Hyperplane

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1. REDIRECT Hyperplane

Introduction

The Template:Short description is an essential MediaWiki template designed to provide concise summaries and descriptions for MediaWiki pages. This template plays an important role in organizing and displaying information on pages related to subjects such as Binary Options, IQ Option, and Pocket Option among others. In this article, we will explore the purpose and utilization of the Template:Short description, with practical examples and a step-by-step guide for beginners. In addition, this article will provide detailed links to pages about Binary Options Trading, including practical examples from Register at IQ Option and Open an account at Pocket Option.

Purpose and Overview

The Template:Short description is used to present a brief, clear description of a page's subject. It helps in managing content and makes navigation easier for readers seeking information about topics such as Binary Options, Trading Platforms, and Binary Option Strategies. The template is particularly useful in SEO as it improves the way your page is indexed, and it supports the overall clarity of your MediaWiki site.

Structure and Syntax

Below is an example of how to format the short description template on a MediaWiki page for a binary options trading article:

Parameter	Description
Description	A brief description of the content of the page.
Example	Template:Short description: "Binary Options Trading: Simple strategies for beginners."

The above table shows the parameters available for Template:Short description. It is important to use this template consistently across all pages to ensure uniformity in the site structure.

Step-by-Step Guide for Beginners

Here is a numbered list of steps explaining how to create and use the Template:Short description in your MediaWiki pages: 1. Create a new page by navigating to the special page for creating a template. 2. Define the template parameters as needed – usually a short text description regarding the page's topic. 3. Insert the template on the desired page with the proper syntax: . Make sure to include internal links to related topics such as Binary Options Trading, Trading Strategies, and Finance. 4. Test your page to ensure that the short description displays correctly in search results and page previews. 5. Update the template as new information or changes in the site’s theme occur. This will help improve SEO and the overall user experience.

Practical Examples

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Example: IQ Option Trading Guide

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• Binary Options
• Binary Options Trading
• Trading Strategies
• Forex Trading
• Finance
• Investment Basics

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Recommendations and Practical Tips

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Conclusion

The Template:Short description provides a powerful tool to improve the structure, organization, and SEO of MediaWiki pages, particularly for content related to binary options trading. Utilizing this template, along with proper internal linking to pages such as Binary Options Trading and incorporating practical examples from platforms like Register at IQ Option and Open an account at Pocket Option, you can effectively guide beginners through the process of binary options trading. Embrace the steps outlined and practical recommendations provided in this article for optimal performance on your MediaWiki platform.

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• • Financial Disclaimer**

The information provided herein is for informational purposes only and does not constitute financial advice. All content, opinions, and recommendations are provided for general informational purposes only and should not be construed as an offer or solicitation to buy or sell any financial instruments.

Any reliance you place on such information is strictly at your own risk. The author, its affiliates, and publishers shall not be liable for any loss or damage, including indirect, incidental, or consequential losses, arising from the use or reliance on the information provided.

Before making any financial decisions, you are strongly advised to consult with a qualified financial advisor and conduct your own research and due diligence. Template:Infobox mathematical concept

Hyperplane

In mathematics, particularly in linear algebra and geometry, a hyperplane is a generalization of a plane to higher dimensions. While a plane is a two-dimensional subspace that divides three-dimensional space into two halves, a hyperplane is a (n-1)-dimensional subspace that divides an n-dimensional space into two halves. Essentially, it's the "one dimension less" analog of the entire space. Understanding hyperplanes is crucial for grasping concepts in various fields, including machine learning, computer graphics, and optimization.

Definition

Formally, let *V* be a vector space of dimension *n* over a field *F*. A hyperplane *H* in *V* is a subspace of dimension *n* − 1. Equivalently, a hyperplane can be defined as the set of all solutions to a single linear equation in *n* variables.

More concretely, consider a vector space *V* = *F<sup>n</sup>*. A hyperplane *H* in *V* can be described by an equation of the form:

a<sub>1</sub>x<sub>1</sub> + a<sub>2</sub>x<sub>2</sub> + ... + a<sub>n</sub>x<sub>n</sub> = b

where *x<sub>1</sub>*, *x<sub>2</sub>*, ..., *x<sub>n</sub>* are the coordinates in *V*, *a<sub>1</sub>*, *a<sub>2</sub>*, ..., *a<sub>n</sub>* are coefficients (not all zero), and *b* is a scalar. The coefficients (a<sub>1</sub>, a<sub>2</sub>, ..., a<sub>n</sub>) define the normal vector to the hyperplane. If all the coefficients are zero, the equation becomes 0 = b, which is only true if b = 0, and this does not define a hyperplane.

Examples

• **In 2D (n=2):** A hyperplane is a line. The equation looks like *ax + by = c*. For example, *x + y = 1* represents a line in the 2D plane. This line divides the plane into two regions.
• **In 3D (n=3):** A hyperplane is a plane. The equation looks like *ax + by + cz = d*. For example, *2x - y + z = 5* represents a plane in 3D space. This plane divides 3D space into two regions.
• **In 4D (n=4):** A hyperplane is a 3-dimensional subspace within a 4-dimensional space. It’s difficult to visualize directly, but we can still describe it mathematically as *ax + by + cz + dw = e*.
• **In general (n>3):** While visualization becomes impossible, the concept remains the same: an (n-1)-dimensional flat subspace within an n-dimensional space.

Geometric Interpretation

A hyperplane divides the vector space it resides in into two *half-spaces*. These half-spaces are defined by the inequality:

a<sub>1</sub>x<sub>1</sub> + a<sub>2</sub>x<sub>2</sub> + ... + a<sub>n</sub>x<sub>n</sub> < b

and

a<sub>1</sub>x<sub>1</sub> + a<sub>2</sub>x<sub>2</sub> + ... + a<sub>n</sub>x<sub>n</sub> > b

Points that satisfy the equation exactly (a<sub>1</sub>x<sub>1</sub> + a<sub>2</sub>x<sub>2</sub> + ... + a<sub>n</sub>x<sub>n</sub> = b) lie *on* the hyperplane.

The vector (a<sub>1</sub>, a<sub>2</sub>, ..., a<sub>n</sub>) is a normal vector to the hyperplane. This means it's orthogonal (perpendicular) to every vector lying in the hyperplane. The normal vector defines the orientation of the hyperplane. The distance from the origin to the hyperplane is determined by *b* and the magnitude of the normal vector.

Relation to Affine Subspaces

A hyperplane is a specific type of affine subspace. An affine subspace is a set of points that can be described by a linear equation. A hyperplane is an affine subspace of dimension *n* − 1. All hyperplanes are affine subspaces, but not all affine subspaces are hyperplanes. An affine subspace could have a dimension less than *n* − 1.

Applications

Hyperplanes find applications in numerous areas. Here are a few key examples:

• **Machine Learning:** In support vector machines (SVMs), hyperplanes are used to separate data points belonging to different classes. The goal is to find the hyperplane that maximizes the margin (distance) between the closest points from each class. This is a fundamental concept in supervised learning. Decision trees can also be conceptually viewed as creating hyperplanes to partition the feature space. Further, Principal Component Analysis (PCA) relies on identifying hyperplanes that capture the most variance in the data.
• **Computer Graphics:** Hyperplanes are used to define clipping planes, which determine what parts of a scene are visible to the camera. They are also used in collision detection algorithms.
• **Optimization:** In linear programming, constraints are often expressed as linear inequalities, which define hyperplanes. The feasible region is the intersection of these hyperplanes. Simplex algorithm utilizes hyperplanes to iteratively improve the solution.
• **Statistics:** Hyperplanes can be used to define decision boundaries in statistical classification problems.
• **Finance:** Hyperplanes are used in portfolio optimization to define efficient frontiers. They can also be used to model risk and return relationships. Value at Risk (VaR) calculations can utilize hyperplane concepts when dealing with multidimensional risk factors.

Hyperplanes and Linear Transformations

Linear transformations can map hyperplanes to other hyperplanes. If *T*: *V* → *W* is a linear transformation, and *H* is a hyperplane in *V*, then *T(H)* is a hyperplane in *W* unless *T* collapses *H* to a lower dimension. This property is important in understanding how geometric objects are transformed under linear operations.

Distance from a Point to a Hyperplane

Given a point *p* = (p<sub>1</sub>, p<sub>2</sub>, ..., p<sub>n</sub>) and a hyperplane defined by *a<sub>1</sub>x<sub>1</sub> + a<sub>2</sub>x<sub>2</sub> + ... + a<sub>n</sub>x<sub>n</sub> = b*, the distance *d* from *p* to the hyperplane is given by:

d = |a<sub>1</sub>p<sub>1</sub> + a<sub>2</sub>p<sub>2</sub> + ... + a<sub>n</sub>p<sub>n</sub> - b| / √(a<sub>1</sub><sup>2</sup> + a<sub>2</sub><sup>2</sup> + ... + a<sub>n</sub><sup>2</sup>)

This formula is a generalization of the distance formula from a point to a line in 2D and a point to a plane in 3D.

Intersecting Hyperplanes

The intersection of two hyperplanes in an *n*-dimensional space is typically a subspace of dimension *n* − 2 (a line if the hyperplanes are distinct, a hyperplane if they are the same). More generally, the intersection of *k* hyperplanes in an *n*-dimensional space is a subspace of dimension *n* − *k*, provided the hyperplanes are in general position (i.e., not all parallel or coincident).

Parametric Representation

A hyperplane can also be represented parametrically. Given a point *p*<sub>0</sub> on the hyperplane and *n* − 1 linearly independent vectors *v*<sub>1</sub>, *v*<sub>2</sub>, ..., *v*<sub>n-1</sub> parallel to the hyperplane, any point *x* on the hyperplane can be written as:

x = p<sub>0</sub> + t<sub>1</sub>v<sub>1</sub> + t<sub>2</sub>v<sub>2</sub> + ... + t<sub>n-1</sub>v<sub>n-1</sub>

where *t<sub>1</sub>*, *t<sub>2</sub>*, ..., *t<sub>n-1</sub>* are scalar parameters.

Hyperplanes in Probability and Statistics

In statistics, hyperplanes are used to define decision boundaries in classification problems. For example, in logistic regression, the decision boundary is a hyperplane that separates the different classes based on the predicted probabilities. Discriminant analysis also employs hyperplanes for classification. Bayesian networks can utilize hyperplane concepts to define conditional independence relationships.

Relationship to Technical Analysis

While not a direct application, the concept of hyperplanes can be metaphorically applied to technical analysis in financial markets. Support and resistance levels can be thought of as "planes" (or hyperplanes in multidimensional analysis of multiple indicators) defining boundaries beyond which price movement is expected to change. Trendlines similarly act as linear boundaries. Fibonacci retracement levels can be visualized as a series of parallel lines (hyperplanes in a higher-dimensional space of price and time). Bollinger Bands create a dynamic "band" around a moving average, effectively defining a region bounded by two curves that can be approximated by hyperplanes. Ichimoku Cloud is a more complex indicator, but its components also define regions bounded by curves that can be related to hyperplane concepts. Elliott Wave Theory relies on identifying patterns that can be geometrically represented using lines and curves, which can be thought of as components of hyperplanes. Candlestick patterns create visual signals, but their underlying price movements can be analyzed using hyperplane-like boundaries. Moving averages smooth out price data, effectively creating a hyperplane-like representation of the average price. Relative Strength Index (RSI) uses boundaries to identify overbought and oversold conditions, which can be visualized as hyperplanes. MACD uses signal lines and histograms, which can be interpreted as components of hyperplanes. Stochastic Oscillator uses boundaries to indicate momentum, akin to hyperplane boundaries. Average True Range (ATR) measures volatility, which influences the width of trading ranges and can be related to the distance between hyperplanes representing support and resistance. Donchian Channels define upper and lower bounds, similar to hyperplane boundaries. Parabolic SAR uses a parabolic curve, which can be approximated by hyperplanes. Volume Weighted Average Price (VWAP) provides a weighted average price, forming a hyperplane-like representation of price. On Balance Volume (OBV) uses volume to confirm trends, creating a visual signal that can be analyzed with hyperplane concepts. Accumulation/Distribution Line (A/D Line) measures buying and selling pressure, forming a line that can be related to hyperplane boundaries. Chaikin Oscillator uses moving averages of A/D Line, creating a signal that can be interpreted with hyperplane concepts. Rate of Change (ROC) measures the percentage change in price, which can be visualized as a line representing a hyperplane. Williams %R identifies overbought and oversold conditions, forming boundaries similar to hyperplanes.

Further Reading

• Linear Algebra
• Vector Space
• Affine Space
• Subspace
• Normal Vector
• Dot Product
• Machine Learning
• Optimization
• Support Vector Machines

File:Hyperplane 3D.svg

A hyperplane (plane) in 3D space.

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