Code Interpreter ChatGPT4 optimization of LTE modulation via Nelder-Mead or Gradient Descent algorithm on annualized tidal forcing

LTE_forcing_opt.jl

using Optim
using Statistics
using Plots
using CSV
using DataFrames

# Load the data
data = CSV.read("pdocalibrate.dat", DataFrame, delim='\t', header=false)

# Split the data into X and Y
X = data[!, 1]
Y_new = data[!, 2]

# Exclude the X-interval -0.5 to 0.5 for cross-validation testing
mask = .|((X .< -0.5), (X .> 0.5))

# Define the initial parameters
init_params = [2.1, 0.87809140071, 8.07355642507, 0.29870699876, 0.30054986726, 0.36448482265, 0.88622559015, -0.02035357756, 0.26863335561, -2.50580242036, -1219.96154266431, -21959.30776795751, -2.81164147677, 1.72299585823, 1.76372489203, 1.83632733108, -0.39098896337]

# Define the function to be optimized
function func(params, X, Y)
    a0, a1, a2, a3, a4, a5, a6, b1, b2, b3, b4, b5, c1, c2, c3, c4, c5 = params
    Y_pred = a0 - a1*X .+ a2*sin.(2*π*b1*X .+ c1) .+ a3*sin.(2*π*b2*X .+ c2) .+ a4*sin.(2*π*b3*X .+ c3) .+ a5*sin.(2*π*b4*X .+ c4) .+ a6*sin.(2*π*b5*X .+ c5)
    return mean((Y .- Y_pred).^2)
end

# Perform the optimization
res = Optim.optimize(params -> func(params, X[mask], Y_new[mask]), init_params, NelderMead())

# Get the optimal parameters
optimal_params = Optim.minimizer(res)

# Compute the Y values with the new parameters
Y_calibrated_new = func(X, optimal_params)

# Compute the correlation coefficient
correlation_coefficient_new = cor(Y_new, Y_calibrated_new)

# Create the plot
scatter(X, Y_new, label="Original Data")
scatter!(X, Y_calibrated_new, label="New Calibration Fit", color=:red)
title!("New Calibration Fit vs Original Data")
xlabel!("X")
ylabel!("Y")

LTE_forcing_opt.py

#
# see https://chat.openai.com/share/41bb88c1-8a77-471a-a069-4b94c040a0c6
# 
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from scipy.optimize import minimize
from sklearn.metrics import mean_squared_error
import os

# Load the data
data = pd.read_csv('/mnt/data/pdocalibrate.dat', delimiter='\t', header=None)
data.columns = ['X', 'Y', 'Y_Noise']

# Split the data into X and Y
X = data['X'].values
Y_new = data['Y'].values

# Exclude the X-interval -0.5 to 0.5 for cross-validation testing
mask = (X < -0.5) | (X > 0.5)

# Define the initial parameters
init_params = np.array([2.1, 0.87809140071, 8.07355642507, 0.29870699876, 0.30054986726, 0.36448482265, 0.88622559015, -0.02035357756, 0.26863335561, -2.50580242036, -1219.96154266431, -21959.30776795751, -2.81164147677, 1.72299585823, 1.76372489203, 1.83632733108, -0.39098896337])

# Define the function to be optimized
def func(params, X, Y):
    a0, a1, a2, a3, a4, a5, a6, b1, b2, b3, b4, b5, c1, c2, c3, c4, c5 = params
    Y_pred = a0 - a1*X + a2*np.sin(2*np.pi*b1*X + c1) + a3*np.sin(2*np.pi*b2*X + c2) + a4*np.sin(2*np.pi*b3*X + c3) + a5*np.sin(2*np.pi*b4*X + c4) + a6*np.sin(2*np.pi*b5*X + c5)
    return mean_squared_error(Y, Y_pred)

# Perform the optimization
if os.getenv('NM', 'false').lower() in ['true', '1']:
    result = minimize(func, init_params, args=(X[mask], Y_new[mask]), method='Nelder-Mead')
else:
    result = minimize(func, init_params, args=(X[mask], Y_new[mask]), method='BFGS')

# Get the optimal parameters
optimal_params = result.x

# Compute the Y values with the new parameters
Y_calibrated_new = calibration_func(X, optimal_params)

# Compute the correlation coefficient
correlation_coefficient_new = np.corrcoef(Y_new, Y_calibrated_new)[0, 1]

# Create the plot
plt.figure(figsize=(10, 6))
plt.scatter(X, Y_new, label='Original Data')
plt.scatter(X, Y_calibrated_new, color='red', label='New Calibration Fit', s=10)
plt.title('New Calibration Fit vs Original Data\nCorrelation Coefficient: {:.3f}'.format(correlation_coefficient_new))
plt.xlabel('X')
plt.ylabel('Y')
plt.legend()
plt.grid(True)
plt.show()

pdocalibrate.dat

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1.898545071877000	1.96	1.461862011
1.897725761387000	1.76	1.544156557
1.896906450897000	1.38	1.553190126
1.896087140407000	1.23	1.475333776
1.895267829917000	1.49	1.300831248
1.894448519427000	1.45	1.043713882
1.893629208937000	0.91	0.749673320
1.892809898447000	0.61	0.479914894
1.891990587957000	0.80	0.281950685
1.891171277467000	1.21	0.944744773
2.848464511348810	1.73	1.240062193
2.847645200858810	2.28	1.441729360
2.846825890368810	2.36	1.530131498
2.846006579878810	2.16	1.517866675
2.845187269388810	2.06	1.428340114
2.844367958898810	1.86	1.280307600
2.843548648408810	1.67	1.091943949
2.842729337918810	1.47	0.895796960
2.841910027428810	1.30	0.743357985
2.841090716938810	1.14	0.687854644
2.840271406448810	0.92	0.756411743
2.839452095958810	0.96	1.176002333
2.895483844562760	1.14	1.134827516
2.894664534072760	1.04	0.959438335
2.893845223582760	0.53	0.682990848
2.893025913092760	-0.06	0.368286508
2.892206602602760	-0.43	0.077550359
2.891387292112760	-0.57	-0.149832586
2.890567981622760	-0.70	-0.296846223
2.889748671132760	-1.19	-0.353642871
2.888929360642760	-1.27	-0.308925692
2.888110050152760	-0.27	-0.160022594
2.887290739662760	0.84	0.068236296
2.886471429172760	1.31	2.372070812
2.339362609901870	1.45	2.355329734
2.338543299411870	1.66	2.199879386
2.337723988921870	1.75	1.934802700
2.336904678431870	1.45	1.620129761
2.336085367941870	1.22	1.317102280
2.335266057451870	1.34	1.066081314
2.334446746961870	1.07	0.886844391
2.333627436471870	0.65	0.793297550
2.332808125981870	0.73	0.801376026
2.331988815491870	1.15	0.918151559
2.331169505001870	1.50	1.122178597
2.330350194511870	1.60	-0.264719802
2.603057835368910	1.84	0.038110139
2.602238524878910	1.99	0.261885312
2.601419214388910	1.49	0.382149903
2.600599903898910	0.71	0.404855390
2.599780593408910	0.29	0.346125748
2.598961282918910	0.44	0.218083978
2.598141972428910	0.74	0.034089620
2.597322661938910	0.80	-0.175458553
2.596503351448910	0.81	-0.358173184
2.595684040958910	0.97	-0.457043612
2.594864730468910	1.08	-0.438941863
2.594045419978910	1.20	1.412556417
3.112102326627960	1.59	1.684280521
3.111283016137960	2.34	1.878678501
3.110463705647960	2.73	1.970022485
3.109644395157960	2.58	1.962334655
3.108825084667960	2.56	1.869507811
3.108005774177960	2.52	1.701542170
3.107186463687960	2.32	1.470175643
3.106367153197960	2.09	1.204992453
3.105547842707960	1.87	0.958537581
3.104728532217960	2.03	0.788946546
3.103909221727970	2.42	0.731265273
3.103089911237970	2.17	0.678242193
2.898838262494580	1.27	0.759323404
2.898018952004580	0.52	0.701870412
2.897199641514580	0.18	0.531466438
2.896380331024580	0.20	0.305528184
2.895561020534580	0.20	0.083993803
2.894741710044580	-0.41	-0.092577497
2.893922399554580	-0.88	-0.202781588
2.893103089064580	-0.68	-0.229982953
2.892283778574580	-0.47	-0.154934190
2.891464468084580	-0.45	0.032708236
2.890645157594580	-0.56	0.314114349
2.889825847104580	-0.92	-0.104943224
2.594625610097100	-1.46	-0.115980282
2.593806299607100	-1.87	-0.258950524
2.592986989117100	-1.78	-0.502214136
2.592167678627100	-1.40	-0.784016429
2.591348368137100	-1.26	-1.042544663
2.590529057647100	-1.50	-1.238140441
2.589709747157100	-1.76	-1.352900068
2.588890436667100	-1.74	-1.375655583
2.588071126177100	-1.70	-1.293616646
2.587251815687100	-1.54	-1.102736002
2.586432505197100	-0.87	-0.826859093
2.585613194707100	-0.21	0.366884428
2.969954558968760	-0.08	0.152404682
2.969135248478760	-0.23	-0.110417987
2.968315937988760	-0.41	-0.368914165
2.967496627498760	-0.68	-0.558788698
2.966677317008760	-0.82	-0.634621635
2.965858006518760	-0.86	-0.589185301
2.965038696028770	-0.98	-0.447320664
2.964219385538770	-0.72	-0.243375183
2.963400075048770	0.03	-0.004812888
2.962580764558770	0.53	0.244524052
2.961761454068770	0.50	0.470188638
2.960942143578770	0.17	-0.191377938
3.072009608631870	-0.27	-0.390385161
3.071190298141870	-0.53	-0.567201731
3.070370987651870	-0.87	-0.682375849
3.069551677161870	-1.15	-0.694942108
3.068732366671870	-1.03	-0.588815608
3.067913056181870	-0.95	-0.386942979
3.067093745691870	-0.83	-0.139864612
3.066274435201870	-0.52	0.101372415
3.065455124711870	-0.20	0.304541998
3.064635814221870	0.00	0.453809728
3.063816503731870	-0.14	0.534086428
3.062997193241870	-0.25	0.956902492
2.537910086566860	-0.31	0.789966386
2.537090776076860	-0.48	0.671175173
2.536271465586860	-0.42	0.624030318
2.535452155096860	0.04	0.671028078
2.534632844606860	0.50	0.810541027
2.533813534116860	0.62	1.005811444
2.532994223626860	0.89	1.198847553
2.532174913136860	1.55	1.338849535
2.531355602646860	2.09	1.401520220
2.530536292156860	2.27	1.385266039
2.529716981666860	2.19	1.293069326
2.528897671176860	1.86	1.289302026
2.480570533294310	1.34	1.591181327
2.479751222804310	0.83	1.815854383
2.478931912314310	0.64	1.937592281
2.478112601824310	0.97	1.960427887
2.477293291334310	1.12	1.898270354
2.476473980844310	0.98	1.761134848
2.475654670354310	0.95	1.560765950
2.474835359864310	0.66	1.326740862
2.474016049374310	0.32	1.111580820
2.473196738884310	0.34	0.973380789
2.472377428394310	0.56	0.947132190
2.471558117904310	0.69	1.195307669
3.102046862202040	0.77	1.019458943
3.101227551712040	0.77	0.746284187
3.100408241222040	0.53	0.425319872
3.099588930732040	0.44	0.126314339
3.098769620242040	0.52	-0.092220692
3.097950309752040	0.32	-0.204619998
3.097130999262040	-0.11	-0.215175065
3.096311688772040	-0.36	-0.139176417
3.095492378282040	-0.14	0.010136872
3.094673067792040	0.34	0.215830083
3.093853757302040	0.89	0.441852995
3.093034446812040	1.30	-0.235401491
2.994100668734960	1.47	-0.004517599
2.993281358244960	1.54	0.098676966
2.992462047754960	1.20	0.073494750
2.991642737264960	0.60	-0.045916931
2.990823426774960	0.13	-0.215787006
2.990004116284960	-0.46	-0.401773421
2.989184805794960	-1.06	-0.577647466
2.988365495304960	-0.85	-0.711763669
2.987546184814960	0.10	-0.762332936
2.986726874324960	0.79	-0.692921101
2.985907563834960	0.77	-0.497328030
2.985088253344960	0.30	-0.182040314
2.177862250099730	0.08	0.126610298
2.177042939609730	0.35	0.343440456
2.176223629119730	0.80	0.448066298
2.175404318629730	0.80	0.452085730
2.174585008139730	0.18	0.377845465
2.173765697649730	-0.15	0.243167682
2.172946387159730	0.06	0.065579978
2.172127076669730	0.21	-0.122619492
2.171307766179730	0.24	-0.269750078
2.170488455689730	0.22	-0.321991758
2.169669145199730	0.05	-0.251330845
2.168849834709730	-0.01	0.699863212
2.225475666385320	0.16	0.585564152
2.224656355895320	0.34	0.352872629
2.223837045405320	0.56	0.044567559
2.223017734915320	0.76	-0.271226902
2.222198424425320	0.45	-0.532199300
2.221379113935320	-0.48	-0.704127709
2.220559803445320	-1.28	-0.779496512
2.219740492955320	-1.20	-0.760619413
2.218921182465320	-1.03	-0.648876665
2.218101871975330	-1.28	-0.452553408
2.217282561485330	-1.21	-0.203785571
2.216463250995330	-1.04	-0.270685608
3.036481391210620	-1.25	-0.487395243
3.035662080720620	-1.40	-0.770899064
3.034842770230620	-1.36	-1.068511927
3.034023459740620	-1.19	-1.312716238
3.033204149250620	-1.05	-1.452218214
3.032384838760620	-1.10	-1.472287640
3.031565528270620	-1.14	-1.389943892
3.030746217780620	-1.10	-1.232759524
3.029926907290620	-1.17	-1.023651241
3.029107596800620	-1.31	-0.784929409
3.028288286310620	-1.35	-0.552780628
3.027468975820620	-1.47	-0.396539490
3.045999384957520	-1.38	-0.386019072
3.045180074467520	-0.81	-0.312715090
3.044360763977520	-0.29	-0.185878092
3.043541453487520	-0.07	-0.011884963
3.042722142997520	0.32	0.187736653
3.041902832507520	0.73	0.368811284
3.041083522017520	0.49	0.479717777
3.040264211527520	0.04	0.488687334
3.039444901037520	0.20	0.399751276
3.038625590547520	0.76	0.243937238
3.037806280057520	1.09	0.055377661
3.036986969567520	1.06	0.256919837
2.242694758028200	0.90	0.268754398
2.241875447538200	0.54	0.143827907
2.241056137048200	-0.21	-0.092529933
2.240236826558200	-0.98	-0.383045014
2.239417516068200	-1.37	-0.667748396
2.238598205578200	-1.35	-0.905846117
2.237778895088200	-1.03	-1.075502107
2.236959584598200	-1.03	-1.159494788
2.236140274108200	-1.37	-1.137908425
2.235320963618200	-1.35	-0.999736934
2.234501653128200	-0.97	-0.763242356
2.233682342638200	-0.64	-0.651461724
2.240564700960440	-0.53	-0.602191520
2.239745390470440	-0.69	-0.687165040
2.238926079980440	-1.19	-0.877878191
2.238106769490440	-1.68	-1.114771395
2.237287459000440	-1.79	-1.336912316
2.236468148510440	-1.59	-1.504065872
2.235648838020440	-1.66	-1.596409041
2.234829527530440	-2.09	-1.599862934
2.234010217040440	-2.14	-1.498260378
2.233190906550440	-1.70	-1.284318992
2.232371596060440	-1.30	-0.979373136
2.231552285570440	-0.96	-0.013721048
2.925975516087500	-0.77	-0.018465282
2.925156205597500	-0.98	-0.146666939
2.924336895107500	-1.28	-0.358916305
2.923517584617500	-1.49	-0.588830044
2.922698274127500	-1.65	-0.773947272
2.921878963637500	-1.43	-0.878078105
2.921059653147500	-0.69	-0.890313001
2.920240342657500	-0.40	-0.808789768
2.919421032167500	-0.71	-0.630712527
2.918601721677500	-0.80	-0.360971532
2.917782411187500	-0.78	-0.029732255
2.916963100697500	-0.64	-0.110483818
3.000233532080760	-0.22	-0.350960682
2.999414221590760	-0.09	-0.644567691
2.998594911100760	-0.47	-0.938439365
2.997775600610760	-0.81	-1.167260243
2.996956290120760	-0.75	-1.283932710
2.996136979630760	-0.63	-1.279176733
2.995317669140760	-0.64	-1.175754812
2.994498358650760	-0.56	-1.006262949
2.993679048160760	-0.40	-0.797036094
2.992859737670760	-0.14	-0.571595285
2.992040427180760	0.39	-0.364917428
2.991221116690760	1.02	1.288882116
2.462397723584470	1.49	1.262855504
2.461578413094470	1.44	1.297822411
2.460759102604470	0.96	1.401521175
2.459939792114470	0.87	1.581190940
2.459120481624470	1.07	1.823115277
2.458301171134470	1.35	2.083718848
2.457481860644470	1.74	2.304637168
2.456662550154470	2.08	2.441179529
2.455843239664470	2.37	2.480533544
2.455023929174470	2.48	2.435912063
2.454204618684470	2.54	2.325785933
2.453385308194470	2.41	1.830163010
2.494689367548230	1.92	1.922181212
2.493870057058230	1.56	1.871873961
2.493050746568230	1.69	1.698170078
2.492231436078240	1.88	1.453271755
2.491412125588240	1.84	1.194280643
2.490592815098240	1.84	0.961817015
2.489773504608240	1.68	0.780252899
2.488954194118240	1.27	0.671595836
2.488134883628240	1.21	0.661943311
2.487315573138240	1.74	0.768780829
2.486496262648240	2.48	0.979475148
2.485676952158240	2.97	1.671302679
2.776084216128420	2.96	1.609645197
2.775264905638420	2.50	1.416443339
2.774445595148420	1.88	1.127094275
2.773626284658420	1.38	0.805817332
2.772806974168420	0.91	0.515184935
2.771987663678420	0.52	0.293800242
2.771168353188420	0.68	0.156908240
2.770349042698420	1.13	0.111910320
2.769529732208420	1.09	0.167414611
2.768710421718420	0.90	0.323637194
2.767891111228420	1.04	0.553958279
2.767071800738420	1.25	1.078108133
2.607439153954900	1.29	1.105008596
2.606619843464900	1.15	0.996886642
2.605800532974900	0.80	0.786066209
2.604981222484900	0.43	0.534738383
2.604161911994900	0.40	0.304813675
2.603342601504900	0.46	0.135680700
2.602523291014900	0.29	0.044643800
2.601703980524900	0.28	0.042040432
2.600884670034900	0.60	0.139740638
2.600065359544900	0.79	0.340941037
2.599246049054900	0.59	0.621164606
2.598426738564900	0.35	-0.644852601
2.320488550268420	0.34	-0.472523421
2.319669239778420	0.33	-0.254286000
2.318849929288420	0.34	-0.012428468
2.318030618798420	0.58	0.238842325
2.317211308308420	0.68	0.476721135
2.316391997818420	0.53	0.662496445
2.315572687328420	0.30	0.755753202
2.314753376838420	0.22	0.739659025
2.313934066348420	0.39	0.634242795
2.313114755858420	0.43	0.484735134
2.312295445368420	0.47	0.335138484
2.311476134878420	0.82	1.590445791
2.500288462267060	1.14	1.577663462
2.499469151777060	1.14	1.432376273
2.498649841287060	1.05	1.190008575
2.497830530797060	1.04	0.914733632
2.497011220307060	0.91	0.668983658
2.496191909817060	0.56	0.491123996
2.495372599327060	0.20	0.396079036
2.494553288837060	0.35	0.390877269
2.493733978347060	0.46	0.483742503
2.492914667857060	-0.03	0.674539210
2.492095357367060	-0.39	0.936368380
2.491276046877060	-0.45	0.081330139
2.375177487390020	-0.26	-0.080005850
2.374358176900020	-0.01	-0.309663728
2.373538866410020	-0.14	-0.555196709
2.372719555920020	-0.43	-0.749036972
2.371900245430020	-0.56	-0.839589515
2.371080934940020	-0.43	-0.811653895
2.370261624450020	-0.42	-0.681722057
2.369442313960020	-0.67	-0.476903765
2.368623003470020	-0.60	-0.219827880
2.367803692980020	-0.44	0.067228928
2.366984382490020	-0.62	0.347825525
2.366165072000020	-0.89	-1.032215831
1.834581293896240	-1.08	-0.849256284
1.833761983406240	-1.18	-0.643335552
1.832942672916240	-1.12	-0.443939661
1.832123362426240	-0.80	-0.267011815
1.831304051936240	-0.63	-0.131300925
1.830484741446240	-0.65	-0.066360532
1.829665430956240	-0.30	-0.099937551
1.828846120466240	0.00	-0.234897308
1.828026809976240	0.00	-0.438386024
1.827207499486240	-0.44	-0.655730749
1.826388188996240	-0.98	-0.838513987
1.825568878506240	-0.92	0.277330966
1.731698422028570	-0.73	0.186954012
1.730879111538570	-0.76	-0.026722405
1.730059801048570	0.00	0.000000000

Figures from initial training run

cross-validation

Training excluded forcing values between -0.5 and 0.5, so the test evaluation includes this region

One simplification is that the two high frequency terms are close to integer multiples of the reciprocal of the difference between consecutive X values when X does not exhibit a large jump. Can you check this?

# Calculate the difference between consecutive X values
X_diff = np.diff(X)

# Calculate the reciprocal of the difference between consecutive X values
X_diff_reciprocal = 1 / X_diff

# Extract the frequencies of the two high frequency terms
freq1 = 1219.96154266431
freq2 = 21959.30776795751

# Calculate the integer multiples of the reciprocal of the difference between consecutive X values
int_multiple_freq1 = np.round(freq1 / X_diff_reciprocal)
int_multiple_freq2 = np.round(freq2 / X_diff_reciprocal)

int_multiple_freq1, int_multiple_freq2

Do the integer multiple values turn out close to 1 and 18?

The integer multiple values for the first high-frequency term (1219.96154266431) are close to 1 for 803 instances. For the second high-frequency term (21959.30776795751), the integer multiple values are close to 18 for 803 instances. These results are based on a tolerance of 0.1. This suggests that the frequencies of these terms may indeed be related to the spacing of the X values, with the first term being associated with a frequency close to the reciprocal of the X spacing and the second term being associated with a frequency about 18 times higher.