In Depth: Gaussian Mixture Models

Jul 29, · One example would be a mixture of pencil lead and diamonds (both carbon). Another example could be a mixture of gold powder and nuggets. Besides being classified as heterogeneous or homogeneous, mixtures may also be described according to the particle size of the components. Sep 21, · An example of a solid and liquid phase suspension is mud. Another type of suspension is called an aerosol, which is where small liquid droplets or small solid particulate is suspended in a gas. When liquids that do not mix form a mixture (weird, I know), it is a heterogeneous mixture .

In chemistry, a mixture forms when two or more substances are combined such that each substance retains its own chemical identity. Chemical bonds between the components are neither broken nor formed. Note ann even though the chemical properties of the components haven't changed, a mixture may exhibit new physical properties, like boiling point and melting point.

For example, mixing together water and alcohol produces a mixture that has a higher boiling point and lower melting point than scuence lower boiling point and higher boiling point than water. Two broad categories of mixtures are heterogeneous and homogeneous mixtures.

Heterogeneous mixtures are not uniform throughout the composition e. The distinction between heterogeneous and homogeneous mixtures is a matter of magnification or scale. For example, even air can appear to be heterogeneous if your sample only contains a few moleculeshow to get cyber akuma a bag of mixed vegetables may appear homogeneous if your sample is an entire truckload full of them.

Also note, even if a sample consists how to get a free ps2 a single element, it may form a heterogeneous mixture.

One example would be a mixture of pencil lead and diamonds both carbon. Another example could be a mixture of gold powder and nuggets. Besides being classified as heterogeneous or homogeneous, mixtures may also be described according to the particle size of the components:. Solution: A chemical solution contains very small particle sizes less than 1 nanometer in diameter. A solution is physically stable and the components cannot be separated by decanting or centrifuging the sample.

Examples of solutions include air gasdissolved oxygen in water liquidand mercury in gold amalgam solidscisnce solidand gelatin solid. Colloid: A colloidal solution appears homogeneous to the naked eye, but particles are apparent under microscope magnification. Particle sizes range from sccience nanometer to 1 micrometer. Like what is a school improvement team, colloids are physically stable.

They exhibit the Tyndall effect. Colloid components can't be separated using decantationbut may be isolated by centrifugation. Examples of colloids include hair spray gassmoke gaswhipped cream liquid foamblood liquid.

Suspension: Particles in a suspension are often large enough that the mixture appears heterogeneous. Stabilizing agents are required to keep the particles from separating. Like colloids, suspensions exhibit the Tyndall effect. Suspensions may be separated using either decantation or centrifugation. Examples of suspensions include dust in air solid in gasvinaigrette liquid in liquidmud solid in liquidsand solids blended togetherand granite blended solids. Just because you mix two chemicals together, don't expect you'll always get a mixture!

If a chemical reaction occurs, the identity of a reactant changes. This is not a mixture. Combining vinegar and baking soda results in a reaction to produce carbon dioxide and water. So, you don't have a mixture. Combining an acid and a base also does not produce a mixture. Share Flipboard Email. Anne Marie Helmenstine, Ph. Chemistry Expert. Helmenstine holds a Ph. Examplw has taught science courses at the high school, college, and graduate levels.

Facebook Facebook Twitter Twitter. Key Takeaways: Mixtures A mixture is defined as the result of combining two or more substances, such that each maintains its chemical identity. In other words, a chemical reaction does not what is a class b felony in ct between components of a mixture. Examples include combinations of salt and sand, sugar and water, and blood. Mixtures are classified based on how uniform they are and on the particle size of components relative to each other.

Homogeneous mixtures have a uniform composition and sciejce throughout their volume, while heterogeneous mixtures do not appear uniform and may consist of different phases e. Examples of types of mixtures defined by particle size scienc colloids, solutions, and suspensions.

Cite this Article Format. Helmenstine, Anne Marie, Ph. What Is a Mixture in Science? Chemistry Scavenger Hunt Clues and Answers. What Is a Heterogeneous Mixture? Definition and Examples. Colloid Definition - Chemistry Glossary. Solutions, Suspensions, Colloids, and Dispersions. Tyndall Effect Definition and Examples. What Are Examples of Pure Substances?

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From intuition to implementation

Apr 08, · Vinegar is an example of a homogenous mixture made up of flavoring and acetic acid used to cook a variety of meals. While vinegar may have different components to it, and be sold in different concentrations, the chemical components which comprise the vinegar are uniformly distributed throughout the mixture so it is a homogenous mixture. In this example the water is the solvent and the salt is the solute. What is the difference between a solution and a mixture? In chemistry a solution is actually a type of mixture. A solution is a mixture that is the same or uniform throughout. Think of the example of salt water. This is also called a "homogenous mixture.". Motivating GMM: Weaknesses of k-Means¶. Let's take a look at some of the weaknesses of k-means and think about how we might improve the cluster datmetopen.com we saw in the previous section, given simple, well-separated data, k-means finds suitable clustering results. For example, if we have simple blobs of data, the k-means algorithm can quickly label those clusters in a way that closely matches.

Sign in. In the world of Machine Learning, we can distinguish two main areas: Supervised and unsupervised learning. The main difference between both lies in the nature of the data as well as the approaches used to deal with it. Clustering is an unsupervised learning problem where we intend to find clusters of points in our dataset that share some common characteristics. Our job is to find sets of poi n ts that appear close together.

In this case, we can clearly identify two clusters of points which we will colour blue and red, respectively:. Please note that we are now introducing some additional notation. A popular clustering algorithm is known as K-means, which will follow an iterative approach to update the parameters of each clusters. More specifically, what it will do is to compute the means or centroids of each cluster, and then calculate their distance to each of the data points.

The latter are then labeled as part of the cluster that is identified by their closest centroid. This process is repeated until some convergence criterion is met, for example when we see no further changes in the cluster assignments.

One important characteristic of K-means is that it is a hard clustering method , which means that it will associate each point to one and only one cluster. A limitation to this approach is that there is no uncertainty measure or probability that tells us how much a data point is associated with a specific cluster. So what about using a soft clustering instead of a hard one?

Each Gaussian k in the mixture is comprised of the following parameters:. Let us now illustrate these parameters graphically:. Each Gaussian explains the data contained in each of the three clusters available. The mixing coefficients are themselves probabilities and must meet this condition:.

Now how do we determine the optimal values for these parameters? To achieve this we must ensure that each Gaussian fits the data points belonging to each cluster.

This is exactly what maximum likelihood does. In general, the Gaussian density function is given by:. Where x represents our data points, D is the number of dimensions of each data point.

For later purposes, we will also find it useful to take the log of this equation, which is given by:. If we differentiate this equation with respect to the mean and covariance and then equate it to zero, then we will be able to find the optimal values for these parameters, and the solutions will correspond to the Maximum Likelihood Estimates MLE for this setting. However, because we are dealing with not just one, but many Gaussians, things will get a bit complicated when time comes for us to find the parameters for the whole mixture.

In this regard, we will need to introduce some additional aspects that we discuss in the next section. We are now going to introduce some additional notation. Just a word of warning. Math is coming on! We can express this as:. It is one when x came from Gaussian k , and zero otherwise. Likewise, we can state the following:. Which means that the overall probability of observing a point that comes from Gaussian k is actually equivalent to the mixing coefficient for that Gaussian.

This makes sense, because the bigger the Gaussian is, the higher we would expect this probability to be. Now let z be the set of all possible latent variables z , hence:. We know beforehand that each z occurs independently of others and that they can only take the value of one when k is equal to the cluster the point comes from.

Now, what about finding the probability of observing our data given that it came from Gaussian k? Turns out to be that it is actually the Gaussian function itself! Following the same logic we used to define p z , we can state:. Ok, now you may be asking, why are we doing all this? Remember our initial aim was to determine what the probability of z given our observation x? Well, it turns out to be that the equations we have just derived, along with the Bayes rule, will help us determine this probability.

From the product rule of probabilities, we know that. Hmm, it seems to be that now we are getting somewhere. The operands on the right are what we have just found. Perhaps some of you may be anticipating that we are going to use the Bayes rule to get the probability we eventually need. However, first we will need p x n , not p x n , z. So how do we get rid of z here?

Yes, you guessed it right. We just need to sum up the terms on z , hence. This is the equation that defines a Gaussian Mixture, and you can clearly see that it depends on all parameters that we mentioned previously! To determine the optimal values for these we need to determine the maximum likelihood of the model. We can find the likelihood as the joint probability of all observations x n , defined by:. Now in order to find the optimal parameters for the Gaussian mixture, all we have to do is to differentiate this equation with respect to the parameters and we are done, right?

Not so fast. We have an issue here. We can see that there is a logarithm that is affecting the second summation. Calculating the derivative of this expression and then solving for the parameters is going to be very hard! What can we do? Well, we need to use an iterative method to estimate the parameters.

But first, remember we were supposed to find the probability of z given x? From Bayes rule, we know that. From our earlier derivations we learned that:. Moving forward we are going to see this expression a lot. Next we will continue our discussion with a method that will help us easily determine the parameters for the Gaussian mixture.

Well, at this point we have derived some expressions for the probabilities that we will find useful in determining the parameters of our model. However, in the past section we could see that simply evaluating 3 to find such parameters would prove to be very hard. Fortunately, there is an iterative method we can use to achieve this purpose. It is called the Expectation — Maximization , or simply EM algorithm. Let the parameters of our model be. For instance, we can use the results obtained by a previous K-Means run as a good starting point for our algorithm.

Step 2 Expectation step : Evaluate. Now if we replace 4 in 5 , we will have:. How can we find it? It is just the complete likelihood of the model, including both X and Z , and we can find it by using the following expression:. Which is the result of calculating the joint probability of all observations and latent variables and is an extension of our initial derivations for p x. The log of this expression is given by.

And we have finally gotten rid of this troublesome logarithm that affected the summation in 3. With all of this in place, it will be much easier for us to estimate the parameters by just maximizing Q with respect to the parameters, but we will deal with this in the maximization step. Besides, remember that the latent variable z will only be 1 once everytime the summation is evaluated.

With that knowledge, we can easily get rid of it as needed for our derivations. Finally, we can replace 7 in 6 to get:. In the maximization step, we will find the revised parameters of the mixture. For this purpose, we will need to make Q a restricted maximization problem and thus we will add a Lagrange multiplier to 8. Which is what we ended up with in the previous step.

To do so, we will need to add a suitable Lagrange multiplier. Therefore, we should rewrite 8 in this way:. And now we can easily determine the parameters by using maximum likelihood. Then, by rearranging the terms and applying a summation over k to both sides of the equation, we obtain:. We can use equation 3 to monitor the log-likelihood in each step and we are always guaranteed to reach a local maximum. Next, we will see parts of the Jupyter notebook I have provided so you can see a working implementation of GMMs in Python.

I have used the Iris dataset for this exercise, mainly for simplicity and fast training. From our previous derivations, we stated that the EM algorithm follows an iterative approach to find the parameters of a Gaussian Mixture Model. Our first step was to initialise our parameters. In this case, we can use the values of K-means to suit this purpose. The Python code for this would look like:.

Next, we execute the expectation step. Here we calculate.

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