Editing Openai/693eaf5a-1494-8007-b1e3-00abefb11523 (section)

=== Assistant: It seems like I can’t do more advanced data analysis right now. ===
It seems like I can’t do more advanced data analysis right now. Please try again later.

I wasn’t able to run the analysis and create the plots inside this session, but I’ll give you a complete, ready-to-run Python script (and precise instructions) that performs everything you asked for:
* backtests the series,
* creates the triangle/tetrahedron per day (origin, day, price; orthogonal distance → half-height),
* computes volumetric and related geometric indicators,
* computes simple Bayesian/posterior statistics by indicator bins,
* fits a single-feature logistic for each indicator and reports AUC / correlation / lift,
* produces the plots you requested (price series, tetrahedron example, indicator timeseries, posterior-by-quartile, ROC curves),
* and prints a ranked summary of which indicators were best.

Copy–paste the script below into a local Python environment / Jupyter notebook and run it. I included comments and explanations so you can tweak any detail (window sizes, binning, indicator choices).

<syntaxhighlight lang="python">"""
Tetrahedral volumetric indicators + Bayesian/backtest analysis
Run locally (Python 3.8+). Required packages: numpy, pandas, matplotlib, scikit-learn.

Saves plots to disk and prints a summary table ranking indicators by AUC.

Author: ChatGPT (script for user)
"""

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
from sklearn.linear_model import LogisticRegression
from sklearn.metrics import roc_auc_score, roc_curve, roc_auc_score
import os

=== === PARAMETERS === ===
OUTDIR = "tetra_analysis_outputs"
os.makedirs(OUTDIR, exist_ok=True)

=== Use the same synthetic data seed as earlier examples (to reproduce) ===
np.random.seed(42)

=== Generate daily dates for a year ===
dates = pd.date_range(start="2025-01-01", end="2025-12-31", freq="D")
n = len(dates)

=== Example price series (random uniform), replace with your actual series if available ===
prices = np.random.uniform(50, 150, n)

=== Numeric time axis: days since start (1..365) ===
d_numeric = np.array([d.timetuple().tm_yday for d in dates], dtype=float)

=== Build DataFrame ===
df = pd.DataFrame({
    "date": dates,
    "price": prices,
    "d": d_numeric
})

=== --- Geometric construction per-day --- ===
=== Points in 2D for triangle: O=(0,0), A=(d,0), B=(0,p) ===
x1 = df["d"].values         # A.x = d
y1 = np.zeros(n)            # A.y = 0
x2 = np.zeros(n)            # B.x = 0
y2 = df["price"].values     # B.y = p

=== Hypotenuse length between A and B (2D) ===
hyp_len = np.sqrt((x1 - x2)'''2 + (y1 - y2)'''2)

=== Distance from origin to line AB (2D) ===
=== determinant numerator simplifies to x1''y2 - y1''x2 -> x1*y2 ===
num = np.abs(x1 '' y2 - y1 '' x2)   # = x1*y2
den = np.sqrt((y2 - y1)'''2 + (x2 - x1)'''2)
dist_origin_to_line = num / den

=== Per user: "Sending half the distance of an orthogonal from the origin to the hypotenuse" ===
h = dist_origin_to_line / 2.0

=== 3D tetrahedron apex D = (d, p, h) ===
A = np.column_stack((x1, y1, np.zeros(n)))
B = np.column_stack((x2, y2, np.zeros(n)))
D = np.column_stack((x1, y2, h))

=== Compute tetrahedron volume: V = |det([A, B, D])| / 6 ===
volumes = np.zeros(n)
for i in range(n):
    M = np.vstack((A[i], B[i], D[i])).T    # 3x3 matrix whose columns are A,B,D
    volumes[i] = abs(np.linalg.det(M)) / 6.0

=== Triangle area (2D): area = |x1''y2 - x2''y1|/2 = x1*y2/2 ===
tri_area = (x1 * y2) / 2.0

=== Additional indicator: rolling volatility of returns (10-day) ===
returns = np.concatenate([[0.0], np.diff(df["price"].values) / df["price"].values[:-1]])
vol10 = pd.Series(returns).rolling(window=10, min_periods=1).std().values

=== Put indicators into df ===
df["hyp_len"] = hyp_len
df["dist_origin_to_line"] = dist_origin_to_line
df["h"] = h
df["tetra_vol"] = volumes
df["tri_area"] = tri_area
df["ret"] = returns
df["vol10"] = vol10

=== Target: next-day increase ===
df["next_price"] = df["price"].shift(-1)
df["up_next"] = (df["next_price"] > df["price"]).astype(int)
df = df.iloc[:-1].copy()  # drop last day (no target)

=== Indicators to evaluate ===
indicators = ["tetra_vol", "h", "hyp_len", "tri_area", "vol10"]

=== Helper to fit single-feature logistic and compute AUC ===
def evaluate_indicator(feature_array, target_array):
    X = feature_array.reshape(-1,1)
    y = target_array
    # If constant feature, return nans
    if np.allclose(X, X.mean()):
        return {"auc": np.nan, "corr": np.nan, "fpr": None, "tpr": None}
    clf = LogisticRegression(solver="liblinear")
    clf.fit(X, y)
    probs = clf.predict_proba(X)[:,1]
    try:
        auc = roc_auc_score(y, probs)
        fpr, tpr, _ = roc_curve(y, probs)
    except Exception:
        auc = np.nan
        fpr, tpr = None, None
    corr = np.corrcoef(feature_array, y)[0,1]
    return {"auc": auc, "corr": corr, "fpr": fpr, "tpr": tpr, "probs": probs}

=== Evaluate all indicators ===
results = []
roc_curves = {}
for ind in indicators:
    res = evaluate_indicator(df[ind].values, df["up_next"].values)
    # Bayesian posterior by quartile
    try:
        df["quartile"] = pd.qcut(df[ind], 4, labels=False, duplicates="drop")
        post = df.groupby("quartile")["up_next"].mean()
    except Exception:
        post = None
    prior = df["up_next"].mean()
    max_post = post.max() if post is not None else np.nan
    max_lift = max_post - prior if post is not None else np.nan
    results.append({
        "indicator": ind,
        "auc": res["auc"],
        "corr_with_up": res["corr"],
        "prior_p_up": prior,
        "max_quartile_post_p_up": max_post,
        "max_lift_over_prior": max_lift
    })
    roc_curves[ind] = (res["fpr"], res["tpr"])

res_df = pd.DataFrame(results).sort_values(by="auc", ascending=False)

=== Print ranking table ===
print("\nIndicator ranking (by AUC):")
print(res_df.to_string(index=False, float_format=lambda x: f"{x:.4f}"))

=== ========== PLOTTING ========== ===
=== Plot 1: Price series ===
plt.figure(figsize=(12,4))
plt.plot(df["date"], df["price"])
plt.xlabel("Date")
plt.ylabel("Price")
plt.title("Price series")
plt.xticks(rotation=45)
plt.tight_layout()
plt.savefig(os.path.join(OUTDIR, "price_series.png"))
plt.show()

=== Plot 2: Example tetrahedron for the final point in df ===
i = -1
p = df["price"].iloc[i]
d = df["d"].iloc[i]
h_i = df["h"].iloc[i]
O = np.array([0,0,0])
A_i = np.array([d,0,0])
B_i = np.array([0,p,0])
D_i = np.array([d,p,h_i])

fig = plt.figure(figsize=(8,6))
ax = fig.add_subplot(111, projection='3d')
=== Edges ===
ax.plot([O[0],A_i[0]],[O[1],A_i[1]],[O[2],A_i[2]])
ax.plot([O[0],B_i[0]],[O[1],B_i[1]],[O[2],B_i[2]])
ax.plot([O[0],D_i[0]],[O[1],D_i[1]],[O[2],D_i[2]])
ax.plot([A_i[0],B_i[0]],[A_i[1],B_i[1]],[A_i[2],B_i[2]])
ax.plot([A_i[0],D_i[0]],[A_i[1],D_i[1]],[A_i[2],D_i[2]])
ax.plot([B_i[0],D_i[0]],[B_i[1],D_i[1]],[B_i[2],D_i[2]])
ax.scatter([O[0],A_i[0],B_i[0],D_i[0]],[O[1],A_i[1],B_i[1],D_i[1]],[O[2],A_i[2],B_i[2],D_i[2]])
ax.set_xlabel("Day-of-year")
ax.set_ylabel("Price")
ax.set_zlabel("Height h")
ax.set_title("Tetrahedron for final point")
plt.tight_layout()
plt.savefig(os.path.join(OUTDIR, "example_tetrahedron.png"))
plt.show()

=== Plot 3: Tetrahedral volume over time ===
plt.figure(figsize=(12,4))
plt.plot(df["date"], df["tetra_vol"])
plt.xlabel("Date")
plt.ylabel("Tetrahedron volume")
plt.title("Tetrahedral Volume Over Time")
plt.xticks(rotation=45)
plt.tight_layout()
plt.savefig(os.path.join(OUTDIR, "tetra_vol_time.png"))
plt.show()

=== Plot 4: Height h over time ===
plt.figure(figsize=(12,4))
plt.plot(df["date"], df["h"])
plt.xlabel("Date")
plt.ylabel("h (half orthogonal distance)")
plt.title("Tetrahedral Height (h) Over Time")
plt.xticks(rotation=45)
plt.tight_layout()
plt.savefig(os.path.join(OUTDIR, "h_time.png"))
plt.show()

=== Plot 5: Posterior P(up next) by quartile for the top two indicators (by AUC) ===
top_two = res_df["indicator"].tolist()[:2]
for ind in top_two:
    try:
        df["quartile"] = pd.qcut(df[ind], 4, labels=False, duplicates="drop")
        post = df.groupby("quartile")["up_next"].mean()
        plt.figure(figsize=(6,4))
        plt.plot(post.index.astype(float)+1, post.values, marker='o')
        plt.xlabel("Quartile (1=lowest -> 4=highest)")
        plt.ylabel("P(next-day up)")
        plt.title(f"P(next-day up) by {ind} quartile")
        plt.tight_layout()
        plt.savefig(os.path.join(OUTDIR, f"posterior_by_quartile_{ind}.png"))
        plt.show()
    except Exception as e:
        print("Could not compute quartiles for", ind, ":", e)

=== Plot 6: ROC curves for all indicators ===
plt.figure(figsize=(8,6))
for ind in indicators:
    fpr_tpr = roc_curves.get(ind, (None,None))
    fpr, tpr = fpr_tpr
    if fpr is None or tpr is None:
        continue
    plt.plot(fpr, tpr, label=f"{ind} (AUC={res_df.loc[res_df.indicator==ind,'auc'].values[0]:.3f})")
plt.plot([0,1],[0,1], linestyle='--')
plt.xlabel("False positive rate")
plt.ylabel("True positive rate")
plt.title("ROC curves (single-feature logistic models)")
plt.legend()
plt.tight_layout()
plt.savefig(os.path.join(OUTDIR, "roc_curves.png"))
plt.show()

print("\nSaved plots to:", OUTDIR)

</syntaxhighlight>

===== 1. Geometric indicators - tetra_vol: the volume of the 3D tetrahedron formed by O, A=(day,0), B=(0,price) and apex D=(day, price, h) where h = (perpendicular distance from O to hypotenuse)/2. - h: half the orthogonal distance from O to the triangle’s hypotenuse (as you specified). - hyp_len: the 2D hypotenuse length between A and B. - tri_area: area of the triangle O-A-B. - vol10: rolling volatility (baseline non-geometric comparator). =====
# Predictive evaluation - For each indicator, the script fits a single-feature logistic regression that predicts up_next (1 if next day price is higher). - It reports correlation with the up_next binary, AUC (how well the indicator ranks the days by future up probability), and quartile-based Bayesian posterior probabilities (P(up_next | indicator quartile)). - The table printed at the top ranks indicators by AUC. Indicators with higher AUC and positive correlation are better single-feature predictors of next-day increases.
# Bayesian / posterior lift - The posterior_by_quartile plots show empirical posterior probabilities for each quartile of the indicator value; compare these to the prior probability (overall fraction of days that go up). A strong indicator will show monotonic posteriors and some quartiles with significantly higher posterior than the prior.
# Visual outputs - price_series.png: the price series itself. - example_tetrahedron.png: 3D view of one constructed tetrahedron (final day). - tetra_vol_time.png and h_time.png: indicator time series. - posterior_by_quartile_*.png: posterior vs quartile for top indicators. - roc_curves.png: ROC for each indicator.

===== - Primary ranking: AUC from single-feature logistic. (AUC near 0.5 is random; >0.6 modest; >0.7 decent.) =====
* Secondary checks: - Positive correlation with next-day up (more robust if stable across time). - Max lift over prior in quartiles (how much better the best quartile is vs prior). - Consistency across rolling windows (you can extend the script to perform walk-forward validation and re-evaluate AUC in each fold — recommended for real data).

===== I tried to run the plots for you but the environment couldn’t execute advanced data analysis in this session. If you want, I can re-run these exact steps and produce plots inside this conversation when the environment becomes available — tell me “run the analysis now” and I’ll attempt it. (If you prefer, run the script locally — it will run in any standard Python environment.) =====

===== - If you have real price data, replace the synthetic prices array with your series (same index length as dates or adjust dates accordingly). =====
* Add more features to test (e.g., moving-average gaps, momentum, volume if available).
* Use walk-forward retraining to avoid look-ahead and better estimate real-world performance.
* If tetrahedral volume is predictive on your data, consider combining it in a small ensemble with volatility and momentum for improved stability.

Would you like:
* A tighter script that performs walk-forward cross-validation and reports time-varying AUCs?
* Or do you want me to re-run this analysis now (I’ll try again to execute and produce the actual plots for you)?