Hybrid fuzzy model for Stock returns forecasting

We present a model for forecasting financial time series. It was created
to forecast stock returns, but can be widely used in other areas. The model
is based on the fuzzy system of Takagi, Sugeno and Kang, which provides
a polynomial dependence with soft switching between equations.
Vector autoregression (VAR) is used to obtain predictions of regressors.
Wavelet decomposition is used as preprocessing and clears time series from noise.
We also use FCM fuzzy clustering to split the data into fuzzy sets and make it ready for
the TSK fuzzy model use.

model structure:

1. Data de-noising with Wavelet decomposition
2. VAR forecast for factors
3. Fuzzy clustering with FCM algorithm
4. Takagi-Sugeno_Kang fuzzy system

Wavelet decomposition

Financial time series have noise that interferes with the identification of trends.
Also, noise, outliers and heteroscedasticity make the time series weakly stationary.
Therefore, we use wavelet decomposition for de-noising. The main idea of this approach is
to decompose the signal into two parts, High-frequency (Wavelet) and Approximating. The
high-frequency part is then removed, thus the signal is cleared from noise

Code documentation for Wavelets

Class

The wavelet decomposition is written as a class: Wavelet_de_noising(signal, plot).

Arguments

signal - Time series or array without NAN values.
plot- Raw and filtered signal graph. plot=True for plotting, default is plot=False

Methods

1. decompose(wavelet, level) - main method for de-noising

   Arguments

   wavelet - name of Wavelet function that is used for decomposition. Default wavelet='db18'
level - level of decomposition. At each level, the signal is decomposed again into the same
   functions. Default wavelet=1
   

2. get_wavelets - additional method to get the names of all available wavelet functions
   (out of 127)

FCM fuzzy clustering

In order to apply fuzzy methods, it is necessary to determine the membership function of elements
to fuzzy sets. In our case, we use the fuzzy clustering technique to determine the membership matrix.
The idea is to use unsupervised learning to split the dataset into fuzzy sets. As a result, each
observation belongs to each of the clusters (sets) with some degree. We use the FCM algorithm for this
task. In fact, it is a modification of the K-means algorithm. In K-means, the element belongs to the
cluster with the nearest centroid, and the centroids themselves are recalculated as the average of the
clusters. In FCM, elements belong to all clusters at the same time. Therefore, we calculate the
membership matrix, and the centroids are updated as the weighted average of the clusters according to
this membership matrix.

Code documentation for FCM

Class

The FCM algorithm is written as a class: FCM(K, max_iters).

Arguments

K - Number of clusters returning as output. Default K=3
max_iters- Maximum number of iterations. Default max_iters=100'

Methods

1. predict(X)

   Arguments

   X - Dataset or variable for clustering. Better to use nupy array.
   Method returns membership matrix as output.
   

Takagi Sugeno Kang fuzzy model

This model uses the consequent equations, each with its own functional dependence. The equations
relate to their clusters. The weight of each equation in the final forecast depends on the degree
to which the observation belongs to these clusters. Gradient descent optimization involves two stages.
At the first stage, models with all possible combinations of clusters and equations are evaluated.
At the second stage, the best combination is selected. Thus, we obtain a polynomial dependence that
dynamically changes depending on the membership matrix of observations to clusters.

Code documentation for TSK model

Class

The TSK is written as a class: TakagiSugeno(cluster_n, lr, n_iters).

Arguments

cluster_n - Number of clusters in membership matrix and also number of consequent equations.
Default cluster_n=2
lr- Learning rate. Default lr=0.01'

n_iters- Maximum number of iterations. Default n_iters=1500'

Methods

1. predict(X)

   Arguments

   X - Dataset or variable for clustering. Better to use nupy array.
   Method returns membership matrix as output.