Articles

Random Walk 🇺🇸

September 14, 2026
Category: Statistics Notes

A random walk models a level that changes by accumulating new shocks over time. Each innovation is temporary as an increment but permanent in the level, so the process can drift far from its starting point even when the expected one-step change is zero...

Difference Equations 🇺🇸

September 14, 2026
Category: Statistics Notes

A difference equation describes how a discrete-time quantity evolves from its earlier values. Rather than specifying every term directly, it gives a recursive rule, so the behavior of the sequence is determined by the recursion together with its initial conditions...

Financial Time Series Models 🇺🇸

September 14, 2026
Category: Statistics Notes

Financial time series often separate naturally into a price level, a return process, and a time-varying scale of uncertainty. Returns may show little predictable movement in their conditional mean while their magnitude clusters over time, producing periods of calm and periods of elevated volatility...

State Space Models 🇺🇸

September 14, 2026
Category: Statistics Notes

State-space models separate a time series into an unobserved state that evolves through time and an observation process that measures that state with noise. This framework is flexible enough to represent latent levels, trends, seasonal components, regression effects, missing observations, and many f...

Autocovariance Function 🇺🇸

September 14, 2026
Category: Statistics Notes

Autocovariance measures how a time series varies jointly with lagged versions of itself. Lag 0 is the variance, while nonzero lags show the direction and scale of linear dependence between observations separated in time, expressed in the squared units of the original series...

Autocorrelation Function 🇺🇸

September 14, 2026
Category: Statistics Notes

The autocorrelation function measures the unit-free linear association between a time series and lagged versions of itself. The partial autocorrelation function asks a narrower question: how much direct linear relationship remains at a given lag after the intervening lags have been accounted for...

Arima Models 🇺🇸

September 14, 2026
Category: Statistics Notes

ARMA models describe stationary linear dependence through a combination of autoregressive terms, which use past observations, and moving-average terms, which use current and past innovations. ARIMA extends that structure by differencing a series with a stochastic trend, while SARIMA adds seasonal di...

Seasonality and Trends 🇺🇸

September 14, 2026
Category: Statistics Notes

Trend and seasonality are systematic forms of time structure that operate on different scales. A trend is a persistent long-run movement in level or slope, while seasonality is a pattern that repeats at a fixed and known period such as day of week, month of year, or quarter...

Regression with Arma Errors 🇺🇸

September 14, 2026
Category: Statistics Notes

Regression with ARMA errors separates two kinds of structure that often appear together in time series. The regression explains how observed predictors shift the conditional mean, while the ARMA component models serial dependence that remains in the unexplained part...

Forecasting 🇺🇸

September 14, 2026
Category: Statistics Notes

A time-series forecast is a conditional statement about a future observation given the information available at a specific forecast origin. The horizon, target functional, and information set are part of the forecast itself, so a point prediction without its timing and uncertainty is an incomplete d...

Moving Average Models 🇺🇸

September 14, 2026
Category: Statistics Notes

A moving-average model represents the current observation as a finite weighted combination of the current innovation and a fixed number of past innovations. Because each shock enters the model for only a limited number of periods, an MA($q$) process has a finite direct shock response and a theoretic...

Time Series 🇺🇸

September 14, 2026
Category: Statistics Notes

A time series is an ordered set of observations together with an information structure: timestamps, sampling rules, and sequence position determine what can depend on what and what information is available for prediction. Treating the same values as an unordered sample would discard much of the stru...

Yule Walker Equations 🇺🇸

September 14, 2026
Category: Statistics Notes

The Yule-Walker equations connect an autoregressive model's coefficients to its autocovariances and autocorrelations. They translate a recursion written in terms of lagged observations into a set of moment relationships that can be studied theoretically or estimated from data...

Backward Shift Operator 🇺🇸

September 14, 2026
Category: Statistics Notes

The backward shift operator is a compact notation for referring to earlier observations in a time series. Instead of writing each lag separately, it lets lagged values, differences, and model equations be expressed as polynomials in a single operator...

Series 🇺🇸

September 14, 2026
Category: Statistics Notes

Sequences and series provide the convergence language behind many time-series representations. A sequence describes an ordered set of terms, while a series asks whether the cumulative effect of those terms approaches a finite limit...

Time Series Modeling 🇺🇸

September 14, 2026
Category: Statistics Notes

Time-series modeling is an iterative process of specifying structure, estimating parameters, diagnosing what remains unexplained, and testing forecasts on future-like data. Choosing an equation is only one step; transformations, initial conditions, residual behavior, and the forecast information set...

Statistical Moments and Time Series 🇺🇸

September 14, 2026
Category: Statistics Notes

Statistical moments summarize features such as the center, spread, and joint variation of a distribution. In time series, those same quantities acquire a time dimension: the mean and variance may change across the record, and covariance becomes a function of lag as well as scale...

Randomness Tests 🇺🇸

September 14, 2026
Category: Statistics Notes

Randomness tests look for specific kinds of structure that should not appear under a stated null model. In time-series work, they are most useful as diagnostics: one test may target linear autocorrelation, another unusual turning-point behavior, another monotone trend, and another dependence in squa...

Typography 🇺🇸

September 13, 2026
Category: Standard Of Iron

Standard of Iron controls its typography explicitly so the game, tools, and promotional output can render the same text consistently on every machine. UI code does not rely on an arbitrary operating-system font lookup for branded text, and the custom title face is built and tested as part of the rep...

Skirmish Bases 🇺🇸

September 13, 2026
Category: Standard Of Iron

Every skirmish map authors its barracks in the map JSON structures array. Originally, the player_id on each barracks determined the starting position completely: player 1 began at p1_barracks, player 2 at p2_barracks, and so on. The setup screen could change which player occupied a seat, but not whi...

Creature Bpat Format 🇺🇸

September 13, 2026
Category: Standard Of Iron

BPAT is the baked runtime animation format used for skinned creatures in Standard of Iron. A BPAT file packages the data the runtime needs to animate a creature without rebuilding authored skeletal animation from source definitions every frame...

Hill Shapes 🇺🇸

September 13, 2026
Category: Standard Of Iron

A terrain entry of type hill uses its shape field to define the tactical footprint of raised ground. Maps can author organic mounds, straight ridges, arcs, elbows, rings, traced paths, or exact painted cell masks...

Windows Code Signing 🇺🇸

September 13, 2026
Category: Standard Of Iron

Windows packaging is implemented in .github/workflows/build-windows.yml. The workflow is reusable and is called by both the weekly packaging job and the release workflow. When the signing secrets are available, it signs standard_of_iron.exe with Authenticode and verifies the result before packaging ...

Battlefield Capture 🇺🇸

September 13, 2026
Category: Standard Of Iron

battlefield_capture is a deterministic, render-free acceptance runner used to validate battlefield behavior during migration work. It advances the simulation on a fixed 30 Hz clock from an explicit seed and writes newline-delimited JSON (JSONL), making its output suitable for repeatable comparisons ...

Frame Pacing 🇺🇸

September 13, 2026
Category: Standard Of Iron

Frame pacing in Standard of Iron is measured from real gameplay runs and reported as a frame_pacing verdict. The gate is designed to answer a broader question than “what was the average FPS?” It measures whether presented frames arrive consistently, whether CPU/GPU frame work fits the active graphic...

Cursed Gold Vein 🇺🇸

September 13, 2026
Category: Standard Of Iron

The cursed gold vein is a capturable world element built around a deliberate trade-off. Visually, it is a crag of dark rock split by an ore seam, with gold crystals growing from the fracture and a claim flag planted beside it. Mechanically, it rewards ownership with a steady income while damaging th...

Unit Balance 🇺🇸

September 13, 2026
Category: Standard Of Iron

Unit balance in Standard of Iron is defined by executable simulation inputs: shipped troop data, nation overrides, combat multipliers, formation/stance behavior, production/economy values, and deterministic matchup fixtures run through tools/balance_sim...

Ui Design System 🇺🇸

September 13, 2026
Category: Standard Of Iron

Standard of Iron's interface uses the StandardOfIron.Design QML module as the shared presentation layer for colour, spacing, typography, motion, iconography, faction styling, notifications, hints, sound hooks, accessibility-derived values, and reusable controls...

Macos Signing 🇺🇸

September 13, 2026
Category: Standard Of Iron

macOS packaging is implemented in .github/workflows/build-macos.yml. The workflow is reusable and is called by both the weekly packaging job and the release workflow...

Multiple Comparisons 🇺🇸

September 13, 2026
Category: Statistics Notes

When conducting multiple hypothesis tests, the probability of making at least one Type I error—falsely rejecting a true null hypothesis—increases. This is known as the multiple comparisons problem or, in some contexts, the look-elsewhere effect...

Type i and Type Ii Errors 🇺🇸

September 13, 2026
Category: Statistics Notes

Hypothesis testing allows researchers to evaluate claims about a population using sample data. We begin with a null hypothesis, $H_0$, which represents the claim being tested, and an alternative hypothesis, $H_a$, which represents a competing claim. Because decisions are based on sample data, errors...

Analysis of Categorical Data 🇺🇸

September 13, 2026
Category: Statistics Notes

The chi-square ($\chi^2$) test is a family of statistical tests for categorical count data. These tests compare observed frequencies with the frequencies expected under a null hypothesis...

Confidence Intervals 🇺🇸

September 13, 2026
Category: Statistics Notes

A confidence interval (CI) is a range of plausible values for a population parameter, such as a mean or proportion, calculated from sample data. It supplements a point estimate by showing the uncertainty around that estimate...

Hypothesis Testing 🇺🇸

September 13, 2026
Category: Statistics Notes

Hypothesis testing is a statistical tool used to draw conclusions about populations based on sample data. It is widely applied in scientific research, from evaluating new treatments in clinical trials to studying customer behavior in business analytics...

Null Hypothesis 🇺🇸

September 13, 2026
Category: Statistics Notes

Statistical hypothesis testing is a method for using sample data to make inferences about a population. Understanding null and alternative hypotheses, along with how p-values are calculated and interpreted, is essential for applying hypothesis tests correctly...