# Multivariate Time Series Analysis of Temperatures in the Archaeological Museum of L’Almoina (Valencia, Spain)

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## Abstract

**:**

## 1. Introduction

**Figure 1.**Plan of the L’Almoina archaeological site, indicating the position of 27 data-loggers for monitoring the air conditions inside the museum. Based on the multivariate analysis of temperatures (see Section 3.3.6), three zones were established: North West (NW, in blue), South East (SE, in green) and Skylight (Sk, in orange). The different observable structures and the construction phases in the museum are indicated: (a) Roman baths; (b) Imperial granary; (c) Portico of the imperial forum; (d) Imperial chapel; (e) Imperial basilica; (f) Byzantine apse and tombs; (g) Byzantine Cathedral Baptistery; (h) Republican and Imperial Asklepieion; (i) Alcázar Aldalusí; (j) Decumani; and (k) Cardus [12].

## 2. Background

#### 2.1. Studies for the Long-Term Preservation of Artworks

#### 2.2. European Standards

#### 2.3. Characteristics of the L’Almoina Museum

## 3. Materials and Methods

#### 3.1. Materials: Description of the Datasets

#### 3.2. Data Calibration

#### 3.3. Statistical Methods

`R`software packages [29] (version 4.3) used to carry out the statistical analyses were:

`mixOmics`[30,31],

`aTSA`[32],

`forecast`[33,34],

`strucchange`[35],

`tseries`[36],

`moments`[37],

`PerformanceAnalytics`[38],

`NbClust`[39] and

`QuantTools`[40].

#### 3.3.1. Identification of Structural Breaks in the Time Series

`Fstats`and

`efp`from the

`strucchange`package [35]. Initially, 5 stages were tentatively established: warm 1 (Wr1, comprising n = 1490 observations), cold (Cd, n = 1703), transition (Tr, n = 2303), hot (Ht, n = 2327) and warm 2 (Wr2, n = 903) (see Figure 3). Wr1 corresponds to 22 October–23 December 2019, Cd to 24 December–3 March 2020, Tr to 4 March–7 June, Ht to 8 June–12 September and Wr2 to 13 September–20 October 2020.

#### 3.3.2. Calculation of Classification Variables—Method M1

`rollrange`from the

`QuantTools`package [40]. The sample ACF and sample PACF values were calculated with the function

`acf`(

`stats`) [47] and

`pacf`(

`tseries`) [36], respectively. The values of the periodogram and their frequencies were obtained with the function

`spectrum`(

`stats`).

#### 3.3.3. Calculation of Classification Variables—Method M2

`HoltWinters`function (

`stats`) was used to fit the Additive SH-W method. The

`shapiro.test`(

`stats`) and

`ks.test`(

`dgof`) [55] were used to apply the normality tests. Values of the sample ACF and sample PACF were computed with the functions

`acf`(

`stats`) and

`pacf`(

`tseries`), respectively. Values of the periodogram and their frequencies were calculated with the function

`spectrum`(

`stats`).

#### 3.3.4. Calculation of Classification Variables—Method M3

`arima`(

`stats`) and

`auto.arima`(

`forecast`) functions [33,34] were used. The ADF test was computed using the

`adf.test`(

`aTSA`) [32]. The LBQ test was applied by means of the

`Box.test`function (

`stats`). The LM test was carried out using the

`arch.test`function (

`aTSA`). The SW and KS normality tests were applied using the

`shapiro.test`(

`stats`) and

`ks.test`functions (

`dgof`), respectively.

#### 3.3.5. Calculation of Classification Variables—Method M4

`ARMAtoMA`(

`stats`). One function was created for reducing a polynomial with AR and SAR components to a polynomial with just AR component. Likewise, another function converted a polynomial with MA and SMA components to one with MA component.

#### 3.3.6. Determination of Number of Classes and Class per Sensor Using PCA and K-Means Algorithm

`kurtosis`and

`skewness`(

`PerformanceAnalytics`) [38] were used to compute the coefficients of kurtosis and skewness, while

`boxcoxfit`(

`geoR`) [64] was employed to apply different Box–Cox transformations. The function

`prcomp`(

`stats`) was used to carry out PCA.

`mixOmics`package [31], which is able to cope with this drawback and returns accurate results [66]. After the data normalization, all variables were mean-centered and scaled to unit variance, which is the common pretreatment in PCA. Next, PCA was carried out to reduce the dimensionality of the TCD matrix. Each observation (sensor) was projected onto the first few principal components to obtain lower-dimensional data, while preserving as much of the data variation as possible.

`NbClust`function of the

`NbClust`package [39] incorporates 30 different indices for determining the number of clusters [67]. This function claims to use the best clustering scheme from the different results obtained, by varying all combinations of the number of clusters, distance measures and clustering methods. It allows the user to identify the value K in which more indices coincide, providing assurance that a good choice is being made.

`kmeans`(

`stats`). It performs better than the algorithms proposed by MacQueen [69], Lloyd [70] and Forgy [71]. However, when the algorithm of Hartigan and Wong is carried out, it is often recommended to try several random starts. In the present study, 100 random starts were employed. This algorithm guarantees that, at each step, the total intra-variance of the clusters is reduced until reaching a local optimum. Results from the k-means algorithm depend on the initial random assignment. For this reason, the algorithm was run 100 times, each with a different initial assignment. The final result was the one leading to a classification with the lowest total variance value. By comparing the classification obtained with the position of sensors in the museum (Figure 8a), the three zones were denoted as North West (NW), South East (SE) and Skylight (Sk).

#### 3.3.7. Sensor Classification Using sPLS-DA

`R`package

`mixOmics`, which provides functions such as

`perf`,

`tune.splsda`and

`splsda`, in order to determine the number of components and elements different to zero in the loading vector before running the final model.

`perf`function outputs the optimal number of components that achieve the best performance based on both types of classification error rate (CER): Balanced Error Rate (BER) and the Overall classification error rate (Overall). BER is the average proportion of wrongly classified sensors in each class, weighted by the number of sensors. In most cases, the results from sPLS-DA were better (or very similar) using Overall than when using BER. However, BER was preferred to Overall because it is less biased towards majority classes during the performance assessment of sPLS-DA. In this step, three different prediction distances were used, maximum, centroid and Mahalanobis [30], in order to determine the predicted class per sensor for each of the test datasets. For each prediction distance, both Overall and BER were computed.

`keepX`) from Component 1 to H, one component at a time. The different grids of values of the number of variables were carefully chosen to achieve a trade-off between resolution and computational time. Firstly, a two coarse tuning grids were assessed before setting a finer grid. The algorithm used the same grids of

`keepX`argument in

`tune.splsda`function to tune each component.

`vip`function (

`mixOmics`). Although the assumption of the sparse loading vectors being orthogonal was considered, in practice, some selected variables were common in two components. Then, a second measure of $VIP$ [88] was employed: $VI{P}_{j}$ denotes the overall importance of variable ${\mathbf{X}}_{j}$ on all responses (one per class) cumulatively over all components. It is defined using the loading vectors and the sum of squares per component. Variables with $VI{P}_{j}>1$ are the most relevant for explaining the classification of sensors. The selected variables were ranked according to both types of VIPs, which are discussed below for each stage of the time series.

#### 3.3.8. Sensor Classification Using Random Forest Algorithm

`randomForest`and

`importance`[93] were used to carried out the RF algorithm.

## 4. Results and Discussion

#### 4.1. Identification of Structural Breaks in the Time Series

#### 4.2. Calculation of Classification Variables—M1–M4

#### 4.3. Determination of Number of Classes and Class per Sensor Using PCA and K-Means Algorithm

`C0`,

`C1`,

`A4`,

`A5`and

`F`in the SE zone. However, by checking their position on the map of the museum (Figure 8b), these sensors could be regarded in the boundary between the NW and SE zones. Hence, it should be discussed whether such classification is appropriate, or if they should be regarded within the NW zone. In order to study this issue, two classifications were analyzed using sPLS-DA: (1) by considering A4, A5 and F in the NW zone; and (2) by locating C0 and C1 in the NW zone. For both cases, the rate of misclassified sensors was computed by means of sPLS-DA. The error rates for the classification from the k-means algorithm were: 0.25 (M1), 0.30 (M2), 0.29 (M3) and 0.42 (M4). For Case (1), the classification error rates were: 0.15 (M1), 0.19 (M2), 0.31 (M3) and 0.35 (M4). For Case (2), the error rates were: 0.21 (M1), 0.23 (M2), 0.30 (M3) and 0.41 (M4). It turns out that the lowest error rates were found for Case (1). Thus, sensors A4, A5 and F were considered as part of the NW zone for the next sections. Finally, 13 sensors were classified in the NW zone (

`A4`,

`A5`,

`A6`,

`B5`,

`B6`,

`C5`,

`C6`,

`D1`,

`D2`,

`D3`,

`D5`,

`D6`and

`F`), eight in the SE zone (

`A`,

`B`,

`B1`,

`C`,

`C0`,

`C1`,

`D`and

`G`) and six in the Sk zone (

`A2`,

`A3`,

`B2`,

`B3`,

`B4`and

`C3`). The proposed classification of sensors is shown in Figure 1a, which depicts an association between T values and the three zones of the museum: NW, SE and Sk.

#### 4.4. Sensor Classification Using sPLS-DA

`acf.m`,

`acf.r`,

`acf.md`and

`acf.v`); PACF at lags 1–6 (

`pacf1`–

`pacf6`); mean of the time series (

`M`); statistics of the KM test (

`kolg.t`); statistics of the SW test (

`shap.t`); maximum values of the periodogram (

`spec.mx`); frequency of the maximum values of the periodogram (

`freq`); variance of the residuals (

`res.v`); SSE (

`sse`); seasonal components (

`s1`–

`s24`); level (

`a`); slope (

`b`); coefficients of the Wold decomposition (

`psi1`–

`psi5`); the first AR term (

`ar1`); the first MA term (

`ma1`); the first SMA term (

`sma1`); the first SAR term (

`sar1`); and the 17th prediction of T, which was denoted as

`pred.17`.

- ☐
- Selected variables from M1: PACF at lags 1–4, MR and parameters of the sample ACF (mean, range, median and mean) (Table 3 (a)).
- ☐
- Selected variables from M2: Level, slope, some seasonal components (5, 8, 10–12, 16–21 and 23), mean and median of sample ACF (residuals), SSE (residuals) and maximum of the periodogram (residuals) (Table 3 (b)).
- ☐
- Selected variables from M3: Parameters of MA, SAR and SMA of seasonal ARIMA models, PACF at lags 1–5 (residuals), mean and median of sample ACF (residuals), statistic of KM normality test (residuals) and frequency for the maximum of the periodogram (residuals) (Table 4 (a)).
- ☐
- Selected variables from M4: The first four Wold coefficients, mean and median of sample ACF (residuals), statistics of the SW and KM normality tests (residuals), PACF at first five lags (residuals), variance (residuals) and maximum of the periodogram (residuals) (see Table 4 (b)).
- ☐
- Selected variables when using the total set of variables from the four methods (some values are highlighted in bold and blue in Table 3 and Table 4): Slope (M2), some seasonal components (2, 3, 5, 8, 11, 12, 14, 16–18 and 21–23) (M2); SSE (M2); Wold Coefficient 1 (M4); mean (M1–M4), median (M2), range (M1 and M2) and variance (M1 and M2) of sample ACF values; and maximum of the periodogram (M2–M4).

**Table 3.**Selected variables (V) per component (C) from sPLS-DA. Variables highlighted in bold and blue correspond to selected variables for Ms.

C1 | C2 | C3 | ||||
---|---|---|---|---|---|---|

Stage | V | Stage | V | Stage | V | |

(a) For Method 1 | ||||||

1 | Tr | pacf2 | Wr2 | acf.r | CdA | pacf3 |

2 | Tr | acf.m | Wr2 | acf.v | Wr1A | pacf4 |

3 | HtA | pacf2 | HtA | acf.m | CdA | pacf4 |

4 | HtA | pacf1 | CdB | pacf1 | Wr1A | acf.r |

5 | HtB | pacf1 | CdB | M | Wr1A | acf.v |

6 | CdB | pacf2 | Wr1B | M | Wr1A | rMh |

7 | Wr2 | pacf1 | HtB | M | Wr1B | rMh |

8 | Wr1B | pacf1 | Wr1A | M | CdA | rMh |

9 | CdA | pacf1 | CdA | pacf1 | CdB | rMh |

10 | HtB | pacf2 | Wr1B | acf.r | Tr | rMh |

11 | Wr1B | pacf3 | Wr1B | acf.v | HtA | rMh |

12 | Tr | M | Wr2 | pacf2 | HtB | rMh |

13 | HtA | pacf4 | CdA | pacf2 | Wr2 | rMh |

14 | Tr | pacf4 | HtA | acf.r | ||

15 | Wr2 | acf.m | HtA | acf.v | ||

(b) For Method 2 | ||||||

1 | HtA | sse | HtA | acf.m | ||

2 | Tr | sse | HtA | acf.md | ||

3 | Tr | s18 | HtA | s12 | ||

4 | Tr | b | HtA | s11 | ||

5 | Tr | s16 | HtA | acf.r | ||

6 | HtB | sse | CdA | acf.v | ||

7 | Tr | s17 | HtA | acf.v | ||

8 | HtB | s21 | HtA | s23 | ||

9 | Tr | spec.mx | CdA | s14 | ||

10 | Wr2 | s20 | CdA | s5 | ||

11 | HtB | acf.m | Tr | a | ||

12 | Wr1B | s12 | CdA | s10 | ||

13 | Wr2 | acf.md | HtA | s8 | ||

14 | HtB | s19 | HtA | s10 | ||

15 | HtB | s23 | Wr1A | acf.m |

**Table 4.**Selected variables (V) per component (C) from sPLS-DA. Variables highlighted in bold and blue correspond to selected variables for Ms.

C1 | C2 | C3 | ||||
---|---|---|---|---|---|---|

Stage | V | Stage | V | Stage | V | |

(a) For Method 3 | ||||||

1 | Tr | acf.m | CdA | acf.m | ||

2 | HtA | res.v | CdA | pacf4 | ||

3 | Tr | ma1 | CdA | pacf5 | ||

4 | HtB | ma1 | Wr2 | pacf4 | ||

5 | Tr | kolg.t | Tr | sar2 | ||

6 | CdB | pacf2 | ||||

7 | HtB | pacf1 | ||||

8 | Wr2 | sar2 | ||||

9 | CdA | sar1 | ||||

10 | HtA | sma1 | ||||

11 | HtA | pacf3 | ||||

12 | HtA | freq | ||||

13 | HtA | pacf1 | ||||

14 | HtB | acf.md | ||||

15 | Wr1A | pacf5 | ||||

(b) For Method 4 | ||||||

1 | HtA | psi1 | HtA | pacf5 | Wr1B | psi1 |

2 | HtA | spec.mx | HtB | pacf1 | CdB | acf.m |

3 | HtA | res.v | Wr2 | acf.v | CdB | spec.mx |

4 | Tr | psi1 | HtA | shap.t | Wr2 | acf.m |

5 | Wr1A | acf.m | HtA | acf.m | Wr2 | shap.t |

6 | Wr2 | acf.r | ||||

7 | Tr | pacf1 | ||||

8 | Tr | acf.md | ||||

9 | Wr1B | psi4 | ||||

10 | CdA | pacf5 | ||||

11 | Tr | psi3 | ||||

12 | CdA | acf.md | ||||

13 | Wr2 | psi2 | ||||

14 | Tr | kolg.t | ||||

15 | Wr2 | kolg.t |

#### 4.5. Sensor Classification Using Random Forest Algorithm

- ☐
- Selected variables from M1: Mean, range and variance of the sample ACF values, the first four values of sample PACF, the maximum value of the periodogram and the mean.
- ☐
- Selected variables from M2: Level, slope and 18 seasonal components (2, 3, 7–12, 14 and 16–24) from sH-W method, SSE, maximum of the periodogram values, statistic of the KS normality test and 17th prediction of T.
- ☐
- Selected variables from M3: First term of the AR, MA and SMA components of the seasonal ARIMA models, sample PACF at lags 1, 2 and 5, mean and median of the sample ACF values, residual variance, maximum of the periodogram values and statistic of the KS normality test.
- ☐
- Selected variables from M4: Coefficients 1, 3, 4 and 5 from the Wold decomposition; the first five values of the sample PACF; the mean, median, range and variance of the sample ACF values; residual variance; maximum of the periodogram values of the residuals; and statistics of both the SH and KS normality tests.
- ☐
- Selected variables when using all classification variables from the four methods (Ms): mean (M1–M3), median (M2 and M3), range (M2) and variance (M2) of the sample ACF values; sample PACF at lags 1 (M1 and M3), 2 (M1) and 3 (M1); first coefficient of the Wold decomposition (M3); and statistic of the KS normality test (M2) (the results from Ms comprise 20% of variables from M1, 60% from M2, 9% from M3 and 11% from M4).

**Table 6.**Selected variables (V) from RF per method and when using the variables from the four methods.

M1 | M2 | M3 | M4 | Ms | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Stage | V | Stage | V | Stage | V | Stage | V | M | Stage | V | |

1 | CdA | pacf1 | HtA | sse | Tr | acf.m | HtA | psi1 | M1 | CdA | pacf1 |

2 | HtA | pacf4 | HtA | s24 | HtA | sma1 | CdB | psi1 | M2 | HtA | sse |

3 | HtA | pacf1 | HtA | s23 | Tr | acf.md | Tr | res.v | M3 | Tr | acf.m |

4 | Tr | pacf2 | HtA | s8 | CdB | pacf2 | HtA | pacf5 | M2 | HtA | s24 |

5 | CdB | pacf2 | HtA | s11 | HtA | res.v | HtA | res.v | M4 | HtA | psi1 |

6 | HtA | acf.m | HtA | s12 | Tr | kolg.t | Tr | acf.m | M1 | HtA | pacf4 |

7 | Tr | acf.m | HtB | acf.m | HtA | spec.mx | Tr | spec.mx | M2 | HtA | s23 |

8 | CdA | acf.m | CdA | acf.v | CdB | ma1 | Tr | psi1 | M3 | HtA | sma1 |

9 | Tr | pacf1 | Tr | sse | Tr | ma1 | Tr | kolg.t | M2 | HtA | s8 |

10 | HtB | pacf2 | HtA | acf.r | HtA | pacf1 | CdA | pacf1 | M2 | HtA | s11 |

11 | HtB | M | HtA | acf.md | CdA | pacf5 | HtA | spec.mx | M2 | HtA | acf.r |

12 | Wr2 | acf.m | Wr2 | s23 | HtA | ar1 | Wr1A | acf.m | M2 | HtA | s12 |

13 | Wr2 | pacf1 | Wr1B | a | Wr1A | sar2 | CdA | acf.md | M2 | HtA | acf.m |

14 | CdB | pacf1 | Tr | b | Wr1A | acf.md | CdA | pacf4 | M2 | CdA | acf.v |

15 | Tr | M | HtA | s22 | Tr | spec.mx | CdA | acf.v | M2 | Wr1B | a |

16 | Wr2 | acf.v | Tr | s16 | CdA | acf.m | HtB | res.v | M2 | HtA | s22 |

17 | Wr2 | pacf2 | Tr | a | Tr | res.v | CdA | kolg.t | M2 | HtB | acf.m |

18 | HtA | pacf2 | Tr | s17 | HtB | ma1 | Wr1A | psi1 | M1 | HtA | pacf1 |

19 | Wr2 | acf.r | CdA | s10 | HtB | sar2 | Wr1B | acf.m | M2 | HtA | acf.md |

20 | CdA | M | CdA | a | HtB | pacf1 | Tr | pacf1 | M3 | Tr | acf.md |

21 | Wr1B | M | HtA | acf.v | CdB | kolg.t | Tr | pacf5 | M2 | HtA | s9 |

22 | HtB | pacf1 | Wr2 | s18 | CdA | sar1 | HtB | psi1 | M2 | Wr2 | s23 |

23 | Wr1B | pacf3 | HtB | kolg.t | HtA | pacf3 | Tr | acf.md | M2 | Tr | s17 |

24 | HtA | M | Wr2 | s19 | HtA | pacf5 | Wr2 | pacf4 | M2 | CdA | a |

25 | CdB | M | HtB | s21 | Tr | freq | CdA | acf.m | M1 | Tr | pacf2 |

26 | Wr1A | pacf4 | HtA | s10 | Wr1B | acf.md | HtA | shap.t | M2 | Tr | sse |

27 | HtB | pacf4 | HtB | s9 | Wr1A | ma1 | HtA | acf.m | M2 | HtA | acf.v |

28 | Wr1B | pacf2 | Tr | spec.mx | Wr1B | psi4 | M4 | CdB | psi1 | ||

29 | HtB | spec.mx | Tr | s18 | Wr2 | res.v | M2 | HtA | s10 | ||

30 | Wr1B | acf.m | CdA | s5 | CdA | pacf5 | M2 | Tr | spec.mx | ||

31 | HtA | pacf3 | Tr | s20 | Wr2 | acf.r | M3 | CdB | pacf2 | ||

32 | Wr1B | pacf1 | CdA | s14 | CdA | shap.t | M2 | Tr | s16 | ||

33 | CdA | pacf3 | CdA | s1 | Wr2 | acf.v | M2 | Tr | b | ||

34 | CdB | pacf4 | HtA | acf.m | HtB | pacf1 | M2 | Wr2 | s20 | ||

35 | Wr1A | pacf2 | HtA | s9 | Wr2 | spec.mx | M3 | CdB | ma1 | ||

36 | Wr1A | M | Wr2 | s17 | Wr1B | psi3 | M2 | CdA | s1 | ||

37 | CdA | pacf2 | All | pred.18 | Wr2 | pacf2 | M2 | CdA | s10 | ||

38 | Tr | acf.v | CdA | s22 | Wr1B | pacf5 | |||||

39 | Wr2 | pacf4 | HtA | spec.mx | CdB | shap.t | |||||

40 | Tr | pacf4 | Wr2 | s3 | HtA | psi5 | |||||

41 | HtA | spec.mx | CdA | s23 | Tr | shap.t | |||||

42 | CdB | acf.m | Wr2 | s2 | Wr1A | pacf5 | |||||

43 | Tr | acf.r | Wr2 | kolg.t | |||||||

44 | HtB | acf.m | CdA | pacf3 |

#### 4.6. Methodology to Select a Subset of Sensors for Future Monitoring Experiments in the Museum

- ☐
- For NW, the number of representative sensors was $15\times 0.35=5.25\approx 6$. The 13 sensors in this zone were classified in each of the three concentric circles, as follows: D2, D5, A6 and C6 for the first area (G1); B6, B5, A5 and A4 for the second area (G2); and D6, D3, F, D1 and C5 for the third area (G3). The number of representative sensors selected per group (G1–G3) in this NW class was decided according to the variance of the distances: one sensor in G1, three in G2 and two in G3. The proposed subset of representative sensors is the following: D2 (for G1); B6, A5 and A4 (for G2); and D6 and F (for G3).
- ☐
- For SE, the number of representative sensors was $15\times 0.4=6$. The eight sensors in this zone were classified according to the concentric circles as: C0 (G1); A and B (G2); and C, C1, B1, D and G (G3). The number of SE sensors selected per group was determined based on the variance of the distances: one sensor in G1, one in G2 and four in G3. The proposed subset of representative sensors is as follows: C0 (for G1); A (for G2); and C, B1, D and G (for G3).
- ☐
- For Sk, the number of representative sensors was established as $15\times 0.25=3.75\approx 4$. The six sensors regarded as Sk were classified as: A2 and A3 (G1); B2, B3 and B4 (G2); and C3 (G3). The number of sensors chosen per group was decided according to the variance of the distances: one in G1, one in G3 and two in G2. The proposed subset of representative sensors was: A3 (for G1); B2 and B4 (for G2); and C3 (for G3).

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Pavlogeorgatos, G. Environmental parameters in museums. Build. Environ.
**2003**, 38, 1457–1462. [Google Scholar] [CrossRef] - Corgnati, S.P.; Filippi, M. Assessment of thermo-hygrometric quality in museums: Method and in-field application to the Duccio di Buoninsegna exhibition at Santa Maria della Scala (Siena, Italy). J. Cult. Herit.
**2010**, 11, 345–349. [Google Scholar] [CrossRef] - Sesana, E.; Gagnon, A.S.; Bertolin, C.; Hughes, J. Adapting cultural heritage to climate change risks: Perspectives of cultural heritage experts in Europe. Geosciences
**2018**, 8, 305. [Google Scholar] [CrossRef] [Green Version] - Fernández-Navajas, A.; Merello, P.; Beltran, P.; García-Diego, F.J. Multivariate thermo-hygrometric characterisation of the archaeological site of Plaza de L’Almoina (Valencia, Spain) for preventive conservation. Sensors
**2013**, 13, 9729. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Merello, P.; Fernandez-Navajas, A.; Curiel-Esparza, J.; Zarzo, M.; García-Diego, F.J. Characterisation of thermo-hygrometric conditions of an archaeological site affected by unlike boundary weather conditions. Build. Environ.
**2014**, 76. [Google Scholar] [CrossRef] - Ministero per i Beni e le Attività Culturali. DM 10/2001. Atto di Indirizzo sui Criteri Tecnico-scientifici e Sugli Standard di Funzionamento e Sviluppo dei Musei. 2001. Available online: https://www.veneto.beniculturali.it/normativa-e-disposizioni/atto-di-indirizzo-sui-criteri-tecnico%E2%80%93scientifici-e-sugli-standard-di (accessed on 8 March 2021).
- UNI Italian Standard 10829. Works of Art of Historical Importance-Ambient Conditions for the Conservation-Measurement and Analysis; Ente nazionale italiano di unificazione (UNI): Milano, Italy, 1999.
- García-Diego, F.J.; Zarzo, M. Microclimate monitoring by multivariate statistical control: The renaissance frescoes of the cathedral of Valencia (Spain). J. Cult. Herit.
**2010**, 11, 339–344. [Google Scholar] [CrossRef] - Zarzo, M.; Fernández-Navajas, A.; García-Diego, F.J. Long-term monitoring of fresco paintings in the cathedral of Valencia (Spain) through humidity and temperature sensors in various locations for preventive conservation. Sensors
**2011**, 11, 8685–8710. [Google Scholar] [CrossRef] - Ramírez, S.; Zarzo, M.; Perles, A.; Garcia-Diego, F.J. Methodology for Discriminant Time Series Analysis Applied to Microclimate Monitoring of Fresco Paintings. Sensors
**2021**, 21, 436. [Google Scholar] [CrossRef] - Merello, P.; García-Diego, F.J.; Zarzo, M. Microclimate monitoring of Ariadne’s house (Pompeii, Italy) for preventive conservation of fresco paintings. Chem. Cent. J.
**2012**, 6. [Google Scholar] [CrossRef] [Green Version] - Falconaumanni. Plan of the L’Amoina Museum Licensed under the Creative Commons Attribution-Share Alike 3.0. 2019. Available online: https://commons.wikimedia.org/wiki/File:Plano_Almoina_recorrido.png (accessed on 8 March 2021).
- Ramírez, S.; Zarzo, M.; Perles, A.; Garcia-Diego, F.J. sPLS-DA to discriminate time series. In JSM Proceedings, Statistics and the Environment; American Statistical Association: Alexandria, VA, USA, 2021; pp. 107–135. [Google Scholar]
- Frasca, F.; Siani, A.M.; Casale, G.R.; Pedone, M.; Bratasz, L.; Strojecki, M.; Mleczkowska, A. Assessment of indoor climate of Mogiła Abbey in Kraków (Poland) and the application of the analogues method to predict microclimate indoor conditions. Environ. Sci. Pollut. Res.
**2017**, 24, 13895–13907. [Google Scholar] [CrossRef] - Huijbregts, Z.; Kramer, R.; Martens, M.; van Schijndel, A.; Schellen, H. A proposed method to assess the damage risk of future climate change to museum objects in historic buildings. Build. Environ.
**2012**, 55, 43–56. [Google Scholar] [CrossRef] - Zítek, P.; Vyhlídal, T. Model-based moisture sorption stabilization in historical buildings. Build. Environ.
**2009**, 44, 1181–1187. [Google Scholar] [CrossRef] - Angelini, E.; Grassini, S.; Corbellini, S.; Parvis, M.; Piantanida, M. A multidisciplinary approach for the conservation of a building of the seventeenth century. Appl. Phys. A
**2010**, 100. [Google Scholar] [CrossRef] - Lourenço, P.B.; Luso, E.; Almeida, M.G. Defects and moisture problems in buildings from historical city centres: A case study in Portugal. Build. Environ.
**2006**, 41. [Google Scholar] [CrossRef] [Green Version] - Stewart, J.; Julien, S.; Staniforth, S. An integrated monitoring strategy at Chedworth Roman Villa (Gloucestershire). In Preserving Archaeological Remains In Situ?, Proceedings of the 2nd Conference, Paris, France, 12–14 September 2001; Museum of London Archaeology Service: London, UK, 2004. [Google Scholar]
- EN16883. Conservation of Cultural Heritage. Guidelines for Improving the Energy Performance of Historic Buildings. 2017. Available online: https://standards.cen.eu/dyn/www/f?p=204:110:0::::FSP_PROJECT:36576&cs=113EECDB855EBEF2097C9F626231290BE (accessed on 8 March 2021).
- EN16141. Conservation of Cultural Heritage. Guidelines for Management of Environmental Conditions. Open Storage Facilities: Definitions and Characteristics of Collection Centres Dedicated to the Preservation and Management of Cultural Heritage. 2012. Available online: https://standards.cen.eu/dyn/www/f?p=204:110:0::::FSP_PROJECT:30978&cs=19D7899D83F3E1FAFF740C53B6D9C068F (accessed on 8 March 2021).
- EN16242. Conservation of Cultural Heritage. Procedures and Instruments for Measuring Humidity in the Air and Moisture Exchanges between Air and Cultural Property. 2012. Available online: https://standards.cen.eu/dyn/www/f?p=204:110:0::::FSP_PROJECT:34048&cs=1000E6B80FEC23200296847848BFD8390 (accessed on 8 March 2021).
- EN15898. Conservation of Cultural Property. Main General Terms and Definitions. 2019. Available online: https://standards.cen.eu/dyn/www/f?p=204:110:0::::FSP_PROJECT:61301&cs=1BFEBDBB425EAF8FEDC7D300B26CB0F1E (accessed on 8 March 2021).
- EN15758. Conservation of Cultural Property. Procedures and Instruments for Measuring Temperatures of the Air and the Surfaces of Objects. 2010. Available online: https://standards.cen.eu/dyn/www/f?p=204:110:0::::FSP_PROJECT:28488&cs=18FBB4BE512863FD1F25E8415D5BA9ACE (accessed on 8 March 2021).
- EN15757. Conservation of Cultural Property. Specifications for Temperature and Relative Humidity to Limit Climate-Induced Mechanical Damage in Organic Hygroscopic Materials. 2010. Available online: https://standards.cen.eu/dyn/www/f?p=204:110:0::::FSP_PROJECT:28487&cs=1CA6AC7E107FE7F852A4F9C8D11CCD217 (accessed on 8 March 2021).
- EN16893. Conservation of Cultural Heritage. Specifications for Location, Construction and Modification of Buildings or Rooms Intended for the Storage or Use of Heritage Collections. 2018. Available online: https://standards.cen.eu/dyn/www/f?p=204:110:0::::FSP_PROJECT:35659&cs=15B18B1A035AB93FB36262CA746F5F7EC (accessed on 8 March 2021).
- Collectioncare.eu. European Horizon 2020 Project Collectioncare: Innovative and Affordable Service for the Preventive Conservation Monitoring of Individual Cultural Artefacts during Display, Storage, Handling and Transport. Available online: https://www.collectioncare.eu/ (accessed on 8 March 2021).
- testo Be Sure. WLAN Data Loggers from Testo for Monitoring Temperature and Ambient Conditions. Available online: https://www.testo.com/en-US/products/wifi-datamonitoring (accessed on 23 June 2021).
- R Core Team. R: A Language and Environment for Statistical Computing. 2014. Available online: https://www.r-project.org/about.html (accessed on 8 March 2021).
- Rohart, F.; Gautier, B.; Singh, A.; Lê Cao, K.A. mixOmics: An R package for omics feature selection and multiple data integration. PLoS Comput. Biol.
**2017**, 13, 1–19. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Lê Cao1, K.A.; Dejean, S. mixOmics: Omics Data Integration Project. 2020. Available online: http://www.bioconductor.org/packages/release/bioc/html/mixOmics.html (accessed on 8 March 2021).
- Qiu, D. Package ‘aTSA’. 2015. Available online: https://cran.r-project.org/web/packages/aTSA/index.html (accessed on 8 March 2021).
- Hyndman, R.J.; Khandakar, Y. Automatic time series forecasting: The forecast package for R. J. Stat. Softw.
**2008**, 26. [Google Scholar] [CrossRef] [Green Version] - Hyndman, R.; Athanasopoulos, G.; Bergmeir, C.; Caceres, G.; Chhay, L.; OHara-Wild, M.; Petropoulos, F.; Razbash, S.; Wang, E.; Yasmeen, F. Package Forecast: Forecasting Functions for Time Series and Linear Models. 2020. Available online: https://cran.r-project.org/web/packages/forecast/ (accessed on 8 March 2021).
- Zeileis, A. Implementing a class of structural change tests: An econometric computing approach. Compt. Stat. Data Anal.
**2006**, 50, 2987–3008. [Google Scholar] [CrossRef] [Green Version] - Trapletti, A.; Hornik, K. Package Tseries: Time Series Analysis and Computational Finance. 2019. Available online: https://cran.r-project.org/web/packages/tseries/index.html (accessed on 8 March 2021).
- Komsta, L.; Novomestky, F. Package ’Moments’. 2015. Available online: https://cran.r-project.org/web/packages/moments/index.html (accessed on 8 March 2021).
- Peterson, B.G.; Peter, C. Package ’PerformanceAnalytics’: Econometric Tools for Performance and Risk Analysis. 2020. Available online: https://cran.r-project.org/web/packages/PerformanceAnalytics/index.html (accessed on 8 March 2021).
- Charrad, M.; Ghazzali, N.; Boiteau, V.; Niknafs, A. Package ’NbClust’: Determining the Best Number of Clusters in a Data Set. 2015. Available online: https://cran.r-project.org/web/packages/NbClust/index.html (accessed on 8 March 2021).
- Kovalevsky, S. Package ‘QuantTools’. 2020. Available online: https://quanttools.bitbucket.io/_site/index.html (accessed on 8 March 2021).
- Palma, W. Time Series Analysis. Wiley Series in Probability and Statistics; John Wiley and Sons Inc.: Hoboken, NJ, USA, 2016. [Google Scholar]
- Leisch, F.; Hornik, K.; Kuan, C.M. Monitoring structural changes with the generalized fluctuation test. Econom. Theory
**2000**, 16, 835–854. [Google Scholar] [CrossRef] [Green Version] - Zeileis, A.; Leisch, F.; Hornik, K.; Kleiber, C. Strucchange: An R package for testing for structural change in linear regression models. J. Stat. Softw.
**2002**, 7, 1–38. [Google Scholar] [CrossRef] [Green Version] - Hamilton, J.D. Time Series Analysis, 5th ed.; Princeton University Press: Princeton, NJ, USA, 1994. [Google Scholar]
- Venables, W.N.; Ripley, B.D. Modern Applied Statistics with S, 4th ed.; Springer: New York, NY, USA, 2002. [Google Scholar]
- Brockwell, P.J.; Davis, R.A. Time Series: Theory and Methods, 2nd ed.; Springer: New York, NY, USA, 1987. [Google Scholar]
- R Core Team and Contributors Worldwide. Package ’Stats’. 2020. Available online: https://www.rdocumentation.org/packages/stats/versions/3.6.2 (accessed on 8 March 2021).
- Holt, C.C. Forecasting seasonals and trends by exponentially weighted moving averages. J. Econ. Soc. Meas.
**2004**, 20, 5–10. [Google Scholar] [CrossRef] - Hyndman, R.; Koehler, A.B.; Ord, J.K.; Snyder, R.D. Forecasting with Exponential Smoothing: The State Space Approach; Springer: Berlin, Germany, 2008. [Google Scholar]
- Hyndman, R.; Athanasopoulos, G. Forecasting: Principles and practice; OTexts: 2013. Available online: http://otexts.org/fpp/ (accessed on 25 June 2021).
- Conover, W.J. Practical Nonparametric Statistics, 3rd ed.; John Wiley and Sons, Inc.: Hoboken, NJ, USA, 1999. [Google Scholar]
- Royston, J.P. Algorithm AS 181: The W test for normality. J. R. Stat. Soc. Ser. C Appl. Stat.
**1982**, 31, 176–180. [Google Scholar] [CrossRef] - Royston, J.P. An extension of Shapiro and Wilk’s W test for normality to large samples. J. R. Stat. Soc. Ser. C Appl. Stat.
**1982**, 31, 115–124. [Google Scholar] [CrossRef] - Royston, P. Remark AS R94: A remark on algorithm AS 181: The W-test for normality. J. R. Stat. Soc. C Appl. Stat
**1995**, 44, 547–551. [Google Scholar] [CrossRef] - Taylor B, A.; Emerson, J.W. Package ’dgof’. 2013. Available online: https://cran.r-project.org/web/packages/dgof/index.html (accessed on 8 March 2021).
- Fuller, W.A. Introduction to Statistical Time Series, 2nd ed.; John Wiley and Sons: New York, NY, USA, 1996. [Google Scholar]
- Engle, R.F. Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica
**1982**, 50, 987–1007. [Google Scholar] [CrossRef] - Ljung, G.M.; Box, G.E.P. On a measure of Lack of fit in time series models. Biometrika
**1978**, 65, 297–303. [Google Scholar] [CrossRef] - Cowpertwait, P.S.P.; Metcalfe, A.V. Introductory Time Series with R; Springer Series: Use R; Springer: New York, NY, USA, 2009. [Google Scholar]
- Tseng, J.J.; Li, S.P. Quantifying volatility clustering in financial time series. Int. Rev. Financ. Anal.
**2012**, 23, 11–19. [Google Scholar] [CrossRef] - McLeod, A.I.; Li, W.K. Diagnostic cheking ARMA time series models using squared-residual autocorrelations. J. Time Ser. Anal.
**1983**, 4, 269–273. [Google Scholar] [CrossRef] - Box, G.E.P.; Cox, D.R. An Analysis of Transformations. J. R. Stat. Soc. Ser. B Methodol.
**1964**, 26, 211–243. [Google Scholar] [CrossRef] - Bacon, C. Practical Portfolio Performance Measurement and Attribution, 2nd ed.; Jhon Wiley and Sons: New Delhi, India, 2008; pp. 84–85. [Google Scholar]
- Ribeiro, P.J., Jr.; Diggle, P.J.; Christensen, O.; Schlather, M.; Bivand, R.; Ripley, B. Package ‘geoR’. 2020. Available online: https://cran.r-project.org/web/packages/geoR/index.html (accessed on 8 March 2021).
- Wold, H. Path models with latent variables: The NIPALS approach. In Quantitative Sociology; Elsevier: Amsterdam, The Netherlands, 1975. [Google Scholar] [CrossRef]
- Dray, S.; Pettorelli, N.; Chessel, D. Multivariate Analysis of Incomplete Mapped Data. Trans. GIS
**2003**, 7, 411–422. [Google Scholar] [CrossRef] - Charrad, M.; Ghazzali, N.; Boiteau, V.; Niknafs, A. NbClust: An R Package for Determining the Relevant Number of Clusters in a Data Set. J. Stat. Softw.
**2014**. [Google Scholar] [CrossRef] [Green Version] - Hartigan, J.A.; Wong, M.A. Algorithm AS 136: A K-Means Clustering Algorithm. J. R. Stat. Soc. Ser. C Appl. Stat.
**1979**, 28, 100–108. [Google Scholar] [CrossRef] - MacQueen, J. Some methods for classification and analysis of multivariate observations. In Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability 1965 and 1966, Volume 1: Statistics, 1967; Fifth Berkeley Symposium on Mathematical Statistics and Probability; University of California Press: Berkeley, CA, USA, 1967; pp. 281–297. [Google Scholar]
- Lloyd, S. Least squares quantization in PCM. IEEE Trans. Inf. Theory
**1982**, 28, 129–136. [Google Scholar] [CrossRef] - Forgy, E. Cluster analysis of multivariate data: Efficiency versus interpretability of classifications. Biometrics
**1965**, 21, 768–769. [Google Scholar] - Wold, H. Multivariate Analysis; Wiley: New York, NY, USA, 1966. [Google Scholar]
- Tibshirani, R. Regression shrinkage and selection via the lasso. J. R. Stat. Soc. Ser. B Methodol.
**1996**, 58, 267–288. [Google Scholar] [CrossRef] - Lê Cao, K.A.; Rossouw, D.; Robert-Granié, C.; Besse, P. A sparse PLS for variable selection when integrating omics data. Stat. Appl. Genet. Mol. Biol.
**2008**, 7, 1–29. [Google Scholar] [CrossRef] [PubMed] - Lê Cao, K.A.; Martin, P.G.P.; Robert-Granie, C.; Besse, P. Sparse canonical methods for biological data integration: Application to a cross-platform study. BiomMed Cent. Bioinform.
**2009**, 10. [Google Scholar] [CrossRef] [PubMed] - Chun, H.; Keleş, S. Sparse partial least squares regression for simultaneous dimension reduction and variable selection. J. R. Stat. Soc. Ser. B Methodol.
**2010**, 72, 3–25. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Chung, D.; Keles, S. Sparse partial least squares classification for high dimensional data. Stat. Appl. Genet. Mol. Biol.
**2010**, 9. [Google Scholar] [CrossRef] [PubMed] - Marx, B.D. Iteratively reweighted partial least squares estimation for generalized linear regression. Technometrics
**1996**, 374–381. [Google Scholar] [CrossRef] - Ding, B.; Gentleman, R. Classification using generalized partial least squares. J. Comput. Graph. Statist.
**2005**, 14, 280–298. [Google Scholar] [CrossRef] [Green Version] - Fort, G.; Lambert-Lacroix, S. Classification using partial least squares with penalized logistic regression. Bioinformatics
**2005**, 21, 1104. [Google Scholar] [CrossRef] [Green Version] - Lê Cao, K.A.; Boitard, S.; Besse, P. Sparse PLS discriminant analysis: Biologically relevant feature selection and graphical displays for multiclass problems. BiomMed Cent. Bioinform.
**2011**, 12, 253. [Google Scholar] [CrossRef] [Green Version] - Wegelin, J.A. A Survey of Partial Least Squares (PLS) Methods, with Emphasis on the Two-Block Case; Technical Report; University of Washington: Seattle, WA, USA, 2000. [Google Scholar]
- De Jong, S. SIMPLS: An alternative approach to partial least squares regression. Chemom. Intell. Lab. Syst.
**1993**, 18, 251–263. [Google Scholar] [CrossRef] - Shen, H.; Huang, J.Z. Sparse principal component analysis via regularized low rank matrix approximation. J. Multivar. Anal.
**2008**, 99, 1015–1034. [Google Scholar] [CrossRef] [Green Version] - Lorber, A.; Wangen, L.E.; Kowalski, B.R. A theoretical foundation for the PLS algorithm. J. Chemom.
**1987**, 1, 19–31. [Google Scholar] [CrossRef] - Tenenhaus, M. La Régression PLS: Thórie et Pratique; Editions Technip: Paris, France, 1998. [Google Scholar]
- Mehmood, T.; Liland, K.H.; Snipen, L.; Sæbø, S. A review of variable selection methods in Partial Least Squares Regression. Chemom. Intell. Lab. Syst.
**2012**, 118, 62–69. [Google Scholar] [CrossRef] - Sartorius Stedim Data Analytics, AB. SIMCA 15 Multivariate Data Analysis Solution User Guide; Sartorius Stedim Data Analytics AB: Umeå, Sweden, 2017. [Google Scholar]
- Breiman, L. Random forests. Mach. Learn.
**2001**, 45. [Google Scholar] [CrossRef] [Green Version] - Han, H.; Guo, X.; Yu, H. Variable selection using Mean Decrease Accuracy and Mean Decrease Gini based on Random Forest. In Proceedings of the 2016 7th IEEE International Conference on Software Engineering and Service Science (ICSESS), Beijing, China, 26–28 August 2016; pp. 219–224. [Google Scholar] [CrossRef]
- Murphy, K.P. Machine Learning a Probabilistic Perspective; The MIT Press: Cambridge, MA, USA; London, UK, 2012. [Google Scholar]
- Menze, B.H.; Kelm, B.M.; Masuch, R.; Himmelreich, U.; Bachert, P.; Petrich, W.; Hamprecht, F.A. A comparison of random forest and its Gini importance with standard chemometric methods for the feature selection and classification of spectral data. BMC Bioinform.
**2009**, 10. [Google Scholar] [CrossRef] [Green Version] - Liaw, A.; Wiener, M. Package randomForest: Breiman and Cutler’s Random Forests for Classification and Regression. 2018. Available online: https://cran.r-project.org/web/packages/randomForest/index.html (accessed on 8 March 2021).
- Gaetano, D. Forecast combinations in the presence of structural breaks: Evidence from U.S. equity markets. Mathematics
**2018**, 6, 34. [Google Scholar] [CrossRef] [Green Version] - Struzik, Z.R.; Siebes, A. The haar wavelet in the time series similarity paradigm. In Principles of Data Mining and Knowledge Discovery PKDD 1999; Lecture Notes in Computer Science; Museum of London Archaeology Service: Prague, Czech Republic, 1999; Volume 1704, pp. 12–22. [Google Scholar]
- Galeano, P.; Peña, D. Multivariate Analysis in Vector Time Series; UC3M Working Papers. Statistics and Econometrics 2001-15; Universidad Carlos III de Madrid: Madrid, Spain, 2000; pp. 1–19. [Google Scholar]
- Caiado, J.; Crato, N.; Peña, D. A periodogram-based metric for time series classification. Compt. Stat. Data Anal.
**2006**, 50, 2668–2684. [Google Scholar] [CrossRef] [Green Version] - Douzal, C.A.; Nagabhushan, P. Adaptive dissimilarity index for measuring time series proximity. Adv. Data Anal. Classif.
**2007**, 1, 5–21. [Google Scholar] [CrossRef] - Piccolo, D. A distance measure for classifying arima models. J. Time Ser. Anal.
**1990**, 11, 153–164. [Google Scholar] [CrossRef] - Maharaj, E.A. A significance test for classifying ARMA models. J. Stat. Comput.
**1996**, 54, 305–331. [Google Scholar] [CrossRef] - Maharaj, E.A. Comparison of non-stationary time series in the frequency domain. Comput. Stat. Data Anal.
**2002**, 40, 131–141. [Google Scholar] [CrossRef] [Green Version] - Kakizawa, Y.; Shumway, R.H.; Taniguchi, M. Discrimination and clustering for multivariate time series. JASA
**1998**, 93, 328–340. [Google Scholar] [CrossRef] - Vilar, J.A.; Pértega, S. Discriminant and cluster analysis for gaussian stationary processes: Local linear fitting approach. J. Nonparametr. Stat.
**2004**, 16, 443–462. [Google Scholar] [CrossRef] - Li, M.; Vit anyi, P. An introduction to kolmogorov complexity and its applications. In Text and Monographs in Computer Science; Springer: New York, NY, USA, 2007. [Google Scholar] [CrossRef]
- Li, M.; Badger, J.H.; Chen, X.; Kwong, S.; Kearney, P.; Zhang, H. An information-based sequence distance and its application to whole mitochondrial genome phylogeny. Bioinformatics
**2001**, 17, 149–154. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Li, M.; Chen, X.; Li, X.; Ma, B.; Vitanyi, P. The similarity metric. IEEE Trans. Inf. Theory
**2004**, 50, 3250–3264. [Google Scholar] [CrossRef] - Cilibrasi, R.C.; Vit anyi, P.M. Clustering by compression. IEEE Trans. Inf. Theory
**2005**, 51, 1523–1545. [Google Scholar] [CrossRef] [Green Version] - Keogh, E.; Lonardi, S.; Ratanamahatana, C.A.; Wei, L.; Lee, S.H.; Handley, J. Compression based data mining of sequential data. Data Min. Knowl. Discov.
**2007**, 14, 99–129. [Google Scholar] [CrossRef] - Brandmaier, A.M. Permutation Distribution Clustering and Structural Equation Model Trees. Ph.D. Thesis, Universitat des Saarlandes, Saarbrücken, Germany, 2011. [Google Scholar]
- Alonso, A.M.; Berrendero, J.R.; Hernández, A.; Justel, A. Time series clustering based on forecast densities. Comput. Stat. Data Anal.
**2006**, 51, 762–776. [Google Scholar] [CrossRef] - Vilar, J.A.; Alonso, A.M.; Vilar, J.M. Non-linear time series clustering based on non-parametric forecast densities. Comput. Stat. Data Anal.
**2010**, 54, 2850–2865. [Google Scholar] [CrossRef] - Batista, G.; Wang, X.; Keogh, E. A complexity-invariant distance measure for time series. In Proceedings of the Eleventh SIAM International Conference on Data Mining, SDM11, Mesa, AZ, USA, 28–30 April 2011; pp. 699–710. [Google Scholar] [CrossRef] [Green Version]
- Vilar, J.; Montero, P. TSclust: An R package for time series clustering. J. Stat. Softw.
**2014**, 62. [Google Scholar] [CrossRef] [Green Version] - Guha, A.; Ho, N.; Nguyen, X. On Posterior Contraction of Parameters and Interpretability in Bayesian Mixture Modeling. 2019. Available online: https://arxiv.org/abs/1901.05078 (accessed on 8 March 2021).

**Figure 2.**View of the skylight covering part of L’Almoina archaeological site: (

**a**) external view from the pedestrian plaza; and (

**b**) internal view.

**Figure 3.**Trajectories of the different time series of T from the 27 sensors located in the museum. Values were recorded between 22 October 2019 and 20 October 2020. The separation of different stages (Wr1, Cd, Tr, Ht and Wr2) is indicated by means of solid vertical lines. Dashed vertical lines indicate the structural breaks identified within the stages Wr1, Cd and Ht.

**Figure 4.**Summary of steps involved in method M1: blue lines, Type 1 classification variables; green lines, Type 2 variables; solid lines, process; dashed line, results. Different boxes contain the name of the stages (i.e., Wr1A, Wr1B, CdA, CdB, Tr, HtA, HtB and Wr2) to indicate that the procedure was applied to all parts of the time series. A structural break was found in Wr1, Cd and Ht, so that the suffixes A and B denote the substages before and after the break, respectively.

**Figure 5.**Summary of steps in method M2: blue lines, Type 3 classification variables; green lines, Type 4 variables; red lines, Type 5 variables; solid lines, process; dashed line, results.

**Figure 6.**Summary of steps in method M3: blue lines, Type 6 classification variables; green lines, Type 7 variables; solid lines, process; dashed line, results.

**Figure 7.**Summary of steps of method M4: red lines, Type 7 variables of M3; green lines, Type 8 variables; solid lines, process; dashed line, results.

**Figure 8.**Results associated with the k-means method. (

**a**) Absolute frequency (number) of indices that indicates the best number of classes in the museum. For example, two classes are selected by seven indices. (

**b**) Classification of sensors installed in the museum, according to the k-means method. Each color (blue, green and orange) corresponds to a different class.

**Figure 9.**Evaluation of the PLS-DA performance for the classification of sensors into three categories. Vertical axes indicate the classification error rate (CER) for each prediction distance as a function of the number of components (horizontal axis) for: M1 (

**a**); M2 (

**b**); M3 (

**c**); M4 (

**d**); and using the variables from all methods (

**e**). Three types of prediction distances were considered: Mahalanobis (green lines), maximum (blue lines) and centroid (red lines). Two types of CER were computed: balanced error rate (dashed lines) and overall error rate (solid lines). PLS-DA was carried out using repeated three-fold CV 1000 times.

**Figure 10.**Evaluation of the PLS-DA performance (considering three components) for the classification of sensors into three categories. Vertical axes indicate the Balance error rate (BER) per component (orange lines, Component 1; green lines, Component 2; and blue lines, Component 3). BER values were computed across all folds using 5, 10 or 15 variable (horizontal axes) for eachmethod: M1 (

**a**);M2 (

**b**);M3 (

**c**);M4 (

**d**); and all methods (Ms) (

**e**). The three-fold CV technique was run 1000 times, using maximum distance prediction. Diamonds highlight the optimal number of variables per component.

**Figure 11.**Error rates derived from the sPLS-DA and RF algorithms. Red points are OOB classification error rates per method, based on different sets of classification variables: M1, M2, M3, M4 and all variables (Ms). Blue points are mean values of BER for all the components, per method, for sPLS-DA.

**Figure 12.**Projection of sensors over the three relevant components (C1–C3) from sPLS-DA, per method (M1–M4) or when using all variables (Ms). Graphs show discrimination of the sensors, according to three classes: North Western (NW), South Eastern (SE) and Skylight (Sk). Color codes: NW sensors in blue, Sk in orange and SE in green. Each graph displays a confidence ellipse for each class (at a confidence level of 95%), in order to highlight the strength of the discrimination.

**Figure 13.**(

**a**) Projection of sensors over components C1 and C2 from sPLS-DA for M1. The graph displays a good discrimination of the sensors according to the classes. They are color coded according to the zone where the sensor is located: NWin blue, SE in green and Sk in orange. The most important variables for (

**b**) C1 (

**c**) and C2, according to the absolute value of their coefficients, are ordered from bottom to top. The color corresponds to the zone in which the variable yields the highest mean component value.

**Table 1.**The most successful models per stage of the different observed time series $\mathbf{r}$ are presented in the second column. Column 3 presents the percentages of the LBQ test on the different lags from $nt$ to $nt+48$ from the 27 sensors that fulfill the assumptions of independence. Column 4 presents the percentages of the LM test from the 27 sensors that fulfill the assumptions of the absence of Arch effect. The significance level used was 0.01.

Stage | Model | LBQ | LM |
---|---|---|---|

(a) For Method 3 | |||

Wr1A | Seasonal ARIMA$(0,1,2){(2,1,0)}_{24}$ | 77.00 | 92.52 |

Wr1B | Seasonal ARIMA$(0,1,0){(2,1,0)}_{24}$ | 3.00 | 44.44 |

CdA | Seasonal ARIMA$(0,1,0){(2,1,0)}_{24}$ | 25.00 | 77.77 |

CdB | Seasonal ARIMA$(0,1,2){(2,1,0)}_{24}$ | 18.00 | 18.52 |

Tr | Seasonal ARIMA$(0,1,3){(2,1,0)}_{24}$ | 11.00 | 3.70 |

HtA | Seasonal ARIMA$(1,1,3){(0,1,1)}_{24}$ | 22.00 | 22.22 |

HtB | Seasonal ARIMA$(0,1,3){(2,1,0)}_{24}$ | 0.00 | 37.04 |

Wr2 | Seasonal ARIMA$(0,1,2){(2,1,0)}_{24}$ | 18.00 | 37.04 |

(b) For Method 4 | |||

Wr1A | A seasonal ARIMA per sensor | 92.59 | 96.30 |

Wr1B | A seasonal ARIMA per sensor | 51.85 | 59.26 |

CdA | A seasonal ARIMA per sensor | 81.48 | 81.48 |

CdB | A seasonal ARIMA per sensor | 48.15 | 25.93 |

Tr | A seasonal ARIMA per sensor | 25.93 | 3.70 |

HtA | A seasonal ARIMA per sensor | 59.26 | 29.63 |

HtB | A seasonal ARIMA per sensor | 25.93 | 44.44 |

Wr2 | A seasonal ARIMA per sensor | 55.55 | 37.04 |

**Table 2.**The most successful seasonal ARIMA $(p,d,q){(P,D,Q)}_{S}$ model per sensor for different stages, with $S=24$ and $D=d=1$.

Sensors | ||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Stage | Order | D | B | A | C | G | B2 | B3 | B4 | A3 | A2 | C3 | B5 | C5 | C6 | D6 | D5 | A4 | A5 | B6 | A6 | F | D1 | D3 | C0 | C1 | B1 | D2 |

Wr1A | p | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 2 | 2 | 0 |

q | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 3 | 1 | 1 | 1 | 2 | 1 | 3 | 2 | 1 | 0 | 0 | 0 | 2 | 1 | 3 | 0 | 0 | 1 | |

P | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |

Q | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

Wr1B | p | 1 | 1 | 1 | 1 | 1 | 0 | 2 | 1 | 0 | 1 | 0 | 1 | 0 | 2 | 2 | 1 | 2 | 3 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 |

q | 1 | 1 | 1 | 1 | 2 | 1 | 0 | 2 | 2 | 2 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | |

P | 2 | 2 | 2 | 2 | 2 | 0 | 2 | 2 | 2 | 2 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |

Q | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

CdA | p | 1 | 0 | 0 | 0 | 1 | 2 | 0 | 2 | 1 | 1 | 2 | 2 | 1 | 0 | 0 | 1 | 3 | 0 | 0 | 2 | 3 | 0 | 1 | 0 | 1 | 0 | 1 |

q | 1 | 1 | 2 | 2 | 3 | 0 | 2 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 2 | 0 | 2 | 0 | 3 | 1 | 1 | 0 | 3 | 1 | |

P | 2 | 2 | 2 | 2 | 0 | 2 | 0 | 2 | 2 | 2 | 0 | 2 | 0 | 2 | 0 | 2 | 0 | 2 | 2 | 0 | 2 | 0 | 1 | 2 | 2 | 2 | 2 | |

Q | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 0 | 2 | 2 | 1 | 0 | 2 | 0 | 0 | 1 | 0 | 2 | 2 | 1 | 2 | 0 | 1 | |

CdB | p | 0 | 0 | 0 | 0 | 3 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 0 | 2 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 3 | 3 | 1 | 3 | 1 | 3 |

q | 2 | 1 | 3 | 3 | 0 | 3 | 2 | 2 | 1 | 2 | 1 | 2 | 1 | 0 | 1 | 1 | 2 | 3 | 3 | 1 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | |

P | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |

Q | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

HtA | p | 2 | 1 | 2 | 1 | 3 | 0 | 0 | 3 | 0 | 2 | 3 | 2 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 2 | 0 | 3 | 1 |

q | 2 | 3 | 0 | 1 | 1 | 1 | 3 | 0 | 4 | 2 | 0 | 1 | 2 | 1 | 3 | 3 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 2 | 2 | 1 | 1 | |

P | 0 | 0 | 2 | 2 | 0 | 2 | 1 | 2 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 2 | 0 | 0 | 2 | 0 | 2 | |

Q | 1 | 1 | 1 | 1 | 1 | 2 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 1 | 0 | 1 | 1 | 1 | 1 | 0 | |

HtB | p | 2 | 3 | 0 | 0 | 1 | 1 | 3 | 0 | 0 | 0 | 1 | 1 | 0 | 3 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 2 | 0 | 2 | 0 | 0 |

q | 1 | 0 | 2 | 1 | 1 | 2 | 1 | 2 | 0 | 2 | 2 | 1 | 1 | 0 | 3 | 0 | 2 | 2 | 2 | 1 | 0 | 1 | 0 | 3 | 0 | 2 | 1 | |

P | 0 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |

Q | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

Tr | p | 2 | 3 | 0 | 0 | 3 | 0 | 1 | 0 | 1 | 0 | 3 | 1 | 3 | 3 | 3 | 3 | 1 | 3 | 0 | 3 | 0 | 3 | 1 | 0 | 3 | 0 | 1 |

q | 0 | 0 | 1 | 3 | 0 | 1 | 3 | 3 | 1 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 3 | 0 | 1 | 3 | 0 | 3 | 2 | |

P | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |

Q | 0 | 0 | 0 | 0 | 0 | 2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

Wr2 | p | 3 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 2 | 0 | 1 | 3 | 2 | 0 | 0 | 2 | 1 | 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

q | 0 | 1 | 3 | 0 | 2 | 3 | 3 | 1 | 3 | 0 | 2 | 2 | 0 | 1 | 3 | 3 | 0 | 2 | 2 | 0 | 0 | 3 | 2 | 2 | 3 | 3 | 2 | |

P | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |

Q | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

**Table 5.**Percentages of selected variables per stage of the time series for each component (C) and each method (M). Values were computed according to the information contained in Table 3 and Table 4 (e.g., the value 60.0% for C2 of M2 means that 9 out of the 15 selected variables correspond to HtA, according to Table 3 (b)). The two highest values of each column are highlighted in bold and blue, but only one is selected in case of a percentage > 50%.

M1 | M2 | M3 | M4 | Ms | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Stage | C1 | C2 | C3 | C1 | C2 | C1 | C2 | C1 | C2 | C3 | C1 | C2 | C3 |

Wr1A | 0.00 | 6.70 | 30.80 | 0.00 | 6.70 | 0.00 | 6.70 | 20.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

Wr1B | 13.30 | 20.00 | 7.70 | 6.70 | 0.00 | 0.00 | 0.00 | 0.00 | 6.70 | 20.00 | 0.00 | 0.00 | 33.30 |

CdA | 6.70 | 13.30 | 23.10 | 0.00 | 26.70 | 0.00 | 26.70 | 0.00 | 13.30 | 0.00 | 0.00 | 20.00 | 6.70 |

CdB | 6.70 | 13.30 | 7.70 | 0.00 | 0.00 | 0.00 | 6.70 | 0.00 | 0.00 | 40.00 | 0.00 | 0.00 | 53.30 |

Tr | 26.70 | 0.00 | 7.70 | 40.00 | 6.70 | 60.00 | 6.70 | 20.00 | 26.70 | 0.00 | 60.00 | 6.70 | 0.00 |

HtA | 20.00 | 20.00 | 7.70 | 6.70 | 60.00 | 20.00 | 26.70 | 60.00 | 20.00 | 0.00 | 20.00 | 60.00 | 6.70 |

HtB | 13.30 | 6.70 | 7.70 | 33.30 | 0.00 | 20.00 | 13.30 | 0.00 | 6.70 | 0.00 | 20.00 | 0.00 | 0.00 |

Wr2 | 13.30 | 20.00 | 7.70 | 13.30 | 0.00 | 0.00 | 13.30 | 0.00 | 26.70 | 40.00 | 0.00 | 13.30 | 0.00 |

**Table 7.**Results from the random forest algorithm: percentages of selected variables per stage and method (M1–M4) or when using all variables from the four methods (Ms). M2 was the only method which used ’all observations’ and the ’time series split per stage’ for computing the prediction of the time series of T. For the first column (Stage), ’All’ refers to ’all observations of a time series’ and this category is only used for M2. The two largest percentages per method are highlighted in bold and blue.

Stage | M1 | M2 | M3 | M4 | Ms |
---|---|---|---|---|---|

Wr1A | 6.80 | 0.00 | 11.10 | 8.80 | 0.00 |

Wr1B | 11.40 | 2.40 | 3.70 | 10.50 | 2.70 |

CdA | 11.40 | 19.00 | 11.10 | 15.80 | 13.50 |

CdB | 11.40 | 0.00 | 11.10 | 7.00 | 8.10 |

Tr | 15.90 | 19.00 | 25.90 | 17.50 | 21.60 |

HtA | 15.90 | 33.30 | 25.90 | 17.50 | 45.90 |

HtB | 13.60 | 9.50 | 11.10 | 7.00 | 2.70 |

Wr2 | 13.60 | 14.30 | 0.00 | 15.80 | 5.40 |

All | 0.00 | 2.40 | 0.00 | 0.00 | 0.00 |

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Ramírez, S.; Zarzo, M.; García-Diego, F.-J.
Multivariate Time Series Analysis of Temperatures in the Archaeological Museum of L’Almoina (Valencia, Spain). *Sensors* **2021**, *21*, 4377.
https://doi.org/10.3390/s21134377

**AMA Style**

Ramírez S, Zarzo M, García-Diego F-J.
Multivariate Time Series Analysis of Temperatures in the Archaeological Museum of L’Almoina (Valencia, Spain). *Sensors*. 2021; 21(13):4377.
https://doi.org/10.3390/s21134377

**Chicago/Turabian Style**

Ramírez, Sandra, Manuel Zarzo, and Fernando-Juan García-Diego.
2021. "Multivariate Time Series Analysis of Temperatures in the Archaeological Museum of L’Almoina (Valencia, Spain)" *Sensors* 21, no. 13: 4377.
https://doi.org/10.3390/s21134377