机器人如何实时确定他们的状态，并从带有噪声的传感器测量量获得周围环境的信息？在这个模块中，你将学习怎样让机器人把不确定性融入估计，并向动态和变化的世界进行学习。特殊专题包括用于定位和绘图的概率生成模型和贝叶斯滤波器。

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来自 University of Pennsylvania 的课程

机器人学：估计和学习

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机器人如何实时确定他们的状态，并从带有噪声的传感器测量量获得周围环境的信息？在这个模块中，你将学习怎样让机器人把不确定性融入估计，并向动态和变化的世界进行学习。特殊专题包括用于定位和绘图的概率生成模型和贝叶斯滤波器。

从本节课中

Gaussian Model Learning

We will learn about the Gaussian distribution for parametric modeling in robotics. The Gaussian distribution is the most widely used continuous distribution and provides a useful way to estimate uncertainty and predict in the world. We will start by discussing the one-dimensional Gaussian distribution, and then move on to the multivariate Gaussian distribution. Finally, we will extend the concept to models that use Mixtures of Gaussians.

- Daniel LeeProfessor of Electrical and Systems Engineering

School of Engineering and Applied Science

In this lecture, we are going to learn how to compute an estimate of

the multivariate Gaussian model parameters from observed data.

Let us remind that a Gaussian model has two parameters,

the mean and covariance matrix.

We are going to use the same definition of likelihood we introduced for

the univariate case.

Likelihood is the probability of one observation

given the model parameters which are unknown.

We are interested in obtaining the mean and

covariance matrix that maximizes the likelihood given a set of observations.

This is the mathematical statements of our goal.

I hope you remember that the likelihood function is a joint

probability of all the data, which can be intractable, in general.

But as we did for univariate Gaussian, if we assume independence of data points,

the joint likelihood can be expressed as the product of individual likelihoods.

With this notation, how can we obtain the maximum likelihood

estimate of the parameters that are now a vector and a matrix.

Again, we can induce the solution analytically, and

the key ideas are the same to the one-d case.

First, we are going to use the properties of log functions.

We have seen that instead of maximizing the likelihood, we may find the parameters

that maximizes the log likelihood, because our maximizers are the same.

Also, remember that the log of products equals the sum of the logs.

Now, we can rewrite the problem as finding mu and sigma, that maximizes

the sum of all the log likelihoods of the individual measurements.

The next step is to apply

the specific form of Gaussian PDF and take the log of it.

As before, we can ignore the constant term C,

because it does not affect the solution.

Then, we can change the formula into a minimization problem.

Finally, we solve the optimality condition for minimizing the cost function j.

This will give us the maximum likelihood estimate of the mean and

covariance matrix.

The full details of mathematics can be found in the supplementary file.

It might look a bit more complicated than the one-d Gaussian case,

because now we have a vectors, a matrix, and a cost function.

But essentially, the principles we apply are the same.

The final solution we get for computation is exactly the sample mean,

the form of vector, and the sample covariance matrix.

Now, we have learned how to estimate the multivariate Gaussian parameters,

the maximum likelihood sense.

Let's get back to our ball color example.

The closer view of the graph shows the ball color distribution,

the blue and red dimensions.

Using the data and the formula we obtained,

we can compute the maximum likelihood estimate of the parameters as shown.

From the contours in the plot, we can visually check that the red and

the blue channels are correlated negatively in our model.

The examples we have created so

far have the nice symmetric shape with a single peak.

However, it is possible that some targets have weird distribution of variables.

In the next lecture, we will learn how we can employ multiple Gaussians to

form a mixure model, which can express diverse distributions.