The previous examples in this thread all had fixed-length inputs. This example has generated rules from variable-length inputs from different assets for simulating trades in another asset.
A short summary of the method is
- Find three value swings of past data for the input assets.
- Fit a curve with a mathematical function to the value swings.
- Create rules for trading a target asset based on parameters of the function and its output.
Here is a more detailed explanation.
Source of Input Data
The input assets are daily close values from the following:
Code:
name yahoo finance symbol prefix
ALL ORDINARIES ^AORD aord
BEL 20 ^BFX bfx
CAC 40 ^FCHI cac40
CBOE Volatility Index ^VIX vix
DAX PERFORMANCE-INDEX ^GDAXI gdaxi
Dow Jones Industrial Average ^DJI dji
Dow Jones Transportation Average ^DJT djt
Dow Jones Utility Average ^DJU dju
HANG SENG INDEX ^HSI hsi
IBEX 35 ^IBEX ibex
IBOVESPA ^BVSP bvsp
IDX COMPOSITE ^JKSE jkse
IPC MEXICO ^MXX mxx
NASDAQ 100 ^NDX nd100
NASDAQ Bank ^BANK ndbank
NASDAQ Biotechnology ^NBI nbi
NASDAQ Composite ^IXIC ixic
NASDAQ Financial 100 ^IXF ndfin
NASDAQ Industrial ^INDS ndind
NASDAQ Insurance ^INSR ndinsr
NASDAQ Transportation ^TRAN ndtran
NYSE ARCA OIL and GAS INDEX ^XOI xoi
NYSE COMPOSITE (DJ) ^NYA nya
Nikkei 225 ^N225 n225
PHLX GOLD and SILVER SECTOR Index ^XAU xau
PHLX Utility Sector ^UTY uty
Russell 1000 ^RUI rui
Russell 2000 ^RUT rut
Russell 3000 ^RUA rua
S&P 100 ^SP100 sp100
S&P 500 Consumer Discretionary (Sector) ^SP500-25 spcd
S&P 500 Consumer Staples (Sector) ^SP500-30 spcst
S&P 500 Energy (Sector) ^GSPE spenrg
S&P 500 Financials (Sector) ^SP500-40 spfin
S&P 500 Health Care (Sector) ^SP500-35 sphe
S&P 500 Industrials (Sector) ^SP500-20 spind
S&P 500 Information Technology (Sector) ^SP500-45 sptech
S&P 500 Materials (Sector) ^SP500-15 spmat
S&P 500 Telecommunication Services (Sector) ^SP500-50 sptel
S&P 500 Utilities (Sector) ^SP500-55 sput
S&P 500 ^GSPC gspc
S&P 600 ^SP600 sp600
S&P MID CAP 400 INDEX ^MID mid
S&P/ASX 200 ^AXJO asx200
S&P/TSX Composite index ^GSPTSE gsptse
TA-125 ^TA125.TA ta125
Treasury Bill 13 Week ^IRX irx
Treasury Yield 10 Years ^TNX tnx
Treasury Yield 30 Years ^TYX tyx
Treasury Yield 5 Years ^FVX fvx
Wilshire 5000 Total Market Index ^W5000 w5000
I chose these assets because data for at least 30 years is available from Yahoo Finance.
For inputs to the method, the data for the local date associated with an asset is grouped together as input for that date. For example, the 2024-01-10 data is grouped with other 2024-01-10 data even though markets in Australia have already closed for January 11. When non-U.S. markets are closed, the previous close values are carried forward until their dates match the U.S. market close date.
Defining Input Value Swings
For each date in a symbol's data, the software finds data for three value swings with algorithm
- The swing direction starts as undefined.
- A lookback period is the past N bars or to the bar just past the previous value swing whichever is larger.
- If no close value in the lookback period is higher than the current close value, the swing direction is up.
- If no close value in the lookback period is lower than the current close value, the swing direction is down.
- If the swing direction is up and down on the same bar (e.g., value hasn't changed since last swing), the swing direction does not change.
- If the swing direction changes from not up to up, a low swing has been detected at the oldest bar with the lowest value in the lookback period.
- If the swing direction changes from not down to down, a high swing has been detected at the oldest bar with the highest value in the lookback period.
- The first swing found is ignored.
Modeling Input Value Swings
The software fits a curve for the data from the current bar to the bar after the value swing that came before the three value swings was detected. The fitted curve has a least squares regression line added to zero or more asymmetric triangle waves.
Here is an example with the close values of the NASDAQ 100 index for 39 trading days from 2023-11-16 through 2024-01-12 with the least squares regression line and fitted curve.
The corresponding fitted curve formula is:
Code:
nd100_20240112_cly = 16881.927734375 + 551.884216308594 * -0.0502413419724309 * x + 209.763610839844 * (
atri(1.1723438501358, 22.5762538909912, 0.833862066268921, 0.565416514873505, x)
+ atri(0.548687338829041, 10.0181884765625, 0.412693917751312, 0.470283389091492, x)
+ atri(0.40699827671051, 7.98540878295898, 0.176088511943817, 0.500997245311737, x)
+ atri(0.368489861488342, 38.861083984375, 0.150145471096039, 0.492421269416809, x)
+ atri(0.325989842414856, 18.3074150085449, 0.756223797798157, 0.466294586658478, x) ) ;
atri(amplitude, period, phase, peakPhase, x) would be evaluated as the asymmetric triangle wave value for x bars before the current bar.
To allow more relevant comparisons of phases like
0.833862066268921, the curve start at the most recent bar and continues backwards in time.
To allow more relevant comparisons of least squares regression line slopes like
-0.0502413419724309, the estimated standard deviation of the data (e.g.,
551.884216308594) is factored out.
To allow more relevant comparisons of amplitudes like
1.1723438501358, the estimated standard deviation of the differences of close values to the least squares regression line (e.g.,
209.763610839844) is factored out.
Creating Inputs for Rules Generator
The inputs for the rules generator are prefixed with the prefix representing a symbol with some having the index of an asymmetric triangle wave (0 has highest amplitude, 1 has the next highest, etc.). The inputs are derived from the fitted curve parameters and/or values from evaluating the fitted curve:
Code:
bars -- number of bars in the three swing pattern
example: aordbars
fitProp -- proportion of cyclic data fitted by the asymmetric triangle waves
example: aordclfitProp
dtwFitProp -- proportion of cyclic data fitted by the asymmetric triangle waves using dynamic time warping
example: aordcldtwFitProp
trendsds -- number of estimated standard deviations of distances to the least squares regression line
in the rise (+) or fall (-) of the regression line in the three swing pattern
example: aordcl_trendsds
bartrendsds -- number of estimated standard deviations of distances to the least squares regression line
in the rise (+) or fall (-) of each bar the regression line in the three swing pattern
example: aordcl_bartrendsds
wsum -- sum of all the asymmetric triangle wave values at the current bar
example: aordcl_wsum
lpr01 -- log of (extrapolated price one bar ahead divided by current price) * 100
example: aordlpr01cl
lpr12 -- log of (extrapolated price two bars ahead divided extrapolated price one bar ahead) * 100
example: aordlpr12cl
wave -- asymmetric triangle wave value at the current bar
example: aordcl_wave_0
wprop -- asymmetric triangle wave value at the current bar divided by the wave amplitude
example: aordcl_wprop_0
amp -- asymmetric triangle wave amplitude
example: aordcl_amp_0
per -- asymmetric triangle wave period
example: aordcl_per_0
ncy -- number of cycles of an asymmetric triangle wave in the three swing pattern
example: aordcl_ncy_0
vel -- absolute vertical movement per bar for an asymmetric triangle wave in the three swing pattern
example: aordcl_vel_0
lvel -- absolute vertical movement per bar from trough to peak for an asymmetric triangle wave in the three swing pattern
example: aordcl_lvel_0
rvel -- absolute vertical movement per bar from peak to trough for an asymmetric triangle wave in the three swing pattern
example: aordcl_rvel_0
phase -- phase of an asymmetric triangle wave in the three swing pattern
example: aordcl_phase_0
offsetphase -- -0.5 + phase of an asymmetric triangle wave in the three swing pattern
example: aordcl_offsetphase_0
peakphase -- peak phase of an asymmetric triangle wave in the three swing pattern
example: aordcl_peakphase_0
offsetpeakphase -- -0.5 + peak phase of an asymmetric triangle wave in the three swing pattern
example: aordcl_offsetpeakphase_0
The target for the rules generator simulates a long trade entering at the next bar's close value and exiting at the following bar's close value (no adjustments for slippage or commission).
This example uses 5253 instances of training data with simulated entries from 1994-02-23 through 2015-01-02 and 2251 instances of evaluation data with simulated entries from 2015-01-05 through 2023-12-21.
Generating Rules
The rules generator uses genetic programming to classify whether to take a trade or not. Each rule in a set of rules has one or more conditions logically ANDed together. Conditions can have inputs compared with constants or other inputs. When a rule returns 1, a long trade at the close of the next bar would be entered, and this trade would exit at the bar following entry.
After the rules generator finishes its runs using the training data, a subset is chosen as the final rules for forward testing.
Rules found for the NASDAQ 100 are:
Code:
if gsptsebars >= 50 if gsptselpr01cl <= 0.180333 if gsptsecl_wprop_0 >= -0.658776 if gsptsecl_rvel_0 > 0.0648147 if gsptsecl_lvel_1 < 0.176615 if gsptsecl_wsum <= 1.3634 if gsptsecl_wave_1 >= -0.192705 if gsptsecl_rvel_0 >= 0.0690005 if gsptsebars > 49 return 1
if sptechcl_ncy_0 < 1.20559 if sptechcl_wprop_1 <= 0.95901 if sptechlpr01cl >= -0.431104 if sptechcl_ncy_0 < 1.20804 if sptechlpr01cl > -0.576447 if sptechcl_ncy_0 >= 1.09133 if sptechcl_amp_0 < 1.24989 if sptechcl_offsetphase_0 >= -0.350003 return 1
if gsptsebars > 49 if gsptsebars >= 50 if gsptselpr01cl <= 0.134077 if gsptsecl_ncy_0 >= 1.10754 if gsptsecl_offsetphase_1 >= -0.286188 if gsptsecl_phase_0 <= 0.917072 if gsptsecl_lvel_1 < 0.174925 if gsptsecl_wsum < 1.26999 return 1
if ndinsrcl_phase_1 < 0.587518 if ndinsrlpr01cl <= 0.0229777 if ndinsrcl_wave_0 <= 0.934144 if ndinsrlpr12cl < 0.296799 if ndinsrcl_lvel_1 <= 0.117298 if ndinsrcl_wprop_0 <= 0.966144 return 1
if sptechcl_lvel_0 <= 0.187625 if sptechcl_phase_0 >= 0.322132 if sptechcl_ncy_0 >= 1.08671 if sptechcl_wprop_1 <= 0.973641 if sptechcl_wprop_0 >= -0.878743 if sptechlpr12cl < 0.660802 if sptechcl_ncy_0 <= 1.20559 if sptechlpr01cl >= -0.637203 return 1
if w5000cl_offsetphase_0 <= 0.299248 if w5000cl_ncy_0 >= 1.61404 if w5000cl_trendsds > -2.94143 if w5000lpr12cl < 0.282786 if w5000cl_lvel_0 > 0.143781 if w5000cl_phase_0 <= 0.80247 if w5000cl_phase_0 <= 0.799542 if w5000cl_wave_1 > -0.0348898 return 1
if asx200cl_wprop_1 < 0.876702 if asx200cl_amp_1 >= 0.451709 if asx200cl_amp_1 < 0.73843 if asx200cl_per_1 < 30.0989 if asx200cl_lvel_1 > 0.0462801 if asx200cl_wave_0 >= 0.326522 if asx200cl_amp_1 >= 0.451709 if asx200cl_rvel_0 > 0.099337 if asx200cl_wsum > 0.294748 return 1
if djulpr12cl >= -0.119748 if djucl_per_0 >= 35.1501 if djucl_lvel_0 > djucl_rvel_0 if djucl_phase_0 > 0.14216 if djulpr12cl >= -0.119748 if djucl_amp_1 < 0.66725 if djucl_per_0 >= 35.1501 if djucl_lvel_0 <= 0.163034 return 1
if nbilpr12cl < 0.199289 if nbicl_wave_0 > -0.383525 if nbicl_trendsds <= 4.66456 if nbicl_offsetphase_0 < -0.0186597 if nbicl_lvel_0 < 0.135501 if nbicl_rvel_1 <= 0.165058 if nbilpr01cl < 0.201862 if nbicl_offsetphase_0 <= 0.00157404 return 1
if sp600cl_wave_0 >= -0.714745 if sp600cl_lvel_0 >= 0.0843772 if sp600cl_wave_0 <= 1.00219 if sp600bars >= 37 if sp600cl_ncy_1 < 2.46636 if sp600lpr01cl <= 0.387207 if sp600cl_ncy_0 >= 1.17258 if sp600bars >= 37 return 1
if nbicl_offsetphase_0 <= nbicl_phase_0 if nbicl_offsetphase_0 <= -0.0201017 if nbicl_ncy_1 <= 3.41247 if nbilpr12cl < 0.193432 if nbicl_trendsds <= 4.1589 if nbicl_wprop_0 >= -0.744252 if nbicl_lvel_0 <= 0.136218 return 1
if bfxcl_lvel_0 > 0.12984 if bfxcl_lvel_0 <= 0.204981 if bfxcl_ncy_1 <= 4.39151 if bfxcl_per_0 > 21.4993 if bfxcl_amp_1 >= 0.538709 if bfxcl_phase_0 > 0.301409 if bfxcl_offsetphase_0 >= -0.27477 return 1
if djicl_offsetphase_0 < -0.00246853 if djicl_rvel_1 >= 0.05449 if djicl_lvel_0 > djicl_rvel_0 if djicl_lvel_0 >= 0.0752994 if djicl_wprop_1 > -0.193109 if djicl_phase_1 < 0.772738 if djicl_rvel_1 >= 0.05449 if djicl_ncy_1 < 3.74095 if djicl_amp_1 <= 1.06974 return 1
if spcdcl_offsetphase_0 < 0.0618283 if spcdcl_bartrendsds < 0.176386 if spcdcl_phase_0 < 0.543865 if spcdcl_wave_1 > -0.637746 if spcdcl_lvel_0 >= 0.128471 if spcdcl_lvel_0 >= 0.123029 if spcdcl_rvel_1 > 0.0738231 if spcdcl_offsetphase_1 > 0.128175 return 1
if sptelcl_offsetphase_1 > -0.0968448 if sptelcl_per_1 <= 15.7406 if sptelcl_lvel_1 < 0.370058 if sptelcl_ncy_0 <= 2.1659 if sptelcl_wprop_1 < 0.876375 if sptelcl_lvel_0 > 0.142592 if sptelcl_lvel_0 > 0.142054 if sptelcl_lvel_0 > 0.142054 return 1
if rutcl_ncy_1 < 3.14879 if rutcl_rvel_0 <= rutcl_lvel_0 if rutlpr12cl < 0.190288 if rutcl_wave_0 > -1.0056 if rutcl_ncy_0 > 1.17494 if rutcl_lvel_1 <= 0.151642 if rutlpr12cl < 0.190288 return 1
if ruacl_ncy_0 >= 1.0421 if ruacl_bartrendsds >= -0.0701428 if rualpr12cl >= -1.87237 if rualpr12cl < -0.169257 if rualpr01cl < -0.3675 if ruacl_wprop_0 < 0.987579 if ruacl_ncy_0 < 3.31658 if rualpr12cl < -0.304058 if ruacl_wave_0 >= -0.246811 return 1
if ruacl_wave_0 > -0.6948 if ruacl_rvel_1 <= ruacl_lvel_1 if ruacl_wprop_0 <= 0.999565 if rualpr12cl <= -0.384271 if ruacl_trendsds > -3.4272 if ruacl_ncy_0 > 1.04294 if rualpr12cl <= -0.384271 if ruacl_trendsds > -3.4272 return 1
if spfincl_per_1 <= 26.8685 if spfincl_wave_1 > -0.189383 if spfincl_wave_1 >= -0.161 if spfincl_wprop_0 <= 0.983375 if spfincl_wave_0 >= -0.58717 if spfincl_per_0 <= 31.6072 if spfincl_amp_0 < 1.02032 return 1
if sp600cl_lvel_0 > 0.0835371 if sp600cl_wprop_0 <= 0.94178 if sp600lpr01cl <= 0.355109 if sp600bars > 40 if sp600cl_per_1 >= 21.0496 if sp600cl_ncy_1 <= 2.53135 if sp600lpr01cl <= 0.355109 if sp600cl_lvel_0 > 0.0832733 if sp600cl_wprop_1 >= -0.505798 return 1
if sp100cl_wprop_0 < 0.684285 if sp100cl_rvel_1 < sp100cl_lvel_1 if sp100cl_wsum > -0.269601 if sp100cl_wave_0 < 0.912238 if sp100cl_wsum > -0.269601 if sp100cl_offsetphase_0 > -0.354417 if sp100cl_trendsds <= 1.95873 if sp100cl_ncy_0 >= 1.22392 if sp100cl_wprop_0 < 0.672633 return 1
if n225cl_ncy_0 <= 1.37116 if n225lpr01cl > -0.512516 if n225cl_rvel_1 <= 0.125867 if n225cl_lvel_1 <= 0.121976 if n225cl_offsetphase_0 <= 0.498048 if n225cl_wprop_0 <= 0.701871 if n225cl_ncy_0 <= 1.37116 if n225cl_bartrendsds < 0.0581231 if n225lpr01cl > -0.512516 return 1
if xoilpr01cl > -0.317378 if xoicl_lvel_1 >= xoicl_rvel_1 if xoicl_amp_0 < 1.19832 if xoicl_rvel_0 <= xoicl_lvel_0 if xoicl_amp_0 <= 1.1726 if xoicl_phase_1 <= 0.664929 return 1
if spfincl_offsetphase_1 < spfincl_phase_1 if spfincl_amp_0 <= 1.01345 if spfincl_per_1 <= 25.1463 if spfincl_wave_0 >= -0.421303 if spfincl_wave_1 >= -0.161709 if spfincl_rvel_0 > 0.0965003 if spfincl_wave_0 < 0.8804 if spfincl_wave_0 >= -0.421303 if spfincl_wave_0 >= -0.421303 return 1
if ndinsrcl_wave_1 >= -0.754112 if ndinsrcl_rvel_1 <= 0.136736 if ndinsrcl_ncy_0 >= 1.68728 if ndinsrcl_amp_1 > 0.390133 if ndinsrlpr12cl < 0.280531 if ndinsrlpr01cl < 0.440081 if ndinsrcl_amp_1 > 0.390133 if ndinsrcl_phase_0 > 0.0562328 return 1
if n225cl_wprop_0 < 0.70024 if n225cl_ncy_0 <= 1.37337 if n225lpr01cl >= -0.51776 if n225cl_wave_1 < 0.593395 if n225lpr12cl <= 0.913939 if n225cl_wprop_0 < 0.792009 if n225lpr01cl >= -0.51776 if n225cl_lvel_1 <= 0.122734 if n225cl_ncy_1 < 6.07028 return 1
if spenrgcl_rvel_0 <= 0.178898 if spenrgcl_wave_1 > 0.160525 if spenrgcl_rvel_0 < 0.139509 if spenrgcl_wave_1 > 0.147221 if spenrgcl_rvel_0 < 0.139509 if spenrgcl_lvel_0 > 0.082349 if spenrgcl_wsum < 1.51997 if spenrgcl_per_1 < 33.2069 return 1
if nyacl_amp_0 <= 1.37292 if nyalpr01cl <= 0.308411 if nyacl_amp_0 < 1.24746 if nyacl_wave_0 >= 0.397509 if nyacl_lvel_0 > nyacl_rvel_0 if nyacl_per_1 <= 28.0804 if nyacl_lvel_1 > 0.0514567 if nyalpr12cl <= 0.929302 if nyacl_wave_0 >= 0.161891 return 1
if nyacl_rvel_0 < nyacl_lvel_0 if nyacl_rvel_0 > 0.0554896 if nyacl_wprop_0 >= 0.550856 if nyacl_phase_0 < 0.461132 if nyalpr12cl <= 0.925539 if nyacl_per_1 <= 31.6198 if nyacl_wave_0 >= 0.411132 if nyacl_phase_0 < 0.461132 return 1
if nd100cl_wsum >= 0.0434539 if nd100cl_wave_1 <= 0.238033 if nd100lpr12cl <= 0.106459 if nd100cl_wave_1 < 0.233823 if nd100cl_lvel_1 > 0.0605134 if nd100cl_wsum < 1.29088 if nd100cl_offsetphase_0 > -0.227007 if nd100cl_rvel_1 > 0.0636087 if nd100cl_ncy_1 > 2.57203 return 1
Performance
For the evaluation period, simulated buy and hold performance was
total return 260.94%, compound annual return 15.40%, maximum drawdown 35.56%, daily mean result 0.0570%, daily mean win 0.9259%, daily mean loss 1.0175%, number of trading days 2251, number of daily wins 1250, daily win rate 55.53%
The simulated performance using the rules for the evaluation period was
total return 1706.05%, compound annual return 38.10%, maximum drawdown 8.83%, mean result 0.1477%, mean win 0.8963%, mean loss 0.8530%, number of trades 1961, number of wins 1126, win rate 57.42%
Other Targets
Potential targets in addition to the NASDAQ 100 based on mapping to ETFs for actual trading and three-year, daily return correlations (
QuantDare method) are in the attached corrmatrix.csv (semicolon-separated).
