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

#### 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 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.

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

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

**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 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.
- \(N\) - Number of Trades - This may be subscripted, e.g. \(N_{ask}\), \(N_{bid}\), etc.
- \(O\) - Opening Price - This may be superscripted, e.g. \(O^{(1)}\), \(O^{(HA)}\), \(O^{(-1)}\), etc.
- \(OI\) - Open Interest
- \(P\) - Price - This may be subscripted, e.g. \(P_{ask}\), \(P_{bid}\), etc.
- \(R\) - +/- Volume - Appears in Volume Zone Oscillator
- \(V\) - 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.

- \(\% B_t(X,n,v)\) - Bollinger Bands: %B
- \(\%Diff_t(X,n,v)\) - Rate of Change - Percentage
- \(\% R_t\left(X^{(High)},X^{(Low)},X^{(Last)},n\right)\) - 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
- \(ADF_t\) - 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
- \(AI^{(Down)}_t\left(X^{(Low)},n\right)\) - Aroon Indicator Down
- \(AI^{(Up)}_t\left(X^{(High)},n\right)\) - Aroon Indicator Up
- \(AO_t(n_1,n_2)\) - Awesome Oscillator, aka Bill Williams Awesome Oscillator
- \(AO_t\left(X^{(High)},X^{(Low)},n\right)\) - Aroon Oscillator
- \(b_t(X,n)\) - Slope of Least Squares Regression Line - Appears in Moving Linear Regression / Moving Average - Linear Regression
- \(BarDiff_t\left(X^{(1)},X^{(2)},k\right)\) or \(BarDiff_t\left(X^{(1)},X^{(2)},k,s\right)\) - Bar Difference
- \(BB_t(n)\) - Bottom Band - Appears in Donchian Channel
- \(BB_t(X,n_K,n_{TR},v_B)\) - Bottom Band - Appears in Keltner Channel
- \(BB_t(X,n_S,n_{TR},v_B)\) - Bottom Band - Appears in Starc Bands
- \(BB_t(X,n,v)\) - Bottom Band - Appears in Bollinger Bands and Standard Deviation Bands
- \(BB_t(X,v)\) 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
- \(BW_t(X,n,v)\) - Bollinger Bands: Bandwidth
- \(c_t(X,n)\) - Smoothing Constant - Appears in Moving Average - Adaptive
- \(CCI_t(X,n,v)\) - Commodity Channel Index
- \(CMF_t(n)\) - Chaikin Money Flow
- \(CMO_t(X,n_{CMO})\) - Chande Momentum Oscillator
- \(CS_{i - n}(X,n)\) - Chikou Span
- \(CumAdjVal_t(X,n)\) - Cumulative Adjusted Value
- \(CVol_t(n)\) - Volatility - Chaikins
- \(D_t(X)\) - Downward Change in \(X\) - Appears in RSI - Also appears in Chande Momentum Oscillator, though defined slightly differently there
- \(D_t(X,n)\) - Downward Change in \(X\) over \(n\) Bars - Appears in Relative Momentum Index
- \(\Delta MA_t(X,n_1,n_2)\) - Moving Average Difference - Also appears in AC/DC Histogram
- \(\Delta MACD_t(X,n_F,n_S,n_M)\) - MACD Difference - Appears in MACD
- \(\Delta MACD^{(3/10)}_t(X,n_F,n_S,n_{3/10})\) - 3/10 Oscillator Difference - Appears in 3/10 Oscillator
- \(\Delta VWMACD_t(X,n_F,n_S,n_M)\) - Volume Weighted MACD Difference - Appears in MACD - Volume Weighted
- \(\Delta X_t(n)\) - Rate of Change - Points
- \(DEMA_t(X,n)\) - 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 Moving Average - Double Exponential, Moving Average - Triple Exponential, T3, and TRIX
- \(Fast\% D_t(X^{(High)},X^{(Low)},X^{(Close)},n_{FastK},n_{FastD})\) - Fast %D (aka Slow %K) - Appears in KD - Fast and KD - Slow
- \(Fast\% K_t(X^{(High)},X^{(Low)},X^{(Close)},n_{FastK})\) - Fast %K - Appears in KD - Fast
- \(FI_t\) - Force Index
- \(\overline{FI}_t(n)\) - Force Index Average
- \(FT_t(X,n)\) - Fisher Transform
- \(HMA_t(X,n)\) - Moving Average - Hull
- \(HVol_t(X,n,N)\) - Volatility - Historical
- \(HVR_t(n_S,n_L)\) - Historical Volatility Ratio
- \(IB_t\) - Inside Bar
- \(IEB_t\) - Inside or Equals Bar
- \(Inertia^{(1)}_t(n_{RVI},n_{LR})\) - Inertia
- \(Inertia^{(2)}_t(n_\sigma,n_{RVI},n_{LR})\) - Inertia 2
- \(Jaw_t(X,n_J)\) - Jaw of Bill Williams Alligator
- \(KS_t\left(X^{(High)},X^{(Low)},n_{KS}\right)\) - Kijun-Sen
- \(Lips_t(X,n_L)\) - Lips of Bill Williams Alligator
- \(LR_t(X)\) - Logarithmic Return - Appears in Volatility - Historical and Historical Volatility Ratio
- \(LRI_t(X,n)\) - Linear Regression Indicator - Appears in Moving Linear Regression / Moving Average - Linear Regression
- \(LRMA_t(X,n)\) - Moving Average - Linear Regression
- \(M_t(X,n)\) - Momentum
- \(\overline{M}_t(X,n,n_{MA})\) - Moving Average of Momentum
- \(MA_t(X,n)\) - Moving Average - Simple
- \(MACD_t(X,n_F,n_S)\) - MACD
- \(MACD^{(3/10)}_t(X,n_F,n_S)\) - 3/10 Oscillator
- \(\overline{MACD}_t(X,n_F,n_S,n_M)\) - MACD Moving Average - Appears in MACD
- \(\overline{{MACD}}^{(3/10)}_t(X,n_F,n_S,n_{3/10})\) - 3/10 Oscillator Moving Average - Appears in 3/10 Oscillator
- \(MACDL_t(X,n_F,n_S)\) - MACD Leader
- \(\max_t(X,n)\) - Moving Maximum - Appears in several studies, such as Highest High/Lowest Low Over N Bars and Donchian Channel
- \(MaxRSI_t(X,n,n_{HL})\) - Maximum RSI - Appears in Stochastic RSI
- \(MB_t(n)\) - Middle Band - Appears in Donchian Channel
- \(\min_t(X,n)\) - Moving Minimum - Appears in several studies, such as Highest High/Lowest Low Over N Bars and Donchian Channel
- \(MinRSI_t(X,n,n_{HL})\) - Minimum RSI - Appears in Stochastic RSI
- \(MFM_t\) - Money Flow Multiplier - Appears in Chaikin Money Flow
- \(MLR_t(X,n)\) - Moving Linear Regression
- \(MMed_t(X,n)\) - Moving Median
- \(NR_t(n)\) - Narrow Range Bar
- \(NVI_t(X,NVI_0)\) - Negative Volume Index
- \(OB_t\) - Outside Bar
- \(OI^{(\pm)}_t(n)\) - Signed Open Interest - Appears in On Balance Open Interest - Short Term
- \(OI^{(OB)}_t\) - On Balance Open Interest
- \(OI^{(OB)}_t(n)\) - On Balance Open Interest - Short Term
- \(\overline{P}_t\) - Average Price - Appears in Average Price For Bar - Also appears in Relative Vigor Index 2 with superscripts: \(\overline{P}^{(C-O)}_t\) and \(\overline{P}^{(H-L)}_t\)
- \(\pi^{(i)}_t(X,n,\mu,DPL_{\max})\), \(i = 1,2,3\) - Period 1, 2, and 3 - Appear in Volatility Trend Indicator
- \(PVI_t(X,PVI_0)\) - Positive Volume Index
- \(PVT_t\) - Price Volume Trend
- \(PPO_t(X,n_L,n_S)\) - Percentage Price Oscillator
- \(PrevBar_t(X)\) - Previous Bar Close
- \(QStick_t(n)\) - Q Stick
- \(R_t(X,n)\) - Rank - Appears in Stochastic - Percentile
- \(Range_t(n_K)\) - Range - Appears in Stochastic Momentum Indicator
- \(RCB_t\) - Range of Current Bar - Appears in Narrow Range Bar and Wide Range Bar
- \(RelRange_t(n_K)\) - Relative Range - Appears in Stochastic Momentum Indicator
- \(Repulse_t(n)\) - Repulse
- \(\rho_t(X,Y,n)\) - Correlation Coefficient
- \(RMI_t(X,n,n_{MA})\) - Relative Momentum Index
- \(RMO_t(X,n_1,n_2,n_4)\) - Rahul Mohindar Oscillator
- \(RSI_t(X,n_{RSI})\) - RSI
- \(\overline{RSI}_t(X,n_{RSI},n)\) - Moving Average of RSI
- \(RV_t\) - Range Volume - Appears in Accumulation/Distribution
- \(RVI_t\) - Relative Vigor Index - Appears in Relative Vigor Index 1 and Relative Vigor Index 2
- \(\overline{RVI}^{(1)}_t(n)\) - Smoothed Relative Vigor Index 1 - Appears in Relative Vigor Index 1
- \(\overline{RVI}^{(2)}_t(n)\) - Smoothed Relative Vigor Index 2 - Appears in Relative Vigor Index 2
- \(RVIX_t(n_\sigma,n_{RVIX})\) - Relative Volatility Index - Appears in Inertia 2
- \(RVIX^{(D)}_t(n_\sigma)\) - Relative Volatility Index Down - Appears in Inertia 2
- \(\overline{RVIX}^{(D)}_t(n_\sigma,n_{RVIX})\) - Smoothed Relative Volatility Index Down - Appears in Inertia 2
- \(RVIX^{(U)}_t(n_\sigma)\) - Relative Volatility Index Up - Appears in Inertia 2
- \(\overline{RVIX}^{(U)}_t(n_\sigma,n_{RVIX})\) - Smoothed Relative Volatility Index Up - Appears in Inertia 2
- \(RWI^{(High)}_t(n)\) - High Random Walk Indicator
- \(RWI^{(Low)}_t(n)\) - Low Random Walk Indicator
- \(Sig^{(1)}_t(n)\) - Signal Line - Appears in Relative Vigor Index 1
- \(Sig^{(2)}_t(n)\) - Signal Line - Appears in Relative Vigor Index 2
- \(\sigma_t(X,n)\) - Standard Deviation
- \(Slow\% D_t(X^{(High)},X^{(Low)},X^{(Close)},n_{FastK},n_{FastD},n_{SlowD})\) - Slow %D - Appears in KD - Slow
- \(SP_t(X,n)\) - Stochastic - Percentile
- \(\overline{SP}_t(X,n,n_{MA})\) - Moving Average of Stochastic - Percentile
- \(SMI_t(n_K,n_D)\) - Stochastic Momentum Indicator
- \(\overline{SMI}_t(n_K,n_D,n_{EMA})\) - Average of Stochastic Momentum Indicator
- \(SMMA_t(X,n,k)\) - Smoothed Moving Average
- \(SSA_t(n_{TS},n_{KS})\) - Senkou Span A
- \(SSB_t\left(X^{(High),}, X^{(Low)}, n\right)\) - Senkou Span B
- \(ST^{(1)}_t(X,n_1,n_2)\) - Swing Trade 1 - Appears in Rahul Mohindar Oscillator
- \(ST^{(2)}_t(X,n_1,n_2,n_3)\) - Swing Trade 2 - Appears in Rahul Mohindar Oscillator
- \(ST^{(3)}_t(X,n_1,n_2,n_3)\) - Swing Trade 3 - Appears in Rahul Mohindar Oscillator
- \(StochRSI_t(n,n_{HL})\) - Stochastic RSI
- \(sum_t(X)\) - Summation
- \(SWWMA_t(X,n)\) - Moving Average - Sine Wave Weighted
- \(SynthVIX_t(n)\) - Synthetic VIX
- \(SZMA_t(X,n)\) - Moving Average - Simple Skip Zeros
- \(T_t^{(Down)}\left(X^{(Low)},n\right)\) - Index of most recent low of \(X^{(Low)}\) - Appears in Aroon Indicator
- \(T_t^{(Up)}\left(X^{(High)},n\right)\) - Index of most recent high of \(X^{(High)}\) - Appears in Aroon Indicator
- \(T3_t(X,n,v)\) - T3
- \(TB_t(n)\) - Top Band - Appears in Donchian Channel
- \(TB_t(X,n_K,n_{TR},v_B)\) - Top Band - Appears in Keltner Channel
- \(TB_t(X,n_S,n_{TR},v_B)\) - Top Band - Appears in Starc Bands
- \(TB_t(X,n,v)\) - Top Band - Appears in Bollinger Bands and Standard Deviation Bands
- \(TB_t(X,v)\) or \(TB_t(X,v,s)\) - Top Band - Appears in Bands/Envelope
- \(Teeth_t(X,n_T)\) - Teeth of Bill Williams Alligator
- \(TEMA_t(X,n_1,n_2)\) - Moving Average - Triple Exponential
- \(TH_t\) - True High - Appears in Ultimate Oscillator
- \(TL_t\) - True Low - Appears in Ultimate Oscillator
- \(TMA_t(X,n)\) - 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
- \(TV_t(n)\) - Total Volume - Appears in Volume Zone Oscillator
- \(U_t(X)\) - Upward Change in \(X\) - Appears in RSI - Also appears in Chande Momentum Oscillator, though defined slightly differently there
- \(U_t(X,n)\) - Upward Change in \(X\) over \(n\) Bars - Appears in Relative Momentum Index
- \(UO_t(n_1,n_2,n_3)\) - Ultimate Oscillator
- \(Var_t(X,n)\) - Variance - Appears in Standard Deviation and Dispersion
- \(V^{(\pm)}_t(n)\) - Signed Volume - Appears in On Balance Volume - Short Term
- \(V^{(OB)}_t\) - On Balance Volume
- \(V^{(OB)}_t(n)\) - On Balance Volume - Short Term
- \(Vol_t(X,n)\) - Volatility - Appears in Moving Average - Adaptive
- \(VP_t(n)\) - Volume Position - Appears in Volume Zone Oscillator
- \(VR_t\) - 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
- \(VZO_t(n)\) - Volume Zone Oscillator
- \(W^{(Bear)}_t(n)\) - Bearish Weighting - Appears in Repulse
- \(W^{(Bull)}_t(n)\) - Bullish Weighting - Appears in Repulse
- \(WAO_t(X,n_F,n_S)\) - Weighted Average Oscillator
- \(WAD_t\) - Accumulation Distribution - Williams
- \(WR_t(n)\) - Wide Range - Appears in Wide Range Bar
- \(WWMA_t(X,n)\) - 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 Wednesday, 18th April, 2018.