# Movement Pattern Analysis Based on Sequence Signatures

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Related Works

#### 2.1. Relationships among MPOs

#### 2.2. Visualization of MPOs

#### 2.3. Clustering of Movements

## 3. Methodology

#### 3.1. Step 1: Converting Raw Trajectory Data into Qualitative Relationships

_{B}) [52], QTC-Double Cross (QTC

_{C}) [53], and QTC for networks (QTC

_{N}) [54]. The first two types address MPOs in a two-dimensional Euclidean space, whereas the latter is employed to describe network-constrained phenomena. In this study, we focus on QTC

_{B}because it is the simplest type from which all of the other types are derived. QTC

_{B}provides a qualitative representation of the two-dimensional movement of a pair of MPOs. Binary relations between two MPOs are evaluated based on the Euclidean distance [21]. In QTC

_{B}, a qualitative relationship between a first object k and a second object l at a time stamp t is defined by a label that is composed of two characters. This label represents the following two relationships [21,55]:

Assume: MPOs k and l, and time stamp t |

k | t denotes the position of k at t |

l | t denotes the position of l at t |

d(u,v) denotes the Euclidean distance between two positions u and v |

t_{1} < t_{2} denotes that t_{1} is temporally before t_{2} |

- A.
- Movement of k with respect to l at t (distance constraint):
- − : k is moving toward l:$$\begin{array}{c}\exists {t}_{1}\left({t}_{1}\prec t\wedge \forall {t}^{-}\left({t}_{1}\prec {t}^{-}\prec t\to d\left(k\left|{t}^{-},l\right|t\right)>d\left(k\left|t,l\right|t\right)\right)\right)\wedge \\ \exists {t}_{2}\left(t\prec {t}_{2}\wedge \forall {t}^{+}\left(t\prec {t}^{+}\prec {t}_{2}\to d\left(k\left|t,l\right|t\right)>d\left(k\left|{t}^{+},l\right|t\right)\right)\right)\end{array}$$
- +: k is moving away from l:$$\begin{array}{c}\exists {t}_{1}\left({t}_{1}\prec t\wedge \forall {t}^{-}\left({t}_{1}\prec {t}^{-}\prec t\to d\left(k\left|{t}^{-},l\right|t\right)<d\left(k\left|t,l\right|t\right)\right)\right)\wedge \\ \exists {t}_{2}\left(t\prec {t}_{2}\wedge \forall {t}^{+}\left(t\prec {t}^{+}\prec {t}_{2}\to d\left(k\left|t,l\right|t\right)<d\left(k\left|{t}^{+},l\right|t\right)\right)\right)\end{array}$$
- 0: k is stable with respect to l (all other cases)

- B.
- Movement of l with respect to k at t (distance constraint) can be described as in A with k and l interchanged, and hence:−: l is moving toward k+: l is moving away from k
- 0: l is stable with respect to k (all other cases)

_{B}is thus the tuple (A B) that considers the distance constraints described above (relationships A and B). In total, this approach yields 9 (3

^{2}) base relations, which are represented in Figure 1. In each of the nine cases that are depicted in this figure, k is represented on the left side, and l is represented on the right. The dashed line segments and crescents delineate potential motion areas. The dots can be stationary only if a dot is filled. The nine represented relationships form a set of jointly exhaustive and pairwise disjoint base relations; meaning that any two intervals stand to each other in exactly one of these relations [56]. Consequently, at each time instant, there is one and only one QTB

_{B}relation for each pair of MPOs.

_{B}relations $\left(0+\right),\text{}\left(0-\right),\text{}\left(+\text{}0\right)$, and $\left(-\text{}0\right)$are explained as follows.

- $\left(0+\right),$: k is stable with respect to l, and l is moving away from k
- $\left(0-\right)$: k is stable with respect to l, and l is moving toward k
- $\left(+\text{}0\right)$: k is moving away from l, and l is stable with respect to k
- $\left(-\text{}0\right)$: k is moving toward l, and l is stable with respect to k

_{B}relations. Suppose that car k is going to overtake car l. This interaction can be expressed in terms of the QTC

_{B}relationships as follows. The video of the overtake event is available as supplementary information.

- State 1.
- car k and car l are both driving in the same traffic lane, and k is driving behind l: $\left(-\text{}+\right)$
- State 2.
- car k is heading out to the second lane: $\left(-\text{}+\right)$
- State 3.
- car k is driving in the second lane and is driving behind l, which is driving in the first lane: $\left(-\text{}+\right)$(see Figure 2).
- State 4.
- car k is driving in the second lane and is driving in front of car l, which is driving in the first lane: $\left(+\text{}-\right)$
- State 5.
- car k is heading back to the first lane: $\left(+\text{}-\right)$
- State 6.
- car k and car l are both driving in the first lane, and l is driving behind k: $\left(+\text{}-\right)$

_{B}movement patterns of MPOs. The next subsection presents a method to transform these patterns into a fractal-based representation.

#### 3.2. Step 2: Summarizing QTC_{B} Movement Patterns in a Sequence Signature

_{B}relationships that constitute a movement pattern). Thus, a SESI of length 1 represents movement patterns that are composed of one QTC

_{B}relation, while, for example, a SESI of length 5 represents movement patterns that are composed of five consecutive QTC

_{B}relations. The resolution of these cells depends on the length of that SESI.

_{B}relationships. For higher lengths, each cell is further subdivided into nine cells, allowing each cell in a SESI of length n to correspond with a unique sequence of n qualitative QTC

_{B}relationships. In this way, each sequence of QTC

_{B}relationships has a specific location in the SESI. For example, the highlighted cells in the SESIs of lengths 1 and 2 in Figure 2 represent the movement patterns $\left\{\left(-\text{}+\right)\right\}$ and $\left\{\left(-\text{}+\right)\to \left(0\text{}0\right)\right\}$ of the above described overtake, respectively. In QTC

_{B}, a SESI of length n contains 9

^{n}cells.

_{B}relationships are possible or significant. First, based on the laws of continuity (see Section 3.1), we can exclude chronologically impossible combinations of QTC

_{B}relationships in SESIs of length 2 or more. Figure 3a demonstrates SESIs of length 2 after imposing the continuity constraint.

_{B}relationship is invariant over time. For example, a transition from $\left(-\text{}+\right)$ into itself is not very meaningful from a qualitative perspective. Figure 3b shows the result of imposing both of the above restrictions on a SESI of length two. For illustrative purposes, we also provide the representations of SESIs of lengths 1 to 3 in Figure 4a along with the SESI representation of the overtake event given in Equation (5) (Figure 4b). The fractal approach outlined above offers an insightful way to address repetitive movement patterns of any length.

#### 3.3. Step 3: Clustering Trajectory Pairs Based on Their Relative Movement Patterns.

_{1}and S

_{2}of length n is defined as:

_{1,ij}and S

_{2,ij}of the mapped QTCB movement patterns in the ij

^{th}cell of S

_{1}and S

_{2}, respectively; ${\alpha}_{n}$ denotes the number of impossible or insignificant cells in a SESI of length n; and U

_{ij}denotes the value of the highest frequency of the ij

^{th}cell among all of the SESIs on which the distance measure is applied.

_{B}movement patterns, to make it possible to compare them. As a result, the normalized frequency of each cell of a SESI is confined to the interval [0, 1]. To calculate the distance between two SESIs, we start from the top left cells in both SESIs to subtract their normalized frequencies from each other. The distance measure in Equation (6) runs over all of the cells in the SESIs. The denominator in Equation (6) represents the number of feasible sequences of QTC relationships in that SESI. By definition, infeasible sequences are assigned a frequency of 0. For example, there are 529 impossible cells in a SESI of length 3. The distance function in Equation (6) ranges from 0 to 1, where 0 indicates that the SESIs are identical and 1 indicates that there is no correspondence at all between the SESIs. In fact, d

_{n}is an indicator expressing to what extent QTC movement patterns of a pair of MPOs (with length n) are different from the QTC

_{B}movement patterns of another pair. As an illustration, Figure 5 depicts two arbitrary trajectory pairs of two objects during two different time intervals. The SESIs of length 3 for both trajectory pairs are shown in Figure 6. The black cells indicate impossible cells, whereas the green cells display the frequency of the movement patterns. In this simple example, the frequency of occurrence of a movement pattern is either 0 or 1. The distance between these two SESIs can be calculated based on Equation (6):

_{B}, qualitatively similar across both time intervals.

**Figure 6.**Movement patterns and SESIs for the trajectory pairs shown in Figure 5.

## 4. Illustration

#### 4.1. Example 1: Overtake Event on a Highway

_{B}movement patterns for all five trajectory pairs. Following our methodology, we must first transform the pairs of trajectories into sequences of qualitative relationships. These sequences are depicted in Table 1. We next convert the sequences into SESIs of lengths 2 and 3 (Figure 8). We then measure the distance between all of the pairs of SESIs of a specific length. These distance measurements are then placed in a (symmetric) distance matrix for each SESI length (Figure 9). We lastly group the trajectory pairs using a hierarchical clustering method (Figure 9). As presented in Table 1, the movement patterns of the first three situations (1, 2, and 3) are similar to each other, and the movement patterns of the last two situations (4 and 5) are the same. In fact, the distance matrix shows that the first three trajectory pairs are identical in terms of qualitative relationships (see Table 1).

**Figure 9.**Distance matrices and hierarchical clustering of five traffic situations (five trajectory pairs) for SESIs of lengths 2 and 3 based on the distance function in Equation (6).

Situation 1 | Situation 2 | Situation 3 | Situation 4 | Situation 5 |
---|---|---|---|---|

{ (+ -) ==> (00)} | { (+ -) ==> (00)} | { (+ -) ==> (00)} | { (- -) ==> (00)} | { (- -) ==> (00)} |

{(0 0) ==> (- +)} | {(0 0) ==> (- +)} | {(0 0) ==> (- +)} | { (0 0) ==> (+ +)} | { (0 0) ==> (+ +)} |

{(+ -) ==> (0 0) ==> (- +)} | {(+ -) ==> (0 0) ==> (- +)} | {(+ -) ==> (0 0) ==> (- +)} | {(- -) ==> (0 0) ==> (+ +)} | {(- -) ==> (0 0) ==> (+ +)} |

_{B}movement patterns are then placed in the same clusters. Thus, trajectory pairs 1, 2 and 3 are taken together in the same cluster, while trajectory pairs 4 and 5 are grouped into another cluster. Here, cluster analysis allows a comparison of the movements of MPOs based on the QTC

_{B}movement patterns extracted from the trajectory pairs. The distance value at each branching in the dendrogram represents the average distance between the SESIs in the branches. Figure 9 represents only the hierarchical clustering of SESIs of lengths 2 and 3, respectively. The reason is that the maximum length of the considered sequences equals 3 in all of the situations presented in Table 1.

#### 4.2. Example 2: Samba Dance

**Figure 10.**(

**a**) An abstracted movement of a samba dancer based on four parts of the body (right finger, left finger, right toe, and left toe), and (

**b**) top view of movements of the samba dancer during four different time intervals, each lasting 0.8 s.

_{B}relations of all possible pairs of MPOs during each time interval, we create hyper-SESIs for four equal time intervals based on the frequency of the QTC

_{B}movement patterns (Figure 11). To enhance the visibility of the transformed QTC

_{B}movement patterns, impossible and insignificant QTC

_{B}movement patterns are not represented.

_{B}movement pattern, between the left toe and left finger. Note that only the upper parts of the main diagonal of the hyper-SESIs must be considered because the interactions of the moving parts are symmetrical. We attempt to discover whether there is a degree of closeness/similarity between the QTC

_{B}movement patterns of different pairs of body parts during different time intervals of movement. Based on the context of the movement data, we can interpret the hyper-SESIs. For example, we know that samba dance is a rhythmical dance that has regularity in the movements. The regularities in the movements can be distinguished on the hyper-SESIs’ cells. In Figure 11, the upper left cell of the hyper-SESIs shows the interactions between the right finger and right toe at four different time intervals. One can observe that there are more repetitive QTC

_{B}movement patterns during the time interval 4 because more frequent patterns are visualized on its hyper-SESI. This means that periodicity or frequency of patterns can be different in time in such fast rhythmical dance.

**Figure 11.**Hyper-SESIs of four equal time intervals of movements based on the frequency of QTC

_{B}movement patterns.

_{B}movement patterns $\left\{\left(+\text{}0\right)\to \left(+\text{}+\right)\right\}$ and $\left\{\left(+\text{}-\right)\to \left(+\text{}0\right)\right\}$ of length two and $\{\left(+\text{}0\right)\to \left(+\text{}+\right))\to \left(0\text{}+\right)\}$ and $\left\{\left(+\text{}-\right)\to \left(+\text{}0\right)\to \left(+\text{}+\right)\right\}$ of length three are observed in the first time interval of movement but not in the other intervals. It makes more sense when we expect certain patterns with a specific frequency. For example, in our case study, a dance tutor can ask the dance amateur to perform some movements. The requested movement patterns should be observed in the corresponding SESIs unless the dance amateur has failed to perform successfully.

_{B}movement patterns (Figure 12). Unlike the frequency, the duration of the QTC

_{B}movement patterns are different, and some of the QTC

_{B}movement patterns have lasted longer while others are shorter. As mentioned earlier, the temporal granularity of capturing dance movement is 0.04 s. For the sake of simplicity, we assume every 0.04 s to be a single time unit. For example, the pattern $\left\{\left(+-\right)\to \left(0-\right)\right\}$ lasting 10 time units is equivalent to$10\times 0.04=0.4s$.

**Figure 12.**Hyper-SESIs of four equal time intervals of movements based on the duration of QTC

_{B}movement patterns.

**Figure 13.**Distance between hyper-SESIs at four time intervals of movement both for (

**a**) frequency and (

**b**) duration alongside the dendrograms that represent the agglomerative hierarchical clustering of pairs of trajectories.

## 5. Discussion

_{B}movement patterns that have occurred is visible in SESIs. The availability of fine-grained movement data at small temporal sampling intervals increases the difficulty of detecting QTC

_{B}movement patterns in SESIs. That is, SESIs are used as a visual-aided approach to perceive the dispersion of QTC

_{B}movement patterns. However, we see the SESI mathematically as a rectangular array of numbers, symbols, or expressions, which are arranged in rows and columns. The individual items in this matrix are referred to as its elements or entries (e.g., the frequency of QTC

_{B}movement patterns). From a computational perspective, the longer the length of a SESI is, the more time is needed to compute the distance between two SESIs. In short, it is possible to calculate distances for any length of SESI, but visually detecting long-movement patterns in SESIs will become challenging.

_{B}(our approach) significantly reduces the accuracy of the result. However, other types of QTC, such as QTC-Double Cross (QTC

_{C}), can incorporate more relevant information about the movement. In addition to the Euclidean distance considered in QTC

_{B}, in QTC

_{C}, directional information between two moving objects is included. Consequently, to visualize the QTC

_{C}movement patterns, a new high resolution SESI is needed because each QTC

_{C}relation is represented by a four-tuple (for more explanation about QTC

_{C}, see ([53]).

## 6. Conclusions and Future Work

_{B}movement patterns were represented in a sequence signature (SESI), which is a fractal way of mapping patterns of interactions between MPOs in an indexed raster space. Then, in the third step, a distance function was used to cluster SESIs and aims to improve the understanding of the movement patterns.

_{B}movement patterns. The proposed methodology could be used in a wide range of research applications. Movement patterns such as walking, running, jumping, lifting, striking and swimming can be investigated for different purposes. For example, the proposed approach can be used in sports sciences to analyze the movement of athletes with the purpose of rehabilitation, physical education and practice. The emphasis of the therapy can be diverse, ranging from upper limb rehabilitation and balance rehabilitation, to the rehabilitation of specific body parts [64].

## Supplementary Files

Supplementary File 1## Acknowledgments

## Author Contributions

## Conflicts of Interest

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## Share and Cite

**MDPI and ACS Style**

Chavoshi, S.H.; De Baets, B.; Neutens, T.; Delafontaine, M.; De Tré, G.; De Weghe, N.V.
Movement Pattern Analysis Based on Sequence Signatures. *ISPRS Int. J. Geo-Inf.* **2015**, *4*, 1605-1626.
https://doi.org/10.3390/ijgi4031605

**AMA Style**

Chavoshi SH, De Baets B, Neutens T, Delafontaine M, De Tré G, De Weghe NV.
Movement Pattern Analysis Based on Sequence Signatures. *ISPRS International Journal of Geo-Information*. 2015; 4(3):1605-1626.
https://doi.org/10.3390/ijgi4031605

**Chicago/Turabian Style**

Chavoshi, Seyed Hossein, Bernard De Baets, Tijs Neutens, Matthias Delafontaine, Guy De Tré, and Nico Van De Weghe.
2015. "Movement Pattern Analysis Based on Sequence Signatures" *ISPRS International Journal of Geo-Information* 4, no. 3: 1605-1626.
https://doi.org/10.3390/ijgi4031605