{
  "source_file": "0930",
  "source_platform": "Medium",
  "author": "James Andrew Godwin",
  "title_listed": "Logistic Regression",
  "keywords": "",
  "domain": "机器学习",
  "content_type": "深度技术解析",
  "is_paid_flag": "否",
  "url": "https://towardsdatascience.com/logistic-regression-cebee0728cbf/",
  "publish_date": "2021-04-14",
  "corpus_stage": "",
  "notes": "",
  "id": "b251a31394c7",
  "collected_at": "2026-10-02T08:34:38",
  "title": "Logistic Regression | Towards Data Science",
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It also helps in adverstising on the website.\n\n\n  * Cookie\nedgebucket\n  * Duration\nsession\n  * Description\nReddit sets this cookie to save the information about a log-on Reddit user, for the purpose of advertisement recommendations and updating the content.\n\n\n  * Cookie\ndid\n  * Duration\n1 year\n  * Description\nArbor sets this cookie to show targeted ads to site visitors.This cookie expires after 2 months or 1 year.\n\n\nUncategorised\nOther uncategorised cookies are those that are being analysed and have not been classified into a category as yet.\nNo cookies to display.\nReject All Save My Preferences Accept All\nToggle Search[](https://towardsdatascience.com/)Toggle Mobile Navigation\n[](https://towardsdatascience.com/)\nPublish AI, ML & data-science insights to a global community of data professionals.\nSign in[Submit an Article](https://write.insightmediagroup.io)\n[Latest](https://towardsdatascience.com/latest/)[Editor's Picks](https://towardsdatascience.com/tag/editors-pick/)[Newsletter](https://towardsdatascience.com/tag/the-variable/)[ShipAI](https://towardsdatascience.com/shipai/)\n* * *\n[Write For TDS](https://towardsdatascience.com/submissions/)\n[LinkedIn](https://www.linkedin.com/company/towards-data-science/)[X](https://x.com/TDataScience)Toggle Search\n[Data Science](https://towardsdatascience.com/category/data-science/)\n# Logistic Regression\nOr how I learned to love the sigmoid \"squishification\" function for categorical data classification\n[James Andrew Godwin](https://towardsdatascience.com/author/james-andrew/)\nApril 13, 20215 min read\n![Logistic Regression | Photo by Denise Bossarte](../images/000.jpeg)\nLogistic Regression | Photo by [Denise Bossarte](https://unsplash.com/@dbossarte)\nThis article is a brief continuation of my [regression](https://towardsdatascience.com/an-introduction-to-linear-regression-9cbb64b52d23) series.\nSo far the regression examples I have been illustrating have been numeric, of numbers: predicting a continuous variable. With the Galton family height dataset, we were predicting children's height - a continuously varying parameter. Yet, note how a regression line fails to fit a binary outcome, a classification example:\n![Line of best fit for binary outcomes](../images/001.png)\nLine of best fit for binary outcomes\nIn the above example, there appears to be a threshold affecting the classification outcomes. It does not make sense to have a continuous regression line for this data. Why does it not make sense? Look at the two dependent (y) variables: they take on one of two values: 0 and 1. There is no value for 0.25 or 0.5, just those two extrema. This means that for this classification problem we want a function that takes as an input, or a series of inputs, and gives us a probability of it being one of the two classes: namely 0 and 1. We can give any arbitrary label to these classes: yes or no, cat or not cat, send a promotional email or do not send a promotional email, sales lead or unlikely sales lead.\nLogistic regression is a powerful tool for data scientists and machine learning engineers alike. I would say, for sanity checks, that it should be the baseline model for classification purposes - one that is compared against more complex models (like PCR, neural networks, etc).\n## History\n  * Developed as a model for population growth, the term logistic came from Belgian mathematician Pierre François Verhulst, who died young and in relative obscurity\n  * The logistic function was independently developed in chemistry as a model of autocatalysis by German chemist Wilhelm Ostwald (1883)\n  * The logistic function was independently rediscovered as a model of population growth in 1920 by Americans Raymond Pearl and Lowell Reed, which led to its use in modern statistics. They were unaware of Verhulst earlier discovery but they gave him little credit and did not adopt his terminology\n  * The probit model was developed in the mid 1930's, generally credited to American biologist Chester Bliss\n  * The multinomial logit model was introduced independently in David Cox (1966) and Henri Thiel (1969)\n\n\n## Logistic regression\nTo get to logistic regression, we first begin with our classic regression example:\n![Classic regression](../images/002.png)\nClassic regression\nTo develop a model for classification we will borrow the sigmoid function from math, commonly expressed below as:\n![Sigmoid formula](../images/003.png)\nSigmoid formula\nRearranging and substituting into the regression:\n![Formula for logistic regression](../images/body-e97ded1ee13f49e385d5.img)\nFormula for logistic regression\nWhat this family of functions does is give a probability of assigning the observation to the target class. Once the logistic regression function assigns a probability, this requires the model to be fit and trained on the dataset, an example further down, the data point is projected into a class based on the probability value. For probability values greater than 0.5 the value is projected onto 1, for us the target class, and for probability values less than 0.5, they are projected onto the 0 class, or not the target class.\nA standard sigmoid function looks as follows:\n![Standard logistic function | Photo source Wikipedia.\\)](../images/004.png)\nStandard logistic function | Photo source [Wikipedia](https://en.wikipedia.org/wiki/Logistic_regression#:~:text=Logistic%20regression%20is%20a%20statistical,a%20form%20of%20binary%20regression).)\nThis transforms our original dataset regression line into:\n![The inflection point is at 0.5](../images/005.png)\nThe inflection point is at 0.5\n### What just happened?\nThis \"s\" shape is a powerful tool for classification purposes. I like to think that given probability value output for a given observation, say 0.36, that the logistic regression is \"squishing\" the probability down to binary value. Interpreting this, the model is saying that there is only a 36% chance that the given observation belongs to the target class the model was trained on - let us say that is 1. Meaning that there is a 1 - .36 or 64% chance that this observation does not belong to the target class, so it is labeled as 0 ->the observation is projected down to the dependent variable outcome label of 0.\nThe logistic regression line is similar to our previous regression examples, but **we interpret the output as a probability of assigning the data point to the target class**.\nThe most common flavor of logistic regression is binary logistic regression, but multinomial logit models do exist for multiclass regression analysis.\nN.B: It is technically possible to achieve a value of 0.5, but vanishingly small. Regardless, the algorithm you are using will have some predefined instruction on how to classify this edge case.\n### Python example: Logistic regression\nFor illustrative purposes, I will be using the famous iris dataset included with `sklean` to demonstrate the classification power in the following Jupyter notebook cells:\ntext\n\n```\nimport numpy as np\nimport pandas as pd\nimport matplotlib.pyplot as plt\nimport seaborn as sns\nfrom sklearn.model_selection import train_test_split\nfrom sklearn.preprocessing import StandardScaler\nfrom sklearn.metrics import accuracy_score\n```\n\ntext\n\n```\n%matplotlib inline\n```\n\nCreate a df and transform all the raw data into a format familiar to `sklearn:`\ntext\n\n```\nfrom sklearn.datasets import load_iris\niris = load_iris()\ndf = pd.DataFrame(data= np.c_[iris['data'], iris['target']],\n columns= [\"sepel_length_cm\", \"sepal_width_cm\",\n \"petal_length_cm\", \"petal_width_cm\", \"target\"])\n```\n\ntext\n\n```\ndf.head()\n```\n\n![Data is well imported](../images/006.png)\nData is well imported\ntext\n\n```\ndf.describe()\n```\n\n![Basic summary statistics](../images/007.png)\nBasic summary statistics\nThe data set contains the true labels, and we can see how nice and separable the three classes are:\ntext\n\n```\nsns.pairplot(df, height=2.5, hue='target')\nplt.show()\n```\n\n![The classifications for the three flower types based on their measurables](../images/008.png)\nThe classifications for the three flower types based on their measurables\ntext\n\n```\ndf['target'].value_counts()\n```\n\n![](../images/body-f1ad29e8f4a4152ebc34.png)\nNote how we have three classes of flowers, each with 50 observations. This dataset is very well balanced.\ntext\n\n```\nX, y = load_iris(return_X_y=True)\n```\n\ntext\n\n```\nX[:5]\n```\n\n![](../images/body-755b16e065c07df4c9a2.png)\ntext\n\n```\ny[:5]\n```\n\n![](../images/body-e93a59add9d75fd7c44b.png)\nThe above five flowers are all of the 0 class.\ntext\n\n```\nfrom sklearn.linear_model import LogisticRegression\n```\n\ntext\n\n```\n# instantiate the model and fit\nlog = LogisticRegression(multi_class='warn', solver='warn')\nlog_reg = log.fit(X, y)\n```\n\ntext\n\n```\nlog_reg\n```\n\n![](../images/body-c605a27841a952f175fd.png)\ntext\n\n```\n# score the model\nlog.score(X, y)\n```\n\n![](../images/body-cc7cb99dbf6f002390bc.png)\nAs we can see, this is a very good score for the model!\nNext, I will talk about [Time Series Analysis](https://towardsdatascience.com/time-series-analysis-7138ec68754a)!\nFind me on [Linkedin](https://www.linkedin.com/in/james-a-w-godwin/)\n_Physicist cum Data Scientist- Available for new opportunity | SaaS | Sports | Start-ups | Scale-ups_\n## Register Now to Continue Reading\n### Unlock Towards Data Science with a Free Account\nAll articles on Towards Data Science are available to registered users. 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With the Galton family height dataset, we were predicting children's height - a continuously varying parameter. Yet, note how a regression line fails to fit a binary outcome, a classification example:\n![Line of best fit for binary outcomes](https://assets.insightmediagroup.io/media/wp-content/uploads/2021/04/1_5vZcCovYSHc1sK8cuc6Ew.png)\nLine of best fit for binary outcomes\nIn the above example, there appears to be a threshold affecting the classification outcomes. It does not make sense to have a continuous regression line for this data. Why does it not make sense? Look at the two dependent (y) variables: they take on one of two values: 0 and 1. There is no value for 0.25 or 0.5, just those two extrema. This means that for this classification problem we want a function that takes as an input, or a series of inputs, and gives us a probability of it being one of the two classes: namely 0 and 1. We can give any arbitrary label to these classes: yes or no, cat or not cat, send a promotional email or do not send a promotional email, sales lead or unlikely sales lead.\nLogistic regression is a powerful tool for data scientists and machine learning engineers alike. I would say, for sanity checks, that it should be the baseline model for classification purposes - one that is compared against more complex models (like PCR, neural networks, etc).\n## History\n  * Developed as a model for population growth, the term logistic came from Belgian mathematician Pierre François Verhulst, who died young and in relative obscurity\n  * The logistic function was independently developed in chemistry as a model of autocatalysis by German chemist Wilhelm Ostwald (1883)\n  * The logistic function was independently rediscovered as a model of population growth in 1920 by Americans Raymond Pearl and Lowell Reed, which led to its use in modern statistics. They were unaware of Verhulst earlier discovery but they gave him little credit and did not adopt his terminology\n  * The probit model was developed in the mid 1930's, generally credited to American biologist Chester Bliss\n  * The multinomial logit model was introduced independently in David Cox (1966) and Henri Thiel (1969)\n\n\n## Logistic regression\nTo get to logistic regression, we first begin with our classic regression example:\n![Classic regression](https://assets.insightmediagroup.io/media/wp-content/uploads/2021/04/0XTYpkb9w8CSLaPVC.png)\nClassic regression\nTo develop a model for classification we will borrow the sigmoid function from math, commonly expressed below as:\n![Sigmoid formula](https://assets.insightmediagroup.io/media/wp-content/uploads/2021/04/0zVc1MM0yfUo7z7Ax.png)\nSigmoid formula\nRearranging and substituting into the regression:\n![Formula for logistic regression](https://assets.insightmediagroup.io/media/0*JN1GwCgjAeV6G3Pw)\nFormula for logistic regression\nWhat this family of functions does is give a probability of assigning the observation to the target class. Once the logistic regression function assigns a probability, this requires the model to be fit and trained on the dataset, an example further down, the data point is projected into a class based on the probability value. For probability values greater than 0.5 the value is projected onto 1, for us the target class, and for probability values less than 0.5, they are projected onto the 0 class, or not the target class.\nA standard sigmoid function looks as follows:\n![Standard logistic function | Photo source Wikipedia.\\)](https://assets.insightmediagroup.io/media/wp-content/uploads/2021/04/15nKWsukS6lPR-7fHtlK2Rg.png)\nStandard logistic function | Photo source [Wikipedia](https://en.wikipedia.org/wiki/Logistic_regression#:~:text=Logistic%20regression%20is%20a%20statistical,a%20form%20of%20binary%20regression).)\nThis transforms our original dataset regression line into:\n![The inflection point is at 0.5](https://assets.insightmediagroup.io/media/wp-content/uploads/2021/04/1xBQdvDqczvLRaOhs4r8-qQ.png)\nThe inflection point is at 0.5\n### What just happened?\nThis \"s\" shape is a powerful tool for classification purposes. I like to think that given probability value output for a given observation, say 0.36, that the logistic regression is \"squishing\" the probability down to binary value. Interpreting this, the model is saying that there is only a 36% chance that the given observation belongs to the target class the model was trained on - let us say that is 1. Meaning that there is a 1 - .36 or 64% chance that this observation does not belong to the target class, so it is labeled as 0 ->the observation is projected down to the dependent variable outcome label of 0.\nThe logistic regression line is similar to our previous regression examples, but **we interpret the output as a probability of assigning the data point to the target class**.\nThe most common flavor of logistic regression is binary logistic regression, but multinomial logit models do exist for multiclass regression analysis.\nN.B: It is technically possible to achieve a value of 0.5, but vanishingly small. Regardless, the algorithm you are using will have some predefined instruction on how to classify this edge case.\n### Python example: Logistic regression\nFor illustrative purposes, I will be using the famous iris dataset included with `sklean` to demonstrate the classification power in the following Jupyter notebook cells:\ntext\n\n```\nimport numpy as np\nimport pandas as pd\nimport matplotlib.pyplot as plt\nimport seaborn as sns\nfrom sklearn.model_selection import train_test_split\nfrom sklearn.preprocessing import StandardScaler\nfrom sklearn.metrics import accuracy_score\n```\n\ntext\n\n```\n%matplotlib inline\n```\n\nCreate a df and transform all the raw data into a format familiar to `sklearn:`\ntext\n\n```\nfrom sklearn.datasets import load_iris\niris = load_iris()\ndf = pd.DataFrame(data= np.c_[iris['data'], iris['target']],\n columns= [\"sepel_length_cm\", \"sepal_width_cm\",\n \"petal_length_cm\", \"petal_width_cm\", \"target\"])\n```\n\ntext\n\n```\ndf.head()\n```\n\n![Data is well imported](https://assets.insightmediagroup.io/media/wp-content/uploads/2021/04/1YAkbwj1ZDhaXpaoMJnJSug.png)\nData is well imported\ntext\n\n```\ndf.describe()\n```\n\n![Basic summary statistics](https://assets.insightmediagroup.io/media/wp-content/uploads/2021/04/1CIWA71YLUQ0yo4U-Maz-Hw.png)\nBasic summary statistics\nThe data set contains the true labels, and we can see how nice and separable the three classes are:\ntext\n\n```\nsns.pairplot(df, height=2.5, hue='target')\nplt.show()\n```\n\n![The classifications for the three flower types based on their measurables](https://assets.insightmediagroup.io/media/wp-content/uploads/2021/04/11Z_ofh9shVH0AjMK4cDuRA.png)\nThe classifications for the three flower types based on their measurables\ntext\n\n```\ndf['target'].value_counts()\n```\n\n![](https://assets.insightmediagroup.io/media/wp-content/uploads/2021/04/1p2jX6HxtvU1AVw5MmqPlcw.png)\nNote how we have three classes of flowers, each with 50 observations. This dataset is very well balanced.\ntext\n\n```\nX, y = load_iris(return_X_y=True)\n```\n\ntext\n\n```\nX[:5]\n```\n\n![](https://assets.insightmediagroup.io/media/wp-content/uploads/2021/04/1cPbqej4xIjN_giFwD0thKw.png)\ntext\n\n```\ny[:5]\n```\n\n![](https://assets.insightmediagroup.io/media/wp-content/uploads/2021/04/1qCB4gqGzE_C_EOxrueanYA.png)\nThe above five flowers are all of the 0 class.\ntext\n\n```\nfrom sklearn.linear_model import LogisticRegression\n```\n\ntext\n\n```\n# instantiate the model and fit\nlog = LogisticRegression(multi_class='warn', solver='warn')\nlog_reg = log.fit(X, y)\n```\n\ntext\n\n```\nlog_reg\n```\n\n![](https://assets.insightmediagroup.io/media/wp-content/uploads/2021/04/1bVOv2W3f4maK9fFhucTlGg.png)\ntext\n\n```\n# score the model\nlog.score(X, y)\n```\n\n![](https://assets.insightmediagroup.io/media/wp-content/uploads/2021/04/1Gn8a8BOg8a0vA_DRINRn3w.png)\nAs we can see, this is a very good score for the model!\nNext, I will talk about [Time Series Analysis](https://towardsdatascience.com/time-series-analysis-7138ec68754a)!\nFind me on [Linkedin](https://www.linkedin.com/in/james-a-w-godwin/)\n_Physicist cum Data Scientist- Available for new opportunity | SaaS | Sports | Start-ups | Scale-ups_\n## Register Now to Continue Reading\n### Unlock Towards Data Science with a Free Account\nAll articles on Towards Data Science are available to registered users. 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