Mining human periodic behaviors via tensor factorization and entropy

Preliminaries

Related works

Mobility intentions extraction

As mentioned earlier, mobility intentions are typically latent but exhibit a strong degree of spatiotemporal regularity. For example, the act of commuting, a fundamental mobility intention found in many spatiotemporal datasets, explains why an individual consistently arrives at the workplace around $9$ am on workdays. Before delving into the analysis of mobility intentions, it is essential to identify the specific mobility intentions that are inherent in a given spatiotemporal dataset.

In this article, we employ the CP decomposition to extract mobility intentions from a given spatiotemporal dataset. To apply the CP decomposition, we construct a three-dimensional tensor $Y$ comprising location, hour, and day. This tensor forms the basis for our analysis. The element $y_{r_i,t_j,d_k}$ of the three-way tensor $Y\in R^{M\times H\times D}$ can be computed as

(2) $$y_{r_i,t_j,d_k}=\frac{Count(r_i,t_j,d_k)}{{\sum }_{q=1}^{L}Count(r_q,t_j,d_k)}$$where $r_i$, $t_j$, and $d_k$ are the index of the location, the time bin and the day of month, respectively; L is the total number of locations, and $Count(r_i,t_j,d_k)$ is the number of users who appeared at location $r_i$ at time $t_j$ on the $d_k$-th day.

In the CP decomposition, the tensor $Y$ is factorized into a sum of rank one component tensors ${\mathbf{Y}}_g$:

(3) $$Y\approx {\sum }_{g=1}^G{\lambda }_g{\mathbf{Y}}_g.$$

The rank one tensor ${\mathbf{Y}}_g$ is outer product of three vectors:

(4) $${\mathbf{Y}}_g={{\mathbf{r}}_g}^{\circ }{{\mathbf{t}}_g}^{\circ }{\mathbf{d}}_g$$

The components of ${\mathbf{r}}_g$, ${\mathbf{t}}_g$ and ${\mathbf{d}}_g$ represent the probability distribution of ${\mathbf{Y}}_g$ on locations, time bins and days of month, respectively. ${\mathbf{Y}}_g$ can be used to explain why a user appeared in one location at a special timestamp. Hence, the ${\mathbf{Y}}_g$ is identified as one mobility intention $m_s$ in this article.

Because CP decomposition can effectively decomposes a tensor into a summation of rank-one tensors, each of these rank-one tensors can be treated as a mobility intention. Therefore, CP decomposition is chosen for extracting mobility intentions from spatiotemporal datasets in this article.

In Fig. 2, we present an example of a rank-one tensor, denoted as ${\mathbf{Y}}_g^{\star }$, extracted from the Beijing Bus Smart Card (BBSC) dataset, which will be used in our later experiment. The figure on the right highlights bus stops with higher values in a deeper shade of red, primarily concentrated in the northwest area of Beijing. These bus stops highlighted in deeper red are the locations where the mobility intention ${\mathbf{Y}}_g^{\star }$ is more likely to appear. Notably, these bus stops are situated close to popular recreation areas such as the Summer Palace, the Yuanmingyuan Imperial Garden and the Fragrant Hills park. Analyzing the day features reveals that this mobility intention exhibits weekly periodicity. Additionally, during the Chinese National Day Golden Week (October $1$ st to $7$ th), we observe a surge in activity at the bus stops highlighted in deeper red. Focusing on the time features reveals a clear peak at around $11:00$, coinciding with the time when most people arrive at the bus stops with a higher probability. Based on this analysis, we can categorize this rank-one tensor shown in Fig. 2 as a mobility intention related to “recreation”.

To reveal the set of mobility intentions hidden within a dataset D, we employ the CP decomposition. To detect the period of a particular mobility intention, we require a function that maps records to those mobility intentions. In this context, each mobility intention is treated as a distinct class. Consequently, every record $f_i$ corresponds to one specific mobility intention $m_s\in M$, and can be thought of as belonging to a particular class. Thus, the process of mapping records to mobility intentions is transformed into a multi-class classification problem. To ensure high performance in multi-class classification, we perform comprehensive feature engineering and model training. After analyzing the vectors of the rank-one tensor and performing feature engineering, we introduce three distinct types of features: spatial features, hourly features, and daily features.

Spatial Features

• 1. Location entropy. Location entropy measures the popularity of a location. For a location $lo{c}_k$, its location entropy is defined as:

  (5) $$H(lo{c}_k^i)=-\sum _i{p}_i\ln p_i$$where $p_i$ is the proportion that the number of times for user $u_i$ to visit location $lo{c}_k$ among all users who visited the same location $lo{c}_k$.

• 2. Location type. Location type indicates the category of location $lo{c}_k$, such as bar, mall and park. Location type can be obtained from LBS’s application program interface (API), such as Google Places API (https://developers.google.com/places/) or Sina Weibo API (http://open.weibo.com/).

• 3. Distance. Here, the distance refers to the Euclidean linear distance between location $lo{c}_k^i$ and its last location $lo{c}_{k-1}^i$ visited by the same user $u_i$.

Hour Features

• 1. Hour. The hour is the hour of time $t_k^i$ in a day which is denoted by $\{0,...,23\}$.

• 2. Stay time. The stay time is time interval between two continuous footprints of an user, i.e., $t_k^i-t_{k-1}^i$.

• 3. Time span. The time span is time interval between two footprints which occurred at the same location $lo{c}_k^i$, i.e., $t_k^i-t_{k^{\mathrm{\prime }}}^i$.

Day Features

• 1. Day of week. The day of week refers weekday of $t_k^i$. It is denoted by $\{0,...,6\}$ which means Sunday to Saturday.

• 2. Day of month. The day of month is day in month of $t_k^i$. It is denoted by $\{1,\cdots ,31\}$.

• 3. Day Type. The day type refers to the category of $t_k^i$. There are three categories in this article, workday, short break holidays and long holidays.

In conclusion, utilizing these three types of features, we utilize the Adaboost classifier from the sklearn (https://scikit-learn.org/) (Pedregosa et al., 2011) to train an Adaboost model, which is responsible for mapping each record into a specific mobility intention.

Through the application of the Adaboost model, the records in our running example from Fig. 1 can be classified into a mixed sequence representing two distinct mobility intention: “recreation” and “shopping”, as depicted in Fig. 3A. Subsequently, as illustrated in Figs. 3B and 3C, we acquire separate time sequences for each mobility intention. The implementation of the Adaboost classification model enables the effective separation of multiple mixed periods.

Period identification

To detect periodicity within noisy, unevenly sampling and incomplete observations, a feasible way involves dividing the sequence X into segments, as proposed by Li, Wang & Han (2012). As depicted in Fig. 4, It’s evident that the majority of observations fall into a specific time interval when the timeline of a given mobility intention is segmented based on the true period.

In general, if we consider a binary sequence X with a length $n$ segmented by a trial period T, we can determine the mobility intention count occurring within each timeslot in T.

(6) $$S_i(T)=\{t|(t,T)=i\wedge \mathrm{I}(t)=1\},t=0,1,\dots ,n-1,i=0,1,\dots ,T-1$$

The probability at each timeslot in T is then

(7) $$p_i(T)=\frac{|S_i|}{\sum _{j=0}^{T-1}|S_j|},i=0,1,\dots ,T-1$$

Apparently, $p_i(T)$ follows ${\sum }_{i=0}^{T-1}{p}_i(T)=1$.

An intuitive method for detecting periodicity involves calculating $p_i(T)$ for each candidate T and select the candidate with the highest probability as a criterion for identifying the period.

(8) $${\gamma }_X(T_0)=max{p}_i(T).$$

The true period under consideration, denoted as $T_0$, corresponds to a candidate period T which yields the maximum value of ${\gamma }_X(T)$, as illustrated in Fig. 4.

However the measurement of ${\gamma }_X(T_0)$ has a drawback of favouring shorter periodicity. In general, when more observations fall into the same timestamps in a candidate T, it results in a lower value of ${\gamma }_X(T)$. Given that the timeline is folded in half, two scenarios arise: either more observations coincide at the same timestamps, resulting in a higher value of ${\gamma }_X(T)$, or the observations’ distribution across timeslots remains unchanged, leading to the same value of ${\gamma }_X(T)$. As shown in Fig. 5, after folding the timeline in half, the value of ${\gamma }_X(T/2)$ is equal to ${\gamma }_X(T)$.

Furthermore, let’s define $r$ as the ratio of the time interval in which most observations fall after segmentation by the true period. The above method’s performance drops drastically and is not robust when the ratio $r$ is smaller than $0.3$, which is just in line with many real-life scenarios. For example, consider the situation where the true period $T_0=24$ and $r=0.125$. In this case, all observations will fall in an interval with length of $3$ after being segmented by the true period $T_0$, and the corresponding accuracy of the method described above is no more than $0.3$. However, in most human mobile behavior scenarios, the value of $r$ is much smaller than $0.125$. Therefore, a new measurement is required to address these limitations.

Suppose we have a binary sequence of behavior X with a true period $T_0$. According to Definition 1, the observations in this sequence should ideally fall within a compact interval of $[t_0-\delta ,t_0+\delta ]$ when X is segmented by its true period $T_0$. However, when X is incorrectly by another candidate period $T_f$, these observations might be dispersed across scattered intervals. For instance, as shown in Fig. 4, when we segment the binary sequence X by its true period of $168$ h (a week), the observations are concentrated around $114$. In contrast, if we segment X by $192$ h, the observations are disordered and spread across a wide time interval. Observations are notably more ordered when X is segmented by the true period $T_0$, in contrast to the disordered pattern that emerges when X is segmented by an incorrect period.

To quantitatively assess this disorder, we turn to the concept of entropy, which is commonly used in physics to measure the chaos or randomness of a system. Consider that different candidate periods can be likened to different systems; we can employ entropy as a metric to quantify the level of disorder across these candidate periods. The entropy of a candidate period is defined as follows:

(9) $$H(T)=-\sum _{i=0}^{T-1}{p}_i(T)\log p_i(T).$$

In our context, a lower entropy value $H(T)$ signifies that observations are more concentrated when segmented by the candidate period T, while a higher entropy value suggests more dispersion. Therefore, entropy can be served as a useful metric for periodicity detection. Notably, the entropy is at its minimum when X is segmented by its true period $T_0$, reflecting a highly ordered and periodic pattern.

However, the entropy metric also exhibits a preference for shorter periodicity, which poses a challenge similar to the intuitive method. For instance, consider a periodic behavior binary sequence X generated from Definition 1 with an candidate period $T_0$ which is even. If $T_0/2$ is not included in the compact interval $[t_0-\delta ,t_0+\delta ]$, then according to Eq. (9), we have $H(T_0)=H(T_0/2)$. Consequently, one might mistakenly conclude that $T_0/2$ is the true period.

This issue arises because the entropy assesses the degree of disorder of a potential period independently. However, our objective is to compare the disorder of various potential periods, and the entropy of different potential periods should not be directly compared. As previously mentioned, a more concentrated distribution of observations segmented by a candidate periodicity T indicates that T is closer to the true period. Conversely, when the distribution of observations approaches uniformity, T is an incorrect period. In an extreme case, a mobility intention might occur at nearly every timestamp with almost equal probability when X is segmented by an incorrect period $T_f$. In such an extreme scenario, no true period exists which means X has no periodicity at all. Therefore, we can employ the discrepancy in entropy between $p_i(T)$ and the uniform distribution on a potential period T as the periodic measurement. The discrepancy of a candidate period T is defined as follows:

$$\begin{gathered} H(U_T)-H(T)=-\sum _{i=0}^{T-1}\frac{1}{T}\log \frac{1}{T}+\sum _{i=0}^{T-1}{p}_i(T)\log p_i(T) \\ =\log T\sum _{i=0}^{T-1}{p}_i(T)+\sum _{i=0}^{T-1}{p}_i(T)\log p_i(T) \\ =\sum _{i=0}^{T-1}{p}_i(T)\log \frac{p_i(T)}{1/T} \end{gathered}$$

The last formula provides the definition of relative entropy between $p_i(T)$ and the uniform distribution on T which is also known as the Kullback-Leibler divergence. Therefore, the above formula can also be written in the following form:

(10) $$\mathrm{K}\mathrm{L}(T)=H(U_T)-H(T)=\sum _{i=0}^{T-1}{p}_i(T)\log \frac{p_i(T)}{1/T}=\log T-H(T)$$

Therefore, when dealing with a periodic behavior binary sequence X with unknown period, if the probability distribution $p_i(T)$ exhibits a sharper peaked and the discrepancy from the uniform distribution on T is more pronounced, T is closer to the true period $T_0$. This leads us to the following lemma, which asserts that the relative entropy reaches its maximum at the true period $T_0$.

• Lemma 1 : If a binary sequence X is generated periodically according to a categorical distribution ${\mathit{\mu }}_0^{\mathrm{\prime }}$ with a period $T_0$, then for any $T\ge 2,T\in \mathbb{N}$, we have

(11) $$\underset{n\to \mathrm{\infty }}{lim}\mathrm{K}\mathrm{L}(T_0)\ge \underset{n\to \mathrm{\infty }}{lim}\mathrm{K}\mathrm{L}(T)$$

The following is proof of Lemma 1. Based on Definition 1, we assume that a periodic time sequence X is generated from a categorical distribution for some period $T_0$, and the parameter of this distribution is ${\mathit{\mu }}_0=({\mu }_0,{\mu }_1,\dots ,{\mu }_{T_0-1})$, where ${\sum }_{i=0}^{T_0-1}{\mu }_i=1$. Let $I_v=[t_0-\delta ,t_0+\delta ]\subseteq [1,T_0]$. From the Definition 1, it’s clear that the probability of ${\mathit{\mu }}_0$ falling in $I_v$ are greater than those falling in $[1,T_0]/I_v$. That is to say that ${\mu }_g\gg {\mu }_h,g\in I_v,h\in [1,T_0]/I_v$,

We use $T_0$ and T to denote the true period and one candidate period. For an interval $[0,T\times T_0-1]$, it’s obvious that this interval contains T periods of $T_0$ or $T_0$ periods of T. Let $p_{i,j}$ be the $i$-th position of period T in the $j$-th segment. Then we have:

(12) $$p_{i,j}={\mu }_{(i+j\cdot T)mod{T}_0}$$

The $i$-th position’s parameter of period T is:

(13) $${\mu }_i^{\mathrm{\prime }}(T)=\frac{1}{T}\sum _{j=0}^{T_0-1}{p}_{i,j}$$

The relative entropy of period T is

(14) $$\mathrm{K}\mathrm{L}(T)=\ln T+\sum _{i=0}^{T-1}{p}_i(T)\ln p_i(T)=-\frac{1}{T}\sum _{i=0}^{T-1}\sum _{j=0}^{T_0-1}{p}_{i,j}\left(\ln \sum _{j=0}^{T_0-1}{p}_{i,j}\right)$$

For ${\mu }_g\gg {\mu }_h,g\in I_v,h\in [0,T_0-1]-I_v$, the relative entropy of period $T_0$ is

(15) $$\mathrm{K}\mathrm{L}(T_0)=\sum _{j=0}^{T_0-1}{\mu }_j\ln {\mu }_j+\ln T_0\ge \sum _{k\in I_v}{\mu }_k\ln {\mu }_k+\ln T_0$$

Then,

$$\begin{gathered} \mathrm{K}\mathrm{L}(T_0)-\mathrm{K}\mathrm{L}(T)\ge \ln T_0+\sum _{k\in I_v}{\mu }_k\ln {\mu }_k-\frac{1}{T}\sum _{i=0}^{T-1}\sum _{j=0}^{T_0-1}{p}_{i,j}\left(\ln \sum _{j=0}^{T_0-1}{p}_{i,j}\right) \\ =\ln T_0+\frac{1}{T}\sum _{i=0}^{T-1}{\sum }_{j=0}^{T_0-1}\left\{\frac{1}{T_0}{\sum }_{k\in I_v}{\mu }_k\ln {\mu }_k-p_{i,j}\left(\ln {\sum }_{j=0}^{T_0-1}{p}_{i,j}\right)\right\} \\ \ge \ln T_0+\frac{1}{T}\sum _{i=0}^{T-1}\sum _{j=0}^{T_0-1}{p}_{i,j}\ln \frac{{\mu }_{\mathrm{m}\mathrm{i}\mathrm{n}}}{\frac{1}{T}{\sum }_{j=0}^{T_0-1}{p}_{i,j}} \\ \ge \ln T_0+\frac{1}{T}{\sum }_{i=0}^{T-1}\sum _{j=0}^{T_0-1}{p}_{i,j}\ln \frac{1}{T_0}=\ln T_0+\ln \frac{1}{T_0}=0 \end{gathered}$$

To mine $T_0$, compute $\mathrm{K}\mathrm{L}(T)$ and select the value of T that maximizes $\mathrm{K}\mathrm{L}(T)$ as the true period. The pseudo code of periodicity detection is shown in Algorithm 1.

The mining of a user’s periodic behaviors allows us to predict the user’s future locations. Specifically, for a given user $u_i$, his history records, and a target time $t_f$, we can identify which periodic behavior $t_f$ belongs to. Subsequently, we use the most frequent location within the target periodic behavior as the prediction result.

The quantitative assessment of periodicity

Human behavior periodicity plays a crucial role in predictive applications, such as forecasting future user activities or movements. The accuracy of these predictions is inherently tied to the quality of the underlying human behavior periodicity. An ideal “good” periodic human behavior, characterized by its strict periodicity, leads to superior predictive performance. Predictions generated from such behavior are precise, resembling specific points on the timeline with high confidence levels. In contrast, a “bad” periodic human behavior exhibits inferior predictive performance, resulting in broader prediction intervals with lower confidence. But how do we measure what makes a periodic human behavior “good” or “bad”?

To assess prediction performance based on human behavior periodicity, we must define quantitative measure for assessment of periodicity. As aforementioned, in a periodic binary sequence, observations are expected to cluster within a narrow interval when partitioned with the true period. In an ideal strictly periodic behavior, which represents the highest quality binary sequence, all observations fall in a interval characterized by a single timestamp. It’s evident that a narrower interval with more observations aligning within it indicates a “good” periodic human behavior. On the contrary, if the interval is wide and contains fewer observations, the periodic behavior is considered “less favorable”. Therefore, a “good” periodic human behavior is one in which the interval is narrow and contains a substantial number of observations.

To quantify and evaluate this distinction, we employ the Coverage Width-Based Criterion (CWC) (Khosravi et al., 2011), a method that doesn’t rely on specific references. The CWC assesses periodic behavior by considering a target interval $I_c$ containing observed data points, the count of observations within this interval $C_f$, and the total number of observations in the sequence $C_o$. The interval coverage probability (ICP) is defined as follows:

(16) $$ICP=\frac{C_f}{C_o}$$

The definition of CWC is

(17) $$CWC=NIW\cdot \left(1+g^{\star }{e}^{-{\eta }^{\star }(ICP-{\mu }^{\star })}\right)$$

The NIW is normalized interval width which refers to the ratio of $C_f$’s width to true periodicity width, i.e.,

(18) $$NIW=\frac{C_f}{T_0}$$

The constants ${\eta }^{\star }$ and ${\mu }^{\star }$ in 17 are two hyper parameters. ${\mu }^{\star }$ is a threshold related with ICP. Some observations should not fall into the target interval $I_c$ after partition for noises or outliers. Therefore, the value of ICP will be less than $1$ in reality. We assumed that all observations from human periodic behavior fall into the target interval $I_c$ when the value of the ICP is greater than the threshold ${\mu }^{\star }$. ${\eta }^{\star }$ is a coefficient determining how much penalty is assigned to ICP with a low value. It magnify the difference between ICP and ${\mu }^{\star }$. In this article, ${\eta }^{\star }$ is set to $2$ and ${\mu }^{\star }$ is set to $0.95$. $g^{\star }$ is a function related with ICP and is given by the following step function:

(19) $$g^{\star }=\{\begin{array}{cc}0,& ICP\ge {\mu }^{\star }\\ 1,& ICP<{\mu }^{\star }.\end{array}$$

This function indicates that CWC will be affected by ICP only when the value of ICP is less than the threshold ${\mu }^{\star }$.

There are two parts in Eq. (17). The first part is NIW which is related with width of target interval $I_c$, and the other part is ICP which is affected by the number of observations that fall into the interval. If ICP is greater than ${\mu }^{\star }$ which means “all observations” fall in the target interval $I_c$, CWC is only decided by the width of $I_c$, i.e., NIW. The narrower the $I_c$ is, the smaller the CWC’s value is. On the other side, the value of CWC is much greater if ICP’s value is far less than ${\mu }^{\star }$. The criteria for a “good” human periodic behavior is narrow target interval with more observations fallen in. Therefore, the smaller the value of CWC, the “better” the human periodic behavior line, that means the human periodic behavior has the more obvious periodicity.

However, for some specific human periodic behavior, these two parts in Eq. (17) can be conflicting. When choosing a narrow interval, ICP is often small. Calculating CWC for a human periodic behavior requires the careful determination of IPC and NIW. To address this issue,we introduce a new CWC computing algorithm outlined in Algorithm 2.

For a strictly periodic behavior, all observation fall in one time stamp. Hence, the value of strictly periodic behavior’s $I_c$ is 1, its ICP is 1 and its CWC is

$$CW{C}_{sp}=\frac{1}{T_0}\cdot \left(1+0\times e^{-2\times (1-0.95)}\right)=\frac{1}{T_0}$$

In the running example presented in this article, the periodicity for the recreation and shopping sequence is $24$ and $168$ respectively, with corresponding CWC values as follows:

$$\begin{gathered} CW{C}_{recreation}=\frac{2}{24}\cdot \left(1+0\times e^{-2\times (1-0.95)}\right)=0.083 \\ CW{C}_{shopping}=\frac{1}{168}\cdot \left(1+1\times e^{-2\times (0.75-0.95)}\right)=0.018 \end{gathered}$$

The result shows that the periodicity of shopping is better than recreation.

Periodicity detection on synthetic time series data

Effectiveness of relative entropy

In previous works, entropy was employed as a measurement for evaluating periodicity, despite its known bias toward favoring the shorter periods. In this section, we will elucidate this issue and demonstrate the effectiveness of the relative entropy as an alternative measure.

Based on the work of Li, Wang & Han (2012) work, the binary sequence X is more concentrated when segmented by the true period $T_0$ compared to an incorrect period $T_f$. Consequently, entropy H serves as a measure of concentration, with the entropy of the true period $T_0$ being lower than any incorrect period $T_f$. To illustrate this, We use a data generator, as described in the “Experimental Settings” section, to generate a periodic behavior binary sequence X consisting of $2,880$ observations with the true period $T_0=24$. In Figs. 6A–6C display the distribution of observations when a binary sequence X with the true period $T_0=24$ is segmented by potential periods $24$, $29$ and $12$. Figure 6D illustrates the entropy for different potential periods. Notably, in Fig. 6D, the entropy corresponding to the true period $T_0=24$ is smaller than the incorrect period $T_f=29$, and the corresponding distribution in Fig. 6A is more concentrated than the one in Fig. 6B. Thus, it seems that entropy can be employed as a criteria for periodicity detection, though it could favor shorter periods.

The distribution $p_i(T)$ in Fig. 6A is identical to that in Fig. 6C. As demonstrated in Fig. 6D, it is evident that the values of $H(T=12)$, $H(T=6)$ and $S(T=3)$ are all equal to the true period $H(T_0=24)$. This could potentially leads to a wrong conclusion that $T=3$ is the true period. This problem arises due to the lack of a reference for comparing $p_i(T)$ among different potential period T. The distributions related to different potential periods can not be directly compared.

In fact, if observations fall in a narrow interval after being segmented by a potential period T, the distribution is more peaked in contrast to the uniform distribution of T. As depicted in Fig. 6, although the entropy of Fig. 6A is equal to entropy of Fig. 6C, the distribution of $p_i(T_0=24)$ is more peaked than $p_i(T=12)$. Thus, it is feasible to utilize the uniform distribution as a reference and employ the entropy discrepancy between $p_i(T)$ and the uniform distribution of a potential period T as criteria for periodicity detection.

In Fig. 7, it is evident that relative entropy reaches it maximum when the potential period T is equal to the true period $T_0$. Therefore, the new measurement effectively mitigates the issue of favoring shorter periods. Although the maximum relative entropy can be also observed when $T=k{T}_0,k\in N$, the true period $T_0$ can still be accurately obtained using Algorithm 1.

Results and analysis

We study the performance of the compared methods on the synthetic dataset byvarying one parameter in each experiment while keeping the others at their default values.

Figure 8 displays the performance of the compared methods on synthetic dataset. It illustrates that, in most case, our method exhibits higher accuracy than the comparison methods. Figure 8 also reveals that most methods perform better when the data is of higher quality, characterized by factors such as a larger number of period repetitions N, a higher sampling rate $\delta$, lower noise rate $\gamma$ and smaller variance $\delta$.

The accuracy of ePeriodicity is consistently reliable with the default parameters only when the observations contain more than $150$ periods, which might not be hold in real spatiotemporal datasets. In contrast, our method achieves close to $100\mathrm{\%}$ accuracy when the observations contains $120$ periods. Impressively, our method can detect more than $80\mathrm{\%}$ of the periods even with as few as $60$ observed periods, a scenario common in many spatiotemporal datasets, such as check-in dataset (Cho, Myers & Leskovec, 2011) and Smart Card Data (SCD) of public transport (Itoh et al., 2014).

The performance of all methods, except FFT, deteriorates as $T_0$ increases, probably due to increased noise interference in longer periods. Due to bias toward shorter periods, ePeriodicity fails to detect period when the true period $T_0$ exceeds $72$.

The center of observed time $t$ has negligible effect on the performance of all methods and can be considered as the default value of parameters. In our experiment scenario, where there is only one mobility intention in a period, the performance of ePeriodicity is no better than FFT, while PRED serves as the best baseline method. However, their method is time-consuming for involving Gibbs sampling.

Figures 8D and 8E demonstrate that as the noise rate $\gamma$ decreases and the sampling rate $\delta$ increases, all methods achieve better results in period detection. Figure 8F illustrates that it becomes considerably more challenging to detect the true period when there is significant oscillation. For instance, when $\delta =2.25,T_0=24$ and $t_0=12$, approximately $99\mathrm{\%}$ of observations fall within the interval $[4,18]$, which covers $60\mathrm{\%}$ of $T_0$. In this extreme case, the distribution generating the time sequence lacks a steep peak, and the performance of MIRE is no better than that of FFT.

In conclusion, these results indicate that the proposed periodicity detection method is better suited for detecting periodicity than the other compared methods.

Performance evaluation using real dataset

We evaluated the performance of our proposed human periodic behavior mining model using two real spatiotemporal datasets: ThaiLandData and Gowalla. We have followed the tensor construction method outlined in Eq. (2) and employed the CP decomposition algorithm provided by TensorLy (https://tensorly.org/) (Kossaifi et al., 2019) to decompose the tensors obtained from the two datasets.

The ThaiLandData dataset is a social media check-in dataset collected from Foursquare Swarm, a mobile app that enables users to share their locations with their friends. The dataset was collected by a researcher interested in tourism data mining as part of the Team for Universal Learning and Intelligent Processing (TULIP) research gruop. The dataset encompasses records primarily located in Thailand and spans from $2014$ to $2018$. It consists of $423,991$ records contributed by $3,046$ users. The majority of the dataset is generated by tourists, and the records appear to be random in both time and place, lacking clear periodicity. However, our methods, as presented in this article, revealed some intriguing insights from this seemingly non-periodic data.

From the ThaiLandData dataset, our proposed method extracted ten mobility intentions. For the parameters of the AdaBoost classifier used for mapping check-ins to mobility intentions, we configured the base estimator as an MLP. The maximum number of estimators was set to 100, and the learning rate was assigned a value of 2.0. Additionally, the Boosting algorithm was set to SAMME.R.

We focused on mobility intentions with more than ten observations to investigate their periodicity. Table 1 presents the ratio of users exhibiting periodic mobility intentions among all users.

The “Home” mobility intention, representing the act of returning home after daily activities, is expected to exhibit periodicity. However, only $4.3\mathrm{\%}$ of user display periodic patterns, likely due to the dataset primarily comprising tourists. In the context of tourism, “Shopping” and “Dining” are the most common periodic behaviors. Nearly half of the users’ shopping and dining behaviors are periodic, with most of these showing a 24-h periodicity, aligning with the characteristics of tourist activities.

In contrast, the periodicity of “Visiting” is less consistent, ranging from 24 h (1 day) to 168 h (1 week). This suggests that sightseeing is more random compared to shopping and dining. “Accommodation”, although an essential part of tourism, doesn’t exhibit strong periodicity for most users, possibly due to irregular resting patterns during their tours.

While users’ mobility intentions display periodicity, the degree of periodicity varies, as shown in Fig. 9. As mentioned earlier, “Accommodation” for most users lacks periodicity, but users following a usual routine, which includes “Entertainment” and “study” (such as going to colleges or libraries), exhibit a more distinct periodic pattern. These mobility intentions show a high degree periodicity. In contrast, the periodicity of “Shopping” and “Dining” appears to be less regular.

The Gowalla dataset is a public available social media check-in dataset (Cho, Myers & Leskovec, 2011). It comprises $6,442,890$ check-ins of $196,591$ users from February 2009 to October 2010. After applying tensor decomposition techniques, we successfully extracted $10$ distinct mobility intentions from this extensive dataset. The parameters for the AdaBoost classifier in the Gowalla dataset closely mirror those in the ThaiLandData dataset. However, there are slight variations: the learning rate is adjusted to $0.5$, and the maximum number of estimators is set to $200$.

To illustrate the process of discovering human periodic behavior, We randomly select a user from Gowalla dataset. Figure 10 presents the distribution of $75$ check-ins for this randomly chosen user, identified as user $\mathrm{\#}11838$. These check-ins were recorded from April to August 2010, with the majority of them concentrated in the north area of Atlanta. It is noteworthy that only four locations have been checked twice, making it challenging, if not impossible, to identify direct periodic behavior based on location data alone.

By adopting the trained Adaboost model, we were able to identify a total of seven distinct mobility intentions for the recorded check-ins, as outlined in Table 2. Subsequently, we employed our proposed periodicity mining algorithm to mine human periodic behavior, utilizing the default sampling rate of one hour. The results of this analysis are displayed in Table 2.

Table 2 reveals that most mobility intentions exhibit a periodicity of $24$ h. The majority of observations are associated with the “Commuting” mobility intention, which aligns with the common daily routine and occurs at a $24$-h interval. The “Daily Routine” periodicity is approximately 53 h, indicative of some routine behaviors, like refueling the car, happening every two days. However, we were unable to detect the periodicity of “Entertainment and Recreation” due to an insufficient number of observations.

Figures 11 and 12 illustrate the locations of “Dining” and “Shopping” mobility intention. “Dining” locations are spread across the north area of Atlanta City, as depicted in 11. Traditional mining methods solely based on locations struggle to mine this type of human periodic behavior. For the user $\mathrm{\#}11838$, most of his “shopping” mobility intention occurred at a shopping center, alghough occasional visits to new places for shopping were also observed. PRED (Yuan et al., 2017b) can identify such human periodic behavior through location clustering, but their results may lack precision since they didn’t consider observations from the northwest and southwest in Fig. 12, which is inconsistent with reality. On the contrary, MIRE offers more accurate and insightful results.

Regarding the “Tour” mobility intention, its $24$-h period is somewhat unreliable. This is due to the numerous check-ins within one “Tour” period, which contradicts the assumption that only one mobility intention should occur in a period. We plan to address this issue in our future work.

Location prediction on real datasets

As aforementioned, mining human period behavior has numerous applications. In this section, we leverage the human period behavior mined from spatiotemporal datasets to predict users’ next locations.

In this context, location prediction refers to forecasting a user’s location at a given time. In the experiments in this subsection, the specified time corresponds to the next timestamp at which a periodic mobility intention is predicted to occur. For our proposed MIRE algorithm, the next location for a user’s periodic mobility intention is the one with the highest historical probability within that periodic mobility intention.

Performance study

Figures 13 and 14 display the performance on all testing records. In general, for the two datasets, MIRE consistently exhibits significantly lower error distances than the baseline methods. PMM, on the other hand, consistently generates the maximum error.

On the Gowalla dataset in Fig. 13, there is notable performance gap between PMM and the other comparison methods. One possible reason for this is that users frequently visit locations beyond just “work” and “home”, which PMM focuses on. However, this gap is not observed in Fig. 14. The reason for this difference is that in the BBSC dataset, around $70\mathrm{\%}$ of users use public transportation to commute between home and work place, aligning well with the assumption of PMM.

Periodica performs better than PMM in Fig. 14. This improvement could be attributed to Periodica’s use of multiple reference spots in addition to “work” and “home”, which aligns more closely with real-world scenarios. PRED, which enhanced the clustering algorithm of Periodica using the Chinese Restaurant Process (CRP) and incorporated a time model as a constrain for clustering, demonstrates the best performance among all baseline models. The proposed MIRE model outperforms the baseline models significantly.

Overall, Figs. 13 and 14 provides a comprehensive and clear comparison of the proposed MIRE model with various baseline methods.

We are also intrigued by the relationship between record density and the performance of human period behavior models. For each dataset, we classify users into five groups based on their record density, which refers to the record count per user per day in the datasets.

Figures 15 and 16 illustrate the error distance of different models in various data groups. As record density increases, the performance of Periodica and PRED, which are based on location cluster, initially improves and then declines. This pattern might be attributed to models obtaining more information at the start and performing better when human period behaviors are simpler. However, as more records accumulate, resulting in a mixture of more complex human period behaviors, Periodica and PRED struggle to handle this situation. In contrast, MIRE consistently achieves better performance with increasing record density.