Notation Guide

Standard notation used throughout temporalcv documentation and code.

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Time Series Variables

Symbol	Python Variable	Meaning
y_t	y[t]	Target value at time t
ŷ_t	predictions[t]	Predicted value at time t
e_t	errors[t]	Forecast error: y_t - ŷ_t
Δy_t	changes[t]	First difference: y_t - y_{t-1}

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Forecast Parameters

Symbol	Python Parameter	Meaning	Typical Values
h	horizon	Forecast horizon (steps ahead)	1, 2, 4, 8
n	n or len(y)	Sample size	50-500
k	n_lags	Number of lagged features	3-10
w	window_size	Training window size	52-104 (weeks)

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Statistical Parameters

Symbol	Python Parameter	Meaning	Typical Values
α	alpha	Significance level (Type I error)	0.05, 0.10
1-α	coverage	Target coverage probability	0.90, 0.95
γ	gamma	Learning rate (adaptive conformal)	0.01-0.1

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AR(1) Process Parameters

Symbol	Python Variable	Meaning	Typical Values
φ	phi	Persistence coefficient	0.9-0.99
σ	sigma	Innovation standard deviation	Data-dependent
ε_t	epsilon[t]	Innovation (white noise)	N(0, 1)

AR(1) Process: y_t = φ × y_{t-1} + σ × ε_t

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Test Statistics

Symbol	Python Variable	Meaning
d_t	d[t]	Loss differential: L(e₁,ₜ) - L(e₂,ₜ)
d̄	d_bar	Mean loss differential
DM	dm_stat	Diebold-Mariano test statistic
PT	pt_stat	Pesaran-Timmermann test statistic
p̂	p_hat	Observed directional accuracy
p*	p_star	Expected accuracy under null

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Variance Estimators

Symbol	Python Variable	Meaning
γ_j	gamma[j]	Autocovariance at lag j
V̂(·)	var_hat	Estimated variance
w(j)	weight	Bartlett kernel weight

HAC Variance: V̂ = (1/n) × [γ₀ + 2 Σⱼ w(j) × γⱼ]

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Regime Classification

Symbol	Python Value	Meaning
τ	threshold	Move threshold
UP	'UP' or MoveDirection.UP	Upward move: value > τ
DOWN	'DOWN' or MoveDirection.DOWN	Downward move: value < -τ
FLAT	'FLAT' or MoveDirection.FLAT	No significant move: |value| ≤ τ
LOW	'LOW'	Low volatility regime
MED	'MED'	Medium volatility regime
HIGH	'HIGH'	High volatility regime

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Skill Scores

Symbol	Python Variable	Formula	Range
SS	skill_score	1 - (model_error / baseline_error)	(-∞, 1]
MC-SS	skill_score	1 - (model_MAE_moves / persistence_MAE_moves)	(-∞, 1]

Interpretation:

• SS > 0: Model beats baseline

• SS = 0: Model equals baseline

• SS < 0: Model worse than baseline

• SS = 1: Perfect forecast

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Conformal Prediction

Symbol	Python Variable	Meaning
s_i	scores[i]	Nonconformity score
q̂	quantile	Empirical quantile of scores
Ĉ(x)	prediction_interval	Conformal prediction set

Prediction Interval: Ĉ(x) = [ŷ(x) - q̂, ŷ(x) + q̂]

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Cross-Validation

Symbol	Python Variable	Meaning
K	n_splits	Number of CV splits
g	gap	Gap between train and test

Gap Requirement: train_end + gap < test_start

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Gate Results

Symbol	Python Value	Meaning	Action
HALT	GateStatus.HALT	Critical failure	Stop and investigate
WARN	GateStatus.WARN	Caution	Continue with verification
PASS	GateStatus.PASS	Validation passed	Proceed
SKIP	GateStatus.SKIP	Insufficient data	Cannot validate

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Common Constants

Value	Meaning	Source
√(2/π) ≈ 0.798	E[|Z|] for Z ~ N(0,1)	Standard result
0.70	Default move threshold percentile	v2 empirical
0.20	Suspicious improvement threshold	v2 empirical
13	Default volatility window (weeks)	Quarterly assumption
30	Minimum n for DM test	Convention
20	Minimum n for PT test	Convention

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Variable Naming Conventions

Pattern	Meaning	Example
*_train	Training data only	X_train, threshold_train
*_test	Test data only	y_test
*_moves	Computed on moves only	mae_moves
*_hat	Estimated quantity	p_hat, var_hat
*_star	Expected under null	p_star
n_*	Count of	n_up, n_splits

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Code-to-Math Mapping

# DM Test
d_bar = np.mean(d)           # d̄
var_d = compute_hac_variance(d)  # V̂(d̄)
dm_stat = d_bar / np.sqrt(var_d)  # DM

# PT Test
p_hat = np.mean(correct)     # p̂
p_star = p_y * p_x + (1-p_y) * (1-p_x)  # p*

# MC-SS
mc_ss = 1 - (model_mae_moves / persistence_mae_moves)  # MC-SS

# AR(1) Optimal MAE
optimal_mae = sigma * np.sqrt(2 / np.pi)  # σ × √(2/π)