Outlier detection for price relatives

Standard cutoff-based methods for detecting outliers with price relatives.

Usage

outliers(
  x,
  upper,
  lower = upper,
  method = c("quartile", "resistant-fences", "kimber", "robust-z", "tukey"),
  scale = 0,
  quantile_type = 7
)

hb_transform(x)

Arguments

x: [numeric > 0] A numeric vector, usually of price relatives. These can be made with, e.g., back_period().
upper, lower: [numeric(1) > 0] A number giving the upper and lower cutoffs for each element of x. By default the lower cutoff is the same as the upper cutoff
method: [character(1)] The outlier detection method, one of "quartile" (the default), "resistant-fences", "kimber", "robust-z", or "tukey".
scale: [0 <= numeric(1) <= 1] A number between 0 and 1 giving the scale factor for the median to establish the minimum dispersion between quartiles for each element of x. The default does not set a minimum dispersion.
quantile_type: [integer(1)] See quantile().

Value

A logical vector, the same length as x, that is TRUE if the corresponding element of x is identified as an outlier, FALSE otherwise.

Details

This function constructs an interval of the form \([b_l(x) - c_l \times l(x), b_u(x) + c_u \times u(x)]\) and assigns a value in x as TRUE if that value does not belong to the interval, FALSE otherwise. The different methods differ in how they construct the values \(b_l(x)\), \(b_u(x)\), \(l(x)\), and \(u(x)\). Any missing values in x are ignored when calculating the interval, but will return NA.

The quartile method and Tukey algorithm are described in paragraphs 5.113 to 5.135 of the CPI manual (2020). The resistant fences method is an alternative to the quartile method, and is described by Rais (2008) and Hutton (2008). The Kimber method is yet another alternative. Quantile-based methods often identify price relatives as outliers because the distribution is concentrated around 1; setting scale > 0 puts a floor on the minimum dispersion between quantiles as a fraction of the median. See the references for more details.

	\(b_l(x)\)	\(b_u(x)\)	\(l(x)\)	\(u(x)\)
Quartile	\(Q_2(x)\)	\(Q_2(x)\)	\(Q_2(x) - Q_1(x)\)	\(Q_3(x) - Q_2(x)\)
Resistant fences	\(Q_1(x)\)	\(Q_3(x)\)	\(Q_3(x) - Q_1(x)\)	\(Q_3(x) - Q_1(x)\)
Kimber	\(Q_1(x)\)	\(Q_3(x)\)	\(Q_2(x) - Q_1(x)\)	\(Q_3(x) - Q_2(x)\)

The robust Z-score is the usual method to identify relatives in the (asymmetric) tails of the distribution, simply replacing the mean with the median, and the standard deviation with the median absolute deviation.

These methods often assume that price relatives are symmetrically distributed (if not Gaussian). As the distribution of price relatives often has a long right tail, the natural logarithm can be used to transform price relative before identifying outliers (sometimes under the assumption that price relatives are distributed log-normal). The Hidiroglou-Berthelot transformation is another approach, described in the CPI manual (par. 5.124). (Sometimes the transformed price relatives are multiplied by \(\max(p_1, p_0)^u\), for some \(0 \le u \le 1\), so that products with a larger price get flagged as outliers (par. 5.128).)

References

Hutton, H. (2008). Dynamic outlier detection in price index surveys. Proceedings of the Survey Methods Section: Statistical Society of Canada Annual Meeting.

IMF, ILO, Eurostat, UNECE, OECD, and World Bank. (2020). Consumer Price Index Manual: Concepts and Methods. International Monetary Fund.

Rais, S. (2008). Outlier detection for the Consumer Price Index. Proceedings of the Survey Methods Section: Statistical Society of Canada Annual Meeting.

Examples

x <- c(1, 10, 15, 100)

outliers(x, upper = 2.5, method = "quartile")
#> [1] FALSE FALSE FALSE  TRUE

# Always identifies fewer outliers than above.
outliers(x, upper = 2.5, method = "resistant-fences")
#> [1] FALSE FALSE FALSE FALSE