---
description: "Predicts a stock's future T-day cumulative return using the K-nearest-neighbor algorithm on normalized price and volume features, then trades based on the predicted return signal."
tags: [stocks, machine-learning, knn, prediction]
---

# Machine Learning — Single-Stock KNN

**Section**: 3.17 | **Asset Class**: Stocks | **Type**: Machine Learning / Prediction

## Overview
This single-stock strategy uses the k-nearest-neighbor (KNN) algorithm to predict future cumulative stock returns based on a set of predictor (feature) variables derived from the stock's own price and volume history. For each stock, the model is trained independently using only that stock's data (no cross-sectional information). The predicted return is then used to generate long/short signals.

## Construction / Signal
**Target variable** — cumulative return over the next T trading days:
```
Y(t) = P(t-T) / P(t) - 1                                  (332)
```
(t ascending corresponds to going back in time; t=0 is today)

**Predictor variables** (moving averages of volume and price over varying windows T_1, T_2, T_3, ...):
```
X_1(t) = (1/T_1) * sum_{s=1}^{T_1} V(t+s)                (333)   [volume MA]
X_2(t) = (1/T_2) * sum_{s=1}^{T_2} P(t+s)                (334)   [price MA 1]
X_3(t) = (1/T_3) * sum_{s=1}^{T_3} P(t+s)                (335)   [price MA 2]
...                                                        (336)
```

Predictor variables are normalized to [0, 1] using the training period's min/max:
```
X_tilde_a(t) = (X_a(t) - X_a^-) / (X_a^+ - X_a^-)       (337)
```
where `X_a^+` and `X_a^-` are the max and min of `X_a(t)` over the training period.

**KNN prediction** — for a given t, find the k nearest neighbors of `X_tilde_a(t)` among training points `t' = t+1, t+2, ..., t+T_*` using Euclidean distance:
```
[D(t, t')]^2 = sum_{a=1}^{m} (X_tilde_a(t) - X_tilde_a(t'))^2   (338)
```

**Predicted return** (simple average):
```
Y(t) = (1/k) * sum_{alpha=1}^{k} Y(t'_alpha(t))           (339)
```

Alternatively, fit a linear model with weights w_alpha and intercept v:
```
Y(t) = sum_{alpha=1}^{k} Y(t'_alpha(t)) w_alpha + v       (340)
```
trained by regressing Y(t) on the k neighbor returns over M values of t.

**Trading signal** (z_1, z_2 are trader-defined thresholds):
```
Signal = { Establish long position if Y > z_1
          { Liquidate long position if Y <= z_2
          { Establish short position if Y < -z_1
          { Liquidate short position if Y >= -z_2           (341)
```

## Entry / Exit Rules
- **Long entry**: Predicted cumulative return `Y = Y(0) > z_1`
- **Long exit**: Predicted return `Y <= z_2` (where z_2 <= z_1)
- **Short entry**: Predicted return `Y < -z_1`
- **Short exit**: Predicted return `Y >= -z_2`
- All thresholds must be backtested out-of-sample.

## Key Parameters
- **Number of neighbors k**: Typically `k = floor(sqrt(T_*))` or `k = ceiling(sqrt(T_*))` (T_* = training sample size)
- **Training sample size T_***: Number of historical time points used for training
- **Prediction horizon T**: Number of trading days for the target return
- **Feature set m**: Number and type of predictor variables (volume MAs, price MAs)
- **Thresholds z_1, z_2**: Entry and exit thresholds for signals (backtested)
- **Train/validation split**: E.g., 60% training, 40% cross-validation
- **Distance metric**: Euclidean (Eq. 338) or Manhattan distance

## Variations
- **Weighted KNN**: Use distance-based weights for the k neighbors instead of uniform averaging (Eq. 340)
- **Cross-sectional extension**: Compute expected returns Y_i for N stocks and use as inputs to cross-sectional mean-reversion or other multi-stock strategies
- **Alternative features**: Fundamental data, earnings surprises, sentiment indicators in addition to price/volume

## Notes
- This is a single-stock strategy: each stock's model is trained on that stock's own price/volume data only.
- The strategy must be backtested strictly out-of-sample; data leakage is a critical risk.
- Simple uniform KNN (Eq. 339) has no parameters to train; the linear model (Eq. 340) requires cross-validation and is prone to out-of-sample instability.
- k can be optimized via backtesting; common heuristic: `k = floor(sqrt(T_*))`.
- Typical holding period: T trading days (matching the prediction horizon).
- Training/cross-validation split: e.g., 60%/40%.