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Saturday, August 31, 2019

Matplotlib Explained - Kite Blog







If Jupyter Notebook is the new Excel, the horsepower of data science (visualizations, presentations and demos), Matplotlib is the engine or the burning core, the source of the power. Matplotlib, often used together with pyplot, can be used to visualize data, graph plots, visualize deep learning and machine learning training loops and loss. 

This article, Matplotlib Explained, was originally written by Hennadii Madan for Kite Blog for Python.  Read the original article on Kite Blog!  It is republished on www.siliconvanity.com with permission from Kite. I recommend this article because it gets down to the details of what Matplotlib is all about. When doing machine learning tutorials, Jupyter Notebook, or Python, sklearn tutorials, matplotlib is usually glossed over, and discussed only as an afterthought. This article talks about the design of Matplotlib. Kite allows developers to code faster in python with line-of-code completion. It is now available for Atom, PyCharm, Sublime, VS Code and Vim. I tried it and like it. You can try it in the browser. What I like the most is to easily auto-complete API syntax without checking the documentation every other second. Recommend! 

This article offers an overview of matplotlib. Once you understand the basics, you may find it very useful to find code snippets for very cool plots and use it in your projects. I usually use a code snippets that I found and customize it for my project rather than compose something from scratch. 

This post features a basic tutorial on matplotlib plotting package for python. In it, we’ll discuss the purpose of data visualization and construct several simple plots to showcase the basic matplotlib functionality. After reading this post you’ll understand what matplotlib is, when and how to use it, when not to use it, and where to find help!

Table of Contents

1. Introduction

  • What is matplotlib?
  • When to use matplotlib
  • When not to use matplotlib
  • Purpose of data visualization

2. Setup

  • Installation
  • Backends and interaction setup
  • Jupyter notebook

3. Visualization techniques

  • We see in 2D
  • 1D data
  • 2D data
  • Multidimensional data

4. Conclusion

1. Introduction

What is matplotlib?


Matplotlib is the most popular plotting library for Python. It was written by John D. Hunter in 2003 as a way of providing a plotting functionality similar to that of MATLAB, which at the time was the most popular programming language in academia.

Matplotlib offers a hierarchy of objects abstracting various elements of a plot. The hierarchy starts with the top-level Figure object that may contain a series of intermediate level objects and Axes – from Scatter, to Line and Marker, and all the way down to Canvas. In order to produce a plot on the screen, the matplotlib Figure instance must be coupled with one of the supported user interface backends such as TkInter, Qt, WxWidgets or MacOs. Outside of matplotlib documentation, user interface backends are typically referred to as “interactive”. In order to produce a file on a disk, matplotlib uses hardcopy backends for a variety of bitmap (png, jpg, gif) and vector (ps, ps, svg) file formats. Hardcopy backends are also called “non-interactive”.

A distinguishing feature of Matplotlib is the pyplot state machine which enables users to write concise procedural code. Pyplot determines the object to apply the relevant method from the context or creates the necessary objects on the fly, if they don’t exist. While this allows for fast experimentation, it can result in less reusable and less maintainable code.

In practice, it’s almost impossible to use matplotlib without pyplot. The Matplotlib user guide recommends using pyplot only to create figures and axes, and, once those are created, use their respective methods to create plots. This is reasonable, and we stick to this style in this tutorial, however I would advise not following it too rigidly when exploring new data. Having to look up which methods belong to which objects interrupts the flow of analytical thought and negatively affects productivity. The initial code can be easily converted to object-oriented style once you have finished exploring the data and know what visualizations you are going to need.

The ability to combine these two styles leads to great flexibility – according to the library maintainers, matplotlib makes easy things easy and hard things possible.

When to use matplotlib


The question is, what is hard and what is easy to implement in matplotlib?
There are two areas where matplotlib is particularly powerful:
  • Exploratory data analysis
  • Scientific plotting for publication

Matplotlib’s strength in exploratory data analysis comes from the pyplot interface. With pyplot you can generate a variety of plots with a small number of keystrokes and interactively augment existing figures with new data. Additionally, the seaborn library built on top of matplotlib provides even more visualizations with some basic data analysis, such as linear regression or kernel density estimation, built in.

The second area of matplotlib’s excellence is data visualization for publication. It can generate vector images in a variety of formats using its hardcopy (non-interactive) backends. When generating bitmap images matplotlib provides aesthetically pleasing rendering using Anti Grain Geometry (Agg). The default selection of axis annotations, fonts and ability to render mathematical notation using LaTeX syntax make it perfect for preparing figures for scientific journals or homework.

When not to use matplotlib


It’s true that you can create interactive graphical user interfaces with realtime updates using matplotlib. But from firsthand experience, I can vouch for a few other, better tools.

I would advise against using matplotlib for:
  • Graphical user interfaces – instead, use pyforms.
  • Interactive visualization for web – instead, use bokeh.
  • Large datasets – instead, use vispy.

Purpose of data visualization


The purpose of data visualization is to give us an insight into the data, so that we can understand it: we don’t understand the data when it’s just a pile of numbers.

I see:





I understand: Nothing.

On the other hand, when we choose a proper visualization technique, the important things become clear.

I see:


I understand: It’s a triangle! (And the top is at 1.00)

It’s worth remembering that what we are after is insight during the entire visualization workflow – starting with data transformations, and ending with the choice of file format to save the images.

2. Setup

Installation


Assuming you have your python development environment set up, install matplotlib using the Python package manager of your choice. If you don’t use one, start now! I highly recommend the Conda package manager that you can get by installing miniconda.

$ conda install matplotlib

in terminal or windows PowerShell will install matplotlib and all the dependencies. If you use pip

$ pip install matplotlib

would do the job.

Backends and interaction setup


Matplotlib supports multiple backends – a concept that might be confusing for new users. Matplotlib can be used for many different things, including saving the visualizations results of long-running calculations for later review. These use cases are non-interactive and use the so-called hardcopy backends. If your matplotlib came preinstalled, it might be using one of the hardcopy backends by default. In this case, you won’t see anything when issuing plotting commands.

In this tutorial, we’ll use matplotlib interactively to see the results of our actions immediately. This means that we need to use a user interface backend. If you installed matplotlib yourself, the default backend would be chosen to match one of the supported GUI frameworks, such as Qt, WxWidgets, or Cacao – which is available on your computer. Tcl/Tk framework and its programming interface TkInter comes alongside with most python installations. To stay on the safe side, we’ll use TkInter backend, as you’re almost guaranteed to have it.

import matplotlib as mpl
mpl.use('TkAgg') #Use TkInter backend with anti-grain geometry renderer

These statements must come before we import pyplot, as otherwise they will have no effect, since the default backend would be chosen during pyplot import.

If we were to use only the commands above, we would have to call pyplot.show() every time we wanted to see our plots. What’s worse, is that we would not be able to enter any python commands until the figure window is closed. To be able to interact both with the plots and with Python, we need to turn on the interactive mode:

import matplotlib.pyplot as plt
plt.ion() # turn on interactive mode
To test the setup type this at python prompt:

>>> plt.text(0.0 , 0.5, 'Hello World!')

This should open a figure window with an Axes and a Text object saying “Hello World!”. Close this window manually using the mouse or enter plt.close() in the interpreter.

Jupyter notebook


If you are using a (properly configured) Jupyter notebook, you may skip the above setup, as you will have your figures rendered in the output cells of the notebook. Just make sure to input all the code from each block in our examples into a single Jupyter cell.


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3. Visualization techniques

We see in 2D


Our eyes’ retina is a thin sheet with light-sensitive photoreceptor cells. The relative positions between photoreceptors change very slowly over our lifetime and can be thought of as pretty much constant. Two numbers and a reference point on the retina are enough to find any given light-sensitive cell, making our sight essentially two-dimensional.












Retinal mosaic: distribution of red, green and blue photoreceptor cells in the center of retina of a person with normal vision (left) and a color-blind person (right). Image by Mark Fairchild under Creative Commons Attribution Share-Alike 3.0 License.

But what about stereo vision? After all, we do live in a three-dimensional world.

While we may live in a 3D world, we never actually see all of it. We don’t see inside objects – otherwise, we wouldn’t we need X-ray or ultrasound machines. What we see with our eyes are just the surfaces of the objects, and those are two dimensional.

Data, on the other hand, can have any number of dimensions. The best way for us, humans, to understand data is to examine its two-dimensional representation. In the rest of this tutorial, we’ll go through the basic techniques to visualize data of different dimensionality: 1D, 2D and multidimensional data.

1D Data
Statistical distributions are a typical example of 1D data. What you want to do is transform your data so that you have another dimension. By far, the most common way to do this is to categorize data and count the frequency of items in the categories. In the case of continuous distributions, categories can be defined by splitting the data range into equally sized intervals. This is the well-known histogram.

Let’s generate some normally distributed data and see which values are most commonly seen. We start by importing the Numpy package: it’s one of matplotlib’s main dependencies and should have been installed by the package manager.

import numpy as np
data = np.random.randn(10000)
fig, ax = plt.subplots()
ax.hist(data,bins=20)
fig.suptitle('Histogram of a sample from standard normal distribution')
ax.set_ylabel('counts')
fig.savefig('1_histogram.png', dpi=200)

I see:



I understand: values around 0 are the most common. Full width at half-maximum is about 3.

The hist() function above calls numpy.histogram() under the hood to count the number of data points in respective bins. For categorical or integer variables you will have to do your own counting and call the bar() function.

For example:

responses = [
 'chocolate', 'chocolate', 'vanilla', 'chocolate', 'strawberry', 'strawberry','chocolate', 'vanilla', 'vanilla', 'chocolate', 'strawberry', 'chocolate', 'strawberry', 'chocolate', 'chocolate','chocolate', 'chocolate', 'strawberry', 'chocolate', 'strawberry', 'vanilla', 'vanilla', 'chocolate', 'chocolate', 'strawberry', 'chocolate', 'strawberry', 'vanilla', 'chocolate', 'chocolate', 'chocolate', 'strawberry'
]
flavors, counts = np.unique(responses, return_counts=True)
fig, ax = plt.subplots()
plt.bar(flavors,counts)
ax.set_ylabel('counts')
fig.suptitle('Ice-cream preference')
fig.savefig('2_bar.png', dpi=200)



I understand: chocolate ice-cream tastes the best.

2D Data


Scatter plot for measurements

For this demo we will use a small real world dataset. Please head to Kite’s Github Repository and download the files ‘data.csv’ and ‘truth.csv’ if you want to follow along!

When measuring a dependence between certain quantities, a scatterplot is a good way to visualize it. scatter() accepts x and y positional arguments representing the coordinates of each marker, followed by an optional size and color arguments that specify appropriate properties for each marker.

# Load data measurements = np.loadtxt('data.csv') print(measurements)



​fig, ax = plt.subplots()
sc = ax.scatter(measurements[:, 0],
                measurements[:, 1],
                measurements[:, 2],
                measurements[:, 3])
plt.colorbar(sc)
plt.title("Axes.scatter() demo")




Joint bivariate distributions

Another type of two dimensional data are bivariate distributions. Density of bivariate distributions can be easily visualized using a scatter with translucent markers.

x = 2*np.random.randn(5000)
y = x+np.random.randn(5000)
fig, ax = plt.subplots()
_=ax.scatter(x,y,alpha = 0.05)



Another way to represent the same data is with a two-dimensional histogram. This might be preferred for smaller samples.

fig, ax = plt.subplots()
_=ax.hist2d(x[::10],y[::10])
ax.set_title('2D Histogram')




Hexbin provides a slightly more aesthetically pleasing result.

fig, ax = plt.subplots()
_=ax.hexbin(x[::10],y[::10],gridsize=20,cmap = 'plasma')
ax.set_title('Hexbin Histogram')




The optional cmap argument sets a colormap for the plot. A list of all built-in colormaps can be found here.

While Matplotlib also supports contour plots, building the contours from the sample requires additional processing. Seaborn and other add-on libraries provide functions that achieve the desired effect in a single line of code.

Images

Matplotlib can display images represented as arrays of shape (n,m), (n,m,3) or (n,m,4). The first case is interpreted as a grayscale image, the second as an RGB image, and the third as an RGB image with an alpha channel. Let’s make some nice gradients:

im = np.zeros((800,600,3))
im[:,:,0] = np.linspace(0,1,800)[:,None]
im[:,:,1] = np.linspace(0,1,600)[None,:]
im[:,:,2] = np.linspace(1,0,600)[None,:]
plt.imshow(im)




Mathematical functions

We have already seen how to set titles, legend, xlabel and ylabel for the axes, and add text annotations. All these functions can render mathematical notations in latex syntax. This is as easy as placing the necessary latex commands within “$” characters. In this example, we will plot a mathematical function and use fill_between to highlight the area under the curve.

# same imports as previous examples
x = np.linspace(-1.,1.,1000)
y = -x*x+1.

fig,ax = plt.subplots()

​ax.plot(x,y)
ax.fill_between(x,y,alpha=0.2,color='cyan') #highlight the area under the curve
ax.xaxis.set_ticks_position('bottom')
ax.yaxis.set_ticks_position('left')
ax.spines['left'].set_position('zero')   # makee x and y axes go through
ax.spines['bottom'].set_position('zero') # the origin
ax.spines['right'].set_color('none')     # hide the unnecessary
ax.spines['top'].set_color('none')       # spines ("the box" around the plot)
ax.set_xlabel('x',fontdict={'size':14})
ax.xaxis.set_label_coords(1.0, 0.0)
ax.set_ylabel('y',rotation=0,fontdict={'size':14})
ax.yaxis.set_label_coords(0.55, 0.95)

#render latex formulas in the title
ax.set_title('$\\int_{-1}^{1}(1-x^2)dx = 1\\frac{1}{3}$',fontdict={'size':28}) 




When using matplotlib to prepare figures for a scientific paper, the default style of mathematical formulas rendered by matplotlib might not match the publisher’s style. To fix this, matplotlib can offload math rendering to an existing TeX installation. This demo shows how to achieve this.




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Multi-dimensional data


With multidimensional data, the task is transforming it into one or several two-dimensional representations. Generally this leads to a loss of information, but that’s actually the point: we want to omit all the irrelevant details and highlight the big picture, or some particular aspect of the data.

Finding the data representation that makes sense for us is at the core of Data Analysis – a vast subject area that goes beyond the scope of this post. However, in certain simple cases, depending on the structure of the data, we might be able to visualize interesting features of the data without transforming it.

For example, the data we loaded previously is actually the result of measuring the same quantity in the same objects using four different measurement methods. The truth.csv file contains reference values for this quantity. So without losing any information, we may plot each column of our data versus the reference values overlaid on top of each other.

Adding overlays on the existing axis is as easy as calling additional plot methods.
# same imports as previous examples
x = np.linspace(-1.,1.,1000)
y = -x*x+1.

fig,ax = plt.subplots()

​ax.plot(x,y)
ax.fill_between(x,y,alpha=0.2,color='cyan') #highlight the area under the curve
ax.xaxis.set_ticks_position('bottom')
ax.yaxis.set_ticks_position('left')
ax.spines['left'].set_position('zero')   # makee x and y axes go through
ax.spines['bottom'].set_position('zero') # the origin
ax.spines['right'].set_color('none')     # hide the unnecessary
ax.spines['top'].set_color('none')       # spines ("the box" around the plot)
ax.set_xlabel('x',fontdict={'size':14})
ax.xaxis.set_label_coords(1.0, 0.0)
ax.set_ylabel('y',rotation=0,fontdict={'size':14})
ax.yaxis.set_label_coords(0.55, 0.95)

#render latex formulas in the title
ax.set_title('$\\int_{-1}^{1}(1-x^2)dx = 1\\frac{1}{3}$',fontdict={'size':28}) 



The third argument in a call to plot() above is the format specifier string. This is a convenient way to set the style of the plot. In this example, the first character ‘o’ tells matplotlib to use circular markers, and the second character ‘:’ tells it to connect the marker with a a dotted line. Other options are ‘:’,’–‘,’-.’ for dotted, dashed, and dot-dashed lines respectively. The list of all marker specifiers can be found here. It’s also possible to specify color this way by adding another character, for instance, ‘r’ for ‘red’. Color options are ‘g’,’b’,’c’,’m’,’y’, and ‘k’ for green, blue, cyan, magenta, yellow, and black, respectively.

The result in the previous example can be obtained by supplying the the entirety of the measurements variable to the plot method. Matplotlib would cycle through the last dimension of the data and overlay the plot using a new color.

fig, ax = plt.subplots()
ax.plot(truth,measurements,'o:')
ax.set_ylabel('Measurements')
ax.set_xlabel('Reference')




The colors are assigned according to the default properties cycle – a property of the Axes object. Below, we use non-default color cycle by setting the property cycle for the axes before calling plot().

fig, ax = plt.subplots()
n = measurements.shape[1]
ax.set_prop_cycle('color',plt.cm.viridis(np.linspace(0, 1, n)))
ax.plot(truth,measurements,'o:')
ax.set_ylabel('Measurements')
ax.set_xlabel('Reference')




The figure above is quite messy, and it would be more understandable if the plots were positioned side by side. This is done with additional arguments to subplots(): we can create several axes arranged in a regular grid within a single figure.

The grid size is specified as integers in the first two arguments to subplots. Alternatively, one can supply a tuple of vertical and horizontal grid dimensions as a first argument. Keep in mind that in this case, subplots() returns an array of axes instead of a single axes object as the second element of its output.

fig, ax_array = plt.subplots(2,2,sharex = 'all', sharey = 'all') #ax_array is 2 by 2
for i in range(measurements.shape[1]):
   ax_index =np.unravel_index(i,ax_array.shape) # unravel index to cycle through subplots
              # with a single loop
   ax_array[ax_index].plot(truth,measurements[:,i],'o',label='method '+str(i))
   ax_array[ax_index].plot(truth,measurements[:,i],':')
   ax_array[ax_index].legend()
plt.suptitle('Subplots demo')




Note the sharex and sharey arguments in the call to subplots() above. This way, we ensure that the limits on the x and y axes are the same between all subplots.

Saving

Saving the rendered visualizations is as simple as a call to savefig() method of the Figure object. Matplotlib will infer the file format from the extension, and you can choose the output resolution for the bitmap formats using the dpi keyword argument:

fig.savefig('Figure.png', dpi=200)
fig.savefig('Figure.svg') # will use SVG vector backend


If you ever happen to lose track of the Figure object, use plt.savefig() to save the active figure.

4. Conclusion


To conclude, matplotlib is an excellent library for exploratory data analysis and publication quality plotting. It’s won its popularity by offering an easy-to-use procedural interface through a pyplot state machine. At the same time, it also allows to control all aspects of plotting for advanced visualizations through its main object-oriented interface, which facilitates the creation of maintainable, modular code.

Because it is so easy to start using matplotlib, it’s almost universally taught as the first graphics library in universities, so it’s safe to say it won’t be going anywhere soon. That being said, matplotlib is quite old and might feel clunky at times. Add-on libraries such as seaborn try to smooth the rough edges of matplotlib by offering an arsenal of advanced visualizations out of the box, better default settings, and extended procedural interfaces to aid with the more common tasks encountered during fine-tuning the appearance of the plots.

To see more examples of what matplotlib and seaborn are capable of, take a look at the galleries on their respective official websites.


The best place to look for answers on matplotlib is in Stack Overflow – it has hundreds of answered questions and you can always ask your own. That said, I personally recommend scanning through the list of all pyplot plotting commands available here before any search, just to know what’s out there. Did you know that you can draw xkcd-style plots with matplotlib?

Happy plotting!

This post is a part of Kite’s new series on Python. You can check out the code from this and other posts on our GitHub repository.

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Friday, August 30, 2019

Understand the Softmax Function in Minutes V2




Understanding Softmax in Minutes by Uniqtech
Learning machine learning? Specifically trying out neural networks for deep learning? You likely have run into the Softmax function, a wonderfulactivation function that turns numbers aka logits into probabilities thatsum to one. Softmax function outputs a vector that represents the probability distributions of a list of potential outcomes. It’s also a core element used in deep learning classification tasks (more on this soon). We will help you understand the Softmax function in a beginner friendly manner by showing you exactly how it works — by coding your very own Softmax function in python.
Reposted with permission. 
If you are implementing Softmax in Pytorch and you already know Pytorch well, scroll down to the Deep Dive section and grab the code.
This article has gotten really popular — 1800 claps and counting and it is updated constantly. Latest update April 4, 2019: updated word choice, listed out assumptions and added advanced usage of Softmax function in Bahtanau Attention for neural machine translation. See full list of updates below. You are welcome to translate it. We would appreciate it if the English version is not reposted elsewhere. If you want a free read, just use incognito mode in your browser. A link back is always appreciated. Comment below and share your links so that we can link to you in this article. Claps on Medium help us earn $$$. Thank you in advance for your support!
Skill pe-requisites: the demonstrative codes are written with Python list comprehension (scroll down to see an entire section explaining list comprehension). The math operations demonstrated are intuitive and code agnostic: it comes down to taking exponentials, sums and division aka the normalization step. This article is for your personal use only, not for production or commercial usage. Please read our disclaimer.
Udacity Deep Learning Slide on Softmax
The above Udacity lecture slide shows that Softmax function turns logits [2.0, 1.0, 0.1] into probabilities [0.7, 0.2, 0.1], and the probabilities sum to 1.
In deep learning, the term logits layer is popularly used for the last neuron layer of neural network for classification task which produces raw prediction values as real numbers ranging from [-infinity, +infinity ]. — Wikipedia
Logits are the raw scores output by the last layer of a neural network. Before activation takes place.
Softmax is not a black box. It has two components: special number e to some power divide by a sum of some sort.
y_i refers to each element in the logits vector y. Python and Numpy code will be used in this article to demonstrate math operations. Let’s see it in code:
logits = [2.0, 1.0, 0.1]
import numpy as npexps = [np.exp(i) for i in logits]
We use numpy.exp(power) to take the special number eto any power we want. We use python list comprehension to iterate through each i of the logits, and compute np.exp(i). If you are not familiar with Python list comprehension, read the explanation in the next section first. Logit is another name for a numeric score. The result is stored in a list called exps. The variable name is short for exponentials.
Replacing i with logit is another verbose way to write outexps = [np.exp(logit) for logit in logits] . Note the use of plural and singular nouns. It’s intentional.
We just computed the top part of the Softmax function. For each logit, we took it to an exponential power of eEach transformed logit j needs to be normalized by another number in order for all the final outputs, which are probabilities, to sum to one. Again, this normalization gives us nice probabilities that sum to one!
We compute the sum of all the transformed logits and store the sum in a single variable sum_of_exps, which we will use to normalize each of the transformed logits.
sum_of_exps = sum(exps)
Now we are ready to write the final part of our Softmax function: each transformed logit jneeds to be normalized by sum_of_exps , which is the sum of all the logits including itself.
softmax = [j/sum_of_exps for j in exps]
Again, we use python list comprehension: we grab each transformed logit using [j for j in exps]divide each j by the sum_of_exps.
List comprehension gives us a list back. When we print the list we get
>>> softmax
[0.6590011388859679, 0.2424329707047139, 0.09856589040931818]
>>> sum(softmax)
1.0
The output rounds to [0.7, 0.2, 0.1] as seen on the slide at the beginning of this article. They sum nicely to one!

Extra — Understanding List Comprehension

This post uses a lot of Python list comprehension which is more concise than Python loops. If you need help understanding Python list comprehension type the following code into your interactive python console (on Mac launch terminal and type python after the dollar sign $ to launch).
sample_list = [1,2,3,4,5] 
# console returns Nonesample_list # console returns [1,2,3,4,5]#print the sample list using list comprehension
[i for i in sample_list] # console returns [1,2,3,4,5]
# note anything before the keyword 'for' will be evaluated
# in this case we just display 'i' each item in the list as is
# for i in sample_list is a short hand for 
# Python for loop used in list comprehension
[i+1 for i in sample_list] # returns [2,3,4,5,6]
# can you guess what the above code does?
# yes, 1) it will iterate through each element of the sample_list
# that is the second half of the list comprehension
# we are reading the second half first
# what do we do with each item in the list?
# 2) we add one to it and then display the value
# 1 becomes 2, 2 becomes 3# note the entire expression 1st half & 2nd half are wrapped in []
# so the final return type of this expression is also a list
# hence the name list comprehension
# my tip to understand list comprehension is
# read the 2nd half of the expression first
# understand what kind of list we are iterating through
# what is the individual item aka 'each'
# then read the 1st half
# what do we do with each item# can you guess the list comprehension for 
# squaring each item in the list?
[i*i     for i in sample_list] #returns [1, 4, 9, 16, 25]

Intuition and Behaviors of Softmax Function

If we hard code our label data to the vectors below, in a format typically used to turn categorical data into numbers, the data will look like this format below.
[[1,0,0], #cat
 [0,1,0], #dog
 [0,0,1],] #bird
Optional Reading: FYI, this is an identity matrix in linear algebra. Note that only the diagonal positions have the value 1 the rest are all zero. This format is useful when the data is not numerical, the data is categorical, each category is independent from others. For example, 1 star yelp review, 2 stars, 3 stars, 4 stars, 5 starscan be one hot coded but note the five are related. They may be better encoded as 1 2 3 4 5 . We can infer that 4 stars is twice as good as 2 stars. Can we say the same about name of dogs?Ginger, Mochi, Sushi, Bacon, Max , is Macon 2x better than Mochi? There’s no such relationship. In this particular encoding, the first column represent cat, second column dog, the third column bird.
The output probabilities are saying 70% sure it is a cat, 20% a dog, 10% a bird. One can see that the initial differences are adjusted to percentages. logits = [2.0, 1.0, 0.1]. It’s not 2:1:0.1. Previously, we cannot say that it’s 2x more likely to be a cat, because the results were not normalized to sum to one.
The output probability vector is [0.7, 0.2, 0.1] . Can we compare this with the ground truth of cat [1,0,0] as in one hot encoding? Yes! That’s what is commonly used in cross entropy loss (We have a cool trick to understand cross entropy loss and will write a tutorial about it. Read it here.). In fact cross entropy loss is the “best friend” of Softmax. It is the most commonly used cost function, aka loss function, aka criterion that is used with Softmax in classification problems. More on that in a different article.
Why do we still need fancy machine learning libraries with fancy Softmax function? The nature of machine learning training requires ten of thousands of samples of training data. Something as concise as the Softmax function needs to be optimized to process each element. Some say that Tensorflow broadcasting is not necessarily faster than numpy’s matrix broadcasting though.

Watch this Softmax tutorial on Youtube

Visual learner? Prefer watching a YouTube video instead? See our tutorial below.

Deeper Dive into Softmax

Softmax is an activation function. Other activation functions include RELU and SigmoidIt is frequently used in classifications. Softmax output is large if the score (input called logit) is large. Its output is small if the score is small. The proportion is not uniform. Softmax is exponential and enlarges differences - push one result closer to 1 while another closer to 0. It turns scores aka logits into probabilities. Cross entropy (cost function) is often computed for output of softmax and true labels (encoded in one hot encoding). Here’s an example of Tensorflow cross entropy computing function. It computes softmax cross entropy between logits and labels. Softmax outputs sum to 1 makes great probability analysis. Remember the takeaway is: the essential goal of softmax is to turn numbers into probabilities.
Thanks. I can now deploy this to production. Uh no. Hold on! Our implementation is meant to help everyone understand what the Softmax function does. It uses for loops and list comprehensions, which are not efficient operations for production environment. That’s why top machine learning frameworks are implemented in C++, such as Tensorflow and Pytorch. These frameworks can offer much faster and efficient computations especially when dimensions of data get large, and can leverage parallel processing. So no, you cannot use this code in production. However, technically if you train on a few thousand examples (generally MLneeds more than 10K records), your machine can still handle it, and inference is possible even on mobile devices! Thanks Apple Core ML. Can I use this softmax on imagenet data? Uh definitely no, there are millions of images. Use Sklearn if you want to prototype. Tensorflow for production. Pytorch 1.0 added support for production as well. For research Pytorch and Sklearn softmax implementations are great.
Best Loss Function / Cost Function / Criterion to Use with Softmax
You have decided to choose Softmax as the final function for classifying your data. What loss function and cost function should you use with Softmax? The theoretical answer is Cross Entropy Loss (let us know if youwant an article on that. We have a full pipeline of topics waiting for your vote).
Tell me more about Cross Entropy Loss. Sure thing! Cross Entropy Loss in this case measures how similar your predictions are to the actual labels. For example if the probabilities are supposed to be [0.7, 0.2, 0.1] but you predicted during the first try [0.3, 0.3, 0.4], during the second try [0.6, 0.2, 0.2] . You can expect the cross entropy loss of the first try , which is totally inaccurate, almost like a random guess to have higher loss than the second scenario where you aren’t too far off from the expected.Read our full Cross Entropy Loss tutorial here.

Deep Dive Softmax

For Softmax deep dive read our article Softmax Beyond the Basics. There you can also find explanation that Softmax and Sigmoid are equivalent for binary classification. Different flavors and implementations of Softmax in Tensorflow and Pytorch. Coming soon in Softmax Beyond the Basics: How to graph Softmax function? Is there a more efficient way to calculate Softmax for big datasets? Stay tuned. Get alerts subscribe@uniqtech.co
Softmax Formula in Pytorch
def softmax(x):
    return torch.exp(x)/torch.sum(torch.exp(x), dim=1).view(-1,1)
dim=1 is for torch.sum() to sum each row across all the columns, .view(-1,1) is for preventing broadcasting. For details of this formula, visit our Softmax Beyond the Basics article. The above is the formula. If you are just looking for an API then use softmax or LogSoftmax see Pytorch documentation.
Bahtanau Attention for Neural Machine Translation — Softmax Function in Real Time
In the neural machine translation architecture, outlined by Dimitry Bahtanau in Neural machine translation by jointly learn to align and translate (2014), uses Softmax output as weights to weigh each of the hidden states right before producing the final output.
Softmax Function BehaviorBecause Softmax function outputs numbers that represent probabilities, each number’s value is between 0 and 1 valid value range of probabilities. The range is denoted as [0,1]The numbers are zero or positive. The entire output vector sums to 1. That is to say when all probabilities are accounted for, that’s 100%.

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