Abstract. The simplest case of clustering is the one dimensional case: when we want to find clusters of a given set of numbers. The simplest 1D clustering is when we know that there is only a single cluster. This document is a comprehensive study of this simplest case. Namely, given a finite set of real numbers that form ``one cluster'', we explain how to find it; that is how to determine its position and its size. This problem has actually many solutions with very different properties, and we try to cover them all.$\newcommand{\R}{\mathbf{R}}$ $\newcommand{\Z}{\mathbf{Z}}$
Our goal is to combine a set of $N$ numbers $x_1,\ldots,x_N$ into a single one. The simplest choice is the average, also called arithmetic mean: \[ \mathrm{avg}(x_1,\ldots,x_N) := \frac{x_1+\cdots+x_N}{N} \] We also have the harmonic mean: \[ \mathrm{har}(x_1,\ldots,x_N) := \frac{N}{\frac{1}{x_1}+\cdots+\frac{1}{x_N}} \] and the geometric mean: \[ \mathrm{geo}(x_1,\ldots,x_N) := \sqrt[N]{x_1\cdot \cdots\cdot x_N} \] The geometric mean is somewhat different, because it is only well-defined (as a function $\R^N\to\R$ with arbitrary $N$) when all the numbers are positive. As we will see, the harmonic mean also really makes sense only when all the numbers are strictly positive. However, the arithmetic mean has good properties for arbitrary input numbers (positive, negative, or zero).
For positive numbers, we also have the quadratic mean, also called root mean square error: \[ \mathrm{rms}(x_1,\ldots,x_N) := \sqrt{\frac{x_1^2+\cdots+x_N^2}{N}} \] These four functions are called Pythagorean means and they are all of fundamental importance. They are related by the following inequalities: \[ \mathrm{min} \le \mathrm{har} \le \mathrm{geo} \le \mathrm{avg} \le \mathrm{rms} \le \mathrm{max} \] These inequalities are elementary to prove for $N=2$, and the general case is a standard exercise.
Finally, the last ``traditional'' aggregator is the median, which is computed by sorting the numbers from low to high, and taking the middle one (if $N$ is odd) or the average of the two middle ones (if $N$ is even). \[ \mathrm{med}(x_1,\ldots,x_N):= \begin{cases} x_{{}_{(M)}} & \textrm{if $N=2M+1$} \\ &\\ \displaystyle\frac{x_{{}_{(M)}} + x_{{}_{(M+1)}}}{2} & \textrm{if $N=2M$} \end{cases} \] where $x_{(1)},\ldots x_{(N)}$ indicates the sorted numbers $x_1,\ldots,x_N$. There is no general inequality relationship between $\mathrm{med}$ and $\mathrm{avg}$, either one can be larger or smaller, depending on the particular set of numbers.
A map $f:\R^N\to\R$ with the following properties is called an aggregator function:
Notice that these properties may only be true on a subdomain of $\R^N$ where $f$ is well-defined (typically, the positive numbers). Properties $P0--P4$ are very natural, and a bit redundant; for example you can prove identity from monotony and bracketing, etc.
The following properties $P5--P7$ are more special and aggregator functions may or may not have them:
This list should contain all the aggregator functions that I know (whether they are useful or useless). Many aggregator functions belong to families that depend on a real-valued parameter, and for extremal values of the parameter they give the min and the max. Thus, all these aggregators can be interpreted as different, data-guided interpolators between min and max.
The power means $M_p$ are defined only for strictly positive numbers: \[ M_p(x_1,\ldots,x_N):=\sqrt[p]{\frac{1}{N}\sum_{i=1}^Nx_i^p} \] Notice that $M_p$ is well defined for $p\neq0$. We extend the definition of $M_p$ to $p=0,\pm\infty$ by taking limits. The resulting definition contains all the Pythagorean means, the minimum and the maximum as particular cases:
power mean | meaning |
$M_{-\infty}$ | minimum |
$M_{-1}$ | harmonic mean |
$M_0$ | geometric mean |
$M_1$ | arithmetic mean |
$M_2$ | quadratic mean |
$M_3$ | cubic mean |
$M_\infty$ | maximum |
Notice that as the parameter $p$ goes from $-\infty$ to $\infty$, the power mean $M_p$ interpolates from the minimum to the maximum sample, passing through the Pythagorean means at the values points $p=-1,0,1,2$.
Power means satisfy properties $P0--P4$ over the positive numbers. Although $M_{-1}$ can be defined also for negative numbers, it fails to satisfy the bracketing property, so it is not considered as such.
Except for the values of $p=1,\pm\infty$, the power means do not satisfy the additivity property $P5$. Thus, they are essentially tied to the position of the zero in the real line.
On the other hand, the power means are composable ($P6$).
The power means are not the only natural interpolation between min and max; there is indeed a more natural one: the order statistics.
Given a set of $N$ numbers $x_1,\ldots,x_N$, they can always be re-indexed so that \[ x_{(1)} \le x_{(2)} \le \cdots \le x_{(N)} \] Thus $\max(x_1,\ldots,x_N)=x_{(N)}$ and $\min(x_1,\ldots,x_N)=x_{(1)}$. In functional notation, we define \[ O_k(x_1,\ldots,x_N) := x_{(k)} \] for $k=1,\ldots,N$. This definition is extended to real-valued $k\in[1,N]$ by interpolating linearly between the two closest integer values of $k$. In that case, we often use the notation $O_{\alpha}$ for $\alpha\in[0,1]$, where $k=(N-1)\alpha+1$. This notation has the advantage of being independent of $N$, so that $O_{\frac{1}{2}}$ is always the median. The following table lists other particular cases
order statistic | meaning |
$O_0$ | minimum |
$O_{0.1}$ | first decile |
$O_{0.25}$ | first quartile |
$O_{0.5}$ | median |
$O_{0.75}$ | third quartile |
$O_{0.9}$ | ninth decile |
$O_1$ | maximum |
Order statistics can be defined for arbitrary numbers (positive, negative and zero) and they have the additivity property P5, thus they are position-invariant. However, except for the min and the max, they are not composable (P6).
A simple yet very robust way to locate a cluster, especially when there is a large number of points, is to build and histogram of the data and find its mode (the position of the bin with higher amount of samples). This method has two parameters, the frequency of the bins, and the phase \[ H_{\varphi,\psi}(x_1,\ldots,x_N) := \psi+\frac{1}{2}\varphi+\varphi\mathrm{argmax}_{k\in\Z} \sum_{i=1}^N\int_{\psi+k\varphi}^{\psi+(k+1)\varphi} \delta(x-x_i)\,\d x \] Notice that the result depends on the two parameters $(\varphi,\psi)$ in a very beautiful and fractal way (see Figure \ref{fig:modefractal}). It is impossible to set reasonable values of these parameters without knowing anything about the nature of the input data.
On a metric space $(M,d)$, the Fréchet centroid of a set of points $p_i\in M$ is the point $c\in M$ that minimizes the sum of squared differences to $p_i$ \[ c := \mathrm{argmin}_{m\in\R}\sum_{i=1}^N d^2(p_i,m). \]
Based on this definition, and using the $L^p$ norms on the real line, we define the $p-$ centroids of a set of numbers: \[ \tilde F_p(x_1,\ldots,x_N) := \mathrm{argmin}_{m\in\R}\sum_{i=1}^N|x_i-m|^p \] This is well-defined for $p>0$, even if the function to minimize is only convex for $p\ge1$. Instead of $\tilde F_p$, we use the following slightly different normalization (which gives the same result for $p>0$). \[ F_p(x_1,\ldots,x_N) := \mathrm{argmin}_{m\in\R}\sqrt[p]{\frac{1}{N}\sum_{i=1}^N|x_i-m|^p} \] This normalization has the advantage that the numerical behavior is somehow independent of $N$ and $p$ (the function to minimize has linear growth at inifinty independently of $p$).
We find the following interesting particular cases
Fréchet centroid | meaning |
$F_2$ | average |
$F_1$ | median |
$F_{\to\infty}$ | midrange $I_{0.5}=(\min+\max)/2$ |
$F_{\to0}$ | ``mode'' |
Notice that the parameter $p$ controls the robustness to outliers. For $p=2$ we have the average, which is not very robust to outliers. Decreasing $p$ down to $1$, we reach the median that is rather robust to outliers. Decreasing $p$ further to 0 we reach the mode, that is extremely robust to outliers (it is independent of outliers, unlike the median). On the other side, increasing $p\to\infty$ we approach the average between the two extremal values, which is the least robust possible aggregator (it depends only on the outliers).
The Fréchet centroids are position independent, but not composable (in fact the ``italian'' theorem says that the only aggregator that has all the properties is the arithmetic mean).
Notice that the effective computation of $F_p$ requires solving an optimization problem. For $p\in[1,4)$ Weiszfeld algorithm is used. For $p>4$ we need to use another algorithm, for example Newton's method. For $p<1$ the function is not convex and some sort of search must be performed (starting from seeds between each pair of data points, for good measure).
Notice that the Fréchet $p$-centroids can be defined over arbitrary metric spaces. In the case of a Riemannian manifold, the Fréchet $\infty$-centroid coincides with the midpoint of the diameter.
For $\alpha\in[0,1]$ we define \[ I_{\alpha}:=\alpha O_1 + (1-\alpha) O_0 \] Thus,
$I_\alpha$ | meaning |
$I_0$ | min |
$I_{0.5}$ | midrange |
$I_1$ | max |
This is just the trivial linear interpolation between min and max. It is an example of $L$-estimator. In general, an $L$ estimator is a function of the form \[ L := \frac{\sum_{k=1}^N{\alpha_kO_k}}{\sum_{k=1}^N{\alpha_k}} \] Some famous $L$ estimators are the midrange, the midhinge (average of first and third quartiles), the trimean, truncated means, and other curiosities
$L$-estimator | name |
$O_0$ | min |
$I_1$ | max |
$(O_0+O_1)/2$ | midrange |
$(O_{0.25}+O_{0.75})/2$ | midhinge (average of quartiles) |
$(O_{0.25}+2O_{0.5}+O_{0.75})/4$ | trimean (average of median and midhinge) |
$\frac{2}{N}\sum_{k>N/4}^{3N/4} O_k$ | midmean (average of central half) |
$\frac{1}{N}\sum_{k=1}^{N} O_k$ | mean (average of everything) |
An advantage of the trimean as a measure of the center (of a distribution) is that it combines the median's emphasis on center values with the midhinge's attention to the extremes.\hfill ---Herbert F. Weisberg, Central Tendency and Variability
The following aggregators are defined for strictly positive numbers \[ L_p(x_1,\ldots,x_N):=\frac{ \displaystyle\sum_{i=1}^N x_i^p }{ \displaystyle\sum_{i=1}^N x_i^{p-1} } \]
This is an increasing family between min and max, different to the power means, but passing through several common points:
Lehmer mean | name |
$L_{-\infty}$ | min |
$L_0$ | harmonic mean |
$L_{0.5}$ | geometric mean (only for $N=2$ ?) |
$L_1$ | arithmetic mean |
$L_2$ | contraharmonic mean $(x_1^2+\cdots+x_N^2)/(x_1+\cdots+x_N)$ |
$L_{\infty}$ | max |
Interestingly, the contraharmonic mean can be defined also for nonzero numbers (not necessarily positive). It is the sum of the average and the variance divided by the average. The contraharmonic mean of positive numbers is always larger or equal than the quadratic mean, but it is otherwise unrelated to the other power means for $p>2$.
The Gini means form a very general family of aggregators that contains the power means and the Lehmer means as particular cases \[ G_{p,q}(x_1,\ldots,x_N) := \begin{cases} \sqrt[p-q]{\displaystyle\frac{\sum x^p}{\sum x^q}} & \textrm{if}\ p>q \\ &\\ \sqrt[\sum x^p]{\prod x^{x^p}} & \textrm{if}\ p=q \end{cases} \] where the sums and products above are performed over $x\in\{x_1,\ldots x_N\}$
Some aggregator functions do not belong to any parametric family, but are particular cases on their own right (for example, the ``MEDIAL''!) Others lie at the intersection of different parametric families:
the arithmetic mean is at the same time a power mean $M_1$ and a Fréchet centroid $F_2$
the median is at the same time an order statistic $U_{\frac{1}{2}}$ and a Fréchet centroid $F_1$
A different, non parametric, generalisation of means that contains the power means and many others as particular cases is the following. It consists in performing the arithmetic mean behind a ``contrast change $f$''.
Let $f:\R\to\R$ be a continuous, strictly monotonic function, then we define \[ M_f(x_1,\ldots,x_N):=f^{-1}\left(\frac{f(x_1)+\cdots+f(x_N)}{N}\right) \] The power means for $p\neq0$ appear as particular cases when $f(x)=x^p$. The geometric mean appears for $f(x)=\log(x)$, and for $f(x)=\exp(x)$ we obtain the ``soft maximum'' LSE (log sum exp).
On the previous section we have described methods to define the position of a cluster, namely to find where a cluster of numbers is located. A different problem is the computation of the size of a cluster.
Many criteria for defining the size---but not all---depend on finding first the position.
Besides axioms P1--P4 above, size measures satisfy the following two axioms:
Some famous size measures
So far I have talked about the common case when there is a single cluster of numbers; or, in statistical parlance, that the data is unimodal. Yet, it is important to be able to identify whether this is this the case, and when the data is not unimodal, how to (god forbid!) find several clusters in it.
A simple criterion for deciding whether a set of numbers forms a single cluster is to compare the geometric and arithmetic means: if they are very different, we say that the data is not unimodal: \[ \frac{M_1(x_1,\ldots,x_N)}{M_0(x_1,\ldots,x_N)} \ge \tau \] for some threshold $\tau>1$, for example $\tau=2$. Notice that this criterion is not shift-invariant.
This ratio is a standard measure for homogeneity detection in radar images.