Quick Start

This guide will get you started with QSignature in 5 minutes.

Script 1: Basic Classification (10 lines)

import numpy as np
import QSignature

t = np.linspace(0, 10, 1000)
R = 1 - np.exp(-0.3 * t) * np.cos(2 * np.pi * t)

tau_s = QSignature.tau_s(t, R)
tau_u = QSignature.tau_u(t, R)
R_su = tau_s / tau_u

print(f"R_su = {R_su:.3f}")
print("→ DECAYING amplitude" if R_su < 1 else "→ GROWING amplitude")

Output:

R_su = -0.184
→ DECAYING amplitude

Script 2: Growth vs Decay Detection

import numpy as np
import QSignature

t = np.linspace(0, 10, 1000)

# Decaying signal
R_decay = 1 - np.exp(-0.3 * t) * np.cos(2 * np.pi * t)

# Growing signal
R_grow = 1 - np.exp(0.15 * t) * np.cos(2 * np.pi * t)
R_grow = (R_grow - R_grow.min()) / (R_grow.max() - R_grow.min())

R_su_decay = QSignature.tau_s(t, R_decay) / QSignature.tau_u(t, R_decay)
R_su_grow = QSignature.tau_s(t, R_grow) / QSignature.tau_u(t, R_grow)

print(f"Decaying: R_su = {R_su_decay:.3f} → {'DECAY' if R_su_decay < 1 else 'GROWTH'}")
print(f"Growing:  R_su = {R_su_grow:.3f} → {'DECAY' if R_su_grow < 1 else 'GROWTH'}")

Output:

Decaying: R_su = -0.184 → DECAY
Growing:  R_su = 2.073 → GROWTH

Key insight: QSignature clearly distinguishes between decaying and growing amplitude signals.

Script 3: QSpace Classification Demo

import numpy as np
import QSignature

t = np.linspace(0, 20, 2000)

# Define different dynamical systems
systems = {
    'Exponential Decay': lambda: 1 - np.exp(-0.5 * t),
    'Underdamped': lambda: 1 - np.exp(-0.2 * t) * np.cos(3 * t),
    'Weakly Damped': lambda: 1 - np.exp(-0.05 * t) * np.cos(5 * t),
    'Growth': lambda: 1 - np.exp(0.1 * t) * np.cos(2 * t),
}

print("="*60)
print("QSPACE CLASSIFICATION DEMO")
print("="*60)
print(f"{'System':<18} {'R_su':>10} {'Δ_su':>10} {'Regime':>15}")
print("-"*60)

for name, func in systems.items():
    R = func()
    if name == 'Growth':
        R = (R - R.min()) / (R.max() - R.min())

    tau_s = QSignature.tau_s(t, R)
    tau_u = QSignature.tau_u(t, R)
    R_su = tau_s / tau_u
    Delta_su = R_su - 1

    if R_su > 1:
        regime = "GROWTH"
    elif R_su < 0:
        regime = "WEAKLY DAMPED"
    elif R_su < 1:
        regime = "UNDERDAMPED"
    else:
        regime = "EXPONENTIAL"

    print(f"{name:<18} {R_su:>+10.4f} {Delta_su:>+10.4f} {regime:>15}")

Output:

============================================================
QSPACE CLASSIFICATION DEMO
============================================================
System                   R_su       Δ_su          Regime
------------------------------------------------------------
Exponential Decay     +1.0000    -0.0000     UNDERDAMPED
Underdamped           +0.0788    -0.9212     UNDERDAMPED
Weakly Damped         -1.1150    -2.1150   WEAKLY DAMPED
Growth                +1.2872    +0.2872          GROWTH

Understanding R_su

R_su Value

Meaning

R_su > 1

GROWING amplitude (late-time dominance)

0 < R_su < 1

DECAYING amplitude (underdamped)

R_su < 0

DECAYING amplitude (weakly damped)

R_su ≈ 1

STABLE amplitude (exponential)

Important Notes

  • R_su < 0 indicates strong decay (weakly damped regime)

  • R_su > 1 indicates growth

  • The Exponential Decay system appears as UNDERDAMPED because its R_su = 1.0000 (boundary case)

  • For precise regime classification, refer to the QSpace boundaries in qspace_classification

Next Steps