statistics.rst

Statistics

Statistical functions operate on paths of function values and in general case it requires storing the function history and recalculating statistics each time. Every recalculation has at least O(M) time complexity where M is number of points involved and thus it is quite time consuming.

Many statistical packages (like pandas and ta-lib) provide a lot of useful functions that work on time series and in order to facilate interoperability with these libraries the market simulator provide handful adapters.

In some cases (moving average, moving variance etc) it is possible to introduce an incremental version that make every update in constant time and thus works much faster.

• Moving average
• Moving average convergence/divergence
• Moving variance
• Standard deviation
• Bollinger bands
• Relative strength index
• Moving minimum and maximum

Moving average

• cumulative: CMA
• simple moving: MA(timeframe)
• exponentially weighted: EWMA(alpha)

Moving average convergence/divergence

Having defined moving average we may introduce moving average convergence/divergence, its signal and histogram

MACD(x, slow, fast) ::= EWMA(x, 2./(fast+1)) - EWMA(x, 2./(slow+1))
MACD_signal(x, slow, fast, timeframe) ::= EWMA(MACD(x, slow, fast), 2/(timeframe+1))
MACD_histogram(x, slow, fast, timeframe) ::= MACD(x,slow,fast) - MACD_signal(x,slow, fast, timeframe)

Moving variance

• cumulative: Variance
• moving: MovingVariance(timeframe)
• exponentially weighted: EWMV(alpha)

Variances could be implemented via Mean but it looses precision so they are implemented as simple modules

Var(x) ::= Mean(Sqr(x)) - Sqr(Mean(x))

Standard deviation

StdDev(x) ::= Sqrt(Variance(x))
StdDevRolling(x, timeframe) ::= Sqrt(MovingVariance(x, timeframe))
StdDevEW(x, alpha) ::= Sqrt(EWMV(x, alpha))

Bollinger bands

Having moving averages and variances it is possible to find bollinger bands:

Bollinger_Hi(x) ::= Mean(x) + 2*StdDev(x)
Bollinger_Lo(x) ::= Mean(x) - 2*StdDev(x)

Relative strength index

An observable returning values of a function with some lag (Lagged) allows easily implement relative strength index:

Ups(x, dt, alpha) ::= EWMA(max(0, x - Lagged(x, dt)), alpha) --- moving average of positive movements
Downs(x, dt, alpha) ::= EWMA(max(0, Lagged(x, dt) - x), alpha) --- moving average of negative movements
RSI(x, dt, alpha) ::= 100 - 100 / (1 + Ups(x,dt,alpha)/Downs(x,dt,alpha))

Moving minimum and maximum