Concentration and Moment Inequalities for Vehicle Sensor Data

Vehicle sensor data is hard to deal with¶

I have been considering how we can apply concentration and moment inequalities to ‘models’ built on vehicle sensor data to determine whether the ‘part’ modeled by the suite of sensors is healthy or not. To do this, we of course use a …

Martingale Construction for Multi-State Battery Health Monitoring with Recovery and Multiple Failure Modes

Multi-State Battery Health Model¶

State Space Definition¶

Define the battery state space $\mathcal{S} = {H, D_1, D_2, \ldots, D_k, F_1, F_2, \ldots, F_m, R}$ where: - $H$: Healthy/Normal operating state - $D_i$: Degradation state $i$ (recoverable) - $F_j$: Failure state $j$ (non-recoverable) - $R$: Recovery/Rejuvenation state (during maintenance)

The transition intensity matrix …

Sequential Testing for Battery Replacement Detection via Remote Telemetry: An E-Process Framework

Problem Formulation: Battery Replacement as a Change-Point Problem¶

Signal Model¶

Consider a device transmitting battery telemetry signals $X_t = (V_t, I_t, T_t, Z_t)$ where: - $V_t$: Voltage measurement at time $t$ - $I_t$: Current draw at time $t$ - $T_t$: Temperature at time $t$ - $Z_t$: Internal impedance at time $t$

The battery replacement problem …

Composition Rules for E-value Based Sequential Testing

1. Product Rule (Intersection of Nulls)¶

For testing the intersection of null hypotheses $H_0 = H_0^{(1)} \cap H_0^{(2)} \cap \cdots \cap H_0^{(k)}$:

Rule: If $E_1, E_2, \ldots, E_k$ are e-values for $H_0^{(1)}, H_0^{(2)}, \ldots, H_0^{(k)}$ respectively, then:

$$E = \prod_{i=1}^k E_i$$

is a …

Martingale Processes & Moment Generating Functions: Mathematical Summary

Core Martingale Framework¶

Definition¶

A stochastic process ${M_t}$ is a martingale with respect to filtration ${\mathcal{F}_t}$ if:

1. Adapted: $M_t$ is $\mathcal{F}_t$-measurable for all $t$
2. Integrable: $\mathbb{E}[|M_t|] < \infty$ for all $t$
3. Martingale Property: $\mathbb{E}[M_t | \mathcal{F}{t-1}] = M$

Wealth Process (Test Martingale …

E-values, Sequential Testing, and Concentrated Differential Privacy: Theory and Practice

Introduction to E-values and E-processes¶

An E-value (evidence value) is a non-negative random variable $E$ with expected value at most 1 under the null hypothesis:

$$ \mathbb{E}_{H_0}[E] \leq 1 $$

E-values generalize likelihood ratios and provide a universal framework for hypothesis testing without p-values. The fundamental property enabling …

My First Post - Test

Mathematical Equations¶

This is inline math: $E = mc^2$.

This is display math: $$ \int_{a}^{b} x^2 \, dx $$ This is part of the doc added later

Diagrams¶

graph TD; A-->B; A-->C; B-->D; C-->D; D-->E; E-->F; D-->F;