{ "source_file": "0930", "source_platform": "Medium", "author": "James Andrew Godwin", "title_listed": "Linear Regression With Bootstrapping", "keywords": "", "domain": "机器学习", "content_type": "深度技术解析", "is_paid_flag": "否", "url": "https://towardsdatascience.com/linear-regression-with-bootstrapping-4924c05d2a9/", "publish_date": "2021-03-13", "corpus_stage": "", "notes": "", "id": "32ce7526cbc6", "collected_at": "2026-10-02T08:41:08", "title": "Linear Regression With Bootstrapping | Towards Data Science", "body_markdown": "![Revisit consent button](../images/body-df29beacd7496220cc94.svg)\nWe value your privacy\nWe use cookies to enhance your browsing experience, serve personalised ads or content, and analyse our traffic. By clicking \"Accept All\", you consent to our use of cookies.\nCustomise Reject All Accept All\nCustomise Consent Preferences ![](../images/body-2159bb67f3de8ad870eb.svg)\nWe use cookies to help you navigate efficiently and perform certain functions. 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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[Machine Learning](https://towardsdatascience.com/category/artificial-intelligence/machine-learning/)\n# Linear Regression With Bootstrapping\nThere were others who had forced their way to the top from the lowest rung by the aid of their bootstraps\n[James Andrew Godwin](https://towardsdatascience.com/author/james-andrew/)\nMarch 13, 20219 min read\n![Bootstrapping Linear Regression | Photo by Ahmad Dirini](../images/000.jpeg)\nBootstrapping Linear Regression | Photo by [Ahmad Dirini](https://unsplash.com/@ahmadirini)\nThis article builds on my [Linear Regression](https://towardsdatascience.com/an-introduction-to-linear-regression-9cbb64b52d23) and [Bootstrap Resampling](https://towardsdatascience.com/bootstrap-resampling-2b453bb036ec) pieces.\nFor the literary-minded among my readers, the subtitle is a quote from 'Ulysses' 1922, by James Joyce! The origin of the term \"bootstrap\" is in literature, though not from Joyce. The usage denotes: to better oneself by one's own efforts - further evolving to encompass metaphors for a series of self-sustaining processes that proceed without external help, the context we are likely most familiar with.\nFor data scientists and machine learning engineers, this bootstrapping context is an important tool for sampling data. For this reason, it is one of the most important to consider what underlies the variation of numbers, the variation of distributions. We use resampling when we have a small amount of data as it allows us to see how much variation there would have been.\nThe following content helps to explain two types of bootstrapping as applied to linear regression models!\nLearn this step by step with the interactive [Machine Learning](https://roadmap.sh/machine-learning) roadmap.\n### Python Jupyter notebook example\nWe treat the data sample we have as the only representation of the population that we have. Then to get more datasets from it, we resample the data **with replacement**.\nFor example, say we polled a classroom of 20 students on how old the teacher is. Their 20 answers are simulated below:\ntext\n\n```\nimport numpy as np\nfrom scipy import stats\nimport matplotlib.pyplot as plt\nfrom sklearn.utils import resample\n```\n\ntext\n\n```\n%matplotlib inline\n```\n\ntext\n\n```\n# 20 student guesses of the teacher's age\nteacher_age = [24, 40, 27, 33, 31, 33, 35, 32, 29, 34, 39, 40, 41, 36, 34, 35, 29, 30, 35, 98]\n```\n\ntext\n\n```\n# expected age could be the average\nteacher_avg = np.mean(teacher_age)\nprint('Average of teacher age guesses: {} years old'.format(teacher_avg))\n```\n\ntext\n\n```\nteacher_std = np.std(teacher_age)\nprint('Std_Dev of teacher age guesses: {0:.2f} years'.format(teacher_std))\nprint(len(teacher_age))\n```\n\n![](../images/001.jpeg)\nConsidering the guesses, only 5 of them are above the average. Visually inspecting the data we see that this is because of one outlier of 98 years. While we may consider dropping this outlier to get a better guess, we can also use bootstrap resampling to get more data that approaches the parent distribution. Here we will repeatedly sample **with replacement** to get a set of subsamples. Each subsample will also have 20 data points, the same as the original.\nN.B: Remember that when sampling with replacement, it is possible - but unlikely, to get a subsample that has the outlier picked several times! We set the `replace=True` flag in `np.random.choice` .\ntext\n\n```\nn_sets = 100\nn_samples = len(teacher_age)\n```\n\ntext\n\n```\ndef generate_samples(dataset, n):\n return list(np.random.choice(dataset, size=n, replace=True))\n```\n\ntext\n\n```\nboot_samples = [generate_samples(teacher_age, n_samples) for _ in range(n_sets)]\n```\n\ntext\n\n```\nprint('Here are the top 3 samples generated:')\nprint('{}, ...'.format(boot_samples[0:3]))\n```\n\n![](../images/body-463eb94c18ae02fb370a.jpeg)\nNow we calculate the mean and standard deviation of each sample and then output the average of the averaged means and the average of the 100 subsampled standard deviations\ntext\n\n```\nsample_means = [np.mean(x) for x in boot_samples]\nsample_stdev = [np.std(x) for x in boot_samples]\n```\n\ntext\n\n```\n# take the average of all the means\nset_mean = np.mean(sample_means)\n# average of all the std_devs\nset_stdev = np.mean(sample_stdev)\n```\n\ntext\n\n```\nprint('Average of the sample averages: {0:.2f}'.format(set_mean))\nprint('Average of the sample st. devs: {0:.2f}'.format(set_stdev))\n```\n\n![A little better](../images/002.jpeg)\nA little better\nHmm - so far bootstrapping has barely changed anything. The sample averages of the sets are very similar (which we expect), but we also have not done much with the 100 subsamples we created.\nLet us create a function that drops the lowest and highest N elements from the subsamples and apply this function to each sample and then investigate the averages and standard deviations of all the bootstrapped samples:\ntext\n\n```\ndef trim_mean(dataset, n):\n # sort dataset in ascending order\n sorted_list = sorted(dataset)\n # trim the list (each list is of size: 20 - 2 = 18)\n trimmed_list = sorted_list[n:(len(sorted_list) - n)]\n return(trimmed_list)\n```\n\ntext\n\n```\n# apply function to all n (100) samples\nboot_trims = [trim_mean(x, 1) for x in boot_samples]\n```\n\ntext\n\n```\n# each sample mean and st. dev.\ntrimmed_means = [np.mean(x) for x in boot_trims]\ntrimmed_stdevs = [np.std(x) for x in boot_trims]\n```\n\ntext\n\n```\n# average of all the means\ntrimmed_mean_avg = np.mean(trimmed_means)\n# And the average of all the standard deviations\ntrimmed_stdev = np.mean(trimmed_stdevs)\n```\n\ntext\n\n```\nprint('Average of the sample averages: {:.2f}'.format(trimmed_mean_avg))\nprint('Average of the sample std_devs: {:.2f}'.format(trimmed_stdev))\n```\n\n![Note the change in numbers!](../images/003.jpeg)\nNote the change in numbers!\n### Histogram of the means\ntext\n\n```\nimport seaborn as sns\n```\n\ntext\n\n```\nbins = [x for x in range(25, 55)]\n```\n\ntext\n\n```\nax = sns.distplot(sample_means, bins, label='Original Bootstrap')\nax = sns.distplot(trimmed_means, bins, label='Trimmed Bootstrap')\nax.legend(loc='upper right')\nplt.show()\n```\n\n![Notice with the Trimmed means we have fewer outlier data to the right](../images/004.jpeg)\nNotice with the Trimmed means we have fewer outlier data to the right\nThe multi-modality (multiple peaks) are an artifact of the subsampling - they are heavily influenced by the random data subsampled, but overall the data appear to be normally distributed.\nHow do the distributions differ if we trim more data points? Let us find out:\ntext\n\n```\n# create different trimmed means\nboot_trim1 = [trimmed_mean(x, 1) for x in boot_samples]\nboot_trim2 = [trimmed_mean(x, 2) for x in boot_samples]\nboot_trim3 = [trimmed_mean(x, 3) for x in boot_samples]\nboot_trim4 = [trimmed_mean(x, 4) for x in boot_samples]\n```\n\ntext\n\n```\n# sample each mean and std dev\ntrimmed_mean1 = [np.mean(x) for x in boot_trim1]\ntrimmed_stdev1 = [np.std(x) for x in boot_trim1]\n```\n\ntext\n\n```\ntrimmed_mean2 = [np.mean(x) for x in boot_trim2]\ntrimmed_stdev2 = [np.std(x) for x in boot_trim2]\n```\n\ntext\n\n```\ntrimmed_mean3 = [np.mean(x) for x in boot_trim3]\ntrimmed_stdev3 = [np.std(x) for x in boot_trim3]\n```\n\ntext\n\n```\ntrimmed_mean4 = [np.mean(x) for x in boot_trim4]\ntrimmed_stdev4 = [np.std(x) for x in boot_trim4]\n```\n\ntext\n\n```\n# plot the means\nbins = [x for x in range(25, 55)]\n```\n\ntext\n\n```\nax = sns.distplot(trimmed_mean1, bins, label='Trimmed 1') #trim most #extreme min and max values\nax = sns.distplot(trimmed_mean2, bins, label='Trimmed 2')\nax = sns.distplot(trimmed_mean3, bins, label='Trimmed 3')\nax = sns.distplot(trimmed_mean4, bins, label='Trimmed 4') # trim 4 #most extreme min and max values\nax.legend(loc='upper right')\nplt.show()\n```\n\n![](../images/body-dfcdea94c2275f743e5e.jpeg)\nNotice how there is less and less multi-modality as more and more data is trimmed. Taken to the limit this trimming function would return only the value closest to the median. The students guesses are now closely distributed around what we would consider the \"true mean,\" which is the teachers age!\n### Bootstrapping Regression\nThe bootstrap method can be applied to regression models. Bootstrapping a regression model gives insight into how variable the model parameters are. It is useful to know how much random variation there is in regression coefficients simply because of small changes in data values.\nAs with most statistics, it is possible to bootstrap almost any regression model. However, since bootstrap resampling uses a large number of subsamples, it can be computationally intensive.\nN.B: For large-scale problems, it is necessary to use other resampling methods like k-fold cross-validation. Happy Data Science Land (nee Physics Land) is not the same grey world of floating-point inaccuracies and compromises need to be made.\nLet us create our data and fit the linear regression:\ntext\n\n```\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport pandas as pd\nimport statsmodels.formula.api as sm\n```\n\ntext\n\n```\n# synthetic sample data\nn_points = 25\nx = np.linspace(0, 10, n_points)\ny = x + (np.random.rand(len(x)) * 5)\ndata_df = pd.DataFrame({'x': x, 'y': y})\n```\n\ntext\n\n```\nols_model = sm.ols(formula = 'y ~ x', data=data_df)\n```\n\ntext\n\n```\nresults = ols_model.fit()\n```\n\ntext\n\n```\n# coefficients\nprint('Intercept, x-Slope : {}'.format(results.params))\ny_pred = ols_model.fit().predict(data_df['x'])\n```\n\ntext\n\n```\n# plot results\nplt.scatter(x, y)\nplt.plot(x, y_pred, linewidth=2)\nplt.grid(True)\nplt.xlabel('x')\nplt.ylabel('y')\nplt.title('x vs y')\nplt.show()\n```\n\n![Least-squares linear regression](../images/005.jpeg)\nLeast-squares linear regression\n### Resample points\nThe first type of bootstrapping to perform is parametric resampling. Parametric because our model has a parameter, the mean, which we are trying to estimate.\nThe pandas `sample` function will help us with this process:\ntext\n\n```\n# resample with replacement each row\nboot_slopes = []\nboot_interc = []\nn_boots = 100\nplt.figure()\nfor _ in range(n_boots):\n # sample the rows, same size, with replacement\n sample_df = data_df.sample(n=n_points, replace=True)\n # fit a linear regression\n ols_model_temp = sm.ols(formula = 'y ~ x', data=sample_df)\n results_temp = ols_model_temp.fit()\n\n # append coefficients\n boot_interc.append(results_temp.params[0])\n boot_slopes.append(results_temp.params[1])\n\n # plot a greyed out line\n y_pred_temp = ols_model_temp.fit().predict(sample_df['x'])\n plt.plot(sample_df['x'], y_pred_temp, color='grey', alpha=0.2)\n```\n\ntext\n\n```\n# add data points\nplt.scatter(x, y)\nplt.plot(x, y_pred, linewidth=2)\nplt.grid(True)\nplt.xlabel('x')\nplt.ylabel('y')\nplt.title('x vs y')\nplt.show()\n```\n\n![Note the shape of the greyed out bands | They are more concentrated at center than the edge](../images/006.jpeg)\nNote the shape of the greyed out bands | They are more concentrated at center than the edge\nThe bands above are positioned more around the center of the data than at the edges because more data are concentrated at the center than at the edges.\nDistribution of slope and intercept coefficients:\ntext\n\n```\nsns.distplot(boot_slopes)\nplt.show()\n```\n\n![Nice normal distribution of slopes](../images/007.jpeg)\nNice normal distribution of slopes\ntext\n\n```\nsns.distplot(boot_interc)\nplt.show()\n```\n\n![Pretty normal distribution of std_dev of slopes](../images/008.jpeg)\nPretty normal distribution of std_dev of slopes\nThe data above paints a pretty picture from parametric bootstrapping. However, if by chance, and this is more likely, we had sparse data, there may be a chance that our random selection of points are entirely in one area and not in another - recall the mention that the outlier can be sampled several times despite being a single outlying point.\nTo fight this, we can apply a different type of bootstrapping, called '**non-parametric bootstrapping** ' whereby we apply bootstrapping on the residuals, and not the parameter itself.\n## Resample Residuals (Non-Parametric Bootstrapping)\nThe idea is that we want to observe a point at every x-position that we have a point in the original data. Therefore, **we fit the original data and re-shuffle the residuals at every point**.\nN.B: The assumption underlying our ability to do this is that the errors are independent and identically distributed (**IID**) among our data.\ntext\n\n```\n# create synthetic sample data\nn_points = 25\nx = np.linspace(0, 10, n_points)\ny = x + (np.random.rand(len(x)) * 5)\ndata_df = pd.DataFrame({'x': x, 'y': y})\n```\n\ntext\n\n```\n# fit linear regression model\nols_model = sm.ols(formula = 'y ~ x', data=data_df)\nresults = ols_model.fit()\n```\n\ntext\n\n```\n# get predicted values\ny_pred = results.predict(data_df['x'])\n```\n\ntext\n\n```\n# get residual values\nresids = results.resid\n```\n\nNow that we fitted the values and residuals from our least-squares fit, we generate **N** (100) new sets of data by **sampling the residuals and adding them to the fitted values** :\ntext\n\n```\nboot2_slopes = []\nboot2_interc = []\nn_boots = 100\nplt.figure()\nfor _ in range(n_boots):\n # create a sampling of the residuals with replacement\n boot_resids = np.random.choice(resids, n_points, replace=True)\n y_temp = [y_pred_i + resid_i for y_pred_i, resid_i in zip(y_pred, boot_resids)]\n\n sample_df = pd.DataFrame({'x': list(x), 'y': y_temp})\n # Fit a linear regression\n ols_model_temp = sm.ols(formula = 'y ~ x', data=sample_df)\n results_temp = ols_model_temp.fit()\n\n # get coefficients\n boot2_interc.append(results_temp.params[0])\n boot2_slopes.append(results_temp.params[1])\n\n # plot a greyed out line\n y_pred_temp = ols_model_temp.fit().predict(sample_df['x'])\n plt.plot(sample_df['x'], y_pred_temp, color='grey', alpha=0.2)\n```\n\ntext\n\n```\n# add data points\nplt.scatter(x, y)\nplt.plot(x, y_pred, linewidth=2)\nplt.grid(True)\nplt.xlabel('x')\nplt.ylabel('y')\nplt.title('x vs y')\nplt.show()\n```\n\n![As before, the lines converge more in the center](../images/009.jpeg)\nAs before, the lines converge more in the center\nAgain we check the distributions of the parameters:\ntext\n\n```\nsns.distplot(boot2_slopes)\nplt.show()\n```\n\n![](../images/body-ee2fc93d55605e18c9ab.jpeg)\ntext\n\n```\nsns.distplot(boot2_interc)\nplt.show()\n```\n\n![](../images/body-e1a2e0854c1a49561c55.jpeg)\nThis helps us understand ther error in our parameters with linear regression. In fact, this type of bootstrapping can be extended to most linear and non-linear models that have a unique, optimal solution. Using this methodology allows us to get an error distribution for every parameter!\n## Conclusion\nBootstrap is not the only resampling method, there are several, but IMO it is the among the best for production models because it makes minimal assumptions about the parent distribution and is well implemented into many languages and packages.\nIn this article we looked at applying bootstrapping techniques to linear regression in two ways:\n**Parametric bootstrapping** - resampling from all of the points:\n 1. Sample the data with replacement numerous times (100)\n 2. Fit a linear regression to each sample\n 3. Store the coefficients (intercept and slopes)\n 4. Plot a histogram of the parameters\n\n\n**Non-parametric boostrapping** - resampling on the residuals with an uneven distribution of feature values:\n 1. Find the optimal linear regression on all the original data\n 2. Extract the residuals from the fit\n 3. Create new y-values using the residual samples\n 4. Fit the linear regression with the new y-values\n 5. Store the slope and intercepts\n 6. Plot a histogram of the parameters\n\n\nLike Ulysses himself in his \"Odessy\" - I hope I have created a simple narrative that helps illustrate bootstrap resampling in the context of linear regressions. Next I will continue writing about regression but with the added wrinkle of talking about overfitting and using regularization help with that!\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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The origin of the term \"bootstrap\" is in literature, though not from Joyce. The usage denotes: to better oneself by one's own efforts - further evolving to encompass metaphors for a series of self-sustaining processes that proceed without external help, the context we are likely most familiar with.\nFor data scientists and machine learning engineers, this bootstrapping context is an important tool for sampling data. For this reason, it is one of the most important to consider what underlies the variation of numbers, the variation of distributions. We use resampling when we have a small amount of data as it allows us to see how much variation there would have been.\nThe following content helps to explain two types of bootstrapping as applied to linear regression models!\nLearn this step by step with the interactive [Machine Learning](https://roadmap.sh/machine-learning) roadmap.\n### Python Jupyter notebook example\nWe treat the data sample we have as the only representation of the population that we have. Then to get more datasets from it, we resample the data **with replacement**.\nFor example, say we polled a classroom of 20 students on how old the teacher is. Their 20 answers are simulated below:\ntext\n\n```\nimport numpy as np\nfrom scipy import stats\nimport matplotlib.pyplot as plt\nfrom sklearn.utils import resample\n```\n\ntext\n\n```\n%matplotlib inline\n```\n\ntext\n\n```\n# 20 student guesses of the teacher's age\nteacher_age = [24, 40, 27, 33, 31, 33, 35, 32, 29, 34, 39, 40, 41, 36, 34, 35, 29, 30, 35, 98]\n```\n\ntext\n\n```\n# expected age could be the average\nteacher_avg = np.mean(teacher_age)\nprint('Average of teacher age guesses: {} years old'.format(teacher_avg))\n```\n\ntext\n\n```\nteacher_std = np.std(teacher_age)\nprint('Std_Dev of teacher age guesses: {0:.2f} years'.format(teacher_std))\nprint(len(teacher_age))\n```\n\n![](https://assets.insightmediagroup.io/media/wp-content/uploads/2021/03/1G_gymZnGWHXBKp5FyZt2VA.jpeg)\nConsidering the guesses, only 5 of them are above the average. Visually inspecting the data we see that this is because of one outlier of 98 years. While we may consider dropping this outlier to get a better guess, we can also use bootstrap resampling to get more data that approaches the parent distribution. Here we will repeatedly sample **with replacement** to get a set of subsamples. Each subsample will also have 20 data points, the same as the original.\nN.B: Remember that when sampling with replacement, it is possible - but unlikely, to get a subsample that has the outlier picked several times! We set the `replace=True` flag in `np.random.choice` .\ntext\n\n```\nn_sets = 100\nn_samples = len(teacher_age)\n```\n\ntext\n\n```\ndef generate_samples(dataset, n):\n return list(np.random.choice(dataset, size=n, replace=True))\n```\n\ntext\n\n```\nboot_samples = [generate_samples(teacher_age, n_samples) for _ in range(n_sets)]\n```\n\ntext\n\n```\nprint('Here are the top 3 samples generated:')\nprint('{}, ...'.format(boot_samples[0:3]))\n```\n\n![](https://assets.insightmediagroup.io/media/wp-content/uploads/2021/03/1RtNT72wzLn5_3sigwqU6qg.jpeg)\nNow we calculate the mean and standard deviation of each sample and then output the average of the averaged means and the average of the 100 subsampled standard deviations\ntext\n\n```\nsample_means = [np.mean(x) for x in boot_samples]\nsample_stdev = [np.std(x) for x in boot_samples]\n```\n\ntext\n\n```\n# take the average of all the means\nset_mean = np.mean(sample_means)\n# average of all the std_devs\nset_stdev = np.mean(sample_stdev)\n```\n\ntext\n\n```\nprint('Average of the sample averages: {0:.2f}'.format(set_mean))\nprint('Average of the sample st. devs: {0:.2f}'.format(set_stdev))\n```\n\n![A little better](https://assets.insightmediagroup.io/media/wp-content/uploads/2021/03/12PBJKqOvSz3XNtdU18giPQ.jpeg)\nA little better\nHmm - so far bootstrapping has barely changed anything. The sample averages of the sets are very similar (which we expect), but we also have not done much with the 100 subsamples we created.\nLet us create a function that drops the lowest and highest N elements from the subsamples and apply this function to each sample and then investigate the averages and standard deviations of all the bootstrapped samples:\ntext\n\n```\ndef trim_mean(dataset, n):\n # sort dataset in ascending order\n sorted_list = sorted(dataset)\n # trim the list (each list is of size: 20 - 2 = 18)\n trimmed_list = sorted_list[n:(len(sorted_list) - n)]\n return(trimmed_list)\n```\n\ntext\n\n```\n# apply function to all n (100) samples\nboot_trims = [trim_mean(x, 1) for x in boot_samples]\n```\n\ntext\n\n```\n# each sample mean and st. dev.\ntrimmed_means = [np.mean(x) for x in boot_trims]\ntrimmed_stdevs = [np.std(x) for x in boot_trims]\n```\n\ntext\n\n```\n# average of all the means\ntrimmed_mean_avg = np.mean(trimmed_means)\n# And the average of all the standard deviations\ntrimmed_stdev = np.mean(trimmed_stdevs)\n```\n\ntext\n\n```\nprint('Average of the sample averages: {:.2f}'.format(trimmed_mean_avg))\nprint('Average of the sample std_devs: {:.2f}'.format(trimmed_stdev))\n```\n\n![Note the change in numbers!](https://assets.insightmediagroup.io/media/wp-content/uploads/2021/03/1FhWUrx8chq817BiVnrK8uw.jpeg)\nNote the change in numbers!\n### Histogram of the means\ntext\n\n```\nimport seaborn as sns\n```\n\ntext\n\n```\nbins = [x for x in range(25, 55)]\n```\n\ntext\n\n```\nax = sns.distplot(sample_means, bins, label='Original Bootstrap')\nax = sns.distplot(trimmed_means, bins, label='Trimmed Bootstrap')\nax.legend(loc='upper right')\nplt.show()\n```\n\n![Notice with the Trimmed means we have fewer outlier data to the right](https://assets.insightmediagroup.io/media/wp-content/uploads/2021/03/1A-94TofwBKgcvcmfkn1y8w.jpeg)\nNotice with the Trimmed means we have fewer outlier data to the right\nThe multi-modality (multiple peaks) are an artifact of the subsampling - they are heavily influenced by the random data subsampled, but overall the data appear to be normally distributed.\nHow do the distributions differ if we trim more data points? Let us find out:\ntext\n\n```\n# create different trimmed means\nboot_trim1 = [trimmed_mean(x, 1) for x in boot_samples]\nboot_trim2 = [trimmed_mean(x, 2) for x in boot_samples]\nboot_trim3 = [trimmed_mean(x, 3) for x in boot_samples]\nboot_trim4 = [trimmed_mean(x, 4) for x in boot_samples]\n```\n\ntext\n\n```\n# sample each mean and std dev\ntrimmed_mean1 = [np.mean(x) for x in boot_trim1]\ntrimmed_stdev1 = [np.std(x) for x in boot_trim1]\n```\n\ntext\n\n```\ntrimmed_mean2 = [np.mean(x) for x in boot_trim2]\ntrimmed_stdev2 = [np.std(x) for x in boot_trim2]\n```\n\ntext\n\n```\ntrimmed_mean3 = [np.mean(x) for x in boot_trim3]\ntrimmed_stdev3 = [np.std(x) for x in boot_trim3]\n```\n\ntext\n\n```\ntrimmed_mean4 = [np.mean(x) for x in boot_trim4]\ntrimmed_stdev4 = [np.std(x) for x in boot_trim4]\n```\n\ntext\n\n```\n# plot the means\nbins = [x for x in range(25, 55)]\n```\n\ntext\n\n```\nax = sns.distplot(trimmed_mean1, bins, label='Trimmed 1') #trim most #extreme min and max values\nax = sns.distplot(trimmed_mean2, bins, label='Trimmed 2')\nax = sns.distplot(trimmed_mean3, bins, label='Trimmed 3')\nax = sns.distplot(trimmed_mean4, bins, label='Trimmed 4') # trim 4 #most extreme min and max values\nax.legend(loc='upper right')\nplt.show()\n```\n\n![](https://assets.insightmediagroup.io/media/wp-content/uploads/2021/03/1OMd_cJDAfyPVeeAB4za2tg.jpeg)\nNotice how there is less and less multi-modality as more and more data is trimmed. Taken to the limit this trimming function would return only the value closest to the median. The students guesses are now closely distributed around what we would consider the \"true mean,\" which is the teachers age!\n### Bootstrapping Regression\nThe bootstrap method can be applied to regression models. Bootstrapping a regression model gives insight into how variable the model parameters are. It is useful to know how much random variation there is in regression coefficients simply because of small changes in data values.\nAs with most statistics, it is possible to bootstrap almost any regression model. However, since bootstrap resampling uses a large number of subsamples, it can be computationally intensive.\nN.B: For large-scale problems, it is necessary to use other resampling methods like k-fold cross-validation. Happy Data Science Land (nee Physics Land) is not the same grey world of floating-point inaccuracies and compromises need to be made.\nLet us create our data and fit the linear regression:\ntext\n\n```\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport pandas as pd\nimport statsmodels.formula.api as sm\n```\n\ntext\n\n```\n# synthetic sample data\nn_points = 25\nx = np.linspace(0, 10, n_points)\ny = x + (np.random.rand(len(x)) * 5)\ndata_df = pd.DataFrame({'x': x, 'y': y})\n```\n\ntext\n\n```\nols_model = sm.ols(formula = 'y ~ x', data=data_df)\n```\n\ntext\n\n```\nresults = ols_model.fit()\n```\n\ntext\n\n```\n# coefficients\nprint('Intercept, x-Slope : {}'.format(results.params))\ny_pred = ols_model.fit().predict(data_df['x'])\n```\n\ntext\n\n```\n# plot results\nplt.scatter(x, y)\nplt.plot(x, y_pred, linewidth=2)\nplt.grid(True)\nplt.xlabel('x')\nplt.ylabel('y')\nplt.title('x vs y')\nplt.show()\n```\n\n![Least-squares linear regression](https://assets.insightmediagroup.io/media/wp-content/uploads/2021/03/1hlNKxbno88HQEk3AlvJUzg.jpeg)\nLeast-squares linear regression\n### Resample points\nThe first type of bootstrapping to perform is parametric resampling. Parametric because our model has a parameter, the mean, which we are trying to estimate.\nThe pandas `sample` function will help us with this process:\ntext\n\n```\n# resample with replacement each row\nboot_slopes = []\nboot_interc = []\nn_boots = 100\nplt.figure()\nfor _ in range(n_boots):\n # sample the rows, same size, with replacement\n sample_df = data_df.sample(n=n_points, replace=True)\n # fit a linear regression\n ols_model_temp = sm.ols(formula = 'y ~ x', data=sample_df)\n results_temp = ols_model_temp.fit()\n\n # append coefficients\n boot_interc.append(results_temp.params[0])\n boot_slopes.append(results_temp.params[1])\n\n # plot a greyed out line\n y_pred_temp = ols_model_temp.fit().predict(sample_df['x'])\n plt.plot(sample_df['x'], y_pred_temp, color='grey', alpha=0.2)\n```\n\ntext\n\n```\n# add data points\nplt.scatter(x, y)\nplt.plot(x, y_pred, linewidth=2)\nplt.grid(True)\nplt.xlabel('x')\nplt.ylabel('y')\nplt.title('x vs y')\nplt.show()\n```\n\n![Note the shape of the greyed out bands | They are more concentrated at center than the edge](https://assets.insightmediagroup.io/media/wp-content/uploads/2021/03/110MY2txDcEOnuzPVQ2v5Zg.jpeg)\nNote the shape of the greyed out bands | They are more concentrated at center than the edge\nThe bands above are positioned more around the center of the data than at the edges because more data are concentrated at the center than at the edges.\nDistribution of slope and intercept coefficients:\ntext\n\n```\nsns.distplot(boot_slopes)\nplt.show()\n```\n\n![Nice normal distribution of slopes](https://assets.insightmediagroup.io/media/wp-content/uploads/2021/03/1TJvdMyhCR9lDVNW_i2HeTw.jpeg)\nNice normal distribution of slopes\ntext\n\n```\nsns.distplot(boot_interc)\nplt.show()\n```\n\n![Pretty normal distribution of std_dev of slopes](https://assets.insightmediagroup.io/media/wp-content/uploads/2021/03/1DbEysr-MJze0AKLlC16uUA.jpeg)\nPretty normal distribution of std_dev of slopes\nThe data above paints a pretty picture from parametric bootstrapping. However, if by chance, and this is more likely, we had sparse data, there may be a chance that our random selection of points are entirely in one area and not in another - recall the mention that the outlier can be sampled several times despite being a single outlying point.\nTo fight this, we can apply a different type of bootstrapping, called '**non-parametric bootstrapping** ' whereby we apply bootstrapping on the residuals, and not the parameter itself.\n## Resample Residuals (Non-Parametric Bootstrapping)\nThe idea is that we want to observe a point at every x-position that we have a point in the original data. Therefore, **we fit the original data and re-shuffle the residuals at every point**.\nN.B: The assumption underlying our ability to do this is that the errors are independent and identically distributed (**IID**) among our data.\ntext\n\n```\n# create synthetic sample data\nn_points = 25\nx = np.linspace(0, 10, n_points)\ny = x + (np.random.rand(len(x)) * 5)\ndata_df = pd.DataFrame({'x': x, 'y': y})\n```\n\ntext\n\n```\n# fit linear regression model\nols_model = sm.ols(formula = 'y ~ x', data=data_df)\nresults = ols_model.fit()\n```\n\ntext\n\n```\n# get predicted values\ny_pred = results.predict(data_df['x'])\n```\n\ntext\n\n```\n# get residual values\nresids = results.resid\n```\n\nNow that we fitted the values and residuals from our least-squares fit, we generate **N** (100) new sets of data by **sampling the residuals and adding them to the fitted values** :\ntext\n\n```\nboot2_slopes = []\nboot2_interc = []\nn_boots = 100\nplt.figure()\nfor _ in range(n_boots):\n # create a sampling of the residuals with replacement\n boot_resids = np.random.choice(resids, n_points, replace=True)\n y_temp = [y_pred_i + resid_i for y_pred_i, resid_i in zip(y_pred, boot_resids)]\n\n sample_df = pd.DataFrame({'x': list(x), 'y': y_temp})\n # Fit a linear regression\n ols_model_temp = sm.ols(formula = 'y ~ x', data=sample_df)\n results_temp = ols_model_temp.fit()\n\n # get coefficients\n boot2_interc.append(results_temp.params[0])\n boot2_slopes.append(results_temp.params[1])\n\n # plot a greyed out line\n y_pred_temp = ols_model_temp.fit().predict(sample_df['x'])\n plt.plot(sample_df['x'], y_pred_temp, color='grey', alpha=0.2)\n```\n\ntext\n\n```\n# add data points\nplt.scatter(x, y)\nplt.plot(x, y_pred, linewidth=2)\nplt.grid(True)\nplt.xlabel('x')\nplt.ylabel('y')\nplt.title('x vs y')\nplt.show()\n```\n\n![As before, the lines converge more in the center](https://assets.insightmediagroup.io/media/wp-content/uploads/2021/03/1o5PJcnhnDocgnvXKu21pKA.jpeg)\nAs before, the lines converge more in the center\nAgain we check the distributions of the parameters:\ntext\n\n```\nsns.distplot(boot2_slopes)\nplt.show()\n```\n\n![](https://assets.insightmediagroup.io/media/wp-content/uploads/2021/03/1QS7vR8EqWVHoMuoaOxR_8w.jpeg)\ntext\n\n```\nsns.distplot(boot2_interc)\nplt.show()\n```\n\n![](https://assets.insightmediagroup.io/media/wp-content/uploads/2021/03/1daarvkBhgdPMkX2A5e19GQ.jpeg)\nThis helps us understand ther error in our parameters with linear regression. In fact, this type of bootstrapping can be extended to most linear and non-linear models that have a unique, optimal solution. Using this methodology allows us to get an error distribution for every parameter!\n## Conclusion\nBootstrap is not the only resampling method, there are several, but IMO it is the among the best for production models because it makes minimal assumptions about the parent distribution and is well implemented into many languages and packages.\nIn this article we looked at applying bootstrapping techniques to linear regression in two ways:\n**Parametric bootstrapping** - resampling from all of the points:\n 1. Sample the data with replacement numerous times (100)\n 2. Fit a linear regression to each sample\n 3. Store the coefficients (intercept and slopes)\n 4. Plot a histogram of the parameters\n\n\n**Non-parametric boostrapping** - resampling on the residuals with an uneven distribution of feature values:\n 1. Find the optimal linear regression on all the original data\n 2. Extract the residuals from the fit\n 3. Create new y-values using the residual samples\n 4. Fit the linear regression with the new y-values\n 5. Store the slope and intercepts\n 6. Plot a histogram of the parameters\n\n\nLike Ulysses himself in his \"Odessy\" - I hope I have created a simple narrative that helps illustrate bootstrap resampling in the context of linear regressions. Next I will continue writing about regression but with the added wrinkle of talking about overfitting and using regularization help with that!\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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