The Currency Times

Field 016 tests

\tilde{s} = \frac{\bar{x}_t - \bar{x}_{t-n}}{n\,\hat{\sigma}}

Trend and persistence

Whether a series is going somewhere, and whether the going can be measured without the answer depending on the units it was recorded in.

Open the field

The first thing anyone claims about a series is that it is rising or falling. Two of the six entries here failed because the claim was made in a way that could not survive a change of scale.

Expression

Test

Status

\tilde{s} = \dfrac{\bar{x}_t - \bar{x}_{t-n}}{n\,\hat{\sigma}}

Scale-normalised slope

Trend strength divided by the dispersion of the series, so it is dimensionless.

In use

\theta = \tfrac{180}{\pi}\arctan\!\big(\tfrac{\Delta \bar{x}}{n}\big)

Slope as an angle

Steepness in degrees. The argument of an inverse tangent has to be dimensionless, and price per bar is not, so the figure reports the price scale rather than the trend.

Not valid

\theta = \frac{180}{\pi}\,\arctan\!\left(\frac{MA_t - MA_{t-n}}{n}\right)

Terms

MA_t

Moving average at time t

n

Lookback in bars

\theta

Claimed trend angle in degrees

Scope

Applies to any attempt to convert the gradient of a price-derived series into an angle, including oscillator slope and the divergence between two slopes computed by the tangent subtraction formula.

The objection

The argument of an inverse tangent must be dimensionless. The quantity $(MA_t - MA_{t-n})/n$ carries units of price per bar, so passing it to $\arctan$ is the same category of error as writing $\sin(5\,\text{metres})$ .

The consequence is not subtle. For a pair quoted near $1.20$ a slope of $5\times10^{-5}$ returns about $0.003^\circ$ . An equivalent trend on an instrument quoted near $50{,}000$ returns about $89.9^\circ$ . The number reports the price scale, not the trend.

There is a second and independent problem. The visual angle of a line on a chart is a property of the axis scaling rather than of the data. Stretch the vertical axis and every line steepens while nothing about the series has changed.

What replaced it

Normalising by average true range restores dimensionlessness, since numerator and denominator are then both in price units. Kaufman's efficiency ratio is bounded on $[0,1]$ by construction and has no dimensional exposure at all.

Finding

Not valid, and the reason is structural rather than empirical. The construction fails on dimensional grounds before any data is involved, which is why no amount of testing could have rescued it. The normalised slope and the efficiency ratio do the same job correctly.

References

Cleveland, W. S. Visualizing Data, 1993, on the dependence of perceived line orientation on physical display units.

CT-01-02Currency Times. Slope as an angle. Note CT-01-02, thecurrencytimes.com.

Close

\alpha = \arctan\!\big(\tfrac{m_1-m_2}{1+m_1m_2}\big)

Angle between two trends

Divergence by tangent subtraction. Inherits the dimensional fault of the angles it is built from, so it fails for the same reason.

Not valid

ER = \dfrac{|\Delta_{\text{net}}|}{\sum|\Delta_i|}

Efficiency ratio

Net travel over total travel. Bounded on zero to one by construction.

In use

S = \sum_{i<j}\mathrm{sgn}(x_j - x_i)

Mann-Kendall test

Distribution-free test for monotonic trend. Makes no assumption of linearity.

In use

\hat{\beta} = \mathrm{med}\left(\tfrac{x_j-x_i}{j-i}\right)

Theil-Sen estimator

Median of pairwise slopes. Unaffected by up to twenty-nine per cent contamination.

In use

Back to the fields

Field 026 tests

\mathrm{d}X_t = \theta(\mu - X_t)\,\mathrm{d}t + \sigma\,\mathrm{d}W_t

Reversion and equilibrium

Whether a system returns to a level it left, and whether the level it appears to return to is real or an accident of where you started looking.

Open the field

Regression to the mean was identified in 1886 and is still the most frequently mistaken effect in applied statistics. Almost every claim of an intervention working is partly this.

Expression

Test

Status

\mathrm{d}X_t = \theta(\mu - X_t)\mathrm{d}t + \sigma\mathrm{d}W_t

Ornstein-Uhlenbeck process

Continuous-time reversion at speed theta toward level mu.

Open

t_{1/2} = \dfrac{\ln 2}{\theta}

Half-life of a deviation

How long a departure takes to decay by half. The practical form of theta.

In use

\Delta x_t = \alpha + \rho x_{t-1} + \varepsilon_t

Augmented Dickey-Fuller

Tests whether a series has a unit root, meaning no level to return to.

In use

z = \dfrac{x-\mu}{\sigma}

Standardised deviation

Extremity in units of dispersion, comparable across unlike series.

In use

\mathbb{E}[Y\mid X=x] \to \mu_Y

Regression to the mean

Galton's effect. Extreme observations are followed by less extreme ones for no reason at all.

In use

y_t - \beta x_t = z_t \sim I(0)

Cointegration

Two wandering series with a stationary combination between them.

Queued

Back to the fields

Field 037 tests

\sigma_t^2 = \omega + \alpha\varepsilon_{t-1}^2 + \beta\sigma_{t-1}^2

Dispersion and clustering

Whether the size of fluctuations is itself predictable, and whether apparent persistence is anything more than a recurring timetable.

Open the field

Turbulent periods follow turbulent periods in rainfall, in electricity demand, in traffic and in markets. Whether that observation forecasts anything is a separate claim from whether it is true.

Expression

Test

Status

\sigma_t^2 = \omega + \alpha\varepsilon_{t-1}^2 + \beta\sigma_{t-1}^2

Conditional variance model

Sound at daily resolution. At intraday frequency the fitted persistence tracks the daily activity cycle rather than the process, so a heterogeneous model took its place.

Superseded

\sigma_t^2 = \omega + \alpha\,\varepsilon_{t-1}^2 + \beta\,\sigma_{t-1}^2, \qquad \alpha + \beta < 1

Terms

\sigma_t^2

Conditional variance at t

\varepsilon_{t-1}

Previous innovation

\omega

Long-run variance term

\alpha,\beta

Shock and persistence weights

Scope

Holds at daily resolution and coarser, where $\alpha+\beta<1$ gives a stationary process with a defined unconditional variance. Does not hold as specified on hourly or finer data.

The objection

Intraday volatility carries a strong deterministic seasonal component. Activity rises on a daily cycle in almost every observed system, from traffic to electricity demand to markets, and again around scheduled events. The model has no term for that, so the seasonality is absorbed into $\alpha$ and $\beta$ , and the fitted persistence is an artefact of the trading day rather than a property of the series.

Fitted to raw intraday returns it therefore produces a model that appears to forecast and is in fact reciting the timetable.

What would settle it

Deseasonalise first by estimating a diurnal factor $s_u$ for each intraday interval, then fit to $\varepsilon_t / s_u$ . If the persistence survives that, it is real.

Finding

Superseded at intraday frequency, pending deseasonalisation. The model is sound where it was designed to work and the fault lies in applying it below daily resolution. A heterogeneous autoregressive model carries the work for now, using daily, weekly and monthly components directly.

CT-03-01Currency Times. Conditional variance model. Note CT-03-01, thecurrencytimes.com.

Close

RV_t^{(d,w,m)}

Heterogeneous autoregression

Daily, weekly and monthly components entered directly rather than through decay.

In use

RV = \sum_i r_i^2

Realised variance

Sum of squared fine-grained changes over a coarser period.

In use

\varepsilon_t / s_u

Seasonal adjustment of variance

Removing the deterministic cycle before fitting anything to the remainder.

Queued

\mathrm{Var}(\sigma_t)

Variance of the variance

Second-moment instability as a regime marker.

Queued

\text{LM} = nR^2

Heteroskedasticity tests

Breusch-Pagan and Levene. Whether dispersion is constant at all.

In use

\hat{\sigma}_{MAD} = 1.4826\,\mathrm{med}|x-\tilde{x}|

Robust scale

Median absolute deviation. The standard deviation is not robust and rarely should be used raw.

In use

Back to the fields

Field 046 tests

P(X > x) \sim x^{-\alpha}

Extremes and rare events

The events that decide outcomes are the ones there is least data on. This field is about reasoning honestly from a handful of observations.

Open the field

Floods, blackouts, epidemics, crashes and record temperatures share a mathematics and share a problem: the estimate that matters most rests on the fewest points.

Expression

Test

Status

G(x) = \exp\!\left(-[1+\xi\tfrac{x-\mu}{\sigma}]^{-1/\xi}\right)

Generalised extreme value

The only possible limiting distribution for block maxima.

In use

P(X>u+y\mid X>u)

Peaks over threshold

Uses every exceedance rather than one maximum per block. More data, harder threshold choice.

In use

\hat{\alpha}_{\text{Hill}} = \left(\tfrac{1}{k}\sum \ln \tfrac{x_{(i)}}{x_{(k)}}\right)^{-1}

Tail index estimation

Hill's estimator. Notoriously sensitive to how many order statistics you keep.

Open

T = \dfrac{1}{1-F(x)}

Return period

The hundred-year event language, and how badly it is usually understood.

In use

\theta_{\text{ext}}

Extremal index

Whether extremes arrive alone or in clusters. Changes the return period entirely.

Queued

\text{blocks} \;\text{vs}\; \text{threshold}

Block maxima against exceedances

Two routes to the same quantity that frequently disagree.

Open

Back to the fields

Field 054 tests

\mathbb{E}[R/S]_n \propto n^{H}

Long memory

Whether a system carries memory of its own past, and whether the estimator that claims to measure it can be trusted on samples of realistic length.

Open the field

A small field with an outsized problem. The standard estimator returns an answer even when there is nothing there, so the interesting work is in the confidence interval rather than the point estimate.

Expression

Test

Status

H = \dfrac{\log(R/S)}{\log T}

Rescaled-range Hurst

Classical estimation. Returns values above one half on series with no memory at all, so the corrected detrended estimator took its place.

Superseded

F(n) \sim n^{H}

Detrended fluctuation analysis

With the Anis-Lloyd correction and a bootstrap interval.

In use

\mathbb{E}[R/S]_n \propto n^{H}

The scaling relation

Above a half is persistence, below is anti-persistence.

Open

\mathbb{E}\!\left[\frac{R(n)}{S(n)}\right] \sim C\,n^{H}, \qquad H \in (0,1)

Terms

R(n)

Range of cumulative deviations

S(n)

Standard deviation over the window

H

Hurst exponent

Scope

Applies to any single series long enough to be split into many windows. Says nothing about the relationship between two series, which is cointegration and a separate field.

Reading it

$H = 0.5$ is a random walk with no memory. $H > 0.5$ indicates persistence, where an up move is more likely to be followed by another. $H < 0.5$ indicates anti-persistence.

The objection

Classical rescaled-range estimation is biased upward in finite samples. A pure random walk of realistic length returns values comfortably above $0.5$ , which means the estimator manufactures the appearance of memory in a series that has none. Short-range dependence and non-stationarity inflate it further.

The practical consequence is that an unqualified $H$ of, say, $0.58$ tells you nothing without knowing what the same estimator returns on synthetic noise of the same length.

What would settle it

Detrended fluctuation analysis with the Anis-Lloyd small-sample correction and a bootstrap confidence interval rather than a point value. If the interval excludes $0.5$ , the memory is real.

Finding

Open. The corrected estimator is in place. Whether any instrument studied shows memory that survives the interval is not yet settled.

CT-05-03Currency Times. The scaling relation. Note CT-05-03, thecurrencytimes.com.

Close

(1-B)^{d} X_t

Fractional differencing

Stationarity at order d between zero and one, memory preserved.

In use

Back to the fields

Field 066 tests

\alpha_{\mathrm{fam}} = 1 - (1-\alpha)^{m}

Dependence and testing

Whether an observed relationship is real, and how many times you may look before the question stops being meaningful.

Open the field

The plumbing beneath every other field. Get this wrong and everything upstream is decoration.

Expression

Test

Status

\rho(x_t, x_{t-k})

Lagged autocorrelation

Serial dependence at lag k. Linear only, which is the limitation.

In use

\widehat{\Omega}_{HAC}

Newey-West standard errors

Valid under heteroskedasticity and autocorrelation. Without it the p-values are wrong before any correction.

In use

p_{(i)} \le \tfrac{i}{m}\alpha

Benjamini-Hochberg

Controls the expected proportion of false positives among rejections.

In use

\alpha_{\mathrm{fam}} = 1-(1-\alpha)^{m}

Family-wise error rate

Probability of at least one false positive across m tests.

Constraint

\alpha_{\mathrm{fam}} = 1 - (1-\alpha)^{m}

Terms

\alpha

Significance level per test

m

Number of tests performed

\alpha_{\mathrm{fam}}

Probability of at least one false positive

Scope

Applies to every screen across instruments, parameters, lookbacks and sessions. It is the same problem as the trial-count problem in validation, arriving through hypothesis testing rather than through strategy search.

The arithmetic

At the conventional $\alpha = 0.05$ , twenty independent tests give $\alpha_{\mathrm{fam}} \approx 0.64$ . At one hundred tests it is $0.994$ . Running a hundred comparisons and reporting the one that cleared the threshold is not a discovery. It is arithmetic behaving exactly as specified.

What is used instead

Controlling the family-wise rate outright is too conservative when hundreds of related signals are screened, because it destroys the power to find anything. The project controls the false discovery rate instead, which bounds the expected proportion of false positives among the rejected hypotheses.

Where the underlying series are autocorrelated, the standard errors feeding the tests are computed with a heteroskedasticity and autocorrelation consistent estimator, since otherwise the p-values are wrong before the correction is even applied.

Finding

Standing constraint. Applied to every screen on the list.

CT-06-04Currency Times. Family-wise error rate. Note CT-06-04, thecurrencytimes.com.

Close

\mathcal{R}(X,Y)

Distance correlation

Zero if and only if independent. Catches non-linear dependence that correlation misses.

Queued

I(X;Y) = \sum p \log \tfrac{p(x,y)}{p(x)p(y)}

Mutual information

Dependence of any form, at the cost of needing far more data.

Queued

Back to the fields

Field 076 tests

P(s_t = j \mid s_{t-1} = i) = p_{ij}

Regime and change points

Whether a system has states, and whether a state can be identified while you are still inside it rather than afterwards.

Open the field

Regime models fit any history beautifully. The whole difficulty is that the fit is retrospective, and a regime you can only name later is a description rather than a tool.

Expression

Test

Status

P(s_t=j\mid s_{t-1}=i)=p_{ij}

Markov regime switching

Hidden states with estimated transition probabilities.

Queued

\hat{T}_1,\dots,\hat{T}_k

Bai-Perron breaks

Multiple structural breaks at unknown dates, chosen by information criterion.

Queued

S_t = \max(0, S_{t-1} + x_t - k)

CUSUM

Cumulative departure from expectation. Detects small persistent shifts early.

In use

F = \dfrac{(RSS_p - RSS_u)/k}{RSS_u/(n-2k)}

Chow test

Whether coefficients differ either side of a known break date.

In use

P(r_t \mid x_{1:t})

Bayesian online change point

Posterior over run length. Updates as observations arrive.

Queued

\mathbb{1}\{\text{break}\}

Pipeline break flagging

Any test spanning a known definitional break is flagged automatically, not silently allowed.

In use

Back to the fields

Field 086 tests

Y \to X \;\text{ or }\; X \to Y \;\text{ or }\; Z \to \text{both}

Coupled systems

Whether one system drives another, whether the direction can be established, and whether a third thing is driving both.

Open the field

The field where the honest answer is usually that you cannot tell. Two series moving together is compatible with three different worlds and the data alone rarely separates them.

Expression

Test

Status

\rho_{xy}(\tau)

Lagged cross-correlation

The first thing anyone computes, and almost never sufficient.

In use

x_t = \sum \alpha_i x_{t-i} + \sum \beta_j y_{t-j}

Granger causality

Whether past y improves prediction of x beyond past x. Not causality despite the name.

Open

T_{Y \to X}

Transfer entropy

Directional information flow. Model-free, and hungry for data.

Queued

\hat{X}\mid M_Y

Convergent cross mapping

Detects coupling in deterministic non-linear systems where Granger fails.

Queued

\arg\max_\tau \rho_{xy}(\tau)

Lead-lag identification

Which moves first, and whether the ordering is stable across the sample.

Open

X \leftarrow Z \rightarrow Y

Common-driver confounding

The default explanation for correlation, and the one least often tested.

Open

Back to the fields

Field 095 tests

d_{ij} = \sqrt{2\left(1 - \rho_{ij}\right)}

Correlation networks

Turning a matrix of pairwise relationships into a map, so that structure can be seen rather than assumed.

Open the field

A correlation matrix of a hundred things contains nearly five thousand numbers and no picture. Converting correlation to distance turns it into a graph you can look at.

Expression

Test

Status

d_{ij} = \sqrt{2(1-\rho_{ij})}

Correlation distance

A proper metric. Satisfies the triangle inequality, which correlation does not.

In use

\text{MST}(G)

Minimum spanning tree

The skeleton of the graph. N minus one edges, no cycles.

In use

\text{PMFG}

Planar filtered graph

Retains more structure than the tree while staying readable.

Queued

c_i = \sum_j \tfrac{1}{d_{ij}}

Centrality

Which nodes sit at the middle of the structure.

Queued

Q = \tfrac{1}{2m}\sum_{ij}\left(A_{ij}-\tfrac{k_ik_j}{2m}\right)\delta_{c_ic_j}

Modularity

Whether detected communities are stronger than chance would produce.

Queued

Back to the fields

Field 107 tests

\lambda_{\pm} = \sigma^2\left(1 \pm \sqrt{N/T}\right)^{2}

Random matrix theory

How much of a correlation matrix estimated from a finite sample is real, and how much is noise wearing the shape of structure.

Open the field

Physics answered this in the 1950s and most fields are still catching up. If you measure relationships between many things from a limited history, most of what you measure is an artefact of the measuring.

Expression

Test

Status

\lambda_{\pm} = \sigma^2(1 \pm \sqrt{N/T})^{2}

Marchenko-Pastur bounds

The eigenvalue band a pure-noise correlation matrix produces.

In use

\lambda_{\pm} = \sigma^{2}\left(1 \pm \sqrt{\frac{N}{T}}\,\right)^{2}, \qquad q = \frac{N}{T}

Opened March 2026 · last revised July 2026 · 4 revisions

Terms

N

Number of series

T

Observations per series

q

Aspect ratio N over T

\lambda_{\pm}

Upper and lower noise bounds

Scope

Applies to any correlation or covariance matrix estimated from a finite sample, which in practice means every one. The result is exact in the limit where $N$ and $T$ both grow with $q$ fixed, and holds well enough at realistic sizes.

The claim

Take $N$ series of pure independent noise, each of length $T$ , and compute the correlation matrix between them. It will not be the identity. It will have a spread of eigenvalues, and that spread is not an accident of your particular sample. It has a known shape, bounded above and below by $\lambda_{\pm}$ .

The consequence is uncomfortable. Any eigenvalue of your real correlation matrix that falls inside that band is indistinguishable from noise. Not weak evidence of structure. No evidence at all.

The arithmetic that should worry you

Fifty instruments and two years of daily data gives $N = 50$ , $T = 500$ , so $q = 0.1$ and the noise band runs from about $0.47$ to about $1.73$ . Most of the eigenvalues of a real correlation matrix over that sample sit inside it.

Push to $N = 200$ over the same period and $q = 0.4$ , with the band running to roughly $2.65$ . At $N > T$ the matrix is singular outright and cannot be inverted at all, which is a problem for anything requiring the inverse covariance, which is most of portfolio construction.

What it is used for here

Eigenvalues inside the band are clipped to their average before the matrix is reused, which leaves the handful of genuine modes and discards the rest. The largest eigenvalue almost always sits far above the band and corresponds to the market moving together, which is real and not noise.

Where it came from

Wigner introduced the approach to model energy levels in heavy atomic nuclei, where the exact Hamiltonian was unknown and unknowable, so he asked what a random one would look like. The question turns out to be the same question.

Finding

In use as a filter rather than as an open question. The Marchenko-Pastur bound is a mathematical result, not a hypothesis about markets. What remains open is how much of the surviving structure is stable out of sample.

References

Marchenko, V. and Pastur, L. Distribution of eigenvalues for some sets of random matrices, 1967.
Laloux, L., Cizeau, P., Bouchaud, J.-P. and Potters, M. Noise dressing of financial correlation matrices, Physical Review Letters, 1999.
Wigner, E. Characteristic vectors of bordered matrices with infinite dimensions, 1955.

CT-10-01Currency Times. Marchenko-Pastur bounds. Note CT-10-01, thecurrencytimes.com.

Close

\lambda_i \in [\lambda_-,\lambda_+] \Rightarrow \bar{\lambda}

Eigenvalue clipping

Replace in-band eigenvalues with their mean, keep the rest.

In use

\lambda_1 \gg \lambda_+

The dominant mode

The largest eigenvalue as common movement. Always well outside the band.

In use

F_{\text{TW}}(\lambda_1)

Tracy-Widom law

Distribution of the largest eigenvalue, giving a significance test.

Queued

P = \left(\sum_i u_i^4\right)^{-1}

Participation ratio

Whether an eigenvector is spread widely or localised in a few components.

Queued

\kappa = \lambda_1/\lambda_N

Condition number

How badly inversion will amplify estimation error.

In use

\hat{\Sigma} = (1-\delta)S + \delta F

Shrinkage estimation

Ledoit-Wolf, tested against the spectral filter.

Queued

Back to the fields

Field 115 tests

g_{ij}(\theta) = \mathbb{E}\!\left[\partial_i \log p \; \partial_j \log p\right]

Information geometry

Treating probability distributions as points on a curved surface, so that the distance between two beliefs becomes something you can compute.

Open the field

The honest answer to how far one estimate sits from another. Euclidean distance between parameters is meaningless because parameters are arbitrary labels. The Fisher metric is not arbitrary, and it is the only one that is not.

Expression

Test

Status

g_{ij} = \mathbb{E}[\partial_i \log p\, \partial_j \log p]

Fisher information metric

The natural distance on a family of distributions.

Open

D(p\,\|\,q) = \sum p \log \tfrac{p}{q}

Kullback-Leibler divergence

Information lost when q is used in place of p. Not symmetric.

In use

\tilde{\nabla} = g^{-1}\nabla

Natural gradient

Steepest descent on the manifold rather than in parameter space.

Queued

\mathrm{Var}(\hat\theta) \ge I(\theta)^{-1}

Cramér-Rao bound

The floor on the variance of any unbiased estimator.

In use

p(x;\theta) = e^{\theta \cdot T(x) - \psi(\theta)}

Exponential family structure

Where the geometry becomes dually flat and tractable.

Queued

Back to the fields

Field 126 tests

\max H(p) \;\text{ s.t. }\; \mathbb{E}[f_k] = c_k

Statistical mechanics

What a system of many interacting participants has in common with a material made of many interacting particles, and where the analogy stops being useful.

Open the field

Borrowed wholesale from physics, and the borrowing is legitimate in some places and lazy in others. Separating the two is most of the work here.

Expression

Test

Status

\max H(p)\ \text{s.t.}\ \mathbb{E}[f_k]=c_k

Maximum entropy

The least committed distribution consistent with what is actually known.

In use

p^{*} = \arg\max_{p} \left(-\sum_x p(x)\log p(x)\right) \quad \text{subject to} \quad \mathbb{E}_p[f_k] = c_k

Opened April 2026 · last revised June 2026 · 2 revisions

Terms

H(p)

Shannon entropy of the distribution

f_k

Constraint functions

c_k

Known values of those constraints

p^{*}

The maximum entropy solution

Scope

Applies wherever a distribution must be assumed and only partial information is available, which covers most of the modelling on this site. Does not licence assuming a distribution when the information to constrain it is absent.

The principle

Of all distributions consistent with what you actually know, choose the one that assumes the least beyond it. Any other choice smuggles in information you do not have.

The results fall out cleanly. Constrain nothing and you get the uniform distribution. Constrain the mean and variance on the real line and you get the Gaussian. Constrain the mean on the positive half line and you get the exponential. Each of these is the honest answer given the stated constraints, and each becomes a dishonest answer the moment the constraints are asserted rather than measured.

Why it matters here

The Gaussian is not a law of markets. It is what maximum entropy returns when you have only claimed to know a mean and a variance. Financial returns very obviously have more structure than that in the tails, so the Gaussian is not wrong because markets are strange. It is wrong because the constraints fed to it were too few.

What is under test

Whether adding tail constraints, or moving to a maximum entropy solution on a heavier-tailed family, produces a distribution that holds out of sample better than fitting a heavy-tailed family directly. Two routes to the same place, and they do not obviously agree.

Finding

In use as a modelling discipline. The specific question of which constraints are justified by measurement rather than convention is open across most of the site.

References

Jaynes, E. T. Information theory and statistical mechanics, Physical Review, 1957.

CT-12-01Currency Times. Maximum entropy. Note CT-12-01, thecurrencytimes.com.

Close

Z = \sum_x e^{-\beta E(x)}

Partition function

The normalising quantity from which everything else follows.

Queued

H = -\sum_{ij} J_{ij}s_i s_j

Interaction models

Participants as coupled spins. Borrowed from magnetism.

Queued

\xi \sim |T-T_c|^{-\nu}

Critical behaviour

Whether collapses resemble phase transitions or merely look like them.

Open

\chi = \beta\,\mathrm{Var}(M)

Fluctuation-dissipation

Response to a shock related to variance at rest.

Queued

p(x) \propto e^{-\beta E(x)}

Canonical form of outcomes

Whether observed outcome distributions take the Boltzmann shape.

Open

Back to the fields

Field 136 tests

u_i(s^{*}) \ge u_i(s_i,\, s^{*}_{-i})

Game theory and strategy

What happens when the thing being predicted knows it is being predicted, and adjusts.

Open the field

The dividing line between forecasting weather and forecasting people. Rain does not read the forecast. Voters, negotiators, competitors and states all do.

Expression

Test

Status

u_i(s^{*}) \ge u_i(s_i, s^{*}_{-i})

Nash equilibrium

No participant gains by unilateral deviation. Existence is guaranteed, uniqueness is not.

Open

\text{sep. vs pool.}

Signalling games

When a costly action credibly reveals private information and when it does not.

Open

\delta \ge \delta^{*}

Repeated games

Cooperation sustainable if the future is weighted heavily enough. The folk theorem.

Queued

\text{IC}: u_i(\theta_i) \ge u_i(\hat\theta_i)

Mechanism design

Designing rules so that truthful behaviour is the participant's best option.

Queued

\dot{x}_i = x_i(f_i - \bar{f})

Evolutionary stability

Which strategies survive repeated selection rather than one-shot reasoning.

Queued

\text{focal } s^{*}

Coordination and focal points

Schelling's problem. How agreement forms without communication.

Queued

Back to the fields

Field 146 tests

\bar{p} = \sum_i w_i\,p_i

Crowds and aggregation

Whether many imperfect forecasts combine into a better one, and the conditions under which they instead reinforce a shared error.

Open the field

The most reliable finding in the whole of forecasting is that the average of several forecasts beats most of the individuals. The second most reliable is that it stops working the moment they are all reading each other.

Expression

Test

Status

\bar{p} = \sum_i w_i p_i

Linear opinion pool

Weighted averaging of probability forecasts. The baseline everything is measured against.

In use

P(\text{correct}) \to 1

Condorcet jury theorem

Majority accuracy rises with size, provided independence and better than chance.

In use

p^{a} / (p^{a} + (1-p)^{a})

Extremising

Averaging underconfident forecasts produces an underconfident average. Pushing it outward improves scores.

Open

\pi \approx P(\text{event})

Prediction market prices

Whether a traded price is a calibrated probability, and where it is biased.

Open

\text{deff} = 1 + (n-1)\rho

Design effect

Why polling error is routinely several times the quoted margin.

Open

x_i \mid x_{1:i-1}

Information cascades

When sequential observers rationally ignore their own evidence and the crowd locks on.

Queued

Back to the fields

Field 158 tests

BS = \frac{1}{n}\sum_{i=1}^{n}\left(p_i - o_i\right)^{2}

Forecast evaluation

Whether a prediction was any good. Not whether it felt right afterwards, and not whether the event happened.

Open the field

Meteorology built all of this because it had to. It is the only field that has issued millions of probabilistic forecasts and been able to check every one. Everything here is borrowed from weather and belongs everywhere.

Expression

Test

Status

BS = \tfrac{1}{n}\sum (p_i - o_i)^2

Brier score

Mean squared error of a probability forecast. Lower is better, zero is perfect.

In use

BS = \frac{1}{n}\sum_{i=1}^{n}\left(p_i - o_i\right)^{2}, \qquad BS \in [0,1]

Opened August 2026 · last revised August 2026 · 1 revisions

Terms

p_i

Probability stated for event i

o_i

Outcome, one if it happened and zero if not

n

Number of forecasts

BS

Brier score, lower is better

Scope

Applies to any forecast expressed as a probability of a binary event, which covers rainfall, elections, defaults, epidemics, equipment failure and match results identically. Does not apply to point forecasts, which cannot be scored honestly at all.

Why it exists

A forecast of seventy per cent is neither right nor wrong when the event happens. It can only be judged across many forecasts, and only against a rule that cannot be gamed. Glenn Brier proposed this one in 1950 for weather, and it remains the standard because of a property that is easy to state and easy to underrate.

The property that matters

The Brier score is $\textit{proper}$ . Your expected score is best when you state exactly what you believe. Say ninety when you believe seventy and you will do worse in the long run, not better. Most intuitive scoring schemes fail this, and any scheme that fails it is quietly paying people to exaggerate.

What it decomposes into

Murphy showed the score splits into three parts. $\textit{Reliability}$ is whether seventy per cent means seventy per cent. $\textit{Resolution}$ is whether you discriminate between cases at all. $\textit{Uncertainty}$ is the difficulty of the question and belongs to nobody.

That decomposition settles a common confusion. A forecaster who says fifty per cent to everything is perfectly reliable and completely useless, because resolution is zero. Reliability alone is not skill.

The reference point

A raw score is meaningless without a baseline. Compare against the base rate. If a model cannot beat simply predicting the historical frequency, it has no skill regardless of how impressive the number looks.

Why it sits at the centre of this project

It is the only honest way to answer whether any prediction about anything was any good, and it applies unchanged across every domain listed on the front page. Almost nothing outside meteorology does this properly.

Finding

Standing method rather than an open question. Every claim on this site that takes the form of a probability is scored this way or it is not recorded.

References

Brier, G. W. Verification of forecasts expressed in terms of probability, Monthly Weather Review, 1950.
Murphy, A. H. A new vector partition of the probability score, 1973.
Gneiting, T. and Raftery, A. Strictly proper scoring rules, prediction and estimation, 2007.

CT-15-01Currency Times. Brier score. Note CT-15-01, thecurrencytimes.com.

Close

L = -\tfrac{1}{n}\sum \log p_i

Logarithmic score

Punishes confident errors without limit. The strictest of the common rules.

In use

BS = \bar{o}(1-\bar{o}) - \text{RES} + \text{REL}

Murphy decomposition

Splits a score into reliability, resolution and the irreducible difficulty of the question.

In use

\mathbb{E}[o \mid p] = p

Calibration

When you say seventy per cent, it happens seventy per cent of the time. Necessary, nowhere near sufficient.

In use

S(p,o) \text{ proper}

Proper scoring rules

A rule where stating your true belief maximises your expected score. Anything else rewards lying.

Constraint

\text{CRPS} = \int (F(y)-\mathbb{1}\{y \ge x\})^2 dy

Continuous ranked probability

The Brier score generalised to a full predictive distribution.

Queued

SS = 1 - \dfrac{S}{S_{\text{ref}}}

Skill score

Performance against the base rate. A forecast that cannot beat climatology has no skill.

In use

\max \text{ sharpness s.t. calibration}

Sharpness subject to calibration

Gneiting's principle. Be as decisive as you can be while remaining honest.

In use

Back to the fields

Field 1612 tests

\mathbb{E}[\max S_N] \approx \sqrt{2\log N}

Validation

Whether a result survives the search that produced it. Everything else here is subject to this field.

Open the field

If one field is read, it should be this one. Nothing in the other sixteen means anything until it has passed through this.

Expression

Test

Status

\mathbb{E}[\max S_N] \approx \sqrt{2\log N}

False strategy theorem

The best of N worthless candidates still scores well.

Constraint

\mathbb{E}\!\left[\max_{i \le N} SR_i\right] \approx (1-\gamma)\,Z^{-1}\!\left[1-\tfrac{1}{N}\right] + \gamma\,Z^{-1}\!\left[1-\tfrac{1}{Ne}\right]

Terms

N

Number of independent trials

SR_i

Sharpe ratio of trial i

Z^{-1}

Inverse standard normal

\gamma

Euler-Mascheroni constant

Scope

Applies to every result on this site without exception, and retroactively, which is the uncomfortable part. A number recorded before the trial count was tracked cannot be deflated afterwards, because the count is unrecoverable.

The claim

If $N$ candidate models are tested and every one of them is pure noise with no true skill at all, the best will still show a positive Sharpe ratio, and the expected size of that false result grows with $\sqrt{2\log N}$ .

The practical form is blunt. Testing roughly forty-five configurations on five years of daily observations is close to guaranteed to produce something that scores as well as a genuinely skilled forecaster in sample and no better than chance out of it.

Why the sample is smaller than it looks

Roughly twenty-five thousand hourly bars over four years sounds ample, but overlapping features destroy independence. A two-hundred-bar moving average means each observation shares one hundred and ninety-nine bars with its neighbour. Effective sample size falls to somewhere between five hundred and two and a half thousand, and it is the number of trades rather than bars that determines power.

What follows

Record the trial count before searching, not after. Deflate for trials, skewness, kurtosis and sample length. Use combinatorial purged cross-validation rather than a single split. Treat $t > 3.0$ rather than $t > 2.0$ as the threshold for a new discovery.

Finding

Standing constraint rather than an open question. It is not being tested. It is the thing every other test is tested against.

References

Bailey, D. and López de Prado, M. The deflated Sharpe ratio, Journal of Portfolio Management, 2014.
Harvey, C., Liu, Y. and Zhu, H. On the threshold for new factor discovery, 2016.
López de Prado, M. Advances in Financial Machine Learning, 2018.

CT-16-01Currency Times. False strategy theorem. Note CT-16-01, thecurrencytimes.com.

Close

\text{PBO}

Probability of overfitting

Measured across combinatorial splits rather than asserted.

Queued

\text{CPCV}

Combinatorial purged cross-validation

Many train and test paths, not one.

In use

\text{embargo} \approx 0.01T

Purging and embargo

Removes overlapping observations and buffers the boundary.

In use

k\text{-fold}

Standard k-fold

Assumes observations are independent. In ordered data they are not, and overlapping windows carry the future into the training set unseen.

Not valid

252\,/\,126

Walk-forward validation

Six or more rolling windows, never a single split.

In use

\text{IS}/\text{OOS}

Single hold-out split

One division into training and testing. The split point itself becomes a parameter to optimise, so the result reports that choice as much as the method.

Not valid

\text{SPA}

Bootstrap significance

Tested against the full candidate set rather than the winner alone.

In use

t > 3.0

Discovery threshold

Raised from two to three for anything claimed as new.

In use

n_{\text{eff}} \ll n

Effective sample size

Overlapping windows collapse the independent count, often by an order of magnitude.

In use

n_{\min}

Minimum sample for a claim

Observations required before a score means anything at all.

Queued

\text{trials recorded}

Trial registry

The count logged before the search, never reconstructed afterwards.

In use

Back to the fields

Field 1712 tests

t_{\text{obs}} \ge t_{\text{release}}

Data integrity

Whether the data tested on could actually have been known at the moment the test claims it was known.

Open the field

The least glamorous field and the one that invalidates more results than every other cause combined. Most of it is a records problem rather than a statistics problem.

Expression

Test

Status

t_{\text{obs}} \ge t_{\text{release}}

Point-in-time discipline

The engine refuses any read dated before the release timestamp.

In use

\text{first print} \ne \text{current}

Revision handling

Most published series are revised. Testing on final values is testing on the future.

In use

r_{\text{exit}}

Survivorship correction

What left the sample, and why, imputed rather than dropped.

In use

\text{ID} \bowtie \text{ID}

Identifier discipline

Permanent identifiers with date-ranged validity. Never a label that gets reused.

In use

U_t \ne U_{\text{today}}

Universe as it was

The population as it stood then, not as it stands now.

In use

t_{\text{release}} + \Delta

Release-lag buffer

Publication date plus a buffer, never the period the figure describes.

In use

\text{UTC},\ \mu s

Single time base

One authoritative clock, converted only at display.

In use

\mathbb{1}\{\text{break}\}

Definitional breaks

When a series changes what it measures without changing its name.

In use

\times 10^{-k}

Unit and base changes

Redenominations, rebasing, and unit revisions that silently rescale history.

In use

\text{fixed} \to \text{free}

Measurement regime changes

When the rule generating the number is replaced.

In use

\rho: + \to -

Relationship regime breaks

When a stable association reverses, dated and flagged.

In use

|x| > \kappa

Extreme-value contamination

Known defect in a derived label set. Hard block until cleared.

Open

Back to the fields

The subject is prediction itself

The methods come from physics

The standard is verification

What is published here

Point-in-time

The search discount

Multiplicity

Seventeen fields, three constraints

Trend and persistence

Terms

Scope

The objection

What replaced it

Finding

References

Reversion and equilibrium

Dispersion and clustering

Terms

Scope

The objection

What would settle it

Finding

Extremes and rare events

Long memory

Terms

Scope

Reading it

The objection

What would settle it

Finding

Dependence and testing

Terms

Scope

The arithmetic

What is used instead

Finding

Regime and change points

Coupled systems

Correlation networks

Random matrix theory

Terms

Scope

The claim

The arithmetic that should worry you

What it is used for here

Where it came from

Finding

References

Information geometry

Statistical mechanics

Terms

Scope

The principle

Why it matters here

What is under test

Finding

References

Game theory and strategy

Crowds and aggregation

Forecast evaluation

Terms

Scope

Why it exists

The property that matters

What it decomposes into

The reference point

Why it sits at the centre of this project

Finding

References

Validation

Terms

Scope

The claim

Why the sample is smaller than it looks

What follows

Finding

References

Data integrity