# Interpolation in Time Series: An Introductive Overview of Existing Methods, Their Performance Criteria and Uncertainty Assessment

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{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.

## 2. Interpolation Methods

#### 2.1. Deterministic Methods

_{B}), and the first one known after the gap (x

_{A}).

#### 2.1.1. Nearest-Neighbor Interpolation

_{A}and x

_{B}.

#### 2.1.2. Polynomial Interpolation

_{A}and x

_{B}. Various equivalent equations exist for this method, such as (2a) given in [16], and (2b) given in [17].

_{A}and x

_{B}, and true values are, in average, underestimated: this affirmation is strongly dependent on the distribution of data (on which side the distribution is tailed—left or right) and should be verified for each data set. [16] demonstrated that this method is efficient, and most of the time it is better than non-linear interpolations for predicting missing values in environmental phenomena with constant rates.

_{MAX}is the maximum degree of the polynomial function, c

_{ij}are coefficients and x

_{i}are existing data. As an interpolation method, the amount of data used to estimate the coefficients is equal to K, in order to force the model to match the original records [9]: this is too strong a constraint for when data are uncertain. Sometimes these methods interpolate outside of the observed range of data, contrary to nearest-neighbor and linear interpolations [9], or with spurious oscillations due to too high a degree of polynomials. Bates et al. [19] proposed to penalize the second derivative function of the polynomial to avoid oscillations. Many types of splines are very popular for interpolation. B-splines and their derivative functions could be summed and weighted in MOMS (Maximal Order and Minimal Support) functions [10].

#### 2.1.3. Methods Based on Distance Weighting

_{i}to the interpolated point (see Equation (4)).

#### 2.1.4. Methods Based on Fourier’s Theory: Use of Signal Analysis Methods

_{D}and C

_{T}are the wavelet coefficients of dilatation and translation [30]. These coefficients can be used to understand phenomena. A lot of wavelet functions are available. WT seems to be useful for data with sudden changes [25].

#### 2.1.5. Other Deterministic Interpolation Method

#### 2.2. Stochastic Methods

#### 2.2.1. Regression Methods

_{0}is the initial value, k is the rate of change, and E(i) represents the stochastic part.

_{MAX}= 1), both methods presented in Section 2.1 are specific cases of regression methods.

_{MAX}can be chosen according to variables and gaps: [16] preferred non-linear interpolations if gaps lasted more than 28 days for water quality time series. The trade-off between significant and noise variability requires the expertise of the physical processes that underline the data regarding the missing data to be interpolated, and probably to check a posteriori the predictions (interpolated values).

#### 2.2.2. AutoRegressive Methods

#### 2.2.3. Machine Learning Methods

#### Artificial Neural Networks (ANN)

_{i}

_{+1}).

_{l}is the number of neurons of the input layer, r is the number of hidden neurons, and g is the hidden-layer transfer function from hidden layer. The logistic function is often used.

#### Kernel Methods

#### Tree Approaches

#### Meta-Algorithms of Machine Learning Methods

#### 2.2.4. Methods Based on Data Dynamics

#### k-Nearest Neighbors (kNN)

#### Box-Jenkins Models

#### Method of Moment Fitting

#### Other Methods Based on Data Dynamic

#### 2.2.5. Methods Based on Kriging

_{i}, such as indicator kriging, disjunctive kriging [65], and Multi-Gaussian kriging.

#### 2.2.6. Other Stochastic Methods

## 3. Criteria Used to Compare Interpolation

_{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.

^{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].

## 4. Evaluation of Uncertainties

#### 4.1. Law of Propagation of Uncertainties

_{i}) is calculated through Equation (18).

#### 4.2. Monte-Carlo Simulation

_{MC}simulations (N

_{MC}is the number of Mont-Carlo simulations), to ascertain the probability distribution of the output y. This technique requires long computation times, but it can be used without an explicit model (as f in Section 4.1) and for every kind of distribution (symmetrical, asymmetrical, etc.). Any statistical property of y can be computed from the Monte Carlo realization of y. For example, the 95% confidence interval of y is calculated in four steps: (i) calculation of N

_{MC}values of y; (ii) ranking of values; (iii) calculation of the width of multiple 95% confidence intervals from the cumulative distribution function [${\mathrm{F}}_{\mathrm{y}}^{-1}\left(0\right)$; ${\mathrm{F}}_{\mathrm{y}}^{-1}\left(0.95\right)$] to [${\mathrm{F}}_{\mathrm{y}}^{-1}\left(0.05\right)$; ${\mathrm{F}}_{\mathrm{y}}^{-1}\left(1\right)$], and finally (iv) the smallest interval is retained as the final 95% confidence interval [6].

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

_{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.

_{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)):

_{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)).

_{i}

_{−1}, τ) = ρ (t

_{i}, τ), resulting in (Equation (24)):

_{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)):

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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Equations | Names | References |
---|---|---|

$\left\{\frac{\left[{{\displaystyle \sum}}_{i=1}^{n}\left({X}_{i}-\overline{X}\right)\left({x}_{i}-\overline{x}\right)\right]}{\sqrt{{{\displaystyle \sum}}_{i=1}^{n}{\left({X}_{i}-\overline{X}\right)}^{2}{{\displaystyle \sum}}_{i=1}^{n}{\left({x}_{i}-\overline{x}\right)}^{2}}}\right\}$ | Coefficient of correlation (r) | [80] |

${\left\{\frac{\left[{{\displaystyle \sum}}_{i=1}^{n}\left({X}_{i}-\overline{X}\right)\left({x}_{i}-\overline{x}\right)\right]}{\sqrt{{{\displaystyle \sum}}_{i=1}^{n}{\left({X}_{i}-\overline{X}\right)}^{2}{{\displaystyle \sum}}_{i=1}^{n}{\left({x}_{i}-\overline{x}\right)}^{2}}}\right\}}^{2}$ | Squared coefficient of correlation (r^{2}) | [16] |

$\left[\frac{{{\displaystyle \sum}}_{i=1}^{n-l}\left({e}_{i}-\mathrm{MBE}\right)\left({e}_{i+l}-\mathrm{MBE}\right)}{{{\displaystyle \sum}}_{i=1}^{n}{\left({e}_{i}-\mathrm{MBE}\right)}^{2}}\right]$ | Mean of autocorrelation function r(l) | [9] ^{1} |

$\frac{{{\displaystyle \sum}}_{i=1}^{n}{X}_{i}-{x}_{i}}{n}$ | Mean Bias Error (MBE) | [9] |

Bias | [80] | |

$\frac{{{\displaystyle \sum}}_{i=1}^{n}{x}_{i}-{X}_{i}}{n}$ | Mean Error (ME) | [12] |

${\displaystyle \sum}_{i=1}^{n}}\left|{X}_{i}-{x}_{i}\right|$ | Absolute differences | [13] |

$\frac{{{\displaystyle \sum}}_{i=1}^{n}\left|{X}_{i}-{x}_{i}\right|}{n}$ | Mean Absolute Error (MAE) | [9,77,80] |

$\frac{{{\displaystyle \sum}}_{i=1}^{n}{x}_{i}-{X}_{i}}{{x}_{i}}$ | Mean Relative Error (MRE) | [77,80] |

$\frac{1}{n}{\displaystyle {\displaystyle \sum}_{i=1}^{n}}\left|\frac{{x}_{i}-{X}_{i}}{{x}_{i}}\right|$ | Mean Absolute Relative Error (MARE) | [77,80] |

$\frac{100}{n}{\displaystyle {\displaystyle \sum}_{i=1}^{n}}\left|\frac{{x}_{i}-{X}_{i}}{{x}_{i}}\right|$ | Mean Absolute Percentage Error (MAPE) | [4,54] |

${\displaystyle \sum}_{i=1}^{n}}{\left({X}_{i}-{x}_{i}\right)}^{2$ | Sum of Squared Errors (SSE) | [45] |

Quadratic differences | [13] | |

$\frac{{{\displaystyle \sum}}_{i=1}^{n}{\left({x}_{i}-{X}_{i}\right)}^{2}}{n}$ | Standard errors | [16] |

Prediction risk—leave one out MSE (R_{LOO}) | [32] | |

Mean Square Error (MSE) | [32,39] | |

nc | [1,81] | |

$\sqrt{\frac{1}{n}{\displaystyle {\displaystyle \sum}_{i=1}^{n}}{\left(\frac{{X}_{i}-{x}_{i}}{{x}_{i}}\right)}^{2}}$ | [82] | |

Root Mean Squares (RMS) | ||

[24] | ||

Root Mean Square Errors of Prediction (RMSEP) | [54] | |

$\frac{{{\displaystyle \sum}}_{i=1}^{n}{\left({x}_{i}-{X}_{i}\right)}^{2}}{{{\displaystyle \sum}}_{i=1}^{n}{\left({x}_{i}-\overline{x}\right)}^{2}}$ | Mean Squares Error (NMSE) | [4] |

$1-\frac{{{\displaystyle \sum}}_{i=1}^{n}{\left({x}_{i}-{X}_{i}\right)}^{2}}{{{\displaystyle \sum}}_{i=1}^{n}{\left({x}_{i}-\overline{x}\right)}^{2}}$ | Reduction of Error (RE) | [81] |

Nash-Sutcliffe coefficient (NS) | [3] | |

$\frac{{{\displaystyle \sum}}_{i=n}^{n+p-h}{\left({x}_{i+h}-{X}_{i+h}\right)}^{2}}{n-h+1}$ | Mean Squared Forecast Error (MFSE(h)) | [45] ^{2} |

$\sqrt{\frac{{{\displaystyle \sum}}_{i=1}^{n}{\left({X}_{i}-{x}_{i}\right)}^{2}}{n}}$ | Root Mean Squares Deviations (RMSD) | [9] |

Root Mean Squares Errors (RMSE) | [12,77,80,83] | |

$100\times \frac{\sqrt{\frac{{{\displaystyle \sum}}_{i=1}^{n}{\left({X}_{i}-{x}_{i}\right)}^{2}}{n}}}{\mathrm{Max}\left({x}_{i}\right)-\mathrm{Min}\left({x}_{i}\right)}$ | Normalized Root Mean Square Deviation | [27] |

$\sqrt{\frac{1}{n}{\displaystyle {\displaystyle \sum}_{i=1}^{n}}{\left(\frac{{X}_{i}-{x}_{i}}{\sigma \left({x}_{i}\right)}\right)}^{2}}$ | Root Mean Square Standardized error (RMSS) | [80] |

$100\times \frac{\left|{M}_{T}-{M}_{R}\right|}{{M}_{R}}$ | Absolute Percent Error (APE) | [13] ^{3} |

$\displaystyle \sum}_{i=1}^{c}}\frac{{\left({f}_{o}-{f}_{e}\right)}^{2}}{{f}_{e$ | Chi-Square (X^{2}) | [9] ^{4} |

$\{\begin{array}{c}=1\text{}\mathrm{if}\text{}\mathrm{L}\mathrm{L}{x}_{i}\mathrm{U}\mathrm{L}\\ =0\text{}\mathrm{Otherwise}\end{array}$ | Coverage | [39] |

$\{\begin{array}{c}=1\text{}\mathrm{if}\text{}{x}_{i}\mathrm{L}\mathrm{L}\\ =0\text{}\mathrm{Otherwise}\end{array}$ | Left Mis-coverage | [39] |

$\{\begin{array}{c}=1\text{}\mathrm{if}\text{}{x}_{i}\mathrm{U}\mathrm{L}\\ =0\text{}\mathrm{Otherwise}\end{array}$ | Right Mis-coverage | [39] |

Various equations | 95% confidence interval | [39] |

$\frac{\mathrm{T}\mathrm{N}+\mathrm{T}\mathrm{P}}{\mathrm{T}\mathrm{N}+\mathrm{F}\mathrm{P}+\mathrm{F}\mathrm{N}+\mathrm{T}\mathrm{P}}$ | Percentage of Correctly Classified observations (PCC) | [18] |

$\frac{\mathrm{T}\mathrm{P}}{\mathrm{T}\mathrm{P}+\mathrm{F}\mathrm{N}}$ | Percentage of correctly classified observations in the positive class (Sensitivity) | [18] |

$\frac{\mathrm{F}\mathrm{P}}{\mathrm{F}\mathrm{P}+\mathrm{T}\mathrm{N}}$ | Percentage of correctly classified observations in the negative class (Specificity) | [18] |

$\frac{\frac{\mathrm{T}\mathrm{P}}{\mathrm{F}\mathrm{P}+\mathrm{T}\mathrm{P}}}{\frac{\mathrm{F}\mathrm{N}+\mathrm{T}\mathrm{P}}{\mathrm{T}\mathrm{N}+\mathrm{F}\mathrm{P}+\mathrm{F}\mathrm{N}+\mathrm{T}\mathrm{P}}}$ | Top 10% lift (Lift) | [18] |

$2\frac{\frac{\mathrm{T}\mathrm{P}}{\mathrm{T}\mathrm{P}+\mathrm{F}\mathrm{P}}\frac{\mathrm{T}\mathrm{P}}{\mathrm{T}\mathrm{P}+\mathrm{F}\mathrm{N}}}{\frac{\mathrm{T}\mathrm{P}}{\mathrm{T}\mathrm{P}+\mathrm{F}\mathrm{P}}+\frac{\mathrm{T}\mathrm{P}}{\mathrm{T}\mathrm{P}+\mathrm{F}\mathrm{N}}}$ | H-measure (H) | [18] |

No equation | Time of computation | [10,12,18,84] |

^{1}Where l is the lag (gap) size, e

_{i}is the residual between observations and interpolated values (i.e., x

_{i}− X

_{i}).

^{2}Where p is the last 20% of the data rounded to the nearest multiple of n

^{2}and h is the number of periods.

_{3}Where M

_{T}is the moment or autocorrelation of the interpolated series, and M

_{R}is the moment or autocorrelation in the existing data.

^{4}Where c is the number of classes, f

_{O}is the observed frequency, and f

_{E}is the expected frequency.

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Lepot, M.; Aubin, J.-B.; Clemens, F.H.L.R.
Interpolation in Time Series: An Introductive Overview of Existing Methods, Their Performance Criteria and Uncertainty Assessment. *Water* **2017**, *9*, 796.
https://doi.org/10.3390/w9100796

**AMA Style**

Lepot M, Aubin J-B, Clemens FHLR.
Interpolation in Time Series: An Introductive Overview of Existing Methods, Their Performance Criteria and Uncertainty Assessment. *Water*. 2017; 9(10):796.
https://doi.org/10.3390/w9100796

**Chicago/Turabian Style**

Lepot, Mathieu, Jean-Baptiste Aubin, and François H.L.R. Clemens.
2017. "Interpolation in Time Series: An Introductive Overview of Existing Methods, Their Performance Criteria and Uncertainty Assessment" *Water* 9, no. 10: 796.
https://doi.org/10.3390/w9100796