# Local Density-Based Outlier Detection using Subsampling with Approximate Nearest Neighbor Search

### Description

This function computes local density-based outlier scores for input data.

### Usage

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### Arguments

`X` |
An n x p data matrix to compute outlier scores |

`k` |
A vector of neighborhood sizes, k must be less than nsub |

`nsub` |
Subsample size, nsub must be greater than k. Usually nsub = 0.10*n or larger is recommended. Default is nsub = n |

`method` |
Character vector specifying the local density-based method(s) to compute. User can specify more than one method. By default all methods are computed |

`ldf.param` |
Vector of parameters for method LDF. h is the positive bandwidth parameter and c is a positive scaling constant. Default values are h=1 and c=0.1 |

`rkof.param` |
Vector of parameters for method RKOF. C is the postive bandwidth paramter, alpha is a sensitiveity parameter in the interval [0,1], and sig2 is the variance parameter. Default values are alpha=1, C=1, sig2=1 |

`lpdf.param` |
Vector of paramters for method LPDF. cov.type is the covariance parameterization type, which users can specifiy as either 'full' or 'diag'. sigma2 is the positive regularization parameter, tmax is the maximum number of updates, and v is the degrees of freedom for the multivariate t distribution. Default values are cov.type = 'full',tmax=1, sigma2=1e-5, and v=1. |

`treetype` |
Character vector specifiying tree method. Either 'kd' or 'bd' tree may be specified. Default is 'kd'. Refer to documentation for RANN package. |

`searchtype` |
Character vector specifiying kNN search type. Default value is "standard". Refer to documentation for RANN package. |

`eps` |
Error bound. Default is 0.0 which implies exact nearest neighgour search. Refer to documentation for RANN package. |

`scale.data` |
Logical value indicating to scale each feature of X using standard noramlization with mean 0 and standard deviation of 1 |

### Details

Computes the local density-based outlier scores for input data, X, referencing a random subsample of X. The subsampled data set is constructed by randomly drawning nsub samples from X without replacement.

Four different methods can be implemented LOF, LDF, RKOF, and LPDF. Each method specified returns densities and relative densities. Methods LDF and RKOF uses guassian kernels, and method LDPF uses multivarite t distribution. Outlier scores returned are positive except for lpde and lpdr which are log scaled densities (natural log). Score lpdr has shown to be highly sensitive to k.

All kNN computations are carried out using the nn2() function from the RANN package. Multivariate t densities are computed using the dmt() function from the mnormt package. Refer to specific packages for more details. Note: all neighborhoods are strickly of size k; therefore, the algorithms for LOP, LDF, and RKOF are not exact implementations, but algorithms are similiar for most situation and are equivalent when distance to k-th nearest neighbor is unique. If there are many many duplicate data points, then implementation of algorithms could lead to dramatically different (positive or negative) results than those that allow neighborhood sizes larger than k, especially if k is relatively small. Removing duplicates is recommended before computing outlier scores unless there is good reason to keep them.

The algorithm can be used to compute an ensemble of unsupervised outlier scores by using multiple k values and/or iterating over multiple subsamples.

### Value

A list of length 9 with the elements:

lrd –An n x length(k) matrix where each column vector represents the local reachabiility denity (LRD) outlier scores for each specifed k value. Smaller values indicate a point in more outlying.

lof –An n x length(k) matrix where each column vector represents the local outlier factor (LOF) outlier scores for each specifed k value. Larger values indicate a point in more outlying.

lde –An n x length(k) matrix where each column vector represents the local density estimate (LDE) outlier scores for each specifed k value. Smaller values indicate a point in more outlying.

ldf –An n x length(k) matrix where each column vector represents the local density factor (LDF) outlier scores for each specifed k value. Larger values indicate a point in more outlying.

kde –An n x length(k) matrix where each column vector represents the kernel density estimate (KDE) outlier scores for each specifed k value. Smaller values indicate a point in more outlying.

rkof –An n x length(k) matrix where each column vector represents the robust kernel density factor (RKOF) outlier scores for each specifed k value. Larger values indicate a point in more outlying.

lpde –An n x length(k) matrix where each column vector represents the local parametric density estimate (LPDE) outlier scores for each specifed k value on log scale. Smaller values indicate a point in more outlying.

lpdf –An n x length(k) matrix where each column vector represents the local parametric density factor (LPDF) outlier scores for each specifed k value. Smaller values indicate a point in more outlying.

lpdr –An n x length(k) matrix where each column vector represents the local parametric density ratio (LPDR) outlier scores for each specifed k value. Smaller values indicate a point in more outlying. LPDR is typically used to detect groups of outliers.

If a method is not specified then returns NULL

### References

M. M. Breunig, H-P. Kriegel, R.T. Ng, and J. Sander (2000). LOF: Identifying density-based local outliers. In Proc. of ACM International Conference on Knowledge Discovery and Data Mining, 93-104.

L. J. Latecki, A. Lazarevic, and D. Pokrajac (2007). Outlier Detection with kernel density funcions. In Proc. of Machine Learning and Data Mining in Pattern Recognition, 61-75

J. Gao, W. Hu, Z. Zhang, X. Zhang, and O. Wu (2011). RKOF: Robust kernel-based local outlier detection. In Proc. of Advances in Knowledge Discovery and Data Mining, 270-283.

K. Williams (2016). Local parametric density-based outlier deteciton and ensemble learning with application to malware detection. (Unpublished doctoral dissertation). The University of Texas at San Antonio, San Antonio, TX.

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 | ```
# 500 x 2 data matrix
X <- matrix(rnorm(1000),500,2)
# five outliers
outliers <- matrix(c(rnorm(2,20),rnorm(2,-12),rnorm(2,-8),rnorm(2,-5),rnorm(2,9)),5,2)
X <- rbind(X,outliers)
# compute outlier scores without subsampling for all methods using neighborhood size of 50
scores <- ldbod(X, k=50)
head(scores$lrd); head(scores$rkof)
# plot data and highlight top 5 outliers retured by lof
plot(X)
top5outliers <- X[order(scores$lof,decreasing=TRUE)[1:5],]
points(top5outliers,col=2)
# plot data and highlight top 5 outliers retured by outlier score lpde
plot(X)
top5outliers <- X[order(scores$lpde,decreasing=FALSE)[1:5],]
points(top5outliers,col=2)
# compute outlier scores for k= 10,20 with 10% subsampling for methods 'lof' and 'lpdf'
scores <- ldbod(X, k = c(10,20), nsub = 0.10*nrow(X), method = c('lof','lpdf'))
# plot data and highlight top 5 outliers retuned by lof for k=20
plot(X)
top5outliers <- X[order(scores$lof[,2],decreasing=TRUE)[1:5],]
points(top5outliers,col=2)
``` |