# Global Accuracy Study

**Study date:** February 2025
**pray-calc version:** 2.0.0
**Reference:** MSC (Moonsighting Committee Worldwide) observation-calibrated model
**Test cases:** 18 city/date combinations
**Latitude range:** 6.2°S (Jakarta) to 51.5°N (London)

See [Methodology](Research-Methodology) for the scoring approach, reference standard, and test infrastructure.

---

## Summary Table

Error is `method_time − MSC_reference` in minutes. Positive = later than MSC, negative = earlier.
`N/A` = sun never reaches the required depression angle (method is undefined for that case).

| #   | City               | Date   | Lat     | Elev  | Sunrise | Maghrib | PCD Fajr° | PCD Isha° | MSC Fajr ref    | MSC Isha ref   |
| --- | ------------------ | ------ | ------- | ----- | ------- | ------- | --------- | --------- | --------------- | -------------- |
| 1   | New York, USA      | Jun 21 | 40.71°N | 10m   | 05:25   | 20:30   | 16.21°    | 12.41°    | 03:35 (110 min) | 21:49 (79 min) |
| 2   | New York, USA      | Dec 21 | 40.71°N | 10m   | 07:16   | 16:32   | 17.48°    | 17.12°    | 05:40 (96 min)  | 18:06 (94 min) |
| 3   | New York, USA      | Mar 21 | 40.71°N | 10m   | 06:56   | 19:09   | 17.48°    | 15.28°    | 05:27 (89 min)  | 20:26 (77 min) |
| 4   | Toronto, Canada    | Jun 21 | 43.65°N | 76m   | 05:36   | 21:02   | 15.33°    | 11.71°    | 03:43 (113 min) | 22:22 (80 min) |
| 5   | London, UK         | Jun 21 | 51.51°N | 11m   | 04:43   | 21:21   | 11.88°    | 10.00°    | 02:43 (120 min) | 22:41 (80 min) |
| 6   | London, UK         | Dec 21 | 51.51°N | 11m   | 08:03   | 15:53   | 14.69°    | 14.24°    | 06:21 (102 min) | 17:32 (99 min) |
| 7   | Istanbul, Turkey   | Jun 21 | 41.01°N | 114m  | 05:32   | 20:39   | 16.28°    | 12.40°    | 03:41 (111 min) | 21:58 (79 min) |
| 8   | Makkah, S. Arabia  | Jun 21 | 21.42°N | 277m  | 05:39   | 19:05   | 19.79°    | 16.56°    | 04:05 (94 min)  | 20:22 (77 min) |
| 9   | Makkah, S. Arabia  | Dec 21 | 21.42°N | 277m  | 06:54   | 17:43   | 19.58°    | 19.36°    | 05:28 (86 min)  | 19:08 (85 min) |
| 10  | Tehran, Iran       | Jun 21 | 35.69°N | 1191m | 04:48   | 19:23   | 17.95°    | 14.09°    | 03:02 (106 min) | 20:42 (79 min) |
| 11  | Cairo, Egypt       | Jun 21 | 30.06°N | 23m   | 04:54   | 18:59   | 18.73°    | 14.97°    | 03:13 (101 min) | 20:17 (78 min) |
| 12  | Cairo, Egypt       | Dec 21 | 30.06°N | 23m   | 06:46   | 16:59   | 19.13°    | 18.72°    | 05:15 (91 min)  | 18:28 (89 min) |
| 13  | Karachi, Pakistan  | Jun 21 | 24.86°N | 8m    | 05:43   | 19:24   | 19.43°    | 16.01°    | 04:06 (97 min)  | 20:42 (78 min) |
| 14  | Dhaka, Bangladesh  | Jun 21 | 23.81°N | 4m    | 05:12   | 18:48   | 19.49°    | 16.19°    | 03:36 (96 min)  | 20:06 (78 min) |
| 15  | Jakarta, Indonesia | Jun 21 | 6.21°S  | 8m    | 06:01   | 17:47   | 18.72°    | 18.72°    | 04:43 (78 min)  | 19:05 (78 min) |
| 16  | Singapore          | Dec 21 | 1.35°N  | 15m   | 07:01   | 19:04   | 18.23°    | 18.23°    | 05:45 (76 min)  | 20:20 (76 min) |
| 17  | Almaty, Kazakhstan | Jun 21 | 43.22°N | 848m  | 04:12   | 19:36   | 15.73°    | 12.05°    | 02:19 (113 min) | 20:56 (80 min) |
| 18  | Riyadh, S. Arabia  | Mar 21 | 24.69°N | 620m  | 05:55   | 18:04   | 20.04°    | 18.25°    | 04:31 (84 min)  | 19:20 (76 min) |

---

## Accuracy Rankings

**Mean Absolute Error vs MSC reference, sorted by combined MAE.**
`n_Fajr` / `n_Isha` = number of valid (non-NaN) results included in MAE calculation.

| Rank  | Method              | Fajr MAE (min) | Isha MAE (min) | Combined MAE | n_Fajr | n_Isha |
| ----- | ------------------- | -------------- | -------------- | ------------ | ------ | ------ |
| 1     | MSC                 | 0.00           | 0.00           | 0.00         | 18     | 18     |
| **2** | **★ PCD (Dynamic)** | **0.64**       | **1.30**       | **0.97**     | **18** | **18** |
| 3     | Qatar               | 10.21          | 10.22          | 10.21        | 17     | 18     |
| 4     | UAQ                 | 11.08          | 10.22          | 10.65        | 17     | 18     |
| 5     | MWL                 | 10.21          | 17.09          | 13.65        | 17     | 17     |
| 6     | DIBT                | 10.21          | 17.09          | 13.65        | 17     | 17     |
| 7     | IGUT                | 9.81           | 17.98          | 13.90        | 17     | 18     |
| 8     | Kuwait              | 10.21          | 18.92          | 14.56        | 17     | 17     |
| 9     | Karachi             | 10.21          | 20.89          | 15.55        | 17     | 17     |
| 10    | SAMR                | 11.54          | 19.94          | 15.74        | 17     | 18     |
| 11    | Egypt               | 14.49          | 18.92          | 16.71        | 17     | 17     |
| 12    | MUIS                | 17.79          | 20.89          | 19.34        | 17     | 17     |
| 13    | ISNA                | 19.44          | 19.94          | 19.69        | 18     | 18     |
| 14    | ISNACA              | 26.96          | 18.45          | 22.71        | 18     | 18     |
| 15    | UOIF                | 31.59          | 20.00          | 25.80        | 18     | 18     |

**Notes on n values:** Methods with `n_Fajr = 17` or `n_Isha = 17` had one NaN result (typically London summer). Methods returning NaN for that case are not penalized in MAE: the actual failure rate is noted in the High-Latitude section below.

---

## Per-Case Detail: Fajr Errors by Method

Error in minutes vs MSC reference (`+` = later, `−` = earlier). `N/A` = sun does not reach required depression.

| City / Date   | PCD      | UOIF  | ISNACA | ISNA  | SAMR  | IGUT  | MWL   | DIBT  | Karachi | Kuwait | UAQ   | Qatar | Egypt | MUIS  |
| ------------- | -------- | ----- | ------ | ----- | ----- | ----- | ----- | ----- | ------- | ------ | ----- | ----- | ----- | ----- |
| NY Jun        | **−0.2** | +34.0 | +26.3  | +10.2 | +1.7  | −13.6 | −16.4 | −16.4 | −16.4   | −16.4  | −21.2 | −16.4 | −31.3 | −36.6 |
| NY Dec        | **+0.1** | +30.6 | +25.0  | +13.8 | +8.3  | −1.1  | −2.7  | −2.7  | −2.7    | −2.7   | −5.5  | −2.7  | −10.9 | −13.7 |
| NY Mar        | **+0.2** | +30.0 | +24.6  | +13.8 | +8.3  | −1.0  | −2.7  | −2.7  | −2.7    | −2.7   | −5.4  | −2.7  | −11.0 | −13.7 |
| Toronto Jun   | **−0.8** | +29.9 | +21.1  | +2.5  | −7.6  | −26.2 | −29.8 | −29.8 | −29.8   | −29.8  | −36.0 | −29.8 | −49.3 | −56.7 |
| London Jun    | **−0.4** | −2.4  | −20.3  | −87.8 | N/A   | N/A   | N/A   | N/A   | N/A     | N/A    | N/A   | N/A   | N/A   | N/A   |
| London Dec    | **+0.1** | +18.6 | +11.6  | −2.1  | −8.8  | −20.2 | −22.2 | −22.2 | −22.2   | −22.2  | −25.6 | −22.2 | −32.2 | −35.5 |
| Istanbul Jun  | **−0.9** | +34.4 | +26.6  | +10.2 | +1.6  | −13.9 | −16.8 | −16.8 | −16.8   | −16.8  | −21.7 | −16.8 | −32.0 | −37.4 |
| Makkah Jun    | **−0.8** | +39.5 | +34.4  | +24.3 | +19.1 | +10.2 | +8.7  | +8.7  | +8.7    | +8.7   | +6.0  | +8.7  | +0.7  | −2.0  |
| Makkah Dec    | **−0.5** | +34.1 | +29.5  | +20.3 | +15.8 | +8.0  | +6.7  | +6.7  | +6.7    | +6.7   | +4.4  | +6.7  | −0.1  | −2.4  |
| Tehran Jun    | **−2.5** | +38.6 | +32.1  | +18.6 | +11.6 | −0.7  | −2.9  | −2.9  | −2.9    | −2.9   | −6.6  | −2.9  | −14.3 | −18.2 |
| Cairo Jun     | **−0.2** | +40.1 | +34.4  | +22.6 | +16.6 | +6.2  | +4.3  | +4.3  | +4.3    | +4.3   | +1.2  | +4.3  | −5.1  | −8.4  |
| Cairo Dec     | **0.0**  | +34.8 | +29.9  | +20.1 | +15.2 | +7.0  | +5.5  | +5.5  | +5.5    | +5.5   | +3.1  | +5.5  | −1.7  | −4.1  |
| Karachi Jun   | **−0.1** | +40.4 | +35.1  | +24.4 | +19.0 | +9.6  | +7.9  | +7.9  | +7.9    | +7.9   | +5.1  | +7.9  | −0.5  | −3.3  |
| Dhaka Jun     | **−0.3** | +39.8 | +34.6  | +24.1 | +18.8 | +9.6  | +7.9  | +7.9  | +7.9    | +7.9   | +5.2  | +7.9  | −0.3  | −3.1  |
| Jakarta Jun   | **−0.3** | +29.0 | +24.7  | +15.9 | +11.6 | +4.2  | +2.9  | +2.9  | +2.9    | +2.9   | +0.7  | +2.9  | −3.7  | −5.9  |
| Singapore Dec | **−0.4** | +26.9 | +22.5  | +13.8 | +9.4  | +2.0  | +0.7  | +0.7  | +0.7    | +0.7   | −1.5  | +0.7  | −5.9  | −8.1  |
| Almaty Jun    | **−3.0** | +30.8 | +22.2  | +4.0  | −5.7  | −23.7 | −27.2 | −27.2 | −27.2   | −27.2  | −33.1 | −27.2 | −45.7 | −52.6 |
| Riyadh Mar    | **−0.9** | +34.8 | +30.4  | +21.5 | +17.1 | +9.5  | +8.2  | +8.2  | +8.2    | +8.2   | +6.0  | +8.2  | +1.5  | −0.7  |

---

## Per-Case Detail: Isha Errors by Method

| City / Date   | PCD       | UOIF  | ISNACA | ISNA   | SAMR   | IGUT  | MWL   | DIBT  | Karachi | Kuwait | UAQ   | Qatar | Egypt | MUIS  |
| ------------- | --------- | ----- | ------ | ------ | ------ | ----- | ----- | ----- | ------- | ------ | ----- | ----- | ----- | ----- |
| NY Jun        | **+0.4**  | −2.8  | +4.9   | +21.1  | +21.1  | +12.9 | +38.4 | +38.4 | +47.6   | +43.0  | +11.0 | +11.0 | +43.0 | +47.6 |
| NY Dec        | **−0.1**  | −28.6 | −23.0  | −11.8  | −11.8  | −17.4 | −0.8  | −0.8  | +4.7    | +2.0   | −4.0  | −4.0  | +2.0  | +4.7  |
| NY Mar        | **−0.9**  | −18.7 | −13.3  | −2.5   | −2.5   | −7.9  | +8.5  | +8.5  | +14.0   | +11.2  | +13.0 | +13.0 | +11.2 | +14.0 |
| Toronto Jun   | **+0.8**  | +3.3  | +12.1  | +30.7  | +30.7  | +21.2 | +51.5 | +51.5 | +63.0   | +57.1  | +10.0 | +10.0 | +57.1 | +63.0 |
| London Jun    | **+13.3** | +42.4 | +60.4  | +127.8 | +127.8 | +83.4 | N/A   | N/A   | N/A     | N/A    | +10.0 | +10.0 | N/A   | N/A   |
| London Dec    | **−0.1**  | −15.6 | −8.6   | +5.1   | +5.1   | −1.8  | +18.6 | +18.6 | +25.2   | +21.9  | −9.0  | −9.0  | +21.9 | +25.2 |
| Istanbul Jun  | **+0.7**  | −2.3  | +5.4   | +21.8  | +21.8  | +13.5 | +39.4 | +39.4 | +48.8   | +44.0  | +11.0 | +11.0 | +44.0 | +48.8 |
| Makkah Jun    | **+0.8**  | −22.5 | −17.4  | −7.2   | −7.2   | −12.4 | +3.1  | +3.1  | +8.3    | +5.7   | +13.0 | +13.0 | +5.7  | +8.3  |
| Makkah Dec    | **+0.5**  | −33.1 | −28.5  | −19.3  | −19.3  | −23.9 | −10.2 | −10.2 | −5.7    | −8.0   | +5.0  | +5.0  | −8.0  | −5.7  |
| Tehran Jun    | **+2.3**  | −11.6 | −5.0   | +8.5   | +8.5   | +1.6  | +22.6 | +22.6 | +29.9   | +26.2  | +11.0 | +11.0 | +26.2 | +29.9 |
| Cairo Jun     | **+0.2**  | −17.1 | −11.4  | +0.4   | +0.4   | −5.5  | +12.5 | +12.5 | +18.7   | +15.6  | +12.0 | +12.0 | +15.6 | +18.7 |
| Cairo Dec     | **0.0**   | −32.8 | −27.9  | −18.1  | −18.1  | −23.0 | −8.4  | −8.4  | −3.5    | −5.9   | +1.0  | +1.0  | −5.9  | −3.5  |
| Karachi Jun   | **+0.1**  | −21.4 | −16.1  | −5.4   | −5.4   | −10.8 | +5.5  | +5.5  | +11.1   | +8.3   | +12.0 | +12.0 | +8.3  | +11.1 |
| Dhaka Jun     | **0.0**   | −22.1 | −16.9  | −6.3   | −6.3   | −11.6 | +4.4  | +4.4  | +9.8    | +7.1   | +12.0 | +12.0 | +7.1  | +9.8  |
| Jakarta Jun   | **+0.1**  | −29.2 | −24.9  | −16.2  | −16.2  | −20.5 | −7.4  | −7.4  | −3.1    | −5.3   | +12.0 | +12.0 | −5.3  | −3.1  |
| Singapore Dec | **−0.1**  | −27.4 | −23.0  | −14.3  | −14.3  | −18.7 | −5.5  | −5.5  | −1.1    | −3.3   | +14.0 | +14.0 | −3.3  | −1.1  |
| Almaty Jun    | **+2.4**  | +2.0  | +10.5  | +28.7  | +28.7  | +19.4 | +48.8 | +48.8 | +59.9   | +54.3  | +10.0 | +10.0 | +54.3 | +59.9 |
| Riyadh Mar    | **+0.5**  | −27.2 | −22.8  | −13.9  | −13.9  | −18.4 | −5.0  | −5.0  | −0.6    | −2.8   | +14.0 | +14.0 | −2.8  | −0.6  |

---

## High-Latitude Edge Case

London, UK on June 21 (51.5°N) is the most extreme test case. As the sun approaches its highest declination, it never reaches deep depression angles.

| Method  | Fixed Isha° | Isha result                            |
| ------- | ----------- | -------------------------------------- |
| UOIF    | 12°         | 23:24 (valid)                          |
| ISNACA  | 13°         | 23:42 (valid)                          |
| ISNA    | 15°         | 24:49 (technically computed, next day) |
| SAMR    | 15°         | 24:49 (same as ISNA Isha)              |
| IGUT    | 14°         | 24:05 (next day)                       |
| UAQ     | +90 min     | 22:51 (valid: not angle-based)         |
| Qatar   | +90 min     | 22:51 (valid: not angle-based)         |
| MSC     | seasonal    | 22:41 (valid: observation-based)       |
| **PCD** | **10.00°**  | **22:54 (valid: adapted)**             |
| MWL     | 17°         | **N/A**                                |
| DIBT    | 17°         | **N/A**                                |
| Karachi | 18°         | **N/A**                                |
| Kuwait  | 17.5°       | **N/A**                                |
| Egypt   | 17.5°       | **N/A**                                |
| MUIS    | 18°         | **N/A**                                |

Six of the 14 methods produce no Isha time at London in midsummer. PCD adapts by clamping to the 10° lower bound and produces 22:54: 13 minutes later than MSC's 22:41 (the largest single error in the study). The error is structural: at 51.5°N in June, the sky never fully darkens, and both the observation reference and any computed method are approximations of a genuinely ambiguous twilight condition.

---

## Dynamic Angle Profile

How the PCD-computed depression angle varies across the test cases.

| City      | Lat    | Season | Fajr°  | Isha°  | Closest Fixed Method |
| --------- | ------ | ------ | ------ | ------ | -------------------- |
| Jakarta   | 6.2°S  | Jun    | 18.72° | 18.72° | UAQ (18.5°)          |
| Singapore | 1.4°N  | Dec    | 18.23° | 18.23° | MWL/DIBT (18°)       |
| Makkah    | 21.4°N | Jun    | 19.79° | 16.56° | MUIS (20°) / Egypt   |
| Makkah    | 21.4°N | Dec    | 19.58° | 19.36° | Egypt (19.5°)        |
| Riyadh    | 24.7°N | Mar    | 20.04° | 18.25° | MUIS (20°)           |
| Karachi   | 24.9°N | Jun    | 19.43° | 16.01° | Egypt (19.5°)        |
| Dhaka     | 23.8°N | Jun    | 19.49° | 16.19° | Egypt (19.5°)        |
| Cairo     | 30.1°N | Jun    | 18.73° | 14.97° | UAQ (18.5°)          |
| Cairo     | 30.1°N | Dec    | 19.13° | 18.72° | Egypt (19.5°)        |
| Tehran    | 35.7°N | Jun    | 17.95° | 14.09° | IGUT (17.7°/14°)     |
| New York  | 40.7°N | Dec    | 17.48° | 17.12° | IGUT (17.7°)         |
| New York  | 40.7°N | Mar    | 17.48° | 15.28° | IGUT (17.7°)         |
| Istanbul  | 41.0°N | Jun    | 16.28° | 12.40° | SAMR (16°)           |
| New York  | 40.7°N | Jun    | 16.21° | 12.41° | SAMR (16°)           |
| Almaty    | 43.2°N | Jun    | 15.73° | 12.05° | SAMR (16°)           |
| Toronto   | 43.7°N | Jun    | 15.33° | 11.71° | ISNA (15°)           |
| London    | 51.5°N | Dec    | 14.69° | 14.24° | ISNA (15°)           |
| London    | 51.5°N | Jun    | 11.88° | 10.00° | UOIF (12°)           |

The angle moves from ~20° at tropical latitudes in summer to ~12° at high-latitude summer. No fixed method tracks this gradient correctly. SAMR (16°) is closest for North American and Central Asian summers. ISNA (15°) handles Toronto and London winter. IGUT (17.7°) is closest for Tehran and mid-latitude winters. But each of these methods fails in other conditions: only PCD adapts dynamically across the full range.

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_[Research](Research) | [Methodology](Research-Methodology) | [Home-Territory Study](Research-Home-Territory) | [Observational Evidence](Research-Observational-Evidence)_