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Introduction to the R Programming Language

(Not a statistics program for pirates.)

Welcome to R! One of the world's most popular data analysis languages that boasts a staggeringly large number of user-created packages (12126 as of last night). CRAN, the central R package repository, requires that all packages be fully documented and that they work in the most current version of R.

R is an open source, object-oriented language with its source code written almost entirely in C. It functions in many similar ways to Python. As you can see, it also works with Jupyter Notebooks.

You can also run R in the terminal, via its own application, or through an IDE like R-Studio.

In this class, I'd like to give you an introduction to the fundamentals of R, before quickly moving on to doing some Bayesian data analysis.

We're going to be using a couple of packages throughout this notebook. Please run the commands below at your earliest convenience.





In [ ]:



    


install.packages("LaplacesDemon")
install.packages("magicaxis")
library(LaplacesDemon)
library(magicaxis)

Fundamentals of R.

R is a fully vectorized language, and supports many different data structures and types; including (but not limited to):

  • vectors (including single scalar elements)
  • matrices (2D structures of the same numeric type; i.e. integer, double, etc.)
  • data frames (2D structures where columns can be different types)
  • lists (ordered collections of objects, which can contain an arbitrary number of other data structures).

As you might expect, R supports integers, doubles (singles are automatically stored as doubles in R), strings, Boolean logic, and factors (categories of data, i.e. hair color ('brown', 'red', etc.).

Very basics.

There is no type assignment necessary in R. You can just do this:





In [38]:



    


a = 10
a = a + 2
a = a^2
a = a*10 + sqrt(a) + a^(1.3) - 1/a
a



    


















    






2091.53824961336

Vectors, matrices & data frames





In [37]:



    


c(1,2,3,4)
c("fee","fi","fo","fum")



    


















    







    
	
  1. 1
    2. 
	
  2. 2
    3. 
	
  3. 3
    4. 
	
  4. 4
    5. 












    







    
	
  1. 'fee'
    2. 
	
  2. 'fi'
    3. 
	
  3. 'fo'
    4. 
	
  4. 'fum'
    5. 



















In [48]:



    


foo1 <- c(1,2,3,4)
foo2 = c(5:8)
foo3 = seq(1,10,by=2) # Do ?seq to see more.



    











In [5]:



    


foo1; foo2; foo3



    


















    







    
	
  1. 1
    2. 
	
  2. 2
    3. 
	
  3. 3
    4. 
	
  4. 4
    5. 












    







    
	
  1. 5
    2. 
	
  2. 6
    3. 
	
  3. 7
    4. 
	
  4. 8
    5. 












    







    
	
  1. 1
    2. 
	
  2. 3
    3. 
	
  3. 5
    4. 
	
  4. 7
    5. 
	
  5. 9
    6.

Vector programming in R is very straightforward.





In [3]:



    


b = c(1:10)
print(b)
for(i in 1:10){
    b[i] = b[i] * 10
}
print(b)



    


















    






 [1]  1  2  3  4  5  6  7  8  9 10
 [1]  10  20  30  40  50  60  70  80  90 100

















In [5]:



    


b = c(1:10)
print(b)
b = b*10
print(b)



    


















    






 [1]  1  2  3  4  5  6  7  8  9 10
 [1]  10  20  30  40  50  60  70  80  90 100

















In [10]:



    


1:10 * c(1,2)^2



    


















    







    
	
  1. 1
    2. 
	
  2. 8
    3. 
	
  3. 3
    4. 
	
  4. 16
    5. 
	
  5. 5
    6. 
	
  6. 24
    7. 
	
  7. 7
    8. 
	
  8. 32
    9. 
	
  9. 9
    10. 
	
  10. 40
    11. 



















In [11]:



    


mat1 = matrix(data=1:9,nrow=3)
mat1



    


















    









	
147

	
258

	
369

The function arg() will tell you all the arguments for another function. To call up the documentation for any function or command, use ?.





In [40]:



    


args(matrix)



    


















    







function (data = NA, nrow = 1, ncol = 1, byrow = FALSE, dimnames = NULL) 
NULL

















In [44]:



    


?matrix

Matrix operations are built in.





In [22]:



    


mat1 * 1:3
mat1 %*% 1:3



    


















    









	
1  4 7

	
4 1016

	
9 1827














    









	
30

	
36

	
42





















In [23]:



    


mat2 = cbind(c(1,2,3),c(4,5,6),c(7,8,9))
mat1 %*% mat2



    


















    









	
30 66 102

	
36 81 126

	
42 96 150





















In [24]:



    


eigen(mat1)



    


















    








eigen() decomposition
$values
[1]  1.611684e+01 -1.116844e+00 -5.700691e-16

$vectors
           [,1]       [,2]       [,3]
[1,] -0.4645473 -0.8829060  0.4082483
[2,] -0.5707955 -0.2395204 -0.8164966
[3,] -0.6770438  0.4038651  0.4082483

R has a number of built in data sets, which can be extremely useful when exploring the language for the first time. For today, we'll be using iris. See more with ?iris.





In [33]:



    


iris



    


















    









Sepal.LengthSepal.WidthPetal.LengthPetal.WidthSpecies


	
5.1   3.5   1.4   0.2   setosa

	
4.9   3.0   1.4   0.2   setosa

	
4.7   3.2   1.3   0.2   setosa

	
4.6   3.1   1.5   0.2   setosa

	
5.0   3.6   1.4   0.2   setosa

	
5.4   3.9   1.7   0.4   setosa

	
4.6   3.4   1.4   0.3   setosa

	
5.0   3.4   1.5   0.2   setosa

	
4.4   2.9   1.4   0.2   setosa

	
4.9   3.1   1.5   0.1   setosa

	
5.4   3.7   1.5   0.2   setosa

	
4.8   3.4   1.6   0.2   setosa

	
4.8   3.0   1.4   0.1   setosa

	
4.3   3.0   1.1   0.1   setosa

	
5.8   4.0   1.2   0.2   setosa

	
5.7   4.4   1.5   0.4   setosa

	
5.4   3.9   1.3   0.4   setosa

	
5.1   3.5   1.4   0.3   setosa

	
5.7   3.8   1.7   0.3   setosa

	
5.1   3.8   1.5   0.3   setosa

	
5.4   3.4   1.7   0.2   setosa

	
5.1   3.7   1.5   0.4   setosa

	
4.6   3.6   1.0   0.2   setosa

	
5.1   3.3   1.7   0.5   setosa

	
4.8   3.4   1.9   0.2   setosa

	
5.0   3.0   1.6   0.2   setosa

	
5.0   3.4   1.6   0.4   setosa

	
5.2   3.5   1.5   0.2   setosa

	
5.2   3.4   1.4   0.2   setosa

	
4.7   3.2   1.6   0.2   setosa

	
⋮⋮⋮⋮⋮

	
6.9      3.2      5.7      2.3      virginica

	
5.6      2.8      4.9      2.0      virginica

	
7.7      2.8      6.7      2.0      virginica

	
6.3      2.7      4.9      1.8      virginica

	
6.7      3.3      5.7      2.1      virginica

	
7.2      3.2      6.0      1.8      virginica

	
6.2      2.8      4.8      1.8      virginica

	
6.1      3.0      4.9      1.8      virginica

	
6.4      2.8      5.6      2.1      virginica

	
7.2      3.0      5.8      1.6      virginica

	
7.4      2.8      6.1      1.9      virginica

	
7.9      3.8      6.4      2.0      virginica

	
6.4      2.8      5.6      2.2      virginica

	
6.3      2.8      5.1      1.5      virginica

	
6.1      2.6      5.6      1.4      virginica

	
7.7      3.0      6.1      2.3      virginica

	
6.3      3.4      5.6      2.4      virginica

	
6.4      3.1      5.5      1.8      virginica

	
6.0      3.0      4.8      1.8      virginica

	
6.9      3.1      5.4      2.1      virginica

	
6.7      3.1      5.6      2.4      virginica

	
6.9      3.1      5.1      2.3      virginica

	
5.8      2.7      5.1      1.9      virginica

	
6.8      3.2      5.9      2.3      virginica

	
6.7      3.3      5.7      2.5      virginica

	
6.7      3.0      5.2      2.3      virginica

	
6.3      2.5      5.0      1.9      virginica

	
6.5      3.0      5.2      2.0      virginica

	
6.2      3.4      5.4      2.3      virginica

	
5.9      3.0      5.1      1.8      virginica

This is an example of a data frame, which is perhaps the most common data type you'll be using in R. Let's take a moment to familiarize ourselves with data frames.

To get a quantitative overview of the contents of a data frame (or, indeed, any other data type), use the summary() function. str() on the other hand, will tell you about the structure of a data type.





In [36]:



    


summary(iris)



    


















    








  Sepal.Length    Sepal.Width     Petal.Length    Petal.Width   
 Min.   :4.300   Min.   :2.000   Min.   :1.000   Min.   :0.100  
 1st Qu.:5.100   1st Qu.:2.800   1st Qu.:1.600   1st Qu.:0.300  
 Median :5.800   Median :3.000   Median :4.350   Median :1.300  
 Mean   :5.843   Mean   :3.057   Mean   :3.758   Mean   :1.199  
 3rd Qu.:6.400   3rd Qu.:3.300   3rd Qu.:5.100   3rd Qu.:1.800  
 Max.   :7.900   Max.   :4.400   Max.   :6.900   Max.   :2.500  
       Species  
 setosa    :50  
 versicolor:50  
 virginica :50  
                
                
                

















In [37]:



    


str(iris)



    


















    






'data.frame':	150 obs. of  5 variables:
 $ Sepal.Length: num  5.1 4.9 4.7 4.6 5 5.4 4.6 5 4.4 4.9 ...
 $ Sepal.Width : num  3.5 3 3.2 3.1 3.6 3.9 3.4 3.4 2.9 3.1 ...
 $ Petal.Length: num  1.4 1.4 1.3 1.5 1.4 1.7 1.4 1.5 1.4 1.5 ...
 $ Petal.Width : num  0.2 0.2 0.2 0.2 0.2 0.4 0.3 0.2 0.2 0.1 ...
 $ Species     : Factor w/ 3 levels "setosa","versicolor",..: 1 1 1 1 1 1 1 1 1 1 ...

Square brackets [] are used to point to individual columns, rows, or cells of a data frame. data[i,j] will point to the element in the ith row and jth column. For a vector, vector[j] will point to the jth element of that vector.





In [38]:



    


iris[1,2]



    


















    






3.5

If you simply want to pull out an entire row or column, use data[i,] or data[,j]. Alternatively, you can use the $ symbol to refer to a column by its name. You can also mix and match as you like; i.e. data$foo[3].





In [46]:



    


iris[,3]
iris[c(5,7),]



    


















    







    
	
  1. 1.4
    2. 
	
  2. 1.4
    3. 
	
  3. 1.3
    4. 
	
  4. 1.5
    5. 
	
  5. 1.4
    6. 
	
  6. 1.7
    7. 
	
  7. 1.4
    8. 
	
  8. 1.5
    9. 
	
  9. 1.4
    10. 
	
  10. 1.5
    11. 
	
  11. 1.5
    12. 
	
  12. 1.6
    13. 
	
  13. 1.4
    14. 
	
  14. 1.1
    15. 
	
  15. 1.2
    16. 
	
  16. 1.5
    17. 
	
  17. 1.3
    18. 
	
  18. 1.4
    19. 
	
  19. 1.7
    20. 
	
  20. 1.5
    21. 
	
  21. 1.7
    22. 
	
  22. 1.5
    23. 
	
  23. 1
    24. 
	
  24. 1.7
    25. 
	
  25. 1.9
    26. 
	
  26. 1.6
    27. 
	
  27. 1.6
    28. 
	
  28. 1.5
    29. 
	
  29. 1.4
    30. 
	
  30. 1.6
    31. 
	
  31. 1.6
    32. 
	
  32. 1.5
    33. 
	
  33. 1.5
    34. 
	
  34. 1.4
    35. 
	
  35. 1.5
    36. 
	
  36. 1.2
    37. 
	
  37. 1.3
    38. 
	
  38. 1.4
    39. 
	
  39. 1.3
    40. 
	
  40. 1.5
    41. 
	
  41. 1.3
    42. 
	
  42. 1.3
    43. 
	
  43. 1.3
    44. 
	
  44. 1.6
    45. 
	
  45. 1.9
    46. 
	
  46. 1.4
    47. 
	
  47. 1.6
    48. 
	
  48. 1.4
    49. 
	
  49. 1.5
    50. 
	
  50. 1.4
    51. 
	
  51. 4.7
    52. 
	
  52. 4.5
    53. 
	
  53. 4.9
    54. 
	
  54. 4
    55. 
	
  55. 4.6
    56. 
	
  56. 4.5
    57. 
	
  57. 4.7
    58. 
	
  58. 3.3
    59. 
	
  59. 4.6
    60. 
	
  60. 3.9
    61. 
	
  61. 3.5
    62. 
	
  62. 4.2
    63. 
	
  63. 4
    64. 
	
  64. 4.7
    65. 
	
  65. 3.6
    66. 
	
  66. 4.4
    67. 
	
  67. 4.5
    68. 
	
  68. 4.1
    69. 
	
  69. 4.5
    70. 
	
  70. 3.9
    71. 
	
  71. 4.8
    72. 
	
  72. 4
    73. 
	
  73. 4.9
    74. 
	
  74. 4.7
    75. 
	
  75. 4.3
    76. 
	
  76. 4.4
    77. 
	
  77. 4.8
    78. 
	
  78. 5
    79. 
	
  79. 4.5
    80. 
	
  80. 3.5
    81. 
	
  81. 3.8
    82. 
	
  82. 3.7
    83. 
	
  83. 3.9
    84. 
	
  84. 5.1
    85. 
	
  85. 4.5
    86. 
	
  86. 4.5
    87. 
	
  87. 4.7
    88. 
	
  88. 4.4
    89. 
	
  89. 4.1
    90. 
	
  90. 4
    91. 
	
  91. 4.4
    92. 
	
  92. 4.6
    93. 
	
  93. 4
    94. 
	
  94. 3.3
    95. 
	
  95. 4.2
    96. 
	
  96. 4.2
    97. 
	
  97. 4.2
    98. 
	
  98. 4.3
    99. 
	
  99. 3
    100. 
	
  100. 4.1
    101. 
	
  101. 6
    102. 
	
  102. 5.1
    103. 
	
  103. 5.9
    104. 
	
  104. 5.6
    105. 
	
  105. 5.8
    106. 
	
  106. 6.6
    107. 
	
  107. 4.5
    108. 
	
  108. 6.3
    109. 
	
  109. 5.8
    110. 
	
  110. 6.1
    111. 
	
  111. 5.1
    112. 
	
  112. 5.3
    113. 
	
  113. 5.5
    114. 
	
  114. 5
    115. 
	
  115. 5.1
    116. 
	
  116. 5.3
    117. 
	
  117. 5.5
    118. 
	
  118. 6.7
    119. 
	
  119. 6.9
    120. 
	
  120. 5
    121. 
	
  121. 5.7
    122. 
	
  122. 4.9
    123. 
	
  123. 6.7
    124. 
	
  124. 4.9
    125. 
	
  125. 5.7
    126. 
	
  126. 6
    127. 
	
  127. 4.8
    128. 
	
  128. 4.9
    129. 
	
  129. 5.6
    130. 
	
  130. 5.8
    131. 
	
  131. 6.1
    132. 
	
  132. 6.4
    133. 
	
  133. 5.6
    134. 
	
  134. 5.1
    135. 
	
  135. 5.6
    136. 
	
  136. 6.1
    137. 
	
  137. 5.6
    138. 
	
  138. 5.5
    139. 
	
  139. 4.8
    140. 
	
  140. 5.4
    141. 
	
  141. 5.6
    142. 
	
  142. 5.1
    143. 
	
  143. 5.1
    144. 
	
  144. 5.9
    145. 
	
  145. 5.7
    146. 
	
  146. 5.2
    147. 
	
  147. 5
    148. 
	
  148. 5.2
    149. 
	
  149. 5.4
    150. 
	
  150. 5.1
    151. 












    









Sepal.LengthSepal.WidthPetal.LengthPetal.WidthSpecies


	
55.0   3.6   1.4   0.2   setosa

	
74.6   3.4   1.4   0.3   setosa





















In [42]:



    


colnames(iris)
iris$Species



    


















    







    
	
  1. 'Sepal.Length'
    2. 
	
  2. 'Sepal.Width'
    3. 
	
  3. 'Petal.Length'
    4. 
	
  4. 'Petal.Width'
    5. 
	
  5. 'Species'
    6. 












    







    
	
  1. setosa
    2. 
	
  2. setosa
    3. 
	
  3. setosa
    4. 
	
  4. setosa
    5. 
	
  5. setosa
    6. 
	
  6. setosa
    7. 
	
  7. setosa
    8. 
	
  8. setosa
    9. 
	
  9. setosa
    10. 
	
  10. setosa
    11. 
	
  11. setosa
    12. 
	
  12. setosa
    13. 
	
  13. setosa
    14. 
	
  14. setosa
    15. 
	
  15. setosa
    16. 
	
  16. setosa
    17. 
	
  17. setosa
    18. 
	
  18. setosa
    19. 
	
  19. setosa
    20. 
	
  20. setosa
    21. 
	
  21. setosa
    22. 
	
  22. setosa
    23. 
	
  23. setosa
    24. 
	
  24. setosa
    25. 
	
  25. setosa
    26. 
	
  26. setosa
    27. 
	
  27. setosa
    28. 
	
  28. setosa
    29. 
	
  29. setosa
    30. 
	
  30. setosa
    31. 
	
  31. setosa
    32. 
	
  32. setosa
    33. 
	
  33. setosa
    34. 
	
  34. setosa
    35. 
	
  35. setosa
    36. 
	
  36. setosa
    37. 
	
  37. setosa
    38. 
	
  38. setosa
    39. 
	
  39. setosa
    40. 
	
  40. setosa
    41. 
	
  41. setosa
    42. 
	
  42. setosa
    43. 
	
  43. setosa
    44. 
	
  44. setosa
    45. 
	
  45. setosa
    46. 
	
  46. setosa
    47. 
	
  47. setosa
    48. 
	
  48. setosa
    49. 
	
  49. setosa
    50. 
	
  50. setosa
    51. 
	
  51. versicolor
    52. 
	
  52. versicolor
    53. 
	
  53. versicolor
    54. 
	
  54. versicolor
    55. 
	
  55. versicolor
    56. 
	
  56. versicolor
    57. 
	
  57. versicolor
    58. 
	
  58. versicolor
    59. 
	
  59. versicolor
    60. 
	
  60. versicolor
    61. 
	
  61. versicolor
    62. 
	
  62. versicolor
    63. 
	
  63. versicolor
    64. 
	
  64. versicolor
    65. 
	
  65. versicolor
    66. 
	
  66. versicolor
    67. 
	
  67. versicolor
    68. 
	
  68. versicolor
    69. 
	
  69. versicolor
    70. 
	
  70. versicolor
    71. 
	
  71. versicolor
    72. 
	
  72. versicolor
    73. 
	
  73. versicolor
    74. 
	
  74. versicolor
    75. 
	
  75. versicolor
    76. 
	
  76. versicolor
    77. 
	
  77. versicolor
    78. 
	
  78. versicolor
    79. 
	
  79. versicolor
    80. 
	
  80. versicolor
    81. 
	
  81. versicolor
    82. 
	
  82. versicolor
    83. 
	
  83. versicolor
    84. 
	
  84. versicolor
    85. 
	
  85. versicolor
    86. 
	
  86. versicolor
    87. 
	
  87. versicolor
    88. 
	
  88. versicolor
    89. 
	
  89. versicolor
    90. 
	
  90. versicolor
    91. 
	
  91. versicolor
    92. 
	
  92. versicolor
    93. 
	
  93. versicolor
    94. 
	
  94. versicolor
    95. 
	
  95. versicolor
    96. 
	
  96. versicolor
    97. 
	
  97. versicolor
    98. 
	
  98. versicolor
    99. 
	
  99. versicolor
    100. 
	
  100. versicolor
    101. 
	
  101. virginica
    102. 
	
  102. virginica
    103. 
	
  103. virginica
    104. 
	
  104. virginica
    105. 
	
  105. virginica
    106. 
	
  106. virginica
    107. 
	
  107. virginica
    108. 
	
  108. virginica
    109. 
	
  109. virginica
    110. 
	
  110. virginica
    111. 
	
  111. virginica
    112. 
	
  112. virginica
    113. 
	
  113. virginica
    114. 
	
  114. virginica
    115. 
	
  115. virginica
    116. 
	
  116. virginica
    117. 
	
  117. virginica
    118. 
	
  118. virginica
    119. 
	
  119. virginica
    120. 
	
  120. virginica
    121. 
	
  121. virginica
    122. 
	
  122. virginica
    123. 
	
  123. virginica
    124. 
	
  124. virginica
    125. 
	
  125. virginica
    126. 
	
  126. virginica
    127. 
	
  127. virginica
    128. 
	
  128. virginica
    129. 
	
  129. virginica
    130. 
	
  130. virginica
    131. 
	
  131. virginica
    132. 
	
  132. virginica
    133. 
	
  133. virginica
    134. 
	
  134. virginica
    135. 
	
  135. virginica
    136. 
	
  136. virginica
    137. 
	
  137. virginica
    138. 
	
  138. virginica
    139. 
	
  139. virginica
    140. 
	
  140. virginica
    141. 
	
  141. virginica
    142. 
	
  142. virginica
    143. 
	
  143. virginica
    144. 
	
  144. virginica
    145. 
	
  145. virginica
    146. 
	
  146. virginica
    147. 
	
  147. virginica
    148. 
	
  148. virginica
    149. 
	
  149. virginica
    150. 
	
  150. virginica
    151.

Mix and match as you like!





In [43]:



    


iris$Species[5]



    


















    






setosa

The which() function is used to identify specific subsets of data. Uses standard Boolean logic, i.e. which(data$foo >= 2 & data$foo2 <= 3).

Quick problem 1:

a) Find the subset of iris flowers with a sepal length greater than 4.3cm, and petal length less than 1.2cm.

b) What species do these correspond to?





In [44]:



    


which(iris$Sepal.Length >= 4.3 & iris$Petal.Length <= 1.2)



    


















    







    
	
  1. 14
    2. 
	
  2. 15
    3. 
	
  3. 23
    4. 
	
  4. 36
    5. 



















In [45]:



    


iris[which(iris$Sepal.Length >= 4.3 & iris$Petal.Length <= 1.2),]



    


















    









Sepal.LengthSepal.WidthPetal.LengthPetal.WidthSpecies


	
144.3   3.0   1.1   0.1   setosa

	
155.8   4.0   1.2   0.2   setosa

	
234.6   3.6   1.0   0.2   setosa

	
365.0   3.2   1.2   0.2   setosa

Lists

Lists are highly flexible, but also the most complicated structures in R.





In [49]:



    


list1 = list(a=a, iris = iris[1:10,], vec=foo3)
list1



    


















    








	
$a

		
2091.53824961336

	
$iris

		


Sepal.LengthSepal.WidthPetal.LengthPetal.WidthSpecies


	
5.1   3.5   1.4   0.2   setosa

	
4.9   3.0   1.4   0.2   setosa

	
4.7   3.2   1.3   0.2   setosa

	
4.6   3.1   1.5   0.2   setosa

	
5.0   3.6   1.4   0.2   setosa

	
5.4   3.9   1.7   0.4   setosa

	
4.6   3.4   1.4   0.3   setosa

	
5.0   3.4   1.5   0.2   setosa

	
4.4   2.9   1.4   0.2   setosa

	
4.9   3.1   1.5   0.1   setosa






	
$vec

		
    
	
  1. 1
    2. 
	
  2. 3
    3. 
	
  3. 5
    4. 
	
  4. 7
    5. 
	
  5. 9
    6. 























In [50]:



    


list1[2]



    


















    






$iris = 


Sepal.LengthSepal.WidthPetal.LengthPetal.WidthSpecies


	
5.1   3.5   1.4   0.2   setosa

	
4.9   3.0   1.4   0.2   setosa

	
4.7   3.2   1.3   0.2   setosa

	
4.6   3.1   1.5   0.2   setosa

	
5.0   3.6   1.4   0.2   setosa

	
5.4   3.9   1.7   0.4   setosa

	
4.6   3.4   1.4   0.3   setosa

	
5.0   3.4   1.5   0.2   setosa

	
4.4   2.9   1.4   0.2   setosa

	
4.9   3.1   1.5   0.1   setosa





















In [51]:



    


list1[[2]]



    


















    









Sepal.LengthSepal.WidthPetal.LengthPetal.WidthSpecies


	
5.1   3.5   1.4   0.2   setosa

	
4.9   3.0   1.4   0.2   setosa

	
4.7   3.2   1.3   0.2   setosa

	
4.6   3.1   1.5   0.2   setosa

	
5.0   3.6   1.4   0.2   setosa

	
5.4   3.9   1.7   0.4   setosa

	
4.6   3.4   1.4   0.3   setosa

	
5.0   3.4   1.5   0.2   setosa

	
4.4   2.9   1.4   0.2   setosa

	
4.9   3.1   1.5   0.1   setosa





















In [52]:



    


list1$vec



    


















    







    
	
  1. 1
    2. 
	
  2. 3
    3. 
	
  3. 5
    4. 
	
  4. 7
    5. 
	
  5. 9
    6. 



















In [55]:



    


list1['a']



    


















    






$a = 2091.53824961336
















In [56]:



    


list1[['a']]



    


















    






2091.53824961336

Lists can get really complicated, and be endlessly recursive -- any element of a list can be another list. Lists can be really useful to keep your workspace tidy; for example, if the output of a function is several different matrices or data frames, they can all be bundled into a single list.

Functions

I've already showed some examples pre-defined R functions that perform certain tasks. R lets you define your own functions as you like.





In [5]:



    


newFunc <- function(x=1, power = 2, addon = 10){
    x ^ power + addon
}
newFunc()
newFunc(1:10)
newFunc(pow = 3, x=23:30)
newFunc(1:10,1:2,1:5)



    


















    






11









    







    
	
  1. 11
    2. 
	
  2. 14
    3. 
	
  3. 19
    4. 
	
  4. 26
    5. 
	
  5. 35
    6. 
	
  6. 46
    7. 
	
  7. 59
    8. 
	
  8. 74
    9. 
	
  9. 91
    10. 
	
  10. 110
    11. 












    







    
	
  1. 12177
    2. 
	
  2. 13834
    3. 
	
  3. 15635
    4. 
	
  4. 17586
    5. 
	
  5. 19693
    6. 
	
  6. 21962
    7. 
	
  7. 24399
    8. 
	
  8. 27010
    9. 












    







    
	
  1. 2
    2. 
	
  2. 6
    3. 
	
  3. 6
    4. 
	
  4. 20
    5. 
	
  5. 10
    6. 
	
  6. 37
    7. 
	
  7. 9
    8. 
	
  8. 67
    9. 
	
  9. 13
    10. 
	
  10. 105
    11. 



















In [6]:



    


newFunc



    


















    







function (x = 1, power = 2, addon = 10) 
{
    x^power + addon
}

Plotting

Plotting is probably my favorite thing to do in R. Even the default plotting packages look decent; but with a little bit of extra work and customization, you can quickly and easily make publication-ready figures on the fly.





In [1]:



    


plot(1:10,1:10)
plot(runif(100),runif(100),pch=19, xlim=c(-1,2))



    


















    

















    
























In [8]:



    


plot(sin,col='red')



    


















    
























In [10]:



    


magplot(1:1e3, newFunc(1:1e3),log='y',type='l')
legend('bottomright',legend='a line', lty=1)

Quick problem 2:

Write your own function, and plot it using the curve() function.

Hint: Remember that you can look up the documentation for any function using ?functionName.





In [9]:



    


myFunc <- function(x,a=1,b=0,c=1){
    a*exp(-(((x-b)^2)/(2*c^2)))
}
curve(myFunc,from=-5,to=5)

Quick problem 3:

Go back to the iris data set, and plot a histogram of all the sepal lengths for the setosa species.

Default histograms in R are probably the worst example of 'out of the box' plotting that the language can do. I prefer convolving my data with a kernel of fixed bandwidth, and plotting the PDF of that instead. You can do all that with the density() function. Give it a shot!

Try displaying both the PDF and the histogram on the same plot.

Hint: Remember to use type='l'!





In [40]:



    


plotDat = iris[which(iris$Species == 'setosa'),'Sepal.Length']
hist(plotDat,freq=F,ylim=c(0,1.5),xlab="Setosa sepal lengths",main="",axes=F,xlim=c(4,6))
myDens = density(iris[which(iris$Species == 'setosa'),'Sepal.Length'],bw=0.3/sqrt(12))
points(myDens$x,myDens$y,type='l',col='red')
magaxis(1:4)

Probability distributions

These are directly built into R. Uniform, normal, chi-squared, binomial, Poisson, all are built into base R. Do ?Distributions to see the full available list.

As a rule of thumb, the following functions interact with these distributions:

  • dDistr gives the instantaneous density at a value.
  • pDistr gives the integrated probability up to a value.
  • qDistr gives the value at an integrated probability (i.e. a quantile).
  • rDistr randomly samples the PDF to generate mock data. This is the one you will likely use most.




In [27]:



    


magplot(dnorm,xlim=c(-5,5))



    


















    
























In [29]:



    


magplot(pnorm,xlim=c(-5,5))



    


















    
























In [31]:



    


magplot(qnorm,xlim=c(0,1))



    


















    
























In [32]:



    


hist(rnorm(1E5))

Problem 4:

Create a function in R to compute the x and y coordinates of an ellipse, just as in the previous problem set.





In [39]:



    


ellipsePoints <- function(x, y, a, b, angle, res=100){
	
	segment = c(0,2*pi)

	z = seq(segment[1],segment[2],length.out=res)
	xx = a * cos(z)
	yy = b * sin(z)

	alpha = atan2(yy,xx)
	rad = sqrt(xx^2 + yy^2)

	xp = rad*cos(alpha + angle) + x
	yp = rad*sin(alpha + angle) + y

	plot(xp,yp)

}

ellipsePoints(1,2,3,0.3,sqrt(2))

Bayesian statistics in R.

Let's look at a common problem in observational astrophysics -- you're putting together a luminosity function based on some galaxy survey data, and have zero counts in a bin. What is the most likely value of $\lambda$ for the underlying Possion distribution that generated this?

$\lambda$ may well be nonzero...





In [6]:



    


plot.new();plot.window(c(0,20),c(0,1))
magaxis(xlab='k',ylab='P(X = k)',side=1:4)
for(i in 1:10){
lines(0:20,dpois(0:20,i-1),col=rainbow(10,end=2/3)[i])
points(0:20,dpois(0:20,i-1),col=rainbow(10,end=2/3)[i])}
legend('topright',legend=paste('Lambda =',0:9),col=rainbow(10,end=2/3),lty=1,pch=1,bty='n')

In Bayesian statistics, the prior $p(\theta)$ is the probability of the model based on prior knowledge of how the input parameters might be distributed. In this case, the input parameter is just $\lambda$.

In this example, we have no prior knowledge: so our prior for $\lambda$ is a uniform distribution from 0 to $\infty$. So let's compute $p( y = 0, \theta)$ for all possible $\lambda$. This goes to 0 quickly, so we can just stop at $\lambda = 5.$





In [7]:



    


xval = seq(0, 5, length.out=1000)
magplot(xval, dpois(0, xval), type='l',col='red',xlab=expression(lambda),ylab='PDF')

Bayesian problems generally require some data, a likelihood model, and some way of producing a posterior distribution.





In [3]:



    


Data = list(data = 0, mon.names = '', parm.names = 'lambda', N = 1)
Model = function(parm, Data){
    parm = interval(parm,0,100)
    val.prior = log(dunif(parm,0,100))
    LL = log(dpois(Data$data,parm))
    LP = LL + val.prior
    Modelout = list(LP=LP, Dev=2*LL, Monitor = 1, yhat=1, parm=parm)
}

It is usually not easy to calculate the posterior directly, so we need a scheme that can integrate the posterior likelihood space in a manner that reflects the parameter distributions.

If we choose a value x for λ, and compare that posterior likelihood to another value for λ (x+dx), the ratio of likelihoods (the BF) tells us the relative densities of the posterior PDF.

To walk through the posterior space we can do the following:

  • If the likelihood at λ(x+dx) is higher than λ(x) we update the posterior.
  • If the likelihood at λ(x+dx) is lower than λ(x) we accept it if a random number between 0 and the likelihood at λ(x) is less than the likelihood at λ(x+dx).

If we keep a chain of all posterior values, the PDF of this chain will reflect the posterior likelihood distribution.

This is a Metropolis-Hastings algorithm.





In [4]:



    


MH=function(Model,Data,start=1,iterations=1e4,thin=100,step=1,burn=0.1){
    posterior={}
    mod=Model(start,Data)
    parm=mod$parm
    currentLP=mod$LP
    keep={} 
    for(i in 1:iterations){
        trial=parm+rnorm(1,sd=step)
        mod=Model(trial,Data)
        trial=mod$parm; newLP=mod$LP
        if(newLP>currentLP){
            parm=trial
            currentLP=newLP
            keep=c(keep,TRUE)
        }else{
            check=runif(1,0,exp(currentLP))<exp(newLP)
            if(check){
                parm=trial
                currentLP=newLP
                keep=c(keep,TRUE)
            }else{
                keep=c(keep,FALSE)
            }
        }

    if(i %% thin ==0){
        posterior=c(posterior,parm)
        }
    }
    posterior=posterior[1:length(posterior)>burn*length(posterior)] 
    return=list(posterior=posterior,acrate=length(which(keep))/iterations) 
}



    











In [8]:



    


testMH=MH(Model, Data, start=1, iterations=1e5, thin=100, step=5, burn=0.1)
magplot(density(testMH$posterior,kern='rect',bw=0.1/sqrt(12)),xlab='Lamba',ylab='Density')
lines(xval,dpois(0,xval),col='red')

Let's examine an astrophysical problem!

The Schechter function is a parametrisation of how galaxies are distributed per unit volume in the Universe. It's an excellent fit to luminosity functions. It generally carries the form:

$dN = \theta(L)dL = (\theta^* / L^*) (L / L*)^{\alpha} \exp(-L/L^*)dL$

Let's generate a mock Schechter function, and then see how well we can recover its parameters. We'll do the generation in a slightly odd way, by creating a random pointing galaxy data in order to use a Monte-Carlo accept-reject technique later.





In [9]:



    


schec=function (m, ns, ms, alpha){ #This is the famous Schechter function
0.4*log(10)*ns*exp(0.4*log(10)*(alpha+1)*(ms-m)-exp(0.4*log(10)*(ms-m)))}

genLF=function(N, ms= -20, alpha= -1, mb= -25, mf= -14){
    maxval = max(schec(seq(mb, mf, by = 0.01), ns = 1, ms = ms, alpha = alpha))
    temp = cbind(runif(N, mb, mf), runif(N, 0, maxval + maxval * 0.01))
    temp = cbind(temp, schec(temp[, 1], ns = 1, ms = ms, alpha = alpha))
    mags = temp[temp[, 2] < temp[, 3], 1]
    gens = 2 * ceiling(N/length(mags)) * N
    temp = cbind(runif(gens, mb, mf), runif(gens, 0, maxval + maxval * 0.01))
    temp = cbind(temp, schec(temp[, 1], ns = 1, ms = ms, alpha = alpha))
    mags = c(mags, temp[temp[, 2] < temp[, 3], 1])
    mags = mags[1:N]
    return(mags)
    }

Data=list(data=genLF(1e4,ms=-20.5,alpha=-1.2),
mon.names='ns',parm.names=c('ms','alpha'),N=1e4)

magplot(density(Data$data,kern='rect',bw=0.2/
sqrt(12)),log='y',ylim=c(1e-4,1),xlab='Magnitude',ylab='Density')

ns=1/integrate(schec,-25,-14,ns=1,ms=-20.5,alpha=-1.2)$value

lines(seq(-24,-14,len=1e3),schec(seq(-24,-14,len=1e3),
ns=ns,ms=-20.5,alpha=-1.2),col='blue')



    


















    






Warning message in xy.coords(x, y, xlabel, ylabel, log):
“6 y values <= 0 omitted from logarithmic plot”









    
























In [18]:



    


Model=function(parm,Data){
ns=1/integrate(schec,-25,-14,ns=1,ms=parm[1],alpha=parm[2])$value
val.prior=dunif(parm[1],-30,-10,log=TRUE)+dunif(parm[2],-3,1,log=TRUE)
LL=sum(log(schec(Data$data,ns=ns,ms=parm[1],alpha=parm[2])))
LP=LL+val.prior
Modelout=list(LP=LP,Dev=2*LL,Monitor=ns,yhat=1,parm=parm)}

Let's use a package called Laplace's Demon to do our fit. This package is named after a hypothetical creature that could potentially intuit everything there is to know about the Universe, provided it knows all the properties of the Universe.

While that doesn't exist, this package does! And it's a great bundle of common Bayesian tools, including many MCMC algorithms.





In [20]:



    


library(LaplacesDemon)
FitLF=LaplacesDemon(Model, Data, Initial.Values=c(-20,-1), Algorithm='HARM', Iterations=1e5)



    


















    






Laplace's Demon was called on Sun Jan 21 23:36:15 2018

Performing initial checks...
Algorithm: Hit-And-Run Metropolis 

Laplace's Demon is beginning to update...
Iteration: 100Iteration: 100,   Proposal: Multivariate,   LP: -18174.7
Iteration: 200Iteration: 200,   Proposal: Multivariate,   LP: -18168.1
Iteration: 300Iteration: 300,   Proposal: Multivariate,   LP: -18168.1
Iteration: 400Iteration: 400,   Proposal: Multivariate,   LP: -18167.9
Iteration: 500Iteration: 500,   Proposal: Multivariate,   LP: -18168.1
Iteration: 600Iteration: 600,   Proposal: Multivariate,   LP: -18167.1
Iteration: 700Iteration: 700,   Proposal: Multivariate,   LP: -18167.4
Iteration: 800Iteration: 800,   Proposal: Multivariate,   LP: -18167.3
Iteration: 900Iteration: 900,   Proposal: Multivariate,   LP: -18167.3
Iteration: 1000Iteration: 1000,   Proposal: Multivariate,   LP: -18167.4
Iteration: 1100Iteration: 1100,   Proposal: Multivariate,   LP: -18167.4
Iteration: 1200Iteration: 1200,   Proposal: Multivariate,   LP: -18167.2
Iteration: 1300Iteration: 1300,   Proposal: Multivariate,   LP: -18167.6
Iteration: 1400Iteration: 1400,   Proposal: Multivariate,   LP: -18167.5
Iteration: 1500Iteration: 1500,   Proposal: Multivariate,   LP: -18167.5
Iteration: 1600Iteration: 1600,   Proposal: Multivariate,   LP: -18167.6
Iteration: 1700Iteration: 1700,   Proposal: Multivariate,   LP: -18168
Iteration: 1800Iteration: 1800,   Proposal: Multivariate,   LP: -18166.9
Iteration: 1900Iteration: 1900,   Proposal: Multivariate,   LP: -18166.9
Iteration: 2000Iteration: 2000,   Proposal: Multivariate,   LP: -18166.9
Iteration: 2100Iteration: 2100,   Proposal: Multivariate,   LP: -18168.4
Iteration: 2200Iteration: 2200,   Proposal: Multivariate,   LP: -18167.8
Iteration: 2300Iteration: 2300,   Proposal: Multivariate,   LP: -18167.5
Iteration: 2400Iteration: 2400,   Proposal: Multivariate,   LP: -18167.8
Iteration: 2500Iteration: 2500,   Proposal: Multivariate,   LP: -18167.1
Iteration: 2600Iteration: 2600,   Proposal: Multivariate,   LP: -18168.6
Iteration: 2700Iteration: 2700,   Proposal: Multivariate,   LP: -18168.5
Iteration: 2800Iteration: 2800,   Proposal: Multivariate,   LP: -18167.6
Iteration: 2900Iteration: 2900,   Proposal: Multivariate,   LP: -18167.7
Iteration: 3000Iteration: 3000,   Proposal: Multivariate,   LP: -18167.6
Iteration: 3100Iteration: 3100,   Proposal: Multivariate,   LP: -18167.6
Iteration: 3200Iteration: 3200,   Proposal: Multivariate,   LP: -18167.6
Iteration: 3300Iteration: 3300,   Proposal: Multivariate,   LP: -18167.6
Iteration: 3400Iteration: 3400,   Proposal: Multivariate,   LP: -18172.6
Iteration: 3500Iteration: 3500,   Proposal: Multivariate,   LP: -18167.4
Iteration: 3600Iteration: 3600,   Proposal: Multivariate,   LP: -18167.4
Iteration: 3700Iteration: 3700,   Proposal: Multivariate,   LP: -18167.9
Iteration: 3800Iteration: 3800,   Proposal: Multivariate,   LP: -18167.3
Iteration: 3900Iteration: 3900,   Proposal: Multivariate,   LP: -18167.5
Iteration: 4000Iteration: 4000,   Proposal: Multivariate,   LP: -18169.1
Iteration: 4100Iteration: 4100,   Proposal: Multivariate,   LP: -18167.7
Iteration: 4200Iteration: 4200,   Proposal: Multivariate,   LP: -18168.3
Iteration: 4300Iteration: 4300,   Proposal: Multivariate,   LP: -18168.4
Iteration: 4400Iteration: 4400,   Proposal: Multivariate,   LP: -18167.7
Iteration: 4500Iteration: 4500,   Proposal: Multivariate,   LP: -18167
Iteration: 4600Iteration: 4600,   Proposal: Multivariate,   LP: -18166.9
Iteration: 4700Iteration: 4700,   Proposal: Multivariate,   LP: -18167.6
Iteration: 4800Iteration: 4800,   Proposal: Multivariate,   LP: -18166.9
Iteration: 4900Iteration: 4900,   Proposal: Multivariate,   LP: -18167.2
Iteration: 5000Iteration: 5000,   Proposal: Multivariate,   LP: -18167.2
Iteration: 5100Iteration: 5100,   Proposal: Multivariate,   LP: -18166.9
Iteration: 5200Iteration: 5200,   Proposal: Multivariate,   LP: -18166.8
Iteration: 5300Iteration: 5300,   Proposal: Multivariate,   LP: -18167
Iteration: 5400Iteration: 5400,   Proposal: Multivariate,   LP: -18167.8
Iteration: 5500Iteration: 5500,   Proposal: Multivariate,   LP: -18167.8
Iteration: 5600Iteration: 5600,   Proposal: Multivariate,   LP: -18167.8
Iteration: 5700Iteration: 5700,   Proposal: Multivariate,   LP: -18169.6
Iteration: 5800Iteration: 5800,   Proposal: Multivariate,   LP: -18168
Iteration: 5900Iteration: 5900,   Proposal: Multivariate,   LP: -18166.9
Iteration: 6000Iteration: 6000,   Proposal: Multivariate,   LP: -18169.5
Iteration: 6100Iteration: 6100,   Proposal: Multivariate,   LP: -18168.6
Iteration: 6200Iteration: 6200,   Proposal: Multivariate,   LP: -18169.4
Iteration: 6300Iteration: 6300,   Proposal: Multivariate,   LP: -18167.9
Iteration: 6400Iteration: 6400,   Proposal: Multivariate,   LP: -18168.7
Iteration: 6500Iteration: 6500,   Proposal: Multivariate,   LP: -18168
Iteration: 6600Iteration: 6600,   Proposal: Multivariate,   LP: -18169.6
Iteration: 6700Iteration: 6700,   Proposal: Multivariate,   LP: -18168.9
Iteration: 6800Iteration: 6800,   Proposal: Multivariate,   LP: -18169.4
Iteration: 6900Iteration: 6900,   Proposal: Multivariate,   LP: -18169.4
Iteration: 7000Iteration: 7000,   Proposal: Multivariate,   LP: -18170
Iteration: 7100Iteration: 7100,   Proposal: Multivariate,   LP: -18170.5
Iteration: 7200Iteration: 7200,   Proposal: Multivariate,   LP: -18168.2
Iteration: 7300Iteration: 7300,   Proposal: Multivariate,   LP: -18167.4
Iteration: 7400Iteration: 7400,   Proposal: Multivariate,   LP: -18168.3
Iteration: 7500Iteration: 7500,   Proposal: Multivariate,   LP: -18167.8
Iteration: 7600Iteration: 7600,   Proposal: Multivariate,   LP: -18167.2
Iteration: 7700Iteration: 7700,   Proposal: Multivariate,   LP: -18167.2
Iteration: 7800Iteration: 7800,   Proposal: Multivariate,   LP: -18167.6
Iteration: 7900Iteration: 7900,   Proposal: Multivariate,   LP: -18167.5
Iteration: 8000Iteration: 8000,   Proposal: Multivariate,   LP: -18167.5
Iteration: 8100Iteration: 8100,   Proposal: Multivariate,   LP: -18167.2
Iteration: 8200Iteration: 8200,   Proposal: Multivariate,   LP: -18167.5
Iteration: 8300Iteration: 8300,   Proposal: Multivariate,   LP: -18169.5
Iteration: 8400Iteration: 8400,   Proposal: Multivariate,   LP: -18169.5
Iteration: 8500Iteration: 8500,   Proposal: Multivariate,   LP: -18167.2
Iteration: 8600Iteration: 8600,   Proposal: Multivariate,   LP: -18167.9
Iteration: 8700Iteration: 8700,   Proposal: Multivariate,   LP: -18167.1
Iteration: 8800Iteration: 8800,   Proposal: Multivariate,   LP: -18167.1
Iteration: 8900Iteration: 8900,   Proposal: Multivariate,   LP: -18167.1
Iteration: 9000Iteration: 9000,   Proposal: Multivariate,   LP: -18167.2
Iteration: 9100Iteration: 9100,   Proposal: Multivariate,   LP: -18167.8
Iteration: 9200Iteration: 9200,   Proposal: Multivariate,   LP: -18167.4
Iteration: 9300Iteration: 9300,   Proposal: Multivariate,   LP: -18167.8
Iteration: 9400Iteration: 9400,   Proposal: Multivariate,   LP: -18167.7
Iteration: 9500Iteration: 9500,   Proposal: Multivariate,   LP: -18167.7
Iteration: 9600Iteration: 9600,   Proposal: Multivariate,   LP: -18167.7
Iteration: 9700Iteration: 9700,   Proposal: Multivariate,   LP: -18168.7
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Iteration: 96400Iteration: 96400,   Proposal: Multivariate,   LP: -18167.6
Iteration: 96500Iteration: 96500,   Proposal: Multivariate,   LP: -18168.2
Iteration: 96600Iteration: 96600,   Proposal: Multivariate,   LP: -18168.5
Iteration: 96700Iteration: 96700,   Proposal: Multivariate,   LP: -18170
Iteration: 96800Iteration: 96800,   Proposal: Multivariate,   LP: -18167.4
Iteration: 96900Iteration: 96900,   Proposal: Multivariate,   LP: -18167.4
Iteration: 97000Iteration: 97000,   Proposal: Multivariate,   LP: -18167.7
Iteration: 97100Iteration: 97100,   Proposal: Multivariate,   LP: -18166.9
Iteration: 97200Iteration: 97200,   Proposal: Multivariate,   LP: -18167.3
Iteration: 97300Iteration: 97300,   Proposal: Multivariate,   LP: -18167
Iteration: 97400Iteration: 97400,   Proposal: Multivariate,   LP: -18167.2
Iteration: 97500Iteration: 97500,   Proposal: Multivariate,   LP: -18167.1
Iteration: 97600Iteration: 97600,   Proposal: Multivariate,   LP: -18170
Iteration: 97700Iteration: 97700,   Proposal: Multivariate,   LP: -18170.5
Iteration: 97800Iteration: 97800,   Proposal: Multivariate,   LP: -18170.5
Iteration: 97900Iteration: 97900,   Proposal: Multivariate,   LP: -18167.3
Iteration: 98000Iteration: 98000,   Proposal: Multivariate,   LP: -18167
Iteration: 98100Iteration: 98100,   Proposal: Multivariate,   LP: -18167
Iteration: 98200Iteration: 98200,   Proposal: Multivariate,   LP: -18167.7
Iteration: 98300Iteration: 98300,   Proposal: Multivariate,   LP: -18167.2
Iteration: 98400Iteration: 98400,   Proposal: Multivariate,   LP: -18167.2
Iteration: 98500Iteration: 98500,   Proposal: Multivariate,   LP: -18167.7
Iteration: 98600Iteration: 98600,   Proposal: Multivariate,   LP: -18168.8
Iteration: 98700Iteration: 98700,   Proposal: Multivariate,   LP: -18169
Iteration: 98800Iteration: 98800,   Proposal: Multivariate,   LP: -18167
Iteration: 98900Iteration: 98900,   Proposal: Multivariate,   LP: -18167.6
Iteration: 99000Iteration: 99000,   Proposal: Multivariate,   LP: -18167.1
Iteration: 99100Iteration: 99100,   Proposal: Multivariate,   LP: -18167
Iteration: 99200Iteration: 99200,   Proposal: Multivariate,   LP: -18167
Iteration: 99300Iteration: 99300,   Proposal: Multivariate,   LP: -18167.6
Iteration: 99400Iteration: 99400,   Proposal: Multivariate,   LP: -18167.6
Iteration: 99500Iteration: 99500,   Proposal: Multivariate,   LP: -18167
Iteration: 99600Iteration: 99600,   Proposal: Multivariate,   LP: -18166.8
Iteration: 99700Iteration: 99700,   Proposal: Multivariate,   LP: -18166.8
Iteration: 99800Iteration: 99800,   Proposal: Multivariate,   LP: -18166.8
Iteration: 99900Iteration: 99900,   Proposal: Multivariate,   LP: -18166.8
Iteration: 100000Iteration: 100000,   Proposal: Multivariate,   LP: -18167.4

Assessing Stationarity
Assessing Thinning and ESS
Creating Summaries
Estimating Log of the Marginal Likelihood
Creating Output

Laplace's Demon has finished.

















In [23]:



    


magplot(density(Data$data,kern='rect',bw=0.2/
sqrt(12)),log='y',ylim=c(1e-4,1),xlab='Magnitude',ylab='Density')

ns=1/integrate(schec,-25,-14,ns=1,ms=-20.5,alpha=-1.2)$value

lines(seq(-24,-14,len=1e3),schec(seq(-24,-14,len=1e3),
ns=ns,ms=-20.5,alpha=-1.2),col='blue')

ns=1/integrate(schec,-25,-14,ns=1, ms=FitLF$Summary2[1,'Mean'],
alpha=FitLF$Summary2[2,'Mean'])$value

lines(seq(-24,-14,len=1e3),schec(seq(-24,-14,len=1e3),
ns=ns,ms=FitLF$Summary2[1,'Mean'],
alpha=FitLF$Summary2[2,'Mean']),col='red')



    


















    






Warning message in xy.coords(x, y, xlabel, ylabel, log):
“8 y values <= 0 omitted from logarithmic plot”

It's a good fit! Let's examine the posterior distribution to see how well the Demon is performing.





In [49]:



    


magcon(FitLF$Posterior2[,1], FitLF$Posterior2[,2],conlevels=c(0.5,0.68,0.95),lty=c(2,1,3),xlim=c(-20.65,-20.2),ylim=c(-1.23,-1.16))
title(xlab='M*',ylab=expression(alpha))
points(FitLF$Summary2[1,'Mean'],FitLF$Summary2[2,'Mean'],col='black',pch=4)
points(-20.5,-1.2,col='blue',pch=4)



    


















    






Warning: weights overwritten by binning

Content source: LSSTC-DSFP/LSSTC-DSFP-Sessions

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