Linear Regression

This program computes linear regression for two paired variable sets, producing the line equation Y = C + MX along with the coefficient of determination (R²). After data entry, the user can correct individual pairs before calculation begins. The computed results are displayed both numerically and graphically: a scatter plot of the data points is drawn using four-pixel clusters, and the regression line is overlaid on the same axes. The program uses a scaling factor derived from the maximum X and Y values to fit all data within the 256×176 pixel screen area, and it guards against division-by-zero conditions by tracking the largest observed values.

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Program Structure

The program is divided into clearly separated phases:

1. Introduction (lines 20–90): Two screens of explanatory text about linear regression and R², separated by a keypress subroutine at line 630.
2. Data entry (lines 100–180): Collects n pairs of X/Y values into a 2D array b(n,2).
3. Correction loop (lines 190–260): Allows the user to overwrite any pair by number; loops back to the correction prompt until the user declines.
4. Calculation (lines 270–360): Accumulates sums and computes slope m, intercept c, and R².
5. Output and graphing (lines 370–610): Prints the equation and R², draws axes with tick marks, plots data points as 2×2 pixel squares, and overlays the regression line.

Statistical Calculations

The standard least-squares formulas are used. Slope and intercept are computed at lines 350–360:

• m = (n*xy - sx*sy) / (n*xx - sx²)
• c = (xx*sy - sx*xy) / (n*xx - sx²)

The coefficient of determination R² is computed at line 390 as the square of the Pearson correlation coefficient, using ABS to avoid negative results from floating-point artifacts. The formula applies SQR to the product of the two variance terms and then squares the whole expression, which is algebraically equivalent to the standard R² formula but structured unusually — the squaring is applied outside the SQR, effectively cancelling it for the denominator product while keeping the numerator squared.

Graphing Technique

Axes are drawn at lines 420–460 with tick marks every 15 pixels on both axes. Scale labels showing the maximum axis values are printed at line 490. Notably, AT PI,1 is used, where PI (≈3.14159) is truncated to row 3, placing the Y-axis maximum label near the top of the graph area — an unconventional but functional use of the automatic integer truncation applied to PRINT AT row and column arguments.

Each data point is plotted as a 2×2 block of pixels (lines 510–540), giving better visibility than a single pixel. The regression line is drawn by iterating x from 0 to 240 in steps of 2 (line 560), with bounds-checking at lines 570–600 to skip points that fall outside the drawable area or below zero.

Scaling

Lines 470–480 compute separate X and Y scale factors based on the maximum observed values (bigx and bigy), then select the smaller of the two to ensure the entire dataset fits within the 240×150 pixel graph area without distortion. This uniform scaling preserves the visual slope of the regression line relative to the data.

Notable Techniques and Idioms

• LET sx=0:LET sy=sx:LET xx=sx:LET yy=sx:LET xy=sx (line 110) — efficient chained zero-initialization, assigning from the just-set variable rather than the literal 0 each time.
• PAUSE NOT PI at line 630 — since PI is non-zero, NOT PI evaluates to 0, making PAUSE 0 which waits indefinitely for a keypress. This is a compact keypress-wait idiom.
• Line 270 uses escape codes \{18}\{1} and \{18}\{0} (BRIGHT 1 and BRIGHT 0 control codes embedded in a string) to highlight the “computing” status message.
• Line 340 contains a REM with similar embedded control codes, serving as a developer annotation for the equation form that never affects execution.

Bugs and Anomalies

Location	Issue
Lines 320, 350, 360, 390	ABS x^2 and ABS sx^2 are used instead of x^2 and sx^2. Due to operator precedence, ABS x^2 is parsed as ABS(x)^2, so ABS has no effect for squared values (squaring always yields a non-negative result). This is harmless but unnecessary.
Line 390	The R² formula applies SQR to the product of two variance terms before squaring; this is unconventional structuring. If either variance term is negative (impossible for valid data but theoretically risky), SQR would error. The outer ABS guards against sign issues in the numerator but not the denominator.
Lines 300–310	bigx and bigy only track the maximum value seen; negative data or data where the maximum is 0 would cause a divide-by-zero at line 470 or produce an inverted/collapsed graph.
Line 240	The correction display uses AT i-1,0, which works correctly only if i equals the pair’s original display row. Since the data was printed sequentially from row 0, this aligns with pair number minus one, but only if fewer than ~21 pairs were entered (screen scroll would misalign rows for larger datasets).