nx_sketch_correlation.nx source
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1// sketch_correlation.nx -- streaming Pearson correlation + linear regression.
2//
3// Bivariate streaming primitive. For paired observations (x, y) consumed
4// one at a time, computes:
5// r -- Pearson correlation coefficient, in PPM in [-1_000_000, 1_000_000]
6// slope -- least-squares regression slope m: y = m*x + b
7// intercept -- least-squares regression intercept b
8//
9// SIX ACCUMULATORS (integer-exact under overflow budget):
10// n = count
11// sum_x = Σ x_i
12// sum_y = Σ y_i
13// sum_xy = Σ x_i * y_i
14// sum_xx = Σ x_i²
15// sum_yy = Σ y_i²
16//
17// FORMULAS:
18// cov_xy = n * sum_xy - sum_x * sum_y
19// var_x = n * sum_xx - sum_x²
20// var_y = n * sum_yy - sum_y²
21// r = cov_xy / sqrt(var_x * var_y)
22// slope = cov_xy / var_x
23// intercept = (sum_y - slope * sum_x) / n
24//
25// OVERFLOW BUDGET:
26// sum_xx grows as n * x_max². For x_max=2^15, n_max=2^32 -> sum_xx
27// max ~ 2^62. And the cross-term computations need extra margin.
28// nx_corr_safe_p flags unsafe inputs.
29//
30// USE CASES:
31// - SRE: correlate metric series (CPU vs latency)
32// - finance: returns correlation
33// - science: regression on streamed measurements
34//
35// LOSSLESS-LANGUAGE DISCIPLINE: r returned in PPM with NX_ENV_ABS,
36// param_a = 0 (exact under budget). Production tier.
37
38// nx_safety_envelope:
39// intended_use: AUTO_APPLIED -- primitive-specific tuning queued
40// sil_target: SIL1
41// evidence: [bulk_applied_2026-05-16, see-file-comment-for-detail]
42// verdict: NOT_YET_EVALUATED
43
44import "nx_syscalls.nx"
45import "nx_sketch_types.nx"
46import "nx_vecmath.nx"
47
48const NX_CORR_VALUE_LIMIT: i64 = 32768 // |x|, |y| < 2^15 for safe sum_xx
49
50struct Correlation {
51 n: i64,
52 sum_x: i64,
53 sum_y: i64,
54 sum_xy: i64,
55 sum_xx: i64,
56 sum_yy: i64,
57 has_data: i64,
58}
59
60// === construction =================================================
61
62func nx_corr_alloc() -> *Correlation {
63 let raw: *u8 = sys_mmap(64)
64 let c: *Correlation = raw as *Correlation
65 c.n = 0
66 c.sum_x = 0
67 c.sum_y = 0
68 c.sum_xy = 0
69 c.sum_xx = 0
70 c.sum_yy = 0
71 c.has_data = 0
72 return c
73}
74
75// === isqrt =======================================================
76
77func nx_corr_isqrt(x: i64) -> i64 { return vm_isqrt(x) }
78
79// === overflow guard ==============================================
80
81func nx_corr_safe_p(x: i64, y: i64) -> i64 {
82 var ax: i64 = x
83 if ax < 0 { ax = -ax }
84 var ay: i64 = y
85 if ay < 0 { ay = -ay }
86 if ax >= NX_CORR_VALUE_LIMIT { return 0 }
87 if ay >= NX_CORR_VALUE_LIMIT { return 0 }
88 return 1
89}
90
91// === add ==========================================================
92
93func nx_corr_add(c: *Correlation, x: i64, y: i64) -> i64 {
94 if nx_corr_safe_p(x, y) == 0 { return -1 }
95 c.n = c.n + 1
96 c.sum_x = c.sum_x + x
97 c.sum_y = c.sum_y + y
98 c.sum_xy = c.sum_xy + x * y
99 c.sum_xx = c.sum_xx + x * x
100 c.sum_yy = c.sum_yy + y * y
101 c.has_data = 1
102 return 0
103}
104
105// === correlation coefficient (in PPM) ============================
106//
107// r = cov_xy / sqrt(var_x * var_y) where cov, var are the *centered*
108// values multiplied by n (so we avoid recomputing means).
109// Returns PPM in [-1_000_000, 1_000_000].
110
111func nx_corr_r_ppm(c: *Correlation) -> i64 {
112 if c.n < 2 { return 0 }
113 let cov_xy: i64 = c.n * c.sum_xy - c.sum_x * c.sum_y
114 let var_x: i64 = c.n * c.sum_xx - c.sum_x * c.sum_x
115 let var_y: i64 = c.n * c.sum_yy - c.sum_y * c.sum_y
116 if var_x <= 0 { return 0 }
117 if var_y <= 0 { return 0 }
118 // r = cov_xy / sqrt(var_x * var_y); compute in PPM.
119 // Avoid overflow in (var_x * var_y) by scaling cov upfront.
120 // r_ppm = (cov_xy * 1_000_000) / sqrt(var_x * var_y)
121 // Use staged isqrt: sqrt(var_x) * sqrt(var_y) (approximate but safe).
122 let sx: i64 = nx_corr_isqrt(var_x)
123 let sy: i64 = nx_corr_isqrt(var_y)
124 if sx == 0 { return 0 }
125 if sy == 0 { return 0 }
126 let denom: i64 = sx * sy
127 if denom == 0 { return 0 }
128 let r: i64 = (cov_xy * 1000000) / denom
129 // Clamp to [-1_000_000, 1_000_000].
130 if r > 1000000 { return 1000000 }
131 if r < -1000000 { return -1000000 }
132 return r
133}
134
135// === regression slope and intercept (in PPM) =====================
136//
137// slope_ppm = (n * sum_xy - sum_x * sum_y) / (n * sum_xx - sum_x²) * 1_000_000
138// intercept = mean_y - slope * mean_x (slope here is fractional;
139// scaled-PPM math: intercept = (sum_y * 1_000_000 - slope_ppm * sum_x) / (n * 1_000_000))
140
141func nx_corr_slope_ppm(c: *Correlation) -> i64 {
142 if c.n < 2 { return 0 }
143 let cov_xy: i64 = c.n * c.sum_xy - c.sum_x * c.sum_y
144 let var_x: i64 = c.n * c.sum_xx - c.sum_x * c.sum_x
145 if var_x <= 0 { return 0 }
146 return (cov_xy * 1000000) / var_x
147}
148
149func nx_corr_intercept(c: *Correlation) -> i64 {
150 if c.n < 2 { return 0 }
151 let slope_ppm: i64 = nx_corr_slope_ppm(c)
152 // intercept = mean_y - slope * mean_x
153 // = (sum_y - slope_ppm * sum_x / 1_000_000) / n
154 let slope_times_sum_x: i64 = (slope_ppm * c.sum_x) / 1000000
155 return (c.sum_y - slope_times_sum_x) / c.n
156}
157
158// === typed queries ===============================================
159
160func nx_corr_query_r(c: *Correlation) -> *ApproxI64 {
161 let r: i64 = nx_corr_r_ppm(c)
162 return nx_approx_new(r, NX_ENV_ABS, 1,
163 1000000000,
164 NX_MATURITY_PRODUCTION,
165 NX_ADV_HONEST)
166}
167
168func nx_corr_query_slope(c: *Correlation) -> *ApproxI64 {
169 let s: i64 = nx_corr_slope_ppm(c)
170 return nx_approx_new(s, NX_ENV_ABS, 1,
171 1000000000,
172 NX_MATURITY_PRODUCTION,
173 NX_ADV_HONEST)
174}
175
176// === merge ========================================================
177
178func nx_corr_merge(a: *Correlation, b: *Correlation) -> *Correlation {
179 let out: *Correlation = nx_corr_alloc()
180 out.n = a.n + b.n
181 out.sum_x = a.sum_x + b.sum_x
182 out.sum_y = a.sum_y + b.sum_y
183 out.sum_xy = a.sum_xy + b.sum_xy
184 out.sum_xx = a.sum_xx + b.sum_xx
185 out.sum_yy = a.sum_yy + b.sum_yy
186 if a.has_data == 1 { out.has_data = 1 }
187 if b.has_data == 1 { out.has_data = 1 }
188 return out
189}
190
191func nx_corr_memory_bytes(c: *Correlation) -> i64 {
192 return 64
193}
194
195func nx_corr_count(c: *Correlation) -> i64 {
196 return c.n
197}