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# 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: \(\lfloor 5.8 \rfloor = 5\)

#### 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.

\(\displaystyle{\sum_{i = 1}^n}X_i = X_1 + X_2 + \cdot\cdot\cdot X_n\)

**summation sign**.

**index of summation**, or simply the

**index**. It functions as a counter from \(1\) to \(n\).

**lower limit of summation**.

**upper limit of summation**.

**summand**.

### 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.

- \(c\) - Smoothing Constant - Appears in several moving averages, such as Moving Average - Adaptive and Moving Average - Exponential. This may be subscripted, e.g. \(c_F\), \(c_S\), etc.
- \(i\) - Variable Chart Bar Index Value - Usually varies from some past value of the Index up to the Current Index Value \(t\).
- \(k\) - Offset
- \(\lambda\) - Lag - Appears in Moving Average - Zero Lag Exponential.
- \(\mu\) - ATR Multiplier - Appears in Volatility Trend Indicator.
- \(n\) - Length - This may be subscripted, e.g. \(n_1\), \(n_{RSI}\).
- \(s\) - Tick Size - This is set through
**Chart >> Chart Settings >> Main Settings**. - \(t\) - Current Chart Bar Index Value
- \(v\) - Multiplier (in T3) or Value (in Bands/Envelope)

### 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 prices.

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. \(C^{(1)}\), \(C^{(HA)}\), etc.
- \(H\) - High Price - This may be superscripted, e.g. \(H^{(1)}\), \(H^{(HA)}\), etc.
- \(L\) - Low Price - This may be superscripted, e.g. \(L^{(1)}\), \(L^{(HA)}\), etc.
- \(N\) - Number of Trades - This may be subscripted, e.g. \(N_{ask}\), \(N_{bid}\), etc.
- \(O\) - Opening Price - This may be superscripted, e.g. \(O^{(1)}\), \(O^{(HA)}\), etc.
- \(P\) - Price - This may be subscripted, e.g. \(P_{ask}\), \(P_{bid}\), etc.
- \(\overline{P}\) - Average Price
- \(V\) - Volume - This may be subscripted, 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.

- \(\% R_t\left(X^{(High)},X^{(Low)},X^{(Last)},n\right)\) - Williams' %R
- \(a_t(X,n)\) - Intercept of Least Squares Regression Line - Appears in Moving Linear Regression / Moving Average - Linear Regression
- \(ACDC_t(X,n_1,n_2,n_3,n_4)\) - AC/DC Histogram
- \(AD_t\) - Accumulation/Distribution
- \(ADF_t\) - Accumulation/Distribution Flow
- \(\overline{ADF}_t(n)\) - Moving Average of Accumulation/Distribution Flow
- \(AdjVal_t(X,n)\) - Adjusted Value - Appears in Cumulative Adjusted Value
- \(AMA_t(X,n,c_F,c_S)\) - Moving Average - Adaptive
- \(AMAHigh_t(X,n,c_F,c_S)\) - Appears in Moving Average - Adaptive Binary Wave
- \(AMALow_t(X,n,c_F,c_S)\) - Appears in Moving Average - Adaptive Binary Wave
- \(AO_t(n_1,n_2)\) - Awesome Oscillator, aka Bill Williams Awesome Oscillator
- \(b_t(X,n)\) - Slope of Least Squares Regression Line - Appears in Moving Linear Regression / Moving Average - Linear Regression
- \(BarDiff_t\left(X^{(1)},X^{(2)},k\right)\) - Bar Difference
- \(BB_t(X,v)\) or \(BB_t(X,v,s)\) - Bottom Band - Appears in Bands/Envelope
- \(BearPow_t(n)\) - Bear Power - Appears in Elder Ray
- \(BullPow_t(n)\) - Bull Power - Appears in Elder Ray
- \(BW_t(X,n,c_F,c_S,f)\) - Binary Wave - Appears in Moving Average - Adaptive Binary Wave
- \(c_t(X,n)\) - Smoothing Constant - Appears in Moving Average - Adaptive
- \(CMO_t(X,n_{CMO})\) - Chande Momentum Oscillator
- \(CumAdjVal_t(X,n)\) - Cumulative Adjusted Value
- \(D_t(X)\) - Downward Change in \(X\) - Appears in RSI
- \(\Delta MA_t(X,n_1,n_2)\) - Moving Average Difference - Also appears in AC/DC Histogram
- \(\Delta MACD_t(X,n_F,n_S,n_M)\) - MACD Difference - Appears in MACD
- \(\Delta VWMACD_t(X,n_F,n_S,n_M)\) - Volume Weighted MACD Difference - Appears in MACD - Volume Weighted
- \(\Delta X_t\) - Change in \(X\) - Appears in Chande Momentum Oscillator
- \(DEMA_t(X,n)\) - Moving Average - Double Exponential
- \(Dir_t(X,n)\) - Direction - Appears in Moving Average - Adaptive
- \(Dir_t(X,n,\mu,DPL_{\max})\) - Direction - Appears in Volatility Trend Indicator - Differs from the function that appears in Moving Average - Adaptive
- \(DPL_t(X,n,\mu,DPL_{\max})\) - Dynamic Period Length - Appears in Volatility Trend Indicator
- \(DR_t\) - Daily Range - Appears in Average Daily Range
- \(\overline{DR}_t(n)\) - Average Daily Range
- \(EMA_t(X,n)\) - Moving Average - Exponential
- \(EMA_t^{(j)}(X,n)\) - \(j-\)fold composition of the Exponential Moving Average with itself - Appears in T3
- \(FI_t\) - Force Index
- \(\overline{FI}_t\) - Force Index Average
- \(FT_t(X,n)\) - Fisher Transform
- \(HMA_t(X,n)\) - Moving Average - Hull
- \(HVR_t(n_S,n_L)\) - Historical Volatility Ratio
- \(IB_t\) - Inside Bar
- \(IEB_t\) - Inside or Equals Bar
- \(KS_t\left(X^{(High)},X^{(Low)},n_{KS}\right)\) - Kijun-Sen
- \(LR_t\) - Logarithmic Return - Appears in Historical Volatility Ratio
- \(LRI_t(X,n)\) - Linear Regression Indicator - Appears in Moving Linear Regression / Moving Average - Linear Regression
- \(LRMA_t(X,n)\) - Linear Regression Moving Average - Appears in Moving Linear Regression / Moving Average - Linear Regression
- \(M_t(X,n)\) - Momentum
- \(MA_t(X,n)\) - Moving Average - Simple
- \(MACD_t(X,n_F,n_S)\) - MACD
- \(\overline{MACD}_t(X,n_F,n_S,n_M)\) - MACD Moving Average - Appears in MACD
- \(\max_t(X,n)\) - Moving Maximum - Appears in several studies, such as Tenkan-Sen and Williams' %R
- \(MaxRSI_t(n,n_{HL})\) - Maximum RSI - Appears in Stochastic RSI
- \(\min_t(X,n)\) - Moving Minimum - Appears in several studies, such as Tenkan-Sen and Williams' %R
- \(MinRSI_t(n,n_{HL})\) - Minimum RSI - Appears in Stochastic RSI
- \(MMed_t(X,n)\) - Moving Median
- \(OB_t\) - Outside Bar
- \(\overline{P}_t\) - Average Price - Appears in Average Price For Bar
- \(\pi^{(i)}_t(X,n,\mu,DPL_{\max})\), \(i = 1,2,3\) - Period 1, 2, and 3 - Appear in Volatility Trend Indicator
- \(QStick_t(n)\) - Q Stick
- \(\rho_t(X,Y,n)\) - Correlation Coefficient
- \(RV_t\) - Range Volume - Appears in Accumulation/Distribution
- \(S_t^{(-)}\) - Down Sum - Appears in Chande Momentum Oscillator
- \(S_t^{(+)}\) - Up Sum - Appears in Chande Momentum Oscillator
- \(\sigma_t(X,n)\) - Standard Deviation
- \(StochRSI_t(n,n_{HL})\) - Stochastic RSI
- \(\mathrm{sum}_t(X,n)\) - Moving Sum - Appears in several studies, such as Ultimate Oscillator
- \(SWWMA_t(X,n)\) - Moving Average - Sine Wave Weighted
- \(SZMA_t(X,n)\) - Moving Average - Simple Skip Zeros
- \(T_3(X,n,v)\) - T3
- \(TB_t(X,v)\) or \(TB_t(X,v,s)\) - Top Band - Appears in Bands/Envelope
- \(TEMA_t(X,n_1,n_2)\) - Moving Average - Triple Exponential
- \(TMA_t(X,n_1,n_2)\) - Moving Average - Triangular
- \(TR_t\) - True Range
- \(\overline{TR}_t(n)\) - Average True Range
- \(TRIX_t(X,n)\) - TRIX
- \(TS_t\left(X^{(High)},X^{(Low)},n_{TS}\right)\) - Tenkan-Sen
- \(U_t(X)\) - Upward Change in \(X\) - Appears in RSI
- \(UO_t(n_1,n_2,n_3)\) - Ultimate Oscillator
- \(Var_t(X,n)\) - Variance - Appears in Standard Deviation
- \(Vol_t(X,n)\) - Volatility - Appears in Moving Average - Adaptive
- \(VR_t\) - Volume Ratio - Appears in Bid Ask Volume Ratio
- \(\overline{VR}_t(n)\) - Average Volume Ratio - Appears in Bid Ask Volume Ratio
- \(VTI_t(X,n,\mu,DPL_{\max})\) - Volatility Trend Indicator
- \(VWMA_t(X,n)\) - Moving Average - Weighted
- \(VWMACD_t(X,n_F,n_S)\) - MACD - Volume Weighted
- \(\overline{VWMACD}_t(X,n_F,n_S,n_M)\) - Volume Weighted MACD Moving Average - Appears in MACD - Volume Weighted
- \(WAO_t(X,n_F,n_S)\) - Weighted Average Oscillator
- \(WMA_t(X,n)\) - Moving Average - Weighted
- \(WWMA_t(X,n)\) - Moving Average - Welles Wilders
- \(\xi_t(X,n)\) - First transformation of Price data - Appears in Fisher Transform
- \(\xi^*_t(X,n)\) - Second transformation of Price data - Appears in Fisher Transform
- \(Z_t(X,n_{\mu},n_{\sigma})\) - Z-Score
- \(ZLEMA_t(X,n)\) - Moving Average - Zero Lag Exponential

*Last modified Saturday, 14th October, 2017.