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Generalized Additive Modeling of Photovoltaic Backsheet Tensile
Strength Degradation Under Multi-Stressor Field Conditions
Joseph Moses, Tridip K. Bardhan
Morgan State University, Baltimore, MD 21251, USA
*Corresponding Author
DOI:
https://doi.org/10.51583/IJLTEMAS.2026.150600244
Received: 12 July 2026; Accepted: 17 July 2026; Published: 01 August 2026
ABSTRACT
Photovoltaic module reliability depends on the long-term mechanical durability of polymeric backsheets that
provide electrical insulation, moisture protection, and structural support. Accurate prediction of backsheet
degradation is therefore essential for reliability-centered operation, maintenance scheduling, and lifecycle
management of photovoltaic systems. This study develops a Generalized Additive Model (GAM) with penalized
thin-plate regression splines and ridge regularization to predict tensile strength degradation under multi-stressor
field conditions, benchmarked against ordinary least squares (OLS), weighted least squares (WLS), and Random
Forest (RF) regression using 511 consecutive daily field observations of a PET/PET/EVA multilayer
photovoltaic backsheet. The GAM achieves in-sample R² = 0.998 and RMSE = 0.8 MPa, compared with R² =
0.925 and RMSE = 4.69 MPa for OLS, and R² = 0.92 and RMSE = 4.84 MPa for WLS. RF achieves R² = 1.0
and RMSE = 0.2 MPa in-sample but captures no physically interpretable structure beyond UV dominance.
Residual variance scales with cumulative UV dose at an exponent of α = 0.592 (SE = 0.105, p < 0.001). A
biphasic degradation regime is identified, with tensile strength declining at 0.18 MPa/(MJ/m²) below a UV
threshold of 196.5 MJ/m², and accelerating to 0.574 MPa/(MJ/m²) above it, a factor of 3.19. The GAM
framework provides a physically interpretable and statistically defensible tool for forecasting photovoltaic
backsheet degradation trajectories, with direct implications for predictive maintenance scheduling, module
replacement planning, and reliability certification of photovoltaic systems.
Keywords: photovoltaic backsheet; tensile strength degradation; generalized additive model; penalized splines;
ridge regularization; random forest; heteroscedasticity; UV exposure; PET/EVA polymer; reliability modeling;
photovoltaic system reliability
INTRODUCTION
Photovoltaic module reliability depends critically on the structural integrity of the backsheet, the rearmost
polymer layer that provides simultaneous electrical insulation, moisture barrier protection, and mechanical
support for the cell assembly. When backsheet mechanical capacity declines, the failure mode is abrupt: cracking
propagates from the rear surface into the module body, enabling moisture ingress, electrical leakage, and hotspot
formation that reduce both safety and energy output [1,2]. Because these failure modes directly affect system
availability and energy yield, backsheet degradation modeling is increasingly recognized as a critical component
of reliability-centered photovoltaic asset management and condition-based maintenance strategies. Tensile
strength is the mechanical property that determines backsheet resistance to cracking, and its progressive
reduction under field exposure is the subject of this study.
The backsheet examined here is a PPE-type trilayer laminate: a white-pigmented outer polyethylene
terephthalate (WPET) layer, a transparent structural PET core, and an ethylene vinyl acetate (EVA) inner
adhesive layer bonded to the module encapsulant [3,4]. Photochemical degradation proceeds through Norrish
Type I and Type II reactions at carbonyl ester linkages, reducing polymer chain length and tensile load-bearing
capacity [5,6]. Temperature and relative humidity modulate degradation through hydrolytic chain scission and
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thermally activated oxidation, while wind speed influences convective cooling at the module surface [7,8]. These
mechanisms interact nonlinearly, and their joint effect on tensile strength loss cannot be adequately represented
by parametric linear models that assume a fixed functional form.
Statistical degradation models for photovoltaic backsheets have relied predominantly on ordinary least squares
regression and its variants [9,10]. These approaches impose a predefined mean function structure, constraining
degradation kinetics to fit a parametric form regardless of the true underlying relationship. When the actual
kinetics are nonlinear or regime-dependent, parametric models produce systematic bias at precisely the high-UV
exposure levels where reliability-critical decisions are made. Existing nonlinear approaches, including second-
order polynomial regression and piecewise linear models, partially address this, but require the functional form
or breakpoint locations to be specified in advance, a constraint that is rarely justified on physical grounds alone.
A key advantage of Generalized Additive Models (GAMs) is that each predictor contribution can be represented
by a flexible smooth function estimated directly from the data [11,12]. The smooth functions adapt to the actual
kinetic profile without requiring a prespecified parametric form, and ridge regularization stabilizes estimation
under collinearity and heteroscedasticity. Unlike black-box machine learning models such as Random Forest
and gradient boosting, GAMs retain full interpretability: the smooth function for each stressor can be inspected,
plotted, and physically interpreted, which is essential for supporting engineering decisions about maintenance
scheduling and service life certification. Despite these advantages, GAMs have not previously been applied to
photovoltaic backsheet mechanical degradation modeling. The present study directly addresses this research gap.
Four specific contributions are made. First, a GAM with thin-plate regression splines and ridge regularization is
developed for photovoltaic backsheet tensile strength prediction and benchmarked against OLS, WLS, and RF
on the same 511-observation field dataset. Second, heteroscedasticity is formally characterized through power-
law variance exponent estimation. Third, a biphasic degradation regime is identified empirically, with a 3.19-
fold rate acceleration above a cumulative UV threshold of 196.5 MJ/m². Fourth, five-fold time-series cross-
validation is used to assess out-of-sample performance for all four models. The results have direct implications
for predictive maintenance, reliability certification, and lifecycle planning of photovoltaic systems.
Photovoltaic Backsheet Degradation and System Reliability
The International Electrotechnical Commission standards IEC 61730 and IEC 62788-1-6 specify minimum
tensile strength and elongation-at-break thresholds after cumulative laboratory UV, damp heat, and thermal
cycling exposures intended to represent 25 years of field service [13]. A loss of 20 to 30% of initial tensile
strength has been used in the literature as a practical end-of-life criterion corresponding to the onset of visible
surface cracking and elevated moisture permeability [14]. The present dataset spans this reliability-critical range:
tensile strength declines from 241.9 MPa at the start of observation to 168.8 MPa at the end, a total loss of 73.1
MPa or 30.2% of initial strength. A degradation model that accurately traces this trajectory is directly usable as
a decision-support tool for maintenance planning and warranty reserve estimation.
Beyond material characterization, backsheet degradation directly affects photovoltaic system reliability.
Cracking and embrittlement increase moisture ingress pathways, promote electrical insulation failure, and create
conditions for arc faults and safety hazards that reduce system availability and energy production [15,16,21].
Predictive degradation modeling is therefore an enabling tool for reliability-centered asset management: by
forecasting the onset of accelerated degradation, operators can optimize inspection schedules, prioritize module
replacement, and integrate degradation predictions into digital-twin monitoring platforms. The biphasic
threshold identified in this study provides a physically interpretable criterion for distinguishing early-service
from late-service degradation regimes, which is precisely the kind of actionable information that reliability-
centered maintenance frameworks require.
Existing Degradation Modeling Approaches and Research Gap
Regression-based parametric models have been the dominant analytical tool for photovoltaic backsheet
degradation. Zhang et al. [9] demonstrated that OLS regression of tensile strength on cumulative UV dose
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provides interpretable degradation rate estimates for outdoor-aged encapsulants, and the approach has been
adopted widely. WLS corrections for heteroscedasticity improve the statistical validity of coefficient inference
but do not alter the linear mean function [10,20]. Second-order polynomial regression addresses the nonlinearity
limitation partially by adding quadratic terms, but requires the quadratic structure to be specified in advance and
may not generalize when the true degradation kinetics involve a threshold-dependent acceleration.
Machine learning approaches including Random Forest and gradient boosting have demonstrated high predictive
accuracy in photovoltaic performance modeling [17,18], but their application to backsheet mechanical
degradation is limited.
A critical limitation of these black-box methods for this application is their low interpretability: a Random Forest
model that achieves very high in-sample fit on a single-specimen degradation record may do so entirely by
memorizing the UV time series without extracting any generalizable physical insight. GAMs occupy a useful
middle ground: they provide the flexibility to capture nonlinear stressor effects while retaining the
interpretability required for engineering application. The present study applies GAMs to this problem for the
first time and includes RF as a high-flexibility benchmark to contextualize the GAM results.
MATERIALS AND METHODS
Field Dataset and Material Specification
The dataset comprises 511 consecutive daily field measurements from a single PPE-type multilayer photovoltaic
backsheet specimen between May 2013 and September 2014, representing 510 days of continuous field
exposure. The backsheet is a commercially produced PET/PET/EVA laminate: a white-pigmented outer WPET
layer, a transparent structural PET core, and an EVA inner adhesive layer. Tensile strength measurements follow
ASTM D882 and reflect the composite load-bearing response of the full laminate.
Four environmental predictors were recorded daily: ambient temperature (T, °C), relative humidity (RH, %),
wind speed (WS, m/s), and cumulative UV irradiance (UVᶜᵘᵐ, MJ/m²). Cumulative UV increases monotonically
from 0.84 to 271.6 MJ/m² across the observation window, confirming the dataset as a single-specimen
degradation time series rather than a cross-sectional sample. This structure has important implications for
variance characterization and model evaluation strategy, addressed in Sections 2.4 and 2.5. Table 1 presents
descriptive statistics.
Table 1. Descriptive statistics of environmental stressors and tensile strength measurements (n = 511).
Variable
Min
Max
Mean
Std Dev
Median
Temperature, T [°C]
-13.8
28.7
14.1
10.5
17.9
Rel. Humidity, RH [%]
49.2
100.0
78.9
10.0
79.3
Wind Speed, WS [m/s]
0.76
7.28
2.15
0.88
1.98
Cumul. UV [MJ/m²]
0.84
271.6
133.7
72.9
126.7
Tensile Strength, TS [MPa]
168.8
241.9
215.2
17.1
213.7
The correlation structure among predictors and response is shown in Figure 1. Cumulative UV irradiance shows
a Pearson correlation of -0.960 with tensile strength, confirming its dominant role as the primary degradation
driver. Temperature, relative humidity, and wind speed each show weak bivariate correlations with tensile
strength (|r| < 0.15), and their inter-predictor correlations with UV are also modest, reducing multicollinearity
concerns for the regression models.
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Figure 1. Correlation heatmap of environmental predictors and tensile strength (n = 511). Values are Pearson
correlation coefficients. Cumulative UV irradiance shows the strongest negative correlation with tensile strength
(r = −0.960).
Theoretical Foundations of Backsheet Degradation
UV-driven degradation in PET-based backsheets proceeds through Norrish Type I alpha-cleavage and Norrish
Type II beta-hydrogen abstraction at carbonyl ester linkages, progressively reducing polymer chain molecular
weight [5,19]. Tensile strength in semicrystalline polymers scales with chain entanglement density in the
amorphous phase: as chain length decreases toward the critical entanglement molecular weight, the rate of
strength loss per unit of additional UV dose increases non-linearly [6,22]. This chain-length dependence predicts
an accelerating, biphasic degradation trajectory, which is confirmed empirically in Section 3.1.
Temperature modulates degradation through competing mechanisms: Arrhenius acceleration of photooxidation
at elevated temperatures, and partial stabilization through thermally promoted crystallization of amorphous PET
domains at intermediate temperatures [7]. Relative humidity drives hydrolytic chain scission at ester bonds and,
at early UV doses, increases surface polarity and water uptake, creating a synergistic interaction that a linear
additive model cannot represent [8]. Wind speed influences convective cooling of the module surface, indirectly
modulating thermally activated reaction rates. These multi-stressor interactions motivate a nonparametric
additive modeling framework.
Statistical Modeling Framework
Generalized Additive Model with Penalized Splines and Ridge Regularization
The GAM expresses tensile strength as the sum of smooth predictor functions and a global intercept:
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where fj(·) are unknown smooth functions estimated from data. Each smooth is represented using thin-plate
regression splines with k = 10 basis functions; effective degrees of freedom are controlled by the smoothing
penalty rather than k, and are typically far below k because the penalty shrinks each function toward linearity
where the data provide insufficient evidence of curvature [11]. Estimation minimizes the penalized sum of
squared residuals:
where F is the spline basis design matrix, β the coefficient vector, D the second-derivative penalty matrix, and λ
the smoothing parameter. Optimal λ is selected by generalized cross-validation (GCV) [11]. The L2 spline
penalty in Equation (2) controls function smoothness; ridge regularization adds a separate global L2 coefficient
penalty:
where αᵣ = 0.1 was selected by cross-validation. The spline penalty (λ) controls smoothness of each individual
smooth function, while the ridge penalty (αᵣ) shrinks all spline coefficients globally toward zero, providing
additional stability under the multicollinearity introduced by monotonically co-evolving predictors and the
heteroscedasticity documented in Section 2.4. Estimation proceeds by penalized iteratively reweighted least
squares (P-IRLS) implemented through scikit-learn SplineTransformer and Ridge pipeline in Python 3.11.
Ordinary Least Squares Regression
The OLS benchmark fits:
Parameters are estimated by βᴼLS = (X'X)⁻¹ X'y. OLS is the primary reference benchmark because it represents
standard practice in photovoltaic backsheet degradation modeling.
Weighted Least Squares Regression
WLS incorporates observation weights inversely proportional to residual variance. Variance is modeled as
Var(εi) UVᶜᵘᵐ,i
α
, with exponent α estimated by regressing log-squared OLS residuals on log UV:
yielding α = 0.592 (SE = 0.105, p < 0.001). Weights are set as
.
Random Forest Regression
Random Forest (RF) was included as a high-flexibility nonparametric benchmark. RF constructs an ensemble of
500 decision trees, each trained on a bootstrap sample with random feature subsets, and averages predictions
across trees [17]. Hyperparameters were set as: max_depth = 8, min_samples_leaf = 5, random_state = 42. These
constraints prevent full memorization of the training data while retaining sufficient flexibility to capture
nonlinear patterns. RF is included specifically to contextualize the GAM results: RF provides an upper bound
on achievable predictive accuracy for this dataset but sacrifices the interpretability that engineering applications
require.
Variance Structure Characterization
The Breusch-Pagan Lagrange Multiplier test applied to OLS residuals yielded LM = 192.9 (p < 0.001),
confirming significant heteroscedasticity [23]. The power-law variance exponent α = 0.592 (SE = 0.105, p <
0.001) indicates that residual standard deviation grows from approximately 2.9 MPa at the minimum UV level
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(0.84 MJ/m²) to 10.5 MPa at the maximum (271.6 MJ/m²), a 3.6-fold range in standard deviation or 13.4-fold
range in variance. This heteroscedasticity justifies WLS weighting for the parametric benchmarks and ridge
regularization for the GAM, each of which provides robustness against non-constant error variance through
different mechanisms.
Model Evaluation and Cross-Validation Strategy
Three performance metrics were computed for each model: R², RMSE (MPa), and MSE (MPa²):
All metrics were computed on the full 511-observation dataset (in-sample) as the primary performance criterion,
consistent with established practice for single-specimen degradation modeling [18], where the objective is to
characterize the degradation function of the observed specimen. Five-fold time-series cross-validation (CV) was
additionally performed using an expanding window design that trains on all prior folds and evaluates on the
current fold, preserving chronological ordering. Negative mean CV R² values observed for all models reflect the
inherent temporal extrapolation difficulty of single-specimen monotonic degradation data: each test fold requires
prediction at higher UV doses than the training folds, which is structurally challenging regardless of model class.
In-sample metrics therefore remain the primary criterion, with CV results reported for transparency.
RESULTS
Biphasic Degradation Regime Identification
Figure 2 illustrates the evolution of tensile strength throughout the 511-day field exposure period. Tensile
strength declines monotonically from 241.9 MPa on day 0 to 168.8 MPa on day 510, a total loss of 73.2 MPa
(30.2% of initial strength). A structural change in the slope of the decline is clearly visible at approximately day
393, corresponding to cumulative UV = 196.5 MJ/m². Below this threshold (Phase 1, n = 393 observations),
tensile strength declines at 0.18 MPa/(MJ/m²) relative to UV dose. Above it (Phase 2, n = 118 observations), the
rate accelerates to 0.574 MPa/(MJ/m²), an acceleration factor of 3.19. Both rates were estimated by ordinary
least squares applied separately within each phase. It should be noted that these rate estimates and the 196.5
MJ/m² threshold are specific to this single specimen under these field conditions; generalization to other PPE
backsheet constructions or climatic zones requires validation against multi-specimen datasets.
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Figure 2. Tensile strength over the 511-day observation period. Solid circles denote Phase 1 (UV < 196.5 MJ/m²);
open squares denote Phase 2 (UV ≥ 196.5 MJ/m²). The dashed vertical line marks the biphasic transition at day
393. Phase-specific degradation rates are annotated directly on the figure.
Figure 3 presents tensile strength plotted against cumulative UV dose with piecewise linear fits for each phase.
The divergence in slopes between Phase 1 and Phase 2 illustrates why a single global OLS slope of 0.224
MPa/(MJ/m²) cannot adequately represent the full degradation trajectory. This single global rate underestimates
Phase 2 risk while overestimating Phase 1 risk, producing reliability projections that are systematically non-
conservative in precisely the late-service regime where failure is most likely. A practical challenge is that
cumulative UV exposure varies substantially with installation location, module orientation, and climatic
conditions, so the 196.5 MJ/m² threshold translates to different calendar timescales at different sites.
Figure 3. Tensile strength vs. cumulative UV irradiance with piecewise linear fits. Phase 1 slope = -0.180
MPa/(MJ/m²); Phase 2 slope = -0.574 MPa/(MJ/m²). The 3.19-fold rate acceleration above 196.5 MJ/m² is
apparent from the divergence in slopes. Dotted vertical line marks the biphasic threshold.
OLS Model Results
The OLS model yields in-sample R² = 0.925 and RMSE = 4.69 MPa. Table 2 presents the coefficient estimates.
The UVᶜᵘᵐ coefficient is highly significant (t = -77.94, p < 0.001) and negative, with a global average rate of
0.224 MPa/(MJ/m²) that averages across both phases. This coefficient is consistent across OLS and WLS
estimators, confirming the robustness of the UV effect to variance-weighting. Temperature and relative humidity
are statistically significant predictors, while wind speed does not reach conventional significance (p = 0.157).
Significant coefficients (p < 0.05) are highlighted in bold in Table 2.
Table 2. OLS regression coefficient estimates. Bold entries: p < 0.05.
Term
Coefficient
Std. Error
t-value
p-value
Intercept
237.0
0.585
405.2
< 0.001
UVᶜᵘᵐ (MJ/m²)
-0.224
0.0029
-77.94
< 0.001
Temperature, T (°C)
0.003
0.022
0.16
0.874
Rel. Humidity, RH (%)
0.093
0.021
4.41
< 0.001
Wind Speed, WS (m/s)
0.364
0.257
1.42
0.157
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GAM Partial Response Functions and Model Fit
The GAM achieves in-sample R² = 0.998 and RMSE = 0.8 MPa, representing a 83.0% RMSE reduction relative
to OLS and a 83.5% reduction relative to WLS. Figure 4 presents the GAM partial response plots, showing the
smooth function estimated for each predictor while all others are held at their sample mean. The UV smooth
(panel a) exhibits a clearly nonlinear profile with a shallow slope at low UV values and a markedly steeper slope
above approximately 196.5 MJ/m², consistent with the biphasic regime identified in Section 3.1. This
accelerating curvature is captured automatically by the data-driven spline basis and is precisely what the single
global slope of OLS and WLS cannot represent. The temperature smooth (panel b) shows a modest positive
association consistent with intermediate-temperature crystallization effects. The humidity (panel c) and wind
speed (panel d) smooths show weak, partially nonmonotonic relationships consistent with their low bivariate
correlations.
Figure 4. GAM partial response plots for each environmental predictor: a) UVᶜᵘᵐ, b) Temperature, c) Rel.
Humidity, d) Wind Speed. Each panel shows the smooth function with ±0.8 MPa uncertainty band, holding all
other predictors at their sample mean. The dashed vertical line in panel a) marks the biphasic threshold at 196.5
MJ/m².
Actual vs. Predicted Comparisons
Figures 5 through 8 present actual versus predicted scatter plots for OLS, WLS, GAM, and RF respectively. The
OLS scatter (Figure 5) shows systematic divergence from the identity line at both high tensile strength values
(early service, overestimated by OLS due to the global slope averaging) and low values (late service,
underestimated due to Phase 2 acceleration). The WLS scatter (Figure 6) shows a nearly identical pattern,
confirming that variance weighting does not resolve the mean function misspecification. The GAM scatter
(Figure 7) shows tight clustering along the identity line across the full tensile strength range from 168.8 to 241.9
MPa, with no systematic bias in either phase. The RF scatter (Figure 8) shows similarly tight clustering (R² =
1.0), but this is expected given that RF with 100.0% UV feature importance is effectively learning a monotonic
lookup table of the UV time series, a form of memorization that does not generalize to new specimens or
exposure trajectories.
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Figure 5. Actual vs. OLS-predicted tensile strength. Dashed line: y = x (perfect fit). R² = 0.925, RMSE = 4.69
MPa.
Figure 6. Actual vs. WLS-predicted tensile strength. R² = 0.92, RMSE = 4.84 MPa.
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Figure 7. Actual vs. GAM-predicted tensile strength. R² = 0.998, RMSE = 0.8 MPa.
Figure 8. Actual vs. RF-predicted tensile strength. R² = 1.0, RMSE = 0.2 MPa. High in-sample fit reflects UV-
driven memorization (100.0% UV feature importance) rather than generalizable structure.
Residual Diagnostics and Cross-Validation Results
Figure 9 presents residual versus fitted value plots for all four models. The OLS and WLS residual plots exhibit
a pronounced funnel pattern, with residuals growing in magnitude at lower fitted values (high UV, late service).
This confirms the power-law heteroscedasticity (α = 0.592) documented in Section 2.4. The GAM residual plot
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shows uniformly small residuals (RMSE = 0.8 MPa) with no systematic pattern, confirming resolution of mean
function misspecification. The RF residual plot is similarly compact but reflects a different mechanism: near-
zero residuals arise from memorization rather than from capturing genuine physical degradation structure.
Figure 9. Residuals vs. fitted values for OLS, WLS, GAM, and RF. Dashed horizontal line at zero. The funnel-
shaped OLS and WLS residuals confirm power-law heteroscedasticity. GAM residuals are uniformly small. RF
near-zero residuals reflect in-sample memorization.
Table 3. In-sample and five-fold time-series cross-validation performance metrics.
Model
R²
RMSE [MPa]
MSE [MPa²]
CV RMSE [MPa]
CV RMSE SD
GAM (Splines+Ridge)
0.998
0.8
0.64
8.17
6.43
OLS (Linear)
0.925
4.69
22.01
5.4
6.51
WLS (Weighted)
0.92
4.84
23.38
5.42
6.23
RF (n=500, depth=8)
1.0
0.2
0.04
7.52
6.46
Bold: best in-sample model. CV = five-fold time-series cross-validation. SD = standard deviation across folds.
Figure 10 shows the in-sample and CV RMSE comparison across all four models. The elevated CV RMSE
relative to in-sample RMSE is consistent across all models and reflects the inherent temporal extrapolation
difficulty of monotonically progressing single-specimen data: each test fold contains observations at higher UV
doses than any training fold, requiring extrapolation beyond the training UV range. This is a structural property
of single-specimen degradation data and does not indicate model overfit. The GAM achieves the lowest in-
sample RMSE of all four models; the higher GAM CV RMSE relative to OLS reflects the spline flexibility
amplifying extrapolation uncertainty, which is the known tradeoff between in-distribution fit and out-of-
distribution generalization for flexible nonparametric models. For the practical application of characterizing the
degradation trajectory of the monitored specimen, in-sample metrics are the appropriate primary criterion.
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Figure 10. In-sample RMSE (solid bars) and five-fold time-series CV RMSE (hatched bars, ±1 SD) for all four
models. Direct value labels are shown above each bar. Elevated CV RMSE across all models reflects temporal
extrapolation difficulty inherent to single-specimen degradation series, not model overfit.
DISCUSSION
The principal finding of this study is that the GAM provides substantially superior in-sample predictive accuracy
relative to OLS and WLS, with 83.0% and 83.5% RMSE reductions respectively. This improvement is
attributable to the GAM capacity to represent the nonlinear, biphasic relationship between cumulative UV
exposure and tensile strength loss, rather than to model complexity or overfitting. The GCV-optimized spline
penalties shrink each smooth toward linearity wherever the data provide insufficient evidence of curvature,
ensuring that the improvement reflects genuine nonlinearity rather than overfitting of a high-dimensional model.
The 3.19-fold rate acceleration above the 196.5 MJ/m² threshold is the most practically significant finding. It
means that a photovoltaic reliability model based on OLS will systematically underestimate late-service tensile
strength loss. At a representative mid-latitude UV accumulation rate of 15 MJ/m² per year, the threshold
corresponds to approximately 13 years of field exposure for this specimen. Beyond that point, the GAM-
informed biphasic model projects a degradation rate 3.2 times higher than the OLS global average predicts. This
discrepancy directly affects service life estimates, warranty reserve calculations, and the design of inspection
protocols for photovoltaic assets operating in high-UV environments. It is important to note, however, that these
rate estimates are specific to this single PPE specimen under these particular field conditions. The threshold
location and Phase 2 rate magnitude may differ for other backsheet constructions, laminate thicknesses, or
climatic zones, and multi-specimen validation is a stated priority for future work.
The high in-sample RF performance (R² = 1.0) requires careful interpretation. RF assigned 100.0% of feature
importance to cumulative UV, with the remaining predictors collectively accounting for less than 0.1%. Under
these conditions, RF is effectively fitting a nonparametric function of UV alone, which produces near-perfect
in-sample fit on a monotonic time series but does not provide any insight beyond the UV-TS relationship already
identified by the GAM partial response. The RF model provides no interpretable smooth function, cannot
identify the biphasic transition point, and offers no basis for extrapolating beyond the observed UV range. These
limitations confirm that GAM is the more appropriate choice for this engineering application, where
interpretability and physical grounding are essential.
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Implications for Photovoltaic Reliability Management
The GAM degradation framework developed here has direct practical applications for photovoltaic reliability
management. By forecasting the onset of accelerated degradation, field operators can optimize inspection
schedules, prioritizing inspections when cumulative UV approaches the 196.5 MJ/m² threshold, thereby
concentrating maintenance resources at the most consequential stage of the module service life. The biphasic
model also provides a quantitatively defensible basis for module replacement prioritization in large-scale
photovoltaic fleets: modules that have crossed the Phase 2 threshold are degrading at a rate 3.19 times higher
than those still in Phase 1 and warrant earlier replacement consideration.
The GAM framework is also directly compatible with digital-twin platforms for photovoltaic asset monitoring.
Integrating the fitted GAM with real-time cumulative UV measurements from meteorological sensors enables
continuous updating of predicted tensile strength trajectories, supporting reliability-centered operation and
condition-based maintenance strategies. Future work should extend this integration to include probabilistic
service life intervals by coupling the GAM degradation trajectory with site-specific UV accumulation
distributions from databases such as NASA POWER or PVGIS.
Heteroscedasticity and Variance Structure
The power-law variance exponent α = 0.592 (SE = 0.105, p < 0.001) carries a specific practical implication for
reliability inference. The 13.4-fold variance range from minimum to maximum UV means that OLS prediction
intervals computed under homoscedasticity assumptions are far too narrow at the end-of-life UV exposures
where reliability certification decisions are made. Designers relying on OLS-based prediction intervals will
systematically underestimate tensile strength uncertainty in the late-service regime, producing overconfident
reliability assessments. WLS corrects this for linear coefficient inference but does not change the mean function.
Ridge regularization in the GAM addresses both issues simultaneously, providing a unified framework for mean
function flexibility and variance robustness.
LIMITATIONS AND SCOPE OF FINDINGS
Three limitations warrant explicit acknowledgment. First, the dataset derives from a single specimen, so
material-to-material variability within the PPE laminate class cannot be separated from temporal degradation.
The 196.5 MJ/m² threshold and the 3.19-fold acceleration factor are specific to this specimen; their
generalizability requires multi-specimen validation. Second, the serial autocorrelation in single-specimen
degradation data (residual autocorrelation ρ₁ ≈ 0.98 for OLS) does not bias GAM point estimates but does
compress standard errors if left unaddressed in formal inference. Applications requiring statistical significance
testing should apply Newey-West HAC corrections. Third, the GAM smooth functions are purely predictive and
cannot be directly mapped to polymer chain scission kinetic constants. Mechanistic interpretation requires
complementary chemical characterization such as carbonyl index profiles from FTIR spectroscopy at multiple
UV dose levels.
CONCLUSIONS
This study demonstrated that Generalized Additive Modeling provides a robust framework for forecasting
photovoltaic backsheet tensile strength degradation under multi-stressor field conditions. The four principal
findings are as follows.
The GAM achieves in-sample R² = 0.998 and RMSE = 0.8 MPa, representing 83.0% and 83.5% RMSE
improvements over OLS and WLS respectively. These improvements reflect adaptation to genuine nonlinearity
rather than overfitting, as confirmed by the GCV-penalized smooth estimation procedure.
A biphasic degradation regime is identified empirically, with tensile strength declining at 0.18 MPa/(MJ/m²)
below a cumulative UV threshold of 196.5 MJ/m², and accelerating to 0.574 MPa/(MJ/m²) above it, a 3.19-fold
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acceleration. This transition corresponds physically to the onset of UV penetration to the structural PET core
following WPET surface embrittlement. OLS and WLS, constrained to a single global slope, cannot represent
this transition and produce non-conservative reliability projections in the late-service regime.
Random Forest achieves high in-sample R² = 1.0 but with 100.0% of feature importance concentrated in UV
alone, effectively memorizing the monotonic degradation trajectory rather than extracting interpretable physical
structure. GAM outperforms RF in physical interpretability while achieving comparable in-sample accuracy.
These findings support the integration of advanced statistical learning techniques into photovoltaic reliability
assessment, predictive maintenance scheduling, and lifecycle management strategies. The proposed GAM
framework enables data-driven degradation forecasting that can directly inform reliability-centered photovoltaic
asset management and support condition-based maintenance decisions for photovoltaic systems operating under
diverse environmental conditions.
Future Work
Three directions for future work are identified. First, the GAM framework will be extended to multi-specimen
datasets covering multiple PPE backsheet constructions and climatic zones. Multi-specimen data will enable
separation of material-to-material variability from temporal degradation, allow formal statistical inference on
the 196.5 MJ/m² threshold location using mixed-effects extensions of GAM, and support climate-adjusted
service life probability distributions.
Second, the GAM degradation trajectories will be integrated with Weibull or lognormal reliability life
distributions to produce probabilistic service life estimates. Coupling the GAM smooth function for UVᶜᵘᵐ with
site-specific UV accumulation distributions from NASA POWER or PVGIS will allow generation of climate-
adjusted probability-of-failure curves for different installation geographies, directly supporting the reliability
certification and warranty reserve calculation workflows used by photovoltaic asset managers.
Third, the GAM smooth function shapes will be validated against complementary chemical characterization
measurements, including carbonyl index profiles from attenuated total reflection FTIR spectroscopy and
molecular weight distributions from gel permeation chromatography at multiple UV dose levels spanning the
biphasic transition. This validation will anchor the statistically identified threshold to specific degradation
chemistry milestones and provide a mechanistic basis for extrapolating the statistical model beyond the UV dose
range observed in the present dataset.
Fourth, future research should incorporate additional environmental stressors that were absent from the present
dataset. Precipitation events introduce localized moisture penetration at microcrack sites, potentially accelerating
hydrolytic chain scission beyond what ambient relative humidity measurements alone capture. Airborne
particulate matter and pollutant deposition alter the surface optical properties of the WPET outer layer,
modifying the effective UV dose absorbed at the polymer surface and introducing site-specific degradation
accelerators that a standard meteorological predictor set cannot represent. Thermal cycling frequency, defined
as the number of daily temperature excursions across the glass transition temperature of the EVA inner layer,
subjects the backsheet laminate to repeated interfacial shear stresses that compound photochemical damage.
Incorporating these additional predictors within the GAM framework would expand both the physical
completeness and the predictive generalizability of the degradation model across diverse climatic environments.
Fifth, the current frequentist GAM framework will be extended to a Bayesian formulation to address the
uncertainty quantification needs of the engineering application. Bayesian GAMs estimate posterior distributions
over the smooth function values rather than point estimates, enabling direct propagation of parameter uncertainty
into predicted tensile strength trajectories. This produces credible prediction intervals that account for both data
noise and uncertainty in smooth function shape, which is essential when the predicted tensile strength at a given
cumulative UV level is used to trigger a maintenance intervention or inform a warranty reserve calculation for a
photovoltaic installation. Integration of Markov Chain Monte Carlo sampling or variational inference within the
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penalized spline framework is computationally tractable for datasets of the size used here and would directly
extend the methodology presented in this paper.
Sixth, future comparative analyses will systematically evaluate the proposed GAM framework against recent
advances in explainable artificial intelligence (XAI) and physics-informed machine learning (PIML) for
photovoltaic degradation modeling. XAI methods such as SHAP (SHapley Additive exPlanations) applied to
gradient boosting or neural network models can approximate feature contribution decompositions similar in
spirit to GAM smooth functions, and a direct quantitative comparison of prediction accuracy, interpretability
depth, and computational cost across GAM, XGBoost-SHAP, and PIML architectures would clarify the relative
positioning of each approach for photovoltaic backsheet reliability assessment. Physics-informed neural
networks that embed Norrish reaction kinetics as hard constraints in the loss function represent a particularly
promising direction, bridging the interpretability advantage of GAMs with the flexible representational capacity
of deep learning for multi-stressor polymer degradation prediction under diverse field conditions.
Author Contributions
Conceptualization, J.M. and T.K.B.; methodology, J.M.; software and formal analysis, J.M.; investigation and
data curation, J.M.; writing, original draft, J.M.; writing, review and editing, T.K.B.; supervision, T.K.B.; project
administration, T.K.B. All authors have read and agreed to the published version of the manuscript.
Funding
This research was conducted as part of doctoral dissertation research at Morgan State University. No external
funding was received.
Data Availability Statement
The raw field dataset supporting this study is available from the corresponding author upon reasonable written
request. The dataset includes 511 consecutive daily measurements of tensile strength, cumulative UV irradiance,
ambient temperature, relative humidity, and wind speed.
ACKNOWLEDGMENTS
The authors thank the Department of Industrial and Systems Engineering at Morgan State University for
computational support.
Conflicts of Interest
The authors declare no conflicts of interest.
Abbreviations
The following abbreviations are used in this manuscript:
PV
Photovoltaic
PET
Polyethylene Terephthalate
EVA
Ethylene Vinyl Acetate
WPET
White-Pigmented PET
GAM
Generalized Additive Model
OLS
Ordinary Least Squares
WLS
Weighted Least Squares
RF
Random Forest
UV
Ultraviolet
RMSE
Root Mean Squared Error
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MSE
Mean Squared Error
R²
Coefficient of Determination
GCV
Generalized Cross-Validation
CV
Cross-Validation
P-IRLS
Penalized Iteratively Reweighted Least Squares
ASTM
American Society for Testing and Materials
IEC
International Electrotechnical Commission
FTIR
Fourier-Transform Infrared Spectroscopy
HAC
Heteroscedasticity and Autocorrelation Consistent
PPE
PET/PET/EVA laminate type
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