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

1. Break of trend line
2. Lower high in uptrend / Higher low in downtrend
3. Moving average cross
4. 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