- \n
- Introduction to the Logistic Sigmoid Function<\/a><\/li>\n
- The syntax for Logistic Sigmoid in Python<\/a><\/li>\n<\/li>\n
- Examples of how to use the Logistic Sigmoid function<\/a><\/li>\n<\/ul>\n
Let’s start with a quick overview of what the function is.<\/p>\n
<\/a><\/p>\n
Introduction to the Logistic Sigmoid Function<\/h2>\n
The logistic sigmoid function is an s-shaped function that’s defined as:<\/p>\n
(1) <\/span> <\/span>
![\"\begin{equation*}](\"https:\/\/www.sharpsightlabs.com\/wp-content\/ql-cache\/quicklatex.com-256dd6d8b1aece5e0e04504d27a3ccf7_l3.png\" "\\"Rendered")
<\/p>\nWhen we plot it, it looks like this:<\/p>\n
![\"A](\"https:\/\/www.sharpsightlabs.com\/wp-content\/uploads\/2022\/02\/logistic-sigmoid_simple-visual-example.png\")
<\/p>\nThis sigmoid function is often used in machine learning<\/a>. In particular, it’s often used as an activation function in deep learning and artificial neural networks.<\/p>\n
Since it’s often used in machine learning and deep learning, it’s potentially useful to know how to implement it in common machine learning programming languages.<\/p>\n
That being the case, let’s look at how we can implement the function in Python<\/a>, one of the most popular programming languages for machine learning.<\/p>\n
<\/a><\/p>\n
The syntax for a Python logistic sigmoid function<\/h2>\n
We can define the logistic sigmoid function in Python as follows:<\/p>\n
![\"An](\"https:\/\/www.sharpsightlabs.com\/wp-content\/uploads\/2022\/02\/logistic-sigmoid_python_syntax-explanation.png\")
<\/p>\n(You can also find the Python code in example 1<\/a>.)<\/p>\n
Here, the
def<\/code> keyword indicates that we’re defining a new Python function.<\/p>\nWe’ve named the function “
logistic_sigmoid<\/code>” (although we could name it something else).<\/p>\nThe input value is called x.<\/p>\n
In the body of the function, we see a return statement and a computation inside of it. This computation is calculating the value:<\/p>\n
(2) <\/span> <\/span>
<\/p>\n… where x is the input value to the function.<\/p>\n
Notice that to perform this computation, we’re calling the Numpy exponential function<\/a>.<\/p>\n
Format of the input values<\/h4>\n
Let’s talk about the possible input values of the function.<\/p>\n
This function will operate on:<\/p>\n
- \n
- single numbers<\/li>\n
- Numpy arrays<\/li>\n
- array-like objects (such as Python lists)<\/li>\n<\/ul>\n
The output of the function<\/h4>\n
The output will vary slightly, depending on the input type.<\/p>\n
If the input is a number, then the output will be a number.<\/p>\n
If the input is an array or array-like object, then the function will output a Numpy array. Importantly, the output array will have the same size and shape as the input. The values of the output array will be the element-wise computation of the input values.<\/p>\n
![\"A](\"https:\/\/www.sharpsightlabs.com\/wp-content\/uploads\/2022\/02\/logistic-sigmoid_array-operation-example_v2.png\")
<\/p>\n<\/a><\/p>\n
Examples: how to implement and use a logistic sigmoid function in Python<\/h2>\n
Now that we’ve looked at the syntax for how to implement the logistic sigmoid function, let’s actually execute the function code and use it on some examples.<\/p>\n
Examples:<\/strong><\/p>\n
- \n
- Define the Numpy logistic sigmoid function<\/a><\/li>\n
- Compute logistic sigmoid of 0<\/a><\/li>\n
- Compute logistic sigmoid of 5<\/a><\/li>\n
- Compute logistic sigmoid of -5<\/a><\/li>\n
- Use logistic sigmoid on an array of numbers<\/a><\/li>\n
- Plot the logistic sigmoid function<\/a><\/li>\n<\/ul>\n
Preliminary code: Import Numpy and Set Up Plotly<\/h3>\n
Before you run the examples, you’ll need to run some setup code.<\/p>\n
Specifically, you need to import some packages, and set up Plotly to render images.<\/p>\n
Import Packages<\/h4>\n
First, we need to import Numpy and Plotly express.<\/p>\n
\r\nimport numpy as np\r\nimport plotly.express as px\r\n<\/pre>\n
We’re going to use a Numpy function \u2013 np.exp \u2013 in our implementation of the function.<\/p>\n
And we’ll use Plotly Express to plot the function in example 6<\/a>.<\/p>\n
Set Up Image Rendering<\/h5>\n
Next, you might need to configure Plotly to render images on your system.<\/p>\n
To be clear: you only need to do this if you’re using an IDE.<\/p>\n
By default, Plotly is set up to render images (i.e., output visualizations) in a browser window. That’s fine if you’re working in a notebook. But if you’re working in an IDE like PyCharm or Spyder, it will cause errors.<\/p>\n
So, you need to tell Plotly to render its output as an svg directly in the IDE.<\/p>\n
(Note: if you’re using Jupyter, you can skip this code<\/em>!)<\/p>\n
\r\nimport plotly.io as pio\r\npio.renderers.default = 'svg'\r\n<\/pre>\n
After you’ve run the setup code, you should be ready to run these examples.<\/p>\n
<\/a><\/p>\n
EXAMPLE 1: Define the Logistic Sigmoid Function using Python<\/h3>\n
First, we’ll define the logistic sigmoid function in Python:<\/p>\n
\r\ndef logistic_sigmoid(x):\r\n return(1\/(1 + np.exp(-x)))\r\n<\/pre>\n
Explanation<\/h4>\n
Here, we’re using Python’s
def<\/code> keyword to define a new function. We’ve named the new function “logistic_sigmoid”.<\/p>\nThe function has one input:
x<\/code>.<\/p>\nThe function will return the following:<\/p>\n
(3) <\/span> <\/span>
![\"\begin{equation*}](\"https:\/\/www.sharpsightlabs.com\/wp-content\/ql-cache\/quicklatex.com-6337bd248aab74d8ab5b731b396ee4a3_l3.png\" "\\"Rendered")
<\/p>\nNotice that we’re computing this output in part by using the Numpy exponential function<\/a>.<\/p>\n
<\/a><\/p>\n
EXAMPLE 2: Compute the logistic sigmoid of 0<\/h3>\n
Now that we have our function defined, let’s compute the sigmoid of 0<\/p>\n
\r\nlogistic_sigmoid(0)\r\n<\/pre>\n
OUT:<\/p>\n
\r\n0.5\r\n<\/pre>\n
<\/a><\/p>\n
EXAMPLE 3: Compute logistic sigmoid of 5<\/h3>\n
Next, let’s compute the sigmoid of 5.<\/p>\n
\r\nlogistic_sigmoid(5)\r\n<\/pre>\n
OUT:<\/p>\n
\r\n0.9933071490757153\r\n<\/pre>\n
Explanation<\/h4>\n
Here, we’ve computed the logistic sigmoid of 5.<\/p>\n
Notice that the value is very close to 1.<\/p>\n
This is expected. As x goes to infinity, the logistic sigmoid function will converge to 1.<\/p>\n
<\/a><\/p>\n
EXAMPLE 4: Compute logistic sigmoid of -5<\/h3>\n
Now, let’s compute the sigmoid of -5.<\/p>\n
\r\nlogistic_sigmoid(-5)\r\n<\/pre>\n
OUT:<\/p>\n
\r\n0.0066928509242848554\r\n<\/pre>\n
Explanation<\/h4>\n
Here, we’ve computed the sigmoid of -5.<\/p>\n
Notice that the value is very close to 0.<\/p>\n
This is as we would expect. As x goes to negative infinity, the function will converge to 0.<\/p>\n
<\/a><\/p>\n
EXAMPLE 5: Use the logistic sigmoid function on an array of numbers<\/h3>\n
Now, we’ll use our sigmoid function on an array<\/em> of numbers.<\/p>\n
Create Numpy Array<\/h4>\n
First, we’ll create a Numpy array<\/a>.<\/p>\n
In particular, we’re going to create an array of evenly spaced values<\/a>.<\/p>\n
To do this, we’ll use Numpy linespace function, and create an array of evenly spaced values from -10 to 10.<\/p>\n
\r\nx_values = np.linspace(start = -10, stop = 10, num = 201)\r\n<\/pre>\n
Let’s print it out to take a look:<\/p>\n
\r\nprint(x_values)\r\n<\/pre>\n
OUT:<\/p>\n
\r\n[-10. -9.9 -9.8 -9.7 -9.6 -9.5 -9.4 -9.3 -9.2 -9.1 -9. -8.9\r\n -8.8 -8.7 -8.6 -8.5 -8.4 -8.3 -8.2 -8.1 -8. -7.9 -7.8 -7.7\r\n -7.6 -7.5 -7.4 -7.3 -7.2 -7.1 -7. -6.9 -6.8 -6.7 -6.6 -6.5\r\n -6.4 -6.3 -6.2 -6.1 -6. -5.9 -5.8 -5.7 -5.6 -5.5 -5.4 -5.3\r\n....\r\n 5.6 5.7 5.8 5.9 6. 6.1 6.2 6.3 6.4 6.5 6.6 6.7\r\n 6.8 6.9 7. 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9\r\n 8. 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 9. 9.1\r\n 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 10. ]\r\n<\/pre>\n
I’ve abbreviated the output somewhat for space.<\/p>\n
But if you look closely, you can see,
x_values<\/code> contains the values from -10 to 10, in increments of .1.<\/p>\nCompute Logistic Sigmoid Values<\/h4>\n
Now, we’re going to use our sigmoid function on
x_values<\/code>.<\/p>\n\r\nlogistic_sigmoid_values = logistic_sigmoid(x_values)\r\n<\/pre>\n
And let’s print the results:<\/p>\n
\r\nprint(logistic_sigmoid_values)\r\n<\/pre>\n
OUT:<\/p>\n
\r\n[4.53978687e-05 5.01721647e-05 5.54485247e-05 6.12797396e-05\r\n 6.77241496e-05 7.48462275e-05 8.27172229e-05 9.14158739e-05\r\n 1.01029194e-04 1.11653341e-04 1.23394576e-04 1.36370327e-04\r\n 1.50710358e-04 1.66558065e-04 1.84071905e-04 2.03426978e-04\r\n....\r\n 9.99849290e-01 9.99863630e-01 9.99876605e-01 9.99888347e-01\r\n 9.99898971e-01 9.99908584e-01 9.99917283e-01 9.99925154e-01\r\n 9.99932276e-01 9.99938720e-01 9.99944551e-01 9.99949828e-01\r\n 9.99954602e-01]\r\n<\/pre>\n
Explanation<\/h4>\n
I’ve abbreviated the results, but here, we can see roughly what’s in the results.<\/p>\n
If you look closely, you can see some values that are very close to 0 and also some values that are very close to 1.<\/p>\n
(We’ll see all of this in a moment when we plot the data.)<\/p>\n
<\/a><\/p>\n
EXAMPLE 6: Plot the Logistic Sigmoid function<\/h3>\n
Finally, let’s plot the logistic sigmoid function.<\/p>\n
Specifically, we’ll plot the output values we just computed in example 5<\/a>.<\/p>\n
To do this, we’ll use the Plotly line function, px.line<\/a>.<\/p>\n
\r\npx.line(x = x_values, y = logistic_sigmoid_values)\r\n<\/pre>\n
OUT:<\/p>\n
![\"A](\"https:\/\/www.sharpsightlabs.com\/wp-content\/uploads\/2022\/02\/logistic-sigmoid_plotly-line-plot.png\")
<\/p>\nExplanation<\/h4>\n
Here, we plotted the logistic sigmoid values that we computed in example 5<\/a>, using the Plotly line function. <\/p>\n
On the x-axis, we mapped the values contained in
x_values<\/code>.<\/p>\nOn the y-axis, we mapped the values contained in the Numpy array,
logistic_sigmoid_values<\/code>.<\/p>\nThe resulting output is a plot of our s-shaped sigmoid function.<\/p>\n
Leave your other questions in the comments below<\/h5>\n
Do you have other questions about how to create or use a logistic sigmoid function in Python? <\/p>\n
If so, leave your questions in the comments section below.<\/p>\n
Join our course to learn more about Numpy<\/h2>\n
In this tutorial, I’ve explained how implement and use a logistic sigmoid in Python, using Numpy.<\/p>\n
If you’re serious about mastering Numpy, and serious about data science in Python, you should consider joining our premium course called Numpy Mastery<\/em>.<\/p>\n
Numpy Mastery will teach you everything you need to know about Numpy, including:<\/p>\n
- \n
- How to create Numpy arrays<\/li>\n
- How to reshape, split, and combine your Numpy arrays<\/li>\n
- What the “Numpy random seed” function does<\/li>\n
- How to use the Numpy random functions<\/li>\n
- How to perform mathematical operations on Numpy arrays<\/li>\n
- and more …<\/li>\n<\/ul>\n
Moreover, this course will show you a practice system that will help you master<\/em> the syntax within a few weeks. We’ll show you a practice system that will enable you to memorize all of the Numpy syntax you learn. If you have trouble remembering Numpy syntax, this is the course you’ve been looking for.<\/p>\n
Find out more here:<\/p>\n
<\/p>\n
\nLearn More About Numpy Mastery<\/a>\n<\/p>\n
<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"
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- Compute logistic sigmoid of 0<\/a><\/li>\n
- Define the Numpy logistic sigmoid function<\/a><\/li>\n
- The syntax for Logistic Sigmoid in Python<\/a><\/li>\n<\/li>\n