Principal component analysis on interval data

Summary

Real world data analysis is often affected by different types of errors as: measurement errors, computation errors, imprecision related to the method adopted for estimating the data.

The uncertainty in the data, which is strictly connected to the above errors, may be treated by considering, rather than a single value for each data, the interval of values in which it may fall: the interval data. Statistical units described by interval data can be assumed as a special case of Symbolic Object (SO). In Symbolic Data Analysis (SDA), these data are represented as boxes. Accordingly, purpose of the present work is the extension of Principal Component analysis (PCA) to obtain a visualisation of such boxes, on a lower dimensional space pointing out of the relationships among the variables, the units, and between both of them. The aim is to use, when possible, the interval algebra instruments to adapt the mathematical models, on the basis of the classical PCA, to the case in which an interval data matrix is given. The proposed method has been tested on a real data set and the numerical results, which are in agreement with the theory, are reported.

Appendix

Given two single-valued variables: X_r = (x_ir), X_s =(x_is), i = 1, …,n, it is known that the correlation between X_r and X_s may be computed as follow:

$$corr\left(X_{r}, X_{s}\right)=h\left(x_{1, r}, \ldots x_{n, r} ; x_{1, s}, \ldots x_{n, s}\right)=\frac{cov\left(X_{r}, X_{s}\right)}{\sqrt{var\left(X_{r}\right)} \sqrt{var\left(X_{s}\right)}}$$

((1))

Let us consider now the following interval-valued variables:

$$X_{r}^{1}=\left(X_{i r}=\left[\underline{x}_{i r}, \overline{x}_{i r}\right]\right) \quad, \quad X_{s}^{1}=\left(X_{i s}=\left[\underline{x}_{is}\ \ , \ \\\overline{x}_{is}\right]\right)_{i} \quad i=1, \ldots, n$$

the interval correlation is computed as follow (Gioia & Lauro 2005):

$$Corr\left(X_{r}^{I}, X_{s}^{I}\right)=\left[\begin{array}{lll}\\ \\\ \ min \quad h\left(x_{l, r}, \ldots, x_{n, r} ; x_{l, s}, \ldots, x_{n, s}\right) \quad, \quad &\ \ max \quad h\left(x_{l, r}, \ldots, x_{n, r} ; x_{l, s}, \ldots, x_{n, s}\right)\\ \ \\ {x_{i r} \in X_{i r}} & {x_{i r} \in X_{i r}} \\ {x_{i s} \in X_{i s}} & {x_{i s} \in X_{i s}} \\ \\ i=1, \ldots, n & i=1, \ldots, n\end{array}\right]$$

where h(x_1,r,…,x_n,r;x_1,s,…,x_n,s) is the function in (1) .

Analogously, given the single-valued variable X_n the standardized S_j=(s_ir)_i, of X_r is given by:

$$s_{i r}=\frac{x_{i r}-\overline{x}_{r}}{\sqrt{n} \cdot \sigma_{r}^{2}}, \quad i=1, \ldots, n$$

((2))

where $\overline{x}_{r}$ and $\sigma_{r}^{2}$ are the mean and the variance of X_r respectively.

When an interval-valued variable $X_{r}^{I}$ is given, following the same approach of (Gioia & Lauro 2005), the component s_ir in (2) , for each i=1,…,n, transforms into the following function:

$$s_{i r}\left(x_{i r}, \ldots x_{n r}\right)=\frac{x_{i r}-\overline{x}_{r}}{\sqrt{n} \cdot \sigma_{r}^{2}}$$

((3))

as x_ir varies in $\lfloor \underline{x}_{i r}, \overline{x}_{i r} \rfloor,$, i=1,…,n. The standardized interval component $s_{i r}^{I}$ of $X_{r}^{I}$ may be computed by minimizing/maximizing function (3) , i.e. calculating the following set:

$$s_{i r}^{I}=\left[\begin{array}{lll} \\ \\ \ min\ \ s_{i r}\left(x_{i r}, \ldots x_{n r}\right) \quad, \quad &\ \ max\ \ s_{i r}\left(x_{i r}, \ldots x_{n r}\right) \\ {x_{i r} \in X_{i r}} &{x_{i r} \in X_{i r}} \\ i=1 ,\ldots n & i=1, \ldots n\end{array}\right]$$

((4))

$s_{i r}^{I}$ in (4) is the interval of the standardized component s_ir that may be computed when each component x_ir ranges in its interval of values. For computing the interval standardized matrix S^I of an n×p matrix X^I, interval (4) may be computed for each i=1,…,n and each r=1, …,p. Given a real matrix X and indicating by S the standardized of X, it is defined the product matrix:$S S^{\prime}=\left(s s^{\prime}_{i j}\right)$. Given an interval matrix X^I, the product of S^I by its transpose will not be computed by the interval matrix product (S’)¹/S¹ but by minimizing/maximizing each component of SS’ when X_ij varies in its interval of values. The interval matrix $\left(S S^{\prime}\right)^{I}=\left(\left(s s_{i j}^{\prime}\right)^{I}\right)$ is:

$$\left(s s_{i j}^{\prime}\right)^{I}=\left[\begin{array}{lll} \\ \\min\ \ ss^{\prime}_{i j}\left(x_{i j}, \ldots x_{n j}\right) \quad, \quad &max\ \ ss^{\prime}_{i j}\left(x_{i j}, \ldots x_{n j}\right) \\ {x_{i j} \in X_{i j}} &{x_{i j} \in X_{i j}} \\ i=1, \ldots n & i=1, \ldots n\end{array}\right]$$