An Optimal Linear Time Algorithm for Quasi-Monotonic Segmentation
International Journal of Computer Mathematics (2007)
- DOI: 10.1080/00207160701694153
- arXiv: cs/0702142
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Abstract
Monotonicity is a simple yet significant qualitative characteristic. We consider the problem of segmenting an array in up to K segments. We want segments to be as monotonic as possible and to alternate signs. We propose a quality metric for this problem, present an optimal linear time algorithm based on novel formalism, and compare experimentally its performance to a linear time top-down regression algorithm. We show that our algorithm is faster and more accurate. Applications include pattern recognition and qualitative modeling.
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An Optimal Linear Time Algorithm for Quasi-Monotonic Segmentation
International Journal of Computer Mathematics
Vol. 00, No. 00, Month 200x, 1{16
An Optimal Linear Time Algorithm for Quasi-Monotonic Segmentation
Daniel Lemirey, Martin Brooks, Yuhong Yanz
yUniversity of Quebec at Montreal (UQAM), 100 Sherbrooke West, Montreal, Qc, H2X 3P2, Canada
z National Research Council of Canada, 1200 Montreal Road, Ottawa, ON, Canada, K1A 0R6
(Received 00 Month 200x; revised 00 Month 200x; in nal form 00 Month 200x)
Monotonicity is a simple yet signi cant qualitative characteristic. We consider the problem of segmenting a sequence in up to K segments.
We want segments to be as monotonic as possible and to alternate signs. We propose a quality metric for this problem using the l1 norm,
and we present an optimal linear time algorithm based on novel formalism. Moreover, given a precomputation in time O(n logn) consisting
of a labeling of all extrema, we compute any optimal segmentation in constant time. We compare experimentally its performance to two
piecewise linear segmentation heuristics (top-down and bottom-up). We show that our algorithm is faster and more accurate. Applications
include pattern recognition and qualitative modeling.
Keywords: Time Series, Segmentation, Monotonicity, Design of Algorithms
AMS Subject Classi cations: H.2.8
1. Introduction
Monotonicity is one of the most natural and important qualitative properties for sequences of data points.
It is easy to determine where the values are strictly going up or down, but we only want to identify
signi cant monotonicity. For example, the drop from 2 to 1.9 in the array 0; 1; 2; 1:9; 3; 4 might not be
signi cant and might even be noise-related. The quasi-monotonic segmentation problem is to determine
where the data is approximately increasing or decreasing.
Corresponding author. Email: lemire@acm.org
This is an expanded version of a conference paper [1].
International Journal of Computer Mathematics
ISSN 0020-7160 print/ISSN 1029-0265 online c
200x Taylor & Francis
http://www.tandf.co.uk/journals
DOI: 10.1080/0003681YYxxxxxxxx
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Vol. 00, No. 00, Month 200x, 1{16
An Optimal Linear Time Algorithm for Quasi-Monotonic Segmentation
Daniel Lemirey, Martin Brooks, Yuhong Yanz
yUniversity of Quebec at Montreal (UQAM), 100 Sherbrooke West, Montreal, Qc, H2X 3P2, Canada
z National Research Council of Canada, 1200 Montreal Road, Ottawa, ON, Canada, K1A 0R6
(Received 00 Month 200x; revised 00 Month 200x; in nal form 00 Month 200x)
Monotonicity is a simple yet signi cant qualitative characteristic. We consider the problem of segmenting a sequence in up to K segments.
We want segments to be as monotonic as possible and to alternate signs. We propose a quality metric for this problem using the l1 norm,
and we present an optimal linear time algorithm based on novel formalism. Moreover, given a precomputation in time O(n logn) consisting
of a labeling of all extrema, we compute any optimal segmentation in constant time. We compare experimentally its performance to two
piecewise linear segmentation heuristics (top-down and bottom-up). We show that our algorithm is faster and more accurate. Applications
include pattern recognition and qualitative modeling.
Keywords: Time Series, Segmentation, Monotonicity, Design of Algorithms
AMS Subject Classi cations: H.2.8
1. Introduction
Monotonicity is one of the most natural and important qualitative properties for sequences of data points.
It is easy to determine where the values are strictly going up or down, but we only want to identify
signi cant monotonicity. For example, the drop from 2 to 1.9 in the array 0; 1; 2; 1:9; 3; 4 might not be
signi cant and might even be noise-related. The quasi-monotonic segmentation problem is to determine
where the data is approximately increasing or decreasing.
Corresponding author. Email: lemire@acm.org
This is an expanded version of a conference paper [1].
International Journal of Computer Mathematics
ISSN 0020-7160 print/ISSN 1029-0265 online c
200x Taylor & Francis
http://www.tandf.co.uk/journals
DOI: 10.1080/0003681YYxxxxxxxx
ar
X
iv
:0
70
9.
11
66
v1
[
cs
.D
B]
7
Se
p 2
00
7
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