In financial markets there are many different types of traders, both human and algorithmic. They are often categorized by their trading time-frame. Alternatively it is possible to categorize based on what kind of strategies they apply.

In general there is one main distinctions to make. There is a difference between technical and fundamental trading. Fundamental trading includes factors outside the price of the security; typically a company’s accounts is used. Technical trading however is only based on the actual price development and its qualities. Technical traders base their strategy on the assumption that historical price data can in some way predict future prices. The degrees to which analysis one employs affect the time-frame, since fundamental analysis is inherently much more static than technical. Price update occur several times per second, while the company results are only published once a quarter. Fundamental data is also more qualitative and thus harder to analyze quickly, e.g., news. Analysis of technical data is purely quantitative, and can thus be readily handled by computers. Generally, the shorter time-frame, the more automated the traders are.

The longest time-frame is often called position trading. It generally consists of all positions held longer than a few months. This is a very reputable strategy, recommended by Warren Buffet himself, but with a limited possibility of return. Since one has plenty of time to do a thorough analysis of the asset, algorithmic methods have less advantage here, but can be used as supplementary guidance. This category also includes more active investors, who are active in a different sense than earlier described. They take an active part in management of the company they’re invested in, like private equity companies.

Positions held over days or weeks include strategies like swing trading and more short-term fundamental analysis strategies. This is generally the shortest time-frame one finds purely fundamental analysis traders. Changes in price on a lower scale than days are often considered noise to these traders. There is some technical analysis done over several days, but it is limited. This is because the primary goal of pure technical analysts is to speculate on price swings. A shorter time frame will give more than enough volatility to do this, due to the fractal nature of the stock market. A fractal is a mathematical concept where roughness exist at whatever resolution you look at. The graph of price a different time periods will tend to have roughly the same kind of volatility.

Intraday trading is trading confined to a single day, most day-traders sell their positions before close. This is mainly speculation and mostly done by technical analysis, though one might supplement with fundamental techniques. Day trading can be a job for human investors, but require much time and is quite close to a zero sum game due to the lack of the long term equity premium. Intraday trading is becoming the realm of machines as they can search through much more data. Most machines are however trading on more or less instant strategies where the aim is to hold an actual position as short as possible; these are called High-frequency traders.

High-frequency trading (HFT) is a poorly defined term. In this paper it is used in its semantic meaning of all trading done at a high frequency instead of meddling with all the contradictory definitions in the literature. With this meaning of the term, HFT will be a broad concept that includes all trading where the trading speed is used as a competitive advantage. In 2010 these systems trade on milli- and microseconds. All HFT is done by algorithms, as the speeds required for this trading type are beyond human capabilities.

Low-latency trading is a subclass of HFT, and can be thought of as a specific strategy of HFT. These are a naive set of simple algorithms which sole advantage is their ultra-low latency, direct market access. An example of a low-latency strategy is to survey the news. The instance good news hits the market, a low-latency trader will buy from slower sellers which have not had time to increase their bid price, and then sell to a new higher price. Most of these tactics revolve around quickly finding and exploiting arbitrage possibilities. There is tough competition in having the fastest algorithms and the lowest latency. This activity is very capital intensive due to the hardware requirements and it needs a certain initial capital. There are few amateurs in this area and most actors are large banks, funds or even specialized algorithmic trading firms.

## Signal Emitter Positioning Using Multilateration

In this post we solve the problem of spatial location of a signal emitter in Euclidian $$\mathbb{R}^3$$ space by means of time difference of arrival (TDOA) at multiple omnidirectional sensors.

## Introduction

Multilateration is a technique that uses multiple omnidirectional sensors in order to isolate the unknown position of a signal emitter in two- or three-dimensional Euclidian space. Two examples of sensors and signals could be microphones listening for sharp noises, or radio receivers listening for radio signals. In any case, these sensors are located at unique known positions where they listen for what is called a signal event. Such events are timestamped based on a synchronized or centralized clock common to all sensors. The signal from an emitter is registered by all sensors only once as the signal wave expands spherically in all directions with constant propagation speed. The time difference between when two sensors register the signal event is called the time difference of arrival (TDOA). Based on the TDOA and the location of each registration, i.e., sensor positions, we can deduce the location of the signal emitter.

## Model definitions

Let sensors $$i=1,\ldots,k$$ at positions $$\vec{p}_{i}$$ be sensors that output a timestamp $$t_{i}$$ for when they register a signal wave with propagation speed $$v$$ from a signal emitter at unknown position $$\vec{x}$$, signal start time $$t$$ and distance $$v\left(t_{i}-t\right)=\|\vec{x}-\vec{p}_{i}\|$$ away. If we define sensor $$c$$ to be the first sensor to register a signal event, then the signal wave has traveled for $$(t_{c}-t)$$ time units and a distance of $$v(t_{c}-t)=\|\vec{x}-\vec{p}_{c}\|$$ before reaching the first sensor. Furthermore, the TDOA between each sensor’s timestamp and the initial sensor’s timestamp is $$(t_{i}-t_{c})$$ yielding the the distance $$v\Delta t_{i}=v(t_{i}-t_{c})$$ traveled by the signal wave since initially being registered and finally reaching the $$i$$th sensor. From the definitions above we have the following important relationships.

\|\vec{x}-\vec{p}_{i}\|=v\left(t_{i}-t\right)\label{eq:main}

and the equivalent formulations

\begin{split}\|\vec{x}-\vec{p}_{i}\|=v(t_{i}-t_{c})+\|\vec{x}-\vec{p}_{c}\|\end{split}
\label{eq:mainSplit}

\|\vec{x}-\vec{p}_{i}\|=v(t_{i}-t_{c})+v(t_{c}-t)\label{eq:mainTime}

The following figure shows the various geometric distance relationships of the multilateration problem.

Distance relationships of the multilateration problem.

## Geometric Interpretation

In general the positions $$\vec{p}_{i},\vec{p}_{c}$$ and TDOA $$t_{i}-t_{c}$$ of a pair of sensors limits the signal emitter’s position to lay on one sheet of a circular two-sheeted hyperboloid with foci at $$\vec{p}_{i}$$ and $$\vec{p}_{c}$$. Given that $$c$$ is the first of the two sensors to register a signal event, then only the sheet at focus $$\vec{p}_{c}$$ is materialized. The associated directrix plane can be found by a known position vector and normal vector. Since the directrix is orthogonal to the vector pointing from sensor $$i$$ to sensor $$c$$, we know that $$p_{c}-p_{i}$$ itself or its unit vector
$\frac{\vec{p}_{c}-\vec{p}_{i}}{\|\vec{p}_{c}-\vec{p}_{i}\|}$
can serve as a normal vector. Next we need a position vector with endpoint on the directrix plane. One such vector can be defined by the point at which the directrix intersects the line segment between $$\vec{p}_{i}$$ and $$\vec{p}_{c}$$. This point lays a length $$v(t_{i}-t_{c})$$ away from $$p_{i}$$ towards $$\vec{p}_{c}$$, that is, it lays at the position vector
$\vec{p}_{i}+v(t_{i}-t_{c})\frac{\vec{p}_{c}-\vec{p}_{i}}{\|\vec{p}_{c}-\vec{p}_{i}\|}$

The following figure illustrates how TDOA can be used to constrain the multilateration problem solution to one sheet of a two-sheeted circular hyperboloid in $$\mathbb{R}^{3}$$.

Given sensor pair i and c the feasible region of emittor position x is constrained to one sheet of a two-sheeted circular hyperboloid.

With enough of these constraints, i.e., having enough sensors, we can isolate the signal emitter position to a single point as in the following figure in $$\mathbb{R}^{2}$$ space.

Given pairings between tre sensors i=1,2,3 with sensor c=4 we can generate three hyperboles which all intersect at emittor possition x.

## Reduction to Linear Form

The most efficient way to solve the multilateration problem is to reduce the problem to linear form. To determining the location of the signal emitter we can use three unique pairs $$i,j$$ of sensors in addtion to an initial discovery sensor $$c$$. This reduces the problem to a linear system of three equations, and requires at least five sensor.

\begin{alignedat}{2}\|\vec{x}-\vec{p}_{i}\| & = & v(t_{i}-t_{c})+\|\vec{x}-\vec{p}_{c}\|\\ \|\vec{x}-\vec{p}_{i}\|^{2} & = & \left(v(t_{i}-t_{c})+\|\vec{x}-\vec{p}_{c}\|\right)^{2}\\ \|\vec{x}-\vec{p}_{i}\|^{2} & = & v^{2}(t_{i}-t_{c})^{2}+2v(t_{i}-t_{c})\|\vec{x}-\vec{p}_{c}\|+\|\vec{x}-\vec{p}_{c}\|^{2} \end{alignedat}

Introducing another sensor $$j$$ we can eliminate the $$\|x-p_{c}\|$$ term.

{2}-2\|\vec{x}-\vec{p}_{c}\|\\ = v(t_{i}-t_{c})+\frac{\|\vec{x}-\vec{p}_{c}\|^{2}-\begin{split}\|\vec{x}-\vec{p}_{i}\|^{2}\end{split}}{v(t_{i}-t_{c})}
\label{eq:tempRes1}

-2\|\vec{x}-\vec{p}_{c}\|\\ = v(t_{j}-t_{c})+\frac{\|\vec{x}-\vec{p}_{c}\|^{2}-\begin{split}\|\vec{x}-\vec{p}_{j}\|^{2}\end{split}}{v(t_{j}-t_{c})}
\label{eq:tempRes2}

v(t_{j}-t_{c})+\frac{\|\vec{x}-\vec{p}_{c}\|^{2}-\begin{split}\|\vec{x}-\vec{p}_{j}\|^{2}\end{split}}{v(t_{j}-t_{c})}\\ = v(t_{i}-t_{c})+\frac{\|\vec{x}-\vec{p}_{c}\|^{2}-\begin{split}\|\vec{x}-\vec{p}_{i}\|^{2}\end{split}}{v(t_{i}-t_{c})}
\label{eq:tempRes3}

Now, expanding the definitions of the squared distances from each of the sensors and the emitter we get the following.

\begin{alignedat}{2}\begin{split}\|\vec{x}-\vec{p}_{i}\|^{2}\end{split}
& = & \left(\vec{x}-\vec{p}_{i}\right)^{T}\left(\vec{x}-\vec{p}_{i}\right)=\vec{x}^{T}\vec{x}-2\vec{p}_{i}^{T}x+\vec{p}_{i}^{T}\vec{p}_{i}\\
\|\vec{x}-\vec{p}_{j}\|^{2} & = & \vec{x}^{T}\vec{x}-2\vec{p}_{j}^{T}\vec{x}+\vec{p}_{j}^{T}\vec{p}_{j}\\
\|\vec{x}-\vec{p}_{c}\|^{2} & = & \vec{x}^{T}\vec{x}-2\vec{p}_{c}^{T}\vec{x}+\vec{p}_{c}^{T}\vec{p}_{c}
\end{alignedat}
\label{eq:xpDefRe}

\begin{alignedat}{2}v(t_{j}-t_{c})+\frac{2\left(\vec{p}_{j}^{T}-\vec{p}_{c}^{T}\right)\vec{x}+\vec{p}_{c}^{T}\vec{p}_{c}-\vec{p}_{j}^{T}\vec{p}_{j}}{v(t_{j}-t_{c})} \\ = v(t_{i}-t_{c})+\frac{2\left(\vec{p}_{i}^{T}-\vec{p}_{c}^{T}\right)\vec{x}+\vec{p}_{c}^{T}\vec{p}_{c}-\vec{p}_{i}^{T}\vec{p}_{i}}{v(t_{i}-t_{c})}\end{alignedat}
\label{eq:systemRaw}

Notice that the quadratic terms have fallen out and we are left with a very solvable linear system of equations

$\vec{a}_{ijc}^{T}\vec{x}=b_{ijc}$

where

\begin{alignedat}{2}\vec{a}_{ijc} & = & 2\left(v(t_{j}-t_{c})(\vec{p}_{i}-\vec{p}_{c})-v(t_{i}-t_{c})(\vec{p}_{j}-\vec{p}_{c})\right)\in\mathbb{R}^{3}\\ b_{ijc} & = & v(t_{i}-t_{c})\left(v^{2}(t_{j}-t_{c})^{2}-\vec{p}_{j}^{T}\vec{p}_{j}\right)\\ & + & \left(v(t_{i}-t_{c})-v(t_{j}-t_{c})\right)\vec{p}_{c}^{T}\vec{p}_{c}\\ & + & v(t_{j}-t_{c})\left(\vec{p}_{i}^{T}\vec{p}_{i}-v^{2}(t_{i}-t_{c})^{2}\right)\in\mathbb{R} \end{alignedat}

This can be expressed as just

$\textbf{A}x=b$
where each pair $$i,j$$ creates the matrix $$\textbf{A}$$ and vector $$b$$ defined by $$b_{ijc}\in b\in\mathbb{R}^{3}$$ and $$\vec{a}_{ijc}\in \textbf{A}\in\mathbb{R}^{3\times3}$$ such that $$\vec{a}_{ijc}^{T}$$ are the rows of $$\textbf{A}$$. This can be solved using LP or some other more or less direct method. We will be simply computing the iverse of $$\textbf{A}$$ such that
$\hat{x}=\textbf{A}^{-1}b=\textbf{A}\backslash b$

## Application

In order to aid intuition it might be helpful to translate the problem in to a real world interpretation. The problem analogy is that of submarine localization using passive acoustic sensors.

The submarine, i.e., the signal emitter, is placed somewhere in a six cubic kilometer body of water. The acoustic sensors are placed randomly in a one cubic kilometer body of water. Both bodies of water has the coordinate system origin in its center.

The following MATLAB code embodies this scenario: multilateration.m.