The Effects of Rounding on the Consumer Price Index

The Bureau of Labor Statistics rounds the Consumer Price Index (CPI) to a single decimal place before releasing it, and the published CPI inflation series is calculated from those rounded index values. While rounding has only a relatively small effect on the level of the CPI series at present, it can have a significant effct on CPI inflation, the monthly percent changes in the CPI. This paper estimates the impact of rounding error on the published CPI inflation for both contemporaneous and historical data. Using an unrounded CPI series from January 1986 to July 2005 as a benchmark, I find that published CPI inflation differs from its full-precision counterpart approximately 25% of the time, and that reporting the CPI levels to three decimal places would reduce these discrepancies to under 0.5%. Further, the variance introduced by rounding error is large when compared to the sampling variation in CPI inflation. I find that the BLS could reduce total CPI inflation error variance by 42% by simply reporting more digits in the CPI index, resulting in a significantly more accurate reflection of monthly inflation. In order to extend these results to the CPI historical series, I derive the distribution of the rounding error component of inflation. From this analysis, it is possible to estimate the probability of large rounding errors for a given CPI level and rounding precision. Three regimes emerge. Before the 1970’s inflation, discrepancies due to rounding were both frequent and frequently large relative to the underlying inflation rate. During the inflationary period of the mid-1970’s to mid-1980’s, both the probability and relative magnitude of discrepancies decrease dramatically. Finally, the last twenty years are characterized by a slowly falling probability of any rounding-induced error, but a roughly constant probability of an error of a given size.Consumer Price Index, Variance, Rounding, Inflation

Unbiased Rounding for HUB Floating-point Addition

Copyright (c) 2018 IEEE doi:10.1109/TC.2018.2807429Half-Unit-Biased (HUB) is an emerging format based on shifting the represented numbers by half Unit in the Last Place. This format simplifies two’s complement and roundto- nearest operations by preventing any carry propagation. This saves power consumption, time and area. Taking into account that the IEEE floating-point standard uses an unbiased rounding as the default mode, this feature is also desirable for HUB approaches. In this paper, we study the unbiased rounding for HUB floating-point addition in both as standalone operation and within FMA. We show two different alternatives to eliminate the bias when rounding the sum results, either partially or totally. We also present an error analysis and the implementation results of the proposed architectures to help the designers to decide what their best option are.TIN2013-42253-P, TIN2016-80920-R, JA2012P12-TIC-169

A COMPARISON OF PRE-ROUNDING AND POST-ROUNDING FLOATING POINT DIVIDE ORDERS

This paper addresses the advantages and disadvantages of pre-rounding vs. post-rounding in a floating-point register. In this day when high speed computers with long registers are the norm, one may think that a rounding scheme is nearly irrelevant, because rounding cannot improve over the addition of a single bit of increased accuracy. However, the scheme is important precisely because of the speed of computation, and the extended computations that speed allows. The propagation of error then becomes a significant issue, and the scheme of rounding is the starting point for making any such subsequent analysis

A note on the acceptability of regression solutions: another look at computational accuracy.

This note examines the experiment performed by Beaton, Rubin, and Barone (1976) to study the effect of rounding errors in published figures, when these data are used in regression analysis. The experiment could be vitiated by the fact that the error introduced in the trend variable is by no means trivial when one measures the data in deviation from the mean. For this reason, the results presented in Beaton, Rubin, and Barone (1976) do not contain enough evidence to suggest "that it is extremely unlikely that the unperturbed solution" (p. 161) of the Longley model is the correct one.Regression analysis; Rounding adjustments;

Measurement rounding errors in an assessment model of project led Engineering Education

This paper analyzes the rounding errors that occur in the assessment of an interdisciplinary Project-Led Education (PLE) process implemented in the Integrated Master degree on Industrial Management and Engineering (IME) at University of Minho. PLE is an innovative educational methodology which makes use of active learning, promoting higher levels of motivation and students’ autonomy. The assessment model is based on multiple evaluation components with different weights. Each component can be evaluated by several teachers involved in different Project Supporting Courses (PSC). This model can be affected by different types of errors, namely: (1) rounding errors, and (2) non-uniform criteria of rounding the grades. A rigorous analysis of the assessment model was made and the rounding errors involved on each project component were characterized and measured. This resulted in a global maximum error of 0.308 on the individual student project grade, in a 0 to 100 scale. This analysis intended to improve not only the reliability of the assessment results, but also teachers’ awareness of this problem. Recommendations are also made in order to improve the assessment model and reduce the rounding errors as much as possible

Project-led Engineering Education : assessment model and rounding errors analysis

This paper presents an analysis of the assessment model used in an interdisciplinary Project-Led Education (PLE) process implemented in the Integrated Master Course on Industrial Management and Engineering (IME) at University of Minho. PLE is an innovative educational methodology which makes use of active learning, promoting higher levels of motivation and students’ autonomy. The assessment model is based on multiple evaluation components with different weights. Each component can be evaluated by several teachers involved in different Project Supporting Courses (PSC). This model can be affected by different types of errors, namely: (1) rounding errors, and (2) non-uniform criteria of rounding the grades. A rigorous analysis of the assessment model was made and the rounding errors involved on each project component characterised. This resulted in a global maximum error of 0.308 on the individual student project grade, in a 0 to 100 scale. This analysis intended to improve not only the reliability of the assessment results, but also teachers’ awareness to this problem. Recommendations are also made in order to improve the assessment model and reduce the rounding errors as much as possible

A radix-independent error analysis of the Cornea-Harrison-Tang method

International audienceAssuming floating-point arithmetic with a fused multiply-add operation and rounding to nearest, the Cornea-Harrison-Tang method aims to evaluate expressions of the form $ab+cd$with high relative accuracy. In this paper we provide a rounding error analysis of this method,which unlike previous studiesis not restricted to binary floating-point arithmetic but holds for any radix $\beta$.We show first that an asymptotically optimal bound on the relative error of this method is$2u + O(u^2)$, where $u= \frac{1}{2}\beta^{1-p}$ is the unit roundoff in radix $\beta$ and precision $p$.Then we show that the possibility of removing the $O(u^2)$ term from this bound is governed bythe radix parity andthe tie-breaking strategy used for rounding: if $\beta$ is odd or rounding is \emph{to nearest even}, then the simpler bound $2u$ is obtained,while if $\beta$ is even and rounding is \emph{to nearest away}, then there exist floating-point inputs $a,b,c,d$ that lead to a relative error larger than $2u + \frac{2}{\beta} u^2 - 4u^3$.All these results hold provided underflows and overflows do not occurand under some mild assumptions on $p$ satisfied by IEEE 754-2008 formats

Oscillating behaviour of the spectrum for a plasmonic problem in a domain with a rounded corner

We investigate the eigenvalue problem $-\text{div}(\sigma \nabla u) = \lambda u\ (\mathscr{P})$ in a 2D domain $\Omega$ divided into two regions $\Omega_{\pm}$. We are interested in situations where $\sigma$ takes positive values on $\Omega_{+}$ and negative ones on $\Omega_{-}$. Such problems appear in time harmonic electromagnetics in the modeling of plasmonic technologies. In a recent work [15], we highlighted an unusual instability phenomenon for the source term problem associated with $(\mathscr{P})$: for certain configurations, when the interface between the subdomains $\Omega_{\pm}$ presents a rounded corner, the solution may depend critically on the value of the rounding parameter. In the present article, we explain this property studying the eigenvalue problem $(\mathscr{P})$. We provide an asymptotic expansion of the eigenvalues and prove error estimates. We establish an oscillatory behaviour of the eigenvalues as the rounding parameter of the corner tends to zero. We end the paper illustrating this phenomenon with numerical experiments.Comment: Mathematical Modelling and Numerical Analysis (ESAIM: M2AN), 09/12/2016. arXiv admin note: text overlap with arXiv:1304.478