Fuzzy Additive Symptom Likelihood (FASL) Explained

• Fuzzy Additive Symptom Likelihood (FASL) is a transparent method that maps home router telemetry into DSM-5 depressive symptom likelihoods using bounded fuzzy membership functions and additive aggregation.
• The approach rigorously normalizes and aggregates network features, incorporating a DSM-style temporal gating mechanism to ensure clinically robust and interpretable inferences.
• FASL supports privacy-preserving behavioral sensing by processing anonymized traffic metrics without intrusive payload inspection, enabling scalable mental health monitoring.

The Fuzzy Additive Symptom Likelihood (FASL) is a transparent, mathematically-defined inference formulation devised to map household router network telemetry into interpretable daily likelihoods for DSM-5 depressive symptom criteria. It operates through bounded fuzzy membership functions over domain-grounded network metrics, an additive (signed) aggregation step, and a DSM-style temporal gate that integrates short-term traffic patterns into clinically resonant evidence. FASL is the inferential core of the CareNet system, enabling explainable behavioral sensing at the home gateway without payload inspection or intrusive device-based monitoring (Nef et al., 16 Nov 2025).

1. Feature Selection and Fuzzy Membership Functions

FASL begins by extracting day-level network features $x_1(t), x_2(t), ..., x_n(t)$ selected for relevance to depressive symptomatology. Example features include:

• $x_1$: digital sleep-onset (minutes past 22:00)
• $x_2$: $|$sleep-duration $z$-score$|$ over the previous 30 days
• $x_3$: daytime idle ratio (08:00–18:00)
• $x_4$: night/day traffic-byte ratio

Each feature is normalized if required and mapped to a bounded fuzzy evidence score via a triangular membership function:

$$\mu_{\mathrm{tri}}(x;\ell,m,h) = \begin{cases} 0, & x \le \ell, \ \dfrac{x - \ell}{m - \ell}, & \ell < x \le m, \ \dfrac{h - x}{h - m}, & m < x < h, \ 0, & x \ge h~. \end{cases}$$

Here, $\ell$ is the threshold for no evidence, $m$ is maximal evidence, and $h$ is the upper support limit. The function is always contained in $[0,1]$ by construction. Orientation is controlled by the argument order: "positive" evidence features use the metric as given, while "negative" orientation (lower $x$ is more evidence) is handled by inverting $x_i \rightarrow \max - x_i$.

2. Additive Aggregation into Signed Behavioral Components

Feature-level fuzzy scores are grouped by DSM criterion-relevant behavioral components ($b$), each of which may represent a cluster of features associated with a specific domain (e.g., sleep timing). For each component within criterion $k$:

$$S_{k,b}(t) = \mathrm{clip}_{[-1,1]} \left( \sum_{i \in \mathcal{I}_b} w_{b,i}\, s_{k,i} \,\mu_{k,i}(x_i(t)) \right)\,,\quad \sum_{i\in\mathcal I_b} w_{b,i} = 1$$

Where:

• $w_{b,i} \geq 0$ are normalized feature weights (reflecting domain importance),
• $s_{k,i} \in \{ +1, -1\}$ encodes the direction of evidence,
• $\mathrm{clip}_{[-1,1]}(\cdot)$ bounds the component evidence sum,
• $S_{k,b}(t) \in [-1, 1]$ represents signed evidence for component $b$ at time $t$.

3. Daily Criterion Likelihood Computation

The signed component scores are shifted from $[-1, 1]$ to $[0, 1]$ via an affine transform:

$\hat{S}_{k,b}(t) = \frac{1}{2}(S_{k,b}(t) + 1)$

These component likelihoods are linearly aggregated per criterion:

$$L_k(t) = \mathrm{clip}_{[0,1]} \left( \sum_{b \in \mathcal{B}_k} v_{k,b} \,\hat{S}_{k,b}(t) \right)\,,\quad \sum_{b\in\mathcal B_k} v_{k,b}=1$$

where $v_{k,b}$ are component weights subject to non-negativity and normalization. The resulting $L_k(t) \in [0,1]$ gives the per-day likelihood that DSM criterion $k$ is met.

4. DSM-style Temporal Gating and Episode Flagging

FASL enforces persistence and clinical timescale requirements via a sliding window temporal gate. For criterion $k$, define:

$I_k(d) = \mathbb{I}[L_k(d) \ge \theta]$

For a chosen threshold $\theta$ (e.g., $\theta = 0.6$), day $d$ is positive if above threshold. Criterion $k$ is considered "present" on day $t$ if positivity occurs on at least $N$ of the past $M$ days:

$$\mathrm{present}_k(t) = \mathbb{I}\left[ \sum_{d = t-M+1}^{t} I_k(d) \ge N \right]$$

Default values are $M=14$, $N=6$, reflecting DSM-5's temporal requirements.

At the episode level, the inference flags a major depressive episode according to:

$$\mathrm{epi}(t) = \mathbb{I}\left[\sum_{k=1}^9 \mathrm{present}_k(t) \ge 5 \wedge (\mathrm{present}_1(t) = 1 \vee \mathrm{present}_2(t) = 1)\right]$$

This implements the DSM-5 algorithmic requirement for at least five symptoms (with at least one core) for episode assignment.

5. Bounding, Normalization, and Interpretability Controls

The FASL pipeline enforces strict boundedness and normalization throughout:

• Each $\mu_i$ is in $[0,1]$ by definition of the triangular function.
• Feature and component weights ($w_{b,i}$, $v_{k,b}$) are non-negative and sum to one.
• The $\mathrm{clip}_{[-1,1]}$ and $\mathrm{clip}_{[0,1]}$ operators guarantee aggregation results are within interval constraints.
• The affine shift $(S + 1)/2$ ensures that additive cancellation cannot yield negative (inverse) evidence for a criterion.

This construction yields outputs that are transparent, monotonic in the domain evidence, and directly traceable to interpretable feature contributions. The explicit inclusion of persistence and thresholding mirrors clinical diagnostic practice.

6. Concrete Numerical Example: Shifted Sleep Timing Domain

Consider DSM-5 Criterion 4, exemplified as "Shifted sleep timing/duration." Using parameterization from Table V of (Nef et al., 16 Nov 2025):

Feature	$w_i$	$\ell_i$	$m_i$	$h_i$	Dir
F2 WakeAfter0400Min	0.65	120	1085	1085	↑
F4 SleepDurationZAbs30d	0.20	0.25	0.80	0.80	↑
F7 DaytimeIdleRatio0818	0.05	0.00	0.08	0.16	↑
F8 NightDayTrafficRatio	0.15	0.20	1.00	1.00	↑

Suppose the following daily observations:

• $x_1 = 300$
• $x_2 = 0.50$
• $x_3 = 0.10$
• $x_4 = 0.50$

Computation proceeds:

1. Memberships:
   • $\mu_1 = (300 - 120)/(1085 - 120) \approx 0.1866$
   • $\mu_2 = (0.50 - 0.25)/(0.80 - 0.25) \approx 0.4545$
   • $\mu_3 = (0.16 - 0.10)/(0.16 - 0.08) = 0.75$ (since $x_3 > m_3$)
   • $\mu_4 = (0.50 - 0.20)/(1.00 - 0.20) = 0.375$
2. Weighted sum (all $s_i = +1$):

$$S_{C4}(t) = 0.65 \cdot 0.1866 + 0.20 \cdot 0.4545 + 0.05 \cdot 0.75 + 0.15 \cdot 0.375 \approx 0.306$$

1. Shifted to $[0,1]$:

$\hat{S}_{C4}(t) = (0.306 + 1)/2 = 0.653$

2. One-component aggregation:

$$L_4(t) = \mathrm{clip}_{[0,1]}(1 \cdot 0.653) = 0.653$$

1. Thresholding ($\theta = 0.6$): $I_4(t) = 1$
2. If at least 6 of the last 14 days are positive, $\mathrm{present}_4(t) = 1$

This output is used in downstream episode detection as described above (Nef et al., 16 Nov 2025).

7. Significance, Clinical Alignment, and Broader Impact

FASL bridges behavioral signal processing and clinical psychiatry by structuring observable network metadata into criterion-aligned, interpretable, and temporally robust inferences. Its workflow enables local, privacy-preserving behavioral sensing with strict alignment to DSM-5 persistence and threshold requirements. Evaluation demonstrates FASL's ability to capture characteristic depressive behavioral phenotypes such as delayed sleep timing and attentional instability from router-side telemetry, without payload inspection or mobile-device dependence. This establishes a precedent for explainable, reproducible mental-health inference in non-intrusive home environments (Nef et al., 16 Nov 2025).