Predicting Poll Trends Using Twitter and Multivariate Time-Series Classification

Abstract

Social media outlets, such as Twitter, provide invaluable information for understanding the social and political climate surrounding particular issues. Millions of people who vary in age, social class, and political beliefs come together in conversation. However, this information poses challenges to making inferences from these tweets. Using the tweets from the 2016 U.S. Presidential campaign, one main research question is addressed in this work. That is, can accurate predictions be made detecting changes in a political candidate’s poll score trends utilizing tweets created during their campaign? The novelty of this work is that we formulate the problem as a multivariate time-series classification problem, which fits the temporal nature of tweets, rather than as a traditional attribute-based classification. Features that represent various aspects of support for (or against) a candidate are tracked on an hour-by-hour basis. Together these form multivariate time-series. One commonly used approach to this problem is based on the majority voting scheme. This method assumes the univariate time-series from different features have equal importance. To alleviate this issue a weighted shapelet transformation model is proposed. Extensive experiments on over 12 million tweets between November 2015 and January 2016 related to the four primary candidates (Bernie Sanders, Hillary Clinton, Donald Trump and Ted Cruz) indicate that the multivariate time-series approach outperforms traditional attribute-based approaches.

Appendix

Learning Time-Series Classification Model (LTS)

LTS [4] is one of the state-of-the-art univariate time-series classification models. The method discovers short time-series sub-sequences known as shapelets [17], which are local discriminative patterns (or sub-sequences) that can be used to characterize the target class, for determining the time-series class membership. In the LTS model, shapelets are learned jointly with a linear classifier rather than searching over all possible time-series segments. More specifically, the algorithm jointly learns the weights of the classifier hyper-plane as well as the generalized shapelets.

A shapelet of length W is a sub-sequence of an instance of the time-series. There can be at most $L~-~W~+~1$ sub-sequences, and each can be represented as $\{f^q_{i,j},...,f^q_{i,j+W-1}\}$. K shapelets are initialized using K-Means centroid of all segments.

Equation 4 represents a linear model, where $M_{i,k}$ is the minimum distance between the i-th series in $T^q$ and the k-th shapelet $S^q_k$.

$$\begin{aligned} \hat{Y^q_i} = \beta _0 + \sum _{k = 1}^K M_{i,k}\beta _k~~~~~~~~~~~~~~~~~ \forall i \in \left\{ {1,...,I}\right\} \end{aligned}$$

(4)

The minimum distance $M_{i,k}$ is the predictor in this framework for shapelet learning and can be defined by a soft-minimum function:

$$\begin{aligned} M_{i,k} = \frac{\sum D_{i,k,j}e^{\alpha D_{i,k,j}}}{\sum e^{\alpha D_{i,k,j^\prime }}} \end{aligned}$$

(5)

where $D_{i,k,j}$ is defined as the distance between the $j^{th}$ segment of series i and the $k^{th}$ shapelet given by the formula

$$\begin{aligned} D_{i,k,j} = \frac{1}{W}\sum _{w=1}^W (T^q_{i,j+w-1} - S^q_{k,w})^2 \end{aligned}$$

(6)

Equation 7 shows the regularized objective function, composed of a logistic loss defined by Eq. 8 and the regularization terms.

$$\begin{aligned} argmin_{S,\beta } F(S,W)=argmin_{S,\beta } \sum _{i=1}^{I} \mathcal {L}(Y^q_i,\hat{Y^q_i})+\lambda _\beta ||\beta ||^2 \end{aligned}$$

(7)

$$\begin{aligned} \mathcal {L}(Y^q_i,\hat{Y^q_i}) = -~Y^q_i~ln(\sigma (\hat{Y^q_i})) - (1-Y^q_i)ln(1-\sigma (\hat{Y^q_i})) \end{aligned}$$

(8)

Equation 7 is optimized using a stochastic gradient descent algorithm. The weights $\beta$ and the shapelet $S^q$ are jointly learned to minimize the objective function. Once the model is learned, classifying an unknown instance is simply computing $\hat{Y^q_t}$ for the t-th test instance of the q-th feature and determining the class label via Eq. 9

$$\begin{aligned} \hat{Y^q_t} \leftarrow argmax_{c \in \left\{ 1, -1 \right\} }~\sigma (\hat{Y^q_{t,c}}), \end{aligned}$$

(9)

where $\sigma (\cdot )$ denotes the sigmoid function.

For more details about individual gradient computation of the objective function, the reader is referred to [4].