Methodology

1📈Trend forecast (linear regression / drift)

For a price series p₀…pₙ₋₁ we fit an ordinary least-squares line of price on index and extrapolate forward by h periods:

slope  β = Σ(xᵢ − x̄)(yᵢ − ȳ) / Σ(xᵢ − x̄)²
intercept α = ȳ − β·x̄
prediction(h) = α + β·(n − 1 + h)
direction = up if predicted > current, down if <, else flat (0.05% dead-band)
confidence = R² = (Σ(xᵢ−x̄)(yᵢ−ȳ))² / [Σ(xᵢ−x̄)²·Σ(yᵢ−ȳ)²]

In plain terms: draw the straight line that best fits recent prices, then follow its slope forward. Direction comes from the trend, not a lagging average — and a jagged, unpredictable series honestly scores a low R².

2📊Confidence band (prediction interval)

The shaded band around the forecast is a real prediction interval:

residual std error  RSE = √( Σ(yᵢ − ŷᵢ)² / (n − 2) )
half-width(h) = 1.96 · RSE · √(1 + h/n)
band = [predicted − half-width, predicted + half-width]   (floored at 0)

In plain terms: the band is how much the line typically missed by, widened for distance. Further out means less certain — so the cone visibly fans open.

3🧪Alternative models & backtesting

Five transparent models are compared: linear-regression drift, EMA trend, momentum, mean-reversion, and a naive baseline (tomorrow = today). Evaluation is walk-forward, out-of-sample: at each point t the model is fit on p₀…pₜ and scored against the real p₍ₜ₊ₕ₎ — it never sees the future point it is judged on (no look-ahead bias).

Directional accuracy = correct up/down/flat calls / total
MAE  = mean |predicted − actual|
MAPE = mean |predicted − actual| / |actual| · 100
Confusion matrix = predicted direction × actual direction

In plain terms: the naive baseline is the bar every model must clear to be worth anything. If a fancy model can't beat "tomorrow = today", it adds nothing.

4💹Strategy backtest (signal → PnL)

Directional forecasts can be evaluated as a trading rule, not just a classification score. The engine walks forward on historical prices: at each point t the model sees only p₀…pₜ, emits a signal for the next h periods, and simulates a position held until t+h. Trades are non-overlapping — the next decision happens only after the prior horizon closes.

Long-only: enter when direction = up and confidence ≥ min_confidence
Long/short:  long on up, short on down (same confidence gate)
Round-trip cost = 2 × (fee_bps + slippage_bps) / 10 000   (default 10 + 5 bps per side)
Metrics: total return, CAGR, Sharpe (per-trade), max drawdown, profit factor, win rate
Benchmark: buy-and-hold over the same entry→exit window, same costs

In plain terms: turn each call into a real trade with fees and slippage, then compare against simply holding. It is not proof of investable alpha — past walk-forward PnL does not guarantee future results.

5🔬Statistical validation

Accuracy alone does not prove edge. Phase B adds inference on top of walk-forward evaluation:

Wilson 95% CI     — directional accuracy vs random (50%)
Bootstrap CI      — mean trade / excess return (2000 resamples)
Permutation test  — sign-flip p-value for positive mean return
Calibration bins  — confidence (R²) vs empirical hit rate
Walk-forward folds — rolling train/test windows with embargo
Bonferroni        — α/5 across models; one declared best

In plain terms: we ask "could this just be luck?" A result marked significant at 95% is unlikely under a random null — but that is still not a guarantee of future profit.

6🎲Portfolio Monte Carlo (GBM)

Each asset follows Geometric Brownian Motion. Daily drift μ and volatility σ are estimated from that asset’s own log-returns (drift shrunk 50% toward zero to avoid over-extrapolating short history):

rᵢ = ln(pᵢ / pᵢ₋₁)          (log-returns)
μ = 0.5 · mean(r) ,  σ = stdev(r)
daily step:  Sₜ₊₁ = Sₜ · exp( (μ − ½σ²) + σ·Z ) ,  Z ~ N(0,1)
We run thousands of seeded paths, then report:
  expected/median return, 95% VaR (5th percentile), P(profit),
  and a percentile fan (p5/p25/p50/p75/p95).

In plain terms: roll the dice thousands of times on each asset, then read off the range of outcomes — the median, the unlucky tail, and the odds of finishing green.

7⚖️Sharpe ratio & weight optimizer

Sharpe = (mean horizon return − rf) / stdev(returns) · √(365/days)
   rf = 4% annual risk-free, prorated to the horizon

In plain terms: reward per unit of bumpiness. The optimizer searches the weight simplex (randomized sampling + local refinement) for the highest-Sharpe mix and returns the best found — an honest heuristic, not a proven global optimum.

8🧬Alpha++ (ensemble v2)

Alpha++ amplifies the investable ensemble with market breadth, walk-forward quality gates, and a unified alpha score that blends signal strength with historical edge:

amplified_vote = 85%·ensemble_v1 + 15%·breadth_tilt(advancing_pct)
composite = (regime_gated_vote + 1) / 2 × 100 × quality_mult × confluence_bonus
alpha_score = 45%·composite + 25%·backtest_accuracy + 20%·strategy_α + 10%·win_rate
            × confluence × (0.85 + 0.15·confidence)

Regime min score: risk_on 68 · chop 72 · risk_off 78 (longs need ≥82 in risk_off)
Grades: A≥85 · B≥75 · C≥65 · D≥50 · F<50

In plain terms: agreement across signals is rewarded, weak walk-forward accuracy and wild volatility are penalised. An empty alpha watchlist in choppy regimes is expected — restraint, not an error.

9📡Forward test (track record)

The forward log is a live, immutable test of the frozen ensemble model. Each worker sync may open new qualifying long signals; trades close automatically at horizon maturity. No retroactive edits.

Open: investable long, composite/alpha ≥ threshold, liquid volume
Hold: fixed horizon (steps × sync interval)
Close: mark-to-market at maturity; apply round-trip costs (fee + slippage bps)
Equity: compound net returns scaled by vol-target position size
Metrics: total return, win rate, Sharpe (per-trade), max drawdown

In plain terms: a public, tamper-proof paper-trading log. Not audited fund performance — early windows are small, so read cumulative stats with caution until the 90-day target elapses.

✦📖Glossary

The terms that recur above, in one place:

R²

Share of price variance the trend line explains (0–1). Used as honest confidence — a noisy series scores low.

Walk-forward

Fit only on the past, score on the next unseen point. No look-ahead, so accuracy isn't inflated by hindsight.

MAE / MAPE

Mean absolute error (in price) and the same as a percentage — how far predictions land from reality on average.

Directional accuracy

How often the up/down/flat call was correct, independent of magnitude.

Sharpe ratio

Return earned per unit of volatility, annualised. Higher means a smoother path to the same return.

VaR (95%)

Value at Risk — the loss the 5% worst simulated outcomes meet or exceed. A tail-risk yardstick.

GBM

Geometric Brownian Motion — the standard random-walk-with-drift model for prices that can't go below zero.

Bootstrap / permutation

Resampling tests that ask: could this result happen by luck? They turn a metric into a p-value.

Bonferroni

A stricter significance bar (α/N) applied when testing many models, so one isn't crowned by chance alone.

Drawdown

The largest peak-to-trough drop in equity — how much pain the strategy put you through along the way.

§References

• Hull, J. — Options, Futures, and Other Derivatives (GBM, risk-neutral pricing).
• Sharpe, W. (1966) — “Mutual Fund Performance” (the Sharpe ratio).
• Hyndman & Athanasopoulos — Forecasting: Principles and Practice (walk-forward evaluation, MAPE).
• Tsay, R. — Analysis of Financial Time Series (log-returns, volatility).