## Thursday, March 5, 2020

### We Can Learn Logistic Regression From the Wrong Introduction

NOTE: This entry is translated from the original entry written in Japanese, at May 23, 2018

I saw this tweet.

He strongly deprecated the following article by Data Science Central. What did he mean?

Why Logistic Regression should be the last thing you learn when becoming a Data Scientist

## What is the point at issue?

Primaly I try to draw the same plot with R. Although the author doesn't mention that what the boundary curve mean, I read between the lines and used the logistic curve1 defined as the following: \begin{aligned} y= & \frac{1}{1+\exp(-x)}. \end{aligned}

On the other hand, You may have seen the following figure, introduced as the logistic regression. the objective variable $y$ is one of two, 0 or 1 because the logistic regression is binary classification algorithm. But the above figure shows diverse $y$ values. It's clearly contradictory.

Making reference to more advanced theory, we can expand expand the target variable from binary to integer, what we call Poisson regression. However this plot displays decimal numbers, not only integer(of course this is a introduction of logistic regression, the author must have been explained that). In addition, the legend on the right says there are two labels, True or False.

Then, I doubt that that y doesn't denotes the target variable, but an explanatory variable, and target variable is exactly binary and identified by two colors, True and False. So we can conclude that this figure shows a scatter between two explanatory variables x and y, identified by two colors which corresponds to binary target values. The 'boundary' curve means the decision boundary. We completely figure out that: The author says logistic regression can draw the decision boundary by a logistic curve.

## Really?

I will check if logistic regression model is really able to classify with the curve like the above figure. I used scikit-learn library to learn the data, and plot the result. The result is:

To compare with the above one, I colored by the prediction values by the logistic model, not the target values, and highlighted by larger triangles if the pairs are not correct. We can see that there are some incorrect predicitons around the left-bottom and right-top area.

## Conclusion

A logistic regression cannot draw its own boundary by a logistic curve. We need somewhat mathematically explanation (No black box!). With two explanatory variables $x,y$ and three coefficients $a,b,c$, and a dependent variable $z$ is defined as \begin{aligned} z= & ax+by+c.\end{aligned} $z$'s range is infinite because it depends on the explanatory variables and coefficients. Next, we convert $z$ by the logistic curve. \begin{aligned} p &= \mathit{logistic}(z)\\ &= \frac{1}{1+\exp(-x)}. \end{aligned} The variable $p$, the output of the curve function is always between zero and one. It's an important characteristic of the curve. Now, we can interpret $p$ as the probability if it is classified as True. So, we can obtain prediction by the following condition: \begin{aligned} \mathit{prediction}= & \begin{cases} \mathit{True} & \text{if }p\geq0.5\\ \mathit{False} & \text{if }p<0.5 \end{cases}.\end{aligned}

The boundary of classification given $x,y$ is $p=0.5$. We obtain the following classification condition expressed by $x, y$ \begin{aligned} \mathit{logit}(0.5)=0= & ax+by+c\end{aligned} This equation means a line. $\mathit{logit}(p)$ is logit function, the inverse of the logistic function, $\mathit{logit}(p)=\log(p/(1-p))$. The boundary is clearly straight, not a curve. Logistic regression consists from the category named linear classification models. Thus the Logistic regression cannot draw nonlinear boundary2.

I submit the above programs as a jupyter notebook on gist. https://gist.github.com/Gedevan-Aleksizde/91be05eb10323aaa234e4351d7ed4db2

p.s. Mr. R Bohn posted a critical comment which contains some topics, which are same with my explanation above. The author dodges the point and responded that how difficult to give mathematical comprehension to the CEO or other non-experts. Ok, I understand the difficulty myself because I am working with non-experts as he say. But I can't understand why he argued we should teach the beginners with such mistaken and nonsense introduction.

1: I adjusted its center and scale

2: For advanced readers: In fact you can have the logistic regression drawing nonlinear boundary with some nonlinear transformation of the explanatory variables (e.g., log, square, and so on). However, the author didn't mention such advanced techniques at all. Besides I cannot find such known transformation that draws logistic-like boundary. In any case, his introduction is misleading.

# Description

• This article targets nonprofessionals or the beginners who are not familiar to the data science1.
• I explain the common mistakes on data-analyzing for nonprofessionals, not to be deceived by wrong data-analyzing reports and for beginners not to acquire bad knowledge.
Some years have passed since the data science, big data, or other words occurred. Nowadays a lot of information about data science is found on the internet and books; statistical inference, causal inference, Bayesian statistics, machine learning, deep learning and Artificial Intellectual. Many people want to learn the data science on his own, and many companies employ the data scientists or ask the data science consultants for analyzing their own data.

However, it is hard and takes much time to understand data science methodologies deeply, so you may misuse the data science methods even without malice, and someone may abuse that with malice. You also may be deceived by malicious someone who abuse the data science methods.

Hence, in this article, I described what is the bad data science practice, and what is the not-bad one. I wrote this article targeting the nonprofessionals who ask the data scientists or data science consultants for analyzing his company’s data. But I also recommend reading this article for those who start to learn the data science, and for those who are using any analytics software yet don’t know theoretical foundations.

The examples of misuses in this articles are from my private/work experience. But I did not write what had actually happened and modified the details, so I did not meant to disparage any person or any organization. I did wrote this article to hope in my very soul that the right data science methodologies spread and contribute to human society.

This article consists of 4 sections: Section 1 is the description. In section 2, I briefly explain the fundamental concept of “analysis.” In section 3, I describe three examples of bad data analyzing and prescription for it. Section 4 is summary and conclusions of this article.

# What Is the Analysis

At first, we need to clarify the meaning of the word “analysis.” A dictionary says the word “analysis” means the followings:
1. the separating of any material or abstract entity into its constituent elements (opposed to synthesis ).
2. this process as a method of studying the nature of something or of determining its essential features and their relations: the grammatical analysis of a sentence.
3. a presentation, usually in writing, of the results of this process: The paper published an analysis of the political situation.
http://www.dictionary.com/browse/analysis
However, my definition of “analysis” is different from the above. In my thought, these definitions refer the concept of “deduction.” My definition of “analysis” is much closer to the concept of “abduction” or “retroduction,” which is introduced by C. S. Peirce than the above definitions. Briefly, I state the definition of the analysis (you must be bored with such abstract and non-practical discussion):
the process of creating hypothesis and validating it, or repeat of this process some times.
In the context of data science (empirical science), It is impossible to separate strictly the problem into constituent elements because the data contains any inseparable error/noise components arising by any reason. Analysis which is in the lexical meaning is suitable for noiseless formal logical or thought experimental problems. On the other hand, This is not suitable for the noise-containing empirical problems. For example, do you know “Hempel’s ravens?” In this universe, some excepts can occur by chance. We cannot state sentences determinedly. This is the reason why I redefine the word “analysis.”

Are we unavailable to analysis the problem and reach the correct answer because we cannot get rid of the noise from the data? No. Actually, analysis isn’t able to lead us to the correct answer. However, analysis is able to says the probability of correctness of the hypothesis. Primary, we make an idea as a hypothesis. Our idea is still just a hypothesis. Secondary, we think the consequences when the hypothesis is true. If the hypothesis is true, this consequences coincide with the reality. If so, we can says “the hypothesis is probably true.” This part is what we call “validation.” The validation part is a essential component of analysis. To strictly validate the hypothesis makes you to get rid of your subjective judgement or sales staffs’ exaggerated reports. However, we still cannot say “the hypothesis is determindly true/false.”

## Two Measurements of Error in Data Science

Q.: Can we know how much probability our hypothesis2 is true/false? A.: Data science makes you to calculate the degrees of error of your hypothesis. This is a great advantage of data science. Now, we can select the model which has less errors and we can say “this hypothesis is more close to the truth.”

Although there are some differences between statistics and machine learning methodologies, there is at least one common point: It is that there are two kind of measurements of error.

The first measurement is the goodness of fit. It denotes how much errors the hypothesis contains when it explains the data. It is also called “internal validity.”

The second is the precision of prediction. It denotes how much errors the hypothesis contains when it explains the future data that we will get next week/month/year, or the data of potential customers. It is also called “external validity” or “generalization ability.”

Both of two measurements make the analysis rigorous. Once you ignore at least one of these essential factors, your analysis report doesn’t become RIGOROUS, but RIDICULOUS, even if your data scientist/consultant uses decision tree/deep neural network/ensemble learning/other intricate methods. But in usually, to improve the precision of prediction is more difficult than to improve the other. https://xkcd.com/605/

# To Keep Yourself not Being Deceived by Ridiculous Analysis

You must carefully listen to the data scientists report. A fair data scientist does validate the hypothesis objectively, does not validate with his subjective judgement common sense. Though, this advise is too abstract for nonprofessionals to do practice. So I explain the following three famous examples.

## Example #1: Cluster Analysis

The cluster analysis provides intuitive graphics such that the tree diagram, and looks persuasive, but this is the very pitfall. The cluster analysis is not able to validate itself, and thus, strictly it is not analysis.

In detail, although there are some indices of goodness of fit on the cluster analysis, these are valid only under some premises, and it is difficult to check them. Beginners often tend to validate it whether it fits his own sense, but such usage is not analysis.
For example, k-means method, one of the most popular cluster methods is assume that each sample point is distributed normally with the mean which is equal to the center of a cluster that it belongs to. This assumption implies that any cluster is circle, but who validated that? Can you assert that the shape of clusters are not oval, torus, vortex, or other complex figure? Thus, if the report consists from only cluster analysis and author’s conclusions, you should better to doubt greatly. I interpret that the cluster analysis only plays a role in path finding of insight to data for further analyzing, or it can only make a hypothesis, and you need to monitor the forthcoming data and validate its hypothesis.

## Example #2: Regression Analysis

The regression analysis is so popular as the cluster analysis, yet unlike the cluster analysis, we can validate the regression with both types of measurements: We can use the coefficient of determination (aka R2, or R-squared) for goodness of fit, and the information criterion (AIC or BIC, or so on) for precision of prediction.

The regression analysis is also able to be abused: Excessive number of variables cause much excellent goodness of fit and very poor precision of prediction simultaneously. This phenomenon is called overfitting or overlearning. The following figure is one of the overfitting examples. if you predict the future (since 1997) stock index by a cubic polynomial, the prediction curve is drawn nearby the actual values, so its precision is not so bad. On the other hand, if you predict that by a polynomial of degree six, the curve fits the actual values better than the other by 1997, but it clearly derails since 1997. To avoid to abuse/misuse the regression analysis, we need to check that the presenter shows the precision of prediction in, for example, AIC or BIC, not only R2.

## Example #3: Hypothesis Testing

The hypothesis testing is also as popular as the above two methods. In particular, testing difference between means is very simple and useful to infer the effect of your sales promotion.

However, this method has two major shortages; (1) it is not able to calculate the precision of prediction, and (2) its theoretical complexity makes beginners to misunderstanding that. The hypothesis testing is an in-sample method, so it judges the hypothesis by only current data and does not guarantee the future. I come up with four typical examples of misunderstanding.
1. To accept the hypothesis (He can misunderstand the significance level and the power).
2. To interpret the p-value as the size of the effect (p-value is not able to show the size of effect).
3. Tuning the sample size or the significance level until he gets a result which he want to show, to repeat testing (It’s cheating!).
4. Without any adjustment, To show multiple testing result simultaneously (He can misunderstand basic joint probability).
However, because of its complexity, its explanation becomes too long to write here. I will write that someday if I get time.

# Summary and Conclusions

It is better to carefully hear how the presenter validate his model. If he only emphasize that we use the super best method of the machine learning, the deep learning or other up-to-date cool techniques ever seen, you should better to be more careful.

There are other check points of sound data science practices, but I cannot enumerate all of them because of profoundness of data science.