Directional changes

What is DC(Directional changes)?

An approach to summarize price changes other than time series

How to?

Market prices are sampled based on the DC greater than PEXTP_{EXT}PEXT​ by at least thetathetatheta

thetathetatheta is the threshold predetermined by the observer

usually expressec as percentage

Formally, a DC event is detected when we come across a price PcP_cPc​ that satisfies below:∣Pc−PEXTPEXT∣≥theta\left|\dfrac{P_c - P_{EXT}}{P_{EXT}}\right|\ge theta∣∣∣∣∣​PEXT​Pc​−PEXT​​∣∣∣∣∣​≥theta

Terminology

Extreme point / DCC point

If the inequality just above holds,

TermDescriptionExample in figureExtreme pointthe time at PEXTP_{EXT}PEXT​points AAA, BBB, CCC, DDD, EEE, FFF, GGGDCC pointthe time at PcP_{c}Pc​points A0.1A^{0.1}A0.1, B0.1B^{0.1}B0.1, C0.1C^{0.1}C0.1, D0.1D^{0.1}D0.1, E0.1E^{0.1}E0.1, F0.1F^{0.1}F0.1, G0.1G^{0.1}G0.1

Note that whilst an extreme point is the end of the one trend, it is also the start point of the next trend

An extreme point is only recognized in hindsight; precisely at the DCC point

DC event / OS event

DC means "Directional Change"and OS does "Overshoot"

TermDescriptionExample in figureDC event-starts with an extreme point-ends with a DCC pointintervals [A,A0.1][A,A^{0.1}][A,A0.1], [B,B0.1][B,B^{0.1}][B,B0.1], …\dots…, [G,G0.1][G,G^{0.1}][G,G0.1]DCC point-starts at the DCC point-ends at the next extreme pointntervals [A0.1,B][A^{0.1},B][A0.1,B], [B0.1,C][B^{0.1},C][B0.1,C], …\dots…, [E0.1,F][E^{0.1},F][E0.1,F]

Note that for a given time series and a predetermined threshold, the DC summary is unique

Implementation in Python

Original version

THETA_bakhach=0.1
data_points = market_index['close_scaled'].tolist()
pext = data_points[0]
vals = []
breakpoints = []
yHat = []

for idx, pc in enumerate(data_points):
    if idx == 0:
        continue

    val = (pc - pext)/pext
    dif = pc - pext

	if abs(val) > THETA_bakhach:
        pext = pc
        vals.append(val)
        breakpoints.append(idx)
        yHat.append(data_points[idx])

# endpoint
breakpoints.append(len(data_points) - 1)
vals.append((data_points[-1] - yHat[-1])/yHat[-1])
yHat.append(data_points[-1])

The right-end point is so ugly!! So, add some correction using amount of change .

Merging adjacent trends

THETA_bakhach=0.1
DELTA_bakhach=(max(market_index['close_scaled'])-min(market_index['close_scaled'])) * 0.08 #0.08
data_points = market_index['close_scaled'].tolist()
pext = data_points[0]
vals = []
breakpoints = []
yHat = []

 
for idx, pc in enumerate(data_points):
    if idx == 0:
        continue

    # val = abs((pc - pext)/pext)
    val = (pc - pext)/pext
    dif = pc - pext

    if abs(val) > THETA_bakhach:
        pext = pc
        vals.append(val)
        breakpoints.append(idx)
        yHat.append(data_points[idx])

# endpoint
breakpoints.append(len(data_points) - 1)
vals.append((data_points[-1] - yHat[-1])/yHat[-1])
yHat.append(data_points[-1])

rm_idx = []
for vi in range(len(vals))[1:]:
    fs = np.sign(vals[vi - 1])
    cs = np.sign(vals[vi])
    if fs == cs:
        rm_idx.append(vi-1)
        
vals_new = [j for i, j in enumerate(vals) if i not in rm_idx]
breakpoints_new = [j for i, j in enumerate(breakpoints) if i not in rm_idx]
yHat_new = [j for i, j in enumerate(yHat) if i not in rm_idx]

The problem in right-end part is solved now! However, it seems that a lot of unnecessary DC has been detected. For this, add a correction merging adjacent trends.

Add δ\deltaδ for observing change amount

THETA_bakhach=0.1
DELTA_bakhach=(max(market_index['close_scaled'])-min(market_index['close_scaled'])) * 0.08
data_points = market_index['close_scaled'].tolist()
pext = data_points[0]
vals = []
breakpoints = []
yHat = []

 
for idx, pc in enumerate(data_points):
    if idx == 0:
        continue

    # val = abs((pc - pext)/pext)
    val = (pc - pext)/pext
    dif = pc - pext

    # if abs(dif) > DELTA_bakhach:
    if abs(val) > THETA_bakhach or abs(dif) > DELTA_bakhach:
        pext = pc
        vals.append(val)
        breakpoints.append(idx)
        yHat.append(data_points[idx])

# endpoint
breakpoints.append(len(data_points) - 1)
vals.append((data_points[-1] - yHat[-1])/yHat[-1])
yHat.append(data_points[-1])

rm_idx = []
for vi in range(len(vals))[1:]:
    fs = np.sign(vals[vi - 1])
    cs = np.sign(vals[vi])
    if fs == cs:
        rm_idx.append(vi-1)
        
vals_new = [j for i, j in enumerate(vals) if i not in rm_idx]
breakpoints_new = [j for i, j in enumerate(breakpoints) if i not in rm_idx]
yHat_new = [j for i, j in enumerate(yHat) if i not in rm_idx]

Finally, the detected trends look great!

[References]

Bakhach, A., Tsang, E.P., & Jalalian, H.R. (2016). Forecasting directional changes in the FX markets. 2016 IEEE Symposium Series on Computational Intelligence (SSCI), 1-8.