Glossary

Oct 16, 2025 · 54 min read · reference HMM statistics finance mathematics trading ·

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Glossary

A comprehensive reference for statistical, mathematical, financial, and trading terms used throughout Hidden Regime. Each entry includes definitions, formulas (where applicable), interpretation guidance, use cases, and cross-references to related concepts.

Updated: October 2025 - Expanded with 51 new trading and risk management terms from Advanced Trading Applications series.

Quick Navigation: A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R | S | T | U | V | W

Section A

Akaike Information Criterion (AIC)

Definition: A measure of statistical model quality that balances goodness-of-fit against model complexity.

Formula:

$$\text{AIC} = 2k - 2\ln(\hat{L})$$

where $k$ is the number of parameters and $\hat{L}$ is the maximum likelihood.

Interpretation: Lower values are better. AIC penalizes model complexity to prevent overfitting. When comparing models, prefer the one with the lowest AIC. AIC tends to favor more complex models than BIC.

Use case: Model selection when prediction accuracy is the priority. In HMM context, use AIC to determine optimal number of states.

See also: BIC, Log Likelihood

Reference: Wikipedia - Akaike Information Criterion

Augmented Dickey-Fuller Test (ADF)

Definition: Statistical test to determine if a time series is stationary (does not have a unit root).

Hypothesis:

Null hypothesis ($H_0$): Series has a unit root (non-stationary)
Alternative ($H_1$): Series is stationary

Interpretation: Low p-value (< 0.05) indicates stationarity. If p < 0.05, reject the null hypothesis and conclude the series is stationary. This is the opposite interpretation of KPSS.

Use case: Must verify stationarity before applying HMMs or other time series models. Log returns should pass this test, raw prices should fail it.

See also: KPSS Test, Stationary, Phillips-Perron

Reference: Wikipedia - Augmented Dickey-Fuller Test

Autocorrelation

Definition: The correlation of a time series with a delayed copy of itself, measuring how current values relate to past values.

Formula:

$$\rho_k = \frac{\sum_{t=k+1}^T (y_t - \bar{y})(y_{t-k} - \bar{y})}{\sum_{t=1}^T (y_t - \bar{y})^2}$$

where $k$ is the lag.

Interpretation: Values range from -1 to +1. High autocorrelation at lag $k$ means past values are predictive of future values. Financial returns typically show low autocorrelation (close to zero), while volatility shows high autocorrelation (volatility clustering).

Use case: Identify patterns in time series data. Significant autocorrelation in returns violates efficient market hypothesis.

See also: Time Series, Stationary

Annualized Return

Definition: The yearly rate of return calculated from returns over a period other than one year, allowing comparison across different time horizons.

Formula:

$$R_{\text{annual}} = R_{\text{daily}} \times 252$$

where 252 is the number of trading days per year. For multi-period returns:

$$R_{\text{annual}} = (1 + R_{\text{total}})^{\frac{252}{n}} - 1$$

where $n$ is the number of days in the period.

Interpretation: Annualization scales returns to a common yearly basis for fair comparison. A daily return of 0.1% annualizes to approximately 25.2% per year (0.1% × 252).

Important: For mean returns, use arithmetic scaling (multiply by 252). For volatility, use square root scaling (multiply by $\sqrt{252}$).

Use case: Essential for comparing strategies with different time horizons. HMM regime returns are typically annualized for interpretability.

See also: Log Returns, Volatility, Sharpe Ratio

Section B

Backtesting

Definition: The process of testing a trading strategy on historical data to evaluate its performance before deploying it with real capital.

Process:

Define strategy rules (entry/exit signals)
Apply rules to historical price data
Calculate hypothetical returns and risk metrics
Analyze performance and drawdowns

Key metrics evaluated:

Total and annualized returns
Sharpe ratio
Maximum drawdown
Win rate
Transaction costs impact

Interpretation: Past performance does not guarantee future results. Backtesting shows how a strategy would have performed, not how it will perform.

Pitfalls:

Overfitting to historical data
Look-ahead bias (using future information)
Survivorship bias (only testing on surviving assets)
Ignoring transaction costs and slippage

Use case: Essential validation step before live trading. For HMM strategies, backtest regime-based signals on out-of-sample data.

Reference: Read our Advanced Trading Applications article

Backward Variable (β)

Definition: In the forward-backward algorithm for HMMs, the backward variable represents the probability of observing the remaining sequence given the current state.

Formula:

$$\beta_t(i) = P(o_{t+1}, o_{t+2}, \ldots, o_T | q_t = s_i, \lambda)$$

where:

$o_{t+1}, \ldots, o_T$ are future observations
$q_t = s_i$ is being in state $i$ at time $t$
$\lambda$ is the HMM model parameters

Recursion:

$$\beta_t(i) = \sum_{j=1}^N a_{ij} b_j(o_{t+1}) \beta_{t+1}(j)$$

Interpretation: Backward variables are computed recursively from the end of the sequence backwards to the beginning. Combined with forward variables, they enable smoothing - estimating state probabilities using the full observation sequence.

Use case: Core component of forward-backward algorithm. Used in Baum-Welch training and for computing state probabilities at each time step.

Basis Points (bps)

Definition: One hundredth of a percentage point (0.01%).

Formula:

$$1 \text{ bp} = 0.0001 = 0.01\%$$

Interpretation: Used in finance to describe small percentage changes. 100 basis points = 1%.

Example: If interest rates rise from 5.00% to 5.25%, they increased by 25 basis points.

Reference: Investopedia - Basis Points

Baum-Welch Algorithm

Definition: An Expectation-Maximization (EM) algorithm for training Hidden Markov Model parameters by maximizing the likelihood of observed data.

Purpose: Estimates the transition matrix $A$, emission parameters $B$, and initial state distribution $\pi$ that best explain the observations.

What it does: Iteratively refines model parameters until convergence. Does NOT determine state sequences - that’s Viterbi’s job.

Process:

E-step: Compute forward and backward probabilities
M-step: Update parameters based on expected state occupancy
Repeat until log-likelihood converges

Interpretation: Convergence typically occurs in < 100 iterations. If not converging, may indicate poor initialization or model misspecification.

Use case: Training HMMs on historical data. Always run Baum-Welch before using Viterbi for prediction.

Reference: Read our HMM Algorithms Explained article

Bayesian Uncertainty Quantification (UQ)

Definition: Characterization of uncertainty in model predictions and parameters using Bayesian probability theory.

Key principle: Uncertainty is expressed as probability distributions over parameters rather than point estimates.

Use case: Quantifying confidence in regime predictions. In HMMs, provides probability distributions over states rather than hard classifications.

Bear Market

Definition: A market condition where prices generally decrease, typically defined as a 20% decline from recent highs.

Characteristics: Negative returns, increasing volatility, risk-off sentiment

Use case: One of the fundamental market regimes detected by HMMs. Often characterized by high volatility and negative mean returns.

See also: Bull Market, Regime, Crisis

Bayesian Information Criterion (BIC)

Definition: A measure of statistical model quality that penalizes complexity more heavily than AIC.

Formula:

$$\text{BIC} = k\ln(n) - 2\ln(\hat{L})$$

where $k$ is the number of parameters, $n$ is sample size, and $\hat{L}$ is maximum likelihood.

Interpretation: Lower values are better. BIC penalizes model complexity more than AIC (logarithmic vs. linear penalty). Tends to favor simpler models with fewer states.

Rule of thumb: Use BIC when model interpretability matters and you want to avoid overfitting. In HMM context, BIC often suggests fewer states than AIC.

Example: For HMM model selection, if 2-state model has BIC = 1500 and 3-state model has BIC = 1520, prefer the 2-state model (lower is better).

See also: AIC, Log Likelihood

Reference: Wikipedia - Bayesian Information Criterion

Bollinger Bands

Definition: A technical indicator consisting of a moving average and two bands (upper and lower) placed at standard deviations above and below the average, used to identify overbought and oversold conditions.

Formula:

$$\text{Middle Band} = SMA_{20}$$

$$\text{Upper Band} = SMA_{20} + 2\sigma_{20}$$

$$\text{Lower Band} = SMA_{20} - 2\sigma_{20}$$

where $SMA_{20}$ is the 20-period simple moving average and $\sigma_{20}$ is the 20-period standard deviation.

Interpretation:

Price near upper band: Potentially overbought, reversal down possible
Price near lower band: Potentially oversold, bounce up possible
Band width: Narrow bands = low volatility, wide bands = high volatility
Bollinger squeeze: Very narrow bands often precede large moves

Trading signals:

Price touching upper band in uptrend → continuation signal
Price touching lower band in downtrend → continuation signal
Mean reversion: Price outside bands tends to return to middle

Use case: Volatility measurement and overbought/oversold identification. In HMM context, Bollinger Band signals can confirm regime transitions or contradict them.

Reference: Investopedia - Bollinger Bands

Bull Market

Definition: A market condition where prices generally increase, characterized by positive returns and investor optimism.

Characteristics: Positive mean returns, relatively low volatility, risk-on sentiment

Use case: One of the fundamental market regimes detected by HMMs. Typically has different emission characteristics than bear or sideways regimes.

See also: Bear Market, Regime, Euphoric

Buy and Hold

Definition: A passive investment strategy where an investor purchases securities and holds them for a long period regardless of market fluctuations.

Philosophy: Based on the belief that markets trend upward over long periods despite short-term volatility. Minimizes trading costs and tax implications.

Advantages:

Low transaction costs (minimal trading)
Simple to implement and maintain
Tax-efficient (long-term capital gains)
Historically effective for broad market indices

Disadvantages:

Full exposure during market downturns
No risk management during crises
Ignores regime changes
Maximum drawdowns can be severe

Performance baseline: Buy and hold typically serves as the benchmark against which active strategies are measured.

Use case: In backtesting HMM strategies, buy-and-hold provides the baseline performance. Regime-based strategies aim to outperform by reducing exposure during bear markets.

Reference: Read our Advanced Trading Applications article

Section C

Confidence Level

Definition: In risk measurement, the probability threshold used to calculate Value at Risk (VaR) and other tail risk metrics.

Common levels:

95% confidence: Captures events that occur 1 in 20 days (once per month)
99% confidence: Captures events that occur 1 in 100 days (2-3 times per year)
99.9% confidence: Captures rare extreme events

Interpretation: A 95% confidence VaR of -$10,000 means: "We expect losses will not exceed $10,000 on 95% of days."

Trade-off: Higher confidence levels capture more extreme events but have wider intervals and less statistical precision.

Use case: Setting risk limits and position sizing. Higher confidence levels appropriate for risk-averse strategies. In HMM context, regime confidence is different - it’s the probability of being in a particular state.

See also: Value at Risk, Expected Shortfall, Tail Risk

Reference: Wikipedia - Confidence Interval

Correlation

Definition: A statistical measure describing the linear relationship between two random variables.

Formula:

$$\rho_{X,Y} = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y} = \frac{\sum_i (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_i (x_i - \bar{x})^2} \sqrt{\sum_i (y_i - \bar{y})^2}}$$

Interpretation:

$\rho = 1$: Perfect positive correlation
$\rho = 0$: No linear correlation
$\rho = -1$: Perfect negative correlation

Important: Correlation measures LINEAR relationships only. Assets can be strongly related nonlinearly while showing zero correlation.

Use case: Measuring co-movement between assets. Correlation often increases during crisis regimes (correlation breakdown).

See also: Standard Deviation, Volatility

Reference: Wikipedia - Correlation

Crisis Regime

Definition: A market condition where prices are decreasing sharply, characterized by extreme volatility and panic selling.

Characteristics: Large negative returns, very high volatility, fat tails in return distribution

Statistical signature: High negative mean, very high standard deviation, often brief duration but severe impact

Use case: One of the most important regimes to detect for risk management. HMMs can identify crisis regimes and trigger defensive portfolio adjustments.

See also: Bear Market, Regime, Volatility

Cumulative Return

Definition: The total return on an investment over a specified period, accounting for compounding of gains and losses.

Formula (from log returns):

$$R_{\text{cumulative}} = \exp\left(\sum_{t=1}^T r_t\right) - 1$$

where $r_t$ are log returns.

Formula (from simple returns):

$$R_{\text{cumulative}} = \prod_{t=1}^T (1 + r_t) - 1$$

Interpretation: A cumulative return of 0.50 (or 50%) means the investment grew by 50% over the period. Unlike individual period returns, cumulative returns show the total wealth creation.

Visualization: Cumulative return charts show portfolio growth over time, making it easy to identify drawdown periods and overall performance.

Use case: Primary metric for backtesting and performance evaluation. Compare cumulative returns of buy-and-hold vs regime-based strategies.

Section D

Downtrend

Definition: A market condition where prices make a series of lower highs and lower lows over time, indicating sustained selling pressure.

Characteristics:

Each price peak is lower than the previous peak
Each price trough is lower than the previous trough
Moving averages slope downward
Negative momentum

Technical identification:

Price consistently below moving averages
Sequence of lower highs and lower lows
Negative MACD and RSI in lower ranges

Trading implications:

Trend-following strategies take short positions
Mean-reversion strategies wait for reversal signals
HMM may identify as Bear regime

Use case: Identifying directional bias in markets. Downtrends often persist longer than expected, making trend-following profitable.

See also: Uptrend, Bear Market, Momentum, Trend-Following

Drawdown

Definition: The decline in value from a peak to a trough in an investment or portfolio, expressed as a percentage.

Formula:

$$\text{DD}_t = \frac{V_t - \max_{\tau \leq t} V_\tau}{\max_{\tau \leq t} V_\tau}$$

where $V_t$ is portfolio value at time $t$ and $\max_{\tau \leq t} V_\tau$ is the running maximum.

Interpretation: A drawdown of -20% means the portfolio has declined 20% from its peak. Drawdowns are always negative or zero.

Recovery time: Number of periods needed to return to previous peak. Long recovery times indicate severe market stress.

Use case: Essential risk metric for portfolio management. Large drawdowns psychologically difficult to tolerate and may force liquidation. HMM regimes help predict drawdown risk.

Reference: Investopedia - Drawdown

Section E

Exponential Moving Average (EMA)

Definition: A type of moving average that gives more weight to recent prices, making it more responsive to new information than a simple moving average.

Formula:

$$EMA_t = \alpha \cdot P_t + (1-\alpha) \cdot EMA_{t-1}$$

where:

$P_t$ is the current price
$\alpha = \frac{2}{n+1}$ is the smoothing factor
$n$ is the period (e.g., 12-day, 26-day)

Characteristics:

More responsive to recent price changes than SMA
Reduces lag compared to simple moving average
Commonly used in MACD indicator (EMA12 - EMA26)

Common periods:

EMA(12): Short-term trend
EMA(26): Medium-term trend
EMA(50), EMA(200): Long-term trends

Interpretation:

Price > EMA → Uptrend
Price < EMA → Downtrend
EMA crossovers generate trading signals

Use case: Trend identification and signal generation. MACD uses difference between EMA(12) and EMA(26). Can confirm HMM regime transitions.

Reference: Investopedia - Exponential Moving Average

Expectation-Maximization (EM)

Definition: An iterative algorithm for finding maximum likelihood estimates of parameters in models with latent (hidden) variables.

Process:

E-step: Compute expected value of log-likelihood with respect to current parameter estimates
M-step: Maximize this expectation to find new parameter estimates
Repeat until convergence

Key property: Guarantees log-likelihood improvement at each iteration (monotonic convergence).

Use case: Baum-Welch is an EM algorithm specialized for HMMs. Used whenever you have hidden variables that need to be marginalized out.

Emission Matrix (B)

Definition: Matrix of probabilities defining how likely each observation is given each hidden state in an HMM.

Formula:

$$b_j(k) = P(o_k | s_j)$$

where $o_k$ is observation $k$ and $s_j$ is state $j$.

For Gaussian emissions:

$$b_j(x) = \frac{1}{\sqrt{2\pi\sigma_j^2}} \exp\left(-\frac{(x-\mu_j)^2}{2\sigma_j^2}\right)$$

Interpretation: Each state has a characteristic distribution of outputs. In financial HMMs, each regime (state) has a typical mean return and volatility (emission parameters).

Example: Bull regime might emit returns from $\mathcal{N}(0.001, 0.01)$ (positive mean, low volatility), while Crisis regime emits from $\mathcal{N}(-0.003, 0.03)$ (negative mean, high volatility).

See also: Transition Matrix, HMM, Baum-Welch Algorithm

Euphoric Regime

Definition: A market condition where prices are increasing sharply, characterized by excessive optimism and potentially unsustainable gains.

Characteristics: Very high positive returns, increasing volatility, often precedes corrections

Statistical signature: High positive mean return, elevated volatility, may show unsustainable trends

Use case: Important regime for risk management - detecting euphoria can signal profit-taking opportunities before reversal.

See also: Bull Market, Crisis, Regime

Expected Shortfall (ES)

Definition: The average loss that occurs beyond the Value at Risk (VaR) threshold, providing a more complete picture of tail risk than VaR alone. Also known as Conditional Value at Risk (CVaR).

Formula:

$$ES_\alpha = E[r \mid r \leq -VaR_\alpha] = -\frac{1}{\alpha} \int_0^\alpha VaR_p \, dp$$

where $\alpha$ is the confidence level (e.g., 0.05 for 95% confidence).

Interpretation: If VaR(95%) = -$10,000, and ES(95%) = -$15,000, this means: on the worst 5% of days, the average loss is $15,000 (not just $10,000).

Why ES > VaR:

ES captures tail severity: VaR only gives threshold, ES gives average beyond
ES is coherent: Satisfies all axioms of coherent risk measures
ES for position sizing: Better than VaR for capital allocation

Example:

VaR(95%) = -2%: Losses exceed 2% on 5% of days
ES(95%) = -3%: When losses exceed VaR, average loss is 3%

Use case: Superior to VaR for risk management and position sizing. In HMM context, calculate regime-specific ES to size positions appropriately.

Reference: Wikipedia - Expected Shortfall

Section F

Forward-Backward Algorithm

Definition: An algorithm that computes the probability of an observation sequence given an HMM, and the probability of being in each state at each time step.

Purpose: Solves the “evaluation problem” - what is $P(\text{observations} | \lambda)$?

Components:

Forward variable $\alpha_t(i) = P(o_1, ..., o_t, q_t = s_i | \lambda)$: Probability of observing sequence up to time $t$ and being in state $i$
Backward variable $\beta_t(i) = P(o_{t+1}, ..., o_T | q_t = s_i, \lambda)$: Probability of observing remaining sequence given state $i$ at time $t$

Use case: Core component of Baum-Welch algorithm. Also used for smoothing - estimating state probabilities at all time points given full observation sequence.

See also: Baum-Welch Algorithm, Viterbi Algorithm, HMM

Reference: Read our HMM Algorithms Explained article

Forward Variable (α)

Definition: In the forward-backward algorithm for HMMs, the forward variable represents the probability of observing the sequence up to time $t$ and being in state $i$ at time $t$.

Formula:

$$\alpha_t(i) = P(o_1, o_2, \ldots, o_t, q_t = s_i | \lambda)$$

where:

$o_1, \ldots, o_t$ are observations up to time $t$
$q_t = s_i$ is being in state $i$ at time $t$
$\lambda$ is the HMM model parameters

Recursion:

$$\alpha_t(i) = \left[\sum_{j=1}^N \alpha_{t-1}(j) a_{ji}\right] b_i(o_t)$$

Initialization:

$$\alpha_1(i) = \pi_i b_i(o_1)$$

Interpretation: Forward variables are computed recursively from the beginning of the sequence forward. They accumulate evidence for each state based on all observations seen so far.

Use case: Core component of forward-backward algorithm. Used to compute the likelihood $P(O|\lambda)$ and for filtering - estimating the current state given observations so far.

Section G

Geometric Brownian Motion (GBM)

Definition: A continuous-time stochastic process where the logarithm of the variable follows a Brownian motion with drift.

Formula:

$$dS_t = \mu S_t dt + \sigma S_t dW_t$$

where $\mu$ is drift, $\sigma$ is volatility, and $W_t$ is a Wiener process.

Key property: Prices $S_t$ are log-normally distributed. This implies log returns are normally distributed (approximately).

Use case: Classical model for stock price movements. Foundation for Black-Scholes option pricing. HMMs extend GBM by allowing parameters to switch between regimes.

See also: Log Returns, Volatility

Section H

Heteroskedasticity

Definition: The property of a time series where the variance (volatility) changes over time rather than remaining constant.

Contrast with homoskedasticity: Constant variance over time.

Formula (ARCH test): Tests if $\sigma_t^2$ varies with time.

Interpretation: Financial returns exhibit strong heteroskedasticity - volatility clusters (high volatility followed by high volatility, low by low). This is why volatility modeling (GARCH) is important.

Use case: Violates assumptions of basic linear regression. HMMs naturally handle heteroskedasticity by allowing different volatility in different states.

See also: Volatility, Standard Deviation

Reference: Investopedia - Heteroskedasticity

Histogram

Definition: A graphical representation showing the frequency distribution of data by dividing it into bins.

Use case: Visual tool for identifying distribution shape. Use to check if returns are approximately normal, or if fat tails/skewness are present.

Interpretation: Bell-shaped histogram suggests normal distribution. Fat tails indicate kurtosis. Asymmetry indicates skewness.

See also: Q-Q Plot, Kurtosis, Skewness

Reference: Tableau - What is a Histogram

Hidden Markov Model (HMM)

Definition: A statistical model for systems with hidden (unobservable) states, where state transitions and observations are probabilistic.

Five components: $\lambda = (S, O, A, B, \pi)$

$S$: Hidden states (e.g., Bull, Bear, Sideways)
$O$: Observations (e.g., daily returns)
$A$: Transition matrix (state-to-state probabilities)
$B$: Emission matrix (observation probabilities given state)
$\pi$: Initial state distribution

Key assumptions:

Markov property: $P(q_t | q_{t-1}, ..., q_1) = P(q_t | q_{t-1})$
Output independence: $P(o_t | q_t, o_{t-1}, ...) = P(o_t | q_t)$

Three fundamental problems:

Evaluation: $P(\text{observations} | \lambda)$ - solved by Forward-Backward
Decoding: Most likely state sequence - solved by Viterbi
Learning: Find best $\lambda$ - solved by Baum-Welch

Use case: Detecting market regimes from price data. The “hidden” part is the market regime (bull/bear/crisis), and the “observations” are daily returns.

Reference: Read our Guide to HMMs and HMM 101 articles

Section I

In-Sample

Definition: The dataset used to train, fit, or optimize a model’s parameters.

Purpose: Develop and calibrate model by finding parameters that best explain historical patterns.

Characteristics:

Used for parameter estimation
Model has “seen” this data
Performance metrics may be optimistic
Risk of overfitting if model too complex

Typical split: 70-80% of data for in-sample training.

Interpretation: In-sample performance represents how well the model explains known data, not how well it predicts new data.

Pitfalls:

Overfitting: High in-sample performance, poor out-of-sample
Data snooping: Repeatedly optimizing on same data
Look-ahead bias: Using future information in training

Use case: For HMMs, train model parameters (transition matrix, emissions) on in-sample data, then validate on out-of-sample.

Initial State Distribution (π)

Definition: The probability distribution over hidden states at the start of the observation sequence.

Formula:

$$\pi_i = P(q_1 = s_i)$$

where $q_1$ is the state at time $t=1$.

Constraint: $\sum_i \pi_i = 1$

Interpretation: Represents prior belief about which state the system starts in. Often initialized uniformly ($\pi_i = 1/N$) if no prior knowledge.

Use case: One of three parameter sets estimated by Baum-Welch. Less critical than transition/emission matrices for long sequences.

See also: Transition Matrix, Emission Matrix, HMM

Inverse CDF (Quantile Function)

Definition: The inverse of the cumulative distribution function (CDF), mapping probabilities to values. Given a probability $p$, returns the value $x$ such that $P(X \leq x) = p$.

Formula:

$$F^{-1}(p) = x \text{ such that } F(x) = p$$

where $F$ is the CDF.

Interpretation: For a 95% confidence level (p=0.95), $F^{-1}(0.95)$ gives the value below which 95% of the data falls.

Common use - VaR calculation:

$$\text{VaR}_\alpha = -F^{-1}(\alpha)$$

For example, VaR at 5% level uses the inverse CDF at p=0.05.

Relationship to quantiles: The inverse CDF is the quantile function. The pth quantile is $F^{-1}(p)$.

Use case: Essential for Value at Risk calculations and generating random samples from distributions. In HMM context, use regime-specific inverse CDF to calculate tail risk.

See also: Value at Risk, Quantile, Confidence Level

Reference: Wikipedia - Quantile Function

Section J

Jarque-Bera Test

Definition: A statistical test for normality based on sample skewness and kurtosis.

Formula:

$$JB = \frac{n}{6}\left(S^2 + \frac{(K-3)^2}{4}\right)$$

where $S$ is skewness, $K$ is kurtosis, and $n$ is sample size.

Hypothesis:

Null ($H_0$): Data is normally distributed
Alternative ($H_1$): Data is not normal

Interpretation: Low p-value (< 0.05) rejects normality. Financial returns typically fail this test due to fat tails and skewness.

Use case: Testing distributional assumptions. HMMs can still work with non-normal data if mixture of Gaussian states approximates the true distribution.

See also: Kurtosis, Skewness

Reference: Wikipedia - Jarque-Bera Test

Section K

KPSS Test

Definition: A statistical test for stationarity (Kwiatkowski-Phillips-Schmidt-Shin test).

Hypothesis:

Null ($H_0$): Series is stationary
Alternative ($H_1$): Series has a unit root (non-stationary)

Interpretation: High p-value (> 0.05) indicates stationarity. This is the OPPOSITE of ADF interpretation. For robust stationarity testing, use both ADF and KPSS - series should reject null in ADF and fail to reject in KPSS.

Use case: Complementary to ADF test. Log returns should pass both tests (stationary), prices should fail both (non-stationary).

See also: ADF Test, Stationary

Reference: Wikipedia - KPSS Test

Kolmogorov-Smirnov Test (KS Test)

Definition: A non-parametric statistical test that compares two probability distributions to determine if they differ significantly. Can compare a sample distribution to a reference distribution, or compare two sample distributions.

Test Statistic:

$$D = \sup_x |F_1(x) - F_2(x)|$$

where $F_1$ and $F_2$ are the empirical cumulative distribution functions (CDFs) of the two samples, and $\sup$ denotes the supremum (maximum distance).

Hypothesis:

Null ($H_0$): The two distributions are the same
Alternative ($H_1$): The distributions are different

Interpretation: Low p-value (< 0.05) indicates distributions are significantly different. The test measures the maximum vertical distance between two CDFs. Larger distances (higher D statistic) indicate more different distributions.

Advantages:

Non-parametric (makes no assumptions about distribution shape)
Sensitive to differences in both location and shape
Easy to interpret (maximum distance between CDFs)

Use case: Common in regime detection to validate that market behavior changed between periods. For example, comparing return distributions before and after a crisis to quantify paradigm shifts. Also used to test if returns follow a normal distribution.

See also: Time Series, Stationary, Jarque-Bera Test

Reference: Wikipedia - Kolmogorov-Smirnov Test

Kurtosis (Fat Tails)

Definition: A measure of the “tailedness” of a probability distribution, indicating the frequency of extreme values.

Formula:

$$\text{Kurt}[X] = \frac{E[(X-\mu)^4]}{(\sigma^2)^2}$$

Excess kurtosis: $\text{Kurt}[X] - 3$ (normal distribution has kurtosis = 3)

Interpretation:

Excess kurtosis = 0: Normal distribution (mesokurtic)
Excess kurtosis > 0: Fatter tails than normal (leptokurtic) - more extreme events
Excess kurtosis < 0: Thinner tails than normal (platykurtic)

Financial reality: Stock returns typically have excess kurtosis between 3-20, meaning extreme events (crashes, rallies) occur much more frequently than normal distribution predicts.

Use case: Assessing tail risk. High kurtosis indicates need for robust risk management. HMMs can capture fat tails through mixture of states.

See also: Jarque-Bera Test, Skewness

Reference: Investopedia - Kurtosis

Section L

Leverage

Definition: The use of borrowed capital or financial derivatives to amplify potential returns (and losses) from an investment.

Formula:

$$\text{Leverage Ratio} = \frac{\text{Total Position Size}}{\text{Equity}}$$

Types:

2x leverage: $100 equity controls $200 position
Margin: Borrowing from broker
Derivatives: Options, futures provide built-in leverage

Risk magnification: Leverage multiplies both gains AND losses.

Example:

Without leverage: 10% gain = $10 profit on $100
With 2x leverage: 10% gain = $20 profit, but 10% loss = -$20 (20% of equity)

Margin call: If losses reduce equity below maintenance requirement, broker may force liquidation.

Use case: Can enhance returns in favorable regimes, but dramatically increases risk. HMM regime detection can inform when to use leverage (Bull regimes) vs when to reduce (Bear/Crisis).

Important: Leverage can lead to total loss of capital. Use cautiously with proper risk management.

Reference: Investopedia - Leverage

Long Position

Definition: Owning an asset with the expectation that its price will increase, profiting from upward price movement.

Mechanics:

Buy asset at price $P_0$
Hold asset
Sell at higher price $P_1$
Profit = $P_1 - P_0$ (minus costs)

Characteristics:

Maximum loss: Purchase price (if asset goes to zero)
Maximum gain: Unlimited (theoretically)
Cash flow: Requires upfront capital
Holding period: Can be indefinite

In returns: Long position profits when returns are positive.

Use case: Default position in buy-and-hold strategies. HMM strategies take long positions in Bull regimes, exit to cash in Bear regimes.

Log Likelihood

Definition: The natural logarithm of the likelihood function, measuring how well a statistical model fits observed data.

Formula:

$$\ell(\theta) = \log L(\theta | x) = \sum_{i=1}^n \log f(x_i | \theta)$$

where $f(x_i | \theta)$ is the probability density function.

Why use log:

Converts products to sums (numerically stable)
Easier to optimize (differentiable)
Comparable across different sample sizes

Interpretation: Higher values indicate better fit. Note: Often reported as negative log-likelihood where lower is better.

Use case: Core metric in Baum-Welch algorithm. Monitors convergence - training stops when log-likelihood increase falls below threshold.

See also: AIC, BIC, Maximum Likelihood Estimation

Reference: StatLect - Log Likelihood

Log Returns

Definition: The logarithmic change in price from one period to the next.

Formula:

$$R_t = \log\left(\frac{P_t}{P_{t-1}}\right) = \log(P_t) - \log(P_{t-1})$$

Key properties:

Time-additive: $R_{1 \to T} = R_1 + R_2 + ... + R_T$
Stationary: Mean and variance don’t change over time
Scale-invariant: Comparable across different price levels
Symmetric: +10% and -10% have equal magnitude in log space
Approximately normal for small returns

Converting to percentage: $\%\text{return} = (e^R - 1) \times 100$

Why use instead of simple returns:

Required for statistical modeling (stationarity)
Mathematically elegant (additivity)
Small return approximation: $\log(1+r) \approx r$

Use case: ALWAYS use log returns for HMMs and time series analysis. Raw prices violate stationarity assumption.

See also: Simple Returns, Stationary, Scale Invariance

Reference: Read our Why Use Log Returns? article

Log Space

Definition: Representing values using their logarithms rather than original scale.

Purpose:

Converts multiplicative relationships to additive
Compresses large ranges (makes visualization easier)
Stabilizes variance

Example: Price $P_t$ in log space is $\log(P_t)$. Returns in log space are $\log(P_t/P_{t-1})$.

Use case: Financial modeling uses log space because price changes are multiplicative (percentage-based) not additive (dollar-based).

See also: Log Returns

Section M

MACD (Moving Average Convergence Divergence)

Definition: A trend-following momentum indicator that shows the relationship between two exponential moving averages of an asset’s price.

Formula:

$$\text{MACD Line} = EMA_{12} - EMA_{26}$$

$$\text{Signal Line} = EMA_9(\text{MACD})$$

$$\text{Histogram} = \text{MACD Line} - \text{Signal Line}$$

Components:

MACD Line: Difference between fast (12) and slow (26) EMAs
Signal Line: 9-period EMA of MACD line
Histogram: Visual representation of divergence

Trading signals:

Bullish: MACD crosses above signal line
Bearish: MACD crosses below signal line
Divergence: Price makes new high/low but MACD doesn’t (reversal signal)

Interpretation:

MACD > 0: Upward momentum
MACD < 0: Downward momentum
Increasing histogram: Strengthening trend
Decreasing histogram: Weakening trend

Use case: Trend identification and momentum measurement. Can confirm HMM regime transitions or provide early warning signals.

See also: EMA, Moving Average, Momentum, Trend-Following

Reference: Investopedia - MACD

Markov Property

Definition: The property that the conditional probability distribution of future states depends only on the present state, not on the sequence of events that preceded it.

Formula:

$$P(X_{t+1} = x | X_t = x_t, X_{t-1} = x_{t-1}, ..., X_0 = x_0) = P(X_{t+1} = x | X_t = x_t)$$

Interpretation: “The future is independent of the past given the present.” No “memory” beyond current state.

First-order Markov: Depends only on previous state (most HMMs) Higher-order Markov: Depends on multiple previous states (more complex)

Use case: Core assumption of HMMs. Makes computation tractable. Reasonable approximation for many financial processes.

See also: HMM, Random Walk

Reference: Wikipedia - Markov Property

Maximum Drawdown (MDD)

Definition: The largest peak-to-trough decline in portfolio value over a specified time period, representing the worst possible loss an investor would have experienced.

Formula:

$$\text{MDD} = \max_{t} \left[ \frac{\max_{\tau \leq t} V_\tau - V_t}{\max_{\tau \leq t} V_\tau} \right]$$

where $V_t$ is portfolio value at time $t$.

Interpretation: An MDD of 30% means the portfolio declined 30% from its peak at some point. This is the worst historical drawdown.

Key metric: MDD shows the maximum pain an investor would have endured, which affects psychology and ability to stay invested.

Recovery factor: $\frac{\text{Total Return}}{\text{MDD}}$ measures return per unit of maximum risk taken.

Acceptable levels:

< 10%: Low risk, conservative
10-20%: Moderate risk
20-30%: High risk
> 30%: Very high risk, may cause panic selling

Use case: Essential for risk assessment and strategy evaluation. Regime-based strategies aim to reduce MDD by exiting during Bear/Crisis regimes.

See also: Drawdown, Sharpe Ratio, Risk Management, Backtesting

Reference: Investopedia - Maximum Drawdown

Mean-Reversion

Definition: A trading strategy based on the theory that prices tend to return to their average over time after deviating from it.

Core hypothesis: Extreme price movements are temporary and will reverse toward the mean.

Mathematical basis: If a time series is stationary with mean $\mu$, deviations from $\mu$ should be temporary.

Trading approach:

Buy when price significantly below mean
Sell when price significantly above mean
Uses indicators like Bollinger Bands, RSI

When it works:

Sideways/ranging markets
Stationary price series
Over-reactions to news
High-frequency trading

When it fails:

Strong trends (Bull or Bear regimes)
Structural changes in asset fundamentals
Regime shifts

Contrast with trend-following: Mean-reversion bets on reversal, trend-following bets on continuation.

Use case: Effective in Sideways regimes detected by HMMs. Switch to trend-following in Bull/Bear regimes.

See also: Trend-Following, Sideways, Reversal, Overbought, Oversold

Reference: Investopedia - Mean Reversion

Momentum

Definition: The rate of acceleration of an asset’s price or volume, representing the strength of a price trend.

Concept: Assets that have performed well recently tend to continue performing well in the near term (momentum effect).

Measurement:

Rate of change: $(P_t - P_{t-n}) / P_{t-n}$
RSI: Relative strength over period
MACD: Difference between moving averages

Momentum trading:

Positive momentum: Buy (expect continuation)
Negative momentum: Sell or short
Momentum reversal: Extreme momentum may signal reversal

Characteristics:

Works in trends: Bull and Bear regimes
Fails in ranges: Sideways markets
Time-dependent: Momentum measured over specific period (e.g., 12-month)

Psychological basis: Herding behavior and delayed reaction to information cause momentum.

Use case: Confirms regime strength. Strong momentum in Bull regime = high confidence. Weakening momentum may signal regime transition.

See also: Trend-Following, RSI, MACD, Reversal

Reference: Investopedia - Momentum

Moving Average (MA)

Definition: The average price of an asset over a specified number of periods, updated as new data becomes available.

Types:

Simple Moving Average (SMA): Arithmetic mean of prices
Exponential Moving Average (EMA): Weighted mean favoring recent prices

Common periods:

MA(20): Short-term trend
MA(50): Medium-term trend
MA(200): Long-term trend

Trading signals:

Price crosses above MA: Bullish signal
Price crosses below MA: Bearish signal
Golden cross: MA(50) crosses above MA(200) - bullish
Death cross: MA(50) crosses below MA(200) - bearish

Characteristics:

Lags price: Moving averages smooth data but delay signals
Trend identification: Direction of MA shows trend
Support/resistance: MA can act as dynamic support or resistance levels

Use case: Simple trend identification. Can confirm HMM regime transitions. Price crossing MA may signal regime change.

See also: SMA, EMA, MACD, Bollinger Bands, Trend-Following

Reference: Investopedia - Moving Average

Maximum Likelihood Estimation (MLE)

Definition: A method for estimating parameters of a statistical model by maximizing the likelihood function.

Goal: Find parameters $\theta^*$ that maximize $P(\text{data} | \theta)$

Formula:

$$\theta^* = \arg\max_\theta L(\theta | x) = \arg\max_\theta \log L(\theta | x)$$

Use case: Baum-Welch is an MLE algorithm for HMMs. Finds transition and emission parameters that best explain observed data.

Advantages: Statistically efficient, asymptotically unbiased Disadvantages: Can overfit with small samples, may need regularization

Section N

Section O

Out-of-Sample Testing

Definition: Evaluating a model’s performance on data that was NOT used during training, providing an unbiased estimate of predictive performance.

Purpose: Test if model generalizes to new data or has merely overfit to training data.

Process:

Split data into training (in-sample) and test (out-of-sample) sets
Train model on in-sample data only
Evaluate model on out-of-sample data
Compare in-sample vs out-of-sample performance

Typical split: 70-30 or 80-20 (training-test)

Red flags:

Large performance gap: In-sample much better than out-of-sample → overfitting
Negative out-of-sample Sharpe: Strategy doesn’t generalize
Increasing gap over time: Model becoming stale

Gold standard: Out-of-sample performance determines real-world viability, not in-sample performance.

Use case: Critical validation for HMM trading strategies. Model must perform reasonably on unseen data before live trading.

Overbought

Definition: A condition where an asset’s price has risen significantly and may be due for a correction or reversal downward.

Technical indicators:

RSI > 70: Traditionally considered overbought
Price > upper Bollinger Band: Extended beyond normal range
Stochastic > 80: Momentum indicator shows overbought

Interpretation: Overbought doesn’t mean “immediately sell”. In strong uptrends (Bull regimes), assets can remain overbought for extended periods.

Two perspectives:

Mean-reversion: Overbought = sell signal (expect reversal)
Trend-following: Overbought in uptrend = strength (continue holding)

Use case: Context matters. In Sideways regimes, overbought is sell signal. In Bull regimes, may be continuation signal.

See also: Oversold, RSI, Bollinger Bands, Mean-Reversion

Reference: Investopedia - Overbought

Oversold

Definition: A condition where an asset’s price has fallen significantly and may be due for a bounce or reversal upward.

Technical indicators:

RSI < 30: Traditionally considered oversold
Price < lower Bollinger Band: Extended below normal range
Stochastic < 20: Momentum indicator shows oversold

Interpretation: Oversold doesn’t mean “immediately buy”. In strong downtrends (Bear regimes), assets can remain oversold for extended periods.

Two perspectives:

Mean-reversion: Oversold = buy signal (expect bounce)
Trend-following: Oversold in downtrend = weakness (stay out or short)

“Catching a falling knife”: Buying oversold assets in Bear regimes can lead to further losses.

Use case: Context matters. In Sideways regimes, oversold is buy signal. In Bear/Crisis regimes, may indicate continued weakness.

See also: Overbought, RSI, Bollinger Bands, Mean-Reversion

Reference: Investopedia - Oversold

Overfitting

Definition: Creating a model that is too complex and fits the training data too closely, capturing noise rather than signal, resulting in poor performance on new data.

Symptoms:

High in-sample performance
Poor out-of-sample performance
Too many parameters relative to data points
Model is overly sensitive to small data changes

Causes:

Too many states in HMM (5+ states often overfit)
Too many features relative to sample size
Over-optimization of parameters
Data snooping: repeatedly testing on same data

Prevention:

Use simpler models (2-4 states for HMMs)
Out-of-sample validation
Walk-forward analysis
Regularization techniques
Larger datasets

Detection: Compare in-sample vs out-of-sample Sharpe ratios. Large gap indicates overfitting.

Trade-off: Balance between underfitting (too simple) and overfitting (too complex).

Use case: Major risk in HMM regime detection. Prefer 3-state models over 5+ state models unless clear evidence supports complexity.

Reference: Wikipedia - Overfitting

Section P

Paper Trading

Definition: Simulated trading using real market prices but virtual (fake) money, allowing traders to test strategies without financial risk.

Purpose: Validate strategy in real-time market conditions before risking capital.

Benefits:

Risk-free learning: Practice without losing money
Real-time validation: Test strategy on live market data
Psychology practice: Experience emotional aspects of trading
System debugging: Identify technical issues

Limitations:

No real money psychology: Fear and greed feel different with real capital
Perfect execution: May not capture slippage and market impact
Selection bias: Tendency to be more conservative or aggressive than with real money

Best practices:

Paper trade for 1-3 months minimum
Use realistic position sizes
Include transaction costs and slippage
Track emotional responses
Don’t cherry-pick results

Use case: Essential step between backtesting and live trading. All HMM strategies should be paper traded before real deployment.

See also: Backtesting, Transaction Costs, Slippage

Reference: Investopedia - Paper Trading

Phillips-Perron Test

Definition: A unit root test for stationarity, similar to ADF but more robust to heteroskedasticity and autocorrelation.

Advantage over ADF: Uses non-parametric correction for serial correlation, making it more robust to volatility clustering.

Interpretation: Same as ADF - low p-value indicates stationarity.

Use case: Alternative to ADF when you suspect heteroskedasticity (which is common in financial data).

See also: ADF Test, KPSS Test, Heteroskedasticity

Reference: Wikipedia - Phillips-Perron Test

Position Sizing

Definition: The process of determining how much capital to allocate to a particular trade or investment, balancing potential returns against risk.

Common methods:

Fixed dollar: Risk same dollar amount per trade
Fixed percentage: Risk same % of capital per trade (e.g., 2%)
Volatility-based: Scale position by $1/\sigma$ (inverse volatility)
Kelly Criterion: $f^* = \frac{p \cdot b - q}{b}$ where $p$ = win rate, $b$ = win/loss ratio, $q = 1-p$

Regime-based sizing:

$$\text{Position} = \text{Base} \times \frac{\sigma_{\text{target}}}{\sigma_{\text{regime}}} \times \text{Confidence} \times f(\text{Sharpe})$$

Key principles:

Never risk more than 1-2% of capital on single trade
Scale positions inversely with volatility
Reduce size in low-confidence situations
Consider correlation across positions

Common mistakes:

Oversizing: Risking too much per trade
Ignoring volatility: Same position size regardless of regime risk
Ignoring correlation: Multiple correlated positions = concentrated risk

Use case: Critical for HMM trading strategies. Adjust position size based on regime volatility and confidence levels.

Reference: Investopedia - Position Sizing

Profit Factor

Definition: The ratio of gross profits to gross losses over a trading period, measuring the profitability of a trading strategy.

Formula:

$$\text{Profit Factor} = \frac{\sum \text{Winning Trades}}{\sum |\text{Losing Trades}|}$$

Interpretation:

PF > 1: Strategy is profitable
PF = 1: Break-even
PF < 1: Losing strategy
PF > 2: Strong strategy
PF > 3: Excellent strategy

Example:

Total wins: $10,000
Total losses: $4,000
Profit factor: 10,000 / 4,000 = 2.5

Relationship to win rate:

High win rate + low profit factor → Many small wins, few large losses
Low win rate + high profit factor → Few large wins, many small losses

Use case: Evaluate trading strategy quality. Compare profit factors across different regimes. High profit factor in Bull regime validates regime-based approach.

See also: Win Rate, Sharpe Ratio, Backtesting

Reference: Investopedia - Profit Factor

Section Q

Q-Q Plot (Quantile-Quantile Plot)

Definition: A graphical tool for assessing if a dataset follows a theoretical distribution (usually normal) by plotting quantiles against each other.

Interpretation:

Points on diagonal line → Data matches theoretical distribution
Points above line → Data has heavier right tail
Points below line → Data has heavier left tail
S-shaped curve → Data has both heavy tails (common for returns)

Use case: Visual test for normality. Complements statistical tests (Jarque-Bera, Shapiro-Wilk). Financial returns typically show S-curve indicating fat tails.

See also: Histogram, Jarque-Bera Test, Kurtosis

Reference: UVA - Understanding Q-Q Plots

Quantile

Definition: A value below which a given percentage of observations in a distribution fall. The pth quantile is the value $x$ such that $P(X \leq x) = p$.

Common quantiles:

0.25 quantile: 25th percentile (Q1, first quartile)
0.50 quantile: 50th percentile (median, Q2)
0.75 quantile: 75th percentile (Q3, third quartile)
0.95 quantile: 95th percentile (used in VaR)

Formula:

$$Q(p) = F^{-1}(p)$$

where $F^{-1}$ is the inverse CDF.

Interpretation: The 95th quantile is the value that separates the bottom 95% from the top 5% of the distribution.

Use in risk management:

VaR at 5% level: Negative of 5th percentile (0.05 quantile)
Tail risk: Focus on extreme quantiles (0.01, 0.05, 0.95, 0.99)

Interquartile range (IQR): $Q(0.75) - Q(0.25)$ measures dispersion.

Use case: Essential for tail risk measurement and VaR calculation. HMM regimes have different quantile structures - Crisis regimes have extreme lower quantiles.

See also: Inverse CDF, Value at Risk, Q-Q Plot

Reference: Wikipedia - Quantile

Section R

Random Walk

Definition: A stochastic process where each step is a random deviation from the previous value.

Formula:

$$X_t = X_{t-1} + \epsilon_t$$

where $\epsilon_t$ is random noise (often $\mathcal{N}(0, \sigma^2)$).

Key property: Non-stationary - variance increases over time. Cumulative sum of stationary process.

Efficient Market Hypothesis: Prices follow random walk because all information is immediately incorporated.

Use case: Null hypothesis for many market models. If HMM detects regime structure, it provides evidence against pure random walk.

See also: Stationary, ADF Test, Markov Property

Reference: Wikipedia - Random Walk

Reversal

Definition: A change in the direction of a price trend, from upward to downward or vice versa.

Types:

Bullish reversal: Change from downtrend to uptrend
Bearish reversal: Change from uptrend to downtrend
V-shaped reversal: Sharp, sudden reversal
Rounded reversal: Gradual trend change

Technical signals:

Overbought/oversold conditions reversing
MACD crossovers in opposite direction
Price breaking through support/resistance
Divergence between price and momentum indicators

vs Retracement: Reversals are trend changes; retracements are temporary pullbacks within ongoing trend.

HMM perspective: Regime transitions (Bull → Bear) are reversals. HMMs can detect regime transitions before traditional indicators.

Use case: Identifying reversals early provides trading opportunities. Exit longs before bearish reversals, enter longs after bullish reversals.

See also: Momentum, Overbought, Oversold, Regime, Trend-Following

Regime

Definition: A hidden state in an HMM representing a distinct market condition with characteristic statistical properties.

Common financial regimes:

Bull: Positive returns, low-moderate volatility
Bear: Negative returns, moderate-high volatility
Sideways: Near-zero returns, low-moderate volatility
Crisis: Large negative returns, very high volatility
Euphoric: Large positive returns, high volatility

Statistical characterization: Each regime has distinct mean return and volatility (emission parameters).

Use case: The hidden states we’re trying to detect with HMMs. Goal is to identify current regime and predict transitions.

See also: HMM, Bull Market, Bear Market, Crisis, Euphoric, Sideways

Risk-Adjusted Return

Definition: Investment returns scaled by the amount of risk taken to achieve them, allowing fair comparison across strategies with different risk profiles.

Common measures:

Sharpe Ratio: Excess return per unit of total risk (volatility)
Sortino Ratio: Excess return per unit of downside risk
Calmar Ratio: Annual return / maximum drawdown
Information Ratio: Excess return / tracking error

Why adjust for risk: A 20% return with 50% volatility is inferior to 15% return with 10% volatility.

Formula (general):

$$\text{Risk-Adjusted Return} = \frac{\text{Return} - \text{Risk-Free Rate}}{\text{Risk Measure}}$$

Use case: Essential for comparing HMM regime-based strategies against buy-and-hold. Focus on Sharpe ratio improvement, not just absolute returns.

See also: Sharpe Ratio, Volatility, Maximum Drawdown

Reference: Investopedia - Risk-Adjusted Return

Risk Budget

Definition: The maximum acceptable amount of risk (measured in dollars, volatility, VaR, or drawdown) that a portfolio or trading strategy is allowed to take.

Common approaches:

Dollar risk: Max $X loss per trade
Volatility budget: Target Y% annual volatility
VaR budget: No more than Z% VaR
Drawdown limit: Stop trading if drawdown exceeds W%

Example: Risk budget of 2% per trade means: with $100,000 capital, max loss per trade = $2,000.

Allocation: Distribute risk budget across:

Multiple strategies
Multiple assets
Multiple regimes

Use case: Core risk management tool. In HMM context, allocate more risk budget to high-Sharpe regimes (Bull), less to low-Sharpe regimes (Bear).

Risk-Free Rate

Definition: The theoretical return on an investment with zero risk, typically approximated by government bonds (e.g., US Treasury bills).

Common proxies:

3-month T-Bill: Short-term risk-free rate
10-year Treasury: Long-term risk-free rate
0%: Often used for simplicity in calculations

Use in Sharpe ratio:

$$\text{Sharpe} = \frac{R_p - R_f}{\sigma_p}$$

where $R_f$ is the risk-free rate.

Reality check: No investment is truly “risk-free” (even governments can default), but T-bills are closest approximation.

Current environment: Risk-free rate changes over time. Higher risk-free rates raise the bar for strategy performance.

Use case: Baseline for evaluating investment returns. Strategies must beat risk-free rate to be worthwhile (after adjusting for risk).

See also: Sharpe Ratio, Risk-Adjusted Return

RSI (Relative Strength Index)

Definition: A momentum oscillator measuring the speed and magnitude of price changes, used to identify overbought and oversold conditions.

Formula:

$$RSI = 100 - \frac{100}{1 + RS}$$

where $RS = \frac{\text{Average Gain over } n \text{ periods}}{\text{Average Loss over } n \text{ periods}}$

Default period: 14 days

Interpretation:

RSI > 70: Overbought (potential reversal down)
RSI < 30: Oversold (potential reversal up)
RSI = 50: Neutral momentum
RSI trending up: Building bullish momentum
RSI trending down: Building bearish momentum

Trading signals:

Divergence: Price makes new high but RSI doesn’t → bearish reversal
Centerline crossover: RSI crosses 50 → trend change
Failure swings: RSI fails to confirm price movement

Limitations: In strong trends (Bull/Bear regimes), RSI can remain overbought/oversold for extended periods.

Use case: Momentum confirmation for HMM regimes. RSI and regime detection can provide complementary signals.

See also: Momentum, Overbought, Oversold, MACD

Reference: Investopedia - RSI

Section S

Sharpe Ratio

Definition: A measure of risk-adjusted return calculated as excess return per unit of volatility, the most widely used metric for comparing investment strategies.

Formula:

$$\text{Sharpe} = \frac{R_p - R_f}{\sigma_p}$$

where:

$R_p$ = Portfolio return (annualized)
$R_f$ = Risk-free rate
$\sigma_p$ = Portfolio volatility (annualized std dev)

Interpretation:

Sharpe < 0: Losing money (worse than risk-free)
Sharpe = 0-1: Poor to acceptable
Sharpe = 1-2: Good
Sharpe > 2: Excellent
Sharpe > 3: Outstanding (rare, scrutinize for errors)

Why it matters: Compares strategies on equal footing. A 30% return with 40% volatility (Sharpe=0.75) is worse than 20% return with 15% volatility (Sharpe=1.33).

Limitations:

Assumes returns are normally distributed (they’re not)
Treats upside and downside volatility equally
Can be gamed with smoothed returns

Use case: PRIMARY metric for evaluating HMM trading strategies. Regime-based strategies aim to improve Sharpe ratio by reducing exposure in Bear regimes.

Reference: Investopedia - Sharpe Ratio

Short Position

Definition: Selling an asset you don’t own with the expectation that its price will decrease, profiting from downward price movement.

Mechanics:

Borrow asset from broker
Sell at current price $P_0$
Buy back at lower price $P_1$
Return asset to broker
Profit = $P_0 - P_1$ (minus costs)

Characteristics:

Maximum gain: Limited to 100% (if price goes to zero)
Maximum loss: Unlimited (price can rise indefinitely)
Costs: Borrowing fees, interest charges
Requirements: Margin account, short interest availability

Risks:

Short squeeze: Rapid price increase forces covering at loss
Margin call: Must post additional collateral if position moves against you
Dividend payments: Responsible for dividends while short

Use case: Profit from Bear regimes. Advanced HMM strategies can short in Bear/Crisis regimes, though many retail traders only go long or cash.

See also: Long Position, Bear Market, Leverage

Reference: Investopedia - Short Selling

Signal Strength

Definition: A numerical value (typically 0.0 to 1.0) representing the confidence or conviction in a trading signal, used for position sizing.

Formula (HMM context):

$$\text{Strength} = f(\text{Confidence}, \text{Agreement}, \text{Regime Quality})$$

Components:

Regime confidence: HMM probability of current state
Indicator agreement: Alignment between HMM and technical indicators
Regime quality: Historical Sharpe ratio of regime

Interpretation:

0.0-0.3: Weak signal, small position or skip
0.3-0.7: Moderate signal, normal position
0.7-1.0: Strong signal, larger position

Use case: Dynamic position sizing based on signal quality. Strong Bull regime with high confidence → high signal strength → larger position.

See also: Position Sizing, Confidence Level

Scale Invariance

Definition: The property where a law, process, or measurement behaves similarly regardless of the scale of the variables.

In finance: Returns are scale-invariant because a $5 move means different things for a $50 stock vs. a $500 stock. Log returns automatically account for this.

Formula relationship:

$$\frac{P_t - P_{t-1}}{P_{t-1}} = \frac{\Delta P}{P}$$

is scale-invariant (percentages), while $\Delta P$ is not.

Use case: Essential property for comparing assets with different price levels. Allows HMM trained on SPY to potentially work on other assets.

See also: Log Returns

Reference: Wikipedia - Scale Invariance

Sideways Regime

Definition: A market condition where prices remain relatively stable, oscillating around a mean without clear trend.

Characteristics: Near-zero mean returns, low-moderate volatility, range-bound behavior

Statistical signature: Mean close to zero, volatility typically lower than crisis but can vary, may show mean-reversion

Use case: Important regime for option sellers and range-trading strategies. HMMs can detect sideways markets and adjust strategy accordingly.

See also: Bull Market, Bear Market, Regime

Simple Returns (Percentage Returns)

Definition: The percentage change in price from one period to the next.

Formula:

$$r_t = \frac{P_t - P_{t-1}}{P_{t-1}} = \frac{P_t}{P_{t-1}} - 1$$

Key properties:

Intuitive interpretation (directly in percentages)
NOT time-additive (must compound)
NOT perfectly symmetric (+10% then -10% ≠ break even)

Converting from log returns: $r = e^R - 1$

Why NOT to use for modeling:

Violates stationarity for long horizons
Non-additive (complex multi-period calculations)
Asymmetric treatment of gains/losses

Use case: Reporting and communication (easier to understand). But ALWAYS use log returns for statistical analysis.

See also: Log Returns, Stationary

Reference: Read our Why Use Log Returns? article

Skewness

Definition: A measure of the asymmetry of a probability distribution.

Formula:

$$\text{Skew}[X] = \frac{E[(X-\mu)^3]}{(\sigma^2)^{3/2}}$$

Interpretation:

Skewness = 0: Symmetric distribution (normal)
Skewness > 0: Right tail is longer (positive outliers more common)
Skewness < 0: Left tail is longer (negative outliers more common)

Financial markets: Often show negative skewness (crash more severe than rallies), though this varies by regime.

Use case: Assessing distribution asymmetry. Combined with kurtosis in Jarque-Bera test. HMMs can capture skewness through asymmetric regime transitions.

See also: Kurtosis, Jarque-Bera Test

Reference: Wikipedia - Skewness

Small Log Approximation

Definition: For small values of $r$, the approximation $\log(1+r) \approx r$.

Mathematical basis: Taylor series expansion of $\log(1+r)$ around $r=0$

Accuracy: Very good for $|r| < 0.01$ (daily returns), breaks down for $|r| > 0.1$

Use case: Explains why log returns ≈ simple returns for daily data. Justifies multiplying log return std dev by 100 for volatility approximation.

Important: Only use for volatility, NOT for mean returns. Mean must always use proper conversion: $(e^{\bar{R}} - 1)$.

See also: Log Returns, Simple Returns

Reference: Read our Why Use Log Returns? article

Standard Deviation

Definition: A measure of the dispersion of data points around their mean.

Formula:

$$\sigma = \sqrt{\frac{1}{N-1}\sum_{i=1}^N (x_i - \bar{x})^2}$$

where $\bar{x}$ is the sample mean.

Population vs sample: Use $N-1$ for sample std (Bessel’s correction for unbiased estimate).

Interpretation:

About 68% of data falls within 1 std dev of mean (for normal distribution)
About 95% within 2 std devs
About 99.7% within 3 std devs

In finance: Standard deviation of returns is called volatility. Higher std dev = higher risk.

Use case: Core measure of risk in portfolio theory. In HMMs, each regime has characteristic std dev (emission parameter).

See also: Volatility, Variance, Correlation

Reference: Wikipedia - Standard Deviation

Stationary Process

Definition: A stochastic process whose statistical properties (mean, variance, autocorrelation) do not change over time.

Weak (covariance) stationarity requires:

Constant mean: $E[X_t] = \mu$ for all $t$
Constant variance: $\text{Var}(X_t) = \sigma^2$ for all $t$
Autocorrelation depends only on lag: $\text{Cov}(X_t, X_{t-k})$ depends on $k$, not $t$

Why it matters: Most statistical models (including HMMs) assume stationarity. Non-stationary data violates assumptions and produces invalid inferences.

Tests: ADF (reject null → stationary), KPSS (fail to reject null → stationary)

Financial data:

Prices: NON-stationary (random walk with drift)
Returns: Approximately stationary (what we use for modeling)
Log returns: More stationary than simple returns

Use case: First step in time series analysis - verify stationarity before modeling. If non-stationary, take differences (returns) or apply transformations.

See also: ADF Test, KPSS Test, Random Walk, Log Returns

Reference: Wikipedia - Stationary Process

Simple Moving Average (SMA)

Definition: The arithmetic mean of an asset’s price over a specified number of periods, giving equal weight to all prices.

Formula:

$$SMA_n = \frac{1}{n}\sum_{i=0}^{n-1} P_{t-i}$$

Common periods:

SMA(20): Short-term trend (~ 1 month)
SMA(50): Medium-term trend (~ 2-3 months)
SMA(200): Long-term trend (~ 1 year)

Characteristics:

Equal weighting: All prices in window treated equally
Lagging indicator: Responds slowly to price changes
Smooth: Reduces noise but delays signals

vs EMA: SMA gives equal weight to all periods; EMA weights recent prices more heavily, making it more responsive.

Golden/Death Cross:

Golden cross: SMA(50) crosses above SMA(200) - bullish
Death cross: SMA(50) crosses below SMA(200) - bearish

Use case: Simple trend identification. Price above SMA → uptrend. Can confirm HMM regime transitions.

See also: EMA, Moving Average, Bollinger Bands

Reference: Investopedia - Simple Moving Average

Slippage

Definition: The difference between the expected execution price of a trade and the actual price received, caused by market conditions and order size.

Types:

Positive slippage: Better price than expected (rare)
Negative slippage: Worse price than expected (common)

Causes:

Market orders: Execute at best available price (may differ from quote)
Low liquidity: Large orders move the market
High volatility: Prices change rapidly between order and execution
Market impact: Your order affects the price

Typical magnitude:

Highly liquid: 0.01-0.05% slippage
Moderate liquidity: 0.1-0.3% slippage
Low liquidity: 0.5%+ slippage

Mitigation:

Use limit orders (but risk non-execution)
Trade liquid assets
Avoid trading during news events
Split large orders

In backtesting: ALWAYS include realistic slippage assumptions (e.g., 0.1-0.2% per trade). Ignoring slippage leads to overly optimistic results.

Use case: Critical cost in frequent-trading strategies. HMM strategies with regime transitions must account for slippage when entering/exiting positions.

See also: Transaction Costs, Backtesting, Paper Trading

Reference: Investopedia - Slippage

Smoothing

Definition: In HMM context, using the full observation sequence (past, present, and future) to estimate state probabilities at each time point, providing the most accurate state estimates.

Forward-Backward Smoothing:

$$P(q_t = s_i | O, \lambda) = \frac{\alpha_t(i) \beta_t(i)}{\sum_{j=1}^N \alpha_t(j) \beta_t(j)}$$

where $\alpha_t$ is forward probability, $\beta_t$ is backward probability.

Smoothing vs Filtering vs Prediction:

Filtering: Estimate state at $t$ using observations up to $t$
Smoothing: Estimate state at $t$ using ALL observations (past, present, future)
Prediction: Estimate state at $t+1$ using observations up to $t$

Accuracy: Smoothing > Filtering > Prediction (because smoothing uses more information).

Use case: When analyzing historical data, smoothing provides best state estimates. For real-time trading, use filtering (can’t see future).

Stop-Loss

Definition: A predetermined price level at which a position is automatically closed to limit losses, a fundamental risk management tool.

Types:

Fixed stop: Set % or $ below entry (e.g., 5% stop-loss)
Trailing stop: Moves with price, locks in gains
Volatility-based: Based on ATR or standard deviation
Time stop: Exit after X days regardless of price

Placement strategies:

Below support: For longs, place below key support level
ATR-based: 2× ATR below entry (adapts to volatility)
Regime-based: Wider stops in high-volatility regimes

Psychology: Prevents hope-based holding of losing positions. Pre-commitment device.

Trade-off: Tighter stops = less risk but more false exits (whipsawed).

Use case: ESSENTIAL for all trading strategies. In HMM context, set wider stops in Crisis regimes (higher volatility), tighter stops in Sideways regimes.

See also: Risk Management, Volatility, Maximum Drawdown

Reference: Investopedia - Stop-Loss

Section T

Tail Risk

Definition: The risk of extreme events that occur in the tails of a probability distribution, representing rare but severe losses.

Characteristics:

Low probability: Occur less than 5% of the time
High impact: Much larger than typical losses
Fat tails: Occur more frequently than normal distribution predicts

Measurement:

VaR: Loss threshold at confidence level
Expected Shortfall: Average loss beyond VaR
Kurtosis: Statistical measure of tail thickness

Financial examples:

2008 financial crisis: -50% market decline
Flash crashes: Sudden extreme moves
Black swan events: Unpredicted large moves

HMM perspective: Crisis regimes are characterized by extreme tail risk. Detecting and avoiding these regimes reduces tail risk exposure.

Use case: Understanding tail risk essential for survival in trading. One extreme event can wipe out years of gains.

See also: Value at Risk, Expected Shortfall, Kurtosis, Crisis

Reference: Investopedia - Tail Risk

Threshold-Based Classification

Definition: A method for labeling HMM states as regime types (Bear/Bull/Sideways) by comparing emission means against data-driven thresholds rather than sorting by state index.

Approach:

$$\text{Regime}_k = \begin{cases} \text{Bear} & \text{if } \mu_k < \theta_{\text{bear}} \\ \text{Bull} & \text{if } \mu_k > \theta_{\text{bull}} \\ \text{Sideways} & \text{otherwise} \end{cases}$$

Why necessary: HMM state indices (0, 1, 2) are arbitrary. State 0 may have highest or lowest mean depending on initialization.

Threshold determination:

Statistical: Based on percentiles of historical returns
Economic: Based on annualized return targets
Volatility-adjusted: Account for risk in classification

Best practice: Let the DATA determine regime labels, not state order.

Use case: Essential for interpreting HMM results correctly. Never assume State 0 = Bear without checking emission means.

See also: Regime, Emission Matrix, HMM

Reference: Read our Full Pipeline Advanced Analysis article

Time Series

Definition: A sequence of data points indexed in time order, typically at equally-spaced intervals.

Examples: Daily stock prices, hourly temperatures, quarterly GDP

Key characteristics:

Temporal ordering matters (unlike cross-sectional data)
Often autocorrelated
May have trends, seasonality, cycles
May be stationary or non-stationary

Analysis goals:

Description: Summarize patterns (trend, seasonality)
Explanation: Model relationships between variables
Prediction: Forecast future values
Control: Use forecasts for decision-making

Use case: Financial data is inherently time series. HMMs are specialized time series models that account for regime changes.

See also: Stationary, Autocorrelation, HMM

Transition Matrix (A)

Definition: Matrix of probabilities defining how likely the system is to transition from one hidden state to another in an HMM.

Formula:

$$a_{ij} = P(q_{t+1} = s_j | q_t = s_i)$$

where $q_t$ is the state at time $t$.

Constraint: Each row sums to 1: $\sum_j a_{ij} = 1$

Interpretation:

Diagonal elements $a_{ii}$: Persistence (staying in same state)
Off-diagonal $a_{ij}$ $(i \neq j)$: Transition probability
High diagonal → States persist (regime stability)
Low diagonal → Rapid switching (regime volatility)

Example (2-state):

       Bull  Bear
Bull  [ 0.95  0.05 ]
Bear  [ 0.10  0.90 ]

This says bull markets persist (95% stay), and bear markets are stickier (90% stay, only 10% exit).

Use case: Captures regime dynamics. Learned by Baum-Welch algorithm. Critical for understanding regime transitions.

Transaction Costs

Definition: The total costs incurred when buying or selling securities, including commissions, fees, spreads, and market impact.

Components:

Commissions: Broker fees per trade
Bid-ask spread: Difference between buy and sell prices
Slippage: Deviation from expected execution price
Exchange fees: Transaction fees charged by exchanges
Market impact: Price movement caused by your order

Typical costs (retail trading):

Commissions: $0-$5 per trade (many brokers now zero-commission)
Spread: 0.01-0.10% for liquid stocks
Slippage: 0.05-0.20% per trade
Total: ~0.1-0.3% per round trip (buy + sell)

Impact on strategies:

High-frequency: Transaction costs can eliminate profits
Low-frequency: Less impactful but still important
Round trips matter: Buy + Sell = 2× costs

In backtesting: MUST include realistic transaction cost assumptions. Common mistake: ignoring costs leads to unrealistic performance.

Use case: HMM strategies with frequent regime transitions must carefully account for transaction costs. May need to add filters to avoid excessive trading.

See also: Slippage, Backtesting, Paper Trading

Reference: Investopedia - Transaction Costs

Trend-Following

Definition: A trading strategy that attempts to capture gains by riding established price trends, buying in uptrends and selling/shorting in downtrends.

Core principle: “The trend is your friend” - established trends tend to continue.

Characteristics:

Wins: Few large winning trades (capture big moves)
Losses: Many small losing trades (whipsawed in ranges)
Win rate: Typically 30-50% (but large wins compensate)

Entry signals:

Price crosses above moving average
MACD turns positive
Price breaks out of range

Exit signals:

Price crosses below moving average
MACD turns negative
Trailing stop hit

When it works:

Bull/Bear regimes: Strong directional markets
High momentum: Sustained price movement
Low mean reversion: Trends persist

When it fails:

Sideways regimes: Range-bound markets
High volatility: Frequent whips awsOut
Trend reversals: Late entry/exit

HMM application: Use trend-following strategies in Bull and Bear regimes, switch to mean-reversion in Sideways regimes.

See also: Mean-Reversion, Momentum, Moving Average, MACD

Reference: Investopedia - Trend Trading

Section U

Uptrend

Definition: A market condition where prices make a series of higher highs and higher lows over time, indicating sustained buying pressure.

Characteristics:

Each price peak is higher than the previous peak
Each price trough is higher than the previous trough
Moving averages slope upward
Positive momentum

Technical identification:

Price consistently above moving averages
Sequence of higher highs and higher lows
Positive MACD and RSI in upper ranges

Trading implications:

Trend-following: Take long positions
Dips as opportunities: Buy pullbacks within trend
HMM may identify as Bull regime

“Higher highs, higher lows”: Classic definition of uptrend structure.

Use case: Identifying directional bias. Uptrends often persist longer than expected, making trend-following profitable.

See also: Downtrend, Bull Market, Momentum, Trend-Following

Section V

Value at Risk (VaR)

Definition: A statistical measure estimating the maximum potential loss over a specified time period at a given confidence level.

Formula:

$$\text{VaR}_\alpha = -F^{-1}(\alpha)$$

where $F^{-1}$ is the inverse CDF and $\alpha$ is the confidence level (e.g., 0.05 for 95% confidence).

Interpretation: A 95% VaR of $10,000 means: "We are 95% confident that losses will not exceed $10,000 over the specified period."

Methods:

Historical: Use empirical distribution of past returns
Parametric: Assume normal distribution (often wrong for finance)
Monte Carlo: Simulate many scenarios

Confidence levels:

95%: Standard for risk management
99%: More conservative, regulatory requirements
99.9%: Extreme tail events

Limitations:

Doesn’t capture severity beyond threshold
Assumes past predicts future
May underestimate tail risk

Use case: Risk budgeting and position sizing. In HMM context, calculate regime-specific VaR to adjust exposure dynamically.

Reference: Investopedia - Value at Risk

Variance

Definition: The expected squared deviation from the mean.

Formula:

$$\sigma^2 = \frac{1}{N-1}\sum_{i=1}^N (x_i - \bar{x})^2$$

Relationship to std dev: Variance is the square of standard deviation: $\text{Var}(X) = \sigma^2$

Use case: Theoretical foundation for volatility. Variance has nice mathematical properties (additive for independent variables), but std dev is easier to interpret (same units as data).

See also: Standard Deviation, Volatility

Viterbi Algorithm

Definition: A dynamic programming algorithm that finds the most likely sequence of hidden states that produced a given observation sequence.

Purpose: Solves the “decoding problem” - given observations and trained HMM, what states did we visit?

Key idea: At each time step, track the most probable path to each state. Then backtrack to find globally optimal path.

Complexity: $O(N^2 T)$ where $N$ is number of states and $T$ is sequence length. Much more efficient than brute force $O(N^T)$.

Viterbi vs Forward-Backward:

Viterbi: Finds single best path (hard decoding)
Forward-Backward: Computes probability of each state at each time (soft decoding)

Use case: After training HMM with Baum-Welch, use Viterbi to predict current regime and historical regime sequence.

Important limitation: Viterbi path may be less probable than any individual state from Forward-Backward at a given time. It’s the most likely PATH, not most likely state at each time.

Reference: Read our HMM Algorithms Explained and HMM 101 articles

Volatility

Definition: A measure of the dispersion of returns for a given security or market index. Represents the degree of variation (risk) in the price.

Formula: Typically measured as annualized standard deviation of returns:

$$\sigma_{\text{annual}} = \sigma_{\text{daily}} \times \sqrt{252}$$

where 252 is the number of trading days per year.

Types:

Historical volatility: Calculated from past returns
Implied volatility: Derived from option prices (market’s expectation)
Realized volatility: Actual volatility measured over a period

Volatility clustering: High volatility tends to follow high volatility (heteroskedasticity). This is why HMMs with state-dependent volatility are useful.

Interpretation:

Low volatility (~10% annual): Stable stocks, defensive sectors
Medium volatility (~15-20% annual): Market average
High volatility (~30%+ annual): Growth stocks, crypto

Use case: Primary risk measure in finance. In HMMs, each regime has characteristic volatility (std dev of emission distribution). Crisis regimes have very high volatility.

Reference: Wikipedia - Volatility (Finance)

Volatility Clustering

Definition: The tendency for periods of high volatility to cluster together and periods of low volatility to cluster together in financial time series.

Observation: “Large changes tend to be followed by large changes, of either sign, and small changes tend to be followed by small changes.”

Statistical evidence:

High autocorrelation in |returns| and returns²
ARCH/GARCH effects
Time-varying volatility

Causes:

Information arrival: News comes in clusters
Market structure: Liquidity varies over time
Behavioral: Herding and momentum effects

HMM perspective: Volatility clustering suggests regime-switching behavior. High-volatility and low-volatility regimes persist before transitioning.

Modeling:

ARCH/GARCH: Explicitly model time-varying volatility
HMMs: Capture clustering via regime persistence (transition matrix)
Stochastic volatility: Volatility follows its own stochastic process

Use case: Critical phenomenon in finance. HMM regime detection naturally captures volatility clustering through state persistence.

Reference: Wikipedia - Volatility Clustering

Section W

Walk-Forward Analysis

Definition: A validation technique that repeatedly trains a model on a rolling window of historical data and tests it on the subsequent out-of-sample period, simulating real-time performance.

Process:

Train on window 1 (e.g., days 1-500)
Test on window 2 (e.g., days 501-600)
Roll forward: Train on days 101-600
Test on days 601-700
Repeat through entire dataset

Advantages:

Realistic: Mimics how model would be used in production
Adaptive: Model parameters update with new data
Robust: Tests if model degrades over time
Out-of-sample: All test periods are truly unseen

Parameters:

Training window: Length of history for fitting (e.g., 500 days)
Test window: Length of forward period (e.g., 100 days)
Step size: How far to roll forward (e.g., 100 days)

Interpretation: If walk-forward results are similar to in-sample results, model is robust. Large degradation indicates overfitting or non-stationarity.

Use case: GOLD STANDARD for validating HMM trading strategies. More rigorous than single train/test split.

Reference: Investopedia - Walk-Forward Analysis

Win Rate

Definition: The percentage of trading periods or trades that result in positive returns.

Formula:

$$\text{Win Rate} = \frac{\text{Number of Winning Trades}}{\text{Total Number of Trades}} \times 100\%$$

Interpretation:

50%: Break-even (random)
> 50%: More wins than losses
> 60%: Good for mean-reversion strategies
< 50%: Can still be profitable if wins are large (trend-following)

Win rate vs Profit factor:

High win rate + low profit factor = Many small wins, few large losses (mean-reversion)
Low win rate + high profit factor = Few large wins, many small losses (trend-following)

Misleading metric: A 90% win rate means nothing if the 10% of losses wipe out all gains. Must consider average win vs average loss size.

Use case: Diagnostic metric for trading strategies. Compare win rates across regimes - expect high win rate in favorable regimes, lower in unfavorable.

See also: Profit Factor, Sharpe Ratio, Backtesting

Reference: Investopedia - Win Rate

Section X

White Noise

Definition: A random signal with constant power spectral density - completely uncorrelated across time.

Properties:

Zero mean: $E[\epsilon_t] = 0$
Constant variance: $\text{Var}(\epsilon_t) = \sigma^2$
No autocorrelation: $\text{Cov}(\epsilon_t, \epsilon_s) = 0$ for $t \neq s$

Gaussian white noise: $\epsilon_t \sim \mathcal{N}(0, \sigma^2)$ and uncorrelated

Use case:

Null model for time series (residuals should be white noise if model is good)
Component of many time series models (noise term)
If returns are white noise → unpredictable (efficient markets)

Testing: Use Ljung-Box test on residuals to check for white noise

See also: Autocorrelation, Stationary

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Last updated: October 12, 2025