Detecting periodicity in a stream of events
View Patent ↗Sequences of discrete events, such as clicks on a website, are evaluated for periodic behavior, a period is calculated, and the sequence is scored to determine the confidence that the sequence really exhibits periodicity. The random variations on the timing of the discrete events due to transmission delays or other factors may be reduced or eliminated from the evaluation. An apparatus for performing the method of evaluation may include a computer programmed to carry out the method.
1. A method of evaluating a likelihood that a sequence of discrete events displays periodic patterns, comprising:
receiving a sequence of discrete events, wherein the sequence of discrete events includes N discrete events, and wherein the sequence of discrete events comprises a stream of mouse clicks received from a remote terminal;
forming a window in time encompassing a temporal extent over which the sequence of discrete events was received;
generating an autocorrelation function of the sequence of discrete events, the autocorrelation function comprising spikes representing pairwise differences in time received for each individual one of the discrete events in the window in time;
generating a convolution between the autocorrelation function and a jitter kernel to form a smoothed autocorrelation function having a set of peaks;
generating a period value corresponding to a highest peak in the smoothed autocorrelation function;
generating a likelihood that the sequence of discrete events is periodic based upon the set of peaks;
generating a period T based on a location of a largest peak of the set of peaks; and
generating a score representing a confidence level that there is a sequence of periodic discrete events having the period T, the score being generated for the next individual discrete event received after the N discrete events, including the last N−1 discrete events of the window in time.
2. The method of claim 1 , wherein generating the convolution includes generating, as the jitter kernel, a probability distribution function for a random variable representing a time delay between discrete events in the sequence of discrete events.
3. The method of claim 2 , wherein generating the probability distribution function includes assigning, to the jitter kernel, a width indicative of noise processes in an environment in which the discrete events occur.
4. The method of claim 2 , wherein generating the jitter kernel includes generating a function including a Gaussian function.
5. The method of claim 1 , wherein generating the autocorrelation function includes forming a linear combination of spikes at each of all possible pairwise time delays between the discrete events.
6. The method of claim 1 wherein generating the convolution includes creating as the smoothed autocorrelation function including peaks at time delays between discrete events in the window in time.
7. The method of claim 1 , wherein:
forming the window in time includes estimating a first period; and
forming the jitter kernel further includes determining a normal distribution of random variations in time of arrival of the discrete event as a measuring location, the jitter kernel having a shape with a finite an area above the time axis.
8. The method of claim 1 , wherein generating the score representing the confidence level includes dividing an area under each of the peaks by a total of the number of pairwise time differences in time received to form a normalized anomaly.
9. The method of claim 8 , wherein the total of the number of pairwise time differences is N(N−1)/2.
10. The method of claim 8 , wherein generating the score representing a confidence level that there is a sequence of periodic discrete events includes multiplying the normalized anomaly by a monotonically increasing function.
11. The method of claim 10 , further including the normalized anomaly by a value equal to (1−(1−normalized anomaly) 2 )×(100).
12. A method of evaluating a likelihood that a sequence of discrete events displays periodic patterns, comprising:
receiving a sequence of discrete events, wherein the sequence of discrete events comprises a stream of mouse clicks received from a remote terminal;
forming a window in time encompassing a temporal extent over which the sequence of discrete events was received;
generating an autocorrelation function of the sequence of discrete events, the autocorrelation function comprising spikes representing pairwise differences in time received for each individual one of the discrete events in the window in time;
generating a convolution between the autocorrelation function and a jitter kernel to form a smoothed autocorrelation function having a set of peaks;
generating a period value corresponding to a highest peak in the smoothed autocorrelation function;
generating a likelihood that the sequence of discrete events is periodic based upon the set of peaks;
generating a period T based on a location of a largest peak of the set of peaks;
forming a second jitter kernel, a width of the second jitter kernel being proportional to the period T;
generating a second convolution between the autocorrelation function and the second jitter kernel to form a second smoothed autocorrelation function; and
generating, from the second smoothed autocorrelation function, a score representing a confidence level that the sequence of discrete events displays periodic patterns.
13. A computer program product which includes a non-transitory computer readable medium storing a set of instructions that, when carried out by a computer, cause the computer to perform a method, comprising:
receiving a sequence of discrete events, wherein the sequence of discrete events includes N discrete events, and wherein the sequence of discrete events comprises a stream of mouse clicks received from a remote terminal;
forming a window in time encompassing a temporal extent over which the sequence of discrete events was received;
generating an autocorrelation function of the sequence of discrete events, the autocorrelation function comprising spikes representing pairwise differences in time received for each individual one of the discrete events in the window in time;
generating a convolution between the autocorrelation function and a jitter kernel to form a smoothed autocorrelation function having a set of peaks;
generating a period value corresponding to a highest peak in the smoothed autocorrelation function;
generating a likelihood that the sequence of discrete events is periodic based upon the set of peaks;
generating a period T based on a location of a largest peak of the set of peaks; and
generating a score representing a confidence level that there is a sequence of periodic discrete events having the period T, the score being generated for the next individual discrete event received after the N discrete events, including the last N−1 discrete events of the window in time.
14. The computer program product of claim 13 ,
wherein generating the convolution includes generating, as the jitter kernel, a probability distribution function for a random variable representing a time delay between discrete events in the sequence of discrete events, where the jitter kernel includes a width indicative of noise processes in an environment in which the discrete events occur; and
wherein generating the autocorrelation function includes forming a linear combination of spikes at each of all possible pairwise time delays between the discrete events.
15. A computing circuit, comprising:
a communications interface connected to a network;
a memory circuit;
a receiving circuit which receives a sequence of discrete events and records a time received of each discrete event in the memory circuit, wherein the sequence of discrete events includes N discrete events, and wherein the sequence of discrete events comprises a stream of mouse clicks received from a remote terminal;
a controller circuit coupled to the communication interface and the memory, wherein the controller circuit:
forms a window in time during which the sequence of discrete events was received;
generates an autocorrelation function of the sequence of discrete events, the autocorrelation function comprising spikes representing pairwise differences in time received for each individual one of the discrete events in the window in time;
generates a convolution between the autocorrelation function and a jitter kernel to form a smoothed autocorrelation function having a set of peaks;
generates a period value corresponding to a highest peak in the smoothed autocorrelation function;
generates a likelihood that the sequence of discrete events is periodic based upon the set of peaks;
generating a period T based on a location of a largest peak of the set of peaks; and
generating a score representing a confidence level that there is a sequence of periodic discrete events having the period T, the score being generated for the next individual discrete event received after the N discrete events, including the last N−1 discrete events of the window in time.
16. The computing circuit of claim 15 , wherein further:
wherein the controller circuit generates the convolution to include, as the jitter kernel, a probability distribution function for a random variable representing a time delay between discrete events in the sequence of discrete events, where the jitter kernel includes a width indicative of noise processes in an environment in which the discrete events occur; and
the controller circuit generates the autocorrelation function to include a linear combination of spikes at each of all possible pairwise time delays between the discrete events.
17. The method of claim 1 , wherein receiving the stream of mouse clicks from the remote terminal includes receiving the stream of mouse clicks by a receiving circuit through a bidirectional communications means communicably connected to a communication network; and
wherein the communication network is located between the receiving circuit and the remote terminal.