They provide the foundation for many popular and effective machine learning algorithms like k-nearest neighbors for supervised learning and k-means clustering for unsupervised learning.

Different distance measures must be chosen and used depending on the types of the data. As such, it is important to know how to implement and calculate a range of different popular distance measures and the intuitions for the resulting scores.

A distance measure is an objective score that summarizes the relative difference between two objects in a problem domain.

## Distance/Similarity Measures in Machine Learning

Most commonly, the two objects are rows of data that describe a subject such as a person, car, or houseor an event such as a purchase, a claim, or a diagnosis. Perhaps the most likely way you will encounter distance measures is when you are using a specific machine learning algorithm that uses distance measures at its core.

The most famous algorithm of this type is the k-nearest neighbors algorithmor KNN for short. In the KNN algorithm, a classification or regression prediction is made for new examples by calculating the distance between the new example row and all examples rows in the training dataset.

The k examples in the training dataset with the smallest distance are then selected and a prediction is made by averaging the outcome mode of the class label or mean of the real value for regression. KNN belongs to a broader field of algorithms called case-based or instance-based learningmost of which use distance measures in a similar manner.

Another popular instance-based algorithm that uses distance measures is the learning vector quantizationor LVQ, algorithm that may also be considered a type of neural network. Related is the self-organizing map algorithm, or SOM, that also uses distance measures and can be used for supervised or unsupervised learning. Another unsupervised learning algorithm that uses distance measures at its core is the K-means clustering algorithm.

In instance-based learning the training examples are stored verbatim, and a distance function is used to determine which member of the training set is closest to an unknown test instance. Once the nearest training instance has been located, its class is predicted for the test instance. A short list of some of the more popular machine learning algorithms that use distance measures at their core is as follows:.

There are many kernel-based methods may also be considered distance-based algorithms. Perhaps the most widely known kernel method is the support vector machine algorithm, or SVM for short. Do you know more algorithms that use distance measures? Let me know in the comments below. When calculating the distance between two examples or rows of data, it is possible that different data types are used for different columns of the examples.

An example might have real values, boolean values, categorical values, and ordinal values.

Different distance measures may be required for each that are summed together into a single distance score. Numerical values may have different scales. This can greatly impact the calculation of distance measure and it is often a good practice to normalize or standardize numerical values prior to calculating the distance measure. Numerical error in regression problems may also be considered a distance. For example, the error between the expected value and the predicted value is a one-dimensional distance measure that can be summed or averaged over all examples in a test set to give a total distance between the expected and predicted outcomes in the dataset.

The calculation of the error, such as the mean squared error or mean absolute error, may resemble a standard distance measure. As we can see, distance measures play an important role in machine learning. Perhaps four of the most commonly used distance measures in machine learning are as follows:.

What are some other distance measures you have used or heard of? You need to know how to calculate each of these distance measures when implementing algorithms from scratch and the intuition for what is being calculated when using algorithms that make use of these distance measures. Hamming distance calculates the distance between two binary vectors, also referred to as binary strings or bitstrings for short.

You are most likely going to encounter bitstrings when you one-hot encode categorical columns of data. The distance between red and green could be calculated as the sum or the average number of bit differences between the two bitstrings.

This is the Hamming distance. For a one-hot encoded string, it might make more sense to summarize to the sum of the bit differences between the strings, which will always be a 0 or 1.Not because I thought it was absurd or something. But because that saying was also very common in Argentina. He always had a website for everything. From me, my home country was far away. So for me, km was far. For some of my friends, km was very far away. I started thinking about how complex the concept of distance was if you see it in a subjective way.

Sometimes you are sitting across someone but that person seems so distant. Other times, a person is kilometers away. And a message is all that it takes to feel that someone very close. In some cases, I guess that close and distant depends on the point of view. In machine learning, many supervised and unsupervised algorithms use Distance Metrics to understand patterns in the input data.

Also, it is used to recognize similarities among the data. Choosing a good distance metric will improve how well a classification or clustering algorithms performed. A Distance Metric employs distance functions that tell us the distance between the elements in the dataset.

Luckily, these distances can be measured with a mathematical formula. If the distance is small, the elements are likely similar. If the distance is large, the degree of similarity will be low.

There are several distance metrics which can be used. It is important to know what they take into account. This will help us choose which one is more appropriated for a model to avoid introducing errors or misinterpretations. If we think of distances between two cities, we think about how many kilometers we have to drive on a highway. These examples of distances that we can think of are examples of Euclidean distance.

Essentially, it measures the length of a segment that connects two points. Does this ring any bell? Do you remember the Pythagorean Theorem from Math classes? The theorem states that that the square of the hypotenuse the side opposite the right angle is equal to the sum of the squares of the other two sides.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. It only takes a minute to sign up. I read that 'Euclidean distance is not a good distance in high dimensions'. I guess this statement has something to do with the curse of dimensionality, but what exactly? Besides, what is 'high dimensions'?

I have been applying hierarchical clustering using Euclidean distance with features. Up to how many features is it 'safe' to use this metric? If a constant number of examples is distributed uniformly in a high-dimensional hypercube, beyond some dimensionality most examples are closer to a face of the hypercube than to their nearest neighbor.

And if we approximate a hypersphere by inscribing it in a hypercube, in high dimensions almost all the volume of the hypercube is outside the hypersphere. This is bad news for machine learning, where shapes of one type are often approximated by shapes of another. Another application, beyond machine learning, is nearest neighbor search: given an observation of interest, find its nearest neighbors in the sense that these are the points with the smallest distance from the query point.

But in high dimensions, a curious phenomenon arises: the ratio between the nearest and farthest points approaches 1, i. This phenomenon can be observed for wide variety of distance metrics, but it is more pronounced for the Euclidean metric than, say, Manhattan distance metric. The premise of nearest neighbor search is that "closer" points are more relevant than "farther" points, but if all points are essentially uniformly distant from each other, the distinction is meaningless.

From Charu C. Aggarwal, Alexander Hinneburg, Daniel A. In such a case, the nearest neighbor problem becomes ill defined, since the contrast between the distances to diferent data points does not exist. In such cases, even the concept of proximity may not be meaningful from a qualitative perspective: a problem which is even more fundamental than the performance degradation of high dimensional algorithms.

Many high-dimensional indexing structures and algorithms use the [E]uclidean distance metric as a natural extension of its traditional use in two- or three-dimensional spatial applications. They produce some results which demonstrate that these "fractional norms" exhibit the property of increasing the contrast between farthest and nearest points.

This may be useful in some contexts, however there is a caveat: these "fractional norms" are not proper distance metrics because they violate the triangle inequality.They supply the inspiration for a lot of standard and efficient machine studying algorithms like k-nearest neighbors for supervised studying and k-means clustering for unsupervised studying. Completely different distance measures should be chosen and used relying on the varieties of the info.

As such, it is very important know learn how to implement and calculate a variety of various standard distance measures and the intuitions for the ensuing scores. A distance measure is an goal rating that summarizes the relative distinction between two objects in an issue area. Mostly, the 2 objects are rows of knowledge that describe a topic similar to an individual, automotive, or homeor an occasion similar to a purchase order, a declare, or a prognosis.

Essentially the most well-known algorithm of this sort is the k-nearest neighbors algorithm, or KNN for brief. Within the KNN algorithm, a classification or regression prediction is made for brand new examples by calculating the gap between the brand new instance row and all examples rows within the coaching dataset. The ok examples within the coaching dataset with the smallest distance are then chosen and a prediction is made by averaging the result mode of the category label or imply of the true worth for regression.

KNN belongs to a broader subject of algorithms referred to as case-based or instance-based studying, most of which use distance measures in an analogous method.

One other standard instance-based algorithm that makes use of distance measures is the training vector quantization, or LVQ, algorithm that will even be thought-about a kind of neural community. Associated is the self-organizing map algorithm, or SOM, that additionally makes use of distance measures and can be utilized for supervised or unsupervised studying.

One other unsupervised studying algorithm that makes use of distance measures at its core is the Ok-means clustering algorithm.

In instance-based studying the coaching examples are saved verbatim, and a distance operate is used to find out which member of the coaching set is closest to an unknown check occasion. As soon as the closest coaching occasion has been situated, its class is predicted for the check occasion.

A brief checklist of a few of the extra standard machine studying algorithms that use distance measures at their core is as follows:. There are numerous kernel-based strategies may additionally be thought-about distance-based algorithms. Maybe essentially the most extensively recognized kernel methodology is the assist vector machine algorithm, or SVM for brief. Have you learnt extra algorithms that use distance measures?

Let me know within the feedback under. An instance may need actual values, boolean values, categorical values, and ordinal values. Completely different distance measures could also be required for every which are summed collectively right into a single distance rating.

Numerical values might have totally different scales. Numerical error in regression issues may additionally be thought-about a distance.

For instance, the error between the anticipated worth and the anticipated worth is a one-dimensional distance measure that may be summed or averaged over all examples in a check set to present a complete distance between the anticipated and predicted outcomes within the dataset.

The calculation of the error, such because the imply squared error or imply absolute error, might resemble a normal distance measure. As we are able to see, distance measures play an essential function in machine studying. Maybe 4 of essentially the most generally used distance measures in machine studying are as follows:.

What are another distance measures you have got used or heard of? Hamming distance calculates the gap between two binary vectors, additionally known as binary strings or bitstrings for brief. The space between crimson and inexperienced could possibly be calculated because the sum or the common variety of bit variations between the 2 bitstrings. That is the Hamming distance. For a one-hot encoded string, it would make extra sense to summarize to the sum of the bit variations between the strings, which is able to all the time be a zero or 1.

We will reveal this with an instance of calculating the Hamming distance between two bitstrings, listed under. We will additionally carry out the identical calculation utilizing the hamming operate from SciPy.

The entire instance is listed under.

Working the instance, we are able to see we get the identical end result, confirming our guide implementation. In any other case, columns which have giant values will dominate the gap measure.

Euclidean distance is calculated because the sq.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. Data Science Stack Exchange is a question and answer site for Data science professionals, Machine Learning specialists, and those interested in learning more about the field.

It only takes a minute to sign up. I am trying to look for a good argument on why one would use the Manhattan distance over the Euclidean distance in machine learning. The closest thing I found to a good argument so far is on this MIT lecture. Shortly after, the professor says that, because the number of legs of a reptile varies from 0 to 4 whereas the other features are binary, only vary from 0 to 1the "number of legs" feature will end up having a much higher weight if the Euclidean distance is used.

Sure enough, that is indeed right. But one would also have that problem if using the Manhattan distance only that the problem would be slightly mitigated because we don't square the difference like we do on the Euclidean distance.

A better way to solve the above problem would be to normalize the "number of legs" feature so its value will always be between 0 and 1. Therefore, since there is a better way to solve the problem, it felt like the argument of using the Manhattan distance in this case lacked a stronger point, at least in my opinion. Does anyone actually know why and when someone would use Manhattan distance over Euclidean?

Can anyone give me an example in which using the Manhattan distance would yield better results? According to this interesting paper, Manhattan distance L1 norm may be preferable to Euclidean distance L2 norm for the case of high dimensional data:. The authors of the paper even go a step further and suggest to use Lk norm distances, with a fractional value of k, for very high dimensional data in order to improve the results of distance-based algorithms, like clustering.

I can suggest a couple ideas, from wikipedia. Both the RMSE and the MAE are ways to measure the distance between two vectors: the vector of predictions and the vector of target values. Various distance measures, or norms, are possible:.

### Different Similarity/Distance Measures in Machine Learning

Computing the root of a sum of squares RMSE corresponds to the Euclidian norm: it is the notion of distance you are familiar with. It is sometimes called the Manhattan norm because it measures the distance between two points in a city if you can only travel along orthogonal city blocks. More generally, The higher the norm index, the more it focuses on large values and neglects small ones. But when outliers are exponentially rare like in a bell-shaped curvethe RMSE performs very well and is generally preferred.

The use of Manhattan distance depends a lot on the kind of co-ordinate system that your dataset is using. While Euclidean distance gives the shortest or minimum distance between two points, Manhattan has specific implementations. For example, if we were to use a Chess dataset, the use of Manhattan distance is more appropriate than Euclidean distance.

Another use would be when are interested in knowing the distance between houses which are few blocks apart. Also, you might want to consider Manhattan distance if the input variables are not similar in type such as age, gender, height, etc.

Due to the curse of dimensionality, we know that Euclidean distance becomes a poor choice as the number of dimensions increases. So in a nutshell: Manhattan distance generally works only if the points are arranged in the form of a grid and the problem which we are working on gives more priority to the distance between the points only along with the grids, but not the geometric distance. Sign up to join this community.

The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. When would one use Manhattan distance as opposed to Euclidean distance?

Ask Question. Asked 2 years, 9 months ago. Active 9 days ago. Viewed 27k times. At you can see on the slides the following statement: "Typically use Euclidean metric; Manhattan may be appropriate if different dimensions are not comparable.

Michael Mior 4 4 bronze badges. Tiago Tiago 1 1 gold badge 2 2 silver badges 10 10 bronze badges.They provide the foundation for many popular and effective machine learning algorithms like k-nearest neighbors for supervised learning and k-means clustering for unsupervised learning. Different distance measures must be chosen and used depending on the types of the data. As such, it is important to know how to implement and calculate a range of different popular distance measures and the intuitions for the resulting scores.

A distance measure is an objective score that summarizes the relative difference between two objects in a problem domain. Most commonly, the two objects are rows of data that describe a subject such as a person, car, or houseor an event such as a purchase, a claim, or a diagnosis.

Perhaps the most likely way you will encounter distance measures is when you are using a specific machine learning algorithm that uses distance measures at its core. The most famous algorithm of this type is the k-nearest neighbors algorithmor KNN for short.

In the KNN algorithm, a classification or regression prediction is made for new examples by calculating the distance between the new example row and all examples rows in the training dataset. The k examples in the training dataset with the smallest distance are then selected and a prediction is made by averaging the outcome mode of the class label or mean of the real value for regression.

KNN belongs to a broader field of algorithms called case-based or instance-based learningmost of which use distance measures in a similar manner. Another popular instance-based algorithm that uses distance measures is the learning vector quantizationor LVQ, algorithm that may also be considered a type of neural network. Related is the self-organizing map algorithm, or SOM, that also uses distance measures and can be used for supervised or unsupervised learning.

Another unsupervised learning algorithm that uses distance measures at its core is the K-means clustering algorithm. In instance-based learning the training examples are stored verbatim, and a distance function is used to determine which member of the training set is closest to an unknown test instance. Once the nearest training instance has been located, its class is predicted for the test instance. A short list of some of the more popular machine learning algorithms that use distance measures at their core is as follows:.

There are many kernel-based methods may also be considered distance-based algorithms. Perhaps the most widely known kernel method is the support vector machine algorithm, or SVM for short. Do you know more algorithms that use distance measures? Let me know in the comments below. When calculating the distance between two examples or rows of data, it is possible that different data types are used for different columns of the examples.

An example might have real values, boolean values, categorical values, and ordinal values. Different distance measures may be required for each that are summed together into a single distance score. Numerical values may have different scales.

This can greatly impact the calculation of distance measure and it is often a good practice to normalize or standardize numerical values prior to calculating the distance measure. Numerical error in regression problems may also be considered a distance. For example, the error between the expected value and the predicted value is a one-dimensional distance measure that can be summed or averaged over all examples in a test set to give a total distance between the expected and predicted outcomes in the dataset.

The calculation of the error, such as the mean squared error or mean absolute error, may resemble a standard distance measure. As we can see, distance measures play an important role in machine learning. Perhaps four of the most commonly used distance measures in machine learning are as follows:.

What are some other distance measures you have used or heard of?In machine learning more often than not you would be dealing with techniques that requires to calculate similarity and distance measure between two data points. Distance between two data points can be interpreted in various ways depending on the context.

If two data points are closer to each other it usually means two data are similar to each other. For e. This denotes that in terms of size both basketball and football are similar. Whereas if we calculate diameter of tennis ball, then its distance with diameter of basketball and football will be larger. This denotes that tennis ball is not similar to either basketball or football in terms of size.

Let us now look at common techniques of calculating distances between data points. The examples would be shown in 2 dimensional scale with two coordinates x,y but the technique can be extended to more dimensions.

In this technique, the data points are considered as vectors that has some direction. On the other hand if the angle theta is more this means the two vectors are more distant apart and has less similarity or may be completely dissimilar.

### Measuring Similarity between Vectors for Machine Learning

To measure this we calculate cosine of theta, hence the name Cosine Similarity. Hope this was a good simple read with some fruitful insight, especially for beginners to understand similarity and distance measure in machine learning. Do share your feed back about this post in the comments section below. If you found this post informative, then please do share this and subscribe to us by clicking on bell icon for quick notifications of new upcoming posts. Sign in. Log into your account. Forgot your password?

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