An empirical study of neural networks for trend detection in time series

Trends can be interpreted as the slopes of a smooth function around which the time series oscillates. The simplest, and probably the closest to human intuition, would be to use piecewise linear functions as in described in [9]. The issue with these filtering approaches is that they tend to be good ex post but slow to detect changes of trends. This is a real problem when the whole time series is not known in advance.
We take a slightly different approach. If the future value of the time series is expected to be higher [respectively lower, equal] than the current one, then the time series is said to be trending up [respectively trending down, not trending]. At each time step, we assign a unique trend value noted ${\delta }_t$, the time-series is:

• trending downward at $t$ if ${\delta }_t<0$

• not trending at $t$ if ${\delta }_t=0$

• trending upward at $t$ if ${\delta }_t>0$

We can directly translate this intuition into mathematical terms. Consider a process $\{Y_t{\}}_{t>0}$ adapted to a filtration $\{F_t{\}}_{t\ge 0}$, under some technical conditions, the Doob-Meyer theorem applies and $\{Y_t{\}}_{t>0}$ can be decomposed in an unique way as

where $\{A_t{\}}_{t>0}$ is a predictable increasing [respectively decreasing, zero] process if $Y_t$ is a sub-martingale [respectively super-martingale, martingale] starting at 0 and $\{M_t{\}}_{t>0}$ is a martingale. Obviously, we can map our intuitive definition to more precise concepts.
$\{Y_t{\}}_{t>0}$ is: trending downward $⟺$ $\{A_t{\}}_{t>0}$ is decreasing not trending $⟺$ $\{A_t{\}}_{t>0}$ is null trending upward $⟺$ $\{A_t{\}}_{t>0}$ is increasing
The monotonicity of the $\{A_t{\}}_{t>0}$ process will be our definition of the trend of $\{Y_t{\}}_{t>0}$ and thus a classification task with three labels $\{-1,0,1\}$ for downward, flat and upward trend. Considering an Itô process $\{Y_t{\}}_{t>0}$

where $\{W_t{\}}_{t>0}$ is a Wiener process. We can track the changing monotonicity of $A_t={\int }_0^t\beta (s,Y_s)ds$ via the sign of $\beta (t,Y_t)$ which will be our practical definition of trend.
The challenge at hand is to build an estimator of the sign of $\beta (t,Y_t)$, which will be our classification label. In the following, we will consider various time series dynamics where we control the sign of $\beta (t,Y_t)$. This gives us a framework to analyse the performance of various estimators, while controlling for the statistical properties of the dataset.
The classification task relies on the labelling of the training set. When using historical data, labelling is not easy to do: the definition of trend is subjective and usually depends on the choice of a time window or of a performance criterion. On the contrary, when using simulated data, labelling of the training set is easy. A general-purpose estimator of trend in a simulated environment is a useful building block for handling more complex real-life cases where no trend labels are available. It gives us a robust starting point on which we can build on¹.

Our idea is to generate as many realistic datasets as possible, and to train trend estimators on those datasets. If we train our estimator on a dataset rich enough to capture all the possible scenarios, we can hope to have an estimator robust to real-life conditions. In the following, we consider three different types of dynamics, hopefully rich and diverse enough to match a lot of the real-life behaviour:

• a noisy piecewise linear process

• a piecewise Ornstein-Uhlenbeck process [16]

• a Markovian switching process [5]

The first two are piecewise meaning that we divide time into intervals on which the time series follows the chosen dynamic. A simple continuity constraint is applied to “glue” together these different periods.
In the rest of the section we define:

• a time interval $[0,T]$

• for piecewise processes, a number $N$ of intervals $[t_i,t_{i+1}],i\in \text{llbracket}1,N\text{rrbracket}$ of possibly different lengths

Training sets are made of 1000 time series containing roughly 1000 data points, randomly drawn:

• from either one of the three previous dynamics (see section 2.2)

• or from all of the previous dynamics. This will be named mixed dynamic in the following

Model selection is made on validation sets composed of 300 time series: 100 samples from each of the three dynamics described in section 2.2. Each sample has between 500 and 1000 points depending on the dynamics and the draw. Figure 4 shows random samples from the validation set. This validation set offers a rich set of scenarios and can be used to assess the ability of an estimator to detect trends. Hyper-parameters are chosen using a separate test set which is a new random draw of the training set.

One important question arising from the chosen approach is the relevance of the simulated data. The dynamics can show behaviours that, even if not designed to simulate market dynamics, can be relatively similar to actual asset prices. As an example on figure 5 we plot real assets daily time-series versus a random sample from our three dynamics.

We see that the trajectories can be visually similar but that the distribution of daily returns may differ greatly. We must bear in mind that our aim is not to simulate market data but to detect trend defined as the sign of the drift term. We think that our dynamics are good enough to simulate this property of real time-series. One general method to get simulated dynamics close to empirical market data is the following :

1. Chose a dynamic

2. Compute the distribution of returns of the market time series of interest

3. Sample time-series of the dynamic and compute the distributions returns

4. Compute the average distance between the sampled distributions and the empirical one³

5. Minimize this function over the dynamic parameters using black-box Bayesian optimization

One of the most common way to detect trends is to adopt a filtering approach, comparing smoothed versions of the initial process. For example, we could aggregate several moving averages like:

with various values of $\alpha \in [0,1]$. Determining the optimal $\alpha$ might be difficult if we want to build an estimator adapted to various dynamics. To circumvent this difficulty, we can aggregate the values for different $\alpha$ as the components of vectors $h_t=(h_t^{{\alpha }_1},\dots ,h_t^{{\alpha }_m})$ through time⁴.
For example, we might want to consider $h_t=(h_t^{{\alpha }_1=0.1},h_t^{{\alpha }_2=0.5},h_t^{{\alpha }_2=0.9})\in R^3$ concatenation of a fast, medium and slow moving averages. We might compare:

• the slow and the fast moving averages by looking at the sign of

  $$h_t^{0.9}-h_t^{0.1}={\left[\begin{array}{c}-1\\ 0\\ 1\end{array}\right]}^{⊺}\cdot \left[\begin{array}{c}{h}_t^{0.1}\\ h_t^{0.5}\\ h_t^{0.9}\end{array}\right]$$

• or maybe the slow versus an average of the medium and slow with the sign of

  $$h_t^{0.9}-\frac{1}{2}(h_t^{0.1}+h_t^{0.5})={\left[\begin{array}{c}-0.5\\ -0.5\\ 1\end{array}\right]}^{⊺}\cdot \left[\begin{array}{c}{h}_t^{0.1}\\ h_t^{0.5}\\ h_t^{0.9}\end{array}\right]$$

• or whatever weighted combination we fancy with the sign of

  $$0.23h_t^{0.9}+1.5h_t^{0.5}-0.96h_t^{0.1}={\left[\begin{array}{c}-0.96\\ 1.5\\ 0.23\end{array}\right]}^{⊺}\cdot \left[\begin{array}{c}{h}_t^{0.1}\\ h_t^{0.5}\\ h_t^{0.9}\end{array}\right]$$

Generally speaking, we look at the signs of components of the vector $W\cdot h_t$ where $W$ is a given⁵ weight matrix. The rows of $W$ define hyperplanes. The half-spaces determined by $W$ are given by the signs of the components of $W\cdot h_t$. Detecting a trend is simply trying to locate $h_t$ with regards to convex polytopes determined by these half-spaces.
Generalizing equation (2) to upper dimensions, we have:

where $W_{hh}\in M_m^{+}\left(R\right)$ is a positive matrix and $w_{ih}\in R_{+}^m$ a positive vector such that

The trend is determined by $\mathrm{sgn}(W\cdot h_t)$ but we could use any other activation function $f$ instead of the sign function.

These equations are exactly equal to the update equation of a RNN composed of

• a vanilla RNN

  • with the identity as activation function

  • with one hidden layer

  • with convex constraints on the weight matrix $\left[\begin{array}{c}{W}_{hh},w_{ih}\end{array}\right]$⁶

• with a simple linear layer and activation function $f=\mathrm{sgn}$

Such a RNN will be called a “convex net” in the following. This shows that RNNs can be considered as generalizations of some basic moving average comparisons. As a working example, we consider the case of the Noisy Line Process $Y_t=Y_0+\mu t+{ϵ}_t$ where ${ϵ}_t$ are independent noise random variables $E\left({ϵ}_t\right)=0$.
For a net with constrained weights it can be shown (see annex B for details):

• without trend, $\mu =0$, then $\left\{h_t\right\}$ becomes centered around a variable of finite variance

• with trend, $\mu \ne 0$ then $\left\{h_t\right\}$ diverges

If we now introduce a hyperbolic tangent activation function instead of identity:

• if $\mu =0$, near zero the cell is in the linear part and we should expect the state to stay bounded around the origin

• if the trend $\mu \ne 0$ then the state should go towards $\mathrm{sgn}\left(\mu \right)\times \infty$ i.e. to navigate near the faces of the $]0,1{[}^n$ hypercube

For a practical illustration see annex C.

We train our triplets as described in subsection 3.2.2 for both Adam and RMSprop and validate each triplet on our 300 validation samples (see section 2.3). The loss is a binary loss on the labels.
Table 1 shows the coefficients of the linear regression of loss against binary variables indicating the training dynamic, the net type, the optimization type and the validation dynamic. Each feature is translated into binary on/off variables with one less modality. The missing modality is on if all others are set to zero. A positive coefficient means that the highlighted feature increases the average loss of the sample, and conversely, a negative coefficient decreases the average loss. Full details can be found in annex D.2. 

From figure 6:

• training on Ornstein-Uhlenbeck dynamic seems to worsen performance

• GRU seems to be the best net type and Vanilla not a great choice

• the optimization algorithm RMSProp has a negative impact on performance. Adam leads to better results

• the validation loss for Markovian Switch is higher than the two other dynamics

Training dynamic has an impact on validation performance. Choosing two dynamics e.g. Noisy Line versus Piecewise Ornstein-Uhlenbeck, we select data from those only and bootstrap. For each bootstrapping iteration, we compute the difference between the medians of losses of one dynamic versus the other. The result can be seen on table 2. Even if all intervals contain zero, and no robust conclusion can be drawn, the median loss seems lower when training using the Noisy Line or Markovian Switch dynamics.

Net structure are compared using the same bootstrapping procedure in table 3. Vanilla RNN is consistently worse than LSTM and GRU at 99% confidence level. As a result, in the following, we will ignore triplets with Vanilla RNN. Vanilla RNN is barely better than a dummy estimator having $\frac{1}{3}$ chance of correctly predicting the trend (see annex D.3).

Optimizer impact: results seem to indicate a slightly better performance of Adam versus RMSprop⁸.
Net structure and training dynamic interaction: using only the triplets where net structure is either GRU or LSTM, we run the same bootstrapping procedure for each datasets on the training dynamic. The results are given in table 4. All the intervals contain 0 and it is difficult to find a combination which does significantly better than the others.

We would like to choose a RNN estimator having a good overall performance on validation data. As we have seen, it is difficult to choose a particular training type or net structure (GRU or LSTM) as being significantly better. A way to build a baseline would be for example to pool the estimated probabilities of the best trained estimators. The pooling function here is a simple average of each estimated probabilities from the selected estimators⁹. And this, indeed, gives good results on validation data as can be seen in table 5. We note little difference in performance when pooling more than five estimators.

Yet, choosing such an estimator would give RNNs an advantage compared to other estimators. To be as fair as possible and favour simplicity over performance we choose to optimize hyper-parameters for a GRU network trained on the Piecewise Noisy Line dynamic using Adam optimization. Some details of the RNN baseline can be found in table 6.

It is interesting to note that adding training epochs¹⁰ seems to slightly increase the median error on the test set but gives a noticeable decrease of the interquartile range by a factor near 25%.

Running the training with hyper-parameters not too far from the ones obtained by optimization gives fairly similar results. The comparison of the RNN baseline versus the pooled estimator is given in table 7 and figure 7 for the loss distributions. Even if our RNN baseline is not the best it still offers good performance.

One of the most intuitive way to detect trend is to compare the speed of two moving averages. We compare our RNN baseline with both the most simple moving average filtering and the convex net generalization approach.

One dimensional CNN is sometimes seen as a good tool to analyse time series. We use a standard CNN structure stacking convolutional layer followed by a pooling layer. To keep nets architecture similar in term of parameters, we use two layers of convolution + pooling. After optimization, we get hyper-parameters shown in table 11. Interestingly, both channel and kernel have taken the maximum value in the range we tested¹¹.

Yet, we are unable to find the supposed general efficiency of CNNs in our setup as seen on figure 9. Actually, CNN performance is barely better than a dummy classifier as seen on table 12.

As a reminder, the training data used for the learning step of the neural networks is comprised of piecewise trajectories of the dynamics and uses randomized model parameters. Taking into account this additional randomness in a MLE estimation framework would make the theory intractable. In order to compare MLE based trend classification with neural networks, we use a sliding window mechanism. For a sliding window $W_i$ of length $\eta$:

• we compute the value of the trend estimator ${\hat{\mu }}_i$

• we map the value of ${\hat{\mu }}_i$ to a label using the sign function ¹² (for a given threshold $ϵ$) and predict this label with probability $1$.

We only need this mechanism for the Noisy Line Process and the Piecewise Ornstein-Uhlenbeck Process.