#!/usr/bin/env python
# coding: utf-8

# <!--BOOK_INFORMATION-->
# <img align="left" style="padding-right:10px;" src="figures/PDSH-cover-small.png">
# *This notebook contains an excerpt from the [Python Data Science Handbook](http://shop.oreilly.com/product/0636920034919.do) by Jake VanderPlas; the content is available [on GitHub](https://github.com/jakevdp/PythonDataScienceHandbook).*
# 
# *The text is released under the [CC-BY-NC-ND license](https://creativecommons.org/licenses/by-nc-nd/3.0/us/legalcode), and code is released under the [MIT license](https://opensource.org/licenses/MIT). If you find this content useful, please consider supporting the work by [buying the book](http://shop.oreilly.com/product/0636920034919.do)!*
# 
# *No changes were made to the contents of this notebook from the original.*

# <!--NAVIGATION-->
# < [Simple Scatter Plots](04.02-Simple-Scatter-Plots.ipynb) | [Contents](Index.ipynb) | [Density and Contour Plots](04.04-Density-and-Contour-Plots.ipynb) >

# # Visualizing Errors

# 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 $\pm$ 2.5 (km/s)/Mpc, and my method has measured a value of 74 $\pm$ 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]:


get_ipython().run_line_magic('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](04.01-Simple-Line-Plots.ipynb) and [Simple Scatter Plots](04.02-Simple-Scatter-Plots.ipynb).
# 
# 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](05.02-Introducing-Scikit-Learn.ipynb) 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](04.14-Visualization-With-Seaborn.ipynb), where we discuss the Seaborn package, which has a more streamlined API for visualizing this type of continuous errorbar.

# <!--NAVIGATION-->
# < [Simple Scatter Plots](04.02-Simple-Scatter-Plots.ipynb) | [Contents](Index.ipynb) | [Density and Contour Plots](04.04-Density-and-Contour-Plots.ipynb) >