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4.1 Reductions

The following operations take arrays as input and produce single output values. The boundary values for size 0 arrays are the unit with respect to the combination operation (min, max, sum, or product).

4.1.1 Minimum and Maximum

real min(real[] x)
The minimum value in x, or \(+\infty\) if x is size 0.

int min(int[] x)
The minimum value in x, or error if x is size 0.

real max(real[] x)
The maximum value in x, or \(-\infty\) if x is size 0.

int max(int[] x)
The maximum value in x, or error if x is size 0.

4.1.2 Sum, Product, and Log Sum of Exp

int sum(int[] x)
The sum of the elements in x, defined for \(x\) of size \(N\) by \[ \text{sum}(x) = \begin{cases} \sum_{n=1}^N x_n & \text{if} N > 0 \\[4pt] 0 & \text{if} N = 0 \end{cases} \]

real sum(real[] x)
The sum of the elements in x; see definition above.

real prod(real[] x)
The product of the elements in x, or 1 if x is size 0.

real prod(int[] x)
The product of the elements in x, \[ \text{product}(x) = \begin{cases} \prod_{n=1}^N x_n & \text{if} N > 0 \\[4pt] 1 & \text{if} N = 0 \end{cases} \]

real log_sum_exp(real[] x)
The natural logarithm of the sum of the exponentials of the elements in x, or \(-\infty\) if the array is empty.

4.1.3 Sample Mean, Variance, and Standard Deviation

The sample mean, variance, and standard deviation are calculated in the usual way. For i.i.d. draws from a distribution of finite mean, the sample mean is an unbiased estimate of the mean of the distribution. Similarly, for i.i.d. draws from a distribution of finite variance, the sample variance is an unbiased estimate of the variance.² The sample deviation is defined as the square root of the sample deviation, but is not unbiased.

real mean(real[] x)
The sample mean of the elements in x. For an array \(x\) of size \(N > 0\), \[ \text{mean}(x) \ = \ \bar{x} \ = \ \frac{1}{N} \sum_{n=1}^N x_n. \] It is an error to the call the mean function with an array of size \(0\).

real variance(real[] x)
The sample variance of the elements in x. For \(N > 0\), \[ \text{variance}(x) \ = \ \begin{cases} \frac{1}{N-1} \sum_{n=1}^N (x_n - \bar{x})^2 & \text{if } N > 1 \\[4pt] 0 & \text{if } N = 1 \end{cases} \] It is an error to call the variance function with an array of size 0.

real sd(real[] x)
The sample standard deviation of elements in x. \[ \text{sd}(x) = \begin{cases} \sqrt{\, \text{variance}(x)} & \text{if } N > 1 \\[4pt] 0 & \text{if } N = 0 \end{cases} \] It is an error to call the sd function with an array of size 0.

4.1.4 Euclidean Distance and Squared Distance

real distance(vector x, vector y)
The Euclidean distance between x and y, defined by \[ \text{distance}(x,y) \ = \ \sqrt{\textstyle \sum_{n=1}^N (x_n - y_n)^2} \] where N is the size of x and y. It is an error to call distance with arguments of unequal size.

real distance(vector x, row_vector y)
The Euclidean distance between x and y

real distance(row_vector x, vector y)
The Euclidean distance between x and y

real distance(row_vector x, row_vector y)
The Euclidean distance between x and y

real squared_distance(vector x, vector y)
The squared Euclidean distance between x and y, defined by \[ \mathrm{squared\_distance}(x,y) \ = \ \text{distance}(x,y)^2 \ = \ \textstyle \sum_{n=1}^N (x_n - y_n)^2, \] where N is the size of x and y. It is an error to call squared_distance with arguments of unequal size.

real squared_distance(vector x, row_vector [] y)
The squared Euclidean distance between x and y

real squared_distance(row_vector x, vector [] y)
The squared Euclidean distance between x and y

real squared_distance(row_vector x, row_vector[] y)
The Euclidean distance between x and y

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2. Dividing by \(N\) rather than \((N-1)\) produces a maximum likelihood estimate of variance, which is biased to underestimate variance.↩