Introduction to R

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From the course by University of California, Santa Cruz

Bayesian Statistics: From Concept to Data Analysis

1287 ratings

University of California, Santa Cruz

Bayesian Statistics: From Concept to Data Analysis

1287 ratings

This course introduces the Bayesian approach to statistics, starting with the concept of probability and moving to the analysis of data. We will learn about the philosophy of the Bayesian approach as well as how to implement it for common types of data. We will compare the Bayesian approach to the more commonly-taught Frequentist approach, and see some of the benefits of the Bayesian approach. In particular, the Bayesian approach allows for better accounting of uncertainty, results that have more intuitive and interpretable meaning, and more explicit statements of assumptions. This course combines lecture videos, computer demonstrations, readings, exercises, and discussion boards to create an active learning experience. For computing, you have the choice of using Microsoft Excel or the open-source, freely available statistical package R, with equivalent content for both options. The lectures provide some of the basic mathematical development as well as explanations of philosophy and interpretation. Completion of this course will give you an understanding of the concepts of the Bayesian approach, understanding the key differences between Bayesian and Frequentist approaches, and the ability to do basic data analyses.

From the lesson

Statistical Inference

This module introduces concepts of statistical inference from both frequentist and Bayesian perspectives. Lesson 4 takes the frequentist view, demonstrating maximum likelihood estimation and confidence intervals for binomial data. Lesson 5 introduces the fundamentals of Bayesian inference. Beginning with a binomial likelihood and prior probabilities for simple hypotheses, you will learn how to use Bayes’ theorem to update the prior with data to obtain posterior probabilities. This framework is extended with the continuous version of Bayes theorem to estimate continuous model parameters, and calculate posterior probabilities and credible intervals.

Lesson 4.1 Confidence intervals4:57

Lesson 4.2 Likelihood function and maximum likelihood7:06

Lesson 4.3 Computing the MLE3:02

Lesson 4.4 Computing the MLE: examples4:21

Introduction to R6:57

Plotting the likelihood in R4:38

Plotting the likelihood in Excel4:48

Meet the Instructors

Herbert Lee
Professor
Applied Mathematics and Statistics

R is an open source statistical software package.

It can be freely downloaded a rproject.org.

There are other Coursera courses that provide detailed lessons on using R.

Here, I will just provide a short introduction to try to get you using R for

the task needed in this course.

First, launch R [SOUND].

Or we'll do basic arithmetic.

You can just type in an expression and hit Enter.

For example, (1+2) x 3.

As you can see, R knows the correct order of operations.

R is an object oriented environment,

meaning that almost everything that it does revolves around objects.

Objects have names and you can assign values to objects.

For example, assigning a number.

We can have an object x, an assign the value 6 to it by typing in x equals 6.

If we want to see the value, we can then enter the name of the object, and

R will display the value.

We can also use descriptive names.

For example, total revenue and

assign that a value of 18.4.

We can copy an object by assigning to another name.

If I have expenses, and that has a value of six, but I already have six in x.

I can just say expenses equal x.

You can also assign the result of an operation to an object.

For example, net profit would

be total revenue minus expenses.

We type in just the name of the object and it will give us the value in it.

And you can see here It has done the subtraction and has saved the result.

You have some flexibility with names, but you can't use certain characters,

such as spaces, in the name.

R also has functions.

To write a function,

you type its name followed by parentheses with any arguments within the parentheses.

For example, the C function, will store a column of data.

Or it will concatenate data.

We might want to assign to a, variable z,

the set of data that goes two, three, four, five, and six.

You could see this data set is now stored in the object Z.

It stores it as a vector.

We can then use functions to get more information or

do analysis on these objects.

We can ask what's the mean or average of the values in Z.

We can ask what's the variance of Z or

what's the standard deviation of Z Which is the square root of the variance.

We can use the function to create a sequence.

The SEQ function creates a sequence.

And we tell it what the starting point is,

the front value what the ending point is, the 2 value.

And what increment we want to use, the buy variable.

This gives us a sequence that goes from 0.1 to 0.9, going in increments of 0.1.

Here we specified three arguments and we've given them all by name.

If we don't name the arguments We can just specify numbers, and

R will fill them in in order.

Any arguments we don't specify will get the default value,

assuming they're at the end of the list.

So for example, if we say sequence from 1 to 10, it's filling them in in order.

The first number we give is the from.

The second number is the to.

And then we haven't specified a by and it will use the default,

which in this case is by once at a time, and so

this will give us a sequence that goes from one to ten.

Let's take a brief look at one of the built in data sets in ARM this

is the trees data set.

This is a set of trees and gives us the diameter the height and

the volume of the tree.

We can learn some more by using the help function.

This opens up a new window, and gives us information about the data set

including references and some examples of analysis.

It gives us some more definition showing that, indeed,

this is a diameter a height and a volume.

As you see here, there are labels for each of these columns, they're named.

We can access these names in our by using the attach command.

This says to look in the trees object for other additional names.

We can ask what are the names, by using the names command.

These are the same ones that we see when we look at the data set itself.

We can then use these names to directly access these variables.

For example, we want to make a histogram of the girth, or diameter, of these trees.

And then we say hist, like it's a histogram function and

it automatically pops up a new window for us.

From the histogram we can see, that most of the trees have diameters between eight

and 18 inches, with one larger tree.

We can also make scatterplots.

Putting the girth on the x axis and

the height on the y axis to see how height relates to diameter.

As you can see here, in general, the wider the tree, the taller the tree.

There's a function in R called pairs,

which allows us to make all possible Pairwise scatter plots for the dataset.

So if we say pairs of trees,

we have each of the three variables plotted against each of the other ones.

So for example,

in this box here we have girth on the x axis and height on the y axis.

And this is exactly the plot that we had previously.

In the lower left here, we have girth on x axis and volume on the y axis.

And we can see there's a very strong relationship between total volume of

timber and the tree, and how wide the tree is.

Similarly, there's a strong relationship between the height of the tree and

the volume of the tree.

In the upper corner,

we have the mirror images of these plots with the variables on the opposite axis.

And so this plot here in the upper right has volume on the x axis and

girth on the y axis.

Finally, we can get numerical summaries using the summary command for

each of the variables this will give us the mean, the median,

the range, and the quantiles.

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