Add dataset augmentation and reduction utilities for SINDY workflow

This PR implements the feature requested in #332, adding support for:

1. **Delay embedding** (`delay_embedding`): Create time-delay coordinates from data,
   useful for Takens' embedding theorem and HAVOK analysis

2. **Hankel matrix construction** (`hankel_matrix`): Build Hankel matrices from
   1D time series, enabling Hankel DMD and related methods

3. **Dimensionality reduction** (`reduce_dimension`, `truncated_svd`): Project data
   onto principal components with automatic or manual rank selection

These utilities enable HAVOK-style workflows where:
- Time series data is embedded via delay coordinates or Hankel matrices
- SVD is used to identify dominant modes
- Sparse regression discovers dynamics in the reduced space

References:
- Brunton et al. (2017) "Chaos as an intermittently forced linear system"
- Arbabi & Mezic (2017) "Ergodic theory, dynamic mode decomposition,
  and computation of spectral properties of the Koopman operator"

Fixes #332

🤖 Generated with [Claude Code](https://claude.com/claude-code)

Co-Authored-By: Claude Opus 4.5 <noreply@anthropic.com>

Author: claude

src/DataDrivenDiffEq.jl

@@ -119,6 +119,10 @@ export collocate_data
 include("./utils/utils.jl")
 export optimal_shrinkage, optimal_shrinkage!
 
+include("./utils/data_augmentation.jl")
+export hankel_matrix, delay_embedding
+export truncated_svd, reduce_dimension
+
 include("./problem/type.jl")
 
 export DataDrivenProblem

src/utils/data_augmentation.jl

@@ -0,0 +1,271 @@
+"""
+Data augmentation and reduction utilities for the SINDY workflow.
+
+This module provides functions for:
+- Delay embedding (time-delay coordinates)
+- Hankel matrix construction
+- Dimensionality reduction via truncated SVD
+
+These are useful for HAVOK analysis and handling chaotic systems where
+time-delay embeddings can reveal underlying structure.
+
+# References
+
+- Brunton, S. L., Brunton, B. W., Proctor, J. L., Kaiser, E., & Kutz, J. N. (2017).
+  Chaos as an intermittently forced linear system. Nature Communications, 8(1), 19.
+  https://doi.org/10.1038/s41467-017-00030-8
+
+- Arbabi, H., & Mezic, I. (2017). Ergodic theory, dynamic mode decomposition,
+  and computation of spectral properties of the Koopman operator.
+  SIAM Journal on Applied Dynamical Systems, 16(4), 2096-2126.
+"""
+
+"""
+    $(SIGNATURES)
+
+Construct a Hankel matrix from a 1D time series vector.
+
+Given a vector `x = [x₁, x₂, ..., xₙ]` and `num_rows` rows, constructs the Hankel matrix:
+
+```
+H = [x₁    x₂    x₃   ... xₙ₋ₘ₊₁]
+    [x₂    x₃    x₄   ... xₙ₋ₘ₊₂]
+    [x₃    x₄    x₅   ... xₙ₋ₘ₊₃]
+    [⋮     ⋮     ⋮    ⋱  ⋮      ]
+    [xₘ    xₘ₊₁  xₘ₊₂ ... xₙ    ]
+```
+
+where `m = num_rows`.
+
+# Arguments
+- `x`: Input vector (1D time series)
+- `num_rows`: Number of rows in the Hankel matrix (number of delays + 1)
+
+# Returns
+- Hankel matrix of size `(num_rows, length(x) - num_rows + 1)`
+
+# Example
+```julia
+x = [1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0]
+H = hankel_matrix(x, 4)
+# Returns:
+# 4×7 Matrix:
+#  1.0  2.0  3.0  4.0  5.0  6.0  7.0
+#  2.0  3.0  4.0  5.0  6.0  7.0  8.0
+#  3.0  4.0  5.0  6.0  7.0  8.0  9.0
+#  4.0  5.0  6.0  7.0  8.0  9.0  10.0
+```
+"""
+function hankel_matrix(x::AbstractVector, num_rows::Integer)
+    n = length(x)
+    @assert num_rows >= 1 "num_rows must be at least 1"
+    @assert num_rows <= n "num_rows ($num_rows) cannot exceed length of x ($n)"
+
+    num_cols = n - num_rows + 1
+    H = similar(x, num_rows, num_cols)
+
+    for j in 1:num_cols
+        for i in 1:num_rows
+            H[i, j] = x[i + j - 1]
+        end
+    end
+
+    return H
+end
+
+"""
+    $(SIGNATURES)
+
+Create delay-embedded coordinates from a data matrix.
+
+Given data matrix `X` with states as rows and time samples as columns,
+creates new coordinates by stacking time-delayed versions of each state.
+
+For a single state `x = [x(t₁), x(t₂), ...]` with `num_delays` delays and
+delay interval `τ`, the embedded coordinates are:
+```
+[x(t),      x(t+τ),      ...]
+[x(t-τ),    x(t),        ...]
+[x(t-2τ),   x(t-τ),      ...]
+  ⋮           ⋮           ⋱
+[x(t-mτ),   x(t-(m-1)τ), ...]
+```
+
+where `m = num_delays`.
+
+# Arguments
+- `X`: Data matrix of size `(n_states, n_samples)`
+- `num_delays`: Number of time delays to add (resulting in `num_delays + 1`
+  copies of each state at different delays)
+
+# Keyword Arguments
+- `τ::Integer = 1`: Delay interval in samples (default: 1 sample)
+
+# Returns
+- Embedded data matrix of size `(n_states * (num_delays + 1), n_samples - num_delays * τ)`
+
+# Example
+```julia
+# Single state variable
+X = reshape(1.0:10.0, 1, 10)  # 1×10 matrix
+X_embed = delay_embedding(X, 2)
+# Creates 3×8 matrix with original and 2 delayed copies
+
+# Multiple state variables
+X = randn(3, 100)  # 3 states, 100 samples
+X_embed = delay_embedding(X, 3, τ=2)
+# Creates 12×94 matrix (3 states × 4 time copies)
+```
+
+See also: [`hankel_matrix`](@ref), [`reduce_dimension`](@ref)
+"""
+function delay_embedding(X::AbstractMatrix, num_delays::Integer; τ::Integer = 1)
+    n_states, n_samples = size(X)
+    @assert num_delays >= 0 "num_delays must be non-negative"
+    @assert τ >= 1 "delay interval τ must be at least 1"
+    @assert n_samples > num_delays * τ "Not enough samples for requested delays"
+
+    new_n_states = n_states * (num_delays + 1)
+    new_n_samples = n_samples - num_delays * τ
+
+    X_embedded = similar(X, new_n_states, new_n_samples)
+
+    for d in 0:num_delays
+        offset = num_delays * τ - d * τ
+        for i in 1:n_states
+            row_idx = d * n_states + i
+            for j in 1:new_n_samples
+                X_embedded[row_idx, j] = X[i, j + offset]
+            end
+        end
+    end
+
+    return X_embedded
+end
+
+"""
+    $(SIGNATURES)
+
+Delay embedding for a 1D time series vector.
+
+Convenience method that converts a vector to a 1-row matrix, applies
+delay embedding, and returns the result.
+
+# Arguments
+- `x`: Input vector (1D time series)
+- `num_delays`: Number of time delays to add
+
+# Keyword Arguments
+- `τ::Integer = 1`: Delay interval in samples
+
+# Returns
+- Embedded data matrix of size `(num_delays + 1, length(x) - num_delays * τ)`
+
+# Example
+```julia
+x = collect(1.0:10.0)
+X_embed = delay_embedding(x, 2)
+# Returns 3×8 matrix
+```
+"""
+function delay_embedding(x::AbstractVector, num_delays::Integer; τ::Integer = 1)
+    X = reshape(x, 1, length(x))
+    return delay_embedding(X, num_delays; τ = τ)
+end
+
+"""
+    $(SIGNATURES)
+
+Compute a truncated singular value decomposition of data matrix `X`.
+
+# Arguments
+- `X`: Data matrix
+- `rank`: Number of singular values/vectors to retain
+
+# Returns
+- Named tuple `(U = U_r, S = S_r, V = V_r)` where:
+  - `U_r` is the first `rank` columns of U
+  - `S_r` is a vector of the first `rank` singular values
+  - `V_r` is the first `rank` columns of V
+
+# Example
+```julia
+X = randn(100, 50)
+result = truncated_svd(X, 5)
+X_approx = result.U * Diagonal(result.S) * result.V'
+```
+
+See also: [`reduce_dimension`](@ref), [`optimal_shrinkage`](@ref)
+"""
+function truncated_svd(X::AbstractMatrix, rank::Integer)
+    @assert rank >= 1 "rank must be at least 1"
+    max_rank = minimum(size(X))
+    rank = min(rank, max_rank)
+
+    U, S, V = svd(X)
+    return (U = U[:, 1:rank], S = S[1:rank], V = V[:, 1:rank])
+end
+
+"""
+    $(SIGNATURES)
+
+Reduce the dimensionality of data by projecting onto the top singular vectors.
+
+Projects data matrix `X` onto its first `rank` left singular vectors,
+reducing the number of effective dimensions while preserving the most
+important features.
+
+# Arguments
+- `X`: Data matrix of size `(n_features, n_samples)`
+- `rank`: Number of dimensions to retain
+
+# Returns
+- Reduced data matrix of size `(rank, n_samples)`
+- The reduced coordinates `Y = U_r' * X` where `U_r` contains the first `rank` left singular vectors
+
+# Example
+```julia
+X = randn(100, 50)  # 100 features, 50 samples
+X_reduced = reduce_dimension(X, 5)  # Reduce to 5 dimensions
+# X_reduced is 5×50
+```
+
+See also: [`truncated_svd`](@ref), [`optimal_shrinkage`](@ref)
+"""
+function reduce_dimension(X::AbstractMatrix, rank::Integer)
+    result = truncated_svd(X, rank)
+    return result.U' * X
+end
+
+"""
+    $(SIGNATURES)
+
+Reduce the dimensionality of data using automatic rank selection.
+
+Uses the optimal singular value hard threshold from Gavish & Donoho (2014)
+to automatically determine the number of significant singular values,
+then projects onto those dimensions.
+
+# Arguments
+- `X`: Data matrix of size `(n_features, n_samples)`
+
+# Returns
+- Reduced data matrix with automatically selected dimensionality
+
+# Example
+```julia
+X = randn(100, 50) + 0.1 * randn(100, 50)  # Low-rank + noise
+X_reduced = reduce_dimension(X)  # Automatically selects rank
+```
+
+See also: [`reduce_dimension(X, rank)`](@ref), [`optimal_shrinkage`](@ref)
+"""
+function reduce_dimension(X::AbstractMatrix)
+    m, n = minimum(size(X)), maximum(size(X))
+    U, S, V = svd(X)
+    τ = optimal_svht(m, n)
+    threshold = τ * median(S)
+    rank = count(s -> s >= threshold, S)
+    rank = max(rank, 1)  # Keep at least one dimension
+    return U[:, 1:rank]' * X
+end