Generalized random shapelet forests

Isak Karlsson; Panagiotis Papapetrou; Henrik Boström

doi:10.1007/s10618-016-0473-y

Data Mining and Knowledge Discovery

September 2016, Volume 30, Issue 5, pp 1053–1085 | Cite as

Generalized random shapelet forests

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Isak Karlsson
Panagiotis Papapetrou
Henrik Boström

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First Online: 12 July 2016

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Abstract

Shapelets are discriminative subsequences of time series, usually embedded in shapelet-based decision trees. The enumeration of time series shapelets is, however, computationally costly, which in addition to the inherent difficulty of the decision tree learning algorithm to effectively handle high-dimensional data, severely limits the applicability of shapelet-based decision tree learning from large (multivariate) time series databases. This paper introduces a novel tree-based ensemble method for univariate and multivariate time series classification using shapelets, called the generalized random shapelet forest algorithm. The algorithm generates a set of shapelet-based decision trees, where both the choice of instances used for building a tree and the choice of shapelets are randomized. For univariate time series, it is demonstrated through an extensive empirical investigation that the proposed algorithm yields predictive performance comparable to the current state-of-the-art and significantly outperforms several alternative algorithms, while being at least an order of magnitude faster. Similarly for multivariate time series, it is shown that the algorithm is significantly less computationally costly and more accurate than the current state-of-the-art.

Keywords

Multivariate time series Time series classification Time series shapelets Decision trees Ensemble methods

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Notes

Acknowledgments

This work was partly supported by the project High-Performance Data Mining for Drug Effect Detection at Stockholm University, funded by Swedish Foundation for Strategic Research under Grant IIS11-0053.

Compliance with ethical standards

Conflicts of interest

The authors declare that they have no conflict of interest.

Source code

Source code (MIT license) for the Generalized Random Shapelet Forest is available at Github.⁸ Instructions and datasets can be found at the supporting website.⁹

Appendix 1: Decomposing the mean squared error in a forest

To investigate whether the observed difference in predictive performance between parameter configurations is mainly due to bias, variance, or both, the mean square error of the forest can be decomposed into those terms. In contrast to regression, where the decomposition of prediction error into bias and variance is well understood and widely used (James 2003), there is no general definition for classification. Hence, several decompositions of the average classification error rate into bias and variance have been proposed (see Friedman 1997; James 2003; Valentini and Dietterich 2004). It is, however, difficult to directly study the effect of bias and variance of randomized algorithms in the context of miss-classification error since a decrease in variability of the predictions can increase the average error, making the averaged model worse than the randomized. Instead, similar to the mean square error (MSE) of a regressor, the mean square error of a forest consisting of trees able to output a conditional class probability estimates for a given example can be computed as follows (Boström 2012). Given the class labels $\{c_1, c_2, \ldots ,c_l\}$, let $\bar{b}^{(i)}_k = (b^{(i)}_{k1}, b^{(i)}_{k2},\ldots ,b^{(i)}_{kl})$ be the probabilities assign by k:th random shapelet tree $ST_k$ in the forest for a labeled time series $z^{(i)}$. Furthermore, let $\bar{c}^{(i)} = (\bar{c}^{(i)}_1, \bar{c}^{(i)}_2, \ldots , \bar{c}^{(i)}_l)$ represent the true class vector for a labeled time series $z^{(i)}$, where $\bar{c}^{(i)}_j$ is 1 if $y^{(i)} = c_j$ and 0 otherwise, then the mean squared error (mse) of the forest can be defined as:

$$\begin{aligned} \mathrm {mse} = \frac{1}{n}\sum _{i=0}^{t} \frac{1}{p}\sum _{k=1}^{p}\left( \bar{b}^{(i)}_k-\bar{c}^{(i)}\right) ^2 \end{aligned}$$

(5)

Given the mean class probability vector $\bar{b}^{(i)}_{\mu }$ for the ith example, the mse can be composed into two parts, the bias (left) and variance (right):

$$\begin{aligned} \mathrm {mse} = \frac{1}{n}\sum _{i=0}^{n} \left( \bar{c}^{(i)} - \bar{b}^{(i)}_{\mu } \right) ^2 + \frac{1}{n}\sum _{1=0}^{n} \frac{1}{T}\left( \bar{p}^{(i)} - \bar{b}^{(i)}_{\mu } \right) ^2 \end{aligned}$$

(6)

Appendix 2: Internal estimates of strength and correlation

In the original Random Forest, Breiman (2001) proposes internal estimates for the strength (i.e., how accurate the individual classifiers are) and correlation (i.e., the dependence between classifiers) of the forest based on the out-of-bag instances not included during training. Using these measures, an upper bound can be derived for the generalization error. More precisely, for the case of random shapelet forests, each random shapelet tree $ST_k \in \mathcal {R}$ can be seen as a base classifier function $f_k$. Hence, we can define a set of p base classifier functions $\{f_1(T), \ldots , f_p(T)\}$ as well as the ensemble classifier function $f_{\mathcal {R}}(T)$. Let us denote the set of out-of-bag instances for a classifier $ST_k$ as $\mathcal {D}_{T_k}$. Furthermore, given a class label $c\in \mathcal {C}$, Q(T, c) is an approximation function for $P(f_{\mathcal {R}}(T) = c)$ corresponding to the out-of-bag proportion of votes for class c for the input time series T. More formally:

$$\begin{aligned} Q(\mathcal {F}, c) = \frac{ \sum _{k=1}^{p} \mathbf {1}(ST_k(T) = c ; T \in \mathcal {D}_{T_k}) }{ |{T \in \mathcal {D}_{T_k}}| } , \end{aligned}$$

(7)

where $\mathbf {1}(\cdot )$ is the indicator function. Then, the margin measures the extent to which the average number of votes for the right class exceeds the average number vote for any other class (Breiman 2001).

Definition 7

(margin function) The empirical margin function for a random shapelet forest, similar to a random forest, is

$$\begin{aligned} \mathrm {mr}(\mathcal {F}, T,c) = P(f_{\mathcal {R}}(T) = c) - \max _{\begin{array}{c} j=1 \\ j \ne c \end{array}}^{|\mathcal {C}|} \{f_{\mathcal {R}}(T) = c_j \} \end{aligned}$$

(8)

where $P(\cdot )$ is estimated using $Q(\cdot )$.

The expectation over the margin function gives a measure of how accurate, or strong, a set of classifiers are.

Definition 8

(strength) The strength of a random shapelet forest is the expected margin, and can be empirically estimated as the average over the training set:

$$\begin{aligned} s = \frac{1}{n}\sum _{i=1}^{n}mr(T_i, y_i) \end{aligned}$$

(9)

By computing the variance of the margin, the correlation and interdependence between the individual classifiers can be estimated as the variance of the margin over the squared standard deviation of the random shapelet forest.

Definition 9

(correlation) The correlation of the random shapelet forest can be empirically estimated as:

$$\begin{aligned} \bar{p}=\frac{\mathrm {var}(mr)}{\mathrm {sd}(\mathcal {F})^2} = \frac{ \frac{1}{n}\sum _{i=1}^{n}mr(T_i, y_i)^2 - s^2 }{ \left( \frac{1}{p}\sum _{k=1}^{p}\sqrt{b_k+\hat{b}_k+(b_k-\hat{b}_k)^2} \right) ^2 } \end{aligned}$$

(10)

where $b_k$ is an out-of-bag estimate of $P(f_k(T = y) = c)$, with

$$\begin{aligned} b_k = \frac{ \sum _{i=1}^{n}\mathbf {1}(f_k(T_i) = y_i ; T_i \in \mathcal {D}_{T_k}) }{ |\{T \in \mathcal {D}_{T_k}\}| }, \end{aligned}$$

(11)

and $\hat{b}_k$ is an out-of-bag estimate of $P(f_{\mathcal {F}}(T) = \hat{c}_j)$

$$\begin{aligned} \hat{b}_k = \frac{ \sum _{i=1}^{n}\mathbf {1}(f_k(T_i) = \hat{c}_i) }{ |\{T\in \mathcal {D}_{T_k}\}| }, \end{aligned}$$

(12)

where $\hat{c}_j$ is estimated for every instance in the training set with Q(T, c) as

$$\begin{aligned} \hat{c}_j = \mathrm {arg} \max _{\begin{array}{c} j=1 \\ j \ne c \end{array}}^{|\mathcal {C}|} Q(T, c_j). \end{aligned}$$

(13)

Assuming that the strength is $s > 0$, Breiman (2001) showed that a (rather loose) upper bound on the generalization error of a random forest, and by similar argument a random shapelet forest, can be given by $\frac{\bar{p}(1-s^2)}{s^2}$. The bound shows that the two ingredients involved in the generalization error for forests of randomized trees are the strength of the individual classifiers, and the dependencies between them in terms of the margin function (Breiman 2001). Furthermore, the correlation divided with the squared strength, $\bar{p}/s^2$, provides a ratio, where smaller is better, that can be used to understand the functioning of the forest (Breiman 2001).

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Isak Karlsson
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Panagiotis Papapetrou
- 1
Henrik Boström
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1.Stockholm UniversityStockholmSweden

Generalized random shapelet forests

Abstract

Keywords

Notes

Acknowledgments

Compliance with ethical standards

Conflicts of interest

Source code

References

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