Stat 5810, Applied Spatial Statistics

Homework Assignment 3 (9/18/00)

25 Points - Due 9/25/00 in class

You may work in groups of 2 people on this assignment and turn in a joint solution.

You have to use S-Plus to answer all of the following questions. Your solutions should consist of the S-Plus commands you have typed, any S-Plus functions you have written, and the output (text and postscript graphics) produced in S-Plus. For example, let us assume we wrote a simple function called jstestfunction in a texteditor:

jstestfunction _ function ()
{
  "a simple function"
}

After copying this function into S-Plus, you should type the function name (without the ``()'') such that S-Plus displays the code of this function. Then you call the function by typing in the function name (this time with the ``()'') and S-Plus will produce the result.

Your output should contain all three parts, i.e., the code of the function, the function calls (if parameters are involved, call the function with at least three different combinations of parameters - to test a function, even more combinations of parameters may be required), and the results. Annotate the output to make things more readable. Here, the answer would be:

>                                ### here is the body of the function
> jstestfunction      
function()
{
        "a simple function"
}
>                                ### here the function is called without
                                 ### any parameter
> jstestfunction()
[1] "a simple function"          ### and this is the result
>

1) Calculate the first 50 powers of 2, i.e., 2, 2 * 2, 2 * 2 * 2, and so on. Calculate the squares of the integers from 1 to 50. Which pairs are equal, i.e., which integers fulfill the condition 2^n = n^2? How many such pairs are there? Answer this question explicitely using S-Plus! (5 Points)
2) Calculate the sine, cosine, and the tangent for the numbers ranging from 0 to 2 * pi with distance 0.1 between them. Remember that tan(x) = sin(x) / cos(x). Now calculate the difference between tan(x) and and sin(x) / cos (x) for the values above. Which values are exactly equal? What is the maximum absolute difference? What is the cause of the differences? (5 Points)
3) In a single figure, plot the functions sin, cos, and sin+cos with different colors and line styles. Use 1000 points in the interval [-10, 10]. Label the figure with a title, subtitle and axis labels. (5 Points)
4) Generate 100 and 1000 random numbers, both samples from a Normal distribution with mean value 3 and variance 5. Draw histograms with bin width 0.5, 1, and 2 for each of the two samples. Plot all histograms in a single graphics window (using the same scale for the x-axis) and label them accordingly. What is visible? (5 Points)
5) Many statistical procedures require that a matrix is symmetric. (If the elements of a matrix A are A[i,j], the matrix is symmetric if A[i,j] = A[j,i] for all i and j.) Using the ncol, nrow, and all functions, write an S-Plus function that will test a matrix to see if it is symmetric.

Test your function with the matrices x0, x1, x2, and x3, which are defined as:
```
x0 _ 1
x1 _ matrix(1:10, 2, 5)
x2 _ matrix(1:9, 3, 3)
x3 _ matrix(c(1, 2, 3, 2, 2, 2, 3, 2, 3), 3, 3)
```
(5 Points)