Trend Analysis: Mathematical Trend Identification

I. Definition of Trend

A trend represents the general direction of price movement over time. Identifying trends is fundamental to profitable trading as it allows traders to align with market momentum.

II. Formal Trend Definitions

Definition 2.1 (Uptrend - Dow Theory)

An uptrend exists when: $H_i > H_{i-1} \land L_i > L_{i-1}$

Higher highs AND higher lows.

Definition 2.2 (Downtrend - Dow Theory)

A downtrend exists when: $H_i < H_{i-1} \land L_i < L_{i-1}$

Lower highs AND lower lows.

Definition 2.3 (Sideways/Range)

$|H_{max} - H_{min}| < \epsilon \land |L_{max} - L_{min}| < \epsilon$

Price contained within horizontal bounds.

III. Trend Measurement Methods

3.1 Linear Regression

Trend Line: $\hat{P}_t = \beta_0 + \beta_1 t$

Where: $\beta_1 = \frac{\sum(t_i - \bar{t})(P_i - \bar{P})}{\sum(t_i - \bar{t})^2}$

$\beta_0 = \bar{P} - \beta_1 \bar{t}$

Interpretation:

β₁ > 0 → Uptrend
β₁ < 0 → Downtrend
β₁ ≈ 0 → Sideways

3.2 Moving Average Slope

Trend Direction: $Trend = sign\left(\frac{dMA_n}{dt}\right) \approx sign(MA_n(t) - MA_n(t-k))$

3.3 Higher Timeframe Confirmation

$TrendStrength = \sum_{TF} w_{TF} \times Trend_{TF}$

Where TF ∈ {1H, 4H, Daily, Weekly}.

IV. Average Directional Index (ADX)

Definition 4.1 (Directional Movement)

$+DM = H_t - H_{t-1} \text{ if } H_t - H_{t-1} > L_{t-1} - L_t \text{ else } 0$ $-DM = L_{t-1} - L_t \text{ if } L_{t-1} - L_t > H_t - H_{t-1} \text{ else } 0$

Definition 4.2 (Directional Indicators)

$+DI = 100 \times \frac{EMA_n(+DM)}{ATR_n}$ $-DI = 100 \times \frac{EMA_n(-DM)}{ATR_n}$

Definition 4.3 (ADX)

$DX = 100 \times \frac{|+DI - -DI|}{+DI + -DI}$ $ADX = EMA_n(DX)$

Interpretation

| ADX Value | Trend Strength | |-----------|----------------| | 0-20 | Weak/No trend | | 20-40 | Moderate trend | | 40-60 | Strong trend | | 60-100 | Very strong trend |

V. Trend Line Construction

5.1 Uptrend Line

Connect two or more swing lows: $TrendLine(t) = L_1 + \frac{L_2 - L_1}{t_2 - t_1}(t - t_1)$

5.2 Downtrend Line

Connect two or more swing highs: $TrendLine(t) = H_1 + \frac{H_2 - H_1}{t_2 - t_1}(t - t_1)$

5.3 Validity Criteria

Minimum 2 touches (3+ preferred)
Angle between 20° and 60°
No penetrations (or brief penetrations only)

VI. Trend Channels

Definition 6.1 (Trend Channel)

Parallel lines containing price action: $Upper = TrendLine + k$ $Lower = TrendLine - k$

Where k is the average distance to the primary trend line.

6.2 Channel Trading

Buy at lower channel boundary in uptrend
Sell at upper channel boundary in downtrend
Breakout signals potential trend acceleration

VII. Mathematical Trend Filters

7.1 Price vs Moving Average

$TrendFilter = sign(P_t - MA_n(t))$

7.2 Moving Average Cross

$TrendFilter = sign(MA_{fast}(t) - MA_{slow}(t))$

7.3 Donchian Channel

$Upper_n = \max_{i=0}^{n-1}(H_{t-i})$ $Lower_n = \min_{i=0}^{n-1}(L_{t-i})$

Breakout above Upper → Uptrend initiation Breakout below Lower → Downtrend initiation

VIII. Trend Reversals

8.1 Reversal Signals

Break of trend line
Lower high in uptrend / Higher low in downtrend
Moving average cross
Divergence on momentum indicators

8.2 Trend Exhaustion Model

$Exhaustion = f(ADX_{declining}, Divergence, Volume_{declining})$

IX. Exercises

Exercise 1: Calculate the slope of linear regression for prices [100, 102, 101, 104, 106].

Exercise 2: Given +DI=25, -DI=15, calculate DX.

Exercise 3: Draw trend lines for swing lows at (1, 100), (5, 102), (10, 105).

Exercise 4: Prove that in a perfect uptrend with constant slope, ADX approaches 100.

X. References

Edwards, R.D. & Magee, J. (1948). Technical Analysis of Stock Trends
Wilder, J.W. (1978). New Concepts in Technical Trading Systems
Murphy, J.J. (1999). Technical Analysis of the Financial Markets