Jakarta is located on the island of Java, which lies along the Pacific Ring of Fire, rendering it susceptible to natural hazards, such as flooding, land subsidence, and earthquakes. As a highly dense metropolis, Jakarta is also prone to non-natural hazards, particularly urban fires. The Indonesian Disaster Risk Index (IRBI) for DKI Jakarta in 2021 was 60.43, placing the province in the medium-risk category (BPBD, 2021). A downward trend in the IRBI has been observed since 2015, reflecting the efforts of the Provincial Government of DKI Jakarta to strengthen regional resilience and stability against disaster risks. The safety of residents and the security of Jakarta remain top priorities, as evidenced by a range of mitigation policies, infrastructure strengthening, and enhanced community preparedness to face potential disasters.

Jakarta is located between 5°19′12″ - 6°23′54″ S and 106°22′42″ and 106°22′42″-106°58′18″ E, with an average elevation of approximately 7 meters above sea level and a total area of 662.33 km². Jakarta is the capital of the Republic of Indonesia and comprises five administrative municipalities and one administrative regency, with a total area of 662.33 km². Administratively, the province comprises five municipalities and one administrative regency, subdivided into 44 districts and 267 urban villages (Bappeda Jakarta, 2018). The southern and eastern boundaries adjoin Depok City, Bogor Regency, Bekasi City, and Bekasi Regency; the western boundary adjoins Tangerang City and Tangerang Regency; and the northern boundary is Java Sea.

This geographic setting is important for the present study because climate-related hazards in Jakarta are spatially heterogeneous. Northern and coastal areas are more strongly affected by tidal inundation, recurrent flooding, and land subsidence, whereas inland areas may experience different combinations of flood, drought, heat, infrastructure constraints, and socioeconomic exposure. Therefore, Jakarta provides an appropriate case for applying a geographically weighted multivariate approach, because household resilience is expected to vary across local spatial contexts rather than following a single global structure.

This study used household survey data collected in 2022 by CCROM-SEAP (IPB University) in collaboration with the Jakarta Environmental Agency (Boer et al., 2022). Survey locations were determined by an urgency assessment derived from the risks of flood, drought, tidal inundation, and extreme temperature (Dinas Lingkungan Hidup Provinsi DKI Jakarta, 2021). The distribution of urban villages across vulnerability levels under each urgency scenario is summarized in Table 1. Based on this urgency framework, 17 mainland urban villages were selected to represent urgency levels and municipalities in Jakarta; the distribution of these selected villages across vulnerability levels is presented in Table 2.

Jakarta’s population reached 10,609,781 in 2021, with an annual growth rate of 0.57%. It is the most densely populated province in Indonesia, with a population density of 15,978 persons per km² (BPS Jakarta, 2022). This survey used stratified random sampling, and the survey sample size was determined using Slovin’s formula based on a population of 2,770,729 households and a margin of error of 5%, resulting in 400 respondents (Boer et al., 2022). The 400 respondents comprised 221 mainland households, 26 island households, and 143 business units drawn from 17 mainland and 4 island villages.

This study is a methodological extension of the previously published HCRI study by (Sundari et al., 2026), who developed a household resilience index using a global Robust PCA-MCD framework. The present study uses the same mainland household dataset but extends the analysis by incorporating a geographically weighted structure into the PCA. Therefore, the sampling design, selected urban villages, and unit of analysis followed those of (Sundari et al., 2026). In the previous study, the selected urban villages were not determined merely by administrative representation but were based on an urgency-based vulnerability stratification derived from Jakarta’s multi-hazard climate risk profile, including flood, drought, tidal inundation, and extreme heat.

This study focuses on the 221 mainland households to ensure adequate local support (i.e., a sufficient number of neighbors) for geographically weighted principal component analysis (GWPCA). The respondents were selected to represent three income groups—lower-middle, middle, and upper-middle—to capture income-based heterogeneity in climate change impacts. The income groups were justified by the enumerators based on the household’s residential location (informal/slum areas, non-exclusive housing estates, and exclusive gated clusters) and the physical characteristics of the dwelling structure (permanent vs. non-permanent).

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The restriction to 221 mainland households is analytically important for two reasons. First, business units were excluded because their exposure pathways, decision-making logic, and adaptation strategies differed from those of households. Second, island households were excluded because their geographic exposure, governance settings, and infrastructure conditions differed from those in mainland Jakarta. Thus, the analytical unit used in this study was the mainland household. However, because the sample represents 221 households from 17 of 267 urban villages in Jakarta, the findings should be interpreted as evidence for the selected mainland urban villages rather than as a full generalization to all households or all urban villages in DKI Jakarta. This representativeness constraint is acknowledged as a structural limitation and is discussed further in the Discussion section.

Household climate resilience is operationalized using four dimensions–exposure, sensitivity, incremental adaptive capacity, and transformational adaptive capacity–implemented through 45 household-level indicators derived from the survey instrument and organized across thematic sectors relevant to climate risk and adaptive capacity (Boer et al., 2022). Following the resilience framework in the literature (Subiyanto et al., 2020), these dimensions are expressed as (i) non-exposure (geographic–demographic conditions related to flood/drought hazards), (ii) low sensitivity (dwelling conditions, health, and water resources), (iii) incremental adaptation (economic conditions and human resources), and (iv) transformational adaptation (personal/household practices). The complete indicator list, sector mapping, coding, and measurement types are presented in Table 3.

In conventional vulnerability analysis, higher exposure and higher sensitivity indicate greater vulnerability. In the present resilience-oriented framework, the directions of these indicators are aligned such that higher transformed values consistently represent more resilient conditions. Thus, exposure indicates lower household exposure to climate-related hazards after indicator orientation, whereas low sensitivity indicates lower internal susceptibility of households, housing conditions, health conditions, and water-resource conditions to climate impacts.

Incremental adaptation refers to household coping and adjustment capacity under current climate stress, such as economic resources, human resources, and actions that reduce immediate losses. Transformational adaptation refers to broader household practices and behavioral changes that may strengthen the long-term adaptive capacity under future climate uncertainty. This distinction is important because the HCRI is not intended to measure exposure alone, material assets alone, or vulnerability alone, but rather the combined household capacity to avoid, withstand, cope with, and adapt to climate-related disturbances.

As shown in Table 3, the final indicator set used to construct the HCRI consists of 30 indicators, comprising 17 numeric variables and 13 categorical variables. The numeric indicators were directly included in the PCA/GWPCA input matrix, whereas the 13 categorical indicators marked as MSI = “Yes” in Table 3 were transformed into interval-scale scores using the Method of Successive Intervals (MSI) prior to covariance-based PCA/GWPCA. Only categorical variables with a meaningful ordinal structure, or variables that had been theoretically recoded into ordered resilience levels, were treated using MSI. Nominal variables were not directly interpreted as ordinal unless their categories had a defensible resilience ordering. This clarification was added to avoid inappropriate treatment of unordered categorical variables in the PCA input matrix.

Before the analysis, the study conducted data pre-processing in three steps. First, because PCA requires numerical inputs, while the questionnaire includes categorical responses (Boer et al., 2022), the 13 ordinal indicators in Table 3 were transformed into interval-scale scores using the MSI (Green, 2017). This transformation produces continuous scores that preserve the original response ordering, allowing ordinal and numeric indicators to be analyzed jointly in the subsequent PCA computation. The MSI was applied only to categorical variables with a meaningful ordinal structure or to variables that had been theoretically recoded into ordered resilience levels. Nominal variables were not treated directly as ordinal unless their categories had a defensible resilience ordering.

Established PCA variants for categorical-only and mixed-type data are available, particularly for categorical PCA/nonlinear PCA via optimal scaling and mixed-data approaches, such as PCAmix or FAMD (Chavent et al., 2022; Pagès, 2014). A preliminary sensitivity analysis using PCAmix showed that the first dimension explained only 6.26% of the total variation, with a cumulative proportion of 6.26%. This result indicates that the first PCAmix dimension did not provide a sufficiently dominant one-dimensional structure for constructing a single HCRI. Therefore, MSI-based intervalization was retained because it produced a coherent numerical input matrix for covariance-based PCA and GWPCA, while preserving the theoretical ordering of ordinal resilience indicators. This choice is consistent with the objective of this study, namely to construct an interpretable one-number HCRI based on the dominant local principal component.

Second, the study retained the final HCRI indicators on their original oriented scales after the required MSI transformation. No additional manual global centering, local centering, or standardization transformation was applied to the full input matrix before the main GWPCA and Robust GWPCA-MCD estimation. This decision was made because the study aimed to preserve the empirical variation in the original household data and facilitate interpretation of the resulting PC scores in relation to household-level differences. Several indicators, such as distance to flood sources, number of flood days, house elevation, building area, land area, health-related proportions, and household expenditures, contain substantively meaningful magnitude information. Standardizing or centering the full input matrix prior to model estimation could reduce the direct interpretability of these original-scale household differences.

This choice means that the main analysis follows a covariance-based GWPCA specification and is scale-dependent. Indicators with larger empirical variance may contribute more strongly to the extracted components. Therefore, the results cannot be interpreted as being free from scale effects. Instead, the covariance-based results are interpreted together with indicator-scale diagnostics, loading interpretation, local support diagnostics, and robustness checks. The dominant indicators are interpreted as statistical contributors to the local covariance structure rather than as causal determinants of household resilience.

Third, multivariate outliers were screened using the robust Mahalanobis distance to reduce the influence of extreme observations on covariance estimation and the resulting principal component directions. In the classical formulation, the Mahalanobis distance is computed from the sample mean vector and covariance matrix and can be compared against a reference cutoff, such as a chi-square quantile with degrees of freedom equal to the number of variables (Filzmoser & Gregorich, 2020; Johnson & Wichern, 2019; Maronna et al., 2019). However, because classical estimates are sensitive to contamination and may lead to masking and swamping when multiple outliers are present, this study employed the MCD estimator to obtain robust estimates of location and scatter and compute robust Mahalanobis distances for outlier flagging.

The robust Mahalanobis distance was calculated using a 13-variable numerical subset selected for outlier detection. This subset was not identical to the full GWPCA indicator set. The full GWPCA model used 30 final indicators after transformation and screening, whereas p = 13 was used only as the degrees of freedom for the robust Mahalanobis distance cutoff. This distinction was added to avoid confusion between the outlier-detection and the final HCRI construction stages.

Observations flagged by robust Mahalanobis distance were not automatically excluded from the analysis. Because 41.6% of households were flagged, the results were interpreted as evidence of strong multivariate heterogeneity rather than simple data contamination. These flagged households may reflect true diversity in household exposure, housing conditions, health burden, economic resources, and adaptation practices. Therefore, robust methods were used to reduce their leverage in the covariance estimation while preserving them as part of the empirical household resilience structure.

Fourth, potential multicollinearity or redundancy among the indicators was assessed using the Variance Inflation Factor (VIF) (Upendra et al., 2023). The VIF quantifies the extent to which a variable can be linearly explained by the remaining variables; values close to 1 indicate negligible collinearity, whereas larger values indicate increasing redundancy and inflated standard errors. Indicators with high VIF values were treated as candidates for further handling, such as removal or consolidation, prior to PCA to improve numerical stability and support a clearer interpretation of the extracted components.

In this study, the VIF was used only as a preliminary redundancy diagnostic, not as a replacement for PCA diagnostics. As VIF is more commonly associated with regression diagnostics, its role in this study was limited to identifying near-linear redundancy among the candidate indicators before robust covariance estimation. The final indicator decision was based on a combination of the VIF results, theoretical relevance, measurement type, data completeness, and interpretability within the HCRI framework.

GWPCA extends global PCA by allowing the multivariate structure of the data to vary across space, thereby capturing spatial heterogeneity in the dominant patterns of covariation. For a data matrix X and a target location (ui,vi), observations are weighted according to their geographic proximity via a kernel function, PCA is performed on the resulting locally weighted variance–covariance structure, and the degree of localization is governed by the bandwidth parameter (Harris et al., 2011; Lu et al., 2025). The geographically weighted variance–covariance matrix at (ui,vi)is defined as Equation 1:

where W(ui,vi) is a diagonal matrix of spatial weights. Local principal components are then obtained by eigendecomposition of Σ(ui,vi), yielding location-specific eigenvalues, loadings, and component scores that can be mapped to interpret geographic variation in the dominant dimensions of the dataset (Harris et al., 2011; Lu et al., 2025).

To reduce sensitivity to multivariate outliers and spatially varying leverage points, this study applies robust GWPCA, replacing the classical locally weighted covariance with a robust estimator. This robust specification is intended to produce more stable local eigenstructures under contamination, thereby strengthening the inference about spatial patterns derived from localized components (Harris et al., 2012). Specifically, both the local mean vector μ(ui,vi) and the local covariance matrix Σ(ui,vi)are estimated using the MCD estimator at each location (Hubert et al., 2018). The MCD seeks a subset of observations with minimal covariance determinants and subsequently applies a standard reweighting step to improve statistical efficiency, reducing the influence of atypical observations on the local covariance matrix used for PCA. In implementation, the MCD solution is computed using the FAST-MCD procedure in standard software, with robustness controlled through the subset size h(or equivalently α) and a chi-square cutoff for robust distances (Hubert et al., 2018; Rousseeuw & Driessen, 1999).

The MCD tuning parameters, including the selected alpha value, corresponding h-subset size, reweighting procedure, and robust-distance cutoff, were reported to improve reproducibility. Because robust local covariance estimation depends on the relationship between the local sample size and the number of variables, local support was also evaluated using the number of neighboring observations and the effective local sample size at each target location.

Spatial data were managed as spatial objects in R, and inter-location distances were derived from household coordinates to construct W(ui,vi)through an explicit distance matrix. Because locations are recorded in longitude–latitude, distances were computed using spherical (great-circle) distance so that geographic separation is measured on the Earth’s surface rather than on a planar projection. A bi-square kernel was then applied to assign higher weights to nearer observations and smoothly downweight more distant observations, thereby emphasizing localized covariance structures while preserving spatial continuity on the weighting surface (Harris et al., 2011; Lu et al., 2025).

The bandwidth in GWPCA may be specified either as a fixed distance or as an adaptive number of nearest neighbors (Harris et al., 2011; Lu et al., 2025). This study adopts a fixed bandwidth because preliminary comparisons indicated that the adaptive specification produced a lower proportion of variance explained by the first component (PC1), which is the primary quantity required for index construction. Bandwidth selection was guided by cross-validation (CV). To align computation with the index-construction objective, only the first two components were retained during bandwidth tuning and estimation, whereas substantive interpretation focused on PC1.

Bandwidth selection was not justified solely by maximizing the percentage of variance explained by PC1 because doing so would create circular reasoning when PC1 is also used as the index basis. Therefore, the revised bandwidth justification combines several criteria: cross-validation, local loading stability, a sufficient effective local sample size, and classification stability across reasonable bandwidth alternatives. The selected bandwidth was reported together with the cross-validation results and the sensitivity of local PC1 variance and dominant loadings.

Robust GWPCA was implemented by replacing the classical locally weighted covariance with an MCD estimator to reduce sensitivity to multivariate outliers and spatially varying leverage points (Hubert et al., 2018; Lu et al., 2014; Rousseeuw & Driessen, 1999). All analyses were performed in R. For reproducibility, the revised manuscript should report the exact R version and package versions used for the analysis, including the GW model for geographically weighted modelling, sp for spatial data handling, and the relevant robust multivariate packages used for MCD estimation.

The central methodological issue in constructing an HCRI is dimensionality reduction: a large set of indicators must be summarized into a single, interpretable measure while retaining the most salient information. From a dimensionality-reduction perspective, prioritizing PC1 with the largest explained variance (e.g., exceeding a threshold such as 70%) for HCRI construction is defensible because PC1 is a one-dimensional projection that captures the maximum variance, making its scores the most informative single-number summary when an index must be expressed as one value per household (Jolliffe & Cadima, 2016). In applied index construction using PCA (e.g., wealth/SES indices), early components, often PC1, are used as index scores when they explain a substantial share of the total variance, indicating that the lower-dimensional representation remains strongly aligned with the dominant information in the original indicators (Howe et al., 2008; Vyas & Kumaranayake, 2006).

The overall workflow, from indicator preparation and robust outlier screening to robust GWPCA estimation and PC1-based index scaling and classification, is summarized in Figure 2.

Thus, the larger the proportion of variance explained by PC1, the smaller the information loss induced by dimensionality reduction, strengthening the interpretability and defensibility of PC1-based scaling and classification of the HCRI (Howe et al., 2008; Jolliffe & Cadima, 2016). Accordingly, this study reports the GWPCA outputs by resilience dimension, focusing on the proportion of variance explained by PC1 and the associated component structure, as summarized in Table 4.

The final dataset used in this study consisted of 221 mainland household observations and 30 HCRI indicators. These indicators represent the four dimensions of household climate resilience: exposure, sensitivity, incremental adaptation, and transformational adaptation. These dimensions are consistent with the broader climate-resilience framework, which views resilience as the capacity of social units to absorb disturbances, maintain function, and adapt to climate-related stresses (Adger, 2000; IPCC, 2014). Before applying the geographically weighted model, the distribution and scale of the indicators were examined to describe the empirical structure of the data and identify potential issues related to unequal variance, extreme values, and differences in measurement units.

The HCRI indicators were measured on heterogeneous empirical scales. Some indicators represented distances to flood sources, number of flood days, house elevation, household expenditure, land area, building area, health-related proportions, and transformed categorical scores. This structure indicates that the final HCRI dataset combines physical exposure, housing characteristics, health burden, economic capacity, and household adaptation responses. Therefore, the indicators are not homogeneous in terms of unit, range, or variability. This issue is important because PCA is directly influenced by the variance–covariance structure of the input variables, particularly when the analysis is performed using covariance-based estimation (Jolliffe, 2002; Jolliffe & Cadima, 2016).

Several indicators exhibited extreme empirical values. These values were not automatically treated as data errors. Instead, they were interpreted as possible indicators of household heterogeneity across the sampled urban villages. In the context of household climate resilience, unusual values may reflect real differences in flood exposure, distance from hazard sources, housing elevation, land ownership, building size, post-disaster expenditure, and household capacity to respond to climate-related disturbances. This interpretation is important because household-level resilience data may contain atypical but substantively meaningful profiles, rather than simple statistical contamination.

The diagnostic assessment of the indicator distribution is also relevant for GWPCA because the method estimates local principal components from geographically weighted covariance structures. Unlike global PCA, which summarizes the overall covariance structure of the full dataset, GWPCA allows the dominant multivariate structure to vary across locations (Fotheringham et al., 2002; Harris et al., 2011). Consequently, extreme or high-leverage observations located within a local neighborhood may affect local eigenvalues, eigenvectors, component loadings, and PC1 scores more strongly than in a global model.

Therefore, the observed variation in the indicator scale and distribution provides an empirical basis for applying a robust covariance approach in the local PCA framework. Robust GWPCA-MCD was used to reduce the influence of high-leverage observations while retaining household profiles that may be substantively meaningful for resilience assessment. The MCD estimator is commonly used to obtain robust estimates of multivariate locations and scatter in the presence of atypical observations (Rousseeuw, 1984; Rousseeuw & Driessen, 1999). Thus, using a robust local covariance estimator is appropriate for household resilience data that may contain heterogeneous and spatially uneven multivariate profiles. The distribution of the 30 final HCRI indicators is shown in Figure 3. This figure provides a visual diagnostic of the differences in spread, range, and potential extreme values across the indicators. The visual pattern is used as an initial indication that the household resilience data contain heterogeneous multivariate structures, which motivates subsequent comparison between the global PCA baselines and the local Robust GWPCA-MCD framework.

Before applying the local Robust GWPCA-MCD model, the global PCA was first examined as a baseline diagnostic. This step was used to evaluate whether the final HCRI indicators formed a dominant multivariate structure at the household level. The global PCA result was not intended to replace the spatially local model but to provide a reference point before the covariance structure was allowed to vary geographically. In the subsequent local model, the final analytical setting consisted of 221 household observations, 30 HCRI indicators, two retained components, and a bi-square geographically weighted kernel.

The global PCA baseline showed that two classical PCA formulations, namely, PCA based on singular value decomposition (PCA-SVD) and PCA based on the covariance matrix, produced the same PC1 variance share of 87.46%. This result indicates that the dominant global component was consistent across the two equivalent PCA computations under the same input structure. In the PCA theory, the SVD formulation and eigendecomposition of the covariance matrix are mathematically connected because both identify the orthogonal directions that maximize the variance of the projected data. This relationship is also described by (Sundari et al., 2026), where PCA is formulated through the eigenvalue problem of the covariance matrix and, equivalently, through the SVD of the column-centered data matrix.

The high PC1 variance from the PCA-SVD and PCA covariance provides an initial indication that the HCRI indicators contain a strong common multivariate structure at the global level. However, this result does not imply that global PCA is sufficient to represent the spatially varying structure of household resilience. Global PCA summarizes one overall covariance structure for all households, whereas household climate resilience may vary across locations owing to differences in exposure, housing conditions, health burden, economic capacity, and adaptation responses. Therefore, the global PCA result is treated as a baseline diagnostic rather than as the final HCRI model.

This covariance-based baseline is also consistent with the previously published robust global HCRI framework of (Sundari et al., 2026). In that study, the HCRI was constructed from 221 household observations across 17 urban villages in Jakarta using four resilience dimensions: exposure, sensitivity, incremental adaptation, and transformational adaptation. The study applied Robust PCA with the MCD estimator to reduce the influence of outliers and obtain a stable component estimation. The published Robust PCA-MCD results showed that PC1 explained 87.38% of the total variance, supporting the use of PC1 as a dominant one-number summary of household resilience, while reducing the influence of atypical multivariate observations.

Based on this global evidence, a covariance-based specification was retained in the present study. This choice was not made to ignore the scale issue but to preserve meaningful magnitude differences among household resilience indicators after indicator orientation and transformation. Several HCRI indicators, such as distance to flood sources, number of flood days, house elevation, building area, land area, health-related proportions, and household expenditures, contain substantive magnitude information. Standardizing all the indicators would place all variables in equal variance but may also reduce the intended role of empirical magnitude differences in index construction. Therefore, the covariance-based approach was considered appropriate because the objective of this study was to extend the previously established robust covariance-based HCRI framework into a geographically weighted setting.

Therefore, the present Robust GWPCA-MCD model is positioned as a spatial extension of the robust global PCA-MCD framework, rather than as a separate index-construction logic. The global PCA baseline confirms that the HCRI indicators have a strong overall multivariate structure, whereas the local Robust GWPCA-MCD model examines whether this structure varies across household locations. This transition from a global to local analysis is important because urban household resilience in Jakarta is not expected to be spatially uniform. Local differences in flood exposure, housing sensitivity, water access, health burden, and adaptation capacity may produce different dominant indicators across neighborhoods.

However, the global PCA baseline was not used as the sole justification for the final HCRI. The covariance-based specification remains sensitive to indicator variance, and this limitation is acknowledged in the interpretation. For this reason, the following sections report additional local diagnostics, including adaptive bandwidth, local support, local PC1 variance, eigenvalue behavior, and top local PC1 loadings. These diagnostics were used to assess whether the local HCRI structure was sufficiently stable and interpretable, rather than relying only on the high global PC1 variance.

Before estimating the Robust GWPCA-MCD model, multivariate household profiles were examined using the robust Mahalanobis distance. This diagnostic step identified observations whose joint indicator patterns differed substantially from the central multivariate structure of the data. In multivariate analysis, the Mahalanobis distance is commonly used to measure the extent to which an observation lies from the multivariate center after accounting for the covariance structure among variables. However, because the classical mean vector and covariance matrix are sensitive to atypical observations, a robust version based on the MCD estimator is preferred when the data may contain multivariate outliers or high-leverage observations (Hubert & Rousseeuw, 2005; Rousseeuw, 1984; Rousseeuw & Driessen, 1999).

The robust Mahalanobis distance was calculated using a 13-variable numerical subset selected specifically for multivariate outlier diagnostics. Therefore, the value p = 13 refers only to the degrees of freedom used in the robust Mahalanobis distance cutoff, not to the full HCRI construction stage. This distinction is important because the final Robust GWPCA-MCD model used 30 HCRI indicators, whereas the outlier diagnostic was conducted on a narrower numerical subset to evaluate multivariate leverage before robust local covariance estimation.

The diagnostic result showed that 95 household observations exceeded the χ² (99%) cutoff value of 27.688 with p = 13, corresponding to 41.6% of households being flagged as multivariate outliers. This proportion is relatively high, indicating that the household resilience data contain substantial multivariate heterogeneity. Therefore, the flagged observations were not interpreted as simple data errors or marginal noise. Instead, they were treated as atypical household profiles that may reflect meaningful differences in exposure, housing conditions, health impacts, economic capacity, and adaptation behavior across the sampled urban villages.

This interpretation is consistent with the logic of robust PCA, in which outlier diagnostics are used not only to remove observations but also to justify the use of robust covariance estimation. (Sundari et al., 2026) emphasized that flagged observations were not mechanically deleted; rather, outlier screening was used to motivate robust covariance estimation such that the component directions were not dominated by a minority of extreme multivariate profiles, while households remained available for scoring and spatial mapping. This principle is also relevant to the present study because household resilience data may contain atypical but substantively meaningful observations.

The high proportion of flagged observations provides an empirical justification for the use of a robust covariance estimator. Classical covariance-based PCA may be affected by high-leverage observations because a small number of atypical profiles can rotate the principal directions and influence the eigenvectors used to define component loadings (Hubert & Rousseeuw, 2005; Jolliffe, 2002). In the published HCRI study, this concern was also noted: classical covariance-based PCA can be rotated by leverage points that disproportionately affect the sample covariance matrix and the resulting eigenvectors. Therefore, MCD-based robust covariance estimation is appropriate for reducing the influence of such observations while preserving the multivariate structure of the data.

In the context of GWPCA, the outlier issue becomes more important because the covariance matrices are estimated locally. Unlike global PCA, which summarizes one covariance structure for all observations, GWPCA estimates local covariance structures using geographically weighted observations (Harris et al., 2011). Consequently, an atypical household located within a local neighborhood may have a stronger influence on the local eigenvalues, eigenvectors, component loadings, and PC1 scores than it would in a global model. This condition strengthens the rationale for using Robust GWPCA-MCD, because the local covariance structure should be protected from excessive leverage while retaining meaningful household variation.

The robust Mahalanobis distance result suggests that household resilience in Jakarta is not represented by a single homogeneous profile. Some households may experience high exposure but possess stronger adaptive capacity, whereas others may face lower exposure but greater sensitivity or weaker coping resources. This variation is consistent with the multidimensional nature of household resilience, which combines exposure, sensitivity, and incremental and transformational adaptation. In the published HCRI framework, these four dimensions were used to construct a household-level resilience index for 221 respondents from 17 urban villages in Jakarta.

Thus, the robust Mahalanobis distance diagnostic serves as a bridge between the global PCA baseline and the local Robust GWPCA-MCD model. The diagnostic indicates that robust treatment is needed because the household data contain heterogeneous multivariate profiles, whereas the geographically weighted framework is needed because the structure of resilience may vary across locations. The robust Mahalanobis distance plot is shown in Figure 4. Observations above the χ² (99%) cutoff are interpreted as atypical multivariate household profiles that require robust covariance treatment, not as observations to be automatically excluded from the HCRI construction.

After the global PCA baseline and robust Mahalanobis distance diagnostics, the local model was estimated using Robust GWPCA-MCD. This model was designed to extend the previously established robust covariance-based HCRI framework into a geographically weighted setting. The objective was not only to obtain a single global resilience gradient but also to examine whether the dominant multivariate structure of household resilience varied across household locations.

The final Robust GWPCA-MCD model was estimated using 221 household observations and 30 final HCRI indicators. Two components were retained in the model, and a bisquare kernel was used to construct the geographical weighting scheme. The selected bandwidth was adaptive, meaning that the local model was calibrated using a fixed number of nearest neighbors rather than a fixed geographical distance. The non-robust GWPCA model selected 120 nearest neighbors, whereas the Robust GWPCA-MCD model selected 96 nearest neighbors. The robust model used the MCD estimator with α = 0.75, resulting in an approximate local h-subset of 72 observations.

The use of an adaptive bandwidth is important because household observations are not uniformly distributed across space. In geographically weighted analysis, adaptive bandwidth helps maintain a comparable number of observations around each target location, especially when the density of the sampled households varies across the study area. This is particularly relevant for local PCA, because each location requires a sufficiently supported local covariance matrix before eigenvalue decomposition can be performed. Therefore, the selected bandwidth reflects a balance between preserving local spatial variation and maintaining numerical stability in the covariance estimation.

In this study, the selected bandwidth of 96 nearest neighbors provides a local neighborhood size approximately 3.20 times the number of final HCRI indicators. With MCD α = 0.75, the approximate robust h-subset is 72 observations, corresponding to 2.40 times the number of indicators. This indicates that the local covariance estimation was supported by a neighborhood size larger than the number of variables. Such local support is important because covariance estimation with a small neighborhood relative to the number of indicators may produce unstable eigenvalues, unstable loading directions, or nearly singular local covariance matrices.

The bisquare kernel assigns larger weights to observations closer to the target location and smaller weights to more distant observations, whereas observations outside the local bandwidth receive zero weight. In the diagnostic procedure, local support was assessed by calculating the number of non-zero-weighted neighbors and the effective local sample size for each location. The effective local sample size was computed from the concentration of geographical weights using the squared sum of weights divided by the sum of squared weights. This diagnostic was used to ensure that the local covariance structure was interpreted not only from the nominal bandwidth but also from the effective contribution of nearby observations.

The use of MCD-based robust covariance estimation further strengthens the local model calibration. The robust Mahalanobis distance diagnostic showed substantial multivariate heterogeneity in the household data; therefore, a classical local covariance matrix may be sensitive to atypical household profiles. In a geographically weighted setting, such profiles may exert a stronger influence because each local covariance matrix is estimated from a neighborhood subset rather than from the full dataset. By combining adaptive geographical weighting with MCD-based robust covariance estimation, Robust GWPCA-MCD mitigates the influence of high-leverage observations while retaining substantively meaningful household variation.

The retained two-component structure follows the general PCA principle that principal components summarize the main directions of variation in a multivariate dataset. Component loadings represent the contribution of indicators, whereas component scores represent the transformed positions of households in the component space. In the present study, PC1 was emphasized as the main local HCRI gradient, whereas PC2 was retained to evaluate additional local variation and cumulative local variance.

Overall, the model calibration results indicate that Robust GWPCA-MCD provides a locally supported framework for estimating spatially varying HCRI structures. However, the model is not interpreted solely based on bandwidth selection or high PC1 variance. The subsequent analysis also considers local PC1 variance, eigenvalue behavior, condition number, and dominant local loadings to evaluate whether the local HCRI structure is stable and substantively interpretable.

After calibrating the Robust GWPCA-MCD model, the local variance explained by the first principal component was evaluated and compared with the classical GWPCA results. This comparison was made to assess whether the robust local covariance specification produced a more informative local PC1 structure than the non-robust covariance-based GWPCA. In PCA, the first principal component represents the dominant linear combination of the original indicators, whereas the proportion of variance explained by PC1 reflects the amount of multivariate information summarized by this component (Jolliffe, 2002; Jolliffe & Cadima, 2016). In the HCRI framework, this diagnostic is important because PC1 scores are used as the basis for constructing household resilience scores, whereas component loadings represent the contribution of indicators to the resilience gradient.

The comparison shows a clear difference between the classical and robust GWPCA-MCD results. The classical GWPCA model, using an adaptive bandwidth of 120 nearest neighbors, produced a local PC1 variance ranging from 12.35% to 79.17%, with a median of only 14.81%. In contrast, the Robust GWPCA-MCD model, using an adaptive bandwidth of 96 nearest neighbors, produced a local PC1 variance ranging from 63.42% to 99.999%, with a median of 83.66%. This indicates that the first component in the classical GWPCA model did not consistently summarize the local covariance structure, whereas the robust model produced a substantially stronger local PC1 structure across household locations.

The low median local PC1 variance in the classical GWPCA results suggests that the local covariance structure was fragmented across several components. In other words, the first component alone was not sufficiently dominant in many local neighborhoods. This condition may occur when the local covariance estimation is affected by heterogeneous household profiles, high-leverage observations, or uneven spatial variation in the indicators. As household resilience indicators combine exposure, sensitivity, health burden, housing characteristics, economic capacity, and adaptation responses, the local covariance structure may be difficult to summarize using a single non-robust principal component.

The Robust GWPCA-MCD results provide stronger evidence for information sufficiency. The robust model produced a local PC1 variance with a minimum of 63.42%, Q1 of 78.09%, a median of 83.66%, Q3 of 85.54%, and a maximum of 99.999%. A median value above 80% indicated that, for most household locations, PC1 captured a large proportion of the local multivariate variation. This finding supports the use of PC1 as the main local HCRI gradient. This result is consistent with the applied PCA criterion that components explaining a substantial proportion of the total variance can be retained for interpretation, with 70% often used as a practical benchmark in index construction.

However, the high local PC1 variance should not be interpreted mechanically. The minimum value of 63.42% indicates that some local neighborhoods still had more complex multivariate structures, whereas PC1 did not capture as much variation as in other locations. Conversely, the maximum value of 99.999% indicates a nearly complete concentration of local variance in PC1. This may reflect a strongly dominant local component but may also indicate a highly concentrated local covariance structure. Therefore, local PC1 variance must be interpreted together with model calibration, bandwidth sensitivity, local support, and loading stability, rather than as a standalone measure of model quality.

Bandwidth sensitivity was then evaluated to examine whether the robust local PC1 structure remained stable under neighboring bandwidth alternatives. The main Robust GWPCA-MCD bandwidth was 96 nearest neighbors. When the bandwidth was reduced to 86, the median local PC1 variance remained high at 80.86%, with 91.86% agreement for top-ranked loading and 85.97% agreement for the HCRI classification. When the bandwidth was increased to 106 and 120, the median local PC1 variance increased slightly, but the HCRI class agreement declined to 61.09% and 44.80%.

The sensitivity results indicated that a higher local PC1 variance did not automatically imply a better local model. Although larger bandwidths produced slightly higher median LPV PC1, they reduced agreement in both the dominant loading and the HCRI classification. This pattern suggests that wider bandwidths may smooth the local differences too strongly, thereby weakening the spatial specificity of the HCRI classification. Therefore, the selected bandwidth of 96 nearest neighbors provides a practical balance between variance explanation, loading stability, and local interpretability.

Overall, the comparison between classical GWPCA and Robust GWPCA-MCD supports the use of the robust local covariance approach. Classical GWPCA produced a low median local PC1 variance, indicating that PC1 was not consistently informative across locations. Robust GWPCA-MCD produced a stronger and more stable PC1 structure while retaining spatial variation in household resilience. Therefore, PC1 from the Robust GWPCA-MCD model was used as the main local HCRI gradient in the subsequent scoring and classification stage.

The classification of the HCRI across selected urban villages reveals clear contrasts among administrative areas and forms a pattern that may be described as a “dual pattern.” This pattern refers to the coexistence of locations that, despite high hazard exposure, remain in high resilience classes alongside other locations with similar exposure levels that are instead concentrated in low-resilience categories. This finding confirms that household resilience is not determined solely by disaster exposure but emerges from the interaction between exposure, adaptive capacity, and enabling conditions, such as access to basic services, infrastructure quality, and local social–institutional support (Cutter et al., 2008; Meerow et al., 2016; Norris et al., 2008). Thus, the spatial distribution of HCRI categories not only identifies “where risks are located” but also reveals “where capacities are able to offset those risks.”

Central Jakarta appears to be the most consistently resilient. Kampung Bali and Karet Tengsin each recorded 100.00% of households in the Very High Resilience category, despite differing vulnerability profiles (Kampung Bali: high vulnerability to drought hazards; Karet Tengsin: low vulnerability to flood hazards). West Jakarta exhibited strong polarization: Duri Selatan (moderate vulnerability, drought, and extreme heat) was dominated by the High category (92.31%), with a small share of Very High (7.69%), whereas Krendang (moderate vulnerability, extreme heat) was entirely High (100.00%). In contrast, highly vulnerable villages in the same municipality were concentrated in the lowest resilience classes: Kamal (flood, tidal flood, and extreme heat) was distributed across Very Low (15.38%) and Extremely Low (84.62%) Resilience, and Semanan (flood, extreme heat) was Extremely Low (100.00%).

South Jakarta also displayed sharp heterogeneity. Cilandak Timur (high vulnerability: flood, drought, and extreme heat) and Ragunan (extremely low vulnerability) both recorded 100.00% in the Very High category, whereas North Petukangan (high vulnerability: flood and extreme heat) was Extremely Low (100.00%). Ulujami (low vulnerability: extreme heat) showed a more layered distribution: Very High (23.08%), High (7.69%), Very Low (23.08%), and Extremely Low (46.15%), indicating internal inequality of resilience within the same village.

In East Jakarta, villages with extremely high vulnerability tend to cluster in the lower classes. Jatinegara (extremely high vulnerability; flood) was dominated by Low (92.31%) and Very Low (7.69%) vulnerability, while Malaka Jaya (Very Low vulnerability) was dominated by Very Low vulnerability (92.30%) and Extremely Low vulnerability (7.69%). Rawa Terate (very high vulnerability; flood and drought) was split between High (53.85%) and Low (46.15%) vulnerability, suggesting that exposure and resilience may coexist in nonlinear ways.

In North Jakarta, the influence of coastal hazards remained heterogeneous. Koja (very high vulnerability: flood, drought, and tidal flood) was dominated by High (76.92%) and Very Low vulnerability (23.08%), while Marunda (very high vulnerability: drought and tidal flood) was dominated by Low (92.31%) and Very Low (7.69%) vulnerability. Pluit (low vulnerability: tidal flood) was dominated by Low (76.92%), High (15.38%), and Very Low vulnerability (7.69%). Interestingly, Tanjung Priok (very high vulnerability: flood and tidal flood) showed 100.00% High, reinforcing the “dual pattern”: high exposure does not necessarily imply low resilience.

These inter-village variations confirm the relevance of a place-based approach for policy formulation. Strategies to enhance resilience must be tailored to the specific combination of risk and capacity at each location. Villages concentrated in the lowest classes (e.g., Kamal, Semanan, and North Petukangan) require prioritized investment in basic services, protective infrastructure, and household-level adaptive support. Conversely, villages that remain resilient under high exposure (e.g., Tanjung Priok and several Central Jakarta cases) can be treated as positive deviance sites to extract effective practices and enabling conditions that may be adapted and replicated elsewhere (Meerow et al., 2016; Norris et al., 2008).

After examining the spatial distribution of the HCRI categories, the local PC1 loadings were analyzed to identify the indicators that contributed most strongly to the Robust GWPCA-MCD-based HCRI structure at each location. While Section 3.7 shows where lower and higher resilience categories are located, this section explains which indicators form the dominant local resilience gradient. In the PCA-based index construction, component loadings represent the contribution of the indicators to the principal component, whereas component scores represent household positions in the component space.

The dominant local indicator was identified from the variable with the largest absolute aligned loading on PC1. The use of aligned loadings was important because the direction of PC1 was adjusted so that higher PC1 scores consistently represented higher resilience. In the diagnostic output, the top local loadings were reported together with the aligned loading values, absolute aligned loadings, and loading signs. The full output also stored the top three loadings for the Robust GWPCA-MCD and classical GWPCA as separate diagnostic sheets, allowing comparison between robust and non-robust local loading structures.

The results showed that the dominant local indicators were not uniform across the sampled urban villages. Several locations were dominated by exposure indicators, particularly C3b and C2b, which represented distance to flood sources. C3b measures the distance to other flood sources, whereas C2b measures the distance of the house to the main flood source. Other locations were dominated by the sensitivity indicators, particularly B3, B4, E14, and I5. B3 and B4 represent building and land areas, whereas E14 and I5 represent medication expenditure and the proportion of sick family members after the drought.

The local loading structure indicates that household resilience in Jakarta is mainly shaped by two dominant dimensions: exposure and sensitivity. Exposure-dominant areas include Jatinegara, Koja, Malaka Jaya, Marunda, Rawa Terate, and Tanjung Priok. In these locations, the local HCRI gradient is primarily associated with proximity to flood-related hazard sources. This suggests that the spatial position of households relative to flood sources is an important component of local resilience differentiation.

Sensitivity-dominant areas include Cilandak Timur, Duri Selatan, Kamal, Kampung Bali, Karet Tengsin, Krendang, Petukangan Utara, Pluit, Ragunan, Semanan, and Ulujami. At these locations, the dominant PC1 loadings were mainly associated with housing characteristics, land or building areas, medication expenditures, and health-related drought impacts. This pattern suggests that household resilience in these areas is more strongly differentiated by internal household conditions and post-hazard burden than by exposure distance alone.

At the household level, the most frequent dominant indicator was C3b, followed by B3, B4, E14, C2b, and I5. This distribution confirms that the local HCRI structure was not driven by a single indicator across the entire study area. Instead, different indicators became dominant in different local contexts. The presence of C2b and C3b as dominant indicators highlights the importance of flood-source proximity, whereas the dominance of B3 and B4 indicates the role of housing and settlement characteristics. Meanwhile, E14 and I5 show that health and post-disaster burdens also contribute to local differences in resilience.

The spatial visualization in Figure 7 provides a local interpretation of the Robust GWPCA-MCD results. Areas dominated by exposure indicators indicate locations where household resilience is primarily structured by spatial hazard proximity. Areas dominated by sensitivity indicators indicate locations where resilience differences are more closely related to household internal conditions, such as housing characteristics, health burdens, and economic consequences after climate-related events.

The dominance of the sensitivity indicators in many urban villages is important. It implies that household resilience is determined not only by whether households are located near or far from hazard sources but also by their capacity to absorb and manage climate-related impacts. In this sense, Robust GWPCA-MCD provides more than a spatial classification of resilience levels; it also identifies the local indicator structure underlying these resilience categories.

Overall, the local dominant loading analysis supports the main argument that household climate resilience is spatially heterogeneous. Different urban villages are characterized by different dominant indicators, indicating that adaptation priorities should not be generalized across all locations. Exposure-dominant areas may require interventions related to flood-risk reduction, drainage improvement, or spatial hazard management, whereas sensitivity-dominant areas may require attention to housing conditions, health protection, water access, and household recovery capacity. This interpretation provides the basis for linking the HCRI results to location-specific adaptation planning.

Robustness and sensitivity diagnostics were used to evaluate whether the Robust GWPCA-MCD-based HCRI was statistically stable and substantively interpretable. These diagnostics were not intended to introduce a new model but to support the reliability of the selected local HCRI specifications. The assessment focused on three aspects: bandwidth sensitivity, classification stability, and consistency of the covariance-based robust specification.

The bandwidth sensitivity results showed that the selected bandwidth of 96 nearest neighbors provided the most balanced specification. At this bandwidth, the median local PC1 variance was 83.66%, with 100% agreement in both the top-ranked local loading and the HCRI class relative to the main model. When the bandwidth was reduced to 86 neighbors, the model still produced a high median local PC1 variance of 80.86%, with 91.86% top-1 loading agreement and 85.97% HCRI class agreement. However, when the bandwidth was increased to 106 and 120 neighbors, the HCRI class agreement declined substantially to 61.09% and 44.80%, respectively.

This pattern indicates that higher local PC1 variance alone is insufficient to determine the best local model. Although wider bandwidths produced a slightly higher median local PC1 variance, they reduced the consistency of the HCRI classes and dominant loading patterns. This suggests that excessive spatial smoothing may weaken the local specificity of household resilience structures. Therefore, the selected bandwidth of 96 nearest neighbors was retained because it provided a balance between local variance explanation, loading stability, and HCRI classification consistency.

The classification stability diagnostic was also used to evaluate whether the final HCRI categories were sensitive to the classification rule. The main classification was based on quartile cutoffs and boxplot fences, whereas equal-interval and six-quantile classifications were computed as sensitivity alternatives. The output stored the main box-fence, equal-interval, and quantile-based classes for each household, allowing us to examine the stability of the resulting HCRI categories. This step is important because the PCA-based index classes are relatively empirical categories that depend on the observed score distribution rather than absolute resilience thresholds.

The covariance-based specification was also examined using a scale-related diagnostic. The main Robust GWPCA-MCD model used the original covariance-based scale, and a standardized/correlation-based diagnostic was generated for comparison. In the available output, both specifications produced the same bandwidth of 96 nearest neighbors, the same local PC1 variance summary, and 100% agreement in both the top-ranked loading and HCRI class. This result suggests that the main HCRI pattern was not altered by the standardized diagnostic in the implemented analysis. Nevertheless, the covariance-based specification remained the main model because this study aimed to preserve meaningful magnitude differences among household indicators after orientation and transformation.

Taken together, these diagnostics support the use of Robust GWPCA-MCD as the final local HCRI model. The bandwidth sensitivity results show that the selected bandwidth preserves local interpretability better than wider neighborhoods. The classification diagnostic confirms that the HCRI classes were evaluated against the alternative grouping rules. The scale-related diagnostic indicates that the main covariance-based HCRI pattern remained consistent under the standardized comparison. Thus, the final HCRI was justified not only by a high local PC1 variance but also by supporting diagnostics related to bandwidth, loading agreement, classification consistency, and scale sensitivity.

Overall, the robustness and sensitivity results strengthened the empirical basis of the Robust GWPCA-MCD-based HCRI. The final model captures spatial heterogeneity in household climate resilience while mitigating the influence of high-leverage multivariate observations through robust local covariance estimation. Therefore, the resulting HCRI classes and local dominant indicators can be interpreted as spatially specific household resilience profiles, rather than as artifacts of a single global covariance structure, arbitrary bandwidth selection, or a single classification rule.