# List of Mathematical Symbols

## List of Mathematical Symbols

This section lists the mathematical symbols that are used in Technical Studies Reference.

### Common Mathematical Operations

This section lists and explains some of the mathematical operations that are frequently used in technical studies.

#### Ceiling Function

The ceiling function returns the least integer that is greater than a given number. The notation for the ceiling function of a number $$X$$ is $$\lceil X \rceil$$.

Example: $$\lceil 3.3 \rceil = 4$$

#### Floor Function

The floor function returns the greatest integer that is less than a given number. The notation for the floor function of a number $$X$$ is $$\lfloor X \rfloor$$.

Example: $$\lfloor 5.8 \rfloor = 5$$

#### Moving Maximum

This operation returns the maximum of a set of values in a moving window of Length $$n$$. Our notation for the Moving Maximum of a Random Variable $$X$$ at Index $$t$$ is $$\max_t(X,n)$$, and it is calculated as follows.

$$\max_t(X,n) = \max\{X_{t - n + 1}, X_{t - n + 2},...,X_t\}$$

In the event that there are not yet $$n$$ values in the moving window (that is, $$t < n - 1$$), the Moving Maximum is calculated as follows.

$$\max_t(X,n) = \max\{X_0, X_1,...,X_t\}$$

#### Moving Minimum

This operation returns the minimum of a set of values in a moving window of Length $$n$$. Our notation for the Moving Minimum of a Random Variable $$X$$ at Index $$t$$ is $$\min_t(X,n)$$, and it is calculated as follows.

$$\min_t(X,n) = \min\{X_{t - n + 1}, X_{t - n + 2},...,X_t\}$$

In the event that there are not yet $$n$$ values in the moving window (that is, $$t < n - 1$$), the Moving Minimum is calculated as follows.

$$\min_t(X,n) = \min\{X_0, X_1,...,X_t\}$$

#### Moving Range

This operation returns the difference of the Moving Maximum of a Random Variable $$X^{(1)}$$ and the Moving Minimum of a Random Variable $$X^{(2)}$$ at Index $$t$$. It is calculated as follows.

$$\textrm{Range}_t\left(X^{(1)},X^{(2)},n\right) = \max_t\left(X^{(1)},n\right) - \min_t\left(X^{(2)},n\right)$$

#### Moving Summation

This operation returns the sum of a set of values in a moving window of Length $$n$$. Our notation for the Moving Sum of a Random Variable $$X$$ at Index $$t$$ is $$\mathrm{sum}_t(X,n)$$, and it is calculated as follows.

$$\mathrm{sum}_t(X,n) = X_{t - n + 1} + X_{t - n + 2} + \cdots + X_t$$

We can express this as a Summation as follows.

$$\displaystyle{\mathrm{sum}_t(X,n) = \sum_{i = t - n + 1}^t X_i}$$

In the event that there are not yet $$n$$ values in the moving window (that is, $$t < n - 1$$), the Moving Summation is calculated as follows.

$$\displaystyle{\mathrm{sum}_t(X,n) = X_0 + X_1 + \cdots + X_t = \sum_{i = 0}^t X_i}$$

#### Multiplication

We make occasional use of Pi ($$\Pi$$) notation for multiplication.

For the list of $$n$$ numbers $$X_1,X_2,...,X_n$$, we denote their product as follows.

$$\displaystyle{\prod_{i = 1}^n}X_i = X_1 \cdot X_2 \cdot \cdots \cdot X_n$$

• $$\Pi$$ is called the product sign.
• $$i$$ is called the index of multiplication, or simply the index. It functions as a counter from $$1$$ to $$n$$.
• $$1$$ is called the lower limit of multiplication.
• $$n$$ is called the upper limit of multiplication.
• $$X_i$$ is called the multiplicand.
• #### Rounding Function

The rounding function returns the integer that is closest to a given number. The notation for the rounding function of a number $$X$$ is $$[X]$$.

Example: $$[5.1] = 5$$

Example: $$[5.8] = 6$$

#### Summation

We make frequent use of Sigma ($$\Sigma$$) notation for summation.

For the list of $$n$$ numbers $$X_1,X_2,...,X_n$$, we denote their sum as follows.

$$\displaystyle{\sum_{i = 1}^n}X_i = X_1 + X_2 + \cdots + X_n$$

• $$\Sigma$$ is called the summation sign.
• $$i$$ is called the index of summation, or simply the index. It functions as a counter from $$1$$ to $$n$$.
• $$1$$ is called the lower limit of summation.
• $$n$$ is called the upper limit of summation.
• $$X_i$$ is called the summand.
• #### Welles Summation

This operation returns the sum of a set of values in a moving window of Length $$n$$. Our notation for the Moving Sum of a Random Variable $$X$$ at Index $$t$$ is $$WS_t(X,n)$$, and it is calculated as follows.

$$WS_t(X,n) = \left\{ \begin{matrix} X_0 & t = 0 \\ WS_{t - 1}(X,n) + X_t & 0 < t < n \\ WS_{t - 1}(X,n) - WS_{t - 1}(X,n)/n + X_t & t \geq n \end{matrix}\right .$$

### Parameters

Parameters are variables whose values are either entered by the user as Inputs, calculated from Input values, automatically generated by Auto Looping, or automatically generated by internal looping.

### Random Variables

Random Variables are variables whose values are determined by the outcome of an experiment. For our purposes, Random Variables are almost always volumes, prices or Statistical Functions of these.

When we refer to the value of a Random Variable at Index $$t$$, we use a subscript to indicate this. For instance, the value of the Random Variable Input Data $$X$$ at Index $$t$$ is denoted as $$X_t$$.

• $$C$$ - Closing Price - This may be superscripted, e.g. $$C^{(1)}$$, $$C^{(HA)}$$, $$C^{(-1)}$$, etc.
• $$H$$ - High Price - This may be superscripted, e.g. $$H^{(1)}$$, $$H^{(HA)}$$, $$H^{(-1)}$$, etc.
• $$L$$ - Low Price - This may be superscripted, e.g. $$L^{(1)}$$, $$L^{(HA)}$$, $$L^{(-1)}$$, etc.
• $$n_t^{(NZ)}$$ - Number of Nonzero values of $$X$$ - Appears in Moving Average - Simple Skip Zeros
• $$N^{(P)}$$ - Number of Prices - Appears in Numbers Bars Avg Volume Per Price Graph
• $$N$$ - Number of Trades - This may be superscripted, e.g. $$N^{(ask)}$$, $$N^{(bid)}$$, etc.
• $$O$$ - Opening Price - This may be superscripted, e.g. $$O^{(1)}$$, $$O^{(HA)}$$, $$O^{(-1)}$$, etc.
• $$OI$$ - Open Interest
• $$P$$ - Price - This may be superscripted, e.g. $$P^{(ask)}$$, $$P^{(bid)}$$, etc.
• $$R$$ - +/- Volume - Appears in Volume Zone Oscillator
• $$S$$ - Study Reference - Appears in Color Bar Based On Above/Below Study and Divergence Detector
• $$V$$ - Volume - This may be superscripted, e.g. $$V^{(ask)}$$, $$V^{(bid)}$$, etc.
• $$X$$ - Input Data - These may be superscripted, e.g. $$X^{(1)}$$, $$X^{(2)}$$.

### Statistical Functions

Statistical Functions take on a value at each Current Index Value $$t$$. Unless otherwise stated, the value of a Statistical Function is 0 prior to the starting value of $$t$$. We refer to the value of a Statistical Function at Index $$t$$ by using a subscript, and we write any Inputs for the Statistical Function in parentheses. For instance, the value of the Statistical Function Moving Average - Simple of Input Data $$X$$ with Length $$n$$ at Index $$t$$ is denoted as $$MA_t(X,n)$$.

When a Statistical Function is used as a Random Variable for another Statistical Function, we indicate this by omitting its subscript. For instance, the value of the Exponential Moving Average of $$X$$ with Length $$n$$ at Index $$t$$ is denoted as $$EMA_t(X,n)$$. If we take the Exponential Moving Average of $$EMA_t(X,n)$$, again with Length $$n$$, we denote its value at Index $$t$$ as $$EMA_t(EMA(X,n),n)$$. Here, $$EMA(X,n)$$ is a random variable corresponding to the first Exponential Moving Average.

When we list the arguments of Statistical Functions, we list only those that have numerical values and that are input by the user. We omit all others. As an example, in the notation for the Bar Difference study, we omit the Input Calculate Difference in Price Ticks from the list of arguments because it is not numerical. As another example, in the notation for the Q Stick study, we omit the random variables $$C$$ and $$O$$ from the list of arguments because these are not input by the user.

When alphabetizing the list of Statistical Functions, we observe the following conventions.

• Symbols take precedence over letters. This is why $$\% B_t(X,n,v)$$ is listed first.
• Subscripts and superscripts are not considered when alphabetizing names of Statistical Functions. This is why $$\max_t(X,n)$$ is listed before $$MaxRSI_t(n,n_{HL})$$.
• When the name of a Statistical Function begins with a Greek letter, the English spelling of the letter is used to determine alphabetization. For example, $$\Delta MA_t(X,n_1,n_2)$$ is treated as though it was spelled "D-E-L-T-A-M-A".
• When two or more statistical functions have the same name, then they are sorted first according to their superscripts, and then according to their parameters in the order in which they appear. For example, $$DeM_t(n)$$ is listed before $$DeM^{(\max)}_t$$.