GAM vs spaMM for prediction, accounting for spatial autocorrelation

Question

I'm modelling a species' response to environmental variables while controlling for spatial autocorrelation and temporal differences in sampling. The goal is to use this model for prediction.

I've been testing the predicitve power of various models by cross validation (train model with 80% of data, predictions on remainder). I initially tried using gam() from the mgcv package but found that the predictions weren't very accurate. I then tried using fitme from the spaMM package and found that this made more accurate predictions (code and example data below).

Why are the predictions from gam less accurate than spaMM? I wonder if I've misspecified the gam model? The diagnostic plots look fine for the gam model (less so for the spaMM model - code below). The only difference in these models is the smooth for the interaction between week, lat and lon - I don't know what the equivalent for this would be in the spaMM model (any ideas?).

Is it just that spaMM is a more appropriate approach with this data? My preference would be to use gam as I find the results easier to understand, although I'd be happy to use spaMM (or any other approach for that matter) if it gives better predictions (i.e. if the difference in predicitve power I've found is real and not due to errors in my approach).

Code:

    library(mgcv)
    library(spaMM)
    
    # function to calculate root mean squared error
    RMSE <- function(f, o){
      sqrt(mean((f - o)^2))
    }
    
    # set up training and validation sets
    set.seed(333)
    fold <- sample(seq_len(nrow(df)),size = floor(0.8*nrow(df)))
    train <- df[fold,]
    validate <- df[-fold,]
    
    # GAM
    m.gam <- gam(species_obs ~ 
                 + temp
                 + rainfall
                 + s(lat, lon, k = 50, m = c(1, 0.5))
                 + s(week, k = 7)
                 + ti(lat, lon, week, d = c(2,1), bs = c('ds'), m = list(c(1, 0.5), NA), k = c(20, 7))
                 + offset(log(duration))
                 , data = train, method = 'REML', family = nb)
    
    # predict
    pred.gam <- as.vector(predict(m.gam, validate))
    
    # inspect prediction vs actual
    data.frame(pred.gam, validate$species_obs)
plot(pred.gam ~ validate$species_obs)
    
    # calculate RMSE
    gam.rmse <- RMSE(f = pred.gam, o = validate$species_obs)
    
    
    # spaMM
    m.spamm <- fitme(species_obs ~ 
                            + temp
                            + rainfall
                            + week
                            + offset(log(duration))
                            + Matern(1|lat+lon)
                            , data=train, family=spaMM::negbin())
    
    # predict
    pred.spaMM <- as.vector(predict(m.spamm, validate))
    
    # inspect prediction vs actual
    data.frame(pred.spaMM, validate$species_obs)
plot(pred.spaMM ~ validate$species_obs)
    
    # calculate RMSE
    spaMM.rmse <- RMSE(f = pred.spaMM, o = validate$species_obs)
    
    # compare
    gam.rmse
    spaMM.rmse
    
    # diagnostic plots
    gratia::appraise(m.gam)
    simulationOutput <- DHARMa::simulateResiduals(m.spamm)
    plot(simulationOutput)

Data:

df <- structure(list(ID = c(398L, 425L, 311L, 66L, 295L, 316L, 2L, 
134L, 67L, 44L, 215L, 359L, 40L, 63L, 161L, 343L, 331L, 346L, 
415L, 326L, 349L, 78L, 228L, 431L, 123L, 406L, 420L, 272L, 419L, 
291L, 113L, 380L, 117L, 26L, 266L, 16L, 324L, 369L, 253L, 333L, 
409L, 265L, 309L, 160L, 109L, 363L, 169L, 105L, 147L, 184L, 204L, 
8L, 286L, 45L, 257L, 111L, 198L, 154L, 58L, 277L, 372L, 362L, 
410L, 385L, 175L, 61L, 304L, 34L, 102L, 149L, 301L, 255L, 407L, 
261L, 17L, 140L, 312L, 345L, 133L, 190L, 354L, 88L, 18L, 285L, 
15L, 314L, 207L, 397L, 336L, 239L, 163L, 315L, 86L, 402L, 387L, 
64L, 90L, 62L, 22L, 247L, 251L, 240L, 292L, 94L, 167L, 80L, 353L, 
394L, 75L, 323L, 427L, 322L, 244L, 356L, 214L, 104L, 373L, 367L, 
408L, 276L, 434L, 55L, 213L, 37L, 379L, 115L, 278L, 317L, 196L, 
5L, 327L, 243L, 318L, 211L, 237L, 186L, 335L, 51L, 32L, 106L, 
70L, 222L, 69L, 125L, 53L, 56L, 191L, 328L, 284L, 126L, 412L, 
1L, 185L, 82L, 194L, 334L, 170L, 360L, 400L, 437L, 224L, 281L, 
413L, 52L, 405L, 101L, 108L, 131L, 201L, 83L, 89L, 159L, 424L, 
216L, 249L, 65L, 283L, 174L, 260L, 57L, 47L, 435L, 297L, 432L, 
337L, 4L, 24L, 152L, 46L, 39L, 289L, 23L, 59L, 98L, 107L, 275L, 
258L, 221L, 296L, 81L, 294L, 195L, 176L, 245L, 230L, 389L, 54L, 
205L, 60L, 377L, 118L, 143L, 3L, 231L, 422L, 332L, 371L, 124L, 
274L, 384L, 130L, 302L, 202L, 13L, 310L, 421L, 110L, 138L, 173L, 
193L, 150L, 352L, 376L, 438L, 429L, 264L, 252L, 73L, 129L, 212L, 
279L, 341L, 430L, 43L, 411L, 181L, 232L, 338L, 114L, 401L), species_obs = c(16, 
5, 33, 3, 61, 7, 2, 4, 12, 72, 21, 25, 3, 31, 34, 59, 28, 381, 
34, 45, 149, 55, 34, 10, 3, 26, 2, 28, 2, 7, 44, 9, 14, 4, 60, 
4, 6, 8, 56, 118, 11, 80, 105, 119, 71, 0, 13, 34, 12, 48, 7, 
3, 133, 34, 8, 69, 127, 125, 7, 9, 50, 5, 9, 80, 11, 11, 51, 
13, 21, 67, 36, 153, 36, 12, 4, 31, 51, 75, 28, 22, 12, 5, 5, 
36, 16, 29, 6, 65, 10, 11, 2, 3, 11, 36, 10, 6, 30, 1, 19, 9, 
55, 9, 18, 16, 19, 18, 31, 388, 54, 5, 65, 39, 54, 5, 7, 14, 
98, 25, 115, 55, 15, 26, 22, 28, 17, 11, 62, 1, 87, 2, 19, 8, 
40, 2, 50, 21, 20, 6, 10, 41, 7, 56, 5, 6, 64, 16, 38, 1, 18, 
5, 8, 4, 48, 7, 66, 19, 7, 12, 21, 263, 22, 16, 14, 37, 39, 14, 
50, 8, 19, 20, 0, 61, 9, 72, 38, 1, 28, 5, 80, 103, 2, 27, 98, 
48, 11, 1, 10, 17, 29, 2, 146, 13, 12, 0, 3, 232, 12, 37, 51, 
29, 25, 38, 4, 42, 27, 18, 13, 7, 16, 15, 12, 35, 5, 14, 33, 
65, 5, 8, 25, 13, 2, 238, 4, 9, 38, 24, 32, 0, 17, 7, 7, 300, 
4, 430, 23, 93, 32, 37, 11, 3, 12, 26, 4, 7, 4, 30, 16, 28, 23, 
11), lat = c(51.451129, 51.502218, 51.489532, 51.495132, 51.511491, 
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51.50229, 51.505993, 51.509689, 51.495543, 51.514119, 51.506038, 
51.482918, 51.448155, 51.472044, 51.44606, 51.495024, 51.523916, 
51.529513, 51.498967, 51.503607, 51.476284, 51.460305, 51.489773, 
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51.527381, 51.478232, 51.496083, 51.475137, 51.495459, 51.503625, 
51.461752, 51.529997, 51.476273, 51.503237, 51.496858, 51.505287, 
51.504638, 51.49083, 51.478764, 51.469812, 51.498601, 51.504588, 
51.504984, 51.472028, 51.503783, 51.507838, 51.476866, 51.489107, 
51.5039, 51.454272, 51.506429, 51.492034, 51.511654, 51.507251, 
51.477878, 51.511867, 51.488981, 51.45982, 51.499169, 51.455187, 
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51.504209, 51.480209, 51.46075, 51.490655, 51.477034, 51.500956, 
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17.339485, 17.297313, 17.282313, 17.335249, 17.358213, 17.348407, 
17.36332, 17.317548, 17.360231, 17.308278, 17.295559, 17.382053, 
17.288016, 17.268009, 17.354437, 17.293116, 17.33772, 17.350742, 
17.29113, 17.319834, 17.346936, 17.329091, 17.290514, 17.342086, 
17.26711, 17.336913, 17.304416, 17.339454, 17.362003, 17.30583, 
17.384091, 17.293661, 17.286183, 17.320187, 17.308189, 17.356476, 
17.287544, 17.34188, 17.326814, 17.35231, 17.320066, 17.31377, 
17.368635, 17.311181, 17.278348, 17.290977, 17.303019, 17.305527, 
17.315307, 17.362876, 17.366789, 17.369126, 17.370542, 17.370688, 
17.325418, 17.359193, 17.322926, 17.314408, 17.366227, 17.364199, 
17.267791, 17.3518, 17.364918, 17.29854, 17.330414, 17.31632, 
17.29293, 17.303164, 17.310034, 17.326356, 17.331351, 17.267669, 
17.320875, 17.326238, 17.344633, 17.297602, 17.342287, 17.296472, 
17.325759, 17.316418, 17.293941, 17.325551, 17.375733, 17.341555, 
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17.345046, 17.336208, 17.28809, 17.350793, 17.326817, 17.287156, 
17.345781, 17.35423, 17.389642, 17.28138, 17.317021, 17.274507, 
17.340184, 17.3138, 17.351766, 17.302942, 17.357397, 17.297564, 
17.304735, 17.318904, 17.317477, 17.291807, 17.33538, 17.287692, 
17.304503, 17.35348, 17.302022, 17.281458, 17.296259, 17.366458, 
17.368954, 17.292115, 17.292487, 17.379349, 17.312119, 17.301971, 
17.362325, 17.305982, 17.312158), duration = c(70.1355555555555, 
50.7836111111112, 121.600833333333, 84.4341666666667, 66.2302777777778, 
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6, 6, 5, 10, 8, 10, 4, 3, 11, 4, 2, 4, 6, 3, 7, 10, 3, 3, 2, 
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Answer 1

This was a simple fix - as pointed out by @GavinSimpson, the issue was the prediction scale: changing the GAM prediction to type = "response" has fixed it. Both approaches now perform similarly.