t_running_centered: Compare data to moments computed over a time sliding window.

Description

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

Usage

t_running_centered(
  v,
  time = NULL,
  time_deltas = NULL,
  window = NULL,
  wts = NULL,
  na_rm = FALSE,
  min_df = 0L,
  used_df = 1,
  lookahead = 0,
  restart_period = 100L,
  variable_win = FALSE,
  wts_as_delta = TRUE,
  check_wts = FALSE,
  normalize_wts = TRUE,
  check_negative_moments = TRUE
)
t_running_scaled(
  v,
  time = NULL,
  time_deltas = NULL,
  window = NULL,
  wts = NULL,
  na_rm = FALSE,
  min_df = 0L,
  used_df = 1,
  lookahead = 0,
  restart_period = 100L,
  variable_win = FALSE,
  wts_as_delta = TRUE,
  check_wts = FALSE,
  normalize_wts = TRUE,
  check_negative_moments = TRUE
)
t_running_zscored(
  v,
  time = NULL,
  time_deltas = NULL,
  window = NULL,
  wts = NULL,
  na_rm = FALSE,
  min_df = 0L,
  used_df = 1,
  lookahead = 0,
  restart_period = 100L,
  variable_win = FALSE,
  wts_as_delta = TRUE,
  check_wts = FALSE,
  normalize_wts = TRUE,
  check_negative_moments = TRUE
)
t_running_sharpe(
  v,
  time = NULL,
  time_deltas = NULL,
  window = NULL,
  wts = NULL,
  lb_time = NULL,
  na_rm = FALSE,
  compute_se = FALSE,
  min_df = 0L,
  used_df = 1,
  restart_period = 100L,
  variable_win = FALSE,
  wts_as_delta = TRUE,
  check_wts = FALSE,
  normalize_wts = TRUE,
  check_negative_moments = TRUE
)
t_running_tstat(
  v,
  time = NULL,
  time_deltas = NULL,
  window = NULL,
  wts = NULL,
  lb_time = NULL,
  na_rm = FALSE,
  compute_se = FALSE,
  min_df = 0L,
  used_df = 1,
  restart_period = 100L,
  variable_win = FALSE,
  wts_as_delta = TRUE,
  check_wts = FALSE,
  normalize_wts = TRUE,
  check_negative_moments = TRUE
)

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.

Arguments

v: a vector of data.
time: an optional vector of the timestamps of v. If given, must be the same length as v. If not given, we try to infer it by summing the time_deltas.
time_deltas: an optional vector of the deltas of timestamps. If given, must be the same length as v. If not given, and wts are given and wts_as_delta is true, we take the wts as the time deltas. The deltas must be positive. We sum them to arrive at the times.
window: the window size, in time units. if given as finite integer or double, passed through. If NULL, NA_integer_, NA_real_ or Inf are given, and variable_win is true, then we infer the window from the lookback times: the first window is infinite, but the remaining is the deltas between lookback times. If variable_win is false, then these undefined values are equivalent to an infinite window. 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. This is in time units, and so should be a real.
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.
variable_win: if true, and the window is not a concrete number, the computation window becomes the time between lookback times.
wts_as_delta: if true and the time and time_deltas are not given, but wts are given, we take wts as the time_deltas.
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.
check_negative_moments: a boolean flag. Normal computation of running moments can result in negative estimates of even order moments due to loss of numerical precision. With this flag active, the computation checks for negative even order moments and restarts the computation when one is detected. This should eliminate the possibility of negative even order moments. The downside is the speed hit of checking on every output step. Note also the code checks for negative moments of every even order tracked, even if they are not output; that is if the kurtosis, say, is being computed, and a negative variance is detected, then the computation is restarted. Defaults to TRUE to avoid negative even moments. Set to FALSE only if you know what you are doing.
lb_time: a vector of the times from which lookback will be performed. The output should be the same size as this vector. If not given, defaults to time.
compute_se: for running_sharpe, return an extra column of the standard error, as computed by Mertens' correction.

Time Windowing

This function supports time (or other counter) based running computation. Here the input are the data $x_i$, and optional weights vectors, $w_i$, defaulting to 1, and a vector of time indices, $t_i$ of the same length as $x$. The times must be non-decreasing: $$t_1 \le t_2 \le \ldots$$ It is assumed that $t_0 = -\infty$. The window, $W$ is now a time-based window. An optional set of lookback times are also given, $b_j$, which may have different length than the $x$ and $w$. The output will correspond to the lookback times, and should be the same length. The $j$th output is computed over indices $i$ such that $$b_j - W < t_i \le b_j.$$

For comparison functions (like Z-score, rescaling, centering), which compare values of $x_i$ to local moments, the lookbacks may not be given, but a lookahead $L$ is admitted. In this case, the $j$th output is computed over indices $i$ such that $$t_j - W + L < t_i \le t_j + L.$$

If the times are not given, ‘deltas’ may be given instead. If $\delta_i$ are the deltas, then we compute the times as $$t_i = \sum_{1 \le j \le i} \delta_j.$$ The deltas must be the same length as $x$. If times and deltas are not given, but weights are given and the ‘weights as deltas’ flag is set true, then the weights are used as the deltas.

Some times it makes sense to have the computational window be the space between lookback times. That is, the $j$th output is to be computed over indices $i$ such that $$b_{j-1} - W < t_i \le b_j.$$ This can be achieved by setting the ‘variable window’ flag true and setting the window to null. This will not make much sense if the lookback times are equal to the times, since each moment computation is over a set of a single index, and most moments are underdefined.

Author

Steven E. Pav shabbychef@gmail.com

Details

Given the length $n$ vector $x$, for a given index $i$, define $x^{(i)}$ as the elements of $x$ defined by the sliding time window (see the section on time windowing). 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.

References

Terriberry, T. "Computing Higher-Order Moments Online." https://web.archive.org/web/20140423031833/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. tools:::Rd_expr_doi("10.1109/CLUSTR.2009.5289161")

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

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