prelminary implementation of static hurst exponent

things still to do:
* extend static hurst to moving hurst
* extend static r/s range to moving r/s range
* write tests
* review assumptions and methodology

src/Indicators.jl

@@ -3,24 +3,26 @@ VERSION >= v"0.4.0" && __precompile__(true)
 module Indicators
 
 export
-    runmean, runsum, runvar, runsd, runcov, runcor, runmax, runmin, runmad, runquantile, runacf,
-    wilder_sum, mode, diffn, #lag,
+    runmean, runsum, runvar, runsd, runcov, runcor, runmax, runmin, runrange, runmad, runquantile, runacf,
+    wilder_sum, mode, diffn,
     sma, trima, wma, ema, mma, kama, mama, hma, swma, dema, tema, alma, zlema,
     mlr_beta, mlr_slope, mlr_intercept, mlr, mlr_se, mlr_ub, mlr_lb, mlr_bands, mlr_rsq,
     aroon, donch, momentum, roc, macd, rsi, adx, psar, kst, wpr, cci, stoch, smi,
     bbands, tr, atr, keltner,
     crossover, crossunder,
     renko,
-    maxima, minima, support, resistance
+    maxima, minima, support, resistance,
+    rescaled_range, hurst_exponent
 
-include("run.jl")
+include("chaos.jl")
 include("ma.jl")
-include("reg.jl")
 include("mom.jl")
-include("vol.jl")
+include("patterns.jl")
+include("reg.jl")
+include("run.jl")
 include("trendy.jl")
 include("utils.jl")
-include("patterns.jl")
+include("vol.jl")
 
 using Temporal
 include("temporal.jl")

src/chaos.jl

@@ -0,0 +1,59 @@
+#=
+Functionality generally related to chaos theory and dynamical systems.
+=#
+
+@doc """
+# Rescaled Range
+
+Compute the rescaled range of a time series.
+
+`rescaled_range(x::Vector{Float64})::Float64`
+""" ->
+function rescaled_range(X::Vector{Float64}; corrected::Bool=false)::Float64
+    return (maximum(X) - minimum(X)) / std(X, corrected=false)
+end
+
+
+@doc """
+# Hurst Exponent
+
+Estimate the Hurst exponent of a time series.
+
+## Arguments
+* `x`: vector of time series data
+* `t`: vector/range of values to use for `t` when estimating the exponent
+* `k`: number of times to resample the time series for each partial time series length nᵢ
+* `c`: constant number corresponding to \$C\$ in Equation 10 [here](http://www.bearcave.com/misl/misl_tech/wavelets/hurst/index.html#WaveletsAndRS)
+* `intercept`: whether an x-intercept should be added when calculating the regression slope
+* `seed`: random number generating seed to set before estimation routine, will skip seed setting if nonpositive value given (default)
+
+## References
+* Qian and Rasheed 2004, "Hurst Exponent and Financial Market Predictability", https://pdfs.semanticscholar.org/0816/a5a989c8d2431a6d20076d27c4295c00fb77.pdf
+* Kaplan 2003, "Estimating the Hurst Exponent", http://www.bearcave.com/misl/misl_tech/wavelets/hurst/index.html#WaveletsAndRS
+""" ->
+function hurst_exponent(X::Vector{Float64};
+                        t::AbstractVector{Int} = [2^i for i in 1:floor(Int, log(size(X,1)) / log(2))],
+                        c::Float64 = 1.0,
+                        intercept::Bool = true,
+                        seed::Int = 0)
+    if seed > 0
+        srand(seed)
+    end
+    x = zeros(Int, length(t))  # the size of the periods
+    y = zeros(length(t))*NaN  # rescaled ranges: the y values, or dependent variable
+    m = mean(X)
+    for (i, ti) in enumerate(t)
+        Y = X[1:ti] - m
+        Z = cumsum(Y)
+        x[i] = ti
+        y[i] = rescaled_range(Z)
+    end
+    # fit the poiiwer law to the data
+    log_x = intercept ? [ones(length(x)) log2.(x)] : log2.(x)
+    log_y = log2.(y)
+    beta = log_x \ log_y
+    if beta[end] < 0 || beta[end] > 1
+        warn("Estimated Hurst exponent ∉ [0, 1].")
+    end
+    return beta, log_x, log_y
+end