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import numpy as np
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data_linear = dugongs[["Length", "Age"]]
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# Big font helper
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plt.style.use("default")  # Revert style to default mpl
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# Constant Model + MSE
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# SLR + 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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drinks = np.array([20, 21, 22, 29, 33])
 drinks

 

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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)
 display(drinks_with_outlier)
 np.mean(drinks_with_outlier), np.median(drinks_with_outlier)

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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)
 drinks_with_additional_observation

 

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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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z = np.log(y)
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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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Data Cleaning and EDA

 
 
 
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import numpy as np
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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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pd.read_csv("data/elections.csv").head(5)

 

 

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