Daron  Moore

Daron Moore

1641276000

DeltaPy⁠⁠ — Tabular Data Augmentation & Feature Engineering

DeltaPy⁠⁠ — Tabular Data Augmentation & Feature Engineering


Finance Quant Machine Learning

Introduction

Tabular augmentation is a new experimental space that makes use of novel and traditional data generation and synthesisation techniques to improve model prediction success. It is in essence a process of modular feature engineering and observation engineering while emphasising the order of augmentation to achieve the best predicted outcome from a given information set. DeltaPy was created with finance applications in mind, but it can be broadly applied to any data-rich environment.

To take full advantage of tabular augmentation for time-series you would perform the techniques in the following order: (1) transforming, (2) interacting, (3) mapping, (4) extracting, and (5) synthesising. What follows is a practical example of how the above methodology can be used. The purpose here is to establish a framework for table augmentation and to point and guide the user to existing packages.

For most the Colab Notebook format might be preferred. I have enabled comments if you want to ask question or address any issues you uncover. For anything pressing use the issues tab. Also have a look at the SSRN report for a more succinct insights.

Data augmentation can be defined as any method that could increase the size or improve the quality of a dataset by generating new features or instances without the collection of additional data-points. Data augmentation is of particular importance in image classification tasks where additional data can be created by cropping, padding, or flipping existing images.

Tabular cross-sectional and time-series prediction tasks can also benefit from augmentation. Here we divide tabular augmentation into columnular and row-wise methods. Row-wise methods are further divided into extraction and data synthesisation techniques, whereas columnular methods are divided into transformation, interaction, and mapping methods.

See the Skeleton Example, for a combination of multiple methods that lead to a halfing of the mean squared error.

Installation & Citation


pip install deltapy
@software{deltapy,
  title = {{DeltaPy}: Tabular Data Augmentation},
  author = {Snow, Derek},
  url = {https://github.com/firmai/deltapy/},
  version = {0.1.0},
  date = {2020-04-11},
}
 Snow, Derek, DeltaPy: A Framework for Tabular Data Augmentation in Python (April 22, 2020). Available at SSRN: https://ssrn.com/abstract=3582219

Function Glossary


Transformation

df_out = transform.robust_scaler(df.copy(), drop=["Close_1"]); df_out.head()
df_out = transform.standard_scaler(df.copy(), drop=["Close"]); df_out.head()           
df_out = transform.fast_fracdiff(df.copy(), ["Close","Open"],0.5); df_out.head()
df_out = transform.windsorization(df.copy(),"Close",para,strategy='both'); df_out.head()
df_out = transform.operations(df.copy(),["Close"]); df_out.head()
df_out = transform.triple_exponential_smoothing(df.copy(),["Close"], 12, .2,.2,.2,0); 
df_out = transform.naive_dec(df.copy(), ["Close","Open"]); df_out.head()
df_out = transform.bkb(df.copy(), ["Close"]); df_out.head()
df_out = transform.butter_lowpass_filter(df.copy(),["Close"],4); df_out.head()
df_out = transform.instantaneous_phases(df.copy(), ["Close"]); df_out.head()
df_out = transform.kalman_feat(df.copy(), ["Close"]); df_out.head()
df_out = transform.perd_feat(df.copy(),["Close"]); df_out.head()
df_out = transform.fft_feat(df.copy(), ["Close"]); df_out.head()
df_out = transform.harmonicradar_cw(df.copy(), ["Close"],0.3,0.2); df_out.head()
df_out = transform.saw(df.copy(),["Close","Open"]); df_out.head()
df_out = transform.modify(df.copy(),["Close"]); df_out.head()
df_out = transform.multiple_rolling(df, columns=["Close"]); df_out.head()
df_out = transform.multiple_lags(df, start=1, end=3, columns=["Close"]); df_out.head()
df_out  = transform.prophet_feat(df.copy().reset_index(),["Close","Open"],"Date", "D"); df_out.head()

Interaction

df_out = interact.lowess(df.copy(), ["Open","Volume"], df["Close"], f=0.25, iter=3); df_out.head()
df_out = interact.autoregression(df.copy()); df_out.head()
df_out = interact.muldiv(df.copy(), ["Close","Open"]); df_out.head()
df_out = interact.decision_tree_disc(df.copy(), ["Close"]); df_out.head()
df_out = interact.quantile_normalize(df.copy(), drop=["Close"]); df_out.head()
df_out = interact.tech(df.copy()); df_out.head()
df_out = interact.genetic_feat(df.copy()); df_out.head()

Mapping

df_out = mapper.pca_feature(df.copy(),variance_or_components=0.80,drop_cols=["Close_1"]); df_out.head()
df_out = mapper.cross_lag(df.copy()); df_out.head()
df_out = mapper.a_chi(df.copy()); df_out.head()
df_out = mapper.encoder_dataset(df.copy(), ["Close_1"], 15); df_out.head()
df_out = mapper.lle_feat(df.copy(),["Close_1"],4); df_out.head()
df_out = mapper.feature_agg(df.copy(),["Close_1"],4 ); df_out.head()
df_out = mapper.neigh_feat(df.copy(),["Close_1"],4 ); df_out.head()

Extraction

extract.abs_energy(df["Close"])
extract.cid_ce(df["Close"], True)
extract.mean_abs_change(df["Close"])
extract.mean_second_derivative_central(df["Close"])
extract.variance_larger_than_standard_deviation(df["Close"])
extract.var_index(df["Close"].values,var_index_param)
extract.symmetry_looking(df["Close"])
extract.has_duplicate_max(df["Close"])
extract.partial_autocorrelation(df["Close"])
extract.augmented_dickey_fuller(df["Close"])
extract.gskew(df["Close"])
extract.stetson_mean(df["Close"])
extract.length(df["Close"])
extract.count_above_mean(df["Close"])
extract.longest_strike_below_mean(df["Close"])
extract.wozniak(df["Close"])
extract.last_location_of_maximum(df["Close"])
extract.fft_coefficient(df["Close"])
extract.ar_coefficient(df["Close"])
extract.index_mass_quantile(df["Close"])
extract.number_cwt_peaks(df["Close"])
extract.spkt_welch_density(df["Close"])
extract.linear_trend_timewise(df["Close"])
extract.c3(df["Close"])
extract.binned_entropy(df["Close"])
extract.svd_entropy(df["Close"].values)
extract.hjorth_complexity(df["Close"])
extract.max_langevin_fixed_point(df["Close"])
extract.percent_amplitude(df["Close"])
extract.cad_prob(df["Close"])
extract.zero_crossing_derivative(df["Close"])
extract.detrended_fluctuation_analysis(df["Close"])
extract.fisher_information(df["Close"])
extract.higuchi_fractal_dimension(df["Close"])
extract.petrosian_fractal_dimension(df["Close"])
extract.hurst_exponent(df["Close"])
extract.largest_lyauponov_exponent(df["Close"])
extract.whelch_method(df["Close"])
extract.find_freq(df["Close"])
extract.flux_perc(df["Close"])
extract.range_cum_s(df["Close"])
extract.structure_func(df["Close"])
extract.kurtosis(df["Close"])
extract.stetson_k(df["Close"])

Test sets should ideally not be preprocessed with the training data, as in such a way one could be peaking ahead in the training data. The preprocessing parameters should be identified on the test set and then applied on the test set, i.e., the test set should not have an impact on the transformation applied. As an example, you would learn the parameters of PCA decomposition on the training set and then apply the parameters to both the train and the test set.

The benefit of pipelines become clear when one wants to apply multiple augmentation methods. It makes it easy to learn the parameters and then apply them widely. For the most part, this notebook does not concern itself with 'peaking ahead' or pipelines, for some functions, one might have to restructure to code and make use of open source packages to create your preferred solution.

Documentation by Example

Notebook Dependencies

pip install deltapy
pip install pykalman
pip install tsaug
pip install ta
pip install tsaug
pip install pandasvault
pip install gplearn
pip install ta
pip install seasonal
pip install pandasvault

Data and Package Load

import pandas as pd
import numpy as np
from deltapy import transform, interact, mapper, extract 
import warnings
warnings.filterwarnings('ignore')

def data_copy():
  df = pd.read_csv("https://github.com/firmai/random-assets-two/raw/master/numpy/tsla.csv")
  df["Close_1"] = df["Close"].shift(-1)
  df = df.dropna()
  df["Date"] = pd.to_datetime(df["Date"])
  df = df.set_index("Date")
  return df
df = data_copy(); df.head()

Some of these categories are fluid and some techniques could fit into multiple buckets. This is an attempt to find an exhaustive number of techniques, but not an exhaustive list of implementations of the techniques. For example, there are thousands of ways to smooth a time-series, but we have only includes 1-2 techniques of interest under each category.

(1) Transformation:


  1. Scaling/Normalisation
  2. Standardisation
  3. Differencing
  4. Capping
  5. Operations
  6. Smoothing
  7. Decomposing
  8. Filtering
  9. Spectral Analysis
  10. Waveforms
  11. Modifications
  12. Rolling
  13. Lagging
  14. Forecast Model

(2) Interaction:


  1. Regressions
  2. Operators
  3. Discretising
  4. Normalising
  5. Distance
  6. Speciality
  7. Genetic

(3) Mapping:


  1. Eigen Decomposition
  2. Cross Decomposition
  3. Kernel Approximation
  4. Autoencoder
  5. Manifold Learning
  6. Clustering
  7. Neighbouring

(4) Extraction:


  1. Energy
  2. Distance
  3. Differencing
  4. Derivative
  5. Volatility
  6. Shape
  7. Occurrence
  8. Autocorrelation
  9. Stochasticity
  10. Averages
  11. Size
  12. Count
  13. Streaks
  14. Location
  15. Model Coefficients
  16. Quantile
  17. Peaks
  18. Density
  19. Linearity
  20. Non-linearity
  21. Entropy
  22. Fixed Points
  23. Amplitude
  24. Probability
  25. Crossings
  26. Fluctuation
  27. Information
  28. Fractals
  29. Exponent
  30. Spectral Analysis
  31. Percentile
  32. Range
  33. Structural
  34. Distribution

 

(1) Transformation

Here transformation is any method that includes only one feature as an input to produce a new feature/s. Transformations can be applied to cross-section and time-series data. Some transformations are exclusive to time-series data (smoothing, filtering), but a handful of functions apply to both.

Where the time series methods has a centred mean, or are forward-looking, there is a need to recalculate the outputed time series on a running basis to ensure that information of the future does not leak into the model. The last value of this recalculated series or an extracted feature from this series can then be used as a running value that is only backward looking, satisfying the no 'peaking' ahead rule.

There are some packaged in Python that dynamically create time series and extracts their features, but none that incoropates the dynamic creation of a time series in combination with a wide application of prespecified list of extractions. Because this technique is expensive, we have a preference for models that only take historical data into account.

In this section we will include a list of all types of transformations, those that only use present information (operations), those that incorporate all values (interpolation methods), those that only include past values (smoothing functions), and those that incorporate a subset window of lagging and leading values (select filters). Only those that use historical values or are turned into prediction methods can be used out of the box. The entire time series can be used in the model development process for historical value methods, and only the forecasted values can be used for prediction models.

Curve fitting can involve either interpolation, where an exact fit to the data is required, or smoothing, in which a "smooth" function is constructed that approximately fits the data. When using an interpolation method, you are taking future information into account e.g, cubic spline. You can use interpolation methods to forecast into the future (extrapolation), and then use those forecasts in a training set. Or you could recalculate the interpolation for each time step and then extract features out of that series (extraction method). Interpolation and other forward-looking methods can be used if they are turned into prediction problems, then the forecasted values can be trained and tested on, and the fitted data can be diregarded. In the list presented below the first five methods can be used for cross-section and time series data, after that the time-series only methods follow.

(1) Scaling/Normalisation

There are a multitude of scaling methods available. Scaling generally gets applied to the entire dataset and is especially necessary for certain algorithms. K-means make use of euclidean distance hence the need for scaling. For PCA because we are trying to identify the feature with maximus variance we also need scaling. Similarly, we need scaled features for gradient descent. Any algorithm that is not based on a distance measure is not affected by feature scaling. Some of the methods include range scalers like minimum-maximum scaler, maximum absolute scaler or even standardisation methods like the standard scaler can be used for scaling. The example used here is robust scaler. Normalisation is a good technique when you don't know the distribution of the data. Scaling looks into the future, so parameters have to be training on a training set and applied to a test set.

(i) Robust Scaler

Scaling according to the interquartile range, making it robust to outliers.

def robust_scaler(df, drop=None,quantile_range=(25, 75) ):
    if drop:
      keep = df[drop]
      df = df.drop(drop, axis=1)
    center = np.median(df, axis=0)
    quantiles = np.percentile(df, quantile_range, axis=0)
    scale = quantiles[1] - quantiles[0]
    df = (df - center) / scale
    if drop:
      df = pd.concat((keep,df),axis=1)
    return df

df_out = transform.robust_scaler(df.copy(), drop=["Close_1"]); df_out.head()

(2) Standardisation

When using a standardisation method, it is often more effective when the attribute itself if Gaussian. It is also useful to apply the technique when the model you want to use makes assumptions of Gaussian distributions like linear regression, logistic regression, and linear discriminant analysis. For most applications, standardisation is recommended.

(i) Standard Scaler

Standardize features by removing the mean and scaling to unit variance

def standard_scaler(df,drop ):
    if drop:
      keep = df[drop]
      df = df.drop(drop, axis=1)
    mean = np.mean(df, axis=0)
    scale = np.std(df, axis=0)
    df = (df - mean) / scale  
    if drop:
      df = pd.concat((keep,df),axis=1)
    return df


df_out = transform.standard_scaler(df.copy(), drop=["Close"]); df_out.head()           

(3) Differencing

Computing the differences between consecutive observation, normally used to obtain a stationary time series.

(i) Fractional Differencing

Fractional differencing, allows us to achieve stationarity while maintaining the maximum amount of memory compared to integer differencing.

import pylab as pl

def fast_fracdiff(x, cols, d):
    for col in cols:
      T = len(x[col])
      np2 = int(2 ** np.ceil(np.log2(2 * T - 1)))
      k = np.arange(1, T)
      b = (1,) + tuple(np.cumprod((k - d - 1) / k))
      z = (0,) * (np2 - T)
      z1 = b + z
      z2 = tuple(x[col]) + z
      dx = pl.ifft(pl.fft(z1) * pl.fft(z2))
      x[col+"_frac"] = np.real(dx[0:T])
    return x 
  
df_out = transform.fast_fracdiff(df.copy(), ["Close","Open"],0.5); df_out.head()

(4) Capping

Any method that provides sets a floor and a cap to a feature's value. Capping can affect the distribution of data, so it should not be exagerated. One can cap values by using the average, by using the max and min values, or by an arbitrary extreme value.

(i) Winzorisation

The transformation of features by limiting extreme values in the statistical data to reduce the effect of possibly spurious outliers by replacing it with a certain percentile value.

def outlier_detect(data,col,threshold=1,method="IQR"):
  
    if method == "IQR":
      IQR = data[col].quantile(0.75) - data[col].quantile(0.25)
      Lower_fence = data[col].quantile(0.25) - (IQR * threshold)
      Upper_fence = data[col].quantile(0.75) + (IQR * threshold)
    if method == "STD":
      Upper_fence = data[col].mean() + threshold * data[col].std()
      Lower_fence = data[col].mean() - threshold * data[col].std()   
    if method == "OWN":
      Upper_fence = data[col].mean() + threshold * data[col].std()
      Lower_fence = data[col].mean() - threshold * data[col].std() 
    if method =="MAD":
      median = data[col].median()
      median_absolute_deviation = np.median([np.abs(y - median) for y in data[col]])
      modified_z_scores = pd.Series([0.6745 * (y - median) / median_absolute_deviation for y in data[col]])
      outlier_index = np.abs(modified_z_scores) > threshold
      print('Num of outlier detected:',outlier_index.value_counts()[1])
      print('Proportion of outlier detected',outlier_index.value_counts()[1]/len(outlier_index))
      return outlier_index, (median_absolute_deviation, median_absolute_deviation)

    para = (Upper_fence, Lower_fence)
    tmp = pd.concat([data[col]>Upper_fence,data[col]<Lower_fence],axis=1)
    outlier_index = tmp.any(axis=1)
    print('Num of outlier detected:',outlier_index.value_counts()[1])
    print('Proportion of outlier detected',outlier_index.value_counts()[1]/len(outlier_index))
    
    return outlier_index, para

def windsorization(data,col,para,strategy='both'):
    """
    top-coding & bottom coding (capping the maximum of a distribution at an arbitrarily set value,vice versa)
    """

    data_copy = data.copy(deep=True)  
    if strategy == 'both':
        data_copy.loc[data_copy[col]>para[0],col] = para[0]
        data_copy.loc[data_copy[col]<para[1],col] = para[1]
    elif strategy == 'top':
        data_copy.loc[data_copy[col]>para[0],col] = para[0]
    elif strategy == 'bottom':
        data_copy.loc[data_copy[col]<para[1],col] = para[1]  
    return data_copy

_, para = transform.outlier_detect(df, "Close")
df_out = transform.windsorization(df.copy(),"Close",para,strategy='both'); df_out.head()

(5) Operations

Operations here are treated like traditional transformations. It is the replacement of a variable by a function of that variable. In a stronger sense, a transformation is a replacement that changes the shape of a distribution or relationship.

(i) Power, Log, Recipricol, Square Root

def operations(df,features):
  df_new = df[features]
  df_new = df_new - df_new.min()

  sqr_name = [str(fa)+"_POWER_2" for fa in df_new.columns]
  log_p_name = [str(fa)+"_LOG_p_one_abs" for fa in df_new.columns]
  rec_p_name = [str(fa)+"_RECIP_p_one" for fa in df_new.columns]
  sqrt_name = [str(fa)+"_SQRT_p_one" for fa in df_new.columns]

  df_sqr = pd.DataFrame(np.power(df_new.values, 2),columns=sqr_name, index=df.index)
  df_log = pd.DataFrame(np.log(df_new.add(1).abs().values),columns=log_p_name, index=df.index)
  df_rec = pd.DataFrame(np.reciprocal(df_new.add(1).values),columns=rec_p_name, index=df.index)
  df_sqrt = pd.DataFrame(np.sqrt(df_new.abs().add(1).values),columns=sqrt_name, index=df.index)

  dfs = [df, df_sqr, df_log, df_rec, df_sqrt]

  df=  pd.concat(dfs, axis=1)

  return df

df_out = transform.operations(df.copy(),["Close"]); df_out.head()

(6) Smoothing

Here we maintain that any method that has a component of historical averaging is a smoothing method such as a simple moving average and single, double and tripple exponential smoothing methods. These forms of non-causal filters are also popular in signal processing and are called filters, where exponential smoothing is called an IIR filter and a moving average a FIR filter with equal weighting factors.

(i) Tripple Exponential Smoothing (Holt-Winters Exponential Smoothing)

The Holt-Winters seasonal method comprises the forecast equation and three smoothing equations — one for the level $ℓt$, one for the trend &bt&, and one for the seasonal component $st$. This particular version is performed by looking at the last 12 periods. For that reason, the first 12 records should be disregarded because they can't make use of the required window size for a fair calculation. The calculation is such that values are still provided for those periods based on whatever data might be available.

def initial_trend(series, slen):
    sum = 0.0
    for i in range(slen):
        sum += float(series[i+slen] - series[i]) / slen
    return sum / slen

def initial_seasonal_components(series, slen):
    seasonals = {}
    season_averages = []
    n_seasons = int(len(series)/slen)
    # compute season averages
    for j in range(n_seasons):
        season_averages.append(sum(series[slen*j:slen*j+slen])/float(slen))
    # compute initial values
    for i in range(slen):
        sum_of_vals_over_avg = 0.0
        for j in range(n_seasons):
            sum_of_vals_over_avg += series[slen*j+i]-season_averages[j]
        seasonals[i] = sum_of_vals_over_avg/n_seasons
    return seasonals

def triple_exponential_smoothing(df,cols, slen, alpha, beta, gamma, n_preds):
    for col in cols:
      result = []
      seasonals = initial_seasonal_components(df[col], slen)
      for i in range(len(df[col])+n_preds):
          if i == 0: # initial values
              smooth = df[col][0]
              trend = initial_trend(df[col], slen)
              result.append(df[col][0])
              continue
          if i >= len(df[col]): # we are forecasting
              m = i - len(df[col]) + 1
              result.append((smooth + m*trend) + seasonals[i%slen])
          else:
              val = df[col][i]
              last_smooth, smooth = smooth, alpha*(val-seasonals[i%slen]) + (1-alpha)*(smooth+trend)
              trend = beta * (smooth-last_smooth) + (1-beta)*trend
              seasonals[i%slen] = gamma*(val-smooth) + (1-gamma)*seasonals[i%slen]
              result.append(smooth+trend+seasonals[i%slen])
      df[col+"_TES"] = result
    #print(seasonals)
    return df

df_out= transform.triple_exponential_smoothing(df.copy(),["Close"], 12, .2,.2,.2,0); df_out.head()

(7) Decomposing

Decomposition procedures are used in time series to describe the trend and seasonal factors in a time series. More extensive decompositions might also include long-run cycles, holiday effects, day of week effects and so on. Here, we’ll only consider trend and seasonal decompositions. A naive decomposition makes use of moving averages, other decomposition methods are available that make use of LOESS.

(i) Naive Decomposition

The base trend takes historical information into account and established moving averages; it does not have to be linear. To estimate the seasonal component for each season, simply average the detrended values for that season. If the seasonal variation looks constant, we should use the additive model. If the magnitude is increasing as a function of time, we will use multiplicative. Here because it is predictive in nature we are using a one sided moving average, as opposed to a two-sided centred average.

import statsmodels.api as sm

def naive_dec(df, columns, freq=2):
  for col in columns:
    decomposition = sm.tsa.seasonal_decompose(df[col], model='additive', freq = freq, two_sided=False)
    df[col+"_NDDT" ] = decomposition.trend
    df[col+"_NDDT"] = decomposition.seasonal
    df[col+"_NDDT"] = decomposition.resid
  return df

df_out = transform.naive_dec(df.copy(), ["Close","Open"]); df_out.head()

(8) Filtering

It is often useful to either low-pass filter (smooth) time series in order to reveal low-frequency features and trends, or to high-pass filter (detrend) time series in order to isolate high frequency transients (e.g. storms). Low pass filters use historical values, high-pass filters detrends with low-pass filters, so also indirectly uses historical values.

There are a few filters available, closely associated with decompositions and smoothing functions. The Hodrick-Prescott filter separates a time-series $yt$ into a trend $τt$ and a cyclical component $ζt$. The Christiano-Fitzgerald filter is a generalization of Baxter-King filter and can be seen as weighted moving average.

(i) Baxter-King Bandpass

The Baxter-King filter is intended to explicitly deal with the periodicity of the business cycle. By applying their band-pass filter to a series, they produce a new series that does not contain fluctuations at higher or lower than those of the business cycle. The parameters are arbitrarily chosen. This method uses a centred moving average that has to be changed to a lagged moving average before it can be used as an input feature. The maximum period of oscillation should be used as the point to truncate the dataset, as that part of the time series does not incorporate all the required datapoints.

import statsmodels.api as sm

def bkb(df, cols):
  for col in cols:
    df[col+"_BPF"] = sm.tsa.filters.bkfilter(df[[col]].values, 2, 10, len(df)-1)
  return df

df_out = transform.bkb(df.copy(), ["Close"]); df_out.head()

(ii) Butter Lowpass (IIR Filter Design)

The Butterworth filter is a type of signal processing filter designed to have a frequency response as flat as possible in the passban. Like other filtersm the first few values have to be disregarded for accurate downstream prediction. Instead of disregarding these values on a per case basis, they can be diregarded in one chunk once the database of transformed features have been developed.

from scipy import signal, integrate
def butter_lowpass(cutoff, fs=20, order=5):
    nyq = 0.5 * fs
    normal_cutoff = cutoff / nyq
    b, a = signal.butter(order, normal_cutoff, btype='low', analog=False)
    return b, a
    
def butter_lowpass_filter(df,cols, cutoff, fs=20, order=5):
    b, a = butter_lowpass(cutoff, fs, order=order)
    for col in cols:
      df[col+"_BUTTER"] = signal.lfilter(b, a, df[col])
    return df

df_out = transform.butter_lowpass_filter(df.copy(),["Close"],4); df_out.head()

(iii) Hilbert Transform Angle

The Hilbert transform is a time-domain to time-domain transformation which shifts the phase of a signal by 90 degrees. It is also a centred measure and would be difficult to use in a time series prediction setting, unless it is recalculated on a per step basis or transformed to be based on historical values only.

from scipy import signal
import numpy as np

def instantaneous_phases(df,cols):
    for col in cols:
      df[col+"_HILLB"] = np.unwrap(np.angle(signal.hilbert(df[col], axis=0)), axis=0)
    return df

df_out = transform.instantaneous_phases(df.copy(), ["Close"]); df_out.head()

(iiiv) Unscented Kalman Filter

The Kalman filter is better suited for estimating things that change over time. The most tangible example is tracking moving objects. A Kalman filter will be very close to the actual trajectory because it says the most recent measurement is more important than the older ones. The Unscented Kalman Filter (UKF) is a model based-techniques that recursively estimates the states (and with some modifications also parameters) of a nonlinear, dynamic, discrete-time system. The UKF is based on the typical prediction-correction style methods. The Kalman Smoother incorporates future values, the Filter doesn't and can be used for online prediction. The normal Kalman filter is a forward filter in the sense that it makes forecast of the current state using only current and past observations, whereas the smoother is based on computing a suitable linear combination of two filters, which are ran in forward and backward directions.

from pykalman import UnscentedKalmanFilter

def kalman_feat(df, cols):
  for col in cols:
    ukf = UnscentedKalmanFilter(lambda x, w: x + np.sin(w), lambda x, v: x + v, observation_covariance=0.1)
    (filtered_state_means, filtered_state_covariances) = ukf.filter(df[col])
    (smoothed_state_means, smoothed_state_covariances) = ukf.smooth(df[col])
    df[col+"_UKFSMOOTH"] = smoothed_state_means.flatten()
    df[col+"_UKFFILTER"] = filtered_state_means.flatten()
  return df 

df_out = transform.kalman_feat(df.copy(), ["Close"]); df_out.head()

(9) Spectral Analysis

There are a range of functions for spectral analysis. You can use periodograms and the welch method to estimate the power spectral density. You can also use the welch method to estimate the cross power spectral density. Other techniques include spectograms, Lomb-Scargle periodograms and, short time fourier transform.

(i) Periodogram

This returns an array of sample frequencies and the power spectrum of x, or the power spectral density of x.

from scipy import signal
def perd_feat(df, cols):
  for col in cols:
    sig = signal.periodogram(df[col],fs=1, return_onesided=False)
    df[col+"_FREQ"] = sig[0]
    df[col+"_POWER"] = sig[1]
  return df

df_out = transform.perd_feat(df.copy(),["Close"]); df_out.head()

(ii) Fast Fourier Transform

The FFT, or fast fourier transform is an algorithm that essentially uses convolution techniques to efficiently find the magnitude and location of the tones that make up the signal of interest. We can often play with the FFT spectrum, by adding and removing successive tones (which is akin to selectively filtering particular tones that make up the signal), in order to obtain a smoothed version of the underlying signal. This takes the entire signal into account, and as a result has to be recalculated on a running basis to avoid peaking into the future.

def fft_feat(df, cols):
  for col in cols:
    fft_df = np.fft.fft(np.asarray(df[col].tolist()))
    fft_df = pd.DataFrame({'fft':fft_df})
    df[col+'_FFTABS'] = fft_df['fft'].apply(lambda x: np.abs(x)).values
    df[col+'_FFTANGLE'] = fft_df['fft'].apply(lambda x: np.angle(x)).values
  return df 

df_out = transform.fft_feat(df.copy(), ["Close"]); df_out.head()

(10) Waveforms

The waveform of a signal is the shape of its graph as a function of time.

(i) Continuous Wave Radar

from scipy import signal
def harmonicradar_cw(df, cols, fs,fc):
    for col in cols:
      ttxt = f'CW: {fc} Hz'
      #%% input
      t = df[col]
      tx = np.sin(2*np.pi*fc*t)
      _,Pxx = signal.welch(tx,fs)
      #%% diode
      d = (signal.square(2*np.pi*fc*t))
      d[d<0] = 0.
      #%% output of diode
      rx = tx * d
      df[col+"_HARRAD"] = rx.values
    return df

df_out = transform.harmonicradar_cw(df.copy(), ["Close"],0.3,0.2); df_out.head()

(ii) Saw Tooth

Return a periodic sawtooth or triangle waveform.

def saw(df, cols):
  for col in cols:
    df[col+" SAW"] = signal.sawtooth(df[col])
  return df

df_out = transform.saw(df.copy(),["Close","Open"]); df_out.head()

(9) Modifications

A range of modification usually applied ot images, these values would have to be recalculate for each time-series.

(i) Various Techniques

from tsaug import *
def modify(df, cols):
  for col in cols:
    series = df[col].values
    df[col+"_magnify"], _ = magnify(series, series)
    df[col+"_affine"], _ = affine(series, series)
    df[col+"_crop"], _ = crop(series, series)
    df[col+"_cross_sum"], _ = cross_sum(series, series)
    df[col+"_resample"], _ = resample(series, series)
    df[col+"_trend"], _ = trend(series, series)

    df[col+"_random_affine"], _ = random_time_warp(series, series)
    df[col+"_random_crop"], _ = random_crop(series, series)
    df[col+"_random_cross_sum"], _ = random_cross_sum(series, series)
    df[col+"_random_sidetrack"], _ = random_sidetrack(series, series)
    df[col+"_random_time_warp"], _ = random_time_warp(series, series)
    df[col+"_random_magnify"], _ = random_magnify(series, series)
    df[col+"_random_jitter"], _ = random_jitter(series, series)
    df[col+"_random_trend"], _ = random_trend(series, series)
  return df

df_out = transform.modify(df.copy(),["Close"]); df_out.head()

(11) Rolling

Features that are calculated on a rolling basis over fixed window size.

(i) Mean, Standard Deviation

def multiple_rolling(df, windows = [1,2], functions=["mean","std"], columns=None):
  windows = [1+a for a in windows]
  if not columns:
    columns = df.columns.to_list()
  rolling_dfs = (df[columns].rolling(i)                                    # 1. Create window
                  .agg(functions)                                # 1. Aggregate
                  .rename({col: '{0}_{1:d}'.format(col, i)
                                for col in columns}, axis=1)  # 2. Rename columns
                for i in windows)                                # For each window
  df_out = pd.concat((df, *rolling_dfs), axis=1)
  da = df_out.iloc[:,len(df.columns):]
  da = [col[0] + "_" + col[1] for col in  da.columns.to_list()]
  df_out.columns = df.columns.to_list() + da 

  return  df_out                      # 3. Concatenate dataframes

df_out = transform.multiple_rolling(df, columns=["Close"]); df_out.head()

(12) Lagging

Lagged values from existing features.

(i) Single Steps

def multiple_lags(df, start=1, end=3,columns=None):
  if not columns:
    columns = df.columns.to_list()
  lags = range(start, end+1)  # Just two lags for demonstration.

  df = df.assign(**{
      '{}_t_{}'.format(col, t): df[col].shift(t)
      for t in lags
      for col in columns
  })
  return df

df_out = transform.multiple_lags(df, start=1, end=3, columns=["Close"]); df_out.head()

(13) Forecast Model

There are a range of time series model that can be implemented like AR, MA, ARMA, ARIMA, SARIMA, SARIMAX, VAR, VARMA, VARMAX, SES, and HWES. The models can be divided into autoregressive models and smoothing models. In an autoregression model, we forecast the variable of interest using a linear combination of past values of the variable. Each method might requre specific tuning and parameters to suit your prediction task. You need to drop a certain amount of historical data that you use during the fitting stage. Models that take seasonality into account need more training data.

(i) Prophet

Prophet is a procedure for forecasting time series data based on an additive model where non-linear trends are fit with yearly, weekly, and daily seasonality. You can apply additive models to your training data but also interactive models like deep learning models. The problem is that because these models have learned from future observations, there would this be a need to recalculate the time series on a running basis, or to only include the predicted as opposed to fitted values in future training and test sets. In this example, I train on 150 data points to illustrate how the remaining or so 100 datapoints can be used in a new prediction problem. You can plot with df["PROPHET"].plot() to see the effect.

You can apply additive models to your training data but also interactive models like deep learning models. The problem is that these models have learned from future observations, there would this be a need to recalculate the time series on a running basis, or to only include the predicted as opposed to fitted values in future training and test sets.

from fbprophet import Prophet

def prophet_feat(df, cols,date, freq,train_size=150):
  def prophet_dataframe(df): 
    df.columns = ['ds','y']
    return df

  def original_dataframe(df, freq, name):
    prophet_pred = pd.DataFrame({"Date" : df['ds'], name : df["yhat"]})
    prophet_pred = prophet_pred.set_index("Date")
    #prophet_pred.index.freq = pd.tseries.frequencies.to_offset(freq)
    return prophet_pred[name].values

  for col in cols:
    model = Prophet(daily_seasonality=True)
    fb = model.fit(prophet_dataframe(df[[date, col]].head(train_size)))
    forecast_len = len(df) - train_size
    future = model.make_future_dataframe(periods=forecast_len,freq=freq)
    future_pred = model.predict(future)
    df[col+"_PROPHET"] = list(original_dataframe(future_pred,freq,col))
  return df

df_out  = transform.prophet_feat(df.copy().reset_index(),["Close","Open"],"Date", "D"); df_out.head()

 

(2) Interaction

Interactions are defined as methods that require more than one feature to create an additional feature. Here we include normalising and discretising techniques that are non-feature specific. Almost all of these method can be applied to cross-section method. The only methods that are time specific is the technical features in the speciality section and the autoregression model.

(1) Regression

Regression analysis is a set of statistical processes for estimating the relationships between a dependent variable (often called the 'outcome variable') and one or more independent variables.

(i) Lowess Smoother

The lowess smoother is a robust locally weighted regression. The function fits a nonparametric regression curve to a scatterplot.

from math import ceil
import numpy as np
from scipy import linalg
import math

def lowess(df, cols, y, f=2. / 3., iter=3):
    for col in cols:
      n = len(df[col])
      r = int(ceil(f * n))
      h = [np.sort(np.abs(df[col] - df[col][i]))[r] for i in range(n)]
      w = np.clip(np.abs((df[col][:, None] - df[col][None, :]) / h), 0.0, 1.0)
      w = (1 - w ** 3) ** 3
      yest = np.zeros(n)
      delta = np.ones(n)
      for iteration in range(iter):
          for i in range(n):
              weights = delta * w[:, i]
              b = np.array([np.sum(weights * y), np.sum(weights * y * df[col])])
              A = np.array([[np.sum(weights), np.sum(weights * df[col])],
                            [np.sum(weights * df[col]), np.sum(weights * df[col] * df[col])]])
              beta = linalg.solve(A, b)
              yest[i] = beta[0] + beta[1] * df[col][i]

          residuals = y - yest
          s = np.median(np.abs(residuals))
          delta = np.clip(residuals / (6.0 * s), -1, 1)
          delta = (1 - delta ** 2) ** 2
      df[col+"_LOWESS"] = yest

    return df

df_out = interact.lowess(df.copy(), ["Open","Volume"], df["Close"], f=0.25, iter=3); df_out.head()

Autoregression

Autoregression is a time series model that uses observations from previous time steps as input to a regression equation to predict the value at the next time step

from statsmodels.tsa.ar_model import AR
from timeit import default_timer as timer
def autoregression(df, drop=None, settings={"autoreg_lag":4}):

    autoreg_lag = settings["autoreg_lag"]
    if drop:
      keep = df[drop]
      df = df.drop([drop],axis=1).values

    n_channels = df.shape[0]
    t = timer()
    channels_regg = np.zeros((n_channels, autoreg_lag + 1))
    for i in range(0, n_channels):
        fitted_model = AR(df.values[i, :]).fit(autoreg_lag)
        # TODO: This is not the same as Matlab's for some reasons!
        # kk = ARMAResults(fitted_model)
        # autore_vals, dummy1, dummy2 = arburg(x[i, :], autoreg_lag) # This looks like Matlab's but slow
        channels_regg[i, 0: len(fitted_model.params)] = np.real(fitted_model.params)

    for i in range(channels_regg.shape[1]):
      df["LAG_"+str(i+1)] = channels_regg[:,i]
    
    if drop:
      df = pd.concat((keep,df),axis=1)

    t = timer() - t
    return df

df_out = interact.autoregression(df.copy()); df_out.head()

(2) Operator

Looking at interaction between different features. Here the methods employed are multiplication and division.

(i) Multiplication and Division

def muldiv(df, feature_list):
  for feat in feature_list:
    for feat_two in feature_list:
      if feat==feat_two:
        continue
      else:
       df[feat+"/"+feat_two] = df[feat]/(df[feat_two]-df[feat_two].min()) #zero division guard
       df[feat+"_X_"+feat_two] = df[feat]*(df[feat_two])

  return df

df_out = interact.muldiv(df.copy(), ["Close","Open"]); df_out.head()

(3) Discretising

In statistics and machine learning, discretization refers to the process of converting or partitioning continuous attributes, features or variables to discretized or nominal attributes

(i) Decision Tree Discretiser

The first method that will be applies here is a supersived discretiser. Discretisation with Decision Trees consists of using a decision tree to identify the optimal splitting points that would determine the bins or contiguous intervals.

from sklearn.tree import DecisionTreeRegressor

def decision_tree_disc(df, cols, depth=4 ):
  for col in cols:
    df[col +"_m1"] = df[col].shift(1)
    df = df.iloc[1:,:]
    tree_model = DecisionTreeRegressor(max_depth=depth,random_state=0)
    tree_model.fit(df[col +"_m1"].to_frame(), df[col])
    df[col+"_Disc"] = tree_model.predict(df[col +"_m1"].to_frame())
  return df

df_out = interact.decision_tree_disc(df.copy(), ["Close"]); df_out.head()

(4) Normalising

Normalising normally pertains to the scaling of data. There are many method available, interacting normalising methods makes use of all the feature's attributes to do the scaling.

(i) Quantile Normalisation

In statistics, quantile normalization is a technique for making two distributions identical in statistical properties.

import numpy as np
import pandas as pd

def quantile_normalize(df, drop):

    if drop:
      keep = df[drop]
      df = df.drop(drop,axis=1)

    #compute rank
    dic = {}
    for col in df:
      dic.update({col : sorted(df[col])})
    sorted_df = pd.DataFrame(dic)
    rank = sorted_df.mean(axis = 1).tolist()
    #sort
    for col in df:
        t = np.searchsorted(np.sort(df[col]), df[col])
        df[col] = [rank[i] for i in t]
    
    if drop:
      df = pd.concat((keep,df),axis=1)
    return df

df_out = interact.quantile_normalize(df.copy(), drop=["Close"]); df_out.head()

(5) Distance

There are multiple types of distance functions like Euclidean, Mahalanobis, and Minkowski distance. Here we are using a contrived example in a location based haversine distance.

(i) Haversine Distance

The Haversine (or great circle) distance is the angular distance between two points on the surface of a sphere.

from math import sin, cos, sqrt, atan2, radians
def haversine_distance(row, lon="Open", lat="Close"):
    c_lat,c_long = radians(52.5200), radians(13.4050)
    R = 6373.0
    long = radians(row['Open'])
    lat = radians(row['Close'])
    
    dlon = long - c_long
    dlat = lat - c_lat
    a = sin(dlat / 2)**2 + cos(lat) * cos(c_lat) * sin(dlon / 2)**2
    c = 2 * atan2(sqrt(a), sqrt(1 - a))
    
    return R * c

df_out['distance_central'] = df.apply(interact.haversine_distance,axis=1); df_out.head()

(6) Speciality

(i) Technical Features

Technical indicators are heuristic or mathematical calculations based on the price, volume, or open interest of a security or contract used by traders who follow technical analysis. By analyzing historical data, technical analysts use indicators to predict future price movements.

import ta

def tech(df):
  return ta.add_all_ta_features(df, open="Open", high="High", low="Low", close="Close", volume="Volume")
  
df_out = interact.tech(df.copy()); df_out.head()

(7) Genetic

Genetic programming has shown promise in constructing feature by osing original features to form high-level ones that can help algorithms achieve better performance.

(i) Symbolic Transformer

A symbolic transformer is a supervised transformer that begins by building a population of naive random formulas to represent a relationship.

df.head()
from gplearn.genetic import SymbolicTransformer

def genetic_feat(df, num_gen=20, num_comp=10):
  function_set = ['add', 'sub', 'mul', 'div',
                  'sqrt', 'log', 'abs', 'neg', 'inv','tan']

  gp = SymbolicTransformer(generations=num_gen, population_size=200,
                          hall_of_fame=100, n_components=num_comp,
                          function_set=function_set,
                          parsimony_coefficient=0.0005,
                          max_samples=0.9, verbose=1,
                          random_state=0, n_jobs=6)

  gen_feats = gp.fit_transform(df.drop("Close_1", axis=1), df["Close_1"]); df.iloc[:,:8]
  gen_feats = pd.DataFrame(gen_feats, columns=["gen_"+str(a) for a in range(gen_feats.shape[1])])
  gen_feats.index = df.index
  return pd.concat((df,gen_feats),axis=1)

df_out = interact.genetic_feat(df.copy()); df_out.head()

 

(3) Mapping

Methods that help with the summarisation of features by remapping them to achieve some aim like the maximisation of variability or class separability. These methods tend to be unsupervised, but can also take an supervised form.

(1) Eigen Decomposition

Eigendecomposition or sometimes spectral decomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Some examples are LDA and PCA.

(i) Principal Component Analysis

Principal component analysis (PCA) is a statistical procedure that uses an orthogonal transformation to convert a set of observations of possibly correlated variables into a set of values of linearly uncorrelated variables called principal components.

def pca_feature(df, memory_issues=False,mem_iss_component=False,variance_or_components=0.80,n_components=5 ,drop_cols=None, non_linear=True):
    
  if non_linear:
    pca = KernelPCA(n_components = n_components, kernel='rbf', fit_inverse_transform=True, random_state = 33, remove_zero_eig= True)
  else:
    if memory_issues:
      if not mem_iss_component:
        raise ValueError("If you have memory issues, you have to preselect mem_iss_component")
      pca = IncrementalPCA(mem_iss_component)
    else:
      if variance_or_components>1:
        pca = PCA(n_components=variance_or_components) 
      else: # automated selection based on variance
        pca = PCA(n_components=variance_or_components,svd_solver="full") 
  if drop_cols:
    X_pca = pca.fit_transform(df.drop(drop_cols,axis=1))
    return pd.concat((df[drop_cols],pd.DataFrame(X_pca, columns=["PCA_"+str(i+1) for i in range(X_pca.shape[1])],index=df.index)),axis=1)

  else:
    X_pca = pca.fit_transform(df)
    return pd.DataFrame(X_pca, columns=["PCA_"+str(i+1) for i in range(X_pca.shape[1])],index=df.index)


  return df

df_out = mapper.pca_feature(df.copy(), variance_or_components=0.9, n_components=8,non_linear=False)

(2) Cross Decomposition

These families of algorithms are useful to find linear relations between two multivariate datasets.

(1) Canonical Correlation Analysis

Canonical-correlation analysis (CCA) is a way of inferring information from cross-covariance matrices.

from sklearn.cross_decomposition import CCA

def cross_lag(df, drop=None, lags=1, components=4 ):

  if drop:
    keep = df[drop]
    df = df.drop([drop],axis=1)

  df_2 = df.shift(lags)
  df = df.iloc[lags:,:]
  df_2 = df_2.dropna().reset_index(drop=True)

  cca = CCA(n_components=components)
  cca.fit(df_2, df)

  X_c, df_2 = cca.transform(df_2, df)
  df_2 = pd.DataFrame(df_2, index=df.index)
  df_2 = df.add_prefix('crd_')

  if drop:
    df = pd.concat([keep,df,df_2],axis=1)
  else:
    df = pd.concat([df,df_2],axis=1)
  return df

df_out = mapper.cross_lag(df.copy()); df_out.head()

(3) Kernel Approximation

Functions that approximate the feature mappings that correspond to certain kernels, as they are used for example in support vector machines.

(i) Additive Chi2 Kernel

Computes the additive chi-squared kernel between observations in X and Y The chi-squared kernel is computed between each pair of rows in X and Y. X and Y have to be non-negative.

from sklearn.kernel_approximation import AdditiveChi2Sampler

def a_chi(df, drop=None, lags=1, sample_steps=2 ):

  if drop:
    keep = df[drop]
    df = df.drop([drop],axis=1)

  df_2 = df.shift(lags)
  df = df.iloc[lags:,:]
  df_2 = df_2.dropna().reset_index(drop=True)

  chi2sampler = AdditiveChi2Sampler(sample_steps=sample_steps)

  df_2 = chi2sampler.fit_transform(df_2, df["Close"])

  df_2 = pd.DataFrame(df_2, index=df.index)
  df_2 = df.add_prefix('achi_')

  if drop:
    df = pd.concat([keep,df,df_2],axis=1)
  else:
    df = pd.concat([df,df_2],axis=1)
  return df

df_out = mapper.a_chi(df.copy()); df_out.head()

(4) Autoencoder

An autoencoder is a type of artificial neural network used to learn efficient data codings in an unsupervised manner. The aim of an autoencoder is to learn a representation (encoding) for a set of data, typically for dimensionality reduction, by training the network to ignore noise.

(i) Feed Forward

The simplest form of an autoencoder is a feedforward, non-recurrent neural network similar to single layer perceptrons that participate in multilayer perceptrons

from sklearn.preprocessing import minmax_scale
import tensorflow as tf
import numpy as np

def encoder_dataset(df, drop=None, dimesions=20):

  if drop:
    train_scaled = minmax_scale(df.drop(drop,axis=1).values, axis = 0)
  else:
    train_scaled = minmax_scale(df.values, axis = 0)

  # define the number of encoding dimensions
  encoding_dim = dimesions
  # define the number of features
  ncol = train_scaled.shape[1]
  input_dim = tf.keras.Input(shape = (ncol, ))

  # Encoder Layers
  encoded1 = tf.keras.layers.Dense(3000, activation = 'relu')(input_dim)
  encoded2 = tf.keras.layers.Dense(2750, activation = 'relu')(encoded1)
  encoded3 = tf.keras.layers.Dense(2500, activation = 'relu')(encoded2)
  encoded4 = tf.keras.layers.Dense(750, activation = 'relu')(encoded3)
  encoded5 = tf.keras.layers.Dense(500, activation = 'relu')(encoded4)
  encoded6 = tf.keras.layers.Dense(250, activation = 'relu')(encoded5)
  encoded7 = tf.keras.layers.Dense(encoding_dim, activation = 'relu')(encoded6)

  encoder = tf.keras.Model(inputs = input_dim, outputs = encoded7)
  encoded_input = tf.keras.Input(shape = (encoding_dim, ))

  encoded_train = pd.DataFrame(encoder.predict(train_scaled),index=df.index)
  encoded_train = encoded_train.add_prefix('encoded_')
  if drop:
    encoded_train = pd.concat((df[drop],encoded_train),axis=1)

  return encoded_train

df_out = mapper.encoder_dataset(df.copy(), ["Close_1"], 15); df_out.head()
df_out.head()

(5) Manifold Learning

Manifold Learning can be thought of as an attempt to generalize linear frameworks like PCA to be sensitive to non-linear structure in data.

(i) Local Linear Embedding

Locally Linear Embedding is a method of non-linear dimensionality reduction. It tries to reduce these n-Dimensions while trying to preserve the geometric features of the original non-linear feature structure.

from sklearn.manifold import LocallyLinearEmbedding

def lle_feat(df, drop=None, components=4):

  if drop:
    keep = df[drop]
    df = df.drop(drop, axis=1)

  embedding = LocallyLinearEmbedding(n_components=components)
  em = embedding.fit_transform(df)
  df = pd.DataFrame(em,index=df.index)
  df = df.add_prefix('lle_')
  if drop:
    df = pd.concat((keep,df),axis=1)
  return df

df_out = mapper.lle_feat(df.copy(),["Close_1"],4); df_out.head()

(6) Clustering

Most clustering techniques start with a bottom up approach: each observation starts in its own cluster, and clusters are successively merged together with some measure. Although these clustering techniques are typically used for observations, it can also be used for feature dimensionality reduction; especially hierarchical clustering techniques.

(i) Feature Agglomeration

Feature agglomerative uses clustering to group together features that look very similar, thus decreasing the number of features.

import numpy as np
from sklearn import datasets, cluster

def feature_agg(df, drop=None, components=4):

  if drop:
    keep = df[drop]
    df = df.drop(drop, axis=1)

  components = min(df.shape[1]-1,components)
  agglo = cluster.FeatureAgglomeration(n_clusters=components)
  agglo.fit(df)
  df = pd.DataFrame(agglo.transform(df),index=df.index)
  df = df.add_prefix('feagg_')

  if drop:
    return pd.concat((keep,df),axis=1)
  else:
    return df


df_out = mapper.feature_agg(df.copy(),["Close_1"],4 ); df_out.head()

(7) Neigbouring

Neighbouring points can be calculated using distance metrics like Hamming, Manhattan, Minkowski distance. The principle behind nearest neighbor methods is to find a predefined number of training samples closest in distance to the new point, and predict the label from these.

(i) Nearest Neighbours

Unsupervised learner for implementing neighbor searches.

from sklearn.neighbors import NearestNeighbors

def neigh_feat(df, drop, neighbors=6):
  
  if drop:
    keep = df[drop]
    df = df.drop(drop, axis=1)

  components = min(df.shape[0]-1,neighbors)
  neigh = NearestNeighbors(n_neighbors=neighbors)
  neigh.fit(df)
  neigh = neigh.kneighbors()[0]
  df = pd.DataFrame(neigh, index=df.index)
  df = df.add_prefix('neigh_')

  if drop:
    return pd.concat((keep,df),axis=1)
  else:
    return df

  return df

df_out = mapper.neigh_feat(df.copy(),["Close_1"],4 ); df_out.head()

 

(4) Extraction

When working with extraction, you have decide the size of the time series history to take into account when calculating a collection of walk-forward feature values. To facilitate our extraction, we use an excellent package called TSfresh, and also some of their default features. For completeness, we also include 12 or so custom features to be added to the extraction pipeline.

The time series methods in the transformation section and the interaction section are similar to the methods we will uncover in the extraction section, however, for transformation and interaction methods the output is an entire new time series, whereas extraction methods takes as input multiple constructed time series and extracts a singular value from each time series to reconstruct an entirely new time series.

Some methods naturally fit better in one format over another, e.g., lags are too expensive for extraction; time series decomposition only has to be performed once, because it has a low level of 'leakage' so is better suited to transformation; and forecast methods attempt to predict multiple future training samples, so won't work with extraction that only delivers one value per time series. Furthermore all non time-series (cross-sectional) transformation and extraction techniques can not make use of extraction as it is solely a time-series method.

Lastly, when we want to double apply specific functions we can apply it as a transformation/interaction then all the extraction methods can be applied to this feature as well. For example, if we calculate a smoothing function (transformation) then all other extraction functions (median, entropy, linearity etc.) can now be applied to that smoothing function, including the application of the smoothing function itself, e.g., a double smooth, double lag, double filter etc. So separating these methods out give us great flexibility.

Decorator

def set_property(key, value):
    """
    This method returns a decorator that sets the property key of the function to value
    """
    def decorate_func(func):
        setattr(func, key, value)
        if func.__doc__ and key == "fctype":
            func.__doc__ = func.__doc__ + "\n\n    *This function is of type: " + value + "*\n"
        return func
    return decorate_func

(1) Energy

You can calculate the linear, non-linear and absolute energy of a time series. In signal processing, the energy $E_S$ of a continuous-time signal $x(t)$ is defined as the area under the squared magnitude of the considered signal. Mathematically, $E_{s}=\langle x(t), x(t)\rangle=\int_{-\infty}^{\infty}|x(t)|^{2} d t$

(i) Absolute Energy

Returns the absolute energy of the time series which is the sum over the squared values

#-> In Package
def abs_energy(x):

    if not isinstance(x, (np.ndarray, pd.Series)):
        x = np.asarray(x)
    return np.dot(x, x)

extract.abs_energy(df["Close"])

(2) Distance

Here we widely define distance measures as those that take a difference between attributes or series of datapoints.

(i) Complexity-Invariant Distance

This function calculator is an estimate for a time series complexity.

#-> In Package
def cid_ce(x, normalize):

    if not isinstance(x, (np.ndarray, pd.Series)):
        x = np.asarray(x)
    if normalize:
        s = np.std(x)
        if s!=0:
            x = (x - np.mean(x))/s
        else:
            return 0.0

    x = np.diff(x)
    return np.sqrt(np.dot(x, x))

extract.cid_ce(df["Close"], True)

(3) Differencing

Many alternatives to differencing exists, one can for example take the difference of every other value, take the squared difference, take the fractional difference, or like our example, take the mean absolute difference.

(i) Mean Absolute Change

Returns the mean over the absolute differences between subsequent time series values.

#-> In Package
def mean_abs_change(x):
    return np.mean(np.abs(np.diff(x)))

extract.mean_abs_change(df["Close"])

(4) Derivative

Features where the emphasis is on the rate of change.

(i) Mean Central Second Derivative

Returns the mean value of a central approximation of the second derivative

#-> In Package
def _roll(a, shift):
    if not isinstance(a, np.ndarray):
        a = np.asarray(a)
    idx = shift % len(a)
    return np.concatenate([a[-idx:], a[:-idx]])

def mean_second_derivative_central(x):

    diff = (_roll(x, 1) - 2 * np.array(x) + _roll(x, -1)) / 2.0
    return np.mean(diff[1:-1])

extract.mean_second_derivative_central(df["Close"])

(5) Volatility

Volatility is a statistical measure of the dispersion of a time-series.

(i) Variance Larger than Standard Deviation

#-> In Package
def variance_larger_than_standard_deviation(x):

    y = np.var(x)
    return y > np.sqrt(y)

extract.variance_larger_than_standard_deviation(df["Close"])

(ii) Variability Index

Variability Index is a way to measure how smooth or 'variable' a time series is.

var_index_param = {"Volume":df["Volume"].values, "Open": df["Open"].values}

@set_property("fctype", "combiner")
@set_property("custom", True)
def var_index(time,param=var_index_param):
    final = []
    keys = []
    for key, magnitude in param.items():
      w = 1.0 / np.power(np.subtract(time[1:], time[:-1]), 2)
      w_mean = np.mean(w)

      N = len(time)
      sigma2 = np.var(magnitude)

      S1 = sum(w * (magnitude[1:] - magnitude[:-1]) ** 2)
      S2 = sum(w)

      eta_e = (w_mean * np.power(time[N - 1] -
                time[0], 2) * S1 / (sigma2 * S2 * N ** 2))
      final.append(eta_e)
      keys.append(key)
    return {"Interact__{}".format(k): eta_e for eta_e, k in zip(final,keys) }

extract.var_index(df["Close"].values,var_index_param)

(6) Shape

Features that emphasises a particular shape not ordinarily considered as a distribution statistic. Extends to derivations of the original time series too For example a feature looking at the sinusoidal shape of an autocorrelation plot.

(i) Symmetrical

Boolean variable denoting if the distribution of x looks symmetric.

#-> In Package
def symmetry_looking(x, param=[{"r": 0.2}]):

    if not isinstance(x, (np.ndarray, pd.Series)):
        x = np.asarray(x)
    mean_median_difference = np.abs(np.mean(x) - np.median(x))
    max_min_difference = np.max(x) - np.min(x)
    return [("r_{}".format(r["r"]), mean_median_difference < (r["r"] * max_min_difference))
            for r in param]
            
extract.symmetry_looking(df["Close"])

(7) Occurrence

Looking at the occurrence, and reoccurence of defined values.

(i) Has Duplicate Max

#-> In Package
def has_duplicate_max(x):
    """
    Checks if the maximum value of x is observed more than once

    :param x: the time series to calculate the feature of
    :type x: numpy.ndarray
    :return: the value of this feature
    :return type: bool
    """
    if not isinstance(x, (np.ndarray, pd.Series)):
        x = np.asarray(x)
    return np.sum(x == np.max(x)) >= 2

extract.has_duplicate_max(df["Close"])

(8) Autocorrelation

Autocorrelation, also known as serial correlation, is the correlation of a signal with a delayed copy of itself as a function of delay.

(i) Partial Autocorrelation

Partial autocorrelation is a summary of the relationship between an observation in a time series with observations at prior time steps with the relationships of intervening observations removed.

#-> In Package
from statsmodels.tsa.stattools import acf, adfuller, pacf

def partial_autocorrelation(x, param=[{"lag": 1}]):

    # Check the difference between demanded lags by param and possible lags to calculate (depends on len(x))
    max_demanded_lag = max([lag["lag"] for lag in param])
    n = len(x)

    # Check if list is too short to make calculations
    if n <= 1:
        pacf_coeffs = [np.nan] * (max_demanded_lag + 1)
    else:
        if (n <= max_demanded_lag):
            max_lag = n - 1
        else:
            max_lag = max_demanded_lag
        pacf_coeffs = list(pacf(x, method="ld", nlags=max_lag))
        pacf_coeffs = pacf_coeffs + [np.nan] * max(0, (max_demanded_lag - max_lag))

    return [("lag_{}".format(lag["lag"]), pacf_coeffs[lag["lag"]]) for lag in param]

extract.partial_autocorrelation(df["Close"])

(9) Stochasticity

Stochastic refers to a randomly determined process. Any features trying to capture stochasticity by degree or type are included under this branch.

(i) Augmented Dickey Fuller

The Augmented Dickey-Fuller test is a hypothesis test which checks whether a unit root is present in a time series sample.

#-> In Package
def augmented_dickey_fuller(x, param=[{"attr": "teststat"}]):

    res = None
    try:
        res = adfuller(x)
    except LinAlgError:
        res = np.NaN, np.NaN, np.NaN
    except ValueError: # occurs if sample size is too small
        res = np.NaN, np.NaN, np.NaN
    except MissingDataError: # is thrown for e.g. inf or nan in the data
        res = np.NaN, np.NaN, np.NaN

    return [('attr_"{}"'.format(config["attr"]),
                  res[0] if config["attr"] == "teststat"
             else res[1] if config["attr"] == "pvalue"
             else res[2] if config["attr"] == "usedlag" else np.NaN)
            for config in param]

extract.augmented_dickey_fuller(df["Close"])

(10) Averages

(i) Median of Magnitudes Skew

@set_property("fctype", "simple")
@set_property("custom", True)
def gskew(x):
    interpolation="nearest"
    median_mag = np.median(x)
    F_3_value = np.percentile(x, 3, interpolation=interpolation)
    F_97_value = np.percentile(x, 97, interpolation=interpolation)

    skew = (np.median(x[x <= F_3_value]) +
            np.median(x[x >= F_97_value]) - 2 * median_mag)

    return skew

extract.gskew(df["Close"])

(ii) Stetson Mean

An iteratively weighted mean used in the Stetson variability index

stestson_param = {"weight":100., "alpha":2., "beta":2., "tol":1.e-6, "nmax":20}

@set_property("fctype", "combiner")
@set_property("custom", True)
def stetson_mean(x, param=stestson_param):
    
    weight= stestson_param["weight"]
    alpha= stestson_param["alpha"]
    beta = stestson_param["beta"]
    tol= stestson_param["tol"]
    nmax= stestson_param["nmax"]
    
    
    mu = np.median(x)
    for i in range(nmax):
        resid = x - mu
        resid_err = np.abs(resid) * np.sqrt(weight)
        weight1 = weight / (1. + (resid_err / alpha)**beta)
        weight1 /= weight1.mean()
        diff = np.mean(x * weight1) - mu
        mu += diff
        if (np.abs(diff) < tol*np.abs(mu) or np.abs(diff) < tol):
            break

    return mu

extract.stetson_mean(df["Close"])

(11) Size

(i) Lenght

#-> In Package
def length(x):
    return len(x)
    
extract.length(df["Close"])

(12) Count

(i) Count Above Mean

Returns the number of values in x that are higher than the mean of x

#-> In Package
def count_above_mean(x):
    m = np.mean(x)
    return np.where(x > m)[0].size

extract.count_above_mean(df["Close"])

(13) Streaks

(i) Longest Strike Below Mean

Returns the length of the longest consecutive subsequence in x that is smaller than the mean of x

#-> In Package
import itertools
def get_length_sequences_where(x):

    if len(x) == 0:
        return [0]
    else:
        res = [len(list(group)) for value, group in itertools.groupby(x) if value == 1]
        return res if len(res) > 0 else [0]

def longest_strike_below_mean(x):

    if not isinstance(x, (np.ndarray, pd.Series)):
        x = np.asarray(x)
    return np.max(get_length_sequences_where(x <= np.mean(x))) if x.size > 0 else 0

extract.longest_strike_below_mean(df["Close"])

(ii) Wozniak

This is an astronomical feature, we count the number of three consecutive data points that are brighter or fainter than $2σ$ and normalize the number by $N−2$

woz_param = [{"consecutiveStar": n} for n in [2, 4]]

@set_property("fctype", "combiner")
@set_property("custom", True)
def wozniak(magnitude, param=woz_param):

    iters = []
    for consecutiveStar in [stars["consecutiveStar"] for stars in param]:
      N = len(magnitude)
      if N < consecutiveStar:
          return 0
      sigma = np.std(magnitude)
      m = np.mean(magnitude)
      count = 0

      for i in range(N - consecutiveStar + 1):
          flag = 0
          for j in range(consecutiveStar):
              if(magnitude[i + j] > m + 2 * sigma or
                  magnitude[i + j] < m - 2 * sigma):
                  flag = 1
              else:
                  flag = 0
                  break
          if flag:
              count = count + 1
      iters.append(count * 1.0 / (N - consecutiveStar + 1))

    return [("consecutiveStar_{}".format(config["consecutiveStar"]), iters[en] )  for en, config in enumerate(param)]

extract.wozniak(df["Close"])

(14) Location

(i) Last location of Maximum

Returns the relative last location of the maximum value of x. last_location_of_minimum(x),

#-> In Package
def last_location_of_maximum(x):

    x = np.asarray(x)
    return 1.0 - np.argmax(x[::-1]) / len(x) if len(x) > 0 else np.NaN

extract.last_location_of_maximum(df["Close"])

(15) Model Coefficients

Any coefficient that are obtained from a model that might help in the prediction problem. For example here we might include coefficients of polynomial $h(x)$, which has been fitted to the deterministic dynamics of Langevin model.

(i) FFT Coefficient

Calculates the fourier coefficients of the one-dimensional discrete Fourier Transform for real input.

#-> In Package
def fft_coefficient(x, param = [{"coeff": 10, "attr": "real"}]):

    assert min([config["coeff"] for config in param]) >= 0, "Coefficients must be positive or zero."
    assert set([config["attr"] for config in param]) <= set(["imag", "real", "abs", "angle"]), \
        'Attribute must be "real", "imag", "angle" or "abs"'

    fft = np.fft.rfft(x)

    def complex_agg(x, agg):
        if agg == "real":
            return x.real
        elif agg == "imag":
            return x.imag
        elif agg == "abs":
            return np.abs(x)
        elif agg == "angle":
            return np.angle(x, deg=True)

    res = [complex_agg(fft[config["coeff"]], config["attr"]) if config["coeff"] < len(fft)
           else np.NaN for config in param]
    index = [('coeff_{}__attr_"{}"'.format(config["coeff"], config["attr"]),res[0]) for config in param]
    return index

extract.fft_coefficient(df["Close"])

(ii) AR Coefficient

This feature calculator fits the unconditional maximum likelihood of an autoregressive AR(k) process.

#-> In Package
from statsmodels.tsa.ar_model import AR

def ar_coefficient(x, param=[{"coeff": 5, "k": 5}]):

    calculated_ar_params = {}

    x_as_list = list(x)
    calculated_AR = AR(x_as_list)

    res = {}

    for parameter_combination in param:
        k = parameter_combination["k"]
        p = parameter_combination["coeff"]

        column_name = "k_{}__coeff_{}".format(k, p)

        if k not in calculated_ar_params:
            try:
                calculated_ar_params[k] = calculated_AR.fit(maxlag=k, solver="mle").params
            except (LinAlgError, ValueError):
                calculated_ar_params[k] = [np.NaN]*k

        mod = calculated_ar_params[k]

        if p <= k:
            try:
                res[column_name] = mod[p]
            except IndexError:
                res[column_name] = 0
        else:
            res[column_name] = np.NaN

    return [(key, value) for key, value in res.items()]

extract.ar_coefficient(df["Close"])

(16) Quantiles

This includes finding normal quantile values in the series, but also quantile derived measures like change quantiles and index max quantiles.

(i) Index Mass Quantile

The relative index $i$ where $q%$ of the mass of the time series $x$ lie left of $i$ .

#-> In Package
def index_mass_quantile(x, param=[{"q": 0.3}]):

    x = np.asarray(x)
    abs_x = np.abs(x)
    s = sum(abs_x)

    if s == 0:
        # all values in x are zero or it has length 0
        return [("q_{}".format(config["q"]), np.NaN) for config in param]
    else:
        # at least one value is not zero
        mass_centralized = np.cumsum(abs_x) / s
        return [("q_{}".format(config["q"]), (np.argmax(mass_centralized >= config["q"])+1)/len(x)) for config in param]

extract.index_mass_quantile(df["Close"])

(17) Peaks

(i) Number of CWT Peaks

This feature calculator searches for different peaks in x.

from scipy.signal import cwt, find_peaks_cwt, ricker, welch

cwt_param = [ka for ka in [2,6,9]]

@set_property("fctype", "combiner")
@set_property("custom", True)
def number_cwt_peaks(x, param=cwt_param):

    return [("CWTPeak_{}".format(n), len(find_peaks_cwt(vector=x, widths=np.array(list(range(1, n + 1))), wavelet=ricker))) for n in param]

extract.number_cwt_peaks(df["Close"])

(18) Density

The density, and more specifically the power spectral density of the signal describes the power present in the signal as a function of frequency, per unit frequency.

(i) Cross Power Spectral Density

This feature calculator estimates the cross power spectral density of the time series $x$ at different frequencies.

#-> In Package
def spkt_welch_density(x, param=[{"coeff": 5}]):
    freq, pxx = welch(x, nperseg=min(len(x), 256))
    coeff = [config["coeff"] for config in param]
    indices = ["coeff_{}".format(i) for i in coeff]

    if len(pxx) <= np.max(coeff):  # There are fewer data points in the time series than requested coefficients

        # filter coefficients that are not contained in pxx
        reduced_coeff = [coefficient for coefficient in coeff if len(pxx) > coefficient]
        not_calculated_coefficients = [coefficient for coefficient in coeff
                                       if coefficient not in reduced_coeff]

        # Fill up the rest of the requested coefficients with np.NaNs
        return zip(indices, list(pxx[reduced_coeff]) + [np.NaN] * len(not_calculated_coefficients))
    else:
        return pxx[coeff].ravel()[0]

extract.spkt_welch_density(df["Close"])

(19) Linearity

Any measure of linearity that might make use of something like the linear least-squares regression for the values of the time series. This can be against the time series minus one and many other alternatives.

(i) Linear Trend Time Wise

Calculate a linear least-squares regression for the values of the time series versus the sequence from 0 to length of the time series minus one.

from scipy.stats import linregress

#-> In Package
def linear_trend_timewise(x, param= [{"attr": "pvalue"}]):

    ix = x.index

    # Get differences between each timestamp and the first timestamp in seconds.
    # Then convert to hours and reshape for linear regression
    times_seconds = (ix - ix[0]).total_seconds()
    times_hours = np.asarray(times_seconds / float(3600))

    linReg = linregress(times_hours, x.values)

    return [("attr_\"{}\"".format(config["attr"]), getattr(linReg, config["attr"]))
            for config in param]

extract.linear_trend_timewise(df["Close"])

(20) Non-Linearity

(i) Schreiber Non-Linearity

#-> In Package
def c3(x, lag=3):
    if not isinstance(x, (np.ndarray, pd.Series)):
        x = np.asarray(x)
    n = x.size
    if 2 * lag >= n:
        return 0
    else:
        return np.mean((_roll(x, 2 * -lag) * _roll(x, -lag) * x)[0:(n - 2 * lag)])

extract.c3(df["Close"])

(21) Entropy

Any feature looking at the complexity of a time series. This is typically used in medical signal disciplines (EEG, EMG). There are multiple types of measures like spectral entropy, permutation entropy, sample entropy, approximate entropy, Lempel-Ziv complexity and other. This includes entropy measures and there derivations.

(i) Binned Entropy

Bins the values of x into max_bins equidistant bins.

#-> In Package
def binned_entropy(x, max_bins=10):
    if not isinstance(x, (np.ndarray, pd.Series)):
        x = np.asarray(x)
    hist, bin_edges = np.histogram(x, bins=max_bins)
    probs = hist / x.size
    return - np.sum(p * np.math.log(p) for p in probs if p != 0)

extract.binned_entropy(df["Close"])

(ii) SVD Entropy

SVD entropy is an indicator of the number of eigenvectors that are needed for an adequate explanation of the data set.

svd_param = [{"Tau": ta, "DE": de}
                      for ta in [4] 
                      for de in [3,6]]
                      
def _embed_seq(X,Tau,D):
  N =len(X)
  if D * Tau > N:
      print("Cannot build such a matrix, because D * Tau > N")
      exit()
  if Tau<1:
      print("Tau has to be at least 1")
      exit()
  Y= np.zeros((N - (D - 1) * Tau, D))

  for i in range(0, N - (D - 1) * Tau):
      for j in range(0, D):
          Y[i][j] = X[i + j * Tau]
  return Y                     

@set_property("fctype", "combiner")
@set_property("custom", True)
def svd_entropy(epochs, param=svd_param):
    axis=0
    
    final = []
    for par in param:

      def svd_entropy_1d(X, Tau, DE):
          Y = _embed_seq(X, Tau, DE)
          W = np.linalg.svd(Y, compute_uv=0)
          W /= sum(W)  # normalize singular values
          return -1 * np.sum(W * np.log(W))

      Tau = par["Tau"]
      DE = par["DE"]

      final.append(np.apply_along_axis(svd_entropy_1d, axis, epochs, Tau, DE).ravel()[0])


    return [("Tau_\"{}\"__De_{}\"".format(par["Tau"], par["DE"]), final[en]) for en, par in enumerate(param)]

extract.svd_entropy(df["Close"].values)

(iii) Hjort

The Complexity parameter represents the change in frequency. The parameter compares the signal's similarity to a pure sine wave, where the value converges to 1 if the signal is more similar.

def _hjorth_mobility(epochs):
    diff = np.diff(epochs, axis=0)
    sigma0 = np.std(epochs, axis=0)
    sigma1 = np.std(diff, axis=0)
    return np.divide(sigma1, sigma0)

@set_property("fctype", "simple")
@set_property("custom", True)
def hjorth_complexity(epochs):
    diff1 = np.diff(epochs, axis=0)
    diff2 = np.diff(diff1, axis=0)
    sigma1 = np.std(diff1, axis=0)
    sigma2 = np.std(diff2, axis=0)
    return np.divide(np.divide(sigma2, sigma1), _hjorth_mobility(epochs))

extract.hjorth_complexity(df["Close"])

(22) Fixed Points

Fixed points and equilibria as identified from fitted models.

(i) Langevin Fixed Points

Largest fixed point of dynamics $max\ {h(x)=0}$ estimated from polynomial $h(x)$ which has been fitted to the deterministic dynamics of Langevin model

#-> In Package
def _estimate_friedrich_coefficients(x, m, r):
    assert m > 0, "Order of polynomial need to be positive integer, found {}".format(m)
    df = pd.DataFrame({'signal': x[:-1], 'delta': np.diff(x)})
    try:
        df['quantiles'] = pd.qcut(df.signal, r)
    except ValueError:
        return [np.NaN] * (m + 1)

    quantiles = df.groupby('quantiles')

    result = pd.DataFrame({'x_mean': quantiles.signal.mean(), 'y_mean': quantiles.delta.mean()})
    result.dropna(inplace=True)

    try:
        return np.polyfit(result.x_mean, result.y_mean, deg=m)
    except (np.linalg.LinAlgError, ValueError):
        return [np.NaN] * (m + 1)


def max_langevin_fixed_point(x, r=3, m=30):
    coeff = _estimate_friedrich_coefficients(x, m, r)

    try:
        max_fixed_point = np.max(np.real(np.roots(coeff)))
    except (np.linalg.LinAlgError, ValueError):
        return np.nan

    return max_fixed_point

extract.max_langevin_fixed_point(df["Close"])

(23) Amplitude

Features derived from peaked values in either the positive or negative direction.

(i) Willison Amplitude

This feature is defined as the amount of times that the change in the signal amplitude exceeds a threshold.

will_param = [ka for ka in [0.2,3]]

@set_property("fctype", "combiner")
@set_property("custom", True)
def willison_amplitude(X, param=will_param):
  return [("Thresh_{}".format(n),np.sum(np.abs(np.diff(X)) >= n)) for n in param]

extract.willison_amplitude(df["Close"])

(ii) Percent Amplitude

Returns the largest distance from the median value, measured as a percentage of the median

perc_param = [{"base":ba, "exponent":exp} for ba in [3,5] for exp in [-0.1,-0.2]]

@set_property("fctype", "combiner")
@set_property("custom", True)
def percent_amplitude(x, param =perc_param):
    final = []
    for par in param:
      linear_scale_data = par["base"] ** (par["exponent"] * x)
      y_max = np.max(linear_scale_data)
      y_min = np.min(linear_scale_data)
      y_med = np.median(linear_scale_data)
      final.append(max(abs((y_max - y_med) / y_med), abs((y_med - y_min) / y_med)))

    return [("Base_{}__Exp{}".format(pa["base"],pa["exponent"]),fin) for fin, pa in zip(final,param)]

extract.percent_amplitude(df["Close"])

(24) Probability

(i) Cadence Probability

Given the observed distribution of time lags cads, compute the probability that the next observation occurs within time minutes of an arbitrary epoch.

#-> fixes required
import scipy.stats as stats

cad_param = [0.1,1000, -234]

@set_property("fctype", "combiner")
@set_property("custom", True)
def cad_prob(cads, param=cad_param):
    return [("time_{}".format(time), stats.percentileofscore(cads, float(time) / (24.0 * 60.0)) / 100.0) for time in param]
    
extract.cad_prob(df["Close"])

(25) Crossings

Calculates the crossing of the series with other defined values or series.

(i) Zero Crossing Derivative

The positioning of the edge point is located at the zero crossing of the first derivative of the filter.

zero_param = [0.01, 8]

@set_property("fctype", "combiner")
@set_property("custom", True)
def zero_crossing_derivative(epochs, param=zero_param):
    diff = np.diff(epochs)
    norm = diff-diff.mean()
    return [("e_{}".format(e), np.apply_along_axis(lambda epoch: np.sum(((epoch[:-5] <= e) & (epoch[5:] > e))), 0, norm).ravel()[0]) for e in param]

extract.zero_crossing_derivative(df["Close"])

(26) Fluctuations

These features are again from medical signal sciences, but under this category we would include values such as fluctuation based entropy measures, fluctuation of correlation dynamics, and co-fluctuations.

(i) Detrended Fluctuation Analysis (DFA)

DFA Calculate the Hurst exponent using DFA analysis.

from scipy.stats import kurtosis as _kurt
from scipy.stats import skew as _skew
import numpy as np

@set_property("fctype", "simple")
@set_property("custom", True)
def detrended_fluctuation_analysis(epochs):
    def dfa_1d(X, Ave=None, L=None):
        X = np.array(X)

        if Ave is None:
            Ave = np.mean(X)

        Y = np.cumsum(X)
        Y -= Ave

        if L is None:
            L = np.floor(len(X) * 1 / (
                    2 ** np.array(list(range(1, int(np.log2(len(X))) - 4))))
                            )
            
        F = np.zeros(len(L))  # F(n) of different given box length n

        for i in range(0, len(L)):
            n = int(L[i])  # for each box length L[i]
            if n == 0:
                print("time series is too short while the box length is too big")
                print("abort")
                exit()
            for j in range(0, len(X), n):  # for each box
                if j + n < len(X):
                    c = list(range(j, j + n))
                    # coordinates of time in the box
                    c = np.vstack([c, np.ones(n)]).T
                    # the value of data in the box
                    y = Y[j:j + n]
                    # add residue in this box
                    F[i] += np.linalg.lstsq(c, y, rcond=None)[1]
            F[i] /= ((len(X) / n) * n)
        F = np.sqrt(F)

        stacked = np.vstack([np.log(L), np.ones(len(L))])
        stacked_t = stacked.T
        Alpha = np.linalg.lstsq(stacked_t, np.log(F), rcond=None)

        return Alpha[0][0]

    return np.apply_along_axis(dfa_1d, 0, epochs).ravel()[0]

extract.detrended_fluctuation_analysis(df["Close"])

(27) Information

Closely related to entropy and complexity measures. Any measure that attempts to measure the amount of information from an observable variable is included here.

(i) Fisher Information

Fisher information is a statistical information concept distinct from, and earlier than, Shannon information in communication theory.

def _embed_seq(X, Tau, D):

    shape = (X.size - Tau * (D - 1), D)
    strides = (X.itemsize, Tau * X.itemsize)
    return np.lib.stride_tricks.as_strided(X, shape=shape, strides=strides)

fisher_param = [{"Tau":ta, "DE":de} for ta in [3,15] for de in [10,5]]

@set_property("fctype", "combiner")
@set_property("custom", True)
def fisher_information(epochs, param=fisher_param):
    def fisher_info_1d(a, tau, de):
        # taken from pyeeg improvements

        mat = _embed_seq(a, tau, de)
        W = np.linalg.svd(mat, compute_uv=False)
        W /= sum(W)  # normalize singular values
        FI_v = (W[1:] - W[:-1]) ** 2 / W[:-1]
        return np.sum(FI_v)

    return [("Tau_{}__DE_{}".format(par["Tau"], par["DE"]),np.apply_along_axis(fisher_info_1d, 0, epochs, par["Tau"], par["DE"]).ravel()[0]) for par in param]

extract.fisher_information(df["Close"])

(28) Fractals

In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is measured.

(i) Highuchi Fractal

Compute a Higuchi Fractal Dimension of a time series

hig_para = [{"Kmax": 3},{"Kmax": 5}]

@set_property("fctype", "combiner")
@set_property("custom", True)
def higuchi_fractal_dimension(epochs, param=hig_para):
    def hfd_1d(X, Kmax):
        
        L = []
        x = []
        N = len(X)
        for k in range(1, Kmax):
            Lk = []
            for m in range(0, k):
                Lmk = 0
                for i in range(1, int(np.floor((N - m) / k))):
                    Lmk += abs(X[m + i * k] - X[m + i * k - k])
                Lmk = Lmk * (N - 1) / np.floor((N - m) / float(k)) / k
                Lk.append(Lmk)
            L.append(np.log(np.mean(Lk)))
            x.append([np.log(float(1) / k), 1])

        (p, r1, r2, s) = np.linalg.lstsq(x, L, rcond=None)
        return p[0]
    
    return [("Kmax_{}".format(config["Kmax"]), np.apply_along_axis(hfd_1d, 0, epochs, config["Kmax"]).ravel()[0] ) for  config in param]
    
extract.higuchi_fractal_dimension(df["Close"])

(ii) Petrosian Fractal

Compute a Petrosian Fractal Dimension of a time series.

@set_property("fctype", "simple")
@set_property("custom", True)
def petrosian_fractal_dimension(epochs):
    def pfd_1d(X, D=None):
        # taken from pyeeg
        """Compute Petrosian Fractal Dimension of a time series from either two
        cases below:
            1. X, the time series of type list (default)
            2. D, the first order differential sequence of X (if D is provided,
               recommended to speed up)
        In case 1, D is computed using Numpy's difference function.
        To speed up, it is recommended to compute D before calling this function
        because D may also be used by other functions whereas computing it here
        again will slow down.
        """
        if D is None:
            D = np.diff(X)
            D = D.tolist()
        N_delta = 0  # number of sign changes in derivative of the signal
        for i in range(1, len(D)):
            if D[i] * D[i - 1] < 0:
                N_delta += 1
        n = len(X)
        return np.log10(n) / (np.log10(n) + np.log10(n / n + 0.4 * N_delta))
    return np.apply_along_axis(pfd_1d, 0, epochs).ravel()[0]

extract.petrosian_fractal_dimension(df["Close"])

(29) Exponent

(i) Hurst Exponent

The Hurst exponent is used as a measure of long-term memory of time series. It relates to the autocorrelations of the time series, and the rate at which these decrease as the lag between pairs of values increases.

@set_property("fctype", "simple")
@set_property("custom", True)
def hurst_exponent(epochs):
    def hurst_1d(X):

        X = np.array(X)
        N = X.size
        T = np.arange(1, N + 1)
        Y = np.cumsum(X)
        Ave_T = Y / T

        S_T = np.zeros(N)
        R_T = np.zeros(N)
        for i in range(N):
            S_T[i] = np.std(X[:i + 1])
            X_T = Y - T * Ave_T[i]
            R_T[i] = np.ptp(X_T[:i + 1])

        for i in range(1, len(S_T)):
            if np.diff(S_T)[i - 1] != 0:
                break
        for j in range(1, len(R_T)):
            if np.diff(R_T)[j - 1] != 0:
                break
        k = max(i, j)
        assert k < 10, "rethink it!"

        R_S = R_T[k:] / S_T[k:]
        R_S = np.log(R_S)

        n = np.log(T)[k:]
        A = np.column_stack((n, np.ones(n.size)))
        [m, c] = np.linalg.lstsq(A, R_S, rcond=None)[0]
        H = m
        return H
    return np.apply_along_axis(hurst_1d, 0, epochs).ravel()[0]

extract.hurst_exponent(df["Close"])

(ii) Largest Lyauponov Exponent

In mathematics the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories.

def _embed_seq(X, Tau, D):
    shape = (X.size - Tau * (D - 1), D)
    strides = (X.itemsize, Tau * X.itemsize)
    return np.lib.stride_tricks.as_strided(X, shape=shape, strides=strides)

lyaup_param = [{"Tau":4, "n":3, "T":10, "fs":9},{"Tau":8, "n":7, "T":15, "fs":6}]

@set_property("fctype", "combiner")
@set_property("custom", True)
def largest_lyauponov_exponent(epochs, param=lyaup_param):
    def LLE_1d(x, tau, n, T, fs):

        Em = _embed_seq(x, tau, n)
        M = len(Em)
        A = np.tile(Em, (len(Em), 1, 1))
        B = np.transpose(A, [1, 0, 2])
        square_dists = (A - B) ** 2  # square_dists[i,j,k] = (Em[i][k]-Em[j][k])^2
        D = np.sqrt(square_dists[:, :, :].sum(axis=2))  # D[i,j] = ||Em[i]-Em[j]||_2

        # Exclude elements within T of the diagonal
        band = np.tri(D.shape[0], k=T) - np.tri(D.shape[0], k=-T - 1)
        band[band == 1] = np.inf
        neighbors = (D + band).argmin(axis=0)  # nearest neighbors more than T steps away

        # in_bounds[i,j] = (i+j <= M-1 and i+neighbors[j] <= M-1)
        inc = np.tile(np.arange(M), (M, 1))
        row_inds = (np.tile(np.arange(M), (M, 1)).T + inc)
        col_inds = (np.tile(neighbors, (M, 1)) + inc.T)
        in_bounds = np.logical_and(row_inds <= M - 1, col_inds <= M - 1)
        # Uncomment for old (miscounted) version
        # in_bounds = numpy.logical_and(row_inds < M - 1, col_inds < M - 1)
        row_inds[~in_bounds] = 0
        col_inds[~in_bounds] = 0

        # neighbor_dists[i,j] = ||Em[i+j]-Em[i+neighbors[j]]||_2
        neighbor_dists = np.ma.MaskedArray(D[row_inds, col_inds], ~in_bounds)
        J = (~neighbor_dists.mask).sum(axis=1)  # number of in-bounds indices by row
        # Set invalid (zero) values to 1; log(1) = 0 so sum is unchanged

        neighbor_dists[neighbor_dists == 0] = 1

        # !!! this fixes the divide by zero in log error !!!
        neighbor_dists.data[neighbor_dists.data == 0] = 1

        d_ij = np.sum(np.log(neighbor_dists.data), axis=1)
        mean_d = d_ij[J > 0] / J[J > 0]

        x = np.arange(len(mean_d))
        X = np.vstack((x, np.ones(len(mean_d)))).T
        [m, c] = np.linalg.lstsq(X, mean_d, rcond=None)[0]
        Lexp = fs * m
        return Lexp

    return [("Tau_{}__n_{}__T_{}__fs_{}".format(par["Tau"], par["n"], par["T"], par["fs"]), np.apply_along_axis(LLE_1d, 0, epochs, par["Tau"], par["n"], par["T"], par["fs"]).ravel()[0]) for par in param]
  
extract.largest_lyauponov_exponent(df["Close"])

(30) Spectral Analysis

Spectral analysis is analysis in terms of a spectrum of frequencies or related quantities such as energies, eigenvalues, etc.

(i) Whelch Method

The Whelch Method is an approach for spectral density estimation. It is used in physics, engineering, and applied mathematics for estimating the power of a signal at different frequencies.

from scipy import signal, integrate

whelch_param = [100,200]

@set_property("fctype", "combiner")
@set_property("custom", True)
def whelch_method(data, param=whelch_param):

  final = []
  for Fs in param:
    f, pxx = signal.welch(data, fs=Fs, nperseg=1024)
    d = {'psd': pxx, 'freqs': f}
    df = pd.DataFrame(data=d)
    dfs = df.sort_values(['psd'], ascending=False)
    rows = dfs.iloc[:10]
    final.append(rows['freqs'].mean())
  
  return [("Fs_{}".format(pa),fin) for pa, fin in zip(param,final)]

extract.whelch_method(df["Close"])
#-> Basically same as above
freq_param = [{"fs":50, "sel":15},{"fs":200, "sel":20}]

@set_property("fctype", "combiner")
@set_property("custom", True)
def find_freq(serie, param=freq_param):

    final = []
    for par in param:
      fft0 = np.fft.rfft(serie*np.hanning(len(serie)))
      freqs = np.fft.rfftfreq(len(serie), d=1.0/par["fs"])
      fftmod = np.array([np.sqrt(fft0[i].real**2 + fft0[i].imag**2) for i in range(0, len(fft0))])
      d = {'fft': fftmod, 'freq': freqs}
      df = pd.DataFrame(d)
      hop = df.sort_values(['fft'], ascending=False)
      rows = hop.iloc[:par["sel"]]
      final.append(rows['freq'].mean())

    return [("Fs_{}__sel{}".format(pa["fs"],pa["sel"]),fin) for pa, fin in zip(param,final)]

extract.find_freq(df["Close"])

(31) Percentile

(i) Flux Percentile

Flux (or radiant flux) is the total amount of energy that crosses a unit area per unit time. Flux is an astronomical value, measured in joules per square metre per second (joules/m2/s), or watts per square metre. Here we provide the ratio of flux percentiles.

#-> In Package

import math
def flux_perc(magnitude):
    sorted_data = np.sort(magnitude)
    lc_length = len(sorted_data)

    F_60_index = int(math.ceil(0.60 * lc_length))
    F_40_index = int(math.ceil(0.40 * lc_length))
    F_5_index = int(math.ceil(0.05 * lc_length))
    F_95_index = int(math.ceil(0.95 * lc_length))

    F_40_60 = sorted_data[F_60_index] - sorted_data[F_40_index]
    F_5_95 = sorted_data[F_95_index] - sorted_data[F_5_index]
    F_mid20 = F_40_60 / F_5_95

    return {"FluxPercentileRatioMid20": F_mid20}

extract.flux_perc(df["Close"])

(32) Range

(i) Range of Cummulative Sum

@set_property("fctype", "simple")
@set_property("custom", True)
def range_cum_s(magnitude):
    sigma = np.std(magnitude)
    N = len(magnitude)
    m = np.mean(magnitude)
    s = np.cumsum(magnitude - m) * 1.0 / (N * sigma)
    R = np.max(s) - np.min(s)
    return {"Rcs": R}

extract.range_cum_s(df["Close"])

(33) Structural

Structural features, potential placeholders for future research.

(i) Structure Function

The structure function of rotation measures (RMs) contains information on electron density and magnetic field fluctuations when used i astronomy. It becomes a custom feature when used with your own unique time series data.

from scipy.interpolate import interp1d

struct_param = {"Volume":df["Volume"].values, "Open": df["Open"].values}

@set_property("fctype", "combiner")
@set_property("custom", True)
def structure_func(time, param=struct_param):

      dict_final = {}
      for key, magnitude in param.items():
        dict_final[key] = []
        Nsf, Np = 100, 100
        sf1, sf2, sf3 = np.zeros(Nsf), np.zeros(Nsf), np.zeros(Nsf)
        f = interp1d(time, magnitude)

        time_int = np.linspace(np.min(time), np.max(time), Np)
        mag_int = f(time_int)

        for tau in np.arange(1, Nsf):
            sf1[tau - 1] = np.mean(
                np.power(np.abs(mag_int[0:Np - tau] - mag_int[tau:Np]), 1.0))
            sf2[tau - 1] = np.mean(
                np.abs(np.power(
                    np.abs(mag_int[0:Np - tau] - mag_int[tau:Np]), 2.0)))
            sf3[tau - 1] = np.mean(
                np.abs(np.power(
                    np.abs(mag_int[0:Np - tau] - mag_int[tau:Np]), 3.0)))
        sf1_log = np.log10(np.trim_zeros(sf1))
        sf2_log = np.log10(np.trim_zeros(sf2))
        sf3_log = np.log10(np.trim_zeros(sf3))

        if len(sf1_log) and len(sf2_log):
            m_21, b_21 = np.polyfit(sf1_log, sf2_log, 1)
        else:

            m_21 = np.nan

        if len(sf1_log) and len(sf3_log):
            m_31, b_31 = np.polyfit(sf1_log, sf3_log, 1)
        else:

            m_31 = np.nan

        if len(sf2_log) and len(sf3_log):
            m_32, b_32 = np.polyfit(sf2_log, sf3_log, 1)
        else:

            m_32 = np.nan
        dict_final[key].append(m_21)
        dict_final[key].append(m_31)
        dict_final[key].append(m_32)

      return [("StructureFunction_{}__m_{}".format(key, name), li)  for key, lis in dict_final.items() for name, li in zip([21,31,32], lis)]

struct_param = {"Volume":df["Volume"].values, "Open": df["Open"].values}

extract.structure_func(df["Close"],struct_param)

(34) Distribution

(i) Kurtosis

#-> In Package
def kurtosis(x):

    if not isinstance(x, pd.Series):
        x = pd.Series(x)
    return pd.Series.kurtosis(x)

extract.kurtosis(df["Close"])

(ii) Stetson Kurtosis

@set_property("fctype", "simple")
@set_property("custom", True)
def stetson_k(x):
    """A robust kurtosis statistic."""
    n = len(x)
    x0 = stetson_mean(x, 1./20**2)
    delta_x = np.sqrt(n / (n - 1.)) * (x - x0) / 20
    ta = 1. / 0.798 * np.mean(np.abs(delta_x)) / np.sqrt(np.mean(delta_x**2))
    return ta
  
extract.stetson_k(df["Close"])

(5) Synthesise

Time-Series synthesisation (TSS) happens before the feature extraction step and Cross Sectional Synthesisation (CSS) happens after the feature extraction step. Currently I will only include a CSS package, in the future, I would further work on developing out this section. This area still has a lot of performance and stability issues. In the future it might be a more viable candidate to improve prediction.

from lightgbm import LGBMRegressor
from sklearn.metrics import mean_squared_error

def model(df_final):
  model = LGBMRegressor()
  test =  df_final.head(int(len(df_final)*0.4))
  train = df_final[~df_final.isin(test)].dropna()
  model = model.fit(train.drop(["Close_1"],axis=1),train["Close_1"])
  preds = model.predict(test.drop(["Close_1"],axis=1))
  test =  df_final.head(int(len(df_final)*0.4))
  train = df_final[~df_final.isin(test)].dropna()
  model = model.fit(train.drop(["Close_1"],axis=1),train["Close_1"])
  val = mean_squared_error(test["Close_1"],preds); 
  return val
pip install ctgan
from ctgan import CTGANSynthesizer

#discrete_columns = [""]
ctgan = CTGANSynthesizer()
ctgan.fit(df,epochs=10) #15

Random Benchmark

np.random.seed(1)
df_in = df.copy()
df_in["Close_1"] = np.random.permutation(df_in["Close_1"].values)
model(df_in)

Generated Performance

df_gen = ctgan.sample(len(df_in)*100)
model(df_gen)

As expected a cross-sectional technique, does not work well on time-series data, in the future, other methods will be investigated.

 

(6) Skeleton Example

Here I will perform tabular agumenting methods on a small dataset single digit features and around 250 instances. This is not necessarily the best sized dataset to highlight the performance of tabular augmentation as some method like extraction would be overkill as it would lead to dimensionality problems. It is also good to know that there are close to infinite number of ways to perform these augmentation methods. In the future, automated augmentation methods can guide the experiment process.

The approach taken in this skeleton is to develop running models that are tested after each augmentation to highlight what methods might work well on this particular dataset. The metric we will use is mean squared error. In this implementation we do not have special hold-out sets.

The above framework of implementation will be consulted, but one still have to be strategic as to when you apply what function, and you have to make sure that you are processing your data with appropriate techniques (drop null values, fill null values) at the appropriate time.

Validation

Develop Model and Define Metric

from lightgbm import LGBMRegressor
from sklearn.metrics import mean_squared_error

def model(df_final):
  model = LGBMRegressor()
  test =  df_final.head(int(len(df_final)*0.4))
  train = df_final[~df_final.isin(test)].dropna()
  model = model.fit(train.drop(["Close_1"],axis=1),train["Close_1"])
  preds = model.predict(test.drop(["Close_1"],axis=1))
  test =  df_final.head(int(len(df_final)*0.4))
  train = df_final[~df_final.isin(test)].dropna()
  model = model.fit(train.drop(["Close_1"],axis=1),train["Close_1"])
  val = mean_squared_error(test["Close_1"],preds); 
  return val

Reload Data

df = data_copy()
model(df)
302.61676570345287

(1) (7) (i) Transformation - Decomposition - Naive

## If Inferred Seasonality is Too Large Default to Five
seasons = transform.infer_seasonality(df["Close"],index=0) 
df_out = transform.naive_dec(df.copy(), ["Close","Open"], freq=5)
model(df_out) #improvement
274.34477082783525

(1) (8) (i) Transformation - Filter - Baxter-King-Bandpass

df_out = transform.bkb(df_out, ["Close","Low"])
df_best = df_out.copy()
model(df_out) #improvement
267.1826850968307

(1) (3) (i) Transformation - Differentiation - Fractional

df_out = transform.fast_fracdiff(df_out, ["Close_BPF"],0.5)
model(df_out) #null
267.7083192402742

(1) (1) (i) Transformation - Scaling - Robust Scaler

df_out = df_out.dropna()
df_out = transform.robust_scaler(df_out, drop=["Close_1"])
model(df_out) #noisy
270.96980399571214

(2) (2) (i) Interactions - Operator - Multiplication/Division

df_out.head()
 Close_1HighLowOpenCloseVolumeAdj CloseClose_NDDTClose_NDDSClose_NDDROpen_NDDTOpen_NDDSOpen_NDDRClose_BPFLow_BPFClose_BPF_frac
Date                
2019-01-08338.5299991.0184130.9640481.0966001.001175-0.1626161.0011750.8322970.8349641.3354330.7587430.6915962.259884-2.534142-2.249135-3.593612
2019-01-09344.9700011.0120681.0233021.0114661.042689-0.5017981.0426890.908963-0.1650361.1113460.8357860.3333611.129783-3.081959-2.776302-2.523465
2019-01-10347.2600101.0355811.0275630.9969691.126762-0.3675761.1267621.0293472.1200260.8536970.9075880.0000000.533777-2.052768-2.543449-0.747382
2019-01-11334.3999941.0731531.1205061.0983131.156658-0.5865711.1566581.109144-5.1560510.5919901.002162-0.6666390.608516-0.694642-0.8316700.414063
2019-01-14344.4299930.9996271.0569911.1021350.988773-0.5417520.9887731.1076330.000000-0.6603501.056302-0.9154910.263025-0.645590-0.116166-0.118012
df_out = interact.muldiv(df_out, ["Close","Open_NDDS","Low_BPF"]) 
model(df_out) #noisy
285.6420643864313
df_r = df_out.copy()

(2) (6) (i) Interactions - Speciality - Technical

import ta
df = interact.tech(df)
df_out = pd.merge(df_out,  df.iloc[:,7:], left_index=True, right_index=True, how="left")

Clean Dataframe and Metric

"""Droping column where missing values are above a threshold"""
df_out = df_out.dropna(thresh = len(df_out)*0.95, axis = "columns") 
df_out = df_out.dropna()
df_out = df_out.replace([np.inf, -np.inf], np.nan).ffill().fillna(0)
close = df_out["Close"].copy()
df_d = df_out.copy()
model(df_out) #improve
592.52971755184

(3) (1) (i) Mapping - Eigen Decomposition - PCA

from sklearn.decomposition import PCA, IncrementalPCA, KernelPCA

df_out = transform.robust_scaler(df_out, drop=["Close_1"])
df_out = df_out.replace([np.inf, -np.inf], np.nan).ffill().fillna(0)
df_out = mapper.pca_feature(df_out, drop_cols=["Close_1"], variance_or_components=0.9, n_components=8,non_linear=False)
model(df_out) #noisy but not too bad given the 10 fold dimensionality reduction
687.158330455884

(4) Extracting

Here at first, I show the functions that have been added to the DeltaPy fork of tsfresh. You have to add your own personal adjustments based on the features you would like to construct. I am using self-developed features, but you can also use TSFresh's community functions.

The following files have been appropriately ammended (Get in contact for advice)

  1. https://github.com/firmai/tsfresh/blob/master/tsfresh/feature_extraction/settings.py
  2. https://github.com/firmai/tsfresh/blob/master/tsfresh/feature_extraction/feature_calculators.py
  3. https://github.com/firmai/tsfresh/blob/master/tsfresh/feature_extraction/extraction.py

(4) (10) (i) Extracting - Averages - GSkew

extract.gskew(df_out["PCA_1"])
-0.7903067336449059

(4) (21) (ii) Extracting - Entropy - SVD Entropy

svd_param = [{"Tau": ta, "DE": de}
                      for ta in [4] 
                      for de in [3,6]]

extract.svd_entropy(df_out["PCA_1"],svd_param)
[('Tau_"4"__De_3"', 0.7234823323374294),
 ('Tau_"4"__De_6"', 1.3014347840145244)]

(4) (13) (ii) Extracting - Streaks - Wozniak

woz_param = [{"consecutiveStar": n} for n in [2, 4]]

extract.wozniak(df_out["PCA_1"],woz_param)
[('consecutiveStar_2', 0.012658227848101266), ('consecutiveStar_4', 0.0)]

(4) (28) (i) Extracting - Fractal - Higuchi

hig_param = [{"Kmax": 3},{"Kmax": 5}]

extract.higuchi_fractal_dimension(df_out["PCA_1"],hig_param)
[('Kmax_3', 0.577913816027104), ('Kmax_5', 0.8176960510304725)]

(4) (5) (ii) Extracting - Volatility - Variability Index

var_index_param = {"Volume":df["Volume"].values, "Open": df["Open"].values}

extract.var_index(df["Close"].values,var_index_param)
{'Interact__Open': 0.00396022538846289,
 'Interact__Volume': 0.20550155114176533}

Time Series Extraction

pip install git+git://github.com/firmai/tsfresh.git
#Construct the preferred input dataframe.
from tsfresh.utilities.dataframe_functions import roll_time_series
df_out["ID"] = 0
periods = 30
df_out = df_out.reset_index()
df_ts = roll_time_series(df_out,"ID","Date",None,1,periods)
counts = df_ts['ID'].value_counts()
df_ts = df_ts[df_ts['ID'].isin(counts[counts > periods].index)]
#Perform extraction
from tsfresh.feature_extraction import extract_features, CustomFCParameters
settings_dict = CustomFCParameters()
settings_dict["var_index"] = {"PCA_1":None, "PCA_2": None}
df_feat = extract_features(df_ts.drop(["Close_1"],axis=1),default_fc_parameters=settings_dict,column_id="ID",column_sort="Date")
Feature Extraction: 100%|██████████| 5/5 [00:10<00:00,  2.14s/it]
# Cleaning operations
import pandasvault as pv
df_feat2 = df_feat.copy()
df_feat = df_feat.dropna(thresh = len(df_feat)*0.50, axis = "columns")
df_feat_cons = pv.constant_feature_detect(data=df_feat,threshold=0.9)
df_feat = df_feat.drop(df_feat_cons, axis=1)
df_feat = df_feat.ffill()
df_feat = pd.merge(df_feat,df[["Close_1"]],left_index=True,right_index=True,how="left")
print(df_feat.shape)
model(df_feat) #noisy
7  variables are found to be almost constant
(208, 48)
2064.7813982935995
from tsfresh import select_features
from tsfresh.utilities.dataframe_functions import impute

impute(df_feat)
df_feat_2 = select_features(df_feat.drop(["Close_1"],axis=1),df_feat["Close_1"],fdr_level=0.05)
df_feat_2["Close_1"] = df_feat["Close_1"]
model(df_feat_2) #improvement (b/ not an augmentation method)
1577.5273071299482

(3) (6) (i) Feature Agglomoration;   (1)(2)(i) Standard Scaler.

Like in this step, after (1), (2), (3), (4) and (5), you can often circle back to the initial steps to normalise the data and dimensionally reduce the data for the final model.

import numpy as np
from sklearn import datasets, cluster

def feature_agg(df, drop, components):
  components = min(df.shape[1]-1,components)
  agglo = cluster.FeatureAgglomeration(n_clusters=components,)
  df = df.drop(drop,axis=1)
  agglo.fit(df)
  df = pd.DataFrame(agglo.transform(df))
  df = df.add_prefix('fe_agg_')

  return df

df_final = transform.standard_scaler(df_feat_2, drop=["Close_1"])
df_final = mapper.feature_agg(df_final,["Close_1"],4)
df_final.index = df_feat.index
df_final["Close_1"] = df_feat["Close_1"]
model(df_final) #noisy
1949.89085894338

Final Model After Applying 13 Arbitrary Augmentation Techniques

model(df_final) #improvement
1949.89085894338

Original Model Before Augmentation

df_org = df.iloc[:,:7][df.index.isin(df_final.index)]
model(df_org)
389.783990984133

Best Model After Developing 8 Augmenting Features

df_best = df_best.replace([np.inf, -np.inf], np.nan).ffill().fillna(0)
model(df_best)
267.1826850968307

Commentary

There are countless ways in which the current model can be improved, this can take on an automated process where all techniques are tested against a hold out set, for example, we can perform the operation below, and even though it improves the score here, there is a need for more robust tests. The skeleton example above is not meant to highlight the performance of the package. It simply serves as an example of how one can go about applying augmentation methods.

Quite naturally this example suffers from dimensionality issues with array shapes reaching (208, 48), furthermore you would need a sample that is at least 50-100 times larger before machine learning methods start to make sense.

Nonetheless, in this example, Transformation, Interactions and Mappings (applied to extraction output) performed fairly well. Extraction augmentation was overkill, but created a reasonable model when dimensionally reduced. A better selection of one of the 50+ augmentation methods and the order of augmentation could further help improve the outcome if robustly tested against development sets.

[1] DeltaPy Development

Author: firmai
Source Code: https://github.com/firmai/deltapy

#engineering 

DeltaPy⁠⁠ — Tabular Data Augmentation & Feature Engineering
Ian  Robinson

Ian Robinson

1629539411

3 Bloody Lessons I Learned As A Big Data Engineer

The Dos and Don’ts of data engineering
Data has become an essential asset in every industry. From healthcare to e-commerce, retail to automobiles. This trend has created an overgrown population of data workers. Their responsibilities are to craft and maintain reliable, stable, and fault-tolerant data pipelines. A data pipeline consists of many phases: data ingestion, data transformation, data storage. At each of those stages, the engineered system must work as its intended purpose. But it is one of the core professionals of technology corporations. For aspiring and junior data engineers, things could be intimidating at first. There are many technical problems when handling a massive amount of data.

#big-data #engineering 

3 Bloody Lessons I Learned As A Big Data Engineer
Siphiwe  Nair

Siphiwe Nair

1629170880

Learn All About Data Engineer: Definition, Responsibilities, Challenge

In this post, I will explain the data roles that exist today and in particular — who is a data engineer? What are the role definition, responsibilities, and challenges contained in it?

Some will confuse it with DevOps, or data analytics, or data science. Some people will think this is the new branding for the mythological role of Database Architect .
 

#database #data #big-data #engineering 

Learn All About Data Engineer: Definition, Responsibilities, Challenge
Siphiwe  Nair

Siphiwe Nair

1629163440

What Data Engineers Do and Skills Needed ( Beginner's Guide )

Data engineering is a fast-growing field in the world of AI and data. In this article, we shine a spotlight on the role of Data Engineer, based on information shared by industry coaches Nana Essuman and Femi Anthony during the Black and Brilliant AI Accelerator program. Nana is the Director of Data Engineering at Condé Nast, and Femi Anthony is a Lead Data Engineer at Capital One. At a high level, Data Engineers play an important role in helping companies make data-driven decisions by collecting, transforming, and publishing data.

Data Engineers work behind the scenes to create the databases that house a company’s data. They build pipelines that transform raw data into formats that are useful for Data Scientists. «Data Engineers are really out there to automate and scale things, to essentially help take this to the next level,» Femi says.

#data #engineering #big-data 

What Data Engineers Do and Skills Needed ( Beginner's Guide )
Mariya James

Mariya James

1628506491

Popular React Native Apps Trends For Enterprise In 2021

Why is there been a massive increase in people showing interest in React Native App in the past few years?

According to Statista, 42 percent of the developers used React Native in 2020 to program mobile apps. React Native is a robust cross-platform mobile app development tool to create apps quickly.

Read here in detail: https://bit.ly/2VFnzFh

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#appdevelopment #xamarin #computerprogramming #reactnative, #reactnativeapplication, #technology #trends, #enterprise, #software, #coding, #engineering #nativeappdevelopment #NativeApp #Android #iphoneapps #crossplatformappdevelopment #mobileappdevelopment

Popular React Native Apps Trends For Enterprise In 2021
Sheldon  Grant

Sheldon Grant

1626122100

GitHub Availability Report: June 2021

In June, we experienced no incidents resulting in service downtime to our core services.

Please follow our  status page for real time updates and watch our blog for next month’s availability report.

#engineering #github

GitHub Availability Report: June 2021

Top Engineering Colleges in Pune

This is image title

Engineering is a highly sought-after career choice in India. There are many career options in engineering for students. DY Institute of Engineering is renowned for its high-quality facilities and is considered to be one of the top engineering colleges in Pune.

Our student-centered culture emphasizes high-quality research and consistent learning. DY Institute of Engineering is the best private engineering college in Pune, India. This is due to its strong alumni network both in India and around the world.

Our students benefit from a holistic learning environment.

Why Choose DY Institute of Engineering - One of the Top 5 Engineering Colleges in Pune

It is important to select the right university for engineering studies. Engineering courses are offered by many universities and colleges in Pune.

DY Institute of Engineering is the best. DY Institute of Engineering is the best engineering college in Pune for many reasons. These are the amazing facts you should know about DY Institute of Engineering.

1. Campus of a state-of-the-art college

Best Engineering college in Pune- State-of-the-art campus

DY Institute of Engineering covers 192 acres of sprawling, eco-friendly land. It boasts an impressive green campus and excellent infrastructure. Our campus has everything necessary to offer the best possible learning experience for students.

Each corner of the campus was designed to provide the best learning environment for students and instructors. Indian Green Building Council has designated the campus as a Gold Rated Campus.

2. The best-in-class infrastructure

Best Engineering college in Pune - Best infrastructure

There are many books and publications available in the central library. There are thousands of books on a variety of subjects and specializations.

This knowledge hub was created for students to research outside of class. Each branch’s study needs are met by our workshops and labs. This allows for more hands-on practice and practical knowledge of various subjects across different branches.

3. A Home Away from Home

Best Engineering college in Pune - A Home Away from Home

DY Institute of Engineering is not only about education, but also about students in the classrooms. Hostels are a home away from home and are made to be comfortable for students who live on campus.

Students can choose between AC or non-AC rooms. Students have access to healthcare facilities 24 hours a day. We also offer food courts and other facilities to ensure a great stay.

Related Post: How to Choose the Best Private Engineering Colleges In India

4. Prestigious tie-ups

Preeminent tie-ups – Best Engineering college in Pune

DY Institute of Engineering’s application-based learning and teaching experience make it stand out from other universities and colleges. This makes DY Institute of Engineering the best college for engineering. We are proud of our partnerships with some of the world’s most respected institutions.

To encourage students to strive for greater, we have 1500+ industry partners, International Transfer Programmes, and Semester Abroad Programmes.

5. Fully flexible credit system

Fully Flexible Credit System – Best Engineering college in Pune
This facility allows students to choose their course, faculty, timing, and place of study. Students also have the opportunity to switch branches during their second year. It makes it easier for students and DY Institute of Engineering is the best college of engineering.

Engineering requires dedication and perseverance. Engineering is a popular choice for many students. However, not everyone is interested in one field.

DY Institute of Engineering provides a variety of specializations within the engineering field to meet this need. This makes DY Institute of Engineering the top-ranked college in engineering. There are many options and flexibility. Here is a list of engineering specializations and fields that DY Institute of Engineering offers.

DY Institute of Engineering UG Engineering Programmes

These Programmes last for four years. These Engineering Programmes provide students with the necessary knowledge in their chosen fields.

B. Tech Civil Engineering
B. Tech Computer Science and Engineering
B. Tech Electrical and Electronics Engineering
B. Tech Electronics and Communication Engineering
B. Tech Electronics and Computer Engineering
B. Tech Fashion Technology
B. Tech Mechanical Engineering
B. Tech Mechatronics and Automation

DY Institute of Engineering PG Engineering Programmes
Best Engineering college in Pune

The PG Programmes last for two years. These Programmes are designed for students who want to go beyond the UG graduation program. The following Programmes are available at DY Institute of Engineering under PG.

M. Tech Computer Science and Engineering
M. Tech Computer Science and Engineering, Specialization in Big Data Analytics
M. Tech Computer Science and Engineering, Specialization in Artificial Intelligence and Machine Learning
M. Tech Computer Science and Engineering, Specialization in Cyber-Physical Systems
M. Tech CAD/CAM
M. Tech Embedded Systems
M. Tech Mechatronics
M. Tech Structural Engineering
M. Tech VLSI Design

PG Integrated Engineering Programmes at DY Institute of Engineering

DY Institute of Engineering Integrated Engineering Programme - Best Engineering College in Pune

Many students choose to study UG or PG programs, but some prefer integrated programs. This allows them to be more focused and saves time as they pursue engineering. This is a list of DY Institute of Engineering’s integrated engineering Programs, which are for 5 years.

M. Tech Software Engineering
M. Tech Computer Science and Engineering, with a specialization in Business Analytics
M. Tech Construction Technology and Management

#engineering #institute #college

Build & Deploy a Telegram Bot with short-term and long-term memory

Create a Chatbot from scratch that remembers and reminds events with Python

Summary

In this article, using Telegram and Python, I will show how to build a friendly Bot with multiple functions that can chat with question-answering conversations (short-term information) and store user data to recall in the future (long-term information).

All this started because a friend of mine yelled at me for not remembering her birthday. I don’t know if that has ever happened to you. So I thought I could pretend I remember birthdays while I actually have a Bot doing it for me. Now I know what you’re thinking, why building something from scratch instead of using one of the millions of calendar apps around? And you’re right, but for nerds like us … what’s the fun in that?

Through this tutorial, I will explain step by step how to build an intelligent Telegram Bot with Python and MongoDB and how to deploy it for free with Heroku and Cron-Job,using my Dates Reminder Bot as an example(link below).

I will present some useful Python code that can be easily applied in other similar cases (just copy, paste, run) and walk through every line of code with comments so that you can replicate this example (link to the full code below).

In particular, I will go through:

  • Setup: architecture overview, new _Telegram _Bot generation, _MongoDB _connection, _Python _environment.
  • Front-end: code the Bot commands for user interaction with pyTelegramBotAPI.
  • Back-end: create the server-side app with _flask _and threading.
  • Deploy the Bot through Heroku and Cron-Job

#artificial-intelligence #programming #web-development #chatbots #engineering #build & deploy a telegram bot with short-term and long-term memory

Build & Deploy a Telegram Bot with short-term and long-term memory
Sheldon  Grant

Sheldon Grant

1623299424

GitHub Availability Report: May 2021

Introduction

In May, we experienced two incidents resulting in significant impact and degraded state of availability for API requests, GitHub Pages, GitHub Actions and the GitHub Packages service, specifically the GitHub Packages Container registry service.

May 8 06:46 UTC (46 minutes)

This  incident was caused by failures in an underlying MySQL database, which caused some operations to time out for the GitHub Container registry service. During this incident, some customers viewing packages in the UI or interacting with the registry through “docker push” and “docker pull” may have experienced failures as the engineering team investigated the incident. After performing a failover to one of our database replicas, the affected systems were properly restored.

Our internal engineering team is now prioritizing work that will help ensure reduced impact to customers should such underlying outages happen again. This work includes creating internal documentation, dashboards, and enhanced alerts to quickly triage the cause of operation failures. We will also continue to actively maintain and increase replicas in different regions and availability zones that serve as a line of defense against unexpected region outages.

#engineering #github

GitHub Availability Report: May 2021
Ruth  Nabimanya

Ruth Nabimanya

1623224340

Relational vs Non-Relational Databases

If we have a look at available databases, we see a clear line, drawn in the middle. On one side, relational systems like MySQL, PostgreSQL, or SQLite. On the other side, non-relational ones like MongoDB or Neo4j. In this article you will learn the difference between them and which ones, you should use when!

Relational data

Non-relational data

When to use what?

Can I mix multiple databases?

#engineering #database #nosql #coding #sql #relational vs non-relational databases

Relational vs Non-Relational Databases
Nigel  Uys

Nigel Uys

1623124123

5 Reasons Why You Should Choose Golang in 2021

Simple, Productive, And Best Programming language to choose

The new language is known as Golang Programming Go Language. Recently Golang has become a topic of excitement for developers around the world because of its features.

And nowadays, many software companies use Golang. It is a powerfully simple and the first language trusted by Tech Heavyweights, Dropbox, Docker, Facebook, Netflix, Uber, and Twitter.

Many tech industries claiming Goland developers. And developers are leaning towards this language. Many programmers think Go is the best programming language.

“The more i think about what a good language should do for us, i am more and more convinced that Go is the best language we have so far.” “Ruby deliberately fights against the SOLID principles, whereas Go encourages them” — Steven Degutis, former Go-skeptic

Yet you may wonder why you will learn Golang. Well, I will give you five reasons why you should choose Golang. So let’s started right now-

1.Simplicity and speed

2.Productivity

3.Reliability

4.Robustness

5.Maturity

#code #programming #developer #engineering

5 Reasons Why You Should Choose Golang in 2021

Addressing Concerns About Being a DevOps Engineer

DevOps engineer is now a popular job, but what is their role in an organization? With the correct mindset and confidence, we can help a company to achieve real DevOps.

In the experience of leading a DevOps team, some team members have the concern about being a DevOps engineer:

Concern #1: Is it better to be a developer because writing programs looks more competitive? However, we seem to be just doing the integration of various tools.

Concern #2: Only developers deliver the value to end user because they face the market requirements, but we not.

Concern #3: Is the DevOps team no longer needed after completing the tool chain?

#devops #engineering

Addressing Concerns About Being a DevOps Engineer
Bailee  Streich

Bailee Streich

1622822400

Building an M&M Colour Classifier

One of my third year Electronic and Electrical Engineering projects was to build a machine which can sort M&Ms according to their colour. The final version could sort approximately 47 sweets per minute, and won our team a few bottles of beers for our work.

Perhaps one day I will write an article detailing that process, but what I want to talk about today is the colour classification; how it classified sweets originally, and two years later, using my new-found data science knowledge to solve this problem. I’ve been using the test data as a proving ground for experimenting with various machine learning techniques as I learn them, so this blog aims to document my learning process.

The data was gathered by running M&Ms through the machine, jotting down the red, green and blue values the colour sensor returned, and logging the colour of the sweet. A long and tedious process indeed. The classification process was entirely manual; I examined where the clusters presided, and set up bounding boxes to classify sweets that fell inside them.

This was functional enough for an electronics project (and enough to win the beer), but it came with a host of problems with hacky workaround solutions.

To begin with, the machine had no idea how to deal with outliers — it would give the sweets a jiggle and rescan them, and if that didn’t work, it threw them in a waste bin. The shapes and orientations of the distributions were also not considered. This became problematic especially for red and orange sweets, as their bounding boxes intersected. If a sweet fell in the intersection region, the machine would jiggle and rescan until it fell into the exclusive red or orange boxes (the red box wins in this demo). The machine also hard classified sweets, which caused many red/orange mixups. Let’s look at the confusion matrix for this technique.

#numpy #python #classification #engineering #machine-learning

Building an M&M Colour Classifier

The evolving role of Operations in DevOps

This is the third blog post in our series of DevOps fundamentals. For a quick intro on what DevOps is, check out part one; for a primer on automation in DevOps, visit part two.

As businesses reorganize for DevOps, the responsibilities of teams throughout the software lifecycle inevitably shift. Operations teams that traditionally measure themselves on uptime and stability—often working in silos separate from business and development teams—become collaborators with new stakeholders throughout the software lifecycle. Development and operations teams begin to work closely together to build and continually improve their delivery and management processes. In this blog post, we’ll share more on what these evolving roles and responsibilities look like for IT teams today, and how operations help drive consistency and success across the entire organization.

The Ops role in DevOps compared to traditional IT operations

To better understand how DevOps changes the responsibilities of operations teams, it will help to recap the traditional, pre-DevOps role of operations. Let’s take a look at a typical organization’s software lifecycle: before DevOps, developers package an application with documentation, and then ship it to a QA team. The QA teams install and test the application, and then hand off to production operations teams. The operations teams are then responsible for deploying and managing the software with little-to-no direct interaction with the development teams.

#engineering #devops

The evolving role of Operations in DevOps
Oral  Brekke

Oral Brekke

1622101260

Why (and How) GitHub Is Adopting OpenTelemetry

Over the years, GitHub engineers have developed many ways to observe how our systems behave. We mostly make use of statsd for metrics, the syslog format for plain text logs and OpenTracing for request traces. While we have somewhat standardized what we emit, we tend to solve the same problems over and over in each new system we develop.

And, while each component serves its individual purpose well, interoperability is a challenge. For example, several pieces of GitHub’s infrastructure use different statsd dialects, which means we have to special-case our telemetry code in different places – a non-trivial amount of work!

Different components can use different vocabularies for similar observability concepts, making investigatory work difficult. For example, there’s quite a bit of complexity around following a GitHub.com request across various telemetry signals, which can include metrics, traces, logs, errors and exceptions. While we emit a lot of telemetry data, it can still be difficult to use in practice.

We needed a solution that would allow us to standardize telemetry usage at GitHub, while making it easy for developers around the organization to instrument their code. The  OpenTelemetry project provided us with exactly that!

Introducing OpenTelemetry

OpenTelemetry introduces a common, vendor-neutral format for telemetry signals: OTLP. It also enables telemetry signals to be easily correlated with each other.

Moreover, we can reduce manual work for our engineers by adopting the OpenTelemetry SDKs – their inherent extensibility allows engineers to avoid re-instrumenting their applications if we need to change our telemetry collection backend, and with only one client SDK per-language we can easily propagate best practices among codebases.

OpenTelemetry empowers us to build integrated and opinionated solutions for our engineers. Designing for observability can be at the forefront of our application engineer’s minds, because we can make it so rewarding.

What we’re working on, and why

At the moment, we’re focusing on building out excellent support for the OpenTelemetry tracing signal. We believe that tracing your application should be the main entry point to observability, because we live in a distributed-systems world and tracing illuminates this in a way that’s almost magical.

We believe that tracing allows us to naturally and easily add/derive additional signals once it’s in place: many metrics, for example, can be calculated by backends automatically, tracing events can be converted to detailed logs automatically, and exceptions can automatically be reported to our tracking systems.

#engineering #github

Why (and How) GitHub Is Adopting OpenTelemetry