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a/docs/case_study_HCE/case_study_HCE.html +++ b/docs/case_study_HCE/case_study_HCE.html @@ -283,6 +283,12 @@ 22  Logistic Regression I + +
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    • @@ -519,7 +525,7 @@

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    Code 
    import numpy as np
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    data_linear = dugongs[["Length", "Age"]]
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    # Constant Model + MSE
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    # Predictions
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    yhats_linear = [theta_0_hat + theta_1_hat * x for x in xs]
    
 
    
 
    
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    # Constant Model Rug Plot
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    # SLR model scatter plot 
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    11.4 Comparing Loss Functions
    
 
    We’ve now tried our hand at fitting a model under both MSE and MAE cost functions. How do the two results compare?
    
 
    Let’s consider a dataset where each entry represents the number of drinks sold at a bubble tea store each day. We’ll fit a constant model to predict the number of drinks that will be sold tomorrow.
    
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    np.mean(drinks), np.median(drinks)
    
 
    
 
    (np.float64(25.0), np.float64(22.0))
    
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    Notice that the MSE above is a smooth function – it is differentiable at all points, making it easy to minimize using numerical methods. The MAE, in contrast, is not differentiable at each of its “kinks.” We’ll explore how the smoothness of the cost function can impact our ability to apply numerical optimization in a few weeks.
    
 
    How do outliers affect each cost function? Imagine we replace the largest value in the dataset with 1000. The mean of the data increases substantially, while the median is nearly unaffected.
    
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    drinks_with_outlier = np.append(drinks, 1033)
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    This means that under the MSE, the optimal model parameter \(\hat{\theta}\) is strongly affected by the presence of outliers. Under the MAE, the optimal parameter is not as influenced by outlying data. We can generalize this by saying that the MSE is sensitive to outliers, while the MAE is robust to outliers.
    
 
    Let’s try another experiment. This time, we’ll add an additional, non-outlying datapoint to the data.
    
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    drinks_with_additional_observation = np.append(drinks, 35)
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    # `corrcoef` computes the correlation coefficient between two variables
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     and "Length". What is making the raw data deviate from a linear relationship? Notice that the data points with "Length" greater than 2.6 have disproportionately high values of "Age" relative to the rest of the data. If we could manipulate these data points to have lower "Age" values, we’d “shift” these points downwards and reduce the curvature in the data. Applying a logarithmic transformation to \(y_i\) (that is, taking \(\log(\) "Age" \()\) ) would achieve just that.
    
 
    An important word on \(\log\): in Data 100 (and most upper-division STEM courses), \(\log\) denotes the natural logarithm with base \(e\). The base-10 logarithm, where relevant, is indicated by \(\log_{10}\).
    
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    \[\log{(y)} = \theta_0 + \theta_1 x\] \[y = e^{\theta_0 + \theta_1 x}\] \[y = (e^{\theta_0})e^{\theta_1 x}\] \[y_i = C e^{k x}\]
    
 
    For some constants \(C\) and \(k\).
    
 
    \(y\) is an exponential function of \(x\). Applying an exponential fit to the untransformed variables corroborates this finding.
    
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    plt.figure(dpi=120, figsize=(4, 3))
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    In sklearn, the train_test_split function (documentation) of the model_selection module allows us to automatically generate train-test splits.
    
 
    We will work with the vehicles dataset from previous lectures. As before, we will attempt to predict the mpg of a vehicle from transformations of its hp. In the cell below, we allocate 20% of the full dataset to testing, and the remaining 80% to training.
    
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    import pandas as pd
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    Y = vehicles["mpg"]
    
 
    
 
    
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    from sklearn.model_selection import train_test_split
 
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    After performing our train-test split, we fit a model to the training set and assess its performance on the test set.
    
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    import sklearn.linear_model as lm
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    \(\lambda\) the regularization penalty hyperparameter; it needs to be determined prior to training the model, so we must find the best value via cross-validation.
    
 
    The process of finding the optimal \(\hat{\theta}\) to minimize our new objective function is called L1 regularization. It is also sometimes known by the acronym “LASSO”, which stands for “Least Absolute Shrinkage and Selection Operator.”
    
 
    Unlike ordinary least squares, which can be solved via the closed-form solution \(\hat{\theta}_{OLS} = (\mathbb{X}^{\top}\mathbb{X})^{-1}\mathbb{X}^{\top}\mathbb{Y}\), there is no closed-form solution for the optimal parameter vector under L1 regularization. Instead, we use the Lasso model class of sklearn.
    
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    import sklearn.linear_model as lm
 
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    16.2.3 Scaling Features for Regularization
    
 
    The regularization procedure we just performed had one subtle issue. To see what it is, let’s take a look at the design matrix for our lasso_model.
    
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    \(\hat{y}\) because it is so much greater than the values of the other features. For hp to have much of an impact at all on the prediction, it must be scaled by a large model parameter.
    
 
    By inspecting the fitted parameters of our model, we see that this is the case – the parameter for hp is much larger in magnitude than the parameter for hp^4.
    
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    pd.DataFrame({"Feature":X_train.columns, "Parameter":lasso_model.coef_})
    
 
    
 
    
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    \[\hat\theta_{\text{ridge}} = (\mathbb{X}^{\top}\mathbb{X} + n\lambda I)^{-1}\mathbb{X}^{\top}\mathbb{Y}\]
    
 
    This solution exists even if \(\mathbb{X}\) is not full column rank. This is a major reason why L2 regularization is often used – it can produce a solution even when there is colinearity in the features. We will discuss the concept of colinearity in a future lecture, but we will not derive this result in Data 100, as it involves a fair bit of matrix calculus.
    
 
    In sklearn, we perform L2 regularization using the Ridge class. It runs gradient descent to minimize the L2 objective function. Notice that we scale the data before regularizing.
    
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    ridge_model = lm.Ridge(alpha=1) # alpha represents the hyperparameter lambda
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    Data Cleaning and EDA
    
 
 
 
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    5.1.1.1 CSV
    
 
    CSVs, which stand for Comma-Separated Values, are a common tabular data format. In the past two pandas lectures, we briefly touched on the idea of file format: the way data is encoded in a file for storage. Specifically, our elections and babynames datasets were stored and loaded as CSVs:
    
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