Python Data Science Handbook
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Introduktion
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IPython: Beyond Normal Python8 Ämnen
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Introduction to NumPy9 Ämnen
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Understanding Data Types in Python
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The Basics of NumPy Arrays
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Computation on NumPy Arrays: Universal Functions
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Aggregations: Min, Max, and Everything In Between
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Computation on Arrays: Broadcasting
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Comparisons, Masks, and Boolean Logic
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Fancy Indexing
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Sorting Arrays
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Structured Data: NumPy's Structured Arrays
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Understanding Data Types in Python
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Data Manipulation with Pandas13 Ämnen
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Introducing Pandas Objects
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Data Indexing and Selection
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Operating on Data in Pandas
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Handling Missing Data
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Hierarchical Indexing
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Combining Datasets: Concat and Append
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Combining Datasets: Merge and Join
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Aggregation and Grouping
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Pivot Tables
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Vectorized String Operations
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Working with Time Series
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High-Performance Pandas: eval() and query()
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Further Resources
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Introducing Pandas Objects
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Visualization with Matplotlib15 Ämnen
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Simple Line Plots
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Simple Scatter Plots
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Visualizing Errors
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Density and Contour Plots
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Histograms, Binnings, and Density
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Customizing Plot Legends
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Customizing Colorbars
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Multiple Subplots
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Text and Annotation
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Customizing Ticks
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Customizing Matplotlib: Configurations and Stylesheets
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Three-Dimensional Plotting in Matplotlib
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Geographic Data with Basemap
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Visualization with Seaborn
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Further Resources
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Simple Line Plots
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Machine Learning15 Ämnen
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What Is Machine Learning?
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Introducing Scikit-Learn
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Hyperparameters and Model Validation
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Feature Engineering
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In Depth: Naive Bayes Classification
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In Depth: Linear Regression
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In-Depth: Support Vector Machines
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In-Depth: Decision Trees and Random Forests
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In Depth: Principal Component Analysis
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In-Depth: Manifold Learning
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In Depth: k-Means Clustering
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In Depth: Gaussian Mixture Models
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In-Depth: Kernel Density Estimation
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Application: A Face Detection Pipeline
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Further Machine Learning Resources
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What Is Machine Learning?
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Appendix: Figure Code
Visualizing Errors
april 18, 2021
For any scientific measurement, accurate accounting for errors is nearly as important, if not more important, than accurate reporting of the number itself. For example, imagine that I am using some astrophysical observations to estimate the Hubble Constant, the local measurement of the expansion rate of the Universe. I know that the current literature suggests a value of around 71 (km/s)/Mpc, and I measure a value of 74 (km/s)/Mpc with my method. Are the values consistent? The only correct answer, given this information, is this: there is no way to know.
Suppose I augment this information with reported uncertainties: the current literature suggests a value of around 71 ±± 2.5 (km/s)/Mpc, and my method has measured a value of 74 ±± 5 (km/s)/Mpc. Now are the values consistent? That is a question that can be quantitatively answered.
In visualization of data and results, showing these errors effectively can make a plot convey much more complete information.
Basic Errorbars
A basic errorbar can be created with a single Matplotlib function call:In [1]:
%matplotlib inline import matplotlib.pyplot as plt plt.style.use('seaborn-whitegrid') import numpy as np
In [2]:
x = np.linspace(0, 10, 50) dy = 0.8 y = np.sin(x) + dy * np.random.randn(50) plt.errorbar(x, y, yerr=dy, fmt='.k');
Here the fmt
is a format code controlling the appearance of lines and points, and has the same syntax as the shorthand used in plt.plot
, outlined in Simple Line Plots and Simple Scatter Plots.
In addition to these basic options, the errorbar
function has many options to fine-tune the outputs. Using these additional options you can easily customize the aesthetics of your errorbar plot. I often find it helpful, especially in crowded plots, to make the errorbars lighter than the points themselves:In [3]:
plt.errorbar(x, y, yerr=dy, fmt='o', color='black', ecolor='lightgray', elinewidth=3, capsize=0);
In addition to these options, you can also specify horizontal errorbars (xerr
), one-sided errorbars, and many other variants. For more information on the options available, refer to the docstring of plt.errorbar
.
Continuous Errors
In some situations it is desirable to show errorbars on continuous quantities. Though Matplotlib does not have a built-in convenience routine for this type of application, it’s relatively easy to combine primitives like plt.plot
and plt.fill_between
for a useful result.
Here we’ll perform a simple Gaussian process regression, using the Scikit-Learn API (see Introducing Scikit-Learn for details). This is a method of fitting a very flexible non-parametric function to data with a continuous measure of the uncertainty. We won’t delve into the details of Gaussian process regression at this point, but will focus instead on how you might visualize such a continuous error measurement:In [4]:
from sklearn.gaussian_process import GaussianProcess # define the model and draw some data model = lambda x: x * np.sin(x) xdata = np.array([1, 3, 5, 6, 8]) ydata = model(xdata) # Compute the Gaussian process fit gp = GaussianProcess(corr='cubic', theta0=1e-2, thetaL=1e-4, thetaU=1E-1, random_start=100) gp.fit(xdata[:, np.newaxis], ydata) xfit = np.linspace(0, 10, 1000) yfit, MSE = gp.predict(xfit[:, np.newaxis], eval_MSE=True) dyfit = 2 * np.sqrt(MSE) # 2*sigma ~ 95% confidence region
We now have xfit
, yfit
, and dyfit
, which sample the continuous fit to our data. We could pass these to the plt.errorbar
function as above, but we don’t really want to plot 1,000 points with 1,000 errorbars. Instead, we can use the plt.fill_between
function with a light color to visualize this continuous error:In [5]:
# Visualize the result plt.plot(xdata, ydata, 'or') plt.plot(xfit, yfit, '-', color='gray') plt.fill_between(xfit, yfit - dyfit, yfit + dyfit, color='gray', alpha=0.2) plt.xlim(0, 10);
Note what we’ve done here with the fill_between
function: we pass an x value, then the lower y-bound, then the upper y-bound, and the result is that the area between these regions is filled.
The resulting figure gives a very intuitive view into what the Gaussian process regression algorithm is doing: in regions near a measured data point, the model is strongly constrained and this is reflected in the small model errors. In regions far from a measured data point, the model is not strongly constrained, and the model errors increase.
For more information on the options available in plt.fill_between()
(and the closely related plt.fill()
function), see the function docstring or the Matplotlib documentation.
Finally, if this seems a bit too low level for your taste, refer to Visualization With Seaborn, where we discuss the Seaborn package, which has a more streamlined API for visualizing this type of continuous errorbar.