data.

    Loss Function

    Linear regression algorithm uses the 'Mean Squared Error’ loss function.’


    MSE = 1/n ∑ (y –ỹ)2


    Here, y-ỹ shows the difference between actual and predicted values.


    The MSE adds the square of the distance between the real and the predicted outcome value for all input samples. After adding, it divides it by the number of input samples. The use of the MSE error function offers accurate results.


    The reason for squaring the distance between the real and the predicted outcome values is to correct the samples whose predicted value is quite far from the real value compared to those whose predicted value is near the actual value.


    Cost Function


    The cost function for logistic regression is a mathematical formula for measuring the error between the expected and predicted values. A logistic regression cost function shows the amount of how incorrect the model is in its capability to assess the relationship between x and y. The cost function returns a value. This value is named ‘cost’ or ‘loss’ or ‘error’.

    The logistic regression cost function is shown with the following equation:

    Cost (hΘ(x), Y(actual)) = -log (hΘ (x)) if y=1

                                     = -log(1-hΘ (x)) if y=0

     

    Here, the negative function represents the ability to maximise the likelihood by minimising the loss function. This happens when we train the logistic regression. Reducing the cost will raise the maximum likelihood, based on the assumption - samples are derived from an indistinguishable independent distribution.

    Bias-Variance tradeoff


    Bias-Variance tradeoff is the model outfitting the training data inefficiently but capable of generating identical results in data exterior to training data. In simple terms, it implies simple models that forecast very far from reality without having significant changes in each dataset.


    For instance, a linear regression model will show high bias when attempting to model a non-linear relationship. The reason is the linear regression model doesn’t properly fit non-linear relationships.


    High bias suggests linear regression implemented in the quadratic relationship.


    Low bias suggests second-degree polynomials implemented in quadratic data.


    There is a trade-off between generalization of pattern and predictive accuracy outside training data, hence the name. Enhancing the accuracy of the model leads to less generalization of pattern exterior to the training data. The bias is inversely proportional to the variance.


    Gradient Descent


    Gradient Descent is an optimization algorithm useful for finding the local or global minima of a differentiable function. This algorithm aims to optimize the value for bias and weight. Therefore, the process of computing the gradients and updating the bias and weight is repeated several times.

    Before thoroughly understanding Gradient Descent, let’s go through certain assumptions and definitions:

    Suppose we are having an independent variable x and a dependent variable y.

    This equation establishes a relationship between these variables:

    y = x * w + b

    here, w: weight (or slope),

    b: bias (or intercept),

    x: independent variable column vector

    y: dependent variable column vector

     

    Loss Function: This function denotes the amount of deviation of predicted values from the actual values of dependent variables.

    The key goal behind the implementation of Gradient Descent is to find b and w for establishing the relationship between x and y variables. This can be accomplished using the Loss Function.

     

    Steps for implementation of Gradient Descent in Linear Regression:

    Step-1: Initialize the bias and weight randomly or with 0

    Step-2: Make logistic regression prediction using the above values of initial bias and weight.

    Step-3: Now compare actual values with the above-predicted values. Define the loss function with the help of both these values.

    Step-4: Based on the differentiation, you need to measure how loss function modifies according to bias and weight.

    Step-5: Update the bias and weight to diminish the loss function.

    Root Mean Squared error


    Root Mean Square Error (RMSE) denotes how efficiently a regression line fits the data points. RMSE can be interpreted as Standard Deviation in residuals. The problem with MSE is that the loss order is higher than the data’s order. Therefore, the corresponding formula takes the root of MSE.


    Its formula is:


    L = √1/N [∑ (Ỹ-Y)2]


    Note: The loss function is unchanged and still the solution is the same. The square root reduces the order of the loss function.


    Recall, Precision, and F1 score:

    To understand Recall, Precision, and F1 score, let’s suppose that we are attempting to assist a rescue shelter for goats and sheep. This rescue shelter has a survey for people who don’t know whether they need a goat or a sheep. Depending on the result of the survey, the appointed agent will determine which animal the person likes more. The shelter aims to have fewer people returning to their adopted pets.


    Precision:


    Precision= tp/(tp + fp)


    When this function predicts the person likes goats, the frequency of being correct can be calculated from the F1 score,


    F1 = (2/(recall-1 + precision-1)) = 2 * ((precision * recall)/ (precision + recall))


    The weighted logistic regression, i.e. the weighted average between recall and precision, is useful when there are unbalanced samples.


    Accuracy = (tp + tn) / (tp + tn+ fp + fn)


    It is the sum of true negatives and true positives divided by the number of samples. It is only accurate when the model is balanced, else it gives wrong results.


    Application

    logistic regression apps

    i. For heart disease prediction:

    Medical researchers intend to know the way weight and exercise influence the probability of heart attack. To comprehend the relationship between the probability of having a heart attack and the predictor variables, medical researchers can use logistic regression. In the heart disease prediction using logistic regression, the response variable in the model would be a heart attack.

    Two probable outcomes: a heart attack occurs or doesn’t occur. The results of the heart disease prediction using logistic regression will inform researchers how modifications in weight and exercise influence the probability that a person has a heart attack.

     

    ii. For email spam detection:

    A business intends to know whether the country of origin and word count influence the probability that an email is detected as spam or not. Researchers can carry out logistic regression to infer the relationship between the probability of an email being spam and these two predictor variables.

    In the model, the response variable will be spam. Two potential outcomes are the email is spam and the email is not spam. The outcome of the model informs the business how modifications in the country of origin and word count influence the likelihood of a given email being spam or not.

     

    iii. Credit card fraud detection:

    Suppose a credit card company intends to know whether credit score and transaction amount influence the probability of a particular transaction is a fraud or not. Credit card fraud detection using logistic regression helps the company to infer the relationship between the probability of a transaction being fraudulent and the relationship between these predictor variables.

    In the model, the response variable will be ‘fraudulent’. It can have any one of the two probable outcomes i.e. the transaction is fraudulent or it's not fraudulent. Based on this outcome of the model, the company can determine how modifications in the credit score and transaction amount can relate to the probability of whether a particular transaction is fraudulent or not.

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    logistic regression online course is full of industry experts in enough numbers. Hence, the students get proper guidance and can conveniently clear their doubts.


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    Frequently Asked Questions about Logistic Regression

    Which regions will present higher business opportunities for the growth of the Tableau services market in the future?

    The Asia Pacific will present more business opportunities to facilitate market growth.  The region features constant development in the business infrastructure of companies. Also, in this region, there is quick digitalization at workplaces.

    Why is logistic regression called regression instead of classification?

    The primary difference between regression and classification is the output variable in the regression is continuous whereas, in classification, it is discrete.

    Logistic regression is a supervised classification algorithm. Its model creates a regression model similar to linear regression to forecast the likelihood for a specific data entry belonging to the category labeled as ‘1’.

    What is the significance of regularized logistic regression?

    Regularisation in logistic regression is a method capable of solving the issue of overfitting a model. This technique is advantageous when a huge number of parameters exist and these parameters help forecast the target function. In these cases, it is challenging to set features for manual operation and features for automatic operation. Hence, the regularized logistic regression helps you determine pertinent parameters and solve the overfitting issue.

    Which is the most popular algorithm for variable selection?

    Lasso is the most popular algorithm for variable selection. The reason is it carries out logistic regression analysis with the help of a shrinkage parameter resulting in data shrinking to a point. The variable selection takes place by pushing the value of coefficients of less significant variables to 0 via a penalty.