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Multivariate Confidence Intervals Multivariate Confidence Intervals∗ Jussi Korpelay Emilia Oikarineny Kai Puolam¨akiy Antti Ukkoneny Abstract a summary. However, most of these approaches focus on describing a single univariate distribution over real Confidence intervals are a popular way to visualize and analyze data distributions. Unlike p-values, they can convey numbers. Also, the precise definition of a confidence information both about statistical significance as well as interval varies slightly across disciplines and application effect size. However, very little work exists on applying domains. confidence intervals to multivariate data. In this paper we define confidence intervals for multivariate data that extend In this paper we focus on confidence areas: a gen- the one-dimensional definition in a natural way. In our eralization of univariate confidence intervals to multi- definition every variable is associated with its own confidence variate data. All our intervals and areas are such that interval as usual, but a data vector can be outside of a few of these, and still be considered to be within the confidence they describe ranges of data and are of minimal width. area. We analyze the problem and show that the resulting In other words, they contain a given fraction of data confidence areas retain the good qualities of their one- within their bounds and are as narrow as possible. By dimensional counterparts: they are informative and easy to interpret. Furthermore, we show that the problem of choosing a confidence area with minimal size we essen- finding multivariate confidence intervals is hard, but provide tially locate the densest part of the distribution. Such efficient approximate algorithms to solve the problem. confidence areas are particularly effective for visually Keywords multivariate statistics; confidence intervals; showing trends, patterns, and outliers. algorithms Considering the usefulness of confidence intervals, it is surprising how little work exists on applying confi- dence intervals on multivariate data [11]. In multivari- 1 Introduction ate statistics confidence regions are a commonly used approach, see e.g., [6], but these methods usually require Confidence intervals are a natural and commonly used making assumptions about the underlying distribution. way to summarize a distribution over real numbers. Moreover, unlike confidence areas, most conference re- In informal terms, a confidence interval is a concise gions cannot be described simply with an upper and way to express what values a given sample mostly a lower bound, e.g., confidence regions for multivariate contains. They are used widely, e.g., to denote ranges Gaussian data are ellipsoids. Thus, these approaches of data, specify accuracies of parameter estimates, or in do not fully capture two useful characteristics of one- Bayesian settings to describe the posterior distribution. dimensional confidence intervals: a) easy interpretabil- A confidence interval is given by two numbers, the lower ity and b) lack of assumptions about the data. and upper bound, and parametrized by the percentage The simplest approach to construct multivariate of probability mass that lies within the bounds. They confidence intervals is to compute one-dimensional in- are easy to interpret, because they can be represented tervals separately for every variable. While this naive visually together with the data, and convey information approach satisfies conditions a) and b) above, it is easy both about the location as well as the variance of a arXiv:1701.05763v1 [stat.AP] 20 Jan 2017 to see how it fails in general. Assume, e.g., that we sample. have 10 independently distributed variables, and have There is a plethora of work on how to estimate computed for each variable a 90% confidence interval. the confidence interval of a distribution based on a This means that when considering every variable indi- finite-sized sample from that distribution, see [7] for vidually, only 10% of the distribution lies outside of the ∗This work was supported by Academy of Finland (decision respective interval, as desired. But taken together, the 288814) and Tekes (Revolution of Knowledge Work). probability that an observation is outside at least one of This document is an extended version of [10] to appear in the the confidence intervals is as much as 1 − 0:910 = 65%. Proceedings of the 2017 SIAM International Conference on Data Mining (SDM 2017). This probability, however, depends strongly on the cor- Implementation of the algorithms (Sec. 5) and code to re- relations between the variables. If the variables were produce Figs. 1 and 5 are provided at https://github.com perfectly correlated with a correlation coefficient of ±1, /jutako/multivariate-ci.git the probability of an observation being outside all con- yFinnish Institute of Occupational Health, Helsinki, Finland fidence intervals would again be 10%. In general the 1 ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ●● ● ● ● ●●● ● ● ●●●● ● ● ● ● ● ● ●●● ●●●● ● ● ● ● ● ● ●● ● ●●● ● ● ● ●●●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ●●●● ● ● ●● ● ●● y ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●●● ● ● ●● ● ● ●●● ● ●● ● ● L=0 ● ● ● ● ● ● ● ● ●● Naive ● ● ● ● ● ● L=25 ● ● ● ● ● L=25 L = 1 Naive L = 0 L=0 naive | | x y x Figure 1: Examples 1 and 2. Please see text for details. LEFT: Local anomalies in time series (solid black line) are easier to spot when computing the confidence area using the proposed method (orange lines) as opposed to existing approaches (green lines). RIGHT: Our approach results in a confidence area (orange \cross") that is a better representation of the underlying distribution than existing approaches (green rectangle). correlation structure of the data affects in a strong and confidence intervals. In the special case l = 0, this new hard-to-predict way on how the naive confidence inter- definition coincides with the MWE problem [11]. The vals should be interpreted in a multivariate setting. following examples illustrate further properties and uses Ideally a multivariate confidence area should retain of the new definition. the simplicity of a one-dimensional confidence interval. It should be representable by upper and lower bounds for each variable and the semantics should be the Example 1: local anomalies in time series. same: each observation is either inside or outside the confidence area, and most observations of a sample The left panel of Fig. 1 presents a number of time series should be inside. Particularly important is specifying over m = 80 time points, shown in light gray, together when an observation in fact is within the confidence with three different types of 90% confidence intervals, area, as we will discuss next. shown in green, purple and orange, respectively. In Confidence areas for time series data have been this example, \normal" behavior of the time series is defined [9, 11] in terms of the minimum width envelope exemplified by the black dash-dotted curve that exhibits (MWE) problem: a time series is within a confidence no noteworthy fluctuation over time. Two other types area if it is within the confidence interval of every of time series are shown also in black: a clear outlier variable. While this has desirable properties, it can, (dashed black line), and a time series that exhibits however, result in very conservative confidence areas normal behavior most of the time, but strongly deviates if there are local deviations from what constitutes from this for a brief moment (solid black line). \normal" behavior. The definition in [11] is in fact too In the situation shown, we would like the confidence strict by requiring an observation to be contained in all interval to only show what constitutes \normal" behav- variable-specific intervals. ior, i.e., not be affected by strong local fluctuations or Thus, here we propose an alternative way to define outliers. Such artifacts can be caused, e.g., by mea- the confidence area: a data vector is within the confi- surement problems, or other types of data corruption. dence area if it is outside the variable-specific confidence Alternatively, such behavior can also reflect some inter- intervals at most l times, where l is a given integer. esting anomaly in the data generating process, and this This formulation preserves easy interpretability: the should not be characterized as \normal" by the confi- user knows that any observation within the confidence dence interval. In Fig. 1 (left) the confidence area based area is guaranteed to violate at most l variable-specific on MWE [11], is shown by the green lines; recall the MWE interval corresponds to setting l = 0. While it is 2 unaffected by the outlier, it strongly reacts to local fluc- Contributions tuations. The naive confidence area, i.e., one where we The basic problem definition we study in this paper have computed confidence intervals for every time point is straightforward: for m-dimensional data and the individually, is shown in purple. It is also affected by parameters α 2 [0; 1] and integer l, find a confidence local peaks, albeit less than the l = 0 confidence area. interval for each of the variables so that the 1 − α Finally, the proposed method is shown in orange. The fraction of the observations lie within the confidence area is computed using l = 25, i.e., a time series is still area, defined so that the sum of the length of the within the confidence area as long as it lies outside in at intervals is minimized, and an observation can break most 25 time points. This variant focuses on what we at most l of the variable-specific confidence intervals. would expect to be normal behavior in this case. Our We make the following contributions in this paper: new definition of a confidence area is thus nicely suited for finding local anomalies in time series data. • We formally define the problem of finding a mul- tivariate confidence area, where observations have to satisfy most but not all of the variable-specific confidence intervals. • We analyze the computational complexity of the Example 2: representing data distribu- problem, and show that it is NP-hard, but admits tions. an approximation algorithm based on a linear pro- gramming relaxation.
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