---
title: "Vignette: Weighted BACON algorithms"
author: "Tobias Schoch"
output:
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        highlight: tango
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  %\VignetteIndexEntry{Weighted BACON algorithms}
  %\VignetteEngine{knitr::rmarkdown}
  \usepackage[utf8]{inputenc}
---

```{r setup, include = FALSE}
knitr::opts_chunk$set(
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)
```

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## 1 Introduction

The package `wbacon` implements a weighted variant of the BACON (blocked
adaptive computationally-efficient outlier nominators) algorithms [Billor et
al.](#biblio) (2000) for multivariate outlier detection and robust linear
regression. The extension of the BACON algorithm for outlier detection to allow
for weighting is due to [Béguin and Hulliger](#biblio) (2008).

The details of the package are discussed in the accompanying paper; see
[Schoch](#biblio) (2021)

First, we attach the package to the search space.

```{r}
library("wbacon")
```


### 1.1 Available methods

* `wBACON()` is for multivariate outlier nomination and robust estimation of
  location/ center and covariance matrix
* `wBACON_reg()` is for robust linear regression (the method is robust against
  outliers in the response variable and the model's design matrix)

### 1.2 Assumptions

The BACON algorithms assume that the underlying model is an appropriate
description of the non-outlying observations; [Billor et al.](#biblio) (2000).
More precisely,

* the outlier nomination method assumes that the "good" data have (roughly) an
  *elliptically contoured* distribution (this includes the Gaussian
  distribution as a special case);
* the regression method assumes that the non-outlying ("good") data are
  described by a *linear* (homoscedastic) regression model and that the
  independent variables (having removed the regression intercept/constant, if
  there is a constant) follow (roughly) an elliptically contoured distribution.

<div class="my-callout">
<p>"Although the algorithms will often do something reasonable even when
these assumptions are violated, it is hard to say what the results mean."
[Billor et al.](#biblio) (2000, p. 290)</p>
</div>

It is strongly recommended that the structure of the data be examined and
whether the assumptions made about the "good" observations are reasonable.

### 1.3 The role of the data analyst

In line with [Billor et al.](#biblio) (2000, p. 290), we use the term outlier
"nomination" rather than "detection" to highlight that algorithms should not go
beyond nominating observations as *potential* outliers; see also [Béguin and
Hulliger](#biblio) (2008). It is left to the analyst to finally label outlying
observations as such.

The software provides the analyst with tools and measures to study potentially
outlying observations. It is strongly recommended to use the tools.

### 1.4 Additional information

Additional information on the BACON algorithms and the implementation can be
found in the documents:

* `methods.pdf`: A mathematical description of the algorithms and their
  implementation;
* `doc_c_functions.pdf`: A documentation of the `C` functions.

Both documents can be found in the package folder `doc`.

## 2 Multivariate outlier detection

In this section, we study multivariate outlier detection for the two datasets

* bushfire data (with sampling weights),
* philips data (without sampling weights).


### 2.1 Bushfire data

The bushfire dataset is on satellite remote sensing. These data were used by
[Campbell](#biblio) (1984) to locate bushfire scars. The data are radiometer
readings from polar-orbiting satellites of the National Oceanic and Atmospheric
Administration (NOAA) which have been collected continuously since 1981. The
measurements are taken on five frequency bands or channels. In the near
infrared band, it is possible to distinguish vegetation types from burned
surface. At visible wavelengths, the vegetation spectra are similar to burned
surface. The spatial resolution is rather low (1.1 km per pixel).

#### 2.1.1 Data preparation

The bushfire data contain radiometer readings for 38 pixels and have been
studied in [Maronna and Yohai](#biblio) (1995), [Béguin and Hulliger](#biblio)
(2002), [Béguin and Hulliger](#biblio) (2008), and [Hulliger and
Schoch](#biblio) (2009). The data can be obtained from the `R` package
`modi`([Hulliger](#biblio), 2023).<sup>[1](#notes)</sup>

```{r}
data(bushfire, package = "modi")
```

The first 6 readings on the five frequency bands (variables) are

```{r}
head(bushfire)
```

[Béguin and Hulliger](#biblio) (2008) generated a set of sampling weights. The
weights can be attached to the current session by

```{r}
data(bushfire.weights, package = "modi")
```

#### 2.1.2 Outlier detection

```{r}
fit <- wBACON(bushfire, w = bushfire.weights, alpha = 0.05)
fit
```

The argument `alpha` determines the $(1-\alpha)$-quantile $\chi_{\alpha,d}^2$
of the chi-square distribution with $d$ degrees of
freedom.<sup>[2](#notes)</sup> All observations whose squared Mahalanobis
distances are smaller than the quantile (times a correction factor) are
selected into the subset of outlier-free data. It is recommended to choose
`alpha` on grounds of an educated guess of the share of "good" observations in
the data. Here, we suppose that 95\% of the observations are not outliers.

By default, the initial subset is determined by the Euclidean norm
(initialization method: `version = "V2"`).

* This initialization method is robust because it is based on the
  coordinate-wise (weighted) median. The resulting estimators of center and
  scatter are *not affine equivariant*. Let $T(\cdot)$ denote an estimator of a
  parameter of interest (e.g., covariance matrix) and let $X$ denote the $(n
  \times p)$ data matrix. An estimator $T$ is affine equivariant if and only if
  $T(A X + b) = A T(X) + b$, for any nonsingular $(m \times n)$ matrix $A$ and
  any $n$-vector $b$. Although version `"V2"` of the BACON method yields an
  estimator that is not affine equivariant in the above sense, [Billor et
  al.](#biblio) (2000) point out that the method is nearly affine equivariant.
* There exists an alternative initialization method (`"version = V1"`) which is
  based on the coordinate-wise (weighted) means; therefore, it is affine
  equivariant but *not robust*.

From the above output, we see that the algorithm converged in three iterations.
In case the algorithm does not converge, we may increase the maximum number of
iterations (default: `maxiter = 50`) and toggle `verbose = TRUE` to (hopefully)
learn more why the method did not converge.

In the next step, we want to study the result in more detail. In particular, we
are interested in the estimated center and scatter (or covariance) matrix. To
this end, we can call the `summary()` method on the object `fit`.

```{r}
summary(fit)
```

#### 2.1.3 Diagnostics

The method has detected `r sum(is_outlier(fit))` *potential* outliers. It is
important to study the diagnostic plot to learn more about the potential
outliers. The robust (Mahalanobis) distances vs. the index of the observations
(`1:n`) can be plotted as follows.

```{r}
plot(fit, 1)
```

The dashed horizontal line shows the cutoff threshold on the robust distances.
Observations above the line are nominated as potential outliers by the BACON
algorithm. It is left to the analyst to finally label outlying observations as
such. In the next section, we introduce an alternative plotting method (see
below).

The method `is_outlier()` returns a vector of logicals whether an observation
has been flagged as a potential outlier.

```{r}
which(is_outlier(fit))
```

The (robust) center and covariance (scatter) matrix can be extracted with the
auxiliary functions, respectively, `center()` and `cov()`.

```{r}
center(fit)
```

The robust Mahalanobis distances can be extracted with the `distance()` method.

### 2.2 Philips data

Old television sets had a cathode ray tube with an electron gun. The emitted
beam runs through a diaphragm that lets pass only a partial beam to the screen.
The diaphragm consists of 9 components. The Philips data set contains $n = 667$
measurements on the $p = 9$ components (variables); see [Rousseeuw and van
Driessen](#biblio) (1999).<sup>[3](#notes)</sup> These data do not have
sampling weights.

```{r}
data(philips)
head(philips)
```

We compute the BACON algorithm but this time with the initialization method
`version = "V1"`.

```{r}
fit <- wBACON(philips, alpha = 0.05, version = "V1")
fit
```

The BACON algorithm detected `r sum(is_outlier(fit))` potential outliers. The
robust (Mahalanobis) distances can be plotted against the univariate projection
of the data, which maximizes the separation criterion of [Qiu and Joe](#biblio)
(2006). This kind of diagnostic graph attempts to separate outlying from
non-outlying observations as much as possible; see [Willems et al.](#biblio)
(2009). It is helpful if the outliers are clustered. The graph is generated as
follows.

```{r}
plot(fit, which = 2)
```

From the visual display, we see a cluster of potential outliers in the top
right corner. The dashed horizontal line indicates the cutoff threshold on the
distances as imposed by the BACON algorithm.

For very large datasets, the plot method can be called with the (additional)
argument `hex = TRUE` to show a hexagonally binned scatter plot; see below.
This plot method uses the functionality of the R package `hexbin` ([Carr et
al.](#biblio), 2023).

```{r}
plot(fit, which = 2, hex = TRUE)
```

## 3 Robust linear regression

The education data is on education expenditures in 50 US states in 1975
([Chatterjee and Hadi](#biblio), 2012, Chap. 5.7). The data can be loaded from
the `robustbase` package.

```{r}
data(education, package = "robustbase")
```

It is convenient to rename the variables.


```{r}
names(education)[3:6] <- c("RES", "INC", "YOUNG", "EXP")
head(education)
```

The measured variables for the 50 states are:

* `State`: State
* `Region`: group variable with outcomes: 1=Northeastern, 2=North central,
  3=Southern, and 4=Western
* `RES`: Number of residents per thousand residing in urban areas in 1970
* `INC`: Per capita personal income in 1973 (\$US)
* `YOUNG`: Number of residents per thousand under 18 years of age in 1974
* `EXP`: Per capita expenditure on public education in a state (\$US),
  projected for 1975

### 3.1 Model fit

We want to regress education expenditures (`EXP`) on the variables `RES`,
`INC`, and `YOUNG` by the BACON algorithm, and obtain

```{r}
reg <- wBACON_reg(EXP ~ RES + INC + YOUNG, data = education)
reg
```

The instance `reg` is an object of the class `wbaconlm`.  The printed output of
`wBACON_reg` is identical with the one of the `lm` function. In addition, we
are told the size of the subset on which the regression has been computed. The
observations not in the subset are considered outliers (here 1 out of 50
observations).

The `summary()` method can be used to obtain a summary of the estimated model.

```{r}
summary(reg)
```

The summary output of `wBACON_reg` is identical with the output of the `lm`
estimate on the subset of outlier-free data,

```{r}
summary(lm(EXP ~ RES + INC + YOUNG, data = education[!is_outlier(reg), ]))
```

where we have used `is_outlier()` to extract the set of declared outliers from
`reg` (the summary output of the `lm` estimate is not shown).

### 3.2 Tuning

By default, `wBACON_reg` uses the parametrization $\alpha = 0.05$, `collect =
4`, and `version = "V2"`. These parameters are used to call the `wBACON`
algorithm on the design matrix. Then, the same parameters are used to compute
the robust regression.

To ensure a high breakdown point, `version = "V2"` should not be changed to
`version = "V1"` unless you have good reasons.  The main "turning knob" to tune
the algorithm is `alpha`, which defines the $(1-$`alpha`$)$ quantile of the
Student $t$-distribution. All observations whose distances/discrepancies [See
document `methods.pdf` in the folder `doc` of the package.] are smaller (in
absolute value) than the quantile are selected into the subset of "good" data.
By choosing smaller values for `alpha` (e.g., 0.2), more observations are
selected (ceteris paribus) into the subset of "good" data (and vice versa).

The parameter `collect` specifies the initial subset size, which is defined as
$m = p \cdot collect$. It can be modified but should be chosen such that $m$ is
considerably smaller than the number of observations $n$.  Otherwise there is a
high risk of selecting too many "bad" observations into the initial subset,
which will eventually bias the regression estimates.

In case the algorithm does not converge, we may increase the maximum number of
iterations (default: `maxiter = 50`) and toggle `verbose = TRUE` to (hopefully)
learn more why the method did not converge.

### 3.3 Model diagnostics

The methods `coef()`, `vcov()`, and `predict()` work exactly the same as their
`lm` counterparts. This is also true for the first three `plot` types, that is

* `which = 1`: Residuals vs Fitted,
* `which = 2`: Normal Q-Q,
* `which = 3`: Scale-Location


The plot types `4:6` of `plot.lm` are not implemented for objects of the class
`wbaconlm` because it is not sensible to study the standard regression
influence diagnostics in the presence of outliers in the model's design space.
Instead, type four (`which = 4`) plots the robust Mahalanobis distances with
respect to the non-constant design variables against the standardized residual.
This plot has been proposed by [Rousseeuw and van Zomeren](#biblio) (1990).

```{r}
plot(reg, 4)
```

The *filled* circle(s) represent the outliers nominated by the BACON algorithm.
The outlier in the top right corner is both a residual outlier and an outlier
in the model's design space.

 * Observation with robust Mahalanobis distances larger than 4.57 (see
   abscissae) are flagged as outliers in the model's design space (leverage
   observations).
 * Observations whose standardized residual falls outside the interval spanned
   by $\pm \, t_{\alpha/(2m+2), m - p}$, where $t_{\alpha, m - p}$ is the
   $(1-\alpha)$ quantile of the Student $t$-distribution with $m-p$ degrees of
   freedom, $m$ denoting the size of the final subset of outlier-free data.
   Here, we have $m=49$, $\alpha = 0.05$ (see argument `alpha` of
   `wBACON_reg`), thus the interval is $[-3.52, \; 3.52]$.

---

## References {#biblio}

Béguin, C. and B. Hulliger (2002). Robust Multivariate Outlier Detection and
Imputation with Incomplete Survey Data, Deliverable D4/5.2.1/2 Part C: EUREDIT
project, https://www.cs.york.ac.uk/euredit/euredit-main.html, research project
funded by the European Commission, IST-1999-10226.

Béguin, C. and B. Hulliger (2008). The BACON-EEM Algorithm for Multivariate
Outlier Detection in Incomplete Survey Data, *Survey Methodology*, **34**,
91--103.

Billor, N., A. S. Hadi, and P. F. Vellemann (2000). BACON: Blocked Adaptive
Computationally-efficient Outlier Nominators, *Computational Statistics and
Data Analysis*, **34**, 279--298.
[DOI 10.1016/S0167-9473(99)00101-2](https://doi.org/10.1016/S0167-9473(99)00101-2)

Campbell, N. A. (1989). Bushfire Mapping using NOAA AVHRR Data. Technical
Report. Commonwealth Scientific and Industrial Research Organisation, North
Ryde.

Carr, D., N. Lewin-Koh, and M. Maechler (2023). hexbin: Hexagonal Binning
Routines. R package version 1.28.3. (The package contains copies of lattice
functions written by Deepayan Sarkar). URL
https://CRAN.R-project.org/package=hexbin

Chatterjee, S. and A. H. Hadi (2012). *Regression Analysis by Example*, 5th
ed., Hoboken (NJ): John Wiley \& Sons.

Hulliger, B. and T. Schoch (2009). Robust multivariate imputation with survey
data, in *Proceedings of the 57th Session of the International Statistical
Institute*, Durban.

Hulliger, B. (2023). modi: Multivariate Outlier Detection and Imputation for
Incomplete Survey Data, R package version 0.1-2. URL
https://CRAN.R-project.org/package=modi

Maechler, M., P. Rousseeuw, C. Croux, V. Todorov, A. Ruckstuhl, M.
Salibian-Barrera, T. Verbeke, M. Koller, E. L. T. Conceicao, and M. Anna di
Palma (2024). robustbase: Basic Robust Statistics, R package version 0.99-2.
URL https://CRAN.R-project.org/package=robustbase

Maronna, R. A. and V. J. Yohai (1995). The Behavior of the Stahel-Donoho Robust
Multivariate Estimator, *Journal of the American Statistical Association*,
**90** 330--341. [DOI 10.2307/2291158](https://doi.org/10.2307/2291158)

Qiu, W. and H. Joe (2006). Separation index and partial membership for
clustering, *Computational Statistics and Data Analysis*, **50**, 585--603.
[DOI 10.1016/j.csda.2004.09.009](https://doi.org/10.1016/j.csda.2004.09.009)

Raymaekers, J. and P. Rousseeuw (2023). cellWise: Analyzing Data with Cellwise
Outliers, R package version 2.5.3. URL
https://CRAN.R-project.org/package=cellWise

Rousseeuw, P. J. and K. van Driessen (1999). A fast algorithm for the Minimum
Covariance Determinant estimator, *Technometrics*, **41**, 212--223.
[DOI 10.2307/1270566](https://doi.org/10.2307/1270566)

Rousseeuw, P. J. and K. van Zomeren (1990). Unmasking Multivariate Outliers and
Leverage Points, *Journal of the American Statistical Association*, **411**,
633--639. [DOI 10.2307/2289995](https://doi.org/10.2307/2289995)

Schoch, T. (2021) wbacon: Weighted BACON algorithms for multivariate outlier
nomination (detection) and robust linear regression, *Journal of Open Source
Software*, **6**, 3238.
[DOI 10.21105/joss.03238](https://doi.org/10.21105/joss.03238)

Willems, G., H. Joe, and R. Zamar (2009). Diagnosing Multivariate Outliers
Detected by Robust Estimators, *Journal of Computational and Graphical
Statistics*, **18**, 73--91.
[DOI 10.1198/jcgs.2009.0005](https://doi.org/10.1198/jcgs.2009.0005)

## Notes {#notes}

<sup>1</sup> The data are also distributed with the `R` package `robustbase`
([Maechler et al.](#biblio), 2023).

<sup>2</sup> The degrees of freedom $d$ is a function of the number of
variables $p$, the number of observations $n$, and the size of the current
subset $m$; see `methods.pdf` in the `inst/doc` folder of the package.

<sup>3</sup> The philips data has been published in the `R` package `cellWise`
([Raymaekers and Rousseeuw](#biblio), 2023).
