From 9b6d595bfbb47f9dbc094253a53e5d7f7735d4e2 Mon Sep 17 00:00:00 2001 From: Ishani Gupta Date: Tue, 10 Oct 2023 15:08:11 -0700 Subject: [PATCH] updating notes 14 --- feature_engineering/feature_engineering.ipynb | 355 ++++++++++++++---- feature_engineering/images/bias.png | Bin 0 -> 38667 bytes feature_engineering/images/remove.png | Bin 0 -> 60260 bytes 3 files changed, 278 insertions(+), 77 deletions(-) create mode 100644 feature_engineering/images/bias.png create mode 100644 feature_engineering/images/remove.png diff --git a/feature_engineering/feature_engineering.ipynb b/feature_engineering/feature_engineering.ipynb index 77b5929c..17c5e370 100644 --- a/feature_engineering/feature_engineering.ipynb +++ b/feature_engineering/feature_engineering.ipynb @@ -51,7 +51,7 @@ }, { "cell_type": "code", - "execution_count": 2, + "execution_count": 57, "metadata": {}, "outputs": [ { @@ -155,7 +155,7 @@ "5 3650.0 Male " ] }, - "execution_count": 2, + "execution_count": 57, "metadata": {}, "output_type": "execute_result" } @@ -179,7 +179,7 @@ }, { "cell_type": "code", - "execution_count": 3, + "execution_count": 58, "metadata": {}, "outputs": [], "source": [ @@ -219,7 +219,7 @@ }, { "cell_type": "code", - "execution_count": 4, + "execution_count": 59, "metadata": {}, "outputs": [ { @@ -230,7 +230,7 @@ " [1.47749591e-03]])" ] }, - "execution_count": 4, + "execution_count": 59, "metadata": {}, "output_type": "execute_result" } @@ -252,7 +252,7 @@ }, { "cell_type": "code", - "execution_count": 5, + "execution_count": 60, "metadata": {}, "outputs": [ { @@ -313,7 +313,7 @@ "4 18.263268" ] }, - "execution_count": 5, + "execution_count": 60, "metadata": {}, "output_type": "execute_result" } @@ -381,7 +381,7 @@ }, { "cell_type": "code", - "execution_count": 17, + "execution_count": 61, "metadata": {}, "outputs": [], "source": [ @@ -405,19 +405,19 @@ }, { "cell_type": "code", - "execution_count": 8, + "execution_count": 62, "metadata": {}, "outputs": [ { "data": { "text/html": [ - "
LinearRegression()
In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook.
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" + "
LinearRegression()
In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook.
On GitHub, the HTML representation is unable to render, please try loading this page with nbviewer.org.
" ], "text/plain": [ "LinearRegression()" ] }, - "execution_count": 8, + "execution_count": 62, "metadata": {}, "output_type": "execute_result" } @@ -448,7 +448,7 @@ }, { "cell_type": "code", - "execution_count": 9, + "execution_count": 63, "metadata": {}, "outputs": [ { @@ -457,7 +457,7 @@ "7.297305899612306" ] }, - "execution_count": 9, + "execution_count": 63, "metadata": {}, "output_type": "execute_result" } @@ -469,7 +469,7 @@ }, { "cell_type": "code", - "execution_count": 10, + "execution_count": 64, "metadata": {}, "outputs": [ { @@ -478,7 +478,7 @@ "array([0.05812622])" ] }, - "execution_count": 10, + "execution_count": 64, "metadata": {}, "output_type": "execute_result" } @@ -501,9 +501,17 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 65, "metadata": {}, - "outputs": [], + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "The RMSE of the model is 1.1549363099239012\n" + ] + } + ], "source": [ "y_hat_one_feature = my_model.predict(penguins[[\"flipper_length_mm\"]])\n", "\n", @@ -523,7 +531,7 @@ }, { "cell_type": "code", - "execution_count": 18, + "execution_count": 66, "metadata": {}, "outputs": [ { @@ -559,9 +567,78 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 67, "metadata": {}, - "outputs": [], + "outputs": [ + { + "data": { + "text/html": [ + "
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" + ], + "text/plain": [ + " Y_hat from OLS Y_hat from sklearn\n", + "0 18.322561 18.322561\n", + "1 18.445578 18.445578\n", + "2 17.721412 17.721412\n", + "3 17.997254 17.997254\n", + "4 18.263268 18.263268" + ] + }, + "execution_count": 67, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "#| code-fold: true\n", "pd.DataFrame({\"Y_hat from OLS\":np.squeeze(y_hat), \"Y_hat from sklearn\":y_hat_two_features}).head()" @@ -616,7 +693,7 @@ }, { "cell_type": "code", - "execution_count": 12, + "execution_count": 83, "metadata": {}, "outputs": [ { @@ -651,69 +728,69 @@ " \n", " \n", " \n", - " 54\n", - " 25.56\n", - " 4.34\n", - " Male\n", + " 0\n", + " 16.99\n", + " 1.01\n", + " Female\n", " No\n", " Sun\n", " Dinner\n", - " 4\n", + " 2\n", " \n", " \n", - " 46\n", - " 22.23\n", - " 5.00\n", + " 1\n", + " 10.34\n", + " 1.66\n", " Male\n", " No\n", " Sun\n", " Dinner\n", - " 2\n", + " 3\n", " \n", " \n", - " 86\n", - " 13.03\n", - " 2.00\n", + " 2\n", + " 21.01\n", + " 3.50\n", " Male\n", " No\n", - " Thur\n", - " Lunch\n", - " 2\n", + " Sun\n", + " Dinner\n", + " 3\n", " \n", " \n", - " 199\n", - " 13.51\n", - " 2.00\n", + " 3\n", + " 23.68\n", + " 3.31\n", " Male\n", - " Yes\n", - " Thur\n", - " Lunch\n", + " No\n", + " Sun\n", + " Dinner\n", " 2\n", " \n", " \n", - " 106\n", - " 20.49\n", - " 4.06\n", - " Male\n", - " Yes\n", - " Sat\n", + " 4\n", + " 24.59\n", + " 3.61\n", + " Female\n", + " No\n", + " Sun\n", " Dinner\n", - " 2\n", + " 4\n", " \n", " \n", "\n", "" ], "text/plain": [ - " total_bill tip sex smoker day time size\n", - "54 25.56 4.34 Male No Sun Dinner 4\n", - "46 22.23 5.00 Male No Sun Dinner 2\n", - "86 13.03 2.00 Male No Thur Lunch 2\n", - "199 13.51 2.00 Male Yes Thur Lunch 2\n", - "106 20.49 4.06 Male Yes Sat Dinner 2" + " total_bill tip sex smoker day time size\n", + "0 16.99 1.01 Female No Sun Dinner 2\n", + "1 10.34 1.66 Male No Sun Dinner 3\n", + "2 21.01 3.50 Male No Sun Dinner 3\n", + "3 23.68 3.31 Male No Sun Dinner 2\n", + "4 24.59 3.61 Female No Sun Dinner 4" ] }, - "execution_count": 12, + "execution_count": 83, "metadata": {}, "output_type": "execute_result" } @@ -721,9 +798,8 @@ "source": [ "#| code-fold: true\n", "import numpy as np\n", - "np.random.seed(1337)\n", - "tips = sns.load_dataset(\"tips\").sample(100)\n", - "tips.head(5)" + "tips = sns.load_dataset(\"tips\")\n", + "tips.head()" ] }, { @@ -736,22 +812,14 @@ "\n", "
ohe
\n", "\n", - "The one-hot encoded features can then be used in the design matrix to train a model:\n", - "\n", - "
ohemodel
\n", - "\n", - "$$\\hat{y} = \\theta_1 (\\text{total}\\textunderscore\\text{bill}) + \\theta_2 (\\text{size}) + \\theta_3 (\\text{day}\\textunderscore\\text{Fri}) + \\theta_4 (\\text{day}\\textunderscore\\text{Sat}) + \\theta_5 (\\text{day}\\textunderscore\\text{Sun}) + \\theta_6 (\\text{day}\\textunderscore\\text{Thur})$$\n", + "
\n", "\n", - "Or in shorthand:\n", - "\n", - "$$\\hat{y} = \\theta_1\\phi_1 + \\theta_2\\phi_2 + \\theta_3\\phi_3 + \\theta_4\\phi_4 + \\theta_5\\phi_5 + \\theta_6\\phi_6$$\n", - "\n", - "The `OneHotEncoder` class of `sklearn` ([documentation](https://scikit-learn.org/stable/modules/generated/sklearn.preprocessing.OneHotEncoder.html#sklearn.preprocessing.OneHotEncoder.get_feature_names_out)) offers a quick way to perform one-hot encoding. You will explore its use in detail in the lab. For now, recognize that we follow a very similar workflow to when we were working with the `LinearRegression` class: we initialize a `OneHotEncoder` object, fit it to our data, then use `.transform` to apply the fitted encoder." + "The `OneHotEncoder` class of `sklearn` ([documentation](https://scikit-learn.org/stable/modules/generated/sklearn.preprocessing.OneHotEncoder.html#sklearn.preprocessing.OneHotEncoder.get_feature_names_out)) offers a quick way to perform this one-hot encoding. You will explore its use in detail in the lab. For now, recognize that we follow a very similar workflow to when we were working with the `LinearRegression` class: we initialize a `OneHotEncoder` object, fit it to our data, then use `.transform` to apply the fitted encoder." ] }, { "cell_type": "code", - "execution_count": 13, + "execution_count": 84, "metadata": {}, "outputs": [ { @@ -800,21 +868,21 @@ " 2\n", " 0.0\n", " 0.0\n", - " 0.0\n", " 1.0\n", + " 0.0\n", " \n", " \n", " 3\n", " 0.0\n", " 0.0\n", - " 0.0\n", " 1.0\n", + " 0.0\n", " \n", " \n", " 4\n", " 0.0\n", - " 1.0\n", " 0.0\n", + " 1.0\n", " 0.0\n", " \n", " \n", @@ -825,12 +893,12 @@ " day_Fri day_Sat day_Sun day_Thur\n", "0 0.0 0.0 1.0 0.0\n", "1 0.0 0.0 1.0 0.0\n", - "2 0.0 0.0 0.0 1.0\n", - "3 0.0 0.0 0.0 1.0\n", - "4 0.0 1.0 0.0 0.0" + "2 0.0 0.0 1.0 0.0\n", + "3 0.0 0.0 1.0 0.0\n", + "4 0.0 0.0 1.0 0.0" ] }, - "execution_count": 13, + "execution_count": 84, "metadata": {}, "output_type": "execute_result" } @@ -853,11 +921,137 @@ ] }, { + "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ + "The one-hot encoded features can then be used in the design matrix to train a model:\n", + "\n", + "
ohemodel
\n", + "\n", + "$$\\hat{y} = \\theta_1 (\\text{total}\\textunderscore\\text{bill}) + \\theta_2 (\\text{size}) + \\theta_3 (\\text{day}\\textunderscore\\text{Fri}) + \\theta_4 (\\text{day}\\textunderscore\\text{Sat}) + \\theta_5 (\\text{day}\\textunderscore\\text{Sun}) + \\theta_6 (\\text{day}\\textunderscore\\text{Thur})$$\n", + "\n", + "Or in shorthand:\n", + "\n", + "$$\\hat{y} = \\theta_1\\phi_1 + \\theta_2\\phi_2 + \\theta_3\\phi_3 + \\theta_4\\phi_4 + \\theta_5\\phi_5 + \\theta_6\\phi_6$$\n", + "\n", "Now, the `\"day\"` feature (or rather, the four new boolean features that represent day) can be used to fit a model.\n", "\n", + "Using `sklearn` to fit the new model, we can determine the model coefficients, allowing us to understand how each feature impacts the predicted tip." + ] + }, + { + "cell_type": "code", + "execution_count": 85, + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "
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FeatureModel Coefficient
0total_bill0.092994
1size0.187132
2day_Fri0.745787
3day_Sat0.621129
4day_Sun0.732289
5day_Thur0.668294
\n", + "
" + ], + "text/plain": [ + " Feature Model Coefficient\n", + "0 total_bill 0.092994\n", + "1 size 0.187132\n", + "2 day_Fri 0.745787\n", + "3 day_Sat 0.621129\n", + "4 day_Sun 0.732289\n", + "5 day_Thur 0.668294" + ] + }, + "execution_count": 85, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "from sklearn.linear_model import LinearRegression\n", + "data_w_ohe = tips[[\"total_bill\", \"size\", \"day\"]].join(encoded_day_df).drop(columns = \"day\")\n", + "ohe_model = lm.LinearRegression(fit_intercept=False) #Tell sklearn to not add an additional bias column. Why?\n", + "ohe_model.fit(data_w_ohe, tips[\"tip\"])\n", + "\n", + "pd.DataFrame({\"Feature\":data_w_ohe.columns, \"Model Coefficient\":ohe_model.coef_})" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "For example, when looking at the coefficient for \"day_Fri\", we can understand how much the fact that it is Friday impacts the predicted tip. \n", + "\n", + "When one-hot encoding, keep in mind that any set of one-hot encoded columns will always sum to a column of all ones, representing the bias column. More formally, the bias column is a linear combination of the OHE columns.\n", + "\n", + "
bias
\n", + "\n", + "We must be careful not to include this bias column in our design matrix. Otherwise, there will be linear dependence in the model, meaning $\\mathbb{X}^T\\mathbb{X}$ would no longer be invertible, and our OLS estimate $\\hat{\\theta} = (\\mathbb{X}^T\\mathbb{X})^{-1}\\mathbb{X}^T\\mathbb{Y}$ fails.\n", + "\n", + "To resolve this issue, we simply omit one of the one-hot encoded columns *or* do not include an intercept term. \n", + "\n", + "
remove
\n", + "\n", + "Either approach works — we still retain the same information as the omitted column being a linear combination of the remaining columns." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ "## Polynomial Features\n", "\n", "We have encountered a few cases now where models with linear features have performed poorly on datasets that show clear non-linear curvature. \n", @@ -1007,6 +1201,13 @@ "\n", "This begs the question: how do we control the complexity of a model? Stay tuned for our Lecture 16 on Cross-Validation and Regularization!" ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] } ], "metadata": { diff --git a/feature_engineering/images/bias.png b/feature_engineering/images/bias.png new file mode 100644 index 0000000000000000000000000000000000000000..e6455ca227e12f25ce92b7362d5697f6b3c68ebb GIT binary patch literal 38667 zcmd?PWmH^C*Qnc#J0ZARf)gaT6C46G&}eXiyE`;NgC@Asc;ha?36?-`2=4BI;CecH z@9&j!&lu{&9ao{u0EIcy9v3;+OttspP04ges)001Br8VWo| zwr8gV0KgKrmXZP~NJ-IvTpTT|?aToH`HxB3s5%|jcH{qV##@YL9*{~d&-Gz+fn zrLCC|>}gs0kw(GL$MWn7eUd((=4Z%tFof}UI7#7qg*VSPA7McL9AsM*HyV#kXN0db zd=LToRC3bF)ydaLl6Dbm7?Q>47_+C(4&Jmy35zF|yN0hESGt-C@+OOKo0s&wruNeRO&?`?BLx8fQVz zC^fMd4>6TE4M@!MkklykS@GBn4a={%q_{rjyhW=}afiWC?BuheXRVvW{mlLAACa9T z2QN;s)Fj>7L{!_DGVsaRM^TM%z5+3k079R-l5rSq7QIS6E?Dhs(($EQ;qFOPyf?ry^%J@_XMApiTLZ1CDGe8dGsiU8sjN&0|i;8*-N#A~!`WVsnmw((G%dyQl~DEU4rv(4v6Gj=B>*k0t_#xjVK 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