## 1. Introduction

For numerous purposes, different time series are recorded and analyzed to understand phenomena or/and behaviors of variables, to try to predict future values, etc. Unfortunately, and for several reasons, there are gaps in data, irregular time steps of recordings, or removed data points that often need to be filled for data analysis, calibration of models, or for data with a regular time step. Generally in practice, incomplete series are the rule [

1]. In order to fill gaps in time series, numerous methods are available, and the choice of the method(s) to apply is not easy for non-mathematicians among data users. This paper presents a general overview, as a general and non-detailed introduction to such methods, especially written for readers without an introduction on interpolation methods.

First to all, a distinction should be clearly stated: the difference between interpolation and extrapolation. A first approach can be defined as the following. To fill gaps and, respectively, to predict future values, often require similar methods [

2], named as interpolation and extrapolation methods respectively. A second approach distinguishes between both cases, while comparing the range of existing and missing data. If there is a model to fill the gaps in a time series, if predictions are similar in content, and there is variability to observations, model errors should be equivalent for the existing and the missing data set (interpolation). Otherwise an extrapolation (i.e., predictions are outside of the observation range) has been performed, and expected errors would be greater for the predictions. An algorithm based on the data depth theory has been coded to make the distinction between interpolation and extrapolation [

3]. Differences between existing and missing data in those methods could cause significant errors [

4]. This paper deals with the interpolation method, in the sense of a first approach without consideration of the potential differences between the ranges of existing and missing data.

Secondly, there are many different kinds of data. Economical, financial, chemical, physical and environmental variables do not have the same characteristics: they can range from random to polycyclic processes. This wide range of characteristics requires a wide range of methods to fill gaps, and poses problems to researchers and practitioners when they choose one of the available methods. One thing is clear: a straightforward calculation of prediction uncertainty does not yield satisfactory results.

Finally, an increasing proportion of data contains values and their uncertainties (standard uncertainties, 95% confidence intervals, etc.), and requires specific methods to properly assess the uncertainties of the interpolated data. Two normalized methods are recommended and presented in a fourth section to assess uncertainties: the law of propagation of uncertainties [

5] and Monte-Carlo simulations [

6]. The estimation of prediction uncertainties can be easily performed with combined procedures. Assessment of the uncertainties of interpolated data is reported only seldomly: a brief review of the literature is presented in

Section 4. Two articles are already mentioned here, due to the relevance of their data and ideas: [

7], where the application of a Markovian arrival process to interpolate daily rainfall amount applied on a wide range of existing distributions is presented, and [

8], where an overview of methods and questions/consequences related to the evaluation of uncertainties in missing data is presented.

According to [

1], a useful interpolation technique should meet four criteria: (i) not a lot of data is required to fill missing values; (ii) estimation of parameters of the model and missing values are permitted at the same time; (iii) computation of large series must be efficient and fast, and (iv) the technique should be applicable to stationary and non-stationary time series. The selected method should also be accurate and robust. Prior to the application of interpolation methods, [

9] mentioned two preliminary steps to apply prior to the determination of a model: (i) to separate the signal (trends of interest, i.e., relevant and dependent on subsequent use of the data) from the noise (variability without interest) as in [

10], or (ii) to understand the past or existing data to better forecast the future, or to fill gaps in missing data. Despite the fact that smoothing algorithms, which may be used to distinguish signals while removing their noise, and interpolation methods are sometimes similar, this article does not deal with smoothing methods.

The following example aims at presenting pending scientific questions on this topic. Suppose rainfall intensity is measured at an hourly basis. The value at 1 p.m. is equal to 10 mm/h with a standard uncertainty of ±1 mm/h at 1 p.m. The instrument produced an error signal at 2 p.m. Finally, at 3 p.m., the instrument again logged 10 ± 1 mm/h. Under the assumption of randomness of the series (covariances and correlation coefficients are equal to 0, direct calculation by law of propagation of uncertainties (an average between 1 p.m. and 3 p.m. values) of the missing value at 2 p.m. yields 10 ± 0.7 mm/h. The standard uncertainty is marked off by calculations that are applied with the minimal (equal to −1) and the maximal (equal to +1) correlation coefficient values. These calculations lead to a standard uncertainty range between ± 0.0 and ± 1 mm/h, which is always smaller or equal to the measurement uncertainties. The smaller uncertainty value, due to the averaging process, reflects the uncertainty of a 2 h averaging period, whereas we are interested in a 1 h averaging period around 2 p.m., similar to the measurement points at 1 p.m. and 3 p.m. It is concluded that the prediction uncertainty cannot be estimated from the law of propagation of uncertainties [

5].

This review paper is structured as follows. Numerous methods (see

Section 2) have been published about filling gaps in data. This paper gives an accessible overview of those methods in

Section 2. In spite of the recommended comparisons [

11] between existing methods, only a few studies have compared methods (e.g., [

11,

12,

13]); these are detailed in the second section. In the third section of this article, criteria applied and published in past studies for comparison of methods are given. Due to the numerous methods and criteria available, a comparison between different studies with numerical values is often impossible. To improve the readability of the present article, a common notation has been applied to present methods:

x_{i} are observed data,

X_{i} are the interpolated value(s) of missing data,

i is an index, and

$\epsilon $ are the residuals. Each of them apply only for univariate time-series. This paper does not deal with multivariate time series.

## 3. Criteria Used to Compare Interpolation

A brief overview of applied criteria is given in

Table 1. Most of them use variance between observed

n values (

x_{i}) and predicted

n values (

X_{i}). Their averages (

$\overline{x}$ or

$\overline{X}$), and standard deviation

$\sigma \left({x}_{i}\right)$ of observed values have been also been used. Equations in

Table 1 have been extracted from articles and, sometimes, from personal conversations with authors. More than 30 criteria have been used in previous studies. Sometimes the same equation has different names in the studies. In [

1], the applied criterion (MSE) has not been explicitly given.

Gnauck [

16] then considered that

r^{2} criteria cannot be used, due to non-normally distributed data and non-linear effects in some phenomena, such as water quality processes. Some criteria could be used only for a specific method, such as AUC or MP, as proposed by [

76], or APE [

13]. Chen et al. [

18] used criteria based on the confusion matrix for binary classification [

76] where TP are the True Positive, FP the False Positive, FN the False Negative, and TN the True Negative values. Those criteria/methods can only be applied if variables have been divided in classes. Alonso and Sipols [

39] introduced criteria about coverage interval of interpolated values: LL for Lower Limit, and UL for Upper Limit of the 95% confidence interval. These values can be calculated with various equations (by the law of propagation of uncertainties) or with numerical methods. This is why no equation has been given for those values. The time of computation has been used in some studies. This is a useful measure for future users, but the characteristics of the computer are not always given: relative comparisons between different studies becomes more complicated with regard to this criterion. Any trade-off between quality and the cost of methods needs to be performed by each user [

10].

## 5. Discussion: From Literature Outcomes to a New Method

As this literature review has pointed out, there are numerous available methods to interpolate time series (Tables 5.1, 5.2 and 5.3 presented in [

20] and other references in this study), and numerous criteria to assess the efficiencies of those methods. A second conclusion from the review is that uncertainties of the interpolated variables are generally not integrated in the interpolation methods, despite the fact that most of the methods can assess the uncertainty. End users of basic interpolation methods (real interpolation, regression methods, (k-) nearest neighbors, etc.) need some guidance for selecting the most appropriate method for their own purposes and data sets. Unfortunately, no study has presented quite an exhaustive comparison between a broad selection of methods, for a number of reasons, which are discussed in the paragraphs hereafter.

The first reason relates to the choice and the nomenclature of performance criteria of interpolation methods. As demonstrated in

Table 1, there are quite a few non-conformities between formulas, names, and authors. In most of the published articles and communications, details of criteria are not given: this makes comparisons between studies more difficult, due to the lack of a common reference.

A second reason, already discussed by [

12,

16], is the typology of gaps that have to be filled. The ranking of desirable methods (obtained through a trade-off of criteria) could be strongly dependent on the size of the gaps, and the nature of recorded phenomena and data. The typologies of gaps should be specified and tested in future studies, to allow a critical review from partial comparisons (only few methods tested).

A third and a last reason is the evaluation of prediction uncertainties, which is the starting point and main question of this present study. During the review process presented, no satisfactory estimation (according to our standards and expected values of uncertainties) has been found to fit the following reflection.

In a given time series x and its standard uncertainty u(x) associated with the vector x, a few values of x (called hereafter x_{REMOVED}) and their standard uncertainties u(x_{REMOVED}) have been intentionally removed to simulate artificial gaps. Interpolation methods have then been applied to estimate the missing values x_{CALCULATED}. Standard uncertainties (u(x_{CALCULATED})) associated to those estimated values should be equal or higher than u(x_{REMOVED}), to take into account the added uncertainties due to the interpolation itself. Listed methods in this paper are at least numerical and often deterministic (i.e., derivatives could be calculated, under differentiability assumptions). Consequently, the law of propagation of uncertainties or Monte Carlo simulations could be applied to assess uncertainties of the interpolation method itself. As shown in the introduction for the law of propagation of uncertainties, those methods are not enough to properly assess prediction uncertainties.

Uncertainty has been estimated in a few studies only. Paulson [

31] calculated uncertainties of predictions with linear interpolation: uncertainties of observations, correlation between them and residuals of the interpolation model have been taken into account in that study. Alonso and Sipols [

39] developed a bootstrap procedure to calculate the lower and upper limit of the confidence interval of the predictions with ARMA models. Athawale and Entezari [

15] presented a method for assessing the probability density function to cross a value between two existing points. Ref. [

7] published a relevant study and review of experimental distribution of daily rain amounts.

To enhance the research on this topic, we propose to explicitly account for the process variance and autocorrelation in the evaluation of the process uncertainty. The following method is proposed here (based on [

84]). Suppose the process state

x represented in a equidistant time series (e.g., water level, discharge, flow velocity) (t

_{i},x

_{i}), with

i = 1, …, N and

t_{i} =

i*

dt at

t =

$\tau $ (with (

i − 1)*

dt <

$\text{}\tau $ <

i*

dt) is obtained from a simple linear interpolation (Equation (19)):

where

${X}_{\tau}$ is the interpolated value,

${x}_{i-1}$ and

${x}_{i}$ are existing (measured) values,

α and

β are weighing factors in the interpolation. The process has a known process variance

σ_{P}, a mean value of

μ_{p}, and an autocorrelation function

ρ(

τ). The mean squared error (MSE, see

Table 1) of the interpolation is calculated (Equation (20)).

The last term in the Equation (20) is the bias term, which vanishes when

α +

β = 1. Minimizing the MSE for

α and

β results in optimal values for the latter parameters. Assuming that

α +

β = 1 and imposing the following condition (Equation (21)):

Then,

α and

β can be derived (Equation (22)):

Which leads to (Equation (23)):

A simple case is obtained when the interpolation is done exactly halfway between the two adjacent samples, in this case,

ρ (

t_{i}_{−1},

τ) =

ρ (

t_{i},

τ), resulting in (Equation (24)):

Assuming that the measuring error

σ_{m} is independent from the process monitored (i.e., the sensor has a measuring error that is not depending on the measuring scale), the total uncertainty at the interpolated point is (Equation (25)):

It can be seen that in a process with an autocorrelation function, |ρ(t)| = 1, the error in the interpolation is equal to the measurement error. For every process with an autocorrelation −1 < ρ(t) < 1 the prediction uncertainty is larger than the measuring error.

The reasoning outlined (incorporating the process variability as well as the measuring uncertainty when interpolating) here can also be applied to more complicated interpolation techniques, as described in the literature review section of this article.

Future research will focus on the differences between several interpolations techniques in terms of prediction uncertainty, taking into account the characteristics of the (physical) process involved.

Figure 1 shows the uncertainties of interpolated values asset by: (i) the law of propagation of uncertainties (top left); (ii) the Monte-Carlo method (top right); (iii) the method proposed in [

17] (bottom left), and (iv) the method proposed above (bottom right). The rain time series recorded in Rotterdam (The Netherlands) has been used for this comparison. On the top left, the law of propagation of uncertainties gave uncertainties lower than, and respectively equal to standard observation uncertainties (0.01 mm/h) under the hypothesis that data are fully negatively (

ρ(

t) = −1) and positively (

ρ(

t) = 1) correlated, respectively. Any additional calculation to estimate partial correlation in the time series will lead to estimations between these two dashed dot lines. The application of this first normalized method always leads to an underestimation of uncertainties, despite calculations of partial autocorrelation in the time series. On the top right, Monte Carlo method results have been plotted with a correlation coefficient of 0.051 (corresponding to the partial correlation of the time series, with a lag of 29 time steps—the lag between the last and the next values known around the gap): the resulting curve is in the area delimited by the law of propagation of uncertainties. The method proposed by [

17] gave standard uncertainties (bottom left) mostly higher than the observation standard uncertainties except at the boundaries: standard uncertainties are lower here. The proposed method (bottom right) seems to give more logical estimations of standard uncertainties, with continuity at the gap boundaries, and the highest value in the middle of the interpolated values (the farthest position from the known data).

## 6. Conclusions

There are numerous methods and criteria for assessing the quality of interpolation methods. In the literature, many redundancies, discrepancies, or subtleties have been found: different names for the same method or criteria, different equations for the same criteria, etc. Future research should be very explicit and detailed, in order to avoid potential misunderstanding due to lexical discrepancies. No comprehensive comparative studies have been published so far: this lack of exhaustive feedback might be problematic for researchers, engineers, and practitioners who need to decide upon choosing interpolation methods for their purposes and data. To the authors’ knowledge, no comparative study published so far has dealt with methods to quantify prediction uncertainties. This can explain why prediction uncertainties are, in practice, only rarely calculated. The combination of the easiest interpolation methods and uncertainty calculation standards leads to mistakes in uncertainty assessments (as demonstrated in the discussion part), and methods that perform both interpolation and uncertainty calculation have not been exhaustively compared.

According to these conclusions, future work should focus on those topics to fill in the gaps in literature, and to give the tools for researchers to decide between the many available methods. In this respect, two kinds of studies could be useful: (i) exhaustive and comparative studies with a special attention for lexical issues, to standardize names of methods and criteria (used as a new reference), and (ii) development of new methods to assess prediction uncertainties.