Home > Articles

This chapter is from the book

3.5 Simple Classifier #1: Nearest Neighbors, Long Distance Relationships, and Assumptions

One of the simpler ideas for making predictions from a labeled dataset is:

  1. Find a way to describe the similarity of two different examples.

  2. When you need to make a prediction on a new, unknown example, simply take the value from the most similar known example.

This process is the nearest-neighbors algorithm in a nutshell. I have three friends Mark, Barb, Ethan for whom I know their favorite snacks. A new friend, Andy, is most like Mark. Mark’s favorite snack is Cheetos. I predict that Andy’s favorite snack is the same as Mark’s: Cheetos.

There are many ways we can modify this basic template. We may consider more than just the single most similar example:

  1. Describe similarity between pairs of examples.

  2. Pick several of the most-similar examples.

  3. Combine those picks to get a single answer.

3.5.1 Defining Similarity

We have complete control over what similar means. We could define it by calculating a distance between pairs of examples: similarity = distance(example_one, example_two). Then, our idea of similarity becomes encoded in the way we calculate the distance. Similar things are close—a small distance apart. Dissimilar things are far away—a large distance apart.

Let’s look at three ways of calculating the similarity of a pair of examples. The first, Euclidean distance, harkens back to high-school geometry or trig. We treat the two examples as points in space. Together, the two points define a line. We let that line be the hypotenuse of a right triangle and, armed with the Pythagorean theorem, use the other two sides of the triangle to calculate a distance (Figure 3.4). You might recall that ineq63_01.jpg. Or, you might just recall it as painful. Don’t worry, we don’t have to do the calculation. scikit-learn can be told, “Do that thing for me.” By now, you might be concerned that my next example can only get worse. Well, frankly, it could. The Minkowski distance would lead us down a path to Einstein and his theory of relativity . . . but we’re going to avoid that black (rabbit) hole.


FIGURE 3.4 Distances from components.

Instead, another option for calculating similarity makes sense when we have examples that consist of simple Yes, No or True, False features. With Boolean data, I can compare two examples very nicely by counting up the number of features that are different. This simple idea is clever enough that it has a name: the Hamming distance. You might recognize this as a close cousin—maybe even a sibling or evil twin—of accuracy. Accuracy is the percent correct—the percent of answers the same as the target—which is ineq63_02.jpg. Hamming distance is the number of differences. The practical implication is that when two sets of answers agree completely, we want the accuracy to be high: 100%. When two sets of features are identical, we want the similarity distance between them to be low: 0.

You might have noticed that these notions of similarity have names—Euclid(-ean), Minkowski, Hamming Distance—that all fit the template of FamousMathDude Distance. Aside from the math dude part, the reason they share the term distance is because they obey the mathematical rules for what constitutes a distance. They are also called metrics by the mathematical wizards-that-be—as in distance metric or, informally, a distance measure. These mathematical terms will sometimes slip through in conversation and documentation. sklearn’s list of possible distance calculators is in the documentation for neighbors.DistanceMetric: there are about twenty metrics defined there.

3.5.2 The k in k-NN

Choices certainly make our lives complicated. After going to the trouble of choosing how to measure our local neighborhood, we have to decide how to combine the different opinions in the neighborhood. We can think about that as determining who gets to vote and how we will combine those votes.

Instead of considering only the nearest neighbor, we might consider some small number of nearby neighbors. Conceptually, expanding our neighborhood gives us more perspectives. From a technical viewpoint, an expanded neighborhood protects us from noise in the data (we’ll come back to this in far more detail later). Common numbers of neighbors are 1, 3, 10, or 20. Incidentally, a common name for this technique, and the abbreviation we’ll use in this book, is k-NN for “k-Nearest Neighbors”. If we’re talking about k-NN for classification and need to clarify that, I’ll tack a C on there: k-NN-C.

3.5.3 Answer Combination

We have one last loose end to tie down. We must decide how we combine the known values (votes) from the close, or similar, neighbors. If we have an animal classification problem, four of our nearest neighbors might vote for cat, cat, dog, and zebra. How do we respond for our test example? It seems like taking the most frequent response, cat, would be a decent method.

In a very cool twist, we can use the exact same neighbor-based technique in regression problems where we try to predict a numerical value. The only thing we have to change is how we combine our neighbors’ targets. If three of our nearest neighbors gave us numerical values of 3.1, 2.2, and 7.1, how do we combine them? We could use any statistic we wanted, but the mean (average) and the median (middle) are two common and useful choices. We’ll come back to k-NN for regression in the next chapter.

3.5.4 k-NN, Parameters, and Nonparametric Methods

Since k-NN is the first model we’re discussing, it is a bit difficult to compare it to other methods. We’ll save some of those comparisons for later. There’s one major difference we can dive into right now. I hope that grabbed your attention.

Recall the analogy of a learning model as a machine with knobs and levers on the side. Unlike many other models, k-NN outputs—the predictions—can’t be computed from an input example and the values of a small, fixed set of adjustable knobs. We need all of the training data to figure out our output value. Really? Imagine that we throw out just one of our training examples. That example might be the nearest neighbor of a new test example. Surely, missing that training example will affect our output. There are other machine learning methods that have a similar requirement. Still others need some, but not all, of the training data when it comes to test time.

Now, you might argue that for a fixed amount of training data there could be a fixed number of knobs: say, 100 examples and 1 knob per example, giving 100 knobs. Fair enough. But then I add one example—and, poof, you now need 101 knobs, and that’s a different machine. In this sense, the number of knobs on the k-NN machine depends on the number of examples in the training data. There is a better way to describe this dependency. Our factory machine had a side tray where we could feed additional information. We can treat the training data as this additional information. Whatever we choose, if we need either (1) a growing number of knobs or (2) the side-input tray, we say the type of machine is nonparametric. k-NN is a nonparametric learning method.

Nonparametric learning methods can have parameters. (Thank you for nothing, formal definitions.) What’s going on here? When we call a method nonparametric, it means that with this method, the relationship between features and targets cannot be captured solely using a fixed number of parameters. For statisticians, this concept is related to the idea of parametric versus nonparametric statistics: nonparametric statistics assume less about a basket of data. However, recall that we are not making any assumptions about the way our black-box factory machine relates to reality. Parametric models (1) make an assumption about the form of the model and then (2) pick a specific model by setting the parameters. This corresponds to the two questions: what knobs are on the machine, and what values are they set to? We don’t make assumptions like that with k-NN. However, k-NN does make and rely on assumptions. The most important assumption is that our similarity calculation is related to the actual example similarity that we want to capture.

3.5.5 Building a k-NN Classification Model

k-NN is our first example of a model. Remember, a supervised model is anything that captures the relationship between our features and our target. We need to discuss a few concepts that swirl around the idea of a model, so let’s provide a bit of context first. Let’s write down a small process we want to walk through:

  1. We want to use 3-NN—three nearest neighbors—as our model.

  2. We want that model to capture the relationship between the iris training features and the iris training target.

  3. We want to use that model to predict—on previously unseen test examples—the iris target species.

  4. Finally, we want to evaluate the quality of those predictions, using accuracy, by comparing predictions against reality. We didn’t peek at these known answers, but we can use them as an answer key for the test.

There’s a diagram of the flow of information in Figure 3.5.


FIGURE 3.5 Workflow of training, testing, and evaluation for 3-NN.

As an aside on sklearn’s terminology, in their documentation an estimator is fit on some data and then used to predict on some data. If we have a training and testing split, we fit the estimator on training data and then use the fit-estimator to predict on the test data. So, let’s

  1. Create a 3-NN model,

  2. Fit that model on the training data,

  3. Use that model to predict on the test data, and

  4. Evaluate those predictions using accuracy.

In [9]:

# default n_neighbors = 5
knn   = neighbors.KNeighborsClassifier(n_neighbors=3)
fit   = knn.fit(iris_train_ftrs, iris_train_tgt)
preds = fit.predict(iris_test_ftrs)

# evaluate our predictions against the held-back testing targets
print("3NN accuracy:",
      metrics.accuracy_score(iris_test_tgt, preds))
3NN accuracy: 1.0

Wow, 100%. We’re doing great! This machine learning stuff seems pretty easy—except when it isn’t. We’ll come back to that shortly. We can abstract away the details of k-NN classification and write a simplified workflow template for building and assessing models in sklearn:

  1. Build the model,

  2. Fit the model using the training data,

  3. Predict using the fit model on the testing data, and

  4. Evaluate the quality of the predictions.

We can connect this workflow back to our conception of a model as a machine. The equivalent steps are:

  1. Construct the machine, including its knobs,

  2. Adjust the knobs and feed the side-inputs appropriately to capture the training data,

  3. Run new examples through the machine to see what the outputs are, and

  4. Evaluate the quality of the outputs.

Here’s one last, quick note. The 3 in our 3-nearest-neighbors is not something that we adjust by training. It is part of the internal machinery of our learning machine. There is no knob on our machine for turning the 3 to a 5. If we want a 5-NN machine, we have to build a completely different machine. The 3 is not something that is adjusted by the k-NN training process. The 3 is a hyperparameter. Hyperparameters are not trained or manipulated by the learning method they help define. An equivalent scenario is agreeing to the rules of a game and then playing the game under that fixed set of rules. Unless we’re playing Calvinball or acting like Neo in The Matrix—where the flux of the rules is the point—the rules are static for the duration of the game. You can think of hyperparameters as being predetermined and fixed in place before we get a chance to do anything with them while learning. Adjusting them involves conceptually, and literally, working outside the learning box or the factory machine. We’ll discuss this topic more in Chapter 11.

InformIT Promotional Mailings & Special Offers

I would like to receive exclusive offers and hear about products from InformIT and its family of brands. I can unsubscribe at any time.


Pearson Education, Inc., 221 River Street, Hoboken, New Jersey 07030, (Pearson) presents this site to provide information about products and services that can be purchased through this site.

This privacy notice provides an overview of our commitment to privacy and describes how we collect, protect, use and share personal information collected through this site. Please note that other Pearson websites and online products and services have their own separate privacy policies.

Collection and Use of Information

To conduct business and deliver products and services, Pearson collects and uses personal information in several ways in connection with this site, including:

Questions and Inquiries

For inquiries and questions, we collect the inquiry or question, together with name, contact details (email address, phone number and mailing address) and any other additional information voluntarily submitted to us through a Contact Us form or an email. We use this information to address the inquiry and respond to the question.

Online Store

For orders and purchases placed through our online store on this site, we collect order details, name, institution name and address (if applicable), email address, phone number, shipping and billing addresses, credit/debit card information, shipping options and any instructions. We use this information to complete transactions, fulfill orders, communicate with individuals placing orders or visiting the online store, and for related purposes.


Pearson may offer opportunities to provide feedback or participate in surveys, including surveys evaluating Pearson products, services or sites. Participation is voluntary. Pearson collects information requested in the survey questions and uses the information to evaluate, support, maintain and improve products, services or sites, develop new products and services, conduct educational research and for other purposes specified in the survey.

Contests and Drawings

Occasionally, we may sponsor a contest or drawing. Participation is optional. Pearson collects name, contact information and other information specified on the entry form for the contest or drawing to conduct the contest or drawing. Pearson may collect additional personal information from the winners of a contest or drawing in order to award the prize and for tax reporting purposes, as required by law.


If you have elected to receive email newsletters or promotional mailings and special offers but want to unsubscribe, simply email information@informit.com.

Service Announcements

On rare occasions it is necessary to send out a strictly service related announcement. For instance, if our service is temporarily suspended for maintenance we might send users an email. Generally, users may not opt-out of these communications, though they can deactivate their account information. However, these communications are not promotional in nature.

Customer Service

We communicate with users on a regular basis to provide requested services and in regard to issues relating to their account we reply via email or phone in accordance with the users' wishes when a user submits their information through our Contact Us form.

Other Collection and Use of Information

Application and System Logs

Pearson automatically collects log data to help ensure the delivery, availability and security of this site. Log data may include technical information about how a user or visitor connected to this site, such as browser type, type of computer/device, operating system, internet service provider and IP address. We use this information for support purposes and to monitor the health of the site, identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents and appropriately scale computing resources.

Web Analytics

Pearson may use third party web trend analytical services, including Google Analytics, to collect visitor information, such as IP addresses, browser types, referring pages, pages visited and time spent on a particular site. While these analytical services collect and report information on an anonymous basis, they may use cookies to gather web trend information. The information gathered may enable Pearson (but not the third party web trend services) to link information with application and system log data. Pearson uses this information for system administration and to identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents, appropriately scale computing resources and otherwise support and deliver this site and its services.

Cookies and Related Technologies

This site uses cookies and similar technologies to personalize content, measure traffic patterns, control security, track use and access of information on this site, and provide interest-based messages and advertising. Users can manage and block the use of cookies through their browser. Disabling or blocking certain cookies may limit the functionality of this site.

Do Not Track

This site currently does not respond to Do Not Track signals.


Pearson uses appropriate physical, administrative and technical security measures to protect personal information from unauthorized access, use and disclosure.


This site is not directed to children under the age of 13.


Pearson may send or direct marketing communications to users, provided that

  • Pearson will not use personal information collected or processed as a K-12 school service provider for the purpose of directed or targeted advertising.
  • Such marketing is consistent with applicable law and Pearson's legal obligations.
  • Pearson will not knowingly direct or send marketing communications to an individual who has expressed a preference not to receive marketing.
  • Where required by applicable law, express or implied consent to marketing exists and has not been withdrawn.

Pearson may provide personal information to a third party service provider on a restricted basis to provide marketing solely on behalf of Pearson or an affiliate or customer for whom Pearson is a service provider. Marketing preferences may be changed at any time.

Correcting/Updating Personal Information

If a user's personally identifiable information changes (such as your postal address or email address), we provide a way to correct or update that user's personal data provided to us. This can be done on the Account page. If a user no longer desires our service and desires to delete his or her account, please contact us at customer-service@informit.com and we will process the deletion of a user's account.


Users can always make an informed choice as to whether they should proceed with certain services offered by InformIT. If you choose to remove yourself from our mailing list(s) simply visit the following page and uncheck any communication you no longer want to receive: www.informit.com/u.aspx.

Sale of Personal Information

Pearson does not rent or sell personal information in exchange for any payment of money.

While Pearson does not sell personal information, as defined in Nevada law, Nevada residents may email a request for no sale of their personal information to NevadaDesignatedRequest@pearson.com.

Supplemental Privacy Statement for California Residents

California residents should read our Supplemental privacy statement for California residents in conjunction with this Privacy Notice. The Supplemental privacy statement for California residents explains Pearson's commitment to comply with California law and applies to personal information of California residents collected in connection with this site and the Services.

Sharing and Disclosure

Pearson may disclose personal information, as follows:

  • As required by law.
  • With the consent of the individual (or their parent, if the individual is a minor)
  • In response to a subpoena, court order or legal process, to the extent permitted or required by law
  • To protect the security and safety of individuals, data, assets and systems, consistent with applicable law
  • In connection the sale, joint venture or other transfer of some or all of its company or assets, subject to the provisions of this Privacy Notice
  • To investigate or address actual or suspected fraud or other illegal activities
  • To exercise its legal rights, including enforcement of the Terms of Use for this site or another contract
  • To affiliated Pearson companies and other companies and organizations who perform work for Pearson and are obligated to protect the privacy of personal information consistent with this Privacy Notice
  • To a school, organization, company or government agency, where Pearson collects or processes the personal information in a school setting or on behalf of such organization, company or government agency.


This web site contains links to other sites. Please be aware that we are not responsible for the privacy practices of such other sites. We encourage our users to be aware when they leave our site and to read the privacy statements of each and every web site that collects Personal Information. This privacy statement applies solely to information collected by this web site.

Requests and Contact

Please contact us about this Privacy Notice or if you have any requests or questions relating to the privacy of your personal information.

Changes to this Privacy Notice

We may revise this Privacy Notice through an updated posting. We will identify the effective date of the revision in the posting. Often, updates are made to provide greater clarity or to comply with changes in regulatory requirements. If the updates involve material changes to the collection, protection, use or disclosure of Personal Information, Pearson will provide notice of the change through a conspicuous notice on this site or other appropriate way. Continued use of the site after the effective date of a posted revision evidences acceptance. Please contact us if you have questions or concerns about the Privacy Notice or any objection to any revisions.

Last Update: November 17, 2020