running_centered {fromo}

Compare data to moments computed over a sliding window.

Description

Computes moments over a sliding window, then adjusts the data accordingly, centering, or scaling, or z-scoring, and so on.

Usage

running_centered(v, window = NULL, wts = NULL, na_rm = FALSE,
  min_df = 0L, used_df = 1, lookahead = 0L, restart_period = 100L,
  check_wts = FALSE, normalize_wts = FALSE)

running_scaled(v, window = NULL, wts = NULL, na_rm = FALSE, min_df = 0L,
  used_df = 1, lookahead = 0L, restart_period = 100L, check_wts = FALSE,
  normalize_wts = TRUE)

running_zscored(v, window = NULL, wts = NULL, na_rm = FALSE,
  min_df = 0L, used_df = 1, lookahead = 0L, restart_period = 100L,
  check_wts = FALSE, normalize_wts = TRUE)

running_sharpe(v, window = NULL, wts = NULL, na_rm = FALSE,
  compute_se = FALSE, min_df = 0L, used_df = 1, restart_period = 100L,
  check_wts = FALSE, normalize_wts = TRUE)

running_tstat(v, window = NULL, wts = NULL, na_rm = FALSE, min_df = 0L,
  used_df = 1, restart_period = 100L, check_wts = FALSE,
  normalize_wts = TRUE)

Arguments

v

a vector

window

the window size. if given as finite integer or double, passed through. If NULL, NA_integer_, NA_real_ or Inf are given, equivalent to an infinite window size. If negative, an error will be thrown.

wts

an optional vector of weights. Weights are ‘replication’ weights, meaning a value of 2 is shorthand for having two observations with the corresponding v value. If NULL, corresponds to equal unit weights, the default. Note that weights are typically only meaningfully defined up to a multiplicative constant, meaning the units of weights are immaterial, with the exception that methods which check for minimum df will, in the weighted case, check against the sum of weights. For this reason, weights less than 1 could cause NA to be returned unexpectedly due to the minimum condition. When weights are NA, the same rules for checking v are applied. That is, the observation will not contribute to the moment if the weight is NA when na_rm is true. When there is no checking, an NA value will cause the output to be NA.

na_rm

whether to remove NA, false by default.

min_df

the minimum df to return a value, otherwise NaN is returned. This can be used to prevent e.g. Z-scores from being computed on only 3 observations. Defaults to zero, meaning no restriction, which can result in infinite Z-scores during the burn-in period.

used_df

the number of degrees of freedom consumed, used in the denominator of the centered moments computation. These are subtracted from the number of observations.

lookahead

for some of the operations, the value is compared to mean and standard deviation possibly using 'future' or 'past' information by means of a non-zero lookahead. Positive values mean data are taken from the future.

restart_period

the recompute period. because subtraction of elements can cause loss of precision, the computation of moments is restarted periodically based on this parameter. Larger values mean fewer restarts and faster, though less accurate results.

check_wts

a boolean for whether the code shall check for negative weights, and throw an error when they are found. Default false for speed.

normalize_wts

a boolean for whether the weights should be renormalized to have a mean value of 1. This mean is computed over elements which contribute to the moments, so if na_rm is set, that means non-NA elements of wts that correspond to non-NA elements of the data vector.

compute_se

for running_sharpe, return an extra column of the standard error, as computed by Mertens' correction.

Details

Given the length n vector x, for a given index i, define x^{(i)} as the vector of x_{i-window+1},x_{i-window+2},...,x_{i}, where we do not run over the 'edge' of the vector. In code, this is essentially x[(max(1,i-window+1)):i]. Then define \mu_i, \sigma_i and n_i as, respectively, the sample mean, standard deviation and number of non-NA elements in x^{(i)}.

We compute output vector m the same size as x. For the 'centered' version of x, we have m_i = x_i - \mu_i. For the 'scaled' version of x, we have m_i = x_i / \sigma_i. For the 'z-scored' version of x, we have m_i = (x_i - \mu_i) / \sigma_i. For the 't-scored' version of x, we have m_i = \sqrt{n_i} \mu_i / \sigma_i.

We also allow a 'lookahead' for some of these operations. If positive, the moments are computed using data from larger indices; if negative, from smaller indices. Letting j = i + lookahead: For the 'centered' version of x, we have m_i = x_i - \mu_j. For the 'scaled' version of x, we have m_i = x_i / \sigma_j. For the 'z-scored' version of x, we have m_i = (x_i - \mu_j) / \sigma_j.

Value

a vector the same size as the input consisting of the adjusted version of the input. When there are not sufficient (non-nan) elements for the computation, NaN are returned.

Note

The moment computations provided by fromo are numerically robust, but will often not provide the same results as the 'standard' implementations, due to differences in roundoff. We make every attempt to balance speed and robustness. User assumes all risk from using the fromo package.

Note that when weights are given, they are treated as replication weights. This can have subtle effects on computations which require minimum degrees of freedom, since the sum of weights will be compared to that minimum, not the number of data points. Weight values (much) less than 1 can cause computations to return NA somewhat unexpectedly due to this condition, while values greater than one might cause the computation to spuriously return a value with little precision.

Author(s)

Steven E. Pav [e-mail]

References

Terriberry, T. "Computing Higher-Order Moments Online." http://people.xiph.org/~tterribe/notes/homs.html

J. Bennett, et. al., "Numerically Stable, Single-Pass, Parallel Statistics Algorithms," Proceedings of IEEE International Conference on Cluster Computing, 2009. https://www.semanticscholar.org/paper/Numerically-stable-single-pass-parallel-statistics-Bennett-Grout/a83ed72a5ba86622d5eb6395299b46d51c901265

Cook, J. D. "Accurately computing running variance." http://www.johndcook.com/standard_deviation.html

Cook, J. D. "Comparing three methods of computing standard deviation." http://www.johndcook.com/blog/2008/09/26/comparing-three-methods-of-computing-standard-deviation

See Also

t_running_centered, scale

Examples

if (require(moments)) {
    set.seed(123)
    x <- rnorm(5e1)
    window <- 10L
    rm1 <- t(sapply(seq_len(length(x)),function(iii) { 
                  xrang <- x[max(1,iii-window+1):iii]
                  c(sd(xrang),mean(xrang),length(xrang)) },
                  simplify=TRUE))
    rcent <- running_centered(x,window=window)
    rscal <- running_scaled(x,window=window)
    rzsco <- running_zscored(x,window=window)
    rshrp <- running_sharpe(x,window=window)
    rtsco <- running_tstat(x,window=window)
    rsrse <- running_sharpe(x,window=window,compute_se=TRUE)
    stopifnot(max(abs(rcent - (x - rm1[,2])),na.rm=TRUE) < 1e-12)
    stopifnot(max(abs(rscal - (x / rm1[,1])),na.rm=TRUE) < 1e-12)
    stopifnot(max(abs(rzsco - ((x - rm1[,2]) / rm1[,1])),na.rm=TRUE) < 1e-12)
    stopifnot(max(abs(rshrp - (rm1[,2] / rm1[,1])),na.rm=TRUE) < 1e-12)
    stopifnot(max(abs(rtsco - ((sqrt(rm1[,3]) * rm1[,2]) / rm1[,1])),na.rm=TRUE) < 1e-12)
    stopifnot(max(abs(rsrse[,1] - rshrp),na.rm=TRUE) < 1e-12)

    rm2 <- t(sapply(seq_len(length(x)),function(iii) { 
                  xrang <- x[max(1,iii-window+1):iii]
                  c(kurtosis(xrang)-3.0,skewness(xrang)) },
                  simplify=TRUE))
    mertens_se <- sqrt((1 + ((2 + rm2[,1])/4) * rshrp^2 - rm2[,2]*rshrp) / rm1[,3])
    stopifnot(max(abs(rsrse[,2] - mertens_se),na.rm=TRUE) < 1e-12)
}