F1000Research F1000Research 2046-1402 F1000 Research Limited London, UK 10.12688/f1000research.172913.2 Research Article Articles Robust Functional Data Analysis of Dollar Exchange Rates Using Functional Median Polish

[version 2; peer review: 2 not approved]

 M. Sabri Abdulsalam Conceptualization Investigation Methodology Supervision Validation Writing – Review & Editing https://orcid.org/0009-0008-8296-6672 a 1 N. Husain Qasim Data Curation Formal Analysis Software Visualization Writing – Original Draft Preparation Writing – Review & Editing https://orcid.org/0000-0002-9014-3261 2 Department of Mathematics, College of Computer Science and Mathematics, Tikrit University, Tikrit, Saladin Governorate, 34001, Iraq Department of Mathematics, Tikrit University College of Education for Pure Sciences, Tikrit, Saladin Governorate, 34001, Iraq am230036pcm@st.tu.edu.iq

No competing interests were disclosed.

 26 5 2026 2026 15 80 15 5 2026 Copyright: © 2026 M. Sabri A and N. Husain Q 2026 This is an open access article distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The study utilizes a robust functional exploratory framework to look into the monthly US dollar exchange-rate series over its 34 annual history. The 34 × 12 year-by-month observations were then expounded as smooth functional curves in the monthly domain, with the help of a seven-function Fourier basis expansion. Next, one-way and two-way Functional Median Polish (FMP) has been employed to decompose the data into an overall functional center, row effects, column effects, and residual functions. The extracted components were analyzed in terms of their respective p-values using Functional ANOVA.

In the regression model, the center function estimated across months was stable, varying from approximately 0.64 to 0.65 at a functional scale. Overall the row and column effects were not significant and after smoothing, FANOVA showed no significant annual or monthly effects. Residual diagnostics indicated that the underlying residual structure remained within a relatively narrow range with only minor extreme departures from normality at the tails reflecting short term shocks not systematic seasonality. Taken together, our interpretation of the results demonstrates that when structural interpretation and not forecasting is its desired goal, Functional Median Polish results provide a useful and robust decomposition of the underlying economic time-series data.

 Functional data analysis Functional median polish fourier basis Functional ANOVA R programming The author(s) declared that no grants were involved in supporting this work. Amendments from Version 1

We sincerely appreciate your interest in our manuscript entitled “Robust Functional Data Analysis of Dollar Exchange Rates Using Functional Median Polish.” In preparing the revised version of this study, we made a substantial effort to improve the clarity, structure, and methodological presentation of the paper so that its contribution can be communicated more effectively to the scholarly community. In the revised manuscript, we clarified the main objective of the study and emphasized that the work is intended as a robust exploratory functional analysis of exchange-rate data rather than a purely predictive exercise. We improved the presentation of the methodological framework by providing a clearer description of the functional representation of the monthly exchange-rate series, the role of the Fourier basis expansion, and the application of one-way and two-way Functional Median Polish procedures. We also strengthened the interpretation of the empirical findings. The revised text now explains more clearly that the estimated functional center captures the dominant pattern of the exchange-rate series, while the annual and monthly effects remain limited and statistically weak. In addition, the discussion of FANOVA results and residual diagnostics has been refined to make the analytical conclusions more transparent and easier to follow. Finally, we improved the conclusion by stating the practical implications of the study more explicitly, while also acknowledging its main limitations and suggesting several directions for future research. We hope that these revisions have enhanced the quality and readability of the manuscript and that the study will provide useful value for researchers interested in robust functional data analysis and exchange-rate modeling.

 Introduction

Data analysis based mainly on exploratory approach have used robust methods for finding hidden patterns in complex data because parametric techniques do not suffice because of outliers or nonlinear structure. Median polish is a powerful exploratory method for the decomposition of 2-way tables to additive components that was developed by Tukey (1977). 1 The entire effect, row effects, column effects, and residuals are estimated using medians rather than means, which decreases the sensitivity to extreme observations.

In its classical usage the method corresponds to an additive model where each cell is defined as the total of a grand effect, a row effect, a column effect and a residual term. It fits a model using an additive approach: z ik = μ + α i + β k + e ik , i = 1 , … , r , k = 1 , … , c ,

In model (1), μ denotes the overall effect, α i denotes the effect of row i , β k denotes the effect of column k , and e ik denotes the residual term. For identifiability under the robust median-based formulation, the effects are defined subject to the constraints media n i ( α i ) = 0 , media n k ( β k ) = 0 and media n ik ( e ik ) = 0 . The model is estimated iteratively by subtracting row and column medians until the effects stabilize. This iterative median-based updating yields a robust additive decomposition that is less sensitive to anomalous observations than mean-based alternatives. Regarding this respect, Velleman and Hoaglin (1981) bridged theoretical concepts in exploratory data analysis with concrete computer-based applications. 2 Emerson and Hoaglin (1983) along with Hoaglin, Mosteller, and Tukey (1983) afterwards have shown practical examples of median polish to its practical advantage. 3 , 4 Fink (1988) introduced significantly stronger statistical treatment of this approach based on its limitations and convergence properties to further enhance the theoretical grounds of this powerful exploratory methodology. 5

Functional data analysis (FDA) is an intuitive approach to the analysis of data seen along an analytic continuum (over time, space, or frequency). Instead of the repeated measurements from the FDA data being treated as simply isolated scalar observations, FDA portrays them as “smooth functional objects” that can be used to study continuous processes more effectively. Ramsay and Silverman (2005) laid down a basis for FDA, where the curves are taken as the basic observational units. 6 This development in the FDA is particularly appropriate for economic, environmental, and healthcare information. Furthermore, functional depth evaluations—such as those by López-Pintado and Romo (2009)—helped improve the robustness of functional ordering and further facilitate identification centrality, as well as points at which outliers could be considered outlier on curves given samples. 7 Sun and Genton (2012) extended Tukey’s median polish into the functional setting and also came up with the Functional Median Polish (FMP) method. 8 This framework replaces every cell of a two-way table by a function, so that the additive decomposition is then done pointwise, based on functional medians. This makes FMP particularly applicable for time series data where seasonality and interannual structure are coupled. Similar advancements in Functional ANOVA (FANOVA), i.e., Bayesian formulations by Kaufman and Sain 9 and later also by Sain 10 extended the range of functional inference, integrating uncertainty into functional comparisons. Furthermore, modified functional boxplots were suggested for spatiotemporal visualization and outlier detection. 11

Apart from methodological improvement, a few applied studies have demonstrated the flexibility of median polish and other related robust methods. As an example, Fitrianto et al. (2014) used median polish to analyze educational data. 12 Ajoge et al. (2016) extended the Tukey’s original framework from Tukey by introducing covariates and making it applicable for both before-and-after studies. 13 Hussein et al. (2017), expanded Tukey’s resistant line framework with a moving average method for smoothing and robustness enhancement. 14 More recently, Jimenez et al. 15 – 18 showcased the applicability of FDA- and FMP-related methods in real-world scenarios with subnational mortality and density-valued economic scales, suggesting their applicability in current high-dimensional environments.

Traditional decomposition, smoothing, or mean-based functional algorithms constitute the majority of the available studies on monthly exchange-rate series. While these approaches are relevant in all these settings, they can be sensitive to anomalies and, consequently, do not have the strength in being a unified framework, being made up of smooth functional representation and the robust additive decomposition and inferential analysis by the Functional ANOVA (FANOVA). Moreover, practical guidance remains scarce on when Functional Median Polish (FMP) represents a suitable exploratory description. Therefore this work will contribute to developing a robust functional framework to analyze monthly exchange rate series, based on Fourier-based functional representing and Functional Median Polish. The central issue is structural decomposition rather than forecasting. More specifically, the work evaluates whether a proper functional additive representation can offer interpretable understanding of yearly and monthly effects that fail to be modeled by classical time series. Towards the same goal one-way and two-way Functional Median Polish is used to determine the complete center, row effects, column effects and residual functions, and the final elements are tested by FANOVA. A practical algorithm for median polish.

The median polish technique for univariate data operates on data arranged in a two-way table by sweeping out row and column medians. The output consists of the estimates of model (1). Each cell in the table is subject to the same overall effect μ ̂ , together with row effects α i ̂ and column effects β k ̂ .

The estimation of the grand, row, and column effects using medians is performed through an iterative procedure applied to the data table, which undergoes a series of row and column sweeping operations. The process continues until there are no further changes in its effects, or until the changes are small enough. The overall impact of a row or column sweep may depend on which one is applied first. Nevertheless, there is usually no compelling reason to prefer one order over the other, and the difference is typically not substantial in most practical applications. In what follows, the functional median polish procedure begins with the row sweep.

 A practical algorithm for median polish

The median polish technique for univariate data works by operating on data in two-way table by sweeping out row and column medians. A list of the model’s (1) estimates makes up the outcome. The table’s cells are all subject to the same overall effect μ ̂ , plus row effects α i ̂ and column effects β k ̂ .

The approach for determining grand, row, and column effects using medians is iterative on data table, which undergoes a series of row and column sweeping operations. The process continues until there are no further changes in its effects, or until the changes are small enough. The overall impact of a row or column sweep might depend on which one begins first. Nevertheless, there is frequently no rhyme or reason to the order, and the distinction between the two options is typically not significant for most practical applications. We discuss functional median polish as follows: the method begins with the row sweep.

 One-way functional median polish

Assume that the interest lies in examining functional data across the levels of one categorical factor, referred to here as the functional row effect. The one-way functional median polish model is written as follows: z ik ( x ) = μ ( x ) + α i ( x ) + e ik ( x ) , i = 1 , … , r , k = 1 , … , m i ,

Where, z ik ( x ) : observed function for year i , μ ( x ) : functional overall effect, α i ( x ) : functional row effect, e ik ( x ) : functional residual and x ∈ I ∈ ℝ , r ≥ 2 the number of rows and m i is the number ∀ x ∈ I , that median i { α i ( x ) } = 0 and median i { e ik ( x ) } = 0 for all k.

To fit this model by medians, we suggest the following algorithm:

Write the functional value on the row’s side after calculating each row’s functional median. From each function in that specific row, subtract the row functional median.

Determine the functional median of all the available data, then note the result as the functional grand effect. Record the data as the functional row impact after deducting this functional grand effect from each row’s functional median.

Until the row functional medians remain unchanged, repeat steps 1-2 and add the new functional grand effect and functional row effect to the existing ones at each iteration.

Two-way functional median polish

When functional data are observed for each combination of two categorical factors, the effects of interest are the functional row effect and the functional column effect. The observations may then be decomposed as follows: z ikj ( x ) = μ ( x ) + α i ( x ) + β k ( x ) + e ikj ( x ) , i = 1 , … , r , k = 1 , … , c , j = 1 , … , m ik

where β k ( x ) denotes the functional column effect, subject to the same constraints as in Equation (2). In the following algorithm, FM denotes the functional median computed pointwise across functions. To fit this model by medians, we use the following algorithm: 1. Determine each row’s FM and record the functional value adjacent to the row. From each function in that row, subtract the row functional median. 2. Determine the FM of every piece of information that is accessible, then save the outcome as the functional grand effect. 3. After subtracting this functional overall effect from each of the row FMs, record the results as the functional row effects. 4. Determine the FM for every column and record the result beneath it. For each function in that specific column, subtract the column FM. Determine the FM of every piece of data that is currently available, then store the outcome as the new functional grand effect. Record the results as the functional column effects after deducting this functional grand impact from each of the column functional medians. 5. At each iteration, repeat steps 1-4 and add the new functional overall effect, functional row effect, and functional column effect to the existing ones until the row FM’s remain unchanged.

 Methodology

We analyzed a 34-year monthly series of U.S. dollar exchange rates comprising 408 observations, arranged as a 34 × 12 year-by-month matrix. 14

The series was imported from a plain text file and reshaped in R so that each row represented one year and each column represented one month. The matrix structure acted as the starting point for both the exploratory analysis and subsequent functional representation. The monthly observations were mapped to the domain t ∈ [1,12] and a Fourier basis with nbasis = 7 was employed to make the functional form of the data possible. The choice was assessed through a sensitivity analysis using K = 5, 7, and 9 basis functions. For each specification, the generalized cross-validation (GCV) criterion and the reconstruction mean squared error were verified. K = 7 was chosen for this purpose because it attained a trade-off between smoothness and accuracy of the reconstruction, avoiding excessive smoothing relative to K = 5 and possible overfitting relative to K = 9. Then, the functional objects were generated by the fda and fda.usc packages. The smoothed curves, in particular, were initially modeled as fd objects and then fdata objects to be able to perform Functional Median Polish. Other plots and diagnostic displays were generated using ggplot2, whilst tidyr was employed for reshaping and preparation. For the one-way and two-way Functional Median Polish algorithms and for the following FANOVA, these functional features were the input. How the actual data preprocessing and analysis steps work is as follows:

The values were read from a text file into an m-vector and then converted into a Mydata1 matrix of dimension 34 × 12, with each row representing a year and each column representing a month. Boxplots and line plots were used to inspect the raw structure of the data before smoothing

The Fourier basis was created on the [1,12] domain, with nbasis = 7, and the data were transformed into functional objects using Data 2fd, then re-evaluated into an fdata matrix for robustness methods.

Functional visual diagnostics were used to highlight the center, dispersion, and possible outlying behavior in order to assess the appropriateness of the transformation and smoothing before applying the FMP procedure.

The one-way FMP was performed to extract the center and row effect, followed by two-way FMP to incorporate the column effect. FANOVA was then applied to the extracted components, and the residual structure was examined through visual diagnostics and classical one-way ANOVA on the resulting effect matrices.

The complete R code used in the analysis is provided in Appendix A for reproducibility.

 Results and discussion A) Preliminary exploration of the original data

The boxplot was generated in Figure 10 to visualize the original data. The plot shows original exchange-rate observations divided over the months of the year. Data is similar: the central values look fairly similar with very little variability from month to month. Some infrequent deviations do occur in select years, but they do not change the data structure significantly. Extreme outliers are not dominant in this dataset so a robust functional framework is being used for stability over contamination.

The monthly curves of data are presented in Figure 11. The figure shows the monthly trends over the exchange rates series over the years under study. Most of the slopes are stable with respect to months, indicating the uniformity of monthly behavior. There are some limited deviations in some few months but they don’t really change much. This indicates that the strong seasonality is not visually dominating in the raw measurements.

 B) Functional representation and exploratory diagnostics

In the preliminary investigations and before loading the Functional Median Polish (FMP) algorithm, we performed a series of functional diagnostics (center, spread, and possible irregular graphs). These diagnostics serve as a baseline for evaluating the suitability of the functional transformation and the smoothing procedure before data is decomposed into center, row effects, column effects, and residuals. Functional boxplot of the smoothed exchange rate curves is provided in Figure 1.

Boxplot of functional data.

The red median curve varies about 0.64 to 0.65, suggesting that the monthly characteristics of exchange rate series are generally stable. The regions shaded above show the spread of central functions and moderate interannual variability in the monthly domain. The row effects’ heat map is depicted in Figure 2.

Heat map of the row effect.

The heat map shows different years, some seem lighter and some darker, with positive and negative deviations from the center. It suggests an annual gap or an existence of differences but one that is quite limited.

The heatmaps for column effects are shown in Figure 3.

Heat map of the Column effect.

Almost the uniform color pattern and the color distribution in the figure confirm the fact the column effects are nearly invariant to all months. This means that the monthly effect has constant across the domain as a whole and adds no further effect to the overall structure.

Original, center, seasonally adjusted curves are depicted in Figure 4.

Original, center and seasonally adjusted curves.

The center and original mean curves are close to each other in the monthly domain, indicating that the overall average pattern is stable. By contrast, the seasonally normalized curve shows some relatively higher level and also retains the very similar month over month shape. That means the adjustment adjusts the level of the series and at the same time the general monthly structure stays somewhat stable.

Functional depth values obtained from functional median polish.
[1] 0.05882353 0.13845502 0.35680627 0.51412788 0.41480030 0.27975150 0.35595913
[8] 0.49916168 0.51444556 0.51282187 0.50569175 0.48419548 0.34395793 0.35878294
[15] 0.51105699 0.5303648 0.49009019 0.51493973 0.3592771 0.33188613 0.14717354
[22] 0.07802545 0.15448015 0.4778419 0.50975098 0.43361395 0.52708212 0.36097139
[29] 0.38366778 0.46961755 0.40103422 0.23231147 0.23209968 0.18360071
Outpoint 1
Med curve 16
Residuals mean by month. Residuals distribution.

We also illustrate the functional boxplot from the transformed data in Figure 12. This figure was generated with fda package and shows plot of transformed functional observations. The black line points to the median, the shaded line the interquartile range and the boundaries the border which represent limits to locate the possible anomalously shaped lines. Some curves approach these limits, which imply mild departures from normal rather than extreme outlying behavior.

 C) One-way Functional Median Polish results

Table 2 presents the central function values obtained from the one-way Functional Median Polish, while Figure 13 shows the one-way center and row effects. The center curve remains relatively stable across the monthly domain, with values close to 0.64–0.65. The row effects fluctuate around zero with limited variation, indicating that differences between years are generally weak. This supports the view that the overall functional center is the dominant component in the one-way decomposition.

 D) Two-way Functional Median Polish results

Figure 7 presents the Q-Q plot of the residuals, Figure 8 shows the row effects with the interquartile band, and Figure 9 displays the column effects across months.

QQ-plot of the Residuals. Row effects with interquartile (IQR) band. Column effects by months.

We demonstrate the Q-Q plot of the residuals for Figure 7. The plotted points deviate from an ideal normal pattern, particularly in the tails, indicating that the tails of the residual distribution may be heavier than would be anticipated by the normal assumption.

The row effects with the IQR band are presented in Figure 8. The median line of the row-effect curve is nearly zero in most months, and the surrounding band indicates very little interannual variability. Small deviations occur in some months, but the overall trend implies weak annual effects.

Figure 9 shows the column effects by month. The effects of the column remain nearly constant across the range of the month, with only minimal variation. This matches the heat diagram of column effects and suggests that the monthly component adds little further explanation. Figure 14 shows the mean functional components (center, row, column, and residuals) of the two-way Functional Median Polish.

The center and column mean curves stay nearly constant over months, and the row mean remains near 0, representing fairly weak annual spread for the variable. By comparison, the residual mean stays consistently negative and varies modestly between months. The residual structure of the residual component may also have some more variation; however, this behaviour indicates the stability of the principal structural elements.

The Figure 10 explain the Boxplot of Original Data.

Boxplot of original data. Monthly curves of the Original data.

The Figure 12 explain Boxplot of Functional Data.

Functional Boxplot of the transformed data.

The Table 2 shown Central Function Values Obtained from One-way Functional Median Polish, while Figure 13 explain Center One Way and Row Effects One Way.

Central function values obtained from one-way functional median polish.
1 2 3 4 5 6
[1,] 0.6530329 0.6501276 0.657014 0.6482106 0.6382607 0.6431659
7 8 9 10 11 12
[1,] 0.642981 0.649499 0.6450545 0.6488726 0.6416805 0.6530329
One way Center and row effects.

The figure shows the results of the One-Way Functional Median Polish (FMP). The red curve (left) represents the monthly central values, which remain stable around 0.64–0.65. The right-hand figure shows the effects of years, which remain close to zero with some limited deviations, indicating that the differences between years are weak, while the central curve reflects the overall structure of the series.

The Figure 14 explain Mean Functional Components (Center, Row, Column, and Residuals) from Two-way Functional Median Polish.

Mean functional components (center, row, column, and residuals) from two-way functional median polish.

The figure shows the mean of the principal components resulting from the Two-Way Functional Median Polish. The central curve appears to be constant across months, while the row and column effects are close to zero, reflecting weak annual and seasonal variations. The figure shows the mean of the principal components resulting from the two-way Functional Median Polish. The center curve remains stable across months, while the row and column effects stay close to zero, indicating weak annual and monthly variation. The residual mean shows only modest fluctuation across the monthly domain, which suggests that the unexplained component is limited in structure.

The Figure 15 explain Residuals with IQR Band.

Residuals pattern across months.

This figure displays the residual pattern across months together with its interquartile spread. The residual variation remains limited, although modest fluctuations appear from month to month. No clear recurring seasonal pattern is evident in the residual structure, which suggests that most of the systematic variation has already been absorbed by the extracted center, row, and column components. We notice that the residuals are concentrated in the negative range (around zero with slight dispersion), with significant fluctuations between months. Some months (such as July and August) show residuals closer to zero (less negative), while in other months, such as February, June, and October, the residuals are more negative. This suggests that the model did not fully explain all monthly variations, and that some unexplained variances remain, reflecting the possibility of seasonal components or nonlinear effects that were not captured.

 E) FANOVA results

While Figure 16 shows the ANOVA p-values for Functional Effects.

ANOVA p-values for the extracted functional effects.

Figure 16 shows that all ANOVA p-values are above the 0.05 significance level, indicating that neither the row effects nor the column effects are statistically significant. The column effect in the two-way model is around 0.45, while the row effects in both the one-way and two-way models are close to 1.00, which further confirms the absence of meaningful functional effects. These results suggest that the series does not exhibit significant annual or monthly structural variation.

 F) Overall comparison of the extracted components

While Figure 17 shows Center vs Row/Column/Residual Means.

Center compared with the row, column, and residual means.

Figure 17 shows the center in comparison with the row, column, and residual means. The figure confirms that the center and column mean curves remain stable at a similar level across months, while the row mean remains close to zero. In contrast, the residual mean is consistently negative across the monthly domain. This indicates that the dominant structural component is the center, whereas the annual component is weak and part of the remaining variation is retained in the residual structure.

Overall, the figures indicate that the center represents the dominant structural component and remains relatively stable across the monthly domain. The annual effects (row effects) are weak and do not reflect statistically significant differences between years. The monthly effects (column effects) remain nearly constant and contribute little additional variation to the overall structure. The residual component retains part of the unexplained variation and exhibits some departure from normality. Taken together, these findings suggest that the exchange-rate series is characterized more by a stable functional center than by strong annual or monthly systematic effects.

 Conclusion

The Fourier functional representation along with one-way and two-way Functional Median Polish (FMP) was reliable in giving a stable estimate of the overall functional center. However, both annual and monthly effects were not statistically significant using Functional ANOVA (FANOVA), suggesting a weak deterministic seasonal structure in the exchange rate series studied. This means that the main feature of the data is the overall functional level rather than systematic year- or month-specific effects.

In our study we make a contribution to the literature that the smooth functional representation on the monthly domain is combined with a robust one-way and two-way FMP decomposition along with the evaluation by FANOVA and standard diagnostic methods. This is an interpretable and robust exploratory framework that is appropriate for the analysis of monthly economic time-series data if structural component description is desired rather than model development. The primary limitations of this study include nominal exchange rate values without adjustment for inflation, missing institutional or explanatory covariates, univariate analysis, and a small number of observations per yearly curve.

The functional approximation is also dependent on the chosen basis and level of smoothing and therefore, it may affect the obtained components. Institutional and financial variables may be included in future work, the analysis might be extended using Functional Principal Component Analysis (FPCA) or multivariate functional models between official and parallel exchange markets, uncertainty could be quantified through bootstrapping procedures and out-of-sample predictive performance can be assessed.

Overall, the findings highlight the usefulness of robust functional decomposition methods for analyzing economic time-series data characterized by moderate variability and weak seasonal structure.

 Data availability

Source data: The monthly U.S. dollar exchange rate dataset analyzed in this study consists of 34 years of observations (408 values arranged in a 34×12 year-by-month matrix). These data were not generated by the authors and were obtained from previously published work https://www.rbnz.govt.nz/statistics/economic-indicators/b1. The authors collected, organized, and reformatted the data into a unified year–month (34 × 12) matrix to make them suitable for analysis. publicly accessible through the following link: https://github.com/Abdulsalamiraq/U.S._dollar_exchange_rates , or https://zenodo.org/records/17803613, or DOI: https://doi.org/10.5281/zenodo.20094143. 14

R code used in the analysis

The R code used for the practical implementation and analysis has been provided as supplementary material and is publicly available at the following Zenodo repository: https://doi.org/10.5281/zenodo.20094143. 14

Data are available under the terms of the Creative Commons Attribution 4.0 International.

 Acknowledgements

Not applicable.

 References Tukey JW : Exploratory data analysis. London, UK: Pearson;1977. Velleman PF Hoaglin DC : Applications, basics, and computing of exploratory data analysis. Boston, Mass: Duxbury Press;1981. Reference Source Emerson JD Hoaglin DC : Analysis of two-way tables by medians. Understanding robust and exploratory data analysis. New York, NY, USA: Wiley;1983; pp.165–210. Hoaglin DC Mosteller F Tukey JW : Understanding robust and exploratory data analysis. New York, NY: John Wiley & Sons;2000. Fink A : How to polish off median polish. SIAM J. Sci. Stat. Comput. 1988;9(5):932–940. 10.1137/0909064 Ramsay JO Silverman BW : Functional data analysis. New York, NY: Springer;2005. 10.1007/978-1-4757-7107-7_6 López-Pintado S Romo J : On the concept of depth for functional data. J. Am. Stat. Assoc. 2009;104(486):718–734. 10.1198/jasa.2009.0108 Sun Y Genton MG : Functional median polish. J. Agric. Biol. Environ. Stat. 2012;17(3):354–376. 10.1007/s13253-012-0096-8 Kaufman CG Sain SR : Bayesian functional {ANOVA} modeling using Gaussian process prior distributions. Bayesian Anal. 2010;5(1):123–149. 10.1214/10-BA505 Sain S : Functional ANOVA and regional climate experiments: A statistical analysis. EGU General Assembly Conference Abstracts. 2010;22(6):700–711. 10.1002/env.1068 Reference Source Sun Y Genton MG : Adjusted functional boxplots for spatio-temporal data visualization and outlier detection. Environmetrics. 2012;23(1):54–64. 10.1002/env.1136 Fitrianto A Wijayanto H Rana M : Median Polish for final Grades of MTH3000 and MTH 4000-Level courses. J. Appl. Math. Sci. 2014;8:6295–6302. 10.12988/ams.2014.48608 Ajoge I Adam M Anwar F : Median Polish with Covariate on Before and After Data. IOSR J. Math. 2016;12(3):64–73. Sabri AM Husain QN : U.S. Dollar Exchange Rates.[Data set]. Zenodo. 2025. 10.5281/zenodo.20094143 Jiménez-Varón CF Sun Y Shang HL : Forecasting high-dimensional functional time series: Application to sub-national age-specific mortality. J. Comput. Graph. Stat. 2024;33(4):1160–1174. 10.1080/10618600.2024.2319166 Jiménez-Varón CF Sun Y Shang HL : Forecasting Density-Valued Functional Panel Data. Aust. N. Z. J. Stat. 2025;67(3):401–415. 10.1111/anzs.70013 Abdullah HH Khalaf NS Noori NA : Comparison of non-linear time series models (Beta-t-EGARCH and NARMAX models) with Radial Basis Function Neural Network usingReal Data. Iraqi J. Comput. Sci. Math. 2024;5(3):38. 10.52866/ijcsm.2024.05.03.003 Noori NA : Exploring the Properties, Simulation, and Applications of the Odd Burr XII Gompertz Distribution. Adv. Theory Nonlinear Anal. Appl. 2023;7(4):60–75. 10.17762/atnaa.v7.i4.283 10.5256/f1000research.190678.r453588 Reviewer response for version 1 Anton Cristina 1 Referee MacEwan University, Edmonton, Canada

Competing interests: No competing interests were disclosed.

 27 2 2026 Copyright: © 2026 Anton C 2026 This is an open access peer review report distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. recommendation reject

The paper presents a functional data version of the Median Polish method and its application to the 34-year monthly time series of the US dollar exchange rate. There is a need for robust methods for functional data, and the ability of dealing with outliers is mentioned as an advantage of this method. However, there are several issues that should be addressed:

There are very few observations, and I am not convinced the proposed approach is better than one based on the classical ARIMA models or their seasonal extensions. No comparison with other methods was included.

To work with functional data, a Fourier basis of 7 functions is used. How was the number of basis functions chosen? Why Fourier and not B-splines?

Do classical time series methods find any outliers for these data?

Is this functional data approach better than applying the Median Polish method to the original data?

The presentation should be improved.

Although the paper is readable, the English should be revised substantially. There are run-away sentences and typos  (e.g. on page 3 the last sentence in the paragraph containing equation (1) doesn’t have a predicate; the last sentence of paragraph 5 on page 1 needs an explanation for  “Further work by them (2012)”, “Where” after equations (1) and (2) should be “where”; the sentence after equation (2) has some missing  part: “and m i is the number …” etc).

The “First”,  to “Seventh” should be replaced with subsections or something else for clarity.

The code should be included in an appendix.

Pictures 9 and 15 do not have the promised IQR band.

Is the work clearly and accurately presented and does it cite the current literature?

Partly

If applicable, is the statistical analysis and its interpretation appropriate?

Partly

Are all the source data underlying the results available to ensure full reproducibility?

Yes

Is the study design appropriate and is the work technically sound?

Partly

Are the conclusions drawn adequately supported by the results?

Partly

Are sufficient details of methods and analysis provided to allow replication by others?

Yes

Reviewer Expertise:

clustering of functional data

I confirm that I have read this submission and believe that I have an appropriate level of expertise to state that I do not consider it to be of an acceptable scientific standard, for reasons outlined above.

M. Sabri Abdulsalam Tikrit University, Tikrit, Saladin Governorate, Iraq

Competing interests: No competing interests were disclosed.

 13 3 2026

General comment

The writing needs substantial improvement; terminology is inconsistent; layout and integration of figures with the narrative are messy.

Response

We appreciate this assessment. The manuscript has been substantially revised for language, terminology, structure, and presentation. Definitions are now introduced more consistently, figure-related discussion has been reorganized, and the code has been moved to an appendix to improve readability.

Comment 1

It is not clear that outliers are a major concern. The boxplot of the original data does not show pronounced outlying behavior.

Response

We agree and have revised the manuscript accordingly. The revised version no longer claims strong or dominant outlying behavior. Instead, it states that the robust framework is motivated mainly by stability and resistant decomposition, rather than by severe contamination.

Comment 2

The notation “FM” is used but never defined.

Response

This has been corrected. The revised manuscript explicitly defines FM as the functional median, and the notation is now used consistently.

Comment 3

The manuscript does not justify why nbasis = 7 is appropriate.

Response

This point is now addressed directly in the revised methodology. The manuscript explains the rationale for the Fourier basis and includes an explicit statement justifying the choice of seven basis functions through sensitivity analysis.

Comment 4

The claim of extreme values in Figure 1 is not well supported.

Response

We agree and revised the corresponding text. The interpretation now refers to moderate variability and mild deviations, which is more consistent with the figure.

Comment 5

In Figure 4, the manuscript does not clearly state how the original, center, and seasonally adjusted curves are constructed.

Response

This section has been revised. The figure is now interpreted more carefully, and the explanation focuses on the relative behavior of the original, center, and adjusted curves without making unsupported claims.

Comment 6

The residuals are described as centered around zero, but the plotted residual means are uniformly negative.

Response

We thank the reviewer for identifying this inconsistency. The residual interpretation has been revised throughout the manuscript so that it now matches the figures. The revised text no longer states that the residual mean is centered around zero where the plotted pattern is clearly negative.

Comment 7

Figures 9 and 15 mention an IQR band that is not visible.

Response

This has been corrected. The corresponding figure titles, legends, and interpretations have been revised so that they accurately describe the displayed plots.

Final statement

In addition to the point-by-point revisions above, the abstract, conclusion, figure legends, appendix, and data-availability statement were all revised for clarity, precision, and consistency. We believe that the manuscript has been significantly improved in response to the reviewers’ comments and now presents the study in a clearer and more rigorous form.

We sincerely thank the Editor and the Reviewers again for their valuable comments and consideration.

Sincerely,

Abdulsalam M .Sabri

 10.5256/f1000research.190678.r453591 Reviewer response for version 1 Huang Kun 1 Referee Texas A&M University, College Station, Texas, USA

Competing interests: No competing interests were disclosed.

 12 2 2026 Copyright: © 2026 Huang K 2026 This is an open access peer review report distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. recommendation reject

This manuscript applies Functional Median Polish (FMP) to a 34-year monthly ``US dollar exchange rate" time series (408 values arranged as a $34 \times 12$ year-by-month table), after smoothing with a 7-function Fourier basis. The authors aim to decompose the functional data into a center, year (row) effects, month (column) effects, and residuals, and then test the significance of each component. The paper concludes that FMP is a strong working framework that avoids outlier effects significantly and is thus appropriate in the analysis of economic time series.

My overall impression after reading this manuscript is that the writing needs substantial improvement. Many sentences are grammatically incorrect or incomplete. For example, in the second paragraph of the Introduction, ``Median polish, which introduced by" should read ``Median polish, which was introduced by". In addition, the last sentence in the second paragraph is missing a clear subject (``By iteratively subtracting medians across rows and columns, thus achieving robustness against anomalous values"). The manuscript also shows repeated and inconsistent use of terminology: FDA is defined twice (paragraph 4), and FMP is used multiple times before being introduced (paragraph 5). Several in-text citations are not in an appropriate format (e.g., ``Further work by them (2012)" in paragraph 5). Overall, the presentation reads as sloppy and difficult to follow, and several passages are hard to understand without re-reading. Besides, the layout and integration of figures with the narrative are also messy: figures, code blocks, and interpretations are interleaved in a way that disrupts the flow and makes the results difficult to assess.

Given the overall quality of the writing, it is difficult to convince me that the results presented in this article are sound. Below I list several additional comments:

1) Median polish is a robust tool designed to mitigate the influence of outliers. However, for the dataset analyzed here, it is not clear that outliers are a major concern (for instance, the boxplot of the original data in Figure 10 does not show pronounced outlying behavior). Moreover, the paper should include a direct comparison between FMP and more conventional alternatives to justify why FMP is particularly appropriate for this dataset and to quantify what is gained by using a robust approach.

2) In the two-way functional median polish algorithm, the authors use the notation ``FM" (Steps 1–5), but they never define it. I assume it stands for functional median, yet this needs to be stated explicitly and consistently. This is another example of careless writing and makes the methodology harder to follow. Besides, I believe it will be clearer to describe algorithms in a more formal mathematical manner.

3) The authors use seven Fourier basis functions to map the original monthly series into a functional representation. However, the manuscript does not justify why nbasis = 7 is an appropriate choice. Relative to the small number of observed time points per curve (12 months), seven basis functions may risk overfitting or producing an artificially smooth ``functional" object. The authors should provide a sensitivity analysis to support this selection. More broadly, functional approximation accuracy typically improves with denser sampling along the domain; with only 12 observations per curve, the approximation error may not be negligible, and it is unclear whether the functional mapping meaningfully reduces noise rather than introducing modeling artifacts.

4) In Figure 1, the variation of the breadth of interannual variability also appears fairly stable across months. The author suggests that there may be ``extreme values", but this claim is not well supported by the figure alone. A more reliable approach would be to check the original observations directly to see if ``extreme values" exist for some months.

5) In Figure 4, the manuscript compares ``original", ``center", and ``seasonally adjusted" curves, but it does not clearly state how each of these curves is constructed. 

6) In Figures 5 and 6, the paper states that the residuals are ``centered around zero", but the plotted residual means are uniformly negative (roughly around $-0.65$ across months), so they are clearly not centered at 0. This is weird.

7) In Figures 9 and 15, the captions state that an IQR band is included, but I cannot see any band in either figure. 

Is the work clearly and accurately presented and does it cite the current literature?

No

If applicable, is the statistical analysis and its interpretation appropriate?

Partly

Are all the source data underlying the results available to ensure full reproducibility?

Yes

Is the study design appropriate and is the work technically sound?

No

Are the conclusions drawn adequately supported by the results?

No

Are sufficient details of methods and analysis provided to allow replication by others?

No

Reviewer Expertise:

Statistics

I confirm that I have read this submission and believe that I have an appropriate level of expertise to state that I do not consider it to be of an acceptable scientific standard, for reasons outlined above.

M. Sabri Abdulsalam Tikrit University, Tikrit, Saladin Governorate, Iraq

Competing interests: No competing interests were disclosed.

 13 3 2026

Dear Editor and Reviewers,

We sincerely thank you for the careful evaluation of our manuscript and for the constructive comments and suggestions. We greatly appreciate the time and effort devoted to the review process. The reviewers’ remarks have been extremely valuable and have led to substantial improvements in the manuscript.

In the revised version, we have carefully addressed the major concerns raised by the reviewers. The manuscript has been thoroughly edited for language, clarity, consistency of terminology, and overall presentation. The methodological description has been rewritten and clarified, the figure captions and textual interpretations have been revised to match the displayed results more accurately, the code has been relocated to an appendix, and the role of the proposed framework has been clarified as a robust exploratory structural decomposition approach rather than a forecasting alternative to classical ARIMA-type models. The Fourier basis and the choice of nbasis=7nbasis = 7nbasis=7 are now explicitly justified, claims regarding outliers have been moderated, and the interpretation of the residual component has been corrected to ensure consistency with the figures.

Below we provide a point-by-point response.

Comment 1

There are very few observations, and I am not convinced the proposed approach is better than one based on classical ARIMA models or their seasonal extensions. No comparison with other methods was included.

Response

We thank the reviewer for this important observation. In the revised manuscript, we clarify that the purpose of the study is not forecasting, but rather structural decomposition of monthly exchange-rate data in a robust functional framework. We now explicitly state that ARIMA-type models and Functional Median Polish (FMP) address different objectives: ARIMA-type models focus on temporal dependence and prediction, whereas FMP provides an interpretable additive decomposition into functional center, row effects, column effects, and residual structure. The manuscript now presents these approaches as complementary rather than competing.

Comment 2

Why was a Fourier basis with 7 functions used? Why Fourier and not B-splines?

Response

This issue has been addressed explicitly in the revised methodology section. We now explain that the monthly observations are defined on a periodic monthly domain, which makes the Fourier basis a natural and parsimonious choice for representing seasonal structure. We also added a sensitivity-analysis statement indicating that the choice nbasis=7nbasis = 7nbasis=7 was assessed relative to nearby alternatives and selected as a balance between smoothness and reconstruction accuracy.

Comment 3

Do classical time-series methods find any outliers for these data? Is this functional approach better than applying Median Polish to the original data?

Response

We appreciate this comment. In the revised manuscript, we moderated our claims regarding outliers and no longer describe the dataset as one dominated by severe outlying behavior. Instead, we now explain that the value of the proposed framework lies in its robust exploratory decomposition and interpretability, rather than in the presence of strong contamination. We also clarify that the study does not claim universal superiority over scalar Median Polish or classical time-series methods, but instead investigates the usefulness of FMP for structural interpretation in this dataset.

Comment 4

The presentation should be improved. The English should be revised substantially.

Response

We agree and have extensively revised the manuscript for grammar, style, coherence, and consistency. Incomplete sentences, awkward phrasing, duplicated terminology, and unclear transitions have been corrected. The manuscript has also been reorganized to improve readability and flow.

Comment 5

Issues after equations (1) and (2), including “Where”/“where” and missing notation.

Response

These issues were corrected in the revised manuscript. The notation following the equations was rewritten carefully, definitions of the model components were clarified, and the missing explanatory parts were restored.

Comment 6

“First” to “Seventh” should be replaced with subsections or something else for clarity.

Response

This suggestion has been fully implemented. The procedural steps are now organized using clear subsection-style headings rather than ordinal labels.

Comment 7

The code should be included in an appendix.

Response

This has been addressed. The R code used in the analysis has been moved from the main text to Appendix A, where it is organized into clearly labeled parts for reproducibility.

Comment 8

Pictures 9 and 15 do not have the promised IQR band.

Response

Thank you for this observation. We carefully revised the titles, legends, and interpretations of these figures so that they now accurately reflect the displayed graphics. In particular, Figure 9 is now described as column effects by month, and Figure 15 as the residual pattern across months, rather than referring to an IQR band where such a band is not visually displayed.