{
  "source_file": "0930",
  "source_platform": "Medium",
  "author": "Tirthajyoti Sarkar",
  "title_listed": "Regression in the face of messy outliers? Try Huber regressor",
  "keywords": "",
  "domain": "机器学习",
  "content_type": "深度技术解析",
  "is_paid_flag": "否",
  "url": "https://towardsdatascience.com/regression-in-the-face-of-messy-outliers-try-huber-regressor-3a54ddc12516/",
  "publish_date": "2021-11-04",
  "corpus_stage": "",
  "notes": "",
  "id": "2c7855b17929",
  "collected_at": "2026-10-02T09:07:49",
  "title": "Regression in the face of messy outliers? Try Huber regressor | 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# Regression in the face of messy outliers? Try Huber regressor\nOutliers in the data are ubiquitous, and they can mess up your regression problem. Try Hubber regressor to tackle this problem.\n[Tirthajyoti Sarkar](https://towardsdatascience.com/author/tirthajyoti/)\nNovember 4, 20214 min read\n![Image source: Created by the author](../images/000.png)\nImage source: Created by the author\n## What's the problem?\nLet's say you have a dataset with two features _X_1 and _X_2, on which you are performing linear regression. However, some noise/outliers got introduced in the dataset.\nSay, **the** _**y**_**-value outliers are exceptionally low** as compared to what they should be. How does that look like?\nThis is the data you got.\n![Image source: Created by the author](../images/001.png)\nImage source: Created by the author\nHowever, if you really think about the slope of the _X_ -_Y_ data, the expected y-values should have been much higher for those _X_ -values. Something like the following,\nLearn this step by step with the interactive [AI and Data Scientist](https://roadmap.sh/ai-data-scientist), [Computer Science](https://roadmap.sh/computer-science) and [Machine Learning](https://roadmap.sh/machine-learning) roadmaps.\n![Image source: Created by the author](../images/002.png)\nImage source: Created by the author\nThese are obvious outliers and you can run a simple exploratory data analysis (EDA) to catch and discard them from the dataset before building the regression model.\nBut you cannot hope to catch all the outliers - _**at scale and in all dimensions**_. Visualization of a dataset with 100 or 1000 dimensions (features) is challenging enough to manually examine the plots and discover outliers.\nA regression algorithm that is _robust to outliers_ sounds like a good bet against those pesky bad data points. The [Huber regressor](https://en.wikipedia.org/wiki/Huber_loss) is one such tool that we will discuss in this article.\n## Huber regression\nIn statistics, Huber loss is a particular loss function (first introduced in 1964 by [**Peter Jost Huber**](https://en.wikipedia.org/wiki/Peter_J._Huber), a Swiss mathematician) that is used widely for robust regression problems - situations where outliers are present that can degrade the performance and accuracy of [least-squared-loss](https://towardsdatascience.com/where-did-the-least-square-come-from-3f1abc7f7caf) error based regression.\n> [**Where did the least-square come from?**](https://towardsdatascience.com/where-did-the-least-square-come-from-3f1abc7f7caf)\n### Technical details\nThe loss is given by,\n![Image source: Wikipedia](../images/003.png)\nImage source: [Wikipedia](https://en.wikipedia.org/wiki/Huber_loss)\nWe can see that the loss is the square of the usual residual (_y_ - _f_(_x_)) only when the absolute value of the residual is smaller than a fixed parameter. The choice and tuning of this parameter are important to get a good estimator. Where the residual is greater than this parameter, the loss is a function of the absolute value of the residual and the Huber parameter.\nNow, you may remember from elementary statistics that the **squared loss comes from the unbiased estimator around the mean whereas an absolute difference loss comes from an unbiased estimator around the median**. Median is much more robust to outliers than mean.\n**Huber loss is a balanced compromise between these two types**. It is robust to the outliers but does not completely ignore them either. The tuning can be done with the free parameter, of course.\n![Image source: Created by the author](../images/004.png)\nImage source: Created by the author\n### A Python demo\nThe demo notebook is here in my Github repo.\nWe created the synthetic data and added some noisy outlier with the following code,\ntext\n\n```\nimport numpy as np\nfrom sklearn.datasets import make_regression\n```\n\ntext\n\n```\nrng = np.random.RandomState(0)\nX, y, coef = make_regression(\n    n_samples=200, n_features=2, noise=4.0, coef=True, random_state=0)\n```\n\ntext\n\n```\n# The first four data points are outlier\nX[:4] = rng.uniform(10, 20, (4, 2))\ny[:4] = rng.uniform(10, 20, 4)\n```\n\nNow, all you have to do is to call Scikit-learn's built-in `HuberRegressor` estimator and fit the data. For comparison, we also have the standard `LinearRegression` method called in.\ntext\n\n```\nfrom sklearn.linear_model import HuberRegressor, LinearRegression\n```\n\ntext\n\n```\nhuber = HuberRegressor().fit(X, y)\nlinear = LinearRegression().fit(X, y)\n```\n\nNow, we know that the first 4 data points are outliers. So, if we try to predict the y-value with the first data point, we will get something like this,\ntext\n\n```\nlinear.predict(X[:1,])\n```\n\ntext\n\n```\n>> array([87.38004436])\n```\n\nAs expected, the linear regression prediction is a low value for _y_. Why? Because the linear fit (based on least-squared loss) has bent towards the outliers due to their large leverage.\n![Image source: Created by the author](../images/005.png)\nImage source: Created by the author\nHowever, the Huber estimator predicts a more reasonable (high) value,\ntext\n\n```\nhuber.predict(X[:1,])\n```\n\ntext\n\n```\n>> array([806.72000092])\n```\n\nTo demonstrate the robustness of the Huber estimator further, we can use the estimated coefficients and plot the best-fitted line,\ntext\n\n```\nhuber_y1 = np.arange(-2.5,20,0.01)*huber.coef_[0] + \n            np.arange(-2.5,20,0.01)*huber.coef_[1] + \n            huber.intercept_\n```\n\ntext\n\n```\nplt.figure(dpi=120)\nplt.scatter(X[:,0],y)\nplt.plot(np.arange(-2.5,20,0.01), \n         huber_y1,\n         color='red',linestyle='--')\nplt.show()\n```\n\n![Image source: Created by the author](../images/006.png)\nImage source: Created by the author\n### Theil-Sen estimator\nAlthough we are not discussing it in this article, readers are encouraged to check the [Theil-Sen estimator](https://en.wikipedia.org/wiki/Theil%E2%80%93Sen_estimator), which is another robust linear regression technique and enjoys being highly insensitive to outliers.\nAs expected, Scikit-learn has a built-in method for this estimator too: [**Scikit-learn Theil-Sen estimator**](https://scikit-learn.org/stable/modules/generated/sklearn.linear_model.TheilSenRegressor.html#sklearn.linear_model.TheilSenRegressor).\nFor multiple linear regression with a large number of features, this is a very efficient and fast regression algorithm among all the robust estimators.\n## Summary\nIn this brief article, we talked about the problem of linear regression estimators in the presence of outliers in the dataset. We demonstrated, with a simple example, that linear estimators based on traditional least-squared loss function, may predict completely wrong values because they are bent towards the outliers.\nWe discussed a couple of robust estimators and demonstrated the Huber regressor in detail. Non-parametric statistics use these robust regression techniques in many places, especially when the data is expected to be particularly noisy.\nData science students and professionals alike should also have a working knowledge of these robust regression methods for automating the modeling of large datasets in the presence of outliers.\n* * *\n_Loved the article? Become a_[ _**Medium member**_](https://medium.com/@tirthajyoti/membership) _to continue_ _**learning without limits**_ _. I'll receive a portion of your membership fee if you use the following link,__**with no extra cost to you**_ _._\n> [**Join Medium with my referral link - Tirthajyoti Sarkar**](https://medium.com/@tirthajyoti/membership)\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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Try Huber regressor\nOutliers in the data are ubiquitous, and they can mess up your regression problem. Try Hubber regressor to tackle this problem.\n[Tirthajyoti Sarkar](https://towardsdatascience.com/author/tirthajyoti/)\nNovember 4, 20214 min read\n![Image source: Created by the author](https://assets.insightmediagroup.io/media/wp-content/uploads/2021/11/1X8YMmvAynEf7nbF5LzHvig.png)\nImage source: Created by the author\n## What's the problem?\nLet's say you have a dataset with two features _X_1 and _X_2, on which you are performing linear regression. However, some noise/outliers got introduced in the dataset.\nSay, **the** _**y**_**-value outliers are exceptionally low** as compared to what they should be. How does that look like?\nThis is the data you got.\n![Image source: Created by the author](https://assets.insightmediagroup.io/media/wp-content/uploads/2021/11/1DoE_1nKPvRFcavUG5j2K0g.png)\nImage source: Created by the author\nHowever, if you really think about the slope of the _X_ -_Y_ data, the expected y-values should have been much higher for those _X_ -values. Something like the following,\nLearn this step by step with the interactive [AI and Data Scientist](https://roadmap.sh/ai-data-scientist), [Computer Science](https://roadmap.sh/computer-science) and [Machine Learning](https://roadmap.sh/machine-learning) roadmaps.\n![Image source: Created by the author](https://assets.insightmediagroup.io/media/wp-content/uploads/2021/11/16he8cMUlfTBPXnz4NoMD4g.png)\nImage source: Created by the author\nThese are obvious outliers and you can run a simple exploratory data analysis (EDA) to catch and discard them from the dataset before building the regression model.\nBut you cannot hope to catch all the outliers - _**at scale and in all dimensions**_. Visualization of a dataset with 100 or 1000 dimensions (features) is challenging enough to manually examine the plots and discover outliers.\nA regression algorithm that is _robust to outliers_ sounds like a good bet against those pesky bad data points. The [Huber regressor](https://en.wikipedia.org/wiki/Huber_loss) is one such tool that we will discuss in this article.\n## Huber regression\nIn statistics, Huber loss is a particular loss function (first introduced in 1964 by [**Peter Jost Huber**](https://en.wikipedia.org/wiki/Peter_J._Huber), a Swiss mathematician) that is used widely for robust regression problems - situations where outliers are present that can degrade the performance and accuracy of [least-squared-loss](https://towardsdatascience.com/where-did-the-least-square-come-from-3f1abc7f7caf) error based regression.\n> [**Where did the least-square come from?**](https://towardsdatascience.com/where-did-the-least-square-come-from-3f1abc7f7caf)\n### Technical details\nThe loss is given by,\n![Image source: Wikipedia](https://assets.insightmediagroup.io/media/wp-content/uploads/2021/11/1ZdCzdOtHoWOMbxNzDF_lXg.png)\nImage source: [Wikipedia](https://en.wikipedia.org/wiki/Huber_loss)\nWe can see that the loss is the square of the usual residual (_y_ - _f_(_x_)) only when the absolute value of the residual is smaller than a fixed parameter. The choice and tuning of this parameter are important to get a good estimator. Where the residual is greater than this parameter, the loss is a function of the absolute value of the residual and the Huber parameter.\nNow, you may remember from elementary statistics that the **squared loss comes from the unbiased estimator around the mean whereas an absolute difference loss comes from an unbiased estimator around the median**. Median is much more robust to outliers than mean.\n**Huber loss is a balanced compromise between these two types**. It is robust to the outliers but does not completely ignore them either. The tuning can be done with the free parameter, of course.\n![Image source: Created by the author](https://assets.insightmediagroup.io/media/wp-content/uploads/2021/11/1HDb-puoKCJcR0WtxRT1jWg.png)\nImage source: Created by the author\n### A Python demo\nThe demo notebook is here in my Github repo.\nWe created the synthetic data and added some noisy outlier with the following code,\ntext\n\n```\nimport numpy as np\nfrom sklearn.datasets import make_regression\n```\n\ntext\n\n```\nrng = np.random.RandomState(0)\nX, y, coef = make_regression(\n    n_samples=200, n_features=2, noise=4.0, coef=True, random_state=0)\n```\n\ntext\n\n```\n# The first four data points are outlier\nX[:4] = rng.uniform(10, 20, (4, 2))\ny[:4] = rng.uniform(10, 20, 4)\n```\n\nNow, all you have to do is to call Scikit-learn's built-in `HuberRegressor` estimator and fit the data. For comparison, we also have the standard `LinearRegression` method called in.\ntext\n\n```\nfrom sklearn.linear_model import HuberRegressor, LinearRegression\n```\n\ntext\n\n```\nhuber = HuberRegressor().fit(X, y)\nlinear = LinearRegression().fit(X, y)\n```\n\nNow, we know that the first 4 data points are outliers. So, if we try to predict the y-value with the first data point, we will get something like this,\ntext\n\n```\nlinear.predict(X[:1,])\n```\n\ntext\n\n```\n>> array([87.38004436])\n```\n\nAs expected, the linear regression prediction is a low value for _y_. Why? Because the linear fit (based on least-squared loss) has bent towards the outliers due to their large leverage.\n![Image source: Created by the author](https://assets.insightmediagroup.io/media/wp-content/uploads/2021/11/19Jq5TEcdfFCjaL7uA_8gOQ.png)\nImage source: Created by the author\nHowever, the Huber estimator predicts a more reasonable (high) value,\ntext\n\n```\nhuber.predict(X[:1,])\n```\n\ntext\n\n```\n>> array([806.72000092])\n```\n\nTo demonstrate the robustness of the Huber estimator further, we can use the estimated coefficients and plot the best-fitted line,\ntext\n\n```\nhuber_y1 = np.arange(-2.5,20,0.01)*huber.coef_[0] + \n            np.arange(-2.5,20,0.01)*huber.coef_[1] + \n            huber.intercept_\n```\n\ntext\n\n```\nplt.figure(dpi=120)\nplt.scatter(X[:,0],y)\nplt.plot(np.arange(-2.5,20,0.01), \n         huber_y1,\n         color='red',linestyle='--')\nplt.show()\n```\n\n![Image source: Created by the author](https://assets.insightmediagroup.io/media/wp-content/uploads/2021/11/1DBwGphqlz1TokehmFPBlBQ.png)\nImage source: Created by the author\n### Theil-Sen estimator\nAlthough we are not discussing it in this article, readers are encouraged to check the [Theil-Sen estimator](https://en.wikipedia.org/wiki/Theil%E2%80%93Sen_estimator), which is another robust linear regression technique and enjoys being highly insensitive to outliers.\nAs expected, Scikit-learn has a built-in method for this estimator too: [**Scikit-learn Theil-Sen estimator**](https://scikit-learn.org/stable/modules/generated/sklearn.linear_model.TheilSenRegressor.html#sklearn.linear_model.TheilSenRegressor).\nFor multiple linear regression with a large number of features, this is a very efficient and fast regression algorithm among all the robust estimators.\n## Summary\nIn this brief article, we talked about the problem of linear regression estimators in the presence of outliers in the dataset. We demonstrated, with a simple example, that linear estimators based on traditional least-squared loss function, may predict completely wrong values because they are bent towards the outliers.\nWe discussed a couple of robust estimators and demonstrated the Huber regressor in detail. Non-parametric statistics use these robust regression techniques in many places, especially when the data is expected to be particularly noisy.\nData science students and professionals alike should also have a working knowledge of these robust regression methods for automating the modeling of large datasets in the presence of outliers.\n* * *\n_Loved the article? Become a_[ _**Medium member**_](https://medium.com/@tirthajyoti/membership) _to continue_ _**learning without limits**_ _. I'll receive a portion of your membership fee if you use the following link,__**with no extra cost to you**_ _._\n> [**Join Medium with my referral link - Tirthajyoti Sarkar**](https://medium.com/@tirthajyoti/membership)\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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