Mean Shift

## 🚀 Clustering Around the Mean

Have you ever noticed how store shelves organize similar products together? Or how online stores recommend items based on your recent browsing? Behind the scenes, algorithms are hard at work grouping similar things together - a process known as clustering.

One popular clustering technique is called Mean Shift. At its core, Mean Shift is all about finding dense areas in a set of data points. It works by shifting points towards regions of higher density until they converge around a center. This center becomes the mean of a cluster. 👉

In this blog, we'll unpack how Mean Shift works its magic to find clusters, advantages and limitations, real-world applications, and key terminology. Let's shift into understanding this concept!

## History
The foundations of Mean Shift trace back to efforts in the 1950s-60s to estimate probability density functions. 🌡️ These functions model the relative likelihood for a random variable to take on different values. Researchers developed non-parametric methods like kernel density estimation to estimate density without assumptions about its shape.

In 1975, Finnish statistician Eero Pekka Tuominen extended this work by introducing the mean shift algorithm for discontinuity-preserving smoothing. 📈 The mean shift procedure shifted points towards regions of higher density by iteratively computing a mean.

Later in 1995, researchers Keith Fukunaga and Larry Hostetler realized mean shift could also be used for clustering! 👥 They described a new mean shift clustering algorithm.

Finally, in 2002, Dorin Comaniciu and Peter Meer published seminal papers expanding on mean shift for robust computer vision applications like image segmentation and tracking. 🖼️ These papers sparked wider interest and usage of mean shift clustering techniques.

Now that we've covered the history, let's shift gears and unpack how mean shift clustering works!

## 🤔 How it Works

At a high level, Mean Shift is an iterative clustering approach that works by shifting data points towards dense regions to locate clusters. The main idea is that data points are attracted to denser areas kind of like moths to a flame! 💡

The algorithm works on a set of data points. We define a window around each data point - this can be thought of as the data point's neighborhood. Using kernel density estimation, we estimate the density around the window. 📈

We then shift the window towards the direction of maximum increase in the density. By iteratively shifting towards denser areas, the windows will converge around local maxima of density. These local maxima become the centers of our clusters!

An analogy is helpful. Imagine colored dots scattered on a paper are shoppers milling around stores in a mall. If you calculate density using a shifting window, shoppers will naturally shift towards crowded shops over time. The crowded shops represent the clusters!

## 📈 The Algorithm

Let's walk through the Mean Shift steps to cluster shoppers in our mall example!

**Step 1) Define windows 🪟**

We start by defining a window around each shopper. This captures their local neighborhood. For example, we'll look at 3 shopper's windows:

Andy's window = the 5 closest shoppers 
Becky's window = the 7 closest shoppers
Chloe's window = the 3 closest shoppers

**Step 2) Estimate density 🌡️** 

Next we estimate density for each window using a kernel density estimator. This gives higher density to windows with more shoppers closer together.  

**Step 3) Shift towards local maximum density 👆**

Now we move the window uphill to where density increases - the local maximum density locations will become our cluster centers.

Andy's window shifts 2 shops to the left  (higher density that way)  
Becky's windows shifts 1 shop to the right (more crowded shop)
Chloe's window stays (she's already in a crowded shop)

**Step 4) Repeat 🔁**

We repeat steps 2-3, iteratively shifting windows uphill until convergence - shoppers end up clustered around the most crowded shops!

Here's a draft section covering some of the key advantages of Mean Shift clustering:

## 👍 Advantages of Shifting Means

As we've seen, Mean Shift offers an intuitive clustering approach by shifting towards areas of higher density. Some key advantages are:

**No assumptions about number of clusters** 🤖 Unlike k-means clustering which requires you set the number of clusters k upfront, mean shift automatically discovers the optimal clusters based on the density landscape. This makes it more flexible.

**Handles arbitrary shapes** 🎨 The kernel density estimator allows Mean Shift to handle arbitrarily shaped clusters rather than constraining to globular shapes. This provides flexibility when cluster shapes are unknown.  

**Robust to outliers** 💪 Outliers don't skew the clustering as significantly compared to centroid-based methods like k-means that optimize for low variance around a mean. Mean shift is less sensitive as it shifts based on local density.

**Smooth convergence** 🚀 The gradual uphill shifts towards local maxima is more stable with lower likelihood of getting stuck in local optima compared to discrete reassignment of data points.

## 👎 Limitations of Shifting Density

While Mean Shift has nice properties, there are some disadvantages:

**Computationally intensive** 💻 All the recomputation of the kernel density estimator and shifting windows makes Mean Shift computationally intensive compared to simpler algorithms like k-means clustering. This can limit application for large datasets.

**Sensitive to bandwidth** 🎚️ The density estimator relies on a bandwidth parameter that controls the size of the shifting windows. The final clusters are sensitive to this choice of bandwidth. Difficult to know the optimal bandwidth a priori without experimentation.

**Prone to local optima** 🥴 Like other hill climbing approaches, Mean Shift can sometimes get stuck around a local density mode rather than the global maximum. This depends on the cluster landscape and initial window locations.

**Metric dependent** 📐 The effectiveness of Mean Shift depends heavily on the choice of similarity metric. The typical Euclidean distance can be ineffective for complex cluster separation.

## 🌎 Clustering Applications

**Image segmentation** 🖼️ Mean Shift can automatically segment images into distinct regions by shifting window locations toward areas of similar color and texture. Handy for detecting objects in images.

**Anomaly detection** 🚨 The clustered model from Mean Shift provides a notion of expected densities. Points in much lower density regions can be flagged as potential anomalies for cybersecurity.

**Social network analysis** 👥 By clustering social graphs based on density, Mean Shift can uncover distinct social communities and see how ideas flow between these communities.

**Recommendation systems** 🛒 As we saw with the mall shopper analogy, Mean Shift clusters can inform recommendation systems on ecommerce sites, recommending items in dense, popular categories that users seem to gravitate towards.

**Visual tracking** 📹 The Mean Shift algorithm shines for tracking objects in video streams. By continuously shifting windows toward areas of highest density, the trajectory of objects through frames can be robustly tracked.

## 💬 TL;DR

Mean Shift is an intuitive clustering technique that works by shifting data points uphill toward areas of higher density. This gradual convergence allows the discovery of clusters without assumptions on shape or number of clusters.

Here's the gist:

- Originated from kernel density estimation and was adapted for clustering
- Defines windows around data points and shifts towards dense areas
- Outperforms k-means for non-globular clusters and is robust against outliers
- Useful for image segmentation, anomaly detection, recommendations, and more
- Limitations include computational complexity and sensitivity to parameters

In the end, Mean Shift leverages density gradients to uncover nuanced groupings in data without getting derailed by stray points or too many assumptions. This intrinsic attraction to density makes shifting means a uniquely intuitive approach!

## 🔤 Vocabulary

**Kernel density estimation** - Non-parametric method to estimate probability density function of random variables without assumptions about the distribution. Helps discover patterns.

Like using a flashlight 🔦 to scan and estimate variation in density of trees in a forest without knowing specifics about tree types.

**Bandwidth** - Parameter that controls the size of the windows in mean shift algorithm. Larger values smooth more.

Bandwidth is like the beam width on the flashlight - broader beam shows less details but general patterns become clearer.

**Hill climbing** - Optimization technique that incrementally moves toward better solutions, akin to walking uphill. Can get stuck in local optima.

Like hiking uphill towards a mountain peak but sometimes getting stuck on a foothill rather than the highest peak.

**k-means clustering** - Classic clustering approach that partitions data into k clusters by minimizing within-cluster variance around cluster means.

Organizing books 📚 on a shelf into k genre sections by similarity so books in each section resemble each other.

**Anomaly detection** - Detecting outlier points in data that are significantly different than expected patterns. Useful for catching issues.

Like identifying a flamingo 🦩 mixed in with a flock of pigeons - the odd one out clearly doesn't fit the expected pattern.

**Image segmentation** - Clustering technique for images that groups pixels by similarity in color and texture to divide image into distinct regions.

Like separating a bowl of mixed fruits 🍎🍌🍇 into groups of apples, bananas, and grapes for making smoothies based on color and shape.

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