Method and apparatus for incremental tracking of multiple quantiles

Abstract

A capability for incremental tracking of multiples quantiles is provided. A method for performing an incremental quantile update using a data value of a received data record includes determining an initial distribution function, updating the initial distribution function to form a new distribution function based on the received data value, generating an approximation of the new distribution function, and determining new quantile estimates from the approximation of the new distribution function. The initial distribution function includes a plurality of initial quantile estimates and a respective plurality of initial probabilities. The new distribution function includes a plurality of quantile points identifying the respective initial quantile estimates and a respective plurality of new probabilities associated with the initial quantile estimates. The approximation of the new distribution function is generated by connecting pairs of adjacent quantile points using linear approximations of regions between the pairs of adjacent quantile points.

Description

FIELD OF THE INVENTION

The invention relates to the field of quantile tracking and, more specifically but not exclusively, to incremental quantile tracking.

BACKGROUND

Quantiles are useful in characterizing the data distribution of evolving data sets. For example, quantiles are useful in many applications, such as in database applications, network monitoring applications, and the like. In many such applications, quantiles need to be tracked dynamically over time. In database applications, for example, operations on records in the database, e.g., insertions, updates, and deletions, change the quantiles of the data distribution. Similarly, in network monitoring applications, for example, anomalies on data streams need to be detected as the data streams change dynamically over time. Computing quantiles on demand is quite expensive, and, similarly, computing quantiles periodically can be prohibitively costly as well. Therefore, it is desirable to incrementally track quantiles of the data distribution.

Most incremental quantile estimation algorithms are based on a summary of the empirical data distribution, using either a representative sample of the distribution or a global approximation of the distribution. In such incremental quantile estimation algorithms, quantiles are computed from summary data. Disadvantageously, however, in order to obtain quantile estimates with good accuracies (especially for tail quantiles, for which the accuracy requirement tends to be higher than for non-tail quantiles), a large amount of summary information must be maintained, which tends to be expensive in terms of memory. Furthermore, for continuous data streams having underlying distributions that change over time, a large bias in quantile estimates may result since most of the summary information is out of date.

By contrast, other incremental quantile estimation algorithms use stochastic approximation (SA) for quantile estimation, in which the data is viewed as being quantities from a random data distribution. The SA-based quantile estimation algorithms do not keep a global approximation of the distribution and, thus, use negligible memory for estimating tail quantiles. Disadvantageously, however, the existing SA-based quantile estimation algorithms derive each quantile estimate individually, in isolation, which causes problems in incremental quantile estimation. First, derivation of the quantile estimates individually often leads to a violation of the monotone property of quantiles (e.g., such as where the value of the 90% quantile is less than the value of the 80% quantile). Second, although this incremental nature is amenable to continuous data updates, use of derivative information renders the SA-based quantile estimation algorithms sensitive to data order and the particular data distribution during intermediate updates. Third, the existing SA-based quantile estimation algorithms cannot handle dynamic underlying data distributions. These and other issues associated with existing SA-based quantile estimation algorithms present challenges for applications in which incremental quantile tracking is performed.

SUMMARY

Various deficiencies in the prior art are addressed via methods, apparatuses, and computer readable mediums for performing incremental quantile tracking of multiple quantiles using stochastic approximation.

In one embodiment, a method for performing an incremental quantile update using a data value of a received data record includes determining an initial distribution function, updating the initial distribution function to form a new distribution function based on the received data value, generating an approximation of the new distribution function, and determining new quantile estimates from the approximation of the new distribution function. The initial distribution function includes a plurality of initial quantile estimates and a respective plurality of initial probabilities associated with the initial quantile estimates. The initial distribution function is updated to form the new distribution function based on the received data value. The new distribution function includes a plurality of quantile points identifying the respective initial quantile estimates and a respective plurality of new probabilities associated with the respective initial quantile estimates. The approximation of the new distribution function is generated by, for each pair of adjacent quantile points in the new distribution function, connecting the adjacent quantile points using a linear approximation of the region between the adjacent quantile points. The new quantile estimates and the new probabilities associated with the new quantile estimates may then be stored.

BRIEF DESCRIPTION OF THE DRAWINGS

The teachings herein can be readily understood by considering the following detailed description in conjunction with the accompanying drawings, in which:

FIG. 1 depicts an exemplary estimated cumulative distribution function (CDF) for use in responding to queries for quantile estimates;

FIG. 2 depicts one embodiment of a method for incrementally tracking estimated quantiles of a data distribution for a single record type;

FIGS. 3A and 3B depicts one embodiment of a method for generating an approximation of a distribution function for use with the method depicted and described with respect to FIG. 2;

FIGS. 4A and 4B depict an approximation of a distribution function for one pair of adjacent quantile points using the method depicted and described with respect to FIGS. 3A and 3B;

FIG. 5 depicts one embodiment of a method for updating an initial distribution function to form a new distribution function for multiple record types;

FIG. 6 depicts one embodiment of a method for responding to queries using a distribution function for which the quantile estimates are incrementally tracked; and

FIG. 7 depicts a high-level block diagram of a general-purpose computer suitable for use in performing the functions described herein.

To facilitate understanding, identical reference numerals have been used, where possible, to designate identical elements that are common to the figures.

DETAILED DESCRIPTION

A capability for incremental tracking of quantiles using stochastic approximation (SA), denoted as an SA-based incremental quantile estimation capability, is depicted and described herein. In general, in incremental quantile estimation, the quantiles for a set of data are updated in real or near-real time as data is received, such that the incremental quantile estimation provides a relatively current estimate of the quantiles for the set of data received up to the current time. The SA-based incremental quantile estimation capability enables incremental tracking of multiple quantiles over time, where each of the quantile estimates is updated for each data record that is received, thereby ensuring that, at any given time, the monotone property of quantiles is maintained. The SA-based incremental quantile estimation capability enables incremental tracking of multiple quantiles for different record types, such as insertions, deletions, and updates. The SA-based incremental quantile estimation capability is adaptive to changes in the data distribution. The SA-based incremental quantile estimation capability only needs to track quantiles of interest and, thus, is memory efficient (as opposed to non-SA-based quantile estimation algorithms in which the memory requirements are dependent on which quantile is being estimated, e.g., tail quantiles require more memory).

FIG. 1 depicts an exemplary estimated cumulative distribution function (CDF) for use in responding to queries for quantile estimates. The exemplary estimated CDF 100, denoted as distribution function F(x), represents a distribution of a set of received data values {x}. The distribution function F(x) is represented using a Cartesian coordinate system in which the x-axis indicates the values of the quantile estimates of the distribution function EQ (which are denoted as quantile estimates S=(S(1), S(2), . . . , S(K))) and the y-axis indicates the values of the probabilities associated with the quantile estimates S (which are denoted as probabilities p=(p(1), p(2), . . . , p(K))). A combination of the quantile estimates S and the associated probabilities p provides a set of quantile points which define the distribution function F(x) (namely, distribution function F(x) is represented using the quantile points {((S(1),p(1)), ((S(2),p(2)), . . . , ((S(K),p(K))). The distribution function F(x) is a strictly monotone function, such that S(1)<S(2)< . . . <S(K), and has positive derivatives on its domain.

The SA-based incremental quantile estimation capability incrementally tracks the estimated quantiles of distribution function F(x) using incremental approximations to distribution function F(x) upon receiving new data values. A current data value of a set of received data values {x} is denoted as data value x_t received at time t. The SA-based incremental quantile estimation capability updates the approximation to the distribution function F(x) based on received data value x_t, such that the quantile estimates are denoted as S_t=(S_t(1), S_t(2), . . . , S_t(K))) and the probabilities associated with the quantile estimates S_t are denoted as probabilities p_t=(p_t(1), p_t(2), . . . , p_t(K))). A method, according to one embodiment, for tracking the estimated quantiles of distribution function F(x) using an incremental approximation to distribution function F(x) upon new data arrivals is depicted and described with respect to FIG. 2-FIG. 5.

FIG. 2 depicts one embodiment of a method for incrementally tracking estimated quantiles of a data distribution for a single record type. The method 200 is performed for incrementally updating the estimated quantiles of a data distribution at a current time t at which a new data record is received. Although depicted and described as being performed serially, at least a portion of the steps of method 200 may be performed contemporaneously, or in a different order than depicted and described with respect to FIG. 2.

At step 202, the method 200 begins.

At step 204, an insertion record is received. The insertion record includes a new data value x_t. The new data value x_t may be any suitable value and may be received in any suitable manner, which may depend, at least in part, on the application for which incremental tracking of estimated quantiles is performed (e.g., receiving a data insertion record for a database, receiving a data value in a data stream in a network, and the like).

At step 206, an initial distribution function (denoted as {circumflex over (F)}_t−1 ) is determined.

The initial distribution function {circumflex over (F)}_t−1 has properties similar to the distribution function F(x) depicted and described with respect to FIG. 1. The initial distribution function {circumflex over (F)}_t−1 includes a plurality of initial quantile estimates (S_t−1(i), 1≦i≦K) and a plurality of initial probabilities (p_t−1(i), 1≦i≦K) associated with the respective initial quantile estimates S_t−1(i). The initial distribution function also includes a plurality of initial derivative estimates (f_t−1(i), 1≦i≦K) associated with the respective initial quantile estimates S_t−1(i). In one embodiment, the initial distribution function {circumflex over (F)}_t−1 may be a distribution function initialized in a certain manner (e.g., the distribution function {circumflex over (F)}₀ at time t=0, where this is the first time that method 200 is being executed).

In one embodiment, the initial distribution function {circumflex over (F)}_t−1 may be the distribution function determined during a previous time (t−1) at which the previous data record was received (e.g., the initial distribution function {circumflex over (F)}_t−1 may be the approximation of the new distribution function determined during the previous execution of method 200 at previous time (t−1), where method 200 has already been executed for one or more previously received data records).

At step 208, the initial distribution function {circumflex over (F)}_t−1 is updated to form a new distribution function (denoted as {circumflex over (F)}_t) based on the new data value x_t.

The new distribution function {circumflex over (F)}_t includes a plurality of new probabilities (p_t(i), 1≦i≦K) associated with the initial quantile estimates S_t−1(i) of the initial distribution function {circumflex over (F)}_t−1.

In one embodiment, the initial distribution function {circumflex over (F)}_t−1 is updated to form new distribution function {circumflex over (F)}_t by determining the new probabilities p_t(i) for the new distribution function {circumflex over (F)}_t using p_t(i)=(1−w_t)p_t−1(i)+w_tI(S_t−1(i)≧x_t). In this equation, x_t is the new data value, w_t is a weight associated with the new data value x_t (which may be chosen in any suitable manner), S_t−1(i) are the initial quantile estimates of initial distribution function {circumflex over (F)}_t−1, p_t−1(i) are the initial probabilities associated with initial quantile estimates S_t−1(i), I(S_t−1(i)≧x_t) is an indicator function and i is a counter over the set of quantile estimates and probabilities (1≦i≦k). This equation follows from updating initial distribution function {circumflex over (F)}_t−I as {circumflex over (F)}_t(x)=(1−w_t){circumflex over (F)}_t−1(x)+w_tI(x≧x_t), evaluating {circumflex over (F)}_t(x) at initial quantile estimates S_t−1(i) at time t−1, and, using the fact that {circumflex over (F)}_t−1(S_t−1(i))≈p(i), thereby giving the equation: {circumflex over (F)}_t(S_t−1(i))≈(1−w_t)p(i)+w_tI(S_t−1(i)≧x_t), which may then be represented as p_t(i)=(1−w_t)p(i)+w_tI(S_t−1(i)≧x_t). The combination of the initial quantile estimates S_t−1(i) and the new probabilities p_t(i) provides a set of quantile points (S_t−1(i), p_t(i)) which defines new distribution function {circumflex over (F)}_t.

At step 210, an approximation of the new distribution function is generated.

In one embodiment, linear interpolation is used to generate the approximation of the new distribution function such that, in the neighborhood of each of the initial quantile estimates S_t−1(i), the approximation of the new distribution function is a linear function with a slope specified by the respective initial derivative estimates f_t−1(i) associated with the initial quantile estimate S_t−1(i), and the linear points around the initial quantile estimates S_t−1(i) are extended under the constraints of monotonicity of the interpolation function.

In one embodiment, generating the approximation of the new distribution function includes, for each pair of adjacent quantile points in the new distribution function {circumflex over (F)}_t (where each pair of adjacent quantile points includes a first quantile point (S_t−1(i), p_t(i)) and a second quantile point (S_t−1(i+1), p_t(i+1)) performing the following: (1) defining a right quantile point to the right of the first quantile point and a left quantile point to the left of the second quantile point; and (2) generating a linear approximation of the new distribution function for the region between the adjacent quantile points by connecting the first quantile point, the right quantile point, the left quantile point, and the second quantile point in a piecewise linear fashion. In one such embodiment, definition of the right quantile points and the left quantile points is performed using the initial quantile estimates S_t−1(i), the initial derivative estimates f_t−1(i), the new probabilities p_t(i), and monotonicity values Δ_t(i). A more detailed description of one such embodiment is depicted and described with respect to FIGS. 3A and 3B.

FIGS. 3A and 3B depicts one embodiment of a method for generating an approximation of a distribution function for use with the method depicted and described with respect to FIG. 2. As indicated above with respect to FIG. 2, in one embodiment method 300 of FIGS. 3A and 3B may be used as step 210 of method 200 of FIG. 2. Although primarily depicted and described as being performed serially, at least a portion of the steps of method 300 may be performed contemporaneously, or in a different order than depicted and described with respect to FIGS. 3A and 3B.

At step 302, method 300 begins.

At step 304, a counter associated with the quantile points is initialized to one (i=1, 1≦i≦K, where K is the number of estimated quantiles of the new distribution function).

At step 306, a pair of adjacent quantile points is determined. The pair of adjacent quantile points is determined based on the current value of the counter i. The pair of adjacent quantile points includes a first quantile point (S_t−1(i), p_t(i)) and a second quantile point (S_t−1(i+1), p_t(i+1)).

At step 308, a monotonicity value (denoted as Δ_t(i)) is computed for the pair of adjacent quantile points.

The monotonicity value Δ_t(i) is computed such that the right quantile point and the left quantile point are non-decreasing, i.e., such that:[S_t−1(i)+Δ_t(i)]≦[S_t−1(i+1)−Δ_t(i)], and[p_t(i)+f_t−1(i)Δ_t(i)]≦[p_t(i+1)−f_t−1(i+1)Δ_t(i)],

which indicate that:

${\Delta }_t\left(i\right)=\min \left(\frac{S_{t-1}\left(i+1\right)-S_{t-1}\left(i\right)}{2},\frac{p_t\left(i+1\right)-p_t\left(i\right))}{f_{t-1}\left(i\right)+f_{t-1}\left(i+1\right)}\right).$

The monotonicity value Δ_t(i) may be selected in any suitable manner. In one embodiment, for example, the monotonicity value Δ_t(i) is selected as the maximum possible value determined from the right-hand side of the above equation for monotonicity value Δ_t(i).

At step 310, the right quantile point (denoted as right_t(i)) and the left quantile point (denoted left_t(i+1)) are defined.

The right quantile point is a point to the right of the first quantile point, and is defined as follows: right_t(i)=(S_t−1(i)+Δ₆(i), p_t(i)+f_t−1(i)Δ_t(i)), which is a point in the new distribution function {circumflex over (F)}_t that is to the right of the first quantile point (S_t−1(i), p_t(i)) with a slope of f_t−1(i).

The left quantile point is a point to the left of the second quantile point, and is defined as follows: left_t(i+1)=(S_t−1(i+1)−Δ_t(i), p_t(i+1)−f_t−1(i+1)Δ_t(i)), which is a point in the new distribution function {circumflex over (F)}_t that is to the left of the second quantile point (S_t−1(i+1), p_t(i+1)) with a slope of f_t−1(i+1).

At step 312, the first quantile point, the right quantile point, the left quantile point, and the second quantile point are connected to form a portion of the approximation of the new distribution function. The first quantile point, the right quantile point, the left quantile point, and the second quantile point are connected in a piecewise linear fashion such that the first quantile point is connected to the right quantile point, the right quantile point is connected to the left quantile point, and the left quantile point is connected to the second quantile point.

At step 314, a determination is made as to whether counter i is equal to K−1. If the counter i is not equal to K−1, method 300 proceeds to step 316. If the counter i is equal to K−1, method 300 proceeds to step 318.

At step 316, the counter i is incremented by one (i=i+1), and, from step 316, method 300 returns to step 304 so that the process can be repeated for the next pair of adjacent quantile points in the new distribution function {circumflex over (F)}_t.

At step 318, the approximation of the new distribution function is extended beyond the two boundary quantile points until it reaches the extreme y-axis values of zero and one (i.e., the approximation of the new distribution function is extended to the left of the quantile point (S_t−1(1), p_t(1)) until it reaches the y-axis value of zero and is extended to the right of quantile point (S_t−1(K),p_t(K)) until it reaches the y-axis value of one).

At step 320, method 300 ends. Although depicted and described as ending (for purposes of clarity), in an embodiment in which method 300 is used as step 210 of method 200 of FIG. 2, processing will proceed to step 212 of method 200 of FIG. 2.

FIGS. 4A and 4B depict an approximation of a distribution function for one pair of adjacent quantile points using method 300 depicted and described with respect to FIGS. 3A and 3B.

As depicted in FIGS. 4A and 4B, the approximation of the new distribution function is depicted for the portion of the new distribution function that is between first quantile point (S_t−1(1), p_t(1)) and second quantile point (S_t−1(2), p_t(2)). As described hereinabove, similar approximations are performed for each of the other adjacent quantile points in the new distribution function (e.g., for second quantile point (S_t−1(2), p_t−1(2)) and third quantile point (S_t−1(3), p_t−1(3)), . . . , for next-to-last quantile point (S_t−1(K−1), p_t−1(K−1)) and last quantile point (S_t−1(K), p_t−1(K))) for purposes of generating the complete approximation of the new distribution function.

In FIGS. 4A and 4B, the depicted portion of the approximation of the new distribution function is plotted using a Cartesian coordinate system in which the x-axis indicates the values of the initial quantile estimates S_t−1 and the y-axis indicates the values of the new probabilities p_t associated with the initial quantile estimates S_t−1.

The curve functions 410A and 410B represent the hypothetical smooth approximation of the data distribution of new distribution function {circumflex over (F)}_t between first quantile point (S_t−1(1), p_t(1)) and second quantile point (S_t−1(2), p_t(2)).

The linear functions 420A and 420B represent the piecewise linear approximations of the new distribution function i; between first quantile point (S_t−1(1), p_t(1)) and second quantile point (S_t−1(2), p_t(2)), determined using first and second quantile points (S_t−1(1), p_t(1)) and (S_t−1(2), p_t(2)), initial derivative estimates f_t−1(1) and f_t−1(2) associated with first and second quantile points (S_t−1(1), p_t(1)) and (S_t−1(2), p_t(2)), respectively, and monotonicity value Δ_t(1).

FIGS. 4A and 4B correspond to two different cases for the monotonicity value Δ_t(1). FIG. 4A illustrates an exemplary case in which monotonicity value Δ_t(1) takes the second value in the monotonicity equation. FIG. 4B illustrates an exemplary case in which monotonicity value Δ_t(1) takes the first value in the monotonicity equation.

Returning now to FIG. 2, new quantile estimates, based upon the new data value x_t, are determined using the approximation of the new distribution function.

At step 212, new quantile estimates (denoted as S_t(i)) are determined from the approximation of the new distribution function. The new quantile estimates S_t(i) are determined from the approximation of the new distribution function as follows: {circumflex over (F)}_t(S_t(i))=p_i.

At step 214, the new quantile estimates S_t(i) and the new probabilities p_t(i) of the approximation of the new distribution function are stored. The new quantile estimates S_t(i) and the new probabilities p_t(i) may be stored in any suitable manner.

In one embodiment, for example, the new quantile estimates S_t(i) and the new probabilities p_t(i) may be stored as respective sets of data values (namely, as a set of new quantile estimates S_t(i)={S_t(1), . . . , S_t(K)} and a set of new probabilities p_t(i)={p_t(1), . . . , p_t(K)}.

In one embodiment, for example, the new quantile estimates S_t(i) and the new probabilities p_t(i) may be stored by storing the approximation of the new distribution function.

The storage of new quantile estimates S_t(i) and new probabilities p_t(i) of the new distribution function enables queries for quantile estimates S_t(i) to be answered. A method according to one embodiment for responding to queries of quantile estimates using the approximation of the new distribution function is depicted and described with respect to FIG. 6.

At step 216, new derivative estimates (denoted as f_t(i)) associated with new quantile estimates S_t(i) are determined.

In one embodiment, new derivative estimates f_t(i) may be determined as follows: f_t(i)=(1−w_t)f_t−1(i)+w_tI(|x_t−S_t(i)|≦c/{2c}), where c is a tunable parameter representing the window size around each of the new quantile estimates S_t(i) for which the respective new derivative estimates f_t(i) are determined. The window sizes c may be set to any suitable values. In one embodiment, for example, the window sizes c each are a fraction of the estimated inter-quantile range, and the window sizes c are the same for all quantiles. In another embodiment, for example, the values of window sizes c are set such that the window sizes c are not uniform across all quantiles.

It will be appreciated that, since the new derivative estimates f_t(i) are not required for use in responding to queries for quantile estimates S_t(i), determining the new derivative estimates may be viewed as an extraneous step performed for purposes of executing method 200 for each received data value. In one embodiment, as depicted in FIG. 2, this step may be considered as a post-processing step (e.g., performed after the approximation of the new distribution function is determined in order to obtain the derivative estimates associated with current time t which are to be used as the initial derivative estimates the next time method 200 is executed). In one embodiment, not depicted in FIG. 2, this step may be considered as a pre-processing step (e.g., performed before the initial distribution function is determined in order to obtain the derivative estimates associated with current time t which are to be used as the initial derivative estimates the next time method 200 is executed).

At step 218, method 200 ends.

Although depicted and described as ending (for purposes of clarity), it will be appreciated that method 200 may be executed for each new insertion record that is received for purposes of incrementally updating quantile estimates.

The SA-based incremental quantile estimation capability depicted and described herein enables incremental tracking of multiple quantiles over time for data with stationary distributions and data with non-stationary distributions. Additionally, the SA-based incremental quantile estimation capability depicted and described herein may utilize multiple types of weights w_t in updating the initial distribution function to form the new distribution function. For example, the weights w_t may be diminishing (e.g., w_t=1/t) or constant (w_t=w), or set in any other suitable manner.

For stationary data (i.e., {circumflex over (F)}_t is stationary), simple SA-based algorithms, in which each of the quantile estimates is updated individually in isolation, will lead to convergence for both of the types of weights w_t described above. For diminishing weights w_t set as w_t=1/t, convergence using simple SA-based algorithms is to the true quantile in probability one. For constant weights w_t set as w_t=w, convergence using simple SA-based algorithms is in distribution to a random variable with mean of the true quantile. These convergence results also are true for the SA-based incremental quantile estimation capability depicted and described herein in which each of the quantile estimates is updated for each received data record. For weights w_t set as w_t=1/t, as t approaches infinity, the SA-based incremental quantile estimations depicted and described herein will converge to true quantiles. For weights w_t set as w_t=w, as t approaches infinity, the SA-based incremental quantile estimations depicted and described herein will converge in distribution to a random variable with mean of the true quantile. In one embodiment, for non-stationary data (i.e., {circumflex over (F)}_t is non-stationary), the SA-based incremental quantile estimations depicted and described herein will use constant weights (w_t=w) as opposed to diminishing weights (w_t=1/t).

It will be appreciated that the weights w_t used in updating the initial distribution function to form the new distribution function, as depicted and described with respect to FIG. 2, may be set in any other suitable manner.

Although primarily depicted and described herein with respect to an embodiment in which estimated quantiles are updated for each new insertion record that is received (i.e., method 200 is executed for each new data value x_t that is received), in other embodiments estimated quantiles may be updated using a batch of M insertion records (i.e., a batch of M data values {x_t}_M). In one such embodiment, steps 204-208 are performed for each of the M data values, and then steps 210-214 are performed once for the batch of M data values using the new distribution function that reflects the M data values. It will be appreciated that method 200 of FIG. 2 may be modified in other ways to support batch processing of data records.

The SA-based incremental quantile estimation capability uses an incremental distribution approximation by interpolating at the updated quantile points. As a result, local to the quantile points the incremental distribution approximation is the same linear function as in existing SA-based quantile estimation algorithms in which each quantile point is updated individually in isolation from other quantile points, whereas globally the incremental distribution approximation is an increasing function.

The SA-based incremental quantile estimation capability opens up the possibility of using other more elaborate interpolation or approximation schemes given the local approximations at the quantile points. The SA-based incremental quantile estimation capability also opens up the possibility of using an asymptotic model to overcome some of the instabilities of SA-based incremental quantile estimation schemes in dealing with extreme tails (e.g., due to very small derivatives associated with extreme tails). It will be appreciated that care must be taken to ensure that utilizing such interpolation or approximation schemes does not lead to biases in quantile estimates (e.g., such as where using linear interpolation by connecting quantile points directly without using the local derivatives provides convergence for stationary data, but with a bias).

The SA-based incremental quantile estimation capability enables the updated quantile estimates to be computed relatively efficiently, while at the same time providing good approximations of quantile estimates.

It will be appreciated that, since the distribution approximation is piecewise linear, finding the quantile points of the function for updating (as in step 212) is relatively simple (e.g., by determining which line segment each probability p(i) falls into and then solving p(i) for that line segment).

It will be further appreciated that the estimated derivative f_t is a vector of estimated derivatives (density) and that it is not crucial to obtain exact values of the derivatives. For example, if estimated derivative f_t is replaced by a vector of fixed positive constants, the quantile estimates derived using the SA-based incremental quantile estimation capability still provide good approximations; however, it is more efficient to use a value of estimated derivative f_t that is close to the actual derivatives of the distribution function since the quantile estimates will stabilize faster around the true value.

Although primarily depicted and described herein with respect to embodiments in which the SA-based incremental quantile estimation capability is utilized for incrementally approximating a distribution function F_t(·) that is a strictly increasing continuous distribution, other embodiments of the SA-based incremental quantile estimation capability may be utilized for incrementally approximating a distribution function F_t(·) that is a discrete distribution. In such embodiments, the SA-based incremental quantile estimation capability may be modified in order to prevent the derivative estimates from becoming infinite. The SA-based incremental quantile estimation capability may be modified in any suitable manner (e.g., by adding a small random noise to the data, where the small random noise may be chosen in a data dependent fashion).

Although primarily depicted and described herein with respect to embodiments in which the SA-based incremental quantile estimation capability is used for a set of data records including only one specific record type (namely, for insertion records), the SA-based incremental quantile estimation capability also may be used for a set of data records including only one specific record type where the one specific record type is different (e.g., using deletion records, update records, and the like) and/or for a set of data records including multiple record types (e.g., using a combination of two or more of insertion records, deletion records, update records, and the like). A description of such embodiments follows.

In one embodiment, the SA-based incremental quantile estimation capability is used for a set of data records including multiple record types. As described herein, the SA-based incremental quantile estimation capability is, in general, based on performing incremental approximations to a distribution function and, thus, the manner depicted and described hereinabove for performing incremental approximations to a distribution function for a set of data records including a single record type (namely, insertion records) is modified to perform incremental approximations to a distribution function for a set of data records including multiple records types. A description of the modification follows.

In this embodiment, assume that the set of data records for which incremental quantile approximation is performed includes insertion records, deletion records, and updated records.

In this embodiment, assume that at time t there is always a data value x_t inserted, but at the same time there also could be: (1) a data value x_t0 from a previous time that will be deleted (i.e., the data value is no longer to be considered), or (2) a data value x_t1 a previous time that will be updated with a new value x′_t1 (i.e., the data sample at time t₁ should be replaced by x′_t1).

In this embodiment, let w_t be a sequence of intended or initial weights for the insertion data value x_t at time t. The weights for the insertion data value x_t are deemed to be intended or initial, because the actual weights for the insertion data value x_t will be modified due to deletion. For deletion data value x_t0, the intended weight at time t is: d_t0(t)=w_t0Π′_s=t0+1(1−w_s).

In this embodiment, assume that the approximation of the distribution function at time t−1 is denoted as {circumflex over (F)}_t−1. Additionally, define total weights value D₀=0. The approximation of the distribution function at time t−1 is the initial distribution function {circumflex over (F)}_t−1 at time t (similar to step 206 described with respect to FIG. 2 in the single-record-type case). The initial distribution function {circumflex over (F)}_t−1 is updated to form the new distribution function {circumflex over (F)}_t by updating initial probabilities p_t−1(i) to form new probabilities p_t(i) (similar to step 208 described with respect to FIG. 2 in the single-record-type case). Additionally, the initial total weights value D_t−1 is updated to form new total weights value D_t.

At time t, with the insertion record including insertion data value x_t, updating of the initial distribution function {circumflex over (F)}_t−1 and the initial total weights value D_t−1 may be represented as follows:

$\mathrm{Insert}\text{:}\{\begin{array}{ccc}{\hat{F}}_t\left(x\right)& \leftarrow & \frac{\left(1-w_t\right)\left(1-D_{t-1}\right){\hat{F}}_{t-1}\left(x\right)+w_{t}I\left(x\ge x_t\right)}{1-D_{t-1}\left(1-w_t\right)}\\ D_t& \leftarrow & \left(1-w_t\right)D_{t-1}\end{array}$

If there are no deletion or update records at time t, the updating of the initial distribution function {circumflex over (F)}_t−1 is complete (because no further update of the initial distribution function {circumflex over (F)}_t−1 is required at time t).

If there is a deletion record or an update record at time t, the updated distribution function {circumflex over (F)}_t that is generated based on the insertion record is further updated to account for the deletion or insertion.

At time t, if there is a deletion record indicating deletion of data value x_t0, then: (a) the updated distribution function {circumflex over (F)}_t that is generated based on the insertion data value from the insertion record is further updated based on the deletion record in order to represent the new distribution function {circumflex over (F)}_t, and (b) the updated total weights value D_t that is generated based on the weight of the insertion data value from the insertion record is further updated based on the deletion record in order to represent the new total weights value D_t. The further updating of the updated distribution function {circumflex over (F)}_t and the updated total weights value D_t based on the deletion record may be represented as follows:

$\mathrm{Delete}\text{:}\{\begin{array}{ccc}{\hat{F}}_t\left(x\right)& \leftarrow & \frac{\left(1-D_t\right)F_t\left(x\right)-d_{t_0}I\left(x\ge x_{t_0}\right)}{1-D_t-d_{t_0}}\\ D_t& \leftarrow & D_t+d_{t_0}\left(t\right)\end{array}$ where d_t0(t) is determined as d_t0(t)=w_t0Π′_s=t0+1(1−w_s). It will be appreciated that these deletion equations remove the influence of deletion data value x_t0 at time t since the weight for x_t0 now reduces to d_t0(t).

At time t, if there is an update record indicating update of data value x_t1 time t with a new value x′_t1, then: (a) the updated distribution function {circumflex over (F)}_t that is generated based on the insertion data value from the insertion record is further updated based on the update record in order to represent the new distribution function {circumflex over (F)}_t, and (b) the updated total weights value D_t that is generated based on the weight of the insertion data value from the insertion record remains unchanged. The further updating of the updated distribution function {circumflex over (F)}_t based on the update record may be represented as follows:

$\mathrm{Update}\text{:}\{\begin{array}{ccc}{\hat{F}}_t\left(x\right)& \leftarrow & \frac{\left(1-D_t\right)F_t\left(x\right)+d_{t_1}\left(I\left(x\ge x_{t_1}\right)-I\left(x\ge x_{t_1}^{\prime }\right)\right)}{1-D_t}\\ D_t& & \mathrm{remains}\mathrm{unchanged}\end{array}$ It will be appreciated from these update equations that an update record is treated as a combination of a deletion record and an insertion record for time t (i.e., the data value to be updated is deleted and replaced with the new value).

In the above-defined equations for insertion, deletion, and update records, the total weights value D_t represents the total of all weights from data values deleted at time t. As such, the total weights of data that contributed to updated distribution function {circumflex over (F)}_t at time t is not one, but, rather, is 1−D_t due to deletions.

For the insertion equations, with the arrival of new data value x_t, the updated distribution function {circumflex over (F)}_t is the weighted sum I(x≧x_t−1) from insertion data value x_t with weight w_t, and initial distribution function {circumflex over (F)}_t−1 with weight (1−w_t)(1−D_t−1), normalized to have a total weight of one. Additionally, the weight of the deleted data in {circumflex over (F)}_t is updated by a factor of (131 w_t).

As described hereinabove, from the above-described equations, the equations adapted for use in updating the initial probabilities p_t−1(i) to form the new probabilities p_t(i) may be derived. Namely, the equations adapted for use in updating the initial probabilities p_t−1(i) to form the new probabilities p_t(i) may be derived by evaluating the new distribution function {circumflex over (F)}_t at each of the initial quantile estimates S_t−1(i) at time t−1.

The initial probabilities p_t−1(i) are updated to form the new probabilities p_t(i) (similar to step 208 described with respect to FIG. 2 in the single record type case) as follows (where an assumption is made that initial total weights value D₀=0).

At time t, with the insertion record including insertion data value x_t: (a) the initial probabilities p_t−1(i) are updated to form intermediate probabilities p_t^INT(i) and (b) and the initial total weights value D_t−1 is updated to form an intermediate total weights value D_t^INT, as follows:

$\mathrm{Insert}\text{:}\{\begin{array}{c}\begin{array}{c}{p}_t^{\mathrm{INT}}\left(i\right)\leftarrow {\left(1-D_{t-1}\left(1-w_t\right)\right)}^{-1}(\left(1-w_t\right)\left(1-D_{t-1}\right)p_{t-1}\left(i\right)+\\ w_{t}I\left(S_{t-1}\left(i\right)\ge x_t\right))\end{array}\\ D_t^{\mathrm{INT}}\leftarrow \left(1-w_t\right)D_{t-1}\end{array}$

If there are no deletion or update records at time t, the intermediate probabilities p_t^INT(i) are denoted as new probabilities p_t(i) (because no further update of the probabilities is required at time t).

If there is a deletion record or an update record at time t, the intermediate probabilities p_t^INT(i) are further updated, based on the deletion or update record, in order to determine new probabilities p_t(i).

At time t, if there is a deletion record indicating deletion of data value x_t0, then: (a) the intermediate probabilities p_t^INT(i) determined based on the insertion record are further updated based on the deletion record in order to determine new probabilities p_t(i), and (b) the intermediate total weights value D_t^INT that is generated based on the weight of the insertion data value from the insertion record is further updated based on the deletion record in order to determine the new total weights value D_t. The further updating of the intermediate probabilities p_t^INT(i) and the intermediate total weights value D_t^INT based on the deletion record is performed as follows:

$\mathrm{Delete}\text{:}\{\begin{array}{c}{p}_t\left(i\right)\leftarrow {\left(1-D_t^{\mathrm{INT}}-d_{t_0}\right)}^{-1}(\left(1-D_t^{\mathrm{INT}}\right)p_t^{\mathrm{INT}}\left(i\right)-\\ d_{t_0}I\left(S_{t-1}\left(i\right)\ge x_{t_0}\right))\\ D_t\leftarrow D_t^{\mathrm{INT}}+d_{t_0}\left(t\right)\end{array}$ where d_t0(t) is d_t0(t)=w_t0Π′_s=t0+1(1−w_s).

At time t, if there is an update record indicating update of data value x_t1 at time t with a new value x′_t1, then: (a) the intermediate probabilities p_t^INT(i) determined based on the insertion record are further updated based on the update record in order to determine new probabilities p_t(i), and (b) the updated total weights value D_t that is generated based on the weight of the insertion data value from the insertion record remains unchanged. The further updating of the intermediate probabilities p_t^INT(i) based on the update record is performed as follows:Update: p_t(i)←(d_t1(I(S_t−1(i)≧x_t1)−I(S_t−1(i)≧x′_t1))+(1−D_t^INT)p_t^INT(i))(1−D_t^INT)⁻¹.

As described herein, the single-record-type case for incrementally tracking estimated quantiles of a data distribution (depicted and described with respect to FIG. 2) may be modified to provide the multiple-record-type case for incrementally tracking estimated quantiles of a data distribution (depicted and described with respect to FIG. 5). In one embodiment, this is achieved by modifying step 208 of FIG. 2 for the multiple-record-type case. An exemplary embodiment for updating the initial distribution function to form the new distribution function, for the multiple-record-type case, is depicted and described with respect to FIG. 5.

FIG. 5 depicts one embodiment of a method for updating an initial distribution function to form a new distribution function in the presence of multiple record types. As noted above, the exemplary method depicted and described with respect to FIG. 5 is suitable for use as step 208 of FIG. 2. Although depicted and described as being performed serially, at least a portion of the steps of method 208 may be performed contemporaneously, or in a different order than depicted and described with respect to FIG. 5.

As depicted in FIG. 5, method 208 is entered from step 206 of method 200 (at which point the initial distribution function is determined) and exits to step 210 of method 200 (at which point an approximation of the determined new distribution function is generated using linear interpolation). A description of method 208 follows.

At step 510, the initial probabilities p_t−1(i) associated with the initial quantile estimates S_t−1(i) of initial distribution function {circumflex over (F)}_t−1 are updated to form intermediate probabilities p_t^INT(i) and the initial total weights value D_t−1 is updated to form an intermediate total weights value D_t^INT. The intermediate probabilities p_t^INT(i) and intermediate total weights value D_t^INT are determined as follows:

$\mathrm{Insert}\text{:}\{\begin{array}{c}\begin{array}{c}{p}_t^{\mathrm{INT}}\left(i\right)\leftarrow {\left(1-D_{t-1}\left(1-w_t\right)\right)}^{-1}(\left(1-w_t\right)\left(1-D_{t-1}\right)p_{t-1}\left(i\right)+\\ w_{t}I\left(S_{t-1}\left(i\right)\ge x_t\right))\end{array}\\ D_t^{\mathrm{INT}}\leftarrow \left(1-w_t\right)D_{t-1}\end{array}$

At step 520, a determination is made as to whether a deletion record or an update record has been received along with the insertion record. If neither a deletion record nor an update record has been received (i.e., only an insertion record was received at time t), method 500 proceeds to step 530. If a deletion record was received at time t, method 500 proceeds to step 540. If an update record was received at time t, method 500 proceeds to step 550.

At step 530, since only an insertion record was received at time t, the intermediate probabilities p_t^INT(i) determined in step 510 become the new probabilities p_t(i) associated with initial quantile estimates S_t−1(i) to form thereby new distribution function {circumflex over (F)}_t, and the intermediate total weights value D_t^INT determined in step 510 becomes the new total weights value D_t.

At step 540, since a deletion record was received in addition to the insertion record: (a) the intermediate probabilities p_t^INT(i) determined in step 510 are updated again to become the new probabilities p_t(i) associated with initial quantile estimates S_t−1(i) to form thereby new distribution function {circumflex over (F)}_t, and (b) the intermediate total weights value D_t^INT determined in step 510 is updated again to become new total weights value D_t. The new probabilities p_t(i) and new total weights value D_t are determined as follows:

$\mathrm{Delete}\text{:}\{\begin{array}{c}{p}_t\left(i\right)\leftarrow {\left(1-D_t^{\mathrm{INT}}-d_{t_0}\right)}^{-1}(\left(1-D_t^{\mathrm{INT}}\right)p_t^{\mathrm{INT}}\left(i\right)-\\ d_{t_0}I\left(S_{t-1}\left(i\right)\ge x_{t_0}\right))\\ D_t\leftarrow D_t^{\mathrm{INT}}+d_{t_0}\left(t\right)\end{array}$ where d_t0(t) is d_t0(t)=w_t0Π′_s=t0+1(1−w_s).

At step 550, since an update record was received in addition to the insertion record, the intermediate probabilities p_t^INT(i) that were determined in step 510 are updated again to become the new probabilities p_t(i) associated with initial quantile estimates S_t−1(i) to form thereby new distribution function {circumflex over (F)}_t. As described hereinabove, the intermediate probabilities p_t^INT(i) are updated based on the update record as follows:Update: p_t(i)←(d_t1(I(S_t−1(i)≧x_t1)−I(S_t−1(i)≧x′_t1))+(1−D_t^INT)p_t^INT(i))(1−D_t^INT)⁻¹.

As depicted in FIG. 5, the result of each of the steps 530, 540, and 550 is the new probabilities p_t(i) associated with initial quantile estimates S_t−1(i). The combination of the initial quantile estimates S_t−1(i) and the associated new probabilities p_t(i) provides the new distribution function {circumflex over (F)}_t. From steps 530, 540, and 550, method 500 ends (i.e., method 500 exits to step 210 of method 200, at which point an approximation of the new distribution function {circumflex over (F)}_t is generated using linear interpolation).

Although primarily depicted and described herein with respect to an embodiment in which the extended version of the SA-based incremental quantile estimation capability supports a set of data records that includes insertion records, deletion records, and updated records, other embodiments of the extended version of the SA-based incremental quantile estimation capability may support sets of data records that include other types and/or combinations of records (e.g., where the set of data records includes insertion records and deletion records, where the set of data records includes insertion records and update records, and the like). In one embodiment, the types of records that are included in the set of data records for which the SA-based incremental quantile estimation capability is implemented may be dependent on the application for which the SA-based incremental quantile estimation capability is used (e.g., database applications, networking applications, and the like).

The SA-based incremental quantile estimation capability depicted and described herein for multiple-record-type implementations may utilize multiple types of weights w_t in updating the initial distribution function to form the new distribution function. For example, the weights w_t may be diminishing (e.g., w_t=1/t) or constant (w_t=w), or set in any other suitable manner.

For diminishing weights w_t set as w_t=1/t, it will be appreciated that D_t is the ratio of deletes in the data. Assuming that this is true for t−1, and further assuming that there are k deletions, then, with the arrival of insertion data value x_t, by 16, {circumflex over (F)}_t(x) is the weighted sum of {circumflex over (F)}_t−1(x) and I(x≧x_t) with weights (t−k−1)/(t−k) and 1/(t−k), and D_t=k/(t+1) is actually the ratio of deletes in the data up to time t. It also will be appreciate that this may be verifies for the deletion and updated equations (17 and 18). In one such embodiment, the actual weight given to x_t is 1/(t−k), not the intended weight 1/t.

For constant weights w_t set as w_t=w (where w is positive), let s₁<s₂< . . . <s_k be the index of the data that are deleted until time t, where k is the total number of deletes before time t. With the arrival of insertion data value x_t, it can be shown that D_t=(1−w)^t−s1−1w+(1−w)^t−s2−1w+ . . . +(1−w)^t−sk−1w.

It will be appreciated that the weights w_t used in updating the initial distribution function to form the new distribution function may be set in any other suitable manner.

With respect to the SA-based incremental quantile estimation capability depicted and described herein for multiple-record-type implementations, in the case of deletions and updates for stationary data that will result in equilibrium, for example, when the deletes occurs at a lag with a stationary random distribution, the estimated quantiles converge to the true quantiles. A heuristic understanding of this convergence is that our insertion, deletion, and update equations depicted and described herein are designed in such a way that the effect of deleted data is diminished in the functional approximation of {circumflex over (F)}_t(x), and thus quantiles of the remaining data will have the correct quantiles.

It will be appreciated that the modified/additional embodiments that are described with respect to the single-record-type implementations of the SA-based incremental quantile estimation capability also apply to the multiple-record-type implementations of the SA-based incremental quantile estimation capability (e.g., batch processing of insertion records, support for both continuous and discrete distribution functions, and the like, as well as various combinations thereof).

FIG. 6 depicts one embodiment of a method for responding to queries using a distribution function for which the quantile estimates are incrementally tracked. Although primarily depicted and described herein as being performed serially, at least a portion of the steps of method 600 may be performed contemporaneously, or in a different order than depicted and described with respect to FIG. 6.

At step 602, method 600 begins.

At step 604, a quantile query request is received.

The quantile query request may be any quantile query request. For example, the quantile query request may be a request for a quantile for a specific value, a request for a quantile for a range of values (e.g., for a portion of a bin, multiple bins, a range of values spanning multiple bins, and the like, as well as various combinations thereof).

The quantile query request may be received from any source. For example, the quantile query request may be received locally at the system performing incremental quantile estimation, received from a remote system in communication with the system performing incremental quantile estimation, and the like, as well as various combinations thereof.

The quantile query request may be initiated in any manner. For example, the quantile query request may be initiated manually by a user, automatically by a system, and the like, as well as various combinations thereof.

At step 606, a quantile query response is determined using a distribution function. As described herein, the distribution function is being updated in real time or near real time as data values are being received and, thus, the distribution function provides an accurate estimate of the current view of the quantile distribution. Thus, the quantile query response provides a current value of the quantile of the data value(s) for which the quantile query request was initiated.

At step 608, method 600 ends.

Although primarily described herein such that the distribution functions are said to include a plurality of quantile estimates and an associated plurality of probabilities, it will be appreciated by those skilled in the art and informed by the teachings herein that the distribution functions also may be said to be represented by a plurality of quantile estimates and an associated plurality of probabilities (as well as the associated derivative estimates associated with the quantile estimates).

Although depicted and described as ending (for purposes of clarity), it will be appreciated that method 600 of FIG. 6 may be executed as often as desired/necessary for the application for which the incremental quantile estimation capability is being used.

FIG. 7 depicts a high-level block diagram of a general-purpose computer suitable for use in performing the functions described herein. As depicted in FIG. 7, system 700 comprises a processor element 702 (e.g., a CPU), a memory 704, e.g., random access memory (RAM) and/or read only memory (ROM), a incremental quantile estimation module 705, and various input/output devices 706 (e.g., storage devices, including but not limited to, a tape drive, a floppy drive, a hard disk drive or a compact disk drive, a receiver, a transmitter, a speaker, a display, an output port, and a user input device (such as a keyboard, a keypad, a mouse, and the like)).

It should be noted that the present invention may be implemented in software and/or in a combination of software and hardware, e.g., using application specific integrated circuits (ASIC), a general purpose computer or any other hardware equivalents. In one embodiment, the incremental quantile estimation process 705 can be loaded into memory 704 and executed by processor 702 to implement the functions as discussed above. As such incremental quantile estimation process 705 (including associated data structures) of the present invention can be stored on a computer readable medium or carrier, e.g., RAM memory, magnetic or optical drive or diskette, and the like.

It is contemplated that some of the steps discussed herein as software methods may be implemented within hardware, for example, as circuitry that cooperates with the processor to perform various method steps. Portions of the functions/elements described herein may be implemented as a computer program product wherein computer instructions, when processed by a computer, adapt the operation of the computer such that the methods and/or techniques described herein are invoked or otherwise provided. Instructions for invoking the inventive methods may be stored in fixed or removable media, transmitted via a data stream in a broadcast or other signal bearing medium, and/or stored within a memory within a computing device operating according to the instructions.

Although various embodiments which incorporate the teachings of the present invention have been shown and described in detail herein, those skilled in the art can readily devise many other varied embodiments that still incorporate these teachings.

Claims

1. A method, comprising: generating, using a processor, an approximation of a quantile distribution function by connecting a pair of adjacent quantile points using linear interpolation;wherein the pair of adjacent quantile points comprises: a first quantile point having a first quantile estimate and a first probability associated therewith; anda second quantile point having a second quantile estimate and a second probability associated therewith; andwherein generating the approximation of the quantile distribution function comprises: defining a right quantile point to the right of the first quantile point in the quantile distribution function;defining a left quantile point to the left of the second quantile point in the quantile distribution function; andconnecting the first quantile point, the right quantile point, the left quantile point, and the second quantile point using linear interpolation.

2. The method of claim 1, wherein the right quantile point is defined using rightt(i)=(St−1(i)+Δt(i), pt(i)+ft−1(i)Δt(i)), wherein: St−1(i) is the first quantile estimate associated with the first quantile point;pt(i) is the first probability associated with the first quantile estimate of the first quantile point;ft−1(i) is a density estimate associated with the first quantile estimate of the first quantile point; andΔt(i) is a monotonicity value.

3. The method of claim 1, wherein the left quantile point is defined using leftt(i+1)=(St−1(i+1)−Δt(i), pt(i+1)−ft−1(i+1)Δt(i)), wherein: St−1(i+1) is the second quantile estimate associated with the second quantile point;pt(i+1) is the second probability associated with the second quantile estimate of the second quantile point;ft−1(i+1) is a density estimate associated with the second quantile estimate of the second quantile point; andΔt(i) is a monotonicity value.

4. The method of claim 1, wherein the first quantile point and the right quantile point are connected linearly, the right quantile point and the left quantile point are connected linearly, and the left quantile point and the second quantile point are connected linearly.

5. The method of claim 1, wherein generating the approximation of the quantile distribution function comprises: computing a monotonicity value Δt(i) using:

6. The method of claim 1, further comprising: prior to generating the approximation of the quantile distribution function: generating the quantile distribution function by updating an initial quantile distribution function based on a received data value.

7. The method of claim 6, wherein the initial quantile distribution function comprises a first initial quantile estimate having a first initial probability associated therewith, wherein updating the initial quantile distribution function comprises: determining a second initial probability associated with the first initial quantile estimate.

8. The method of claim 7, wherein determining the second initial probability associated with the first initial quantile estimate comprises: evaluating pt(i)=(1−wt) pt−1(i)+wtI(St−1(i)≧xt) for the first initial quantile estimate, wherein: xt is the received data value;wt is a weight associated with the received data value;St−1(i) is the first initial quantile estimate;pt−1(i) is the first initial probability associated with the first initial quantile estimate;I(St−1(i)≧xt) is an indicator function; andpt(i) is the second initial probability associated with the first initial quantile estimate.

9. The method of claim 1, further comprising: determining an updated first quantile estimate associated with the first probability based on the approximation of the quantile distribution function.

10. The method of claim 9, wherein determining the updated first quantile estimate associated with the first probability based on the approximation of the quantile distribution function comprises: evaluating the approximation of the quantile distribution function using the first probability as an input.

11. The method of claim 9, wherein determining the updated first quantile estimate associated with the first probability based on the approximation of the quantile distribution function comprises: evaluating {circumflex over (F)}t(St(i))=pi, wherein{circumflex over (F)}tis the approximation of the quantile distribution function;St(i) is the updated first quantile estimate; andpi is the first probability.

12. The method of claim 1, wherein the quantile distribution function has a first density estimate associated therewith, wherein the first density estimate is associated with the first quantile point, wherein the method further comprises: updating the first density estimate of the quantile distribution function to form a second density estimate using ft(i)=(1−wt)ft−1(i)+wtI(|xt−St(i)|≦c /{2c}), wherein:xt is a received data value;wt is a weight associated with the received data value;St(i) is the first quantile estimate associated with the first quantile point;ft−1(i) is the first density estimate associated with the first quantile point;I(|xt−St−1(i)≦c) is an indicator function;c is a tunable parameter representing a window size; andft(i) is the second density estimate.

13. The method of claim 1, further comprising: prior to generating the approximation of the quantile distribution function: generating the quantile distribution function by updating an initial quantile distribution function based on a received data value, wherein the initial quantile distribution function comprises a first initial quantile estimate having a first initial probability associated therewith, wherein updating the initial quantile distribution function comprises determining a second initial probability associated with the first initial quantile estimate.

14. A non-transitory computer-readable storage medium storing instructions which, when executed by a computer, cause the computer to perform a method, the method comprising: generating an approximation of a quantile distribution function by connecting a pair of adjacent quantile points using linear interpolation;wherein the pair of adjacent quantile points comprises: a first quantile point having a first quantile estimate and a first probability associated therewith; anda second quantile point having a second quantile estimate and a second probability associated therewith; andwherein generating the approximation of the quantile distribution function comprises: defining a right quantile point to the right of the first quantile point in the quantile distribution function;defining a left quantile point to the left of the second quantile point in the quantile distribution function; andconnecting the first quantile point, the right quantile point, the left quantile point, and the second quantile point using linear interpolation.

15. An apparatus, comprising: a processor and a memory communicatively connected to the processor, the processor configured to: generate an approximation of a quantile distribution function by connecting a pair of adjacent quantile points using linear interpolation;wherein the pair of adjacent quantile points comprises: a first quantile point having a first quantile estimate and a first probability associated therewith; anda second quantile point having a second quantile estimate and a second probability associated therewith; andwherein the processor is configured to generate the approximation of the quantile distribution function by: defining a right quantile point to the right of the first quantile point in the quantile distribution function;defining a left quantile point to the left of the second quantile point in the quantile distribution function; andconnecting the first quantile point, the right quantile point, the left quantile point, and the second quantile point using linear interpolation.