Descriptive Statistics - Untitled
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### 24.1 Descriptive Statistics

Octave can compute various statistics such as the moments of a data set.

— Function File: mean (x, dim, opt)

> If x is a vector, compute the mean of the elements of x

>               mean (x) = SUM_i x(i) / N

> If x is a matrix, compute the mean for each column and return them in a row vector.

> With the optional argument opt, the kind of mean computed can be selected. The following options are recognized:

> `"a"`
>   Compute the (ordinary) arithmetic mean. This is the default.
> `"g"`
>   Compute the geometric mean.
> `"h"`
>   Compute the harmonic mean.

> If the optional argument dim is supplied, work along dimension dim.

> Both dim and opt are optional. If both are supplied, either may appear first.

— Function File: median (x, dim)

> If x is a vector, compute the median value of the elements of x. If the elements of x are sorted, the median is defined as

>                           x(ceil(N/2)),             N odd
>               median(x) =
>                           (x(N/2) + x((N/2)+1))/2,  N even

> If x is a matrix, compute the median value for each column and return them in a row vector. If the optional dim argument is given, operate along this dimension.

> See also: std, mean.

— Function File: q = quantile (x, p)
— Function File: q = quantile (x, p, dim)
— Function File: q = quantile (x, p, dim, method )

> For a sample, x, calculate the quantiles, q, corresponding to the cumulative probability values in p. All non-numeric values (NaNs) of x are ignored.

> If x is a matrix, compute the quantiles for each column and return them in a matrix, such that the i-th row of q contains the p(i)th quantiles of each column of x.

> The optional argument dim determines the dimension along which the percentiles are calculated. If dim is omitted, and x is a vector or matrix, it defaults to 1 (column wise quantiles). In the instance that x is a N-d array, dim defaults to the first dimension whose size greater than unity.

> The methods available to calculate sample quantiles are the nine methods used by R (http://www.r-project.org/). The default value is METHOD = 5.

> Discontinuous sample quantile methods 1, 2, and 3

> 1. Method 1: Inverse of empirical distribution function.
> 2. Method 2: Similar to method 1 but with averaging at discontinuities.
> 3. Method 3: SAS definition: nearest even order statistic.

> Continuous sample quantile methods 4 through 9, where p(k) is the linear interpolation function respecting each methods' representative cdf.

> 4. Method 4: p(k) = k / n. That is, linear interpolation of the empirical cdf.
> 5. Method 5: p(k) = (k - 0.5) / n. That is a piecewise linear function where the knots are the values midway through the steps of the empirical cdf.
> 6. Method 6: p(k) = k / (n + 1).
> 7. Method 7: p(k) = (k - 1) / (n - 1).
> 8. Method 8: p(k) = (k - 1/3) / (n + 1/3). The resulting quantile estimates are approximately median-unbiased regardless of the distribution of x.
> 9. Method 9: p(k) = (k - 3/8) / (n + 1/4). The resulting quantile estimates are approximately unbiased for the expected order statistics if x is normally distributed.

> Hyndman and Fan (1996) recommend method 8. Maxima, S, and R (versions prior to 2.0.0) use 7 as their default. Minitab and SPSS use method 6. Matlab uses method 5.

> References:

> - Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole.
> - Hyndman, R. J. and Fan, Y. (1996) Sample quantiles in statistical packages, American Statistician, 50, 361-365.
> - R: A Language and Environment for Statistical Computing; http://cran.r-project.org/doc/manuals/fullrefman.pdf.

— Function File: y = prctile (x, p)
— Function File: q = prctile (x, p, dim)

> For a sample x, compute the the quantiles, y, corresponding to the cumulative probability values, P, in percent. All non-numeric values (NaNs) of X are ignored.

> If x is a matrix, compute the the percentiles for each column and return them in a matrix, such that the i-th row of y contains the p(i)th percentiles of each column of x.

> The optional argument dim determines the dimension along which the percentiles are calculated. If dim is omitted, and x is a vector or matrix, it defaults to 1 (column wise quantiles). In the instance that x is a N-d array, dim defaults to the first dimension whose size greater than unity.

— Function File: meansq (x)
— Function File: meansq (x, dim)

> For vector arguments, return the mean square of the values. For matrix arguments, return a row vector containing the mean square of each column. With the optional dim argument, returns the mean squared of the values along this dimension.

— Function File: std (x)
— Function File: std (x, opt)
— Function File: std (x, opt, dim)

> If x is a vector, compute the standard deviation of the elements of x.

>               std (x) = sqrt (sumsq (x - mean (x)) / (n - 1))

> If x is a matrix, compute the standard deviation for each column and return them in a row vector.

> The argument opt determines the type of normalization to use. Valid values are

> 0:
>   normalizes with N-1, provides the square root of best unbiased estimator of the variance [default]
> 1:
>   normalizes with N, this provides the square root of the second moment around the mean

> The third argument dim determines the dimension along which the standard deviation is calculated.

> See also: mean, median.

— Function File: var (x)

> For vector arguments, return the (real) variance of the values. For matrix arguments, return a row vector containing the variance for each column.

> The argument opt determines the type of normalization to use. Valid values are

> 0:
>   Normalizes with N-1, provides the best unbiased estimator of the variance [default].
> 1:
>   Normalizes with N, this provides the second moment around the mean.

> The third argument dim determines the dimension along which the variance is calculated.

— Function File: [m, f, c] = mode (x, dim)

> Count the most frequently appearing value. `mode` counts the frequency along the first non-singleton dimension and if two or more values have te same frequency returns the smallest of the two in m. The dimension along which to count can be specified by the dim parameter.

> The variable f counts the frequency of each of the most frequently occurring elements. The cell array c contains all of the elements with the maximum frequency .

— Function File: cov (x, y)

> Compute covariance.

> If each row of x and y is an observation and each column is a variable, the (i, j)-th entry of `cov (`x`,` y`)` is the covariance between the i-th variable in x and the j-th variable in y. If called with one argument, compute `cov (`x`,` x`)`.

— Function File: cor (x, y)

> Compute correlation.

> The (i, j)-th entry of `cor (`x`,` y`)` is the correlation between the i-th variable in x and the j-th variable in y.

>               corrcoef(x,y) = cov(x,y)/(std(x)*std(y))

> For matrices, each row is an observation and each column a variable; vectors are always observations and may be row or column vectors.

> `cor (`x`)` is equivalent to `cor (`x`,` x`)`.

> Note that the `corrcoef` function does the same as `cor`.

— Function File: corrcoef (x, y)

> Compute correlation.

> If each row of x and y is an observation and each column is a variable, the (i, j)-th entry of `corrcoef (`x`,` y`)` is the correlation between the i-th variable in x and the j-th variable in y.

>               corrcoef(x,y) = cov(x,y)/(std(x)*std(y))

> If called with one argument, compute `corrcoef (`x`,` x`)`.

— Function File: kurtosis (x, dim)

> If x is a vector of length N, return the kurtosis

>               kurtosis (x) = N^(-1) std(x)^(-4) sum ((x - mean(x)).^4) - 3

> of x. If x is a matrix, return the kurtosis over the first non-singleton dimension. The optional argument dim can be given to force the kurtosis to be given over that dimension.

— Function File: skewness (x, dim)

> If x is a vector of length n, return the skewness

>               skewness (x) = N^(-1) std(x)^(-3) sum ((x - mean(x)).^3)

> of x. If x is a matrix, return the skewness along the first non-singleton dimension of the matrix. If the optional dim argument is given, operate along this dimension.

— Function File: statistics (x)

> If x is a matrix, return a matrix with the minimum, first quartile, median, third quartile, maximum, mean, standard deviation, skewness and kurtosis of the columns of x as its columns.

> If x is a vector, calculate the statistics along the non-singleton dimension.

— Function File: moment (x, p, opt, dim)

> If x is a vector, compute the p-th moment of x.

> If x is a matrix, return the row vector containing the p-th moment of each column.

> With the optional string opt, the kind of moment to be computed can be specified. If opt contains `"c"` or `"a"`, central and/or absolute moments are returned. For example,

>               moment (x, 3, "ac")

> computes the third central absolute moment of x.

> If the optional argument dim is supplied, work along dimension dim.