सेक्टर ETF PCA: फ़ैक्टर समझना और लोडिंग स्थिरता जाँचना

# PCA on Sector ETFs with Bootstrap Loading Stability


# PCA on Sector ETFs with Bootstrap Loading Stability

**Docker image**: `ml4t`

**Chapter 14: Latent Factors**
**Section Reference**: See Section 14.2 for PCA theory and Section 14.3 for eigenportfolios

## Purpose
This notebook applies PCA to sector ETFs to extract latent risk factors and quantifies
loading stability using bootstrap resampling. We demonstrate how PCA captures market
and rotation factors, and how to assess whether factor loadings are statistically reliable.

## Where this fits in the framework

PCA is the simplest realisation of **Stage 1** in the two-step latent-factor
framework (Figure 14.9): it compresses an $(T \times N)$ returns panel to
a $(T \times K)$ factor history via SVD. This notebook focuses on the
Stage 1 outputs - variance decomposition, loadings, bootstrap stability -
and the structural interpretation of the principal components. The full
Stage 1 + 2 + 3 pipeline (with a Stage 2 forecaster turning factor history
into asset signals) is exercised in [`04_ipca`](04_ipca.ipynb) and
[`05_rp_pca`](05_rp_pca.ipynb).

## Learning Objectives
- LO1: Apply PCA to sector ETF returns and interpret variance decomposition
- LO2: Quantify loading stability with bootstrap confidence intervals
- LO3: Interpret sector factor exposures (market vs rotation)
- LO4: Analyze temporal stability of factor structure using rolling PCA

## Cross-References
- **Upstream**: Chapter 3 (ETF data loading)
- **Downstream**: Chapter 16 (factor investing), Chapter 18 (factor-based backtesting)
- **Related**: [`02_eigenportfolios`](02_eigenportfolios.ipynb) (stock-level PCA), Section 9.5 (Regime features)

## Data Source
ETF Universe parquet (canonical data, no API calls)

**Prerequisites**: Requires ETF Universe data (see Chapter 3).

## 1. Setup and Imports

```python
"""PCA on Sector ETFs - Variance decomposition and bootstrap loading stability."""

from datetime import date

import matplotlib.pyplot as plt
import numpy as np
import pandas as pd
import plotly.graph_objects as go
import polars as pl
from matplotlib.ticker import PercentFormatter
from plotly.subplots import make_subplots
from scipy.optimize import linear_sum_assignment
from sklearn.decomposition import PCA
from sklearn.preprocessing import StandardScaler

from data import load_etfs
from utils.reproducibility import set_global_seeds
from utils.style import (
    COLORS,
    FIGSIZE,
    add_message_title,
    show_plotly_with_alt,
    show_with_alt,
    zero_line,
)
```

```python
# Production defaults (Papermill overrides for CI testing)
START_DATE = "2010-01-01"
END_DATE = "2024-12-01"
N_BOOTSTRAP = 100
BLOCK_LENGTH = 20
SEED = 42
```

```python
set_global_seeds(SEED)
rng = np.random.default_rng(SEED)

SECTOR_ETFS = {
    "XLB": "Materials",
    "XLE": "Energy",
    "XLF": "Financials",
    "XLI": "Industrials",
    "XLK": "Technology",
    "XLP": "Staples",
    "XLU": "Utilities",
    "XLV": "Healthcare",
    "XLY": "Discretionary",
}

print(
    f"PCA Sector: {len(SECTOR_ETFS)} ETFs, {START_DATE} to {END_DATE}, {N_BOOTSTRAP} bootstrap samples"
)
```

## 2. Load Sector ETF Data

Load sector ETFs from the canonical ETF Universe parquet file.

```python
etf_data = load_etfs()

# Filter to sector ETFs and date range
tickers = list(SECTOR_ETFS.keys())
start_dt = date.fromisoformat(START_DATE)
end_dt = date.fromisoformat(END_DATE)
sector_data = (
    etf_data.filter(pl.col("symbol").is_in(tickers))
    .filter(pl.col("timestamp") >= start_dt)
    .filter(pl.col("timestamp") <= end_dt)
    .select(["timestamp", "symbol", "close"])
    .sort(["timestamp", "symbol"])
)

# Pivot to wide format
prices = (
    sector_data.pivot(on="symbol", index="timestamp", values="close")
    .sort("timestamp")
    .to_pandas()
    .set_index("timestamp")
)

# Ensure column order matches SECTOR_ETFS
prices = prices[[t for t in tickers if t in prices.columns]]

# Clean data
prices = prices.dropna(how="all").ffill()

# Calculate returns
returns = prices.pct_change().dropna()
print(
    f"Loaded: {prices.shape[0]} days, {prices.shape[1]} sectors, {returns.shape[0]} return observations"
)
```

Daily return distributions reveal whether a few volatile sectors could dominate covariance-PCA.
The interquartile ranges and whiskers make the scale differences visible without a printed
`describe()` table.

```python
fig, ax = plt.subplots(figsize=FIGSIZE["single_wide"], constrained_layout=True)
ax.boxplot(
    (returns * 100).to_numpy(),
    tick_labels=returns.columns,
    showfliers=False,
    boxprops={"color": COLORS["blue"]},
    medianprops={"color": COLORS["amber"], "linewidth": 1.5},
    whiskerprops={"color": COLORS["neutral"]},
    capprops={"color": COLORS["neutral"]},
)
zero_line(ax)
ax.set_xlabel("Sector ETF")
ax.set_ylabel("Daily return (%)")
add_message_title(
    ax,
    "Daily return distribution by sector ETF",
    subtitle="Outliers hidden so the central ranges are comparable",
)
show_with_alt(
    fig,
    "A vertical box plot with one box per sector ETF along the horizontal axis and "
    "daily return in percent on the vertical, with outliers hidden and a dashed line "
    "at zero. The height of each box is that sector's interquartile range and the "
    "whiskers give its wider central spread, so the boxes can be compared against each "
    "other for scale.",
)
```

## 3. PCA on Sector Returns

We use **correlation-PCA** (standardized returns) rather than covariance-PCA. Standardizing
to unit variance prevents high-volatility sectors (e.g., Energy) from dominating the first
component. For cross-sectional equity analysis this is the standard choice - see the Scale
Sensitivity discussion in Section 14.2.

PCA decomposes the covariance matrix as:

$$\Sigma = V \Lambda V^T$$

where $V$ contains eigenvectors (loadings) and $\Lambda$ is the diagonal matrix of
eigenvalues.

```python
# Standardize returns (correlation-PCA)
scaler = StandardScaler()
returns_scaled = scaler.fit_transform(returns)

# Fit PCA - default solver is appropriate for small N (9 sectors);
# use svd_solver='randomized' for N > 500
n_components = min(len(SECTOR_ETFS), 5)
pca = PCA(n_components=n_components, svd_solver="full")
factors = pca.fit_transform(returns_scaled)

# Create factor DataFrame
factor_cols = [f"PC{i + 1}" for i in range(n_components)]
factors_df = pd.DataFrame(factors, index=returns.index, columns=factor_cols)
```

### Variance Decomposition

The fraction of total variance explained by component $k$ is:

$$\text{VE}_k = \frac{\lambda_k}{\sum_{i=1}^{N} \lambda_i}$$

```python
cum_var = np.cumsum(pca.explained_variance_ratio_)
for i, (share, cumulative) in enumerate(
    zip(pca.explained_variance_ratio_, cum_var, strict=True), 1
):
    print(f"PC{i}: {share:6.1%} of variance, {cumulative:6.1%} cumulative")
```

### Scree Plot

The scree plot (Figure 14.2 in the text) shows eigenvalues in descending order. The
"elbow" where explained variance levels off suggests how many components to retain.

```python
fig, ax = plt.subplots(figsize=FIGSIZE["single"], constrained_layout=True)

# Bar chart of individual variance shares
ax.bar(
    range(1, n_components + 1),
    pca.explained_variance_ratio_,
    alpha=0.7,
    color=COLORS["blue"],
    label="Individual",
)

# Cumulative line on same axis
ax.plot(
    range(1, n_components + 1),
    cum_var,
    "o-",
    color=COLORS["amber"],
    label="Cumulative",
)

ax.set_xlabel("Principal Component")
ax.set_ylabel("Variance Explained")
ax.yaxis.set_major_formatter(PercentFormatter(1.0))
ax.set_xticks(range(1, n_components + 1))
ax.legend()
add_message_title(
    ax,
    "Variance explained per principal component",
    subtitle="Correlation-PCA on the sector ETF panel",
)
show_with_alt(
    fig,
    "A bar chart with component number on the horizontal axis and share of variance "
    "explained, in percent, on the vertical. Each bar is one component's own share, "
    "ordered from largest to smallest, and a line over the bars gives the running "
    "cumulative share.",
)
```

**Reading it**: the height of the first bar against the rest is the question. One
component carrying most of the variance is what a market factor looks like in a
sector panel - the sectors move together, and how much they move together is the
first bar.

Where the cumulative line flattens, each further component is adding little to the
variance already accounted for, and the "elbow" is a retention heuristic built on
that. It is a heuristic and not a finding: a component explaining little variance can
still carry systematic structure, and variance explained says nothing about whether a
direction is priced or predictive. Nothing in this notebook tests either. The printed
shares above the chart are the numbers; the shape is what the figure is for.

## 4. Loadings Interpretation

Each eigenvector defines a principal component as a linear combination of sector returns.
The loadings can be interpreted as portfolio weights.

```python
loadings = pd.DataFrame(
    pca.components_.T,
    index=returns.columns,
    columns=factor_cols,
)
loadings["Sector"] = loadings.index.map(SECTOR_ETFS)
```

**Interpretation**: PC1 loadings are uniformly positive - the classic "market factor" where
all sectors move together. PC2 separates defensive sectors (positive loadings: Utilities,
Staples) from cyclical sectors (negative loadings: Energy, Financials, Discretionary),
capturing the sector rotation dimension.

## 5. Bootstrap Loading Stability

Are the loadings statistically reliable, or could they be estimation noise? We use a moving-block
bootstrap to construct 95% confidence intervals while preserving short-run return dependence.
The default `N_BOOTSTRAP=100` is a teaching budget; increase it for final inference.

### Component Matching

PCA signs are arbitrary, and nearby eigenvalues can swap component order across resamples. We use
maximum absolute loading similarity to match each bootstrap component to the full-sample basis,
then align its sign.

```python
def align_components(loadings_new: np.ndarray, reference: np.ndarray) -> np.ndarray:
    """Match component order and signs to a reference loading basis."""
    if loadings_new.shape != reference.shape:
        raise ValueError("Candidate and reference loading bases must have the same shape")
    similarity = np.abs(reference.T @ loadings_new)
    reference_idx, new_idx = linear_sum_assignment(-similarity)
    aligned = np.zeros_like(loadings_new)
    for ref_col, candidate_col in zip(reference_idx, new_idx, strict=True):
        candidate = loadings_new[:, candidate_col]
        sign = np.sign(reference[:, ref_col] @ candidate) or 1.0
        aligned[:, ref_col] = sign * candidate
    return aligned
```

### Moving-Block Resamples

Twenty-day blocks retain local dependence while resampling the historical sequence. Each draw
concatenates random contiguous blocks until it reaches the original sample length.

```python
def moving_block_indices(n_obs: int, block_length: int, rng: np.random.Generator) -> np.ndarray:
    """Draw one moving-block bootstrap index of length `n_obs`."""
    n_blocks = int(np.ceil(n_obs / block_length))
    starts = rng.integers(0, n_obs - block_length + 1, size=n_blocks)
    return np.concatenate([np.arange(start, start + block_length) for start in starts])[:n_obs]
```

### Bootstrap Confidence Intervals

Each resample receives its own scaler and PCA fit. The component-matching step prevents a PC2/PC3
swap from being misread as loading uncertainty.

```python
def bootstrap_pca(
    returns_df: pd.DataFrame,
    reference_components: np.ndarray,
    n_bootstrap: int = 100,
    n_components: int = 2,
    block_length: int = 20,
    rng: np.random.Generator | None = None,
) -> np.ndarray:
    """Estimate loading uncertainty with a moving-block bootstrap."""
    if rng is None:
        rng = np.random.default_rng()
    n_obs = len(returns_df)
    n_features = returns_df.shape[1]
    bootstrap_loadings = np.zeros((n_bootstrap, n_features, n_components))

    for b in range(n_bootstrap):
        idx = moving_block_indices(n_obs, block_length, rng)
        sample = returns_df.iloc[idx]

        sample_scaled = StandardScaler().fit_transform(sample)
        pca_boot = PCA(n_components=n_components, svd_solver="full")
        pca_boot.fit(sample_scaled)

        loadings_boot = pca_boot.components_.T
        reference_loadings = reference_components[:n_components].T
        bootstrap_loadings[b] = align_components(loadings_boot, reference_loadings)

    return bootstrap_loadings
```

```python
bootstrap_results = bootstrap_pca(
    returns,
    reference_components=pca.components_,
    n_bootstrap=N_BOOTSTRAP,
    n_components=2,
    block_length=BLOCK_LENGTH,
    rng=rng,
)

# Confidence intervals
loading_mean = bootstrap_results.mean(axis=0)
loading_lower = np.percentile(bootstrap_results, 2.5, axis=0)
loading_upper = np.percentile(bootstrap_results, 97.5, axis=0)
```

## 6. Visualization: Loading Confidence Intervals

```python
fig = make_subplots(rows=1, cols=2, subplot_titles=("PC1: market", "PC2: sector rotation"))
for component_idx, color in enumerate((COLORS["blue"], COLORS["copper"])):
    order = loading_mean[:, component_idx].argsort()
    sector_names = [SECTOR_ETFS.get(returns.columns[i], returns.columns[i]) for i in order]
    fig.add_trace(
        go.Scatter(
            x=loading_mean[order, component_idx],
            y=sector_names,
            mode="markers",
            marker={"size": 10, "color": color},
            error_x={
                "type": "data",
                "symmetric": False,
                "array": loading_upper[order, component_idx] - loading_mean[order, component_idx],
                "arrayminus": loading_mean[order, component_idx]
                - loading_lower[order, component_idx],
                "color": COLORS["neutral"],
                "thickness": 1.5,
            },
            showlegend=False,
        ),
        row=1,
        col=component_idx + 1,
    )
    fig.add_vline(
        x=0,
        line_dash="dash",
        line_color=COLORS["neutral"],
        row=1,
        col=component_idx + 1,
    )
```

Both panels get the same loading range, so an interval's width means the same thing
on each and the two components can be compared by eye.

```python
fig.update_layout(
    title="Bootstrap loading intervals for the first two components",
    height=500,
)
loading_limit = 1.1 * max(abs(loading_lower.min()), abs(loading_upper.max()))
fig.update_xaxes(title_text="Loading", range=[-loading_limit, loading_limit])
fig.update_yaxes(title_text="Sector")
show_plotly_with_alt(
    fig,
    "Two panels, one per component, sharing a loading axis. Each row is a sector, "
    "drawn as its bootstrap point estimate with a horizontal confidence interval, "
    "sorted by loading, against a dashed vertical line at zero. The width of an "
    "interval is how much the resamples disagreed about that sector's loading.",
)
```

**Reading it**: two things are visible at once. Whether every sector falls on the same
side of zero on a component says whether that component is a common direction or a
contrast between groups, and which sectors sit at the two ends of a contrast says what
the contrast is between. Separately, the width of an interval says how much the
resamples disagreed - a loading whose interval spans zero is one the data does not
pin down, and building a portfolio on it is building on an estimate the bootstrap
cannot distinguish from no exposure at all.

## 7. Temporal Stability: Rolling PCA

Factor structure is not constant. Correlations often rise during stress, increasing PC1's share,
and relax in calm markets. Each point below fits a fresh correlation-PCA on the trailing 252
observations ending strictly before the plotted date.

```python
window = 252

rolling_var_explained = []

for i in range(window, len(returns)):
    window_returns = returns.iloc[i - window : i]

    scaled = StandardScaler().fit_transform(window_returns)
    pca_roll = PCA(n_components=2, svd_solver="full")
    pca_roll.fit(scaled)
    rolling_var_explained.append(pca_roll.explained_variance_ratio_)

rolling_var_explained = np.array(rolling_var_explained)

roll_dates = returns.index[window:]
```

Plot the two leading variance shares on the same scale so their relative importance remains
visually honest throughout the sample.

```python
fig = go.Figure()
fig.add_trace(
    go.Scatter(
        x=roll_dates, y=rolling_var_explained[:, 0], name="PC1", line=dict(color=COLORS["blue"])
    )
)
fig.add_trace(
    go.Scatter(
        x=roll_dates, y=rolling_var_explained[:, 1], name="PC2", line=dict(color=COLORS["copper"])
    )
)
fig.update_layout(
    title="Variance explained by the first two components, rolling window",
    xaxis_title="Date",
    yaxis_title="Variance Explained",
    yaxis_tickformat=".0%",
    height=400,
)
show_plotly_with_alt(
    fig,
    "Two lines against date, one per component, giving the share of variance each "
    "explains within a trailing window. The vertical axis is in percent. The first "
    "component's line moves over a wide range across the sample while the second's "
    "stays much lower.",
)
print(
    f"PC1 rolling variance explained: {rolling_var_explained[:, 0].min():.1%} to "
    f"{rolling_var_explained[:, 0].max():.1%}; PC2: "
    f"{rolling_var_explained[:, 1].min():.1%} to {rolling_var_explained[:, 1].max():.1%}"
)
```

**Reading it**: the first component's share is how much of the panel's variance sits
in a single linear direction, so the line is a picture of concentration over time rather
than of performance. That is not the same as the sectors moving the same way. A window
in which one group of sectors rises whenever another falls is still one direction, with
opposite-signed loadings, and the share stays high; what pulls the share down is
variance spreading across several independent directions. Telling the two apart needs
the loadings of the window itself, and this loop keeps only the variance shares - the
loadings in section 4 are fitted on the whole sample and cannot settle what any one
window was doing. So read a rise as concentration, not as co-movement, and note that
stress has no one signature here: a synchronized sell-off and a rotation can produce
the same share.

Every point is computed from the window ending at that date, so the line describes
what had already happened. Nothing here forecasts the next window's value.

## 8. Factor Score Analysis

PC scores from correlation-PCA are mean-zero standardized projections - they cannot be
compounded as portfolio returns because PCA centers the input. What they *can* do is
expose the cross-sectional structure each component captures. We rescale all PC scores by
a single constant so PC1's daily standard deviation matches the equal-weight sector
portfolio; this preserves the eigenvalue hierarchy ($\sigma_{\text{PC2}}/\sigma_{\text{PC1}} =
\sqrt{\lambda_2/\lambda_1}$) while putting PC1 on the same daily scale as the broad market.

```python
ew_daily_std = returns.mean(axis=1).std()
score_scale = ew_daily_std / factors_df["PC1"].std()
factor_returns = factors_df * score_scale
```

A component's daily volatility is $\sqrt{\lambda_k}$, so the ratio
$\sqrt{\lambda_k / \lambda_1}$ says how large each later component is beside the first.
The values are printed below rather than stated here, because they follow the
eigenvalues and change with the sample.

```python
volatility_ratio = np.sqrt(pca.explained_variance_ / pca.explained_variance_[0])
for i, ratio in enumerate(volatility_ratio, 1):
    print(f"PC{i} daily volatility as a share of PC1's: {ratio:.0%}")
```

Prepare two diagnostic series: PC1 vs the equal-weight market (both standardized to unit
variance so the scatter slope is the correlation), and the rolling 63-day PC1-PC2
correlation (in-sample orthogonality means the long-run mean is zero by construction).

```python
ew_returns = returns.mean(axis=1)
pc1_z = (factor_returns["PC1"] - factor_returns["PC1"].mean()) / factor_returns["PC1"].std()
ew_z = (ew_returns - ew_returns.mean()) / ew_returns.std()
roll_corr = factor_returns["PC1"].rolling(63).corr(factor_returns["PC2"])
pc1_market_corr = factor_returns["PC1"].corr(ew_returns)
scatter_min = min(ew_z.min(), pc1_z.min())
scatter_max = max(ew_z.max(), pc1_z.max())
```

Build the two-panel subplot grid: scatter on the left, time series on the right.

```python
fig = make_subplots(
    rows=1,
    cols=2,
    subplot_titles=(
        "PC1 vs Equal-Weight Market (standardized)",
        "Rolling 63d PC1-PC2 Correlation",
    ),
    column_widths=[0.45, 0.55],
    horizontal_spacing=0.15,
)
```

Add the scatter, 45° reference line, and rolling-correlation traces to the grid.

```python
fig.add_trace(
    go.Scatter(
        x=ew_z,
        y=pc1_z,
        mode="markers",
        marker=dict(size=3, color=COLORS["blue"], opacity=0.4),
        name="Daily",
        showlegend=False,
    ),
    row=1,
    col=1,
)
fig.add_trace(
    go.Scatter(
        x=[scatter_min, scatter_max],
        y=[scatter_min, scatter_max],
        mode="lines",
        line=dict(color=COLORS["neutral"], dash="dash"),
        name="45°",
        showlegend=False,
    ),
    row=1,
    col=1,
)
_ = fig.add_trace(
    go.Scatter(
        x=roll_corr.index,
        y=roll_corr,
        line=dict(color=COLORS["slate"]),
        name="PC1-PC2",
        showlegend=False,
    ),
    row=1,
    col=2,
)
```

Apply axis labels and honest correlation limits before rendering.

```python
fig.add_hline(y=0, line_dash="dash", line_color=COLORS["neutral"], row=1, col=2)
fig.update_xaxes(
    title_text="Equal-weight return (z)", range=[scatter_min, scatter_max], row=1, col=1
)
fig.update_yaxes(title_text="PC1 score (z)", range=[scatter_min, scatter_max], row=1, col=1)
fig.update_xaxes(title_text="Date", row=1, col=2)
fig.update_yaxes(title_text="63-day correlation", range=[-1, 1], row=1, col=2)
fig.update_layout(
    height=420,
    title_text="PC1 against the equal-weight market, and PC1-PC2 correlation over time",
)
show_plotly_with_alt(
    fig,
    "Two panels. The left scatters the first component's daily score against the "
    "equal-weight sector return, both standardized, on equal axes with a dashed "
    "diagonal; a cloud lying on that diagonal means the two are the same series up to "
    "scale. The right plots the rolling correlation between the first two components "
    "against date, on an axis from minus one to one with a line at zero.",
)
print(
    f"PC1 against equal-weight market: correlation {pc1_market_corr:.3f}. "
    f"Rolling PC1-PC2 correlation: mean {roll_corr.mean():+.3f}, "
    f"range {roll_corr.min():+.2f} to {roll_corr.max():+.2f}"
)
```

**Reading it**: the left panel asks what the first component *is*. If its score is
nearly the equal-weight return, then PCA has recovered the market and given it a name,
rather than found something the equal-weight portfolio does not already capture. The
printed correlation is the number; the diagonal is what makes it legible.

The right panel is about a property that is easy to over-trust. PCA makes the
components orthogonal *over the sample it was fitted on*, and that is a statement
about the whole period, not about any window inside it. The rolling correlation shows
how far short windows depart from zero. Anything that relies on the components being
uncorrelated - a risk decomposition, a hedge built from one against the other - needs
them re-fitted on the window it will be used in, because a single in-sample
decomposition does not deliver orthogonality inside that window.

## Key Takeaways

1. **Standardize before decomposing, or the loudest sector wins.** Correlation-PCA
   puts every sector on unit variance first, so the components describe co-movement
   rather than which sector happens to be most volatile. The box plot at the top is
   why: the sectors differ enough in spread for that choice to change the answer.
2. **A loading is an estimate, and the bootstrap says how good.** The interval, not
   the point, is what a portfolio decision should read. An interval spanning zero is
   an exposure the data does not establish.
3. **The first component's share measures concentration in one direction, and it
   moves.** A single full-sample number hides the movement, and the rolling window is
   what shows it. The share does not say which direction: a synchronized sell-off and a
   rotation that sets one group of sectors against another can both leave most of the
   variance in one component, and separating them needs that window's own loadings,
   which this notebook does not keep. Neither is "the stress signature".
4. **Orthogonality is a property of the fitting sample, not of every window in it.**
   The rolling correlation between the first two components shows how far a short
   window departs from zero, which is what anything built on their independence has to
   contend with.
5. **This is a description, not a forecast.** PCA on the full sample says what the
   covariance structure was. Turning that into a return prediction needs a second
   stage, which is the subject of `04_ipca` and `05_rp_pca`.

### PCA in the Two-Step Framework

Everything above is **Stage 1**. To turn it into a return forecast, a
Stage 2 factor-premium forecaster is required (see Figure 14.10 for the
full catalog). PCA with a sample-mean Stage 2 reduces to a per-asset
historical-mean predictor - a useful sanity-check baseline but not a
forecaster in any meaningful sense. Non-trivial cross-sectional ranking
emerges only when Stage 2 conditions on the factor path (AR(1), EWMA, or
richer ML forecasters). The IPCA notebook demonstrates the full pipeline
end-to-end.

**Next Steps**:
- For eigenportfolio construction on a broader stock universe, see [`02_eigenportfolios`](02_eigenportfolios.ipynb)
- For Stage 1 + 2 + 3 end-to-end with characteristic-driven loadings, see [`04_ipca`](04_ipca.ipynb)
- For production PCA with walk-forward CV on ETFs, see [`11_latent_factors`](../case_studies/etfs/11_latent_factors.ipynb)
![notebook output](figures/p1_1.png)
![notebook output](figures/p1_2.png)
![notebook output](figures/p1_3.png)
![notebook output](figures/p1_4.png)
![notebook output](figures/p1_5.png)