1560831560

When starting with a new framework it might be hard to get to know all the things about it. Vue.js is an amazing framework that is powerful and easy to learn. Today I want to share with you a few tips and tricks that can be useful for your Vue projects.

It is not documented in Vue documentation, but each Vue instance has access to $createElement method, which can create and return virtual nodes. You could, for example, use it if you would like to prepare Markup in your methods and then pass it to ‘v-html’ directive. You also have access to this method in functional components, which receive it as the first parameter in the render function.

Imagine you have a news application and you have a component to display an article. This component could fetch an article for a route like “www.news.com/article/:articleId”. Usually, you would initialize an API call in “created” life-cycle hook to fetch article details.

You also have next and previous article functionality, which will let users go to other articles. When a user changes an article, nothing would happen, so that’s why we need a watcher to fetch data for a new article.

However, in this case we are calling ‘fetchArticle’ method in both “created” hook and watcher. Fortunately, we can change that by using ‘immediate’ property for the watcher.

This will result in the handler being invoked immediately when the component is created. Be aware, though, that immediate watcher handlers will be run just after anything in the ‘created’ hook. So, if for any reason you need to fetch data before something else happens in the created hook, then you will need to have it twice and resign from the ‘immediate’ prop.

In some instances, you might have a few different routes that are using the same component. However, by default if you change between those routes, the component will not be re-rendered. This is a normal thing as Vue is reusing the already existing component for performance reasons. However, if you want the component to be re-created, you can provide “:key” prop to the “” component.

This is another nicety which I think is still is not documented yet. Often if you are using a third-party plugin or need to add a custom event listener, you first define it in created or mounted hook and then remove it in “beforeDestroy” hook. It is a very important to clear out event listeners, so your code doesn’t cause memory leaks. Sometimes plugins might have a ‘destroy’ method.

With use of $on(‘hook:) you can avoid a definition of another life-cycle hook.

By default, v-model is the syntactic sugar feature over “@input” event listener and “:value” prop. Did you know that you can actually specify what event and value prop should be used? You can easily do that by specifying ‘model’ property in your Vue component.

Most of the time you might be fine with validating your props by providing String, Object, etc. However, props can also be validated with use of custom validators. For instance, if you expect to get a string that should match any string from a list of string, then you can do something like that:

Vue is using double curly brackets “{{ }}” for expressions in HTML files and templates. Unfortunately, it can collide with other engines — for example if you are using Jinja templates, which also do use double curly braces. Fortunately, Vue offers a way to change delimiters used in your templates, so you could use double brackets “[[ ]]” instead.

This is not really a tip, but something you should be using in your projects. Usually, if you have a component that is only accepting props, rendering markup and not using anything from Vue instance like life-cycle hooks, computed properties, methods, or data model, then you can provide the “functional” option and set it to true to indicate that this component should not have Vue instance. You can also do it by providing ‘functional’ prop on the template.

The benefit of functional components is they are much cheaper to re-render than stateful components. However, be careful when you wrap stateful components with functional components, as functional components are always re-rendered and will cause stateful components to be re-rendered as well.

Something for React lovers. Since release of Vue CLI 3, JSX is supported by default in Vue, and if your project is on an earlier version of Vue CLI, then you can use babel-plugin-transform-vue-jsx plugin. I often use it, especially in functional components since writing pure render functions can be quite tedious.

Snippets can be a real time saver as you can write code quickly. For instance, in Visual Studio Code with these two snippets configured, I can create base code for stateful and functional components by typing “vtemp” or “vfcomp”.

Not sure about other code editors, but if you are using Visual Studio Code, then you should certainly check out Vetur plugin. It provides quite useful features out of the box, like syntax-highlighting, linting and error checking, formatting, emmet, etc.

Most of the time there are components in a project which are used over and over and importing them in almost every component is quite tedious. Therefore, as they are used almost everywhere, they could be imported and registered just once, globally. You can think of them as ‘Base’ components of your application. This script can register all base components automatically.

Next, import this method in your main.js files and initialize it.

These are just a few tips and tricks that I hope you will find useful. Don’t forget to share with others any tips you have! Feel free to leave your own favorites in the comments below.

☞ Build a Basic CRUD App with Laravel and Vue

☞ Build a Simple CRUD App with Spring Boot and Vue.js

☞ ES5 to ESNext — here’s every feature added to JavaScript since 2015

☞ 5 Javascript (ES6+) features that you should be using in 2019

☞ Vue JS 2 - The Complete Guide (incl. Vue Router & Vuex)

☞ The Complete JavaScript Course 2018: Build Real Projects!

☞ Become a JavaScript developer - Learn (React, Node,Angular)

☞ JavaScript: Understanding the Weird Parts

☞ MERN Stack Front To Back: Full Stack React, Redux & Node.js

*Originally published by Thomas Findlay at **https://www.telerik.com*

#vue-js #javascript #web-development

1619518440

Welcome to my Blog , In this article, you are going to learn the top 10 python tips and tricks.

…

#python #python hacks tricks #python learning tips #python programming tricks #python tips #python tips and tricks #python tips and tricks advanced #python tips and tricks for beginners #python tips tricks and techniques #python tutorial #tips and tricks in python #tips to learn python #top 30 python tips and tricks for beginners

1593867420

Android Projects with Source Code – Your entry pass into the world of Android

Hello Everyone, welcome to this article, which is going to be really important to all those who’re in dilemma for their projects and the project submissions. This article is also going to help you if you’re an enthusiast looking forward to explore and enhance your Android skills. The reason is that we’re here to provide you the best ideas of Android Project with source code that you can choose as per your choice.

These project ideas are simple suggestions to help you deal with the difficulty of choosing the correct projects. In this article, we’ll see the project ideas from beginners level and later we’ll move on to intermediate to advance.

Before working on real-time projects, it is recommended to create a sample hello world project in android studio and get a flavor of project creation as well as execution: Create your first android project

Android Project: A calculator will be an easy application if you have just learned Android and coding for Java. This Application will simply take the input values and the operation to be performed from the users. After taking the input it’ll return the results to them on the screen. This is a really easy application and doesn’t need use of any particular package.

To make a calculator you’d need Android IDE, Kotlin/Java for coding, and for layout of your application, you’d need XML or JSON. For this, coding would be the same as that in any language, but in the form of an application. Not to forget creating a calculator initially will increase your logical thinking.

Once the user installs the calculator, they’re ready to use it even without the internet. They’ll enter the values, and the application will show them the value after performing the given operations on the entered operands.

Source Code: Simple Calculator Project

Android Project: This is a good project for beginners. A Reminder App can help you set reminders for different events that you have throughout the day. It’ll help you stay updated with all your tasks for the day. It can be useful for all those who are not so good at organizing their plans and forget easily. This would be a simple application just whose task would be just to remind you of something at a particular time.

To make a Reminder App you need to code in Kotlin/Java and design the layout using XML or JSON. For the functionality of the app, you’d need to make use of AlarmManager Class and Notifications in Android.

In this, the user would be able to set reminders and time in the application. Users can schedule reminders that would remind them to drink water again and again throughout the day. Or to remind them of their medications.

Android Project: Another beginner’s level project Idea can be a Quiz Application in android. Here you can provide the users with Quiz on various general knowledge topics. These practices will ensure that you’re able to set the layouts properly and slowly increase your pace of learning the Android application development. In this you’ll learn to use various Layout components at the same time understanding them better.

To make a quiz application you’ll need to code in Java and set layouts using xml or java whichever you prefer. You can also use JSON for the layouts whichever preferable.

In the app, questions would be asked and answers would be shown as multiple choices. The user selects the answer and gets shown on the screen if the answers are correct. In the end the final marks would be shown to the users.

Android Project: Tic-Tac-Toe is a nice game, I guess most of you all are well aware of it. This will be a game for two players. In this android game, users would be putting X and O in the given 9 parts of a box one by one. The first player to arrange X or O in an adjacent line of three wins.

To build this game, you’d need Java and XML for Android Studio. And simply apply the logic on that. This game will have a set of three matches. So, it’ll also have a scoreboard. This scoreboard will show the final result at the end of one complete set.

Upon entering the game they’ll enter their names. And that’s when the game begins. They’ll touch one of the empty boxes present there and get their turn one by one. At the end of the game, there would be a winner declared.

Source Code: Tic Tac Toe Game Project

Android Project: A stopwatch is another simple android project idea that will work the same as a normal handheld timepiece that measures the time elapsed between its activation and deactivation. This application will have three buttons that are: start, stop, and hold.

This application would need to use Java and XML. For this application, we need to set the timer properly as it is initially set to milliseconds, and that should be converted to minutes and then hours properly. The users can use this application and all they’d need to do is, start the stopwatch and then stop it when they are done. They can also pause the timer and continue it again when they like.

Android Project: This is another very simple project idea for you as a beginner. This application as the name suggests will be a To-Do list holding app. It’ll store the users schedules and their upcoming meetings or events. In this application, users will be enabled to write their important notes as well. To make it safe, provide a login page before the user can access it.

So, this app will have a login page, sign-up page, logout system, and the area to write their tasks, events, or important notes. You can build it in android studio using Java and XML at ease. Using XML you can build the user interface as user-friendly as you can. And to store the users’ data, you can use SQLite enabling the users to even delete the data permanently.

Now for users, they will sign up and get access to the write section. Here the users can note down the things and store them permanently. Users can also alter the data or delete them. Finally, they can logout and also, login again and again whenever they like.

Android Project: This app is aimed at the conversion of Roman numbers to their significant decimal number. It’ll help to check the meaning of the roman numbers. Moreover, it will be easy to develop and will help you get your hands on coding and Android.

You need to use Android Studio, Java for coding and XML for interface. The application will take input from the users and convert them to decimal. Once it converts the Roman no. into decimal, it will show the results on the screen.

The users are supposed to just enter the Roman Number and they’ll get the decimal values on the screen. This can be a good android project for final year students.

Android Project: Well, coming to this part that is Virtual Dice or a random no. generator. It is another simple but interesting app for computer science students. The only task that it would need to do would be to generate a number randomly. This can help people who’re often confused between two or more things.

Using a simple random number generator you can actually create something as good as this. All you’d need to do is get you hands-on OnClick listeners. And a good layout would be cherry on the cake.

The user’s task would be to set the range of the numbers and then click on the roll button. And the app will show them a randomly generated number. Isn’t it interesting ? Try soon!

Android Project: This application is very important for you as a beginner as it will let you use your logical thinking and improve your programming skills. This is a scientific calculator that will help the users to do various calculations at ease.

To make this application you’d need to use Android Studio. Here you’d need to use arithmetic logics for the calculations. The user would need to give input to the application that will be in terms of numbers. After that, the user will give the operator as an input. Then the Application will calculate and generate the result on the user screen.

Android Project: An SMS app is another easy but effective idea. It will let you send the SMS to various no. just in the same way as you use the default messaging application in your phone. This project will help you with better understanding of SMSManager in Android.

For this application, you would need to implement Java class SMSManager in Android. For the Layout you can use XML or JSON. Implementing SMSManager into the app is an easy task, so you would love this.

The user would be provided with the facility to text to whichever number they wish also, they’d be able to choose the numbers from the contact list. Another thing would be the Textbox, where they’ll enter their message. Once the message is entered they can happily click on the send button.

#android tutorials #android application final year project #android mini projects #android project for beginners #android project ideas #android project ideas for beginners #android projects #android projects for students #android projects with source code #android topics list #intermediate android projects #real-time android projects

1600583123

In this article, we are going to list out the most popular websites using Vue JS as their frontend framework.

Vue JS is one of those elite progressive JavaScript frameworks that has huge demand in the web development industry. Many popular websites are developed using Vue in their frontend development because of its imperative features.

This framework was created by Evan You and still it is maintained by his private team members. Vue is of course an open-source framework which is based on MVVM concept (Model-view view-Model) and used extensively in building sublime user-interfaces and also considered a prime choice for developing single-page heavy applications.

Released in February 2014, Vue JS has gained 64,828 stars on Github, making it very popular in recent times.

Evan used Angular JS on many operations while working for Google and integrated many features in Vue to cover the flaws of Angular.

“I figured, what if I could just extract the part that I really liked about Angular and build something really lightweight." - Evan You

#vuejs #vue #vue-with-laravel #vue-top-story #vue-3 #build-vue-frontend #vue-in-laravel #vue.js

1595547778

Developing a mobile application can often be more challenging than it seems at first glance. Whether you’re a developer, UI designer, project lead or CEO of a mobile-based startup, writing good project briefs prior to development is pivotal. According to **Tech Jury**, 87% of smartphone users spend time exclusively on mobile apps, with 18-24-year-olds spending 66% of total digital time on mobile apps. Of that, 89% of the time is spent on just 18 apps depending on individual users’ preferences, making proper app planning crucial for success.

Today’s audiences know what they want and don’t want in their mobile apps, encouraging teams to carefully write their project plans before they approach development. But how do you properly write a **mobile app development** brief without sacrificing your vision and staying within the initial budget? Why should you do so in the first place? Let’s discuss that and more in greater detail.

It’s worth discussing the significance of mobile app project briefs before we tackle the writing process itself. In practice, a project brief is used as a reference tool for developers to remain focused on the client’s deliverables. Approaching the development process without written and approved documentation can lead to drastic, last-minute changes, misunderstanding, as well as a loss of resources and brand reputation.

For example, developing a mobile app that filters restaurants based on food type, such as **Happy Cow**, means that developers should stay focused on it. Knowing that such and such features, UI elements, and API are necessary will help team members collaborate better in order to meet certain expectations. Whether you develop an app under your brand’s banner or outsource coding and design services to would-be clients, briefs can provide you with several benefits:

- Clarity on what your mobile app project “is” and “isn’t” early in development
- Point of reference for developers, project leads, and clients throughout the cycle
- Smart allocation of available time and resources based on objective development criteria
- Streamlined project data storage for further app updates and iterations

Depending on how “open” your project is to the public, you will want to write a detailed section about who the developers are. Elements such as company name, address, project lead, project title, as well as contact information, should be included in this introductory segment. Regardless of whether you build an in-house app or outsource developers to a client, this section is used for easy document storage and access.

#android app #ios app #minimum viable product (mvp) #mobile app development #web development #how do you write a project design #how to write a brief #how to write a project summary #how to write project summary #program brief example #project brief #project brief example #project brief template #project proposal brief #simple project brief template

1641276000

- ML-Quant.com - Automated Research Repository

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.

```
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
```

**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.

**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
```

```
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.

- Scaling/Normalisation
- Standardisation
- Differencing
- Capping
- Operations
- Smoothing
- Decomposing
- Filtering
- Spectral Analysis
- Waveforms
- Modifications
- Rolling
- Lagging
- Forecast Model

- Regressions
- Operators
- Discretising
- Normalising
- Distance
- Speciality
- Genetic

- Eigen Decomposition
- Cross Decomposition
- Kernel Approximation
- Autoencoder
- Manifold Learning
- Clustering
- Neighbouring

- Energy
- Distance
- Differencing
- Derivative
- Volatility
- Shape
- Occurrence
- Autocorrelation
- Stochasticity
- Averages
- Size
- Count
- Streaks
- Location
- Model Coefficients
- Quantile
- Peaks
- Density
- Linearity
- Non-linearity
- Entropy
- Fixed Points
- Amplitude
- Probability
- Crossings
- Fluctuation
- Information
- Fractals
- Exponent
- Spectral Analysis
- Percentile
- Range
- Structural
- Distribution

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.

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()
```

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()
```

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()
```

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()
```

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()
```

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()
```

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()
```

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()
```

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()
```

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()
```

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()
```

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()
```

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()
```

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.

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()
```

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()
```

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()
```

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()
```

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()
```

(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()
```

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()
```

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.

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)
```

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()
```

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()
```

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()
```

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()
```

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()
```

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()
```

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
```

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"])
```

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)
```

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"])
```

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"])
```

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)
```

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"])
```

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"])
```

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"])
```

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"])
```

(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"])
```

(i) Lenght

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

(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"])
```

(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"])
```

(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"])
```

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"])
```

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"])
```

(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"])
```

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"])
```

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"])
```

(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"])
```

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"])
```

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"])
```

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"])
```

(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"])
```

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"])
```

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"])
```

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"])
```

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"])
```

(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"])
```

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"])
```

(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"])
```

(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"])
```

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)
```

(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"])
```

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.

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.

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_1 | High | Low | Open | Close | Volume | Adj Close | Close_NDDT | Close_NDDS | Close_NDDR | Open_NDDT | Open_NDDS | Open_NDDR | Close_BPF | Low_BPF | Close_BPF_frac | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Date | ||||||||||||||||

2019-01-08 | 338.529999 | 1.018413 | 0.964048 | 1.096600 | 1.001175 | -0.162616 | 1.001175 | 0.832297 | 0.834964 | 1.335433 | 0.758743 | 0.691596 | 2.259884 | -2.534142 | -2.249135 | -3.593612 |

2019-01-09 | 344.970001 | 1.012068 | 1.023302 | 1.011466 | 1.042689 | -0.501798 | 1.042689 | 0.908963 | -0.165036 | 1.111346 | 0.835786 | 0.333361 | 1.129783 | -3.081959 | -2.776302 | -2.523465 |

2019-01-10 | 347.260010 | 1.035581 | 1.027563 | 0.996969 | 1.126762 | -0.367576 | 1.126762 | 1.029347 | 2.120026 | 0.853697 | 0.907588 | 0.000000 | 0.533777 | -2.052768 | -2.543449 | -0.747382 |

2019-01-11 | 334.399994 | 1.073153 | 1.120506 | 1.098313 | 1.156658 | -0.586571 | 1.156658 | 1.109144 | -5.156051 | 0.591990 | 1.002162 | -0.666639 | 0.608516 | -0.694642 | -0.831670 | 0.414063 |

2019-01-14 | 344.429993 | 0.999627 | 1.056991 | 1.102135 | 0.988773 | -0.541752 | 0.988773 | 1.107633 | 0.000000 | -0.660350 | 1.056302 | -0.915491 | 0.263025 | -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)*

- https://github.com/firmai/tsfresh/blob/master/tsfresh/feature_extraction/settings.py
- https://github.com/firmai/tsfresh/blob/master/tsfresh/feature_extraction/feature_calculators.py
- 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 **