Functional datasets are those in which each datum takes the form of a continuous curve or surface which is a function of at least one other variable. A recent development in FDA includes a method for calculating scalar-valued summary statistics, specifically the variance and correlation, for functional datasets . The method is described briefly here, and a simple example is illustrated schematically in Fig. . Full details and proofs are given in . As background to the formulation below, functional data are represented as linear combinations of orthogonal basis functions, with associated weights known as basis coefficients. Basis functions are smooth reference curves that act as building blocks for reconstructing profiles. Orthogonality implies that each basis function captures an independent component of the data that cannot be reproduced by combining the others. Changes in the basis coefficients modify the values of the functional data and may also change its overall shape. Measuring the variability of the basis coefficients therefore provides a way to quantify variability in a functional dataset. Extending this to two datasets, together with the covariance of their basis coefficients, yields a natural description of dependence between the datasets.

Suppose that X and Y are two datasets each containing n profiles (for example, paired chlorophyll and temperature profiles). Decompose each profile from X and Y into p orthogonal basis functions with ai,j and bi,j denoting the coefficient of the jth basis function of the ith profile in sets X and Y respectively. Calculation of the variability of the basis coefficients requires first computing their mean values, which for sets X and Y are respectively given by 1Aj‾=1n∑i=1nai,j,Bj‾=1n∑i=1nbi,j.

Using the mean basis coefficients with the basis functions represents a mean function for each set, describing a characteristic profile shape. Using these mean basis coefficients, the variance of the coefficients can be calculated for each of the basis functions (denoted Vaj and Vbj respectively). The variances of sets X and Y are then simply the sums of the basis coefficient variances Vaj and Vbj respectively. 2Var(X)=∑j=1pVaj=∑j=1p1n∑i=1n(ai,j-Aj‾)23Var(Y)=∑j=1pVbj=∑j=1p1n∑i=1n(bi,j-Bj‾)2 Similarly, the covariance between sets X and Y is defined as the sum of the covariances of each basis coefficient Cj. 4Cov(X,Y)=∑j=1pCj=∑j=1p1n∑i=1n(ai,j-Aj‾)(bi,j-Bj‾) The equation to calculate the correlation between the sets X and Y is the same for scalar data. 5Cor(X,Y)=Cov(X,Y)Var(X)Var(Y)

The correlation for functional data is restricted to the closed interval [-1,1], identical to that for scalar data. Mathematically, a correlation of +1 implies that corresponding basis coefficients are perfectly positively linearly related, such that a positive change in a basis coefficient in one dataset results in a proportional positive change in the same coefficient in the corresponding observation. A deviation (from the mean function) in a specific basis component in set X corresponds to a deviation in the same direction in the equivalent component in the set Y. In contrast, a correlation of -1 represents a case where each basis is perfectly linear and negatively correlated. Specifically, this implies that any deviation in a particular component in set X corresponds to a deviation in the opposite direction from the mean function in set Y. Note that this does not imply that each pair of functions is a pointwise negative of the other, but instead they vary in opposite directions along the structural features captured by the basis coefficient. Figure  shows several examples of datasets containing pairs of oceanographic profiles, illustrating a range of correlation structures. In Examples 1 and 2, the profiles exhibit strong positive correlation, meaning that positive deviations from the mean in one dataset are mirrored by similar deviations in the other. Notably, Example 2 highlights that this correlation coefficient reflects dependence between deviations, rather than similarity in the overall shape of the mean profiles. In contrast, Examples 3 and 4 show negligible correlation, with variations in one profile occurring independently of the other. Finally, Examples 5 and 6 demonstrate negative correlation, where positive deviations in one dataset are associated with corresponding negative deviations in the other.

It is worth noting that this method could, in principle, be extended to compare sets of profiles measured over different depth ranges, although an example is not provided in the present study. This could be achieved by rescaling each range to a common interval (e.g., [0,1]) and representing the profiles using regularly spaced measurements with a consistent number of points and basis functions. The interpretation would then differ slightly, in that a deviation in one set of profiles may correspond to a deviation at a different physical depth in another set. This approach could also help identify depth ranges over which two variables are correlated, such as within the mixed layer, by iteratively repeating the analysis with modified profile segments to detect where the dependence in profile shape breaks down.

In practice, the profiling data from each set were stored as a matrix, with each column representing a profile and each row representing a depth level. Each profile was converted to basis function coefficients using a fast Fourier transform (FFT), where the number of basis functions equals the number of depth levels in the profile (and therefore the maximum number of basis functions is equal to the number of depth levels). Fourier bases were chosen because many oceanographic profiles vary smoothly with depth, allowing compact representation by sinusoidal functions, and because they performed well in previous work on temperature time series . This transformation requires complete data vectors; therefore, profiles with missing values were removed or gap-filled prior to analysis. Details of the preprocessing steps are provided in the case studies in Sect. 3. It is worth noting that this study does not explore the effect of smoothing on the computed correlation coefficients. However, the approach is expected to be robust, provided that the general shape of the profiles is sufficiently clear to allow identification of meaningful water column features. Where profiles exhibit substantial noise relative to the underlying signal, smoothing is recommended prior to computing correlation coefficients.

The FFT produces complex-valued coefficients, stored as a matrix where each row now contains coefficients associated with a different basis function. To preserve linearity for subsequent statistical calculations, the variance of each coefficient was defined as the sum of the variances of its real and imaginary components, and the covariance between coefficients as the sum of the covariances of the corresponding real and imaginary parts. This enables direct computation of variance and covariance in the transformed space, whereas some alternative basis expansions yield real-valued coefficients only. Row-wise variances were computed for each coefficient matrix and summed across coefficients to obtain dataset-level variances. Covariances were computed similarly by summing row-wise covariances between paired matrices.

To compute correlations between oceanographic variables, each data matrix contained profiles of a single variable, with simultaneously measured profiles aligned in corresponding columns. Temporal ACFs were estimated by identifying all profile pairs separated by a specified time lag. For each lag, earlier profiles formed one matrix and later profiles the other, ensuring matched dimensions and indexing. Time lags up to two years (730 d) were evaluated. All analyses were conducted in R version 4.4.1 . To improve computational efficiency, the search for profile pairs at each temporal lag was implemented in C++, which was substantially faster than the equivalent R implementation, and executed on a workstation with an Intel Core i5 processor. The data for both case studies and code used in the analysis are publicly available at 10.5281/zenodo.18164579 .

The proposed method extends existing statistical tools for oceanographic profile analysis by summarising the similarity between two sets of profiles with a single correlation coefficient, analogous to the scalar-valued case. Importantly, the coefficient reflects both profile shape as well as the absolute magnitude, so dependencies in either characteristic are captured. Given the widespread familiarity with correlation coefficients, this approach provides a natural extension of scalar correlation analyses to full vertical profiles.

In an exploratory data analysis context, the method enables rapid identification of potential dependencies among multiple oceanographic variables and can inform subsequent modelling decisions. In the first case study, simultaneously measured profiles of temperature, salinity, potential density, and DO were analysed. The resulting correlation matrix (Table ) suggests that salinity-driven density variability is a primary driver of DO variability at the mooring site. The high correlations among hydrographic variables, notably between salinity and potential density, indicate multicollinearity, implying that a model predicting DO profiles may not require all three predictors. These high correlations may be expected given that the variables are connected through well established physical relationships . The second case study shows that the correlation between chlorophyll and temperature varies spatially (Table ). The difference in correlation coefficients suggests that region specific analyses are required, as a single relationship may not hold across locations. In some regions, factors other than water column structure – such as nutrient availability or light – may exert stronger control on chlorophyll profiles .

Oceanographers frequently investigate the temporal scales of variability in oceanographic time series, for which ACFs are a standard diagnostic tool. Previous approaches have produced ACFs for measurements collected at the surface and within specific depth bins . In addition, modes of variability have been identified using empirical orthogonal functions (EOFs) or by computing seasonal climatologies . The advantage of this approach is the quantification of temporal autocorrelation at specific time lags for time series of entire profiles. For example, in the first case study, it is shown that the autocorrelation of DO decays significantly faster than hydrographic variables for the shortest time lags (Fig. ). When combined with weaker annual autocorrelation, this suggests that variability over short time scales is relatively more important for DO profiles, as has been shown for surface chlorophyll measurements . The second case study builds on this by showing that the similarity between the ACFs of two oceanographic variables can vary spatially. Furthermore, the fact that BGC-Argo floats can move substantial distances over time can lead to temporal variability being dominated by spatial variability. This is demonstrated in Fig. e–g where the ACFs do not display a sinusoidal curve. Visual inspection of the corresponding time series (Fig. e–g) confirms that that seasonal cycles are not present and sinusoidal ACFs would not be expected.

Several caveats accompany this approach. First, it quantifies only linear dependence between sets of profiles, so non linear relationships will not be reflected in the coefficient. Second, as in the analysis of scalar-valued data, the correlation coefficient presented here does not quantify the magnitude of the causal effects between functional variables in the way that the slope of a linear regression does. Consequently, functional regression models are required to obtain this type of information . Interpreting correlation scores can also be challenging because the coefficient alone gives no indication of which depths contribute to the relationship, so this method should be used alongside other tools when investigating drivers of profile variability.

The FDA approach presented here assumes that observations represent smooth functions of the indexing variable and that the sampling resolution is sufficient to capture the main vertical structure of the profile, which oceanographic profiles typically satisfy. An appropriate set of basis functions is therefore required; Fourier bases are recommended because they produce smooth function estimates while capturing variation across a range of spatial scales. However, as with any basis expansion, artefacts can arise when profiles contain sharp gradients or substantial noise. For example, truncated Fourier representations may exhibit oscillatory behaviour near sharp transitions (similar to the Gibbs phenomenon), and noisy observations can introduce small-scale fluctuations in the fitted curves. Consequently, some smoothing may be necessary prior to analysis, although excessive smoothing can artificially inflate correlation estimates. In cases where profiles from different platforms have different vertical resolutions, it is important to ensure that the analysis resolution is adequate to resolve the main differences in shape. This may require either reducing the resolution of higher-resolution profiles or interpolating lower-resolution profiles to a finer grid, depending on the vertical scale of the features being investigated.

In addition, long time lag estimates may be affected by sensor drift, which is well documented for autonomous platforms . Finally, the method becomes less effective for identifying long-term autocorrelation when profiling platforms move large distances, particularly between distinct oceanic regions with different hydrographic or ecological characteristics (Fig. ).

The scalar correlation framework has broad potential across oceanographic research. It can be applied to the growing collection of datasets from moorings, profiling floats and gliders, and to model output or reanalyses to evaluate coupled physical–biogeochemical variability. As profiling data from autonomous platforms continue to expand, this method offers a way to quantify spatio-temporal autocorrelation for multiple variables and across a wide range of scales. The approach is well suited to parallelisation, making it feasible for analysing high-resolution ocean model output or global observational archives.

This framework could support observing system design for programs such as the (BGC-)Argo array , and extend previous decorrelation length scale studies by incorporating full vertical profile structure. It could also be applied to high resolution glider datasets to evaluate efficient sampling strategies or to compare and cross calibrate nearby platforms. Combined with satellite products such as geostrophic velocities , it may help assess the role of advection in driving profile variability. Alternatively, it could be beneficial to assess small scale processes through the autocorrelation of climatological anomalies.

The method could also, in principle, be used to assess reconstructed profile shapes from machine learning predictions . Although such models are often evaluated using pointwise metrics (e.g., RMSE), the proposed approach provides a complementary means of comparing predicted and observed profiles, potentially offering further insight into the representation of vertical water column structure. Similarly, outputs from a general circulation model (GCM) can be validated by pairing predicted profiles with the observations used to generate the model , while similar analyses across multiple GCMs can help assess inter-model variability. In all cases, differences in vertical resolution between products may need to be accounted for. Another possible use is exploring potential relationships between the vertical distributions of different phytoplankton communities and their preferred environmental conditions.

Future developments could include extending this approach to quantify spatial dependence in vertical profiles within spatial or spatio-temporal models . The framework could also be integrated with functional regression and clustering methods to identify coherent oceanic regimes . In addition, the autocorrelation framework could be used to fit functional time series models, such as functional autoregressive models .