GAM and LOESS models

Andrew Irwin, a.irwin@dal.ca

2026-02-12

GAM and LOESS models

• GAM: generalized additive models

• LOESS: locally estimated scatterplot smoothing (quadratic, distance weighted)

• Describing, fitting, and using models

• No (simple, easily understood) model coefficients

• Calculating residuals

• Making predictions: point estimates and standard errors or confidence intervals

Start with a basic graph

Fit a cubic spline

library(mgcv)
m1 <- gam( mean_temp ~ s(year_day, bs="cc"), data = hfx)
summary(m1)

Family: gaussian 
Link function: identity 

Formula:
mean_temp ~ s(year_day, bs = "cc")

Parametric coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)    7.059      0.142   49.71   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Approximate significance of smooth terms:
              edf Ref.df   F p-value    
s(year_day) 7.829      8 478  <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

R-sq.(adj) =  0.843   Deviance explained = 84.4%
GCV = 14.637  Scale est. = 14.457    n = 717

Fit a cubic spline

library(broom)
glance(m1)  |> kable(digits = 2) |> kable_styling(full_width = FALSE)

df	logLik	AIC	BIC	deviance	df.residual	nobs	adj.r.squared	npar
8.83	-1970.55	3960.75	4005.72	10237.84	708.17	717	0.84	9

tidy(m1) |> kable(digits = 2) |> kable_styling(full_width = FALSE)

term	edf	ref.df	statistic	p.value
s(year_day)	7.83	8	477.99	0

Find predicted values and residuals

library(gratia)
hfx |> add_fitted(m1) |> select(year_day, mean_temp, .fitted) |>
  add_residuals(m1)  |> kable()

year_day	mean_temp	.fitted	.residual
1	-11.5	-3.0403174	-8.4596826
2	-12.5	-3.1093521	-9.3906479
3	-9.0	-3.1747773	-5.8252227
4	2.2	-3.2366173	5.4366173
5	-2.7	-3.2948963	0.5948963
6	-13.8	-3.3496383	-10.4503617
7	-14.1	-3.4008676	-10.6991324
8	-4.1	-3.4486084	-0.6513916
9	-2.0	-3.4928848	1.4928848
10	-5.8	-3.5337209	-2.2662791
11	1.3	-3.5711410	4.8711410
12	9.6	-3.6051692	13.2051692
13	6.7	-3.6358296	10.3358296
14	-5.0	-3.6631465	-1.3368535
15	-6.4	-3.6871440	-2.7128560
16	-5.0	-3.7078463	-1.2921537
17	-4.3	-3.7252775	-0.5747225
18	-4.3	-3.7394619	-0.5605381
19	-9.1	-3.7504235	-5.3495765
20	-2.9	-3.7581865	0.8581865
21	-1.7	-3.7627752	2.0627752
22	-9.3	-3.7642136	-5.5357864
23	-0.4	-3.7625260	3.3625260
24	2.0	-3.7577365	5.7577365
25	-7.4	-3.7498692	-3.6501308
26	-10.8	-3.7389484	-7.0610516
27	-4.0	-3.7249983	-0.2750017
28	4.3	-3.7080429	8.0080429
29	-3.0	-3.6881064	0.6881064
30	-7.0	-3.6652131	-3.3347869
31	-9.1	-3.6393870	-5.4606130
32	-2.7	-3.6106524	0.9106524
33	-2.9	-3.5790334	0.6790334
34	-14.0	-3.5445541	-10.4554459
36	3.2	-3.4671117	6.6671117
37	-4.8	-3.4241968	-1.3758032
38	0.7	-3.3785184	4.0785184
39	1.4	-3.3301006	4.7301006
40	-7.4	-3.2789675	-4.1210325
41	1.1	-3.2251435	4.3251435
42	3.2	-3.1686525	6.3686525
43	2.2	-3.1095188	5.3095188
44	-7.9	-3.0477666	-4.8522334
46	3.0	-2.9165031	5.9165031
47	1.7	-2.8470402	4.5470402
48	-5.8	-2.7750555	-3.0249445
49	-4.4	-2.7005730	-1.6994270
50	-1.7	-2.6236170	0.9236170
51	2.7	-2.5442116	5.2442116
52	5.6	-2.4623810	8.0623810
53	-0.1	-2.3781493	2.2781493
54	-4.4	-2.2915408	-2.1084592
55	0.7	-2.2025795	2.9025795
56	-3.5	-2.1112897	-1.3887103
57	0.8	-2.0176956	2.8176956
58	0.4	-1.9218212	2.3218212
59	2.4	-1.8236907	4.2236907
60	0.1	-1.7233284	1.8233284
61	3.0	-1.6207584	4.6207584
62	4.3	-1.5160049	5.8160049
63	0.8	-1.4090919	2.2090919
64	2.0	-1.3000438	3.3000438
65	1.0	-1.1888846	2.1888846
66	1.3	-1.0756385	2.3756385
67	1.4	-0.9603298	2.3603298
68	1.1	-0.8429825	1.9429825
69	0.6	-0.7236208	1.3236208
70	0.3	-0.6022689	0.9022689
71	-1.4	-0.4789510	-0.9210490
72	-0.8	-0.3536912	-0.4463088
73	1.8	-0.2265137	2.0265137
74	1.4	-0.0974426	1.4974426
75	-0.7	0.0334978	-0.7334978
76	-7.3	0.1662834	-7.4662834
77	-6.7	0.3008901	-7.0008901
78	-6.2	0.4372936	-6.6372936
79	-2.9	0.5754698	-3.4754698
80	-1.4	0.7153946	-2.1153946
81	0.5	0.8570437	-0.3570437
82	-0.5	1.0003931	-1.5003931
84	-3.5	1.2920942	-4.7920942
85	-2.0	1.4403864	-3.4403864
86	1.1	1.5902584	-0.4902584
87	3.4	1.7416732	1.6583268
88	4.2	1.8945942	2.3054058
89	5.7	2.0489844	3.6510156
90	2.9	2.2048072	0.6951928
91	3.1	2.3620257	0.7379743
92	0.1	2.5206032	-2.4206032
93	2.2	2.6805028	-0.4805028
94	5.9	2.8416878	3.0583122
95	2.3	3.0041214	-0.7041214
96	-1.5	3.1677668	-4.6677668
97	1.5	3.3325872	-1.8325872
98	-0.5	3.4985457	-3.9985457
99	-2.8	3.6656057	-6.4656057
100	-0.4	3.8337303	-4.2337303
101	2.2	4.0028828	-1.8028828
102	2.6	4.1730263	-1.5730263
103	5.7	4.3441240	1.3558760
104	1.1	4.5161392	-3.4161392
105	-1.7	4.6890351	-6.3890351
106	-0.7	4.8627749	-5.5627749
107	5.8	5.0373218	0.7626782
108	5.2	5.2126390	-0.0126390
109	5.3	5.3886896	-0.0886896
110	1.5	5.5654370	-4.0654370
111	3.8	5.7428444	-1.9428444
112	5.4	5.9208749	-0.5208749
113	7.9	6.0994917	1.8005083
114	9.8	6.2786581	3.5213419
115	9.0	6.4583373	2.5416627
116	10.0	6.6384924	3.3615076
117	11.6	6.8190868	4.7809132
118	10.7	7.0000835	3.6999165
119	11.4	7.1814459	4.2185541
120	7.9	7.3631370	0.5368630
121	6.4	7.5451202	-1.1451202
122	11.7	7.7273587	3.9726413
123	10.1	7.9098156	2.1901844
124	7.7	8.0924541	-0.3924541
126	14.5	8.4581290	6.0418710
127	8.8	8.6410918	0.1589082
128	8.8	8.8240891	-0.0240891
130	7.5	9.1900400	-1.6900400
131	10.5	9.3729200	1.1270800
132	7.2	9.5556874	-2.3556874
133	10.8	9.7383053	1.0616947
134	9.7	9.9207369	-0.2207369
135	12.2	10.1029456	2.0970544
136	9.1	10.2848944	-1.1848944
137	12.8	10.4665466	2.3334534
138	9.1	10.6478654	-1.5478654
139	9.8	10.8288140	-1.0288140
140	14.5	11.0093556	3.4906444
141	14.2	11.1894534	3.0105466
142	11.4	11.3690707	0.0309293
143	10.4	11.5481706	-1.1481706
144	7.0	11.7267163	-4.7267163
145	12.5	11.9046712	0.5953288
146	13.7	12.0819983	1.6180017
147	8.4	12.2586608	-3.8586608
148	9.2	12.4346221	-3.2346221
149	13.9	12.6098453	1.2901547
150	12.2	12.7842935	-0.5842935
151	17.5	12.9579301	4.5420699
152	21.9	13.1307183	8.7692817
153	12.2	13.3026212	-1.1026212
154	5.0	13.4736020	-8.4736020
155	6.1	13.6436240	-7.5436240
156	5.4	13.8126503	-8.4126503
157	6.7	13.9806442	-7.2806442
158	11.0	14.1475689	-3.1475689
159	13.3	14.3133877	-1.0133877
160	14.5	14.4780636	0.0219364
161	11.9	14.6415599	-2.7415599
162	9.1	14.8038399	-5.7038399
163	12.2	14.9648667	-2.7648667
164	17.3	15.1246035	2.1753965
165	8.2	15.2830137	-7.0830137
166	13.1	15.4400580	-2.3400580
167	16.9	15.5956852	1.3043148
168	13.9	15.7498395	-1.8498395
169	11.5	15.9024652	-4.4024652
170	17.2	16.0535065	1.1464935
171	18.8	16.2029077	2.5970923
173	14.7	16.4965668	-1.7965668
174	14.6	16.6407134	-2.0407134
175	18.0	16.7829968	1.2170032
176	11.2	16.9233616	-5.7233616
177	12.5	17.0617518	-4.5617518
178	18.6	17.1981118	1.4018882
179	16.6	17.3323858	-0.7323858
180	20.7	17.4645181	3.2354819
181	22.2	17.5944530	4.6055470
182	19.5	17.7221348	1.7778652
183	17.3	17.8475076	-0.5475076
184	21.7	17.9705158	3.7294842
185	25.2	18.0911036	7.1088964
186	23.8	18.2092154	5.5907846
187	23.3	18.3247953	4.9752047
188	17.0	18.4377876	-1.4377876
189	19.4	18.5481367	0.8518633
190	20.1	18.6557867	1.4442133
191	18.9	18.7606819	0.1393181
192	19.3	18.8627667	0.4372333
193	18.3	18.9619852	-0.6619852
194	17.4	19.0582817	-1.6582817
195	15.6	19.1516006	-3.5516006
196	19.8	19.2418860	0.5581140
197	22.1	19.3290822	2.7709178
198	20.9	19.4131335	1.4868665
199	20.2	19.4939843	0.7060157
200	19.0	19.5715786	-0.5715786
201	20.0	19.6458608	0.3541392
202	18.0	19.7167752	-1.7167752
203	20.0	19.7842661	0.2157339
204	24.1	19.8482776	4.2517224
205	24.2	19.9087541	4.2912459
206	24.4	19.9656399	4.4343601
207	22.3	20.0188791	2.2811209
208	22.3	20.0684161	2.2315839
209	21.9	20.1141951	1.7858049
210	22.7	20.1561604	2.5438396
211	22.4	20.1942563	2.2057437
212	22.7	20.2284271	2.4715729
213	23.6	20.2586169	3.3413831
214	21.7	20.2847701	1.4152299
215	25.2	20.3068309	4.8931691
216	24.4	20.3247436	4.0752564
217	20.8	20.3384524	0.4615476
218	23.3	20.3479018	2.9520982
219	25.4	20.3530358	5.0469642
220	22.8	20.3537987	2.4462013
221	21.2	20.3501349	0.8498651
222	21.9	20.3419886	1.5580114
223	20.6	20.3293040	0.2706960
224	22.2	20.3120255	1.8879745
225	21.3	20.2900973	1.0099027
226	20.9	20.2634636	0.6365364
227	22.4	20.2320688	2.1679312
228	21.2	20.1958571	1.0041429
229	18.8	20.1547727	-1.3547727
230	17.2	20.1087599	-2.9087599
231	18.5	20.0577631	-1.5577631
232	17.4	20.0017264	-2.6017264
233	16.7	19.9405941	-3.2405941
234	19.5	19.8743106	-0.3743106
235	19.9	19.8028199	0.0971801
236	19.6	19.7260666	-0.1260666
237	20.5	19.6439947	0.8560053
238	18.3	19.5565486	-1.2565486
239	21.0	19.4636724	1.5363276
240	22.9	19.3653106	3.5346894
241	23.6	19.2614074	4.3385926
242	20.5	19.1519070	1.3480930
243	17.2	19.0367536	-1.8367536
244	16.5	18.9158917	-2.4158917
245	18.5	18.7892653	-0.2892653
246	19.9	18.6568200	1.2431800
247	19.2	18.5185680	0.6814320
248	19.9	18.3746256	1.5253744
249	22.1	18.2251179	3.8748821
250	16.7	18.0701699	-1.3701699
251	13.2	17.9099068	-4.7099068
252	12.5	17.7444536	-5.2444536
253	12.1	17.5739356	-5.4739356
254	15.3	17.3984778	-2.0984778
255	16.5	17.2182052	-0.7182052
256	17.8	17.0332431	0.7667569
257	19.0	16.8437166	2.1562834
258	20.4	16.6497506	3.7502494
259	19.2	16.4514705	2.7485295
260	21.0	16.2490011	4.7509989
261	19.3	16.0424678	3.2575322
262	11.3	15.8319955	-4.5319955
263	10.5	15.6177094	-5.1177094
264	11.6	15.3997346	-3.7997346
265	14.8	15.1781962	-0.3781962
266	11.0	14.9532193	-3.9532193
267	7.0	14.7249290	-7.7249290
268	7.4	14.4934504	-7.0934504
269	15.6	14.2589087	1.3410913
270	15.2	14.0214290	1.1785710
271	14.5	13.7811363	0.7188637
272	13.4	13.5381557	-0.1381557
273	11.6	13.2926124	-1.6926124
274	9.6	13.0446316	-3.4446316
275	7.7	12.7943382	-5.0943382
276	14.3	12.5418574	1.7581426
277	15.7	12.2873143	3.4126857
278	9.5	12.0308340	-2.5308340
279	8.9	11.7725417	-2.8725417
280	9.1	11.5125624	-2.4125624
281	6.7	11.2510212	-4.5510212
282	12.1	10.9880433	1.1119567
283	12.5	10.7237538	1.7762462
284	7.5	10.4582777	-2.9582777
285	10.6	10.1917402	0.4082598
286	6.2	9.9242664	-3.7242664
287	6.0	9.6559814	-3.6559814
288	8.8	9.3870104	-0.5870104
289	9.9	9.1174783	0.7825217
290	5.7	8.8475103	-3.1475103
291	2.5	8.5772316	-6.0772316
292	2.9	8.3067673	-5.4067673
293	10.9	8.0362424	2.8637576
294	7.0	7.7657820	-0.7657820
295	2.2	7.4955114	-5.2955114
296	4.1	7.2255555	-3.1255555
297	10.0	6.9560395	3.0439605
298	3.4	6.6870885	-3.2870885
299	0.8	6.4188276	-5.6188276
300	1.6	6.1513819	-4.5513819
301	11.4	5.8848765	5.5151235
302	12.1	5.6194366	6.4805634
303	8.5	5.3551872	3.1448128
304	5.7	5.0922535	0.6077465
305	7.7	4.8307605	2.8692395
306	11.3	4.5708335	6.7291665
307	13.2	4.3125973	8.8874027
308	4.6	4.0561773	0.5438227
309	3.1	3.8016985	-0.7016985
310	10.6	3.5492860	7.0507140
311	11.7	3.2990649	8.4009351
312	7.3	3.0511603	4.2488397
313	5.1	2.8056973	2.2943027
314	6.8	2.5628011	4.2371989
315	0.8	2.3225968	-1.5225968
316	-0.7	2.0852094	-2.7852094
317	4.2	1.8507641	2.3492359
318	1.2	1.6193859	-0.4193859
320	-1.9	1.1663316	-3.0663316
321	0.1	0.9449057	-0.8449057
322	-2.4	0.7270473	-3.1270473
323	-2.6	0.5128817	-3.1128817
324	0.8	0.3025339	0.4974661
325	-0.1	0.0961291	-0.1961291
326	-7.6	-0.1062077	-7.4937923
327	-9.1	-0.3043632	-8.7956368
328	-1.4	-0.4983176	-0.9016824
329	1.1	-0.6880961	1.7880961
330	0.3	-0.8737241	1.1737241
331	3.0	-1.0552267	4.0552267
332	3.8	-1.2326295	5.0326295
333	0.2	-1.4059576	1.6059576
334	-0.1	-1.5752365	1.4752365
335	-0.4	-1.7404914	1.3404914
336	2.4	-1.9017478	4.3017478
337	6.6	-2.0590308	8.6590308
338	-1.3	-2.2123660	0.9123660
339	-6.0	-2.3617785	-3.6382215
341	-2.5	-2.6489371	0.1489371
342	-8.5	-2.7867338	-5.7132662
343	-9.5	-2.9207092	-6.5792908
344	-7.7	-3.0508887	-4.6491113
345	-9.0	-3.1772976	-5.8227024
346	-10.3	-3.2999612	-7.0000388
347	-8.7	-3.4189048	-5.2810952
348	-4.0	-3.5341539	-0.4658461
349	2.8	-3.6457336	6.4457336
350	-0.1	-3.7536694	3.6536694
351	-0.2	-3.8579866	3.6579866
352	-2.4	-3.9587105	1.5587105
353	-4.5	-4.0558664	-0.4441336
354	-2.3	-4.1494798	1.8494798
355	4.2	-4.2395758	8.4395758
356	9.5	-4.3261799	13.8261799
357	-0.4	-4.4093173	4.0093173
358	-6.5	-4.4890135	-2.0109865
359	-10.3	-4.5652937	-5.7347063
360	-7.8	-4.6381833	-3.1618167
361	-10.7	-4.7077075	-5.9922925
362	-5.1	-4.7738918	-0.3261082
363	1.5	-4.8367615	6.3367615
364	-8.7	-4.8963419	-3.8036581
365	-8.3	-4.9526583	-3.3473417
366	-0.9	-5.0057361	4.1057361
367	-10.3	-5.0556006	-5.2443994
368	-10.6	-5.1022771	-5.4977229
369	-3.1	-5.1457909	2.0457909
370	1.4	-5.1861675	6.5861675
371	-0.6	-5.2234321	4.6234321
372	-7.3	-5.2576100	-2.0423900
373	-6.7	-5.2887266	-1.4112734
374	2.4	-5.3168073	7.7168073
375	-0.4	-5.3418773	4.9418773
376	-6.9	-5.3639620	-1.5360380
377	-10.9	-5.3830867	-5.5169133
378	-9.7	-5.3992767	-4.3007233
379	-11.1	-5.4125575	-5.6874425
380	-7.9	-5.4229542	-2.4770458
381	-4.1	-5.4304924	1.3304924
382	-8.3	-5.4351971	-2.8648029
383	-8.6	-5.4370939	-3.1629061
384	-7.5	-5.4362081	-2.0637919
385	-1.6	-5.4325649	3.8325649
386	0.3	-5.4261897	5.7261897
387	-10.9	-5.4171079	-5.4828921
388	-4.4	-5.4053447	1.0053447
389	7.5	-5.3909256	12.8909256
390	4.0	-5.3738758	9.3738758
391	-2.5	-5.3542206	2.8542206
392	-0.7	-5.3319855	4.6319855
393	-3.4	-5.3071957	1.9071957
394	-8.0	-5.2798765	-2.7201235
395	-1.9	-5.2500534	3.3500534
396	-7.3	-5.2177515	-2.0822485
397	-11.4	-5.1829964	-6.2170036
398	-9.0	-5.1458132	-3.8541868
399	-8.5	-5.1062274	-3.3937726
400	-1.9	-5.0642642	3.1642642
401	4.9	-5.0199490	9.9199490
402	-2.0	-4.9733072	2.9733072
403	-4.3	-4.9243640	0.6243640
404	1.7	-4.8731447	6.5731447
405	-5.0	-4.8196748	-0.1803252
406	-7.6	-4.7639796	-2.8360204
407	-10.3	-4.7060843	-5.5939157
408	-7.9	-4.6460095	-3.2539905
409	-3.4	-4.5837659	1.1837659
410	-4.9	-4.5193628	-0.3806372
411	-3.5	-4.4528096	0.9528096
412	2.4	-4.3841157	6.7841157
413	-5.7	-4.3132905	-1.3867095
414	-7.6	-4.2403433	-3.3596567
415	-9.9	-4.1652835	-5.7347165
416	-11.5	-4.0881206	-7.4118794
417	-8.2	-4.0088638	-4.1911362
418	-6.3	-3.9275225	-2.3724775
419	-7.2	-3.8441062	-3.3558938
420	-5.8	-3.7586242	-2.0413758
421	-2.0	-3.6710860	1.6710860
422	-11.1	-3.5815007	-7.5184993
423	-12.6	-3.4898780	-9.1101220
424	-10.8	-3.3962270	-7.4037730
425	-6.2	-3.3005573	-2.8994427
426	-3.7	-3.2028781	-0.4971219
427	-4.9	-3.1031990	-1.7968010
428	-2.3	-3.0015291	0.7015291
429	-6.2	-2.8978780	-3.3021220
430	-8.2	-2.7922550	-5.4077450
431	-11.7	-2.6846694	-9.0153306
432	-9.1	-2.5751308	-6.5248692
433	-4.2	-2.4636483	-1.7363517
434	-5.1	-2.3502315	-2.7497685
435	2.9	-2.2348896	5.1348896
436	-0.2	-2.1176322	1.9176322
437	-0.1	-1.9984685	1.8984685
438	1.8	-1.8774079	3.6774079
439	3.1	-1.7544598	4.8544598
440	4.1	-1.6296336	5.7296336
441	-2.4	-1.5029387	-0.8970613
442	-3.6	-1.3743844	-2.2256156
443	-3.4	-1.2439801	-2.1560199
444	-1.7	-1.1117353	-0.5882647
445	0.8	-0.9776592	1.7776592
446	3.7	-0.8417613	4.5417613
447	2.4	-0.7040509	3.1040509
448	4.4	-0.5645374	4.9645374
449	-0.1	-0.4232302	0.3232302
450	-2.4	-0.2801387	-2.1198613
451	-3.3	-0.1352723	-3.1647277
452	-1.5	0.0113598	-1.5113598
453	5.0	0.1597480	4.8402520
454	4.1	0.3098830	3.7901170
455	7.7	0.4617555	7.2382445
456	2.7	0.6153560	2.0846440
457	2.0	0.7706752	1.2293248
458	3.7	0.9277037	2.7722963
460	0.3	1.2468511	-0.9468511
461	2.1	1.4089512	0.6910488
462	3.5	1.5727231	1.9272769
463	-1.9	1.7381574	-3.6381574
464	-2.4	1.9052448	-4.3052448
465	-1.8	2.0739758	-3.8739758
466	3.2	2.2443410	0.9556590
467	6.7	2.4163312	4.2836688
468	8.1	2.5899368	5.5100632
469	11.8	2.7651486	9.0348514
470	7.8	2.9419572	4.8580428
471	3.2	3.1203531	0.0796469
472	0.4	3.3003271	-2.9003271
473	3.6	3.4818696	0.1181304
474	6.7	3.6649714	3.0350286
475	11.9	3.8496231	8.0503769
476	12.8	4.0358152	8.7641848
477	10.3	4.2235384	6.0764616
478	6.8	4.4127834	2.3872166
479	7.8	4.6035407	3.1964593
480	2.4	4.7958010	-2.3958010
481	5.7	4.9895549	0.7104451
482	5.7	5.1847930	0.5152070
483	7.7	5.3815059	2.3184941
484	5.0	5.5796843	-0.5796843
485	5.1	5.7793188	-0.6793188
486	5.6	5.9803999	-0.3803999
487	6.4	6.1829184	0.2170816
488	7.3	6.3868610	0.9131390
489	4.8	6.5921698	-1.7921698
490	10.2	6.7987580	3.4012420
491	8.7	7.0065386	1.6934614
492	11.1	7.2154242	3.8845758
493	8.2	7.4253277	0.7746723
494	6.2	7.6361619	-1.4361619
495	5.8	7.8478396	-2.0478396
496	9.2	8.0602736	1.1397264
497	7.1	8.2733767	-1.1733767
498	6.3	8.4870616	-2.1870616
499	4.9	8.7012413	-3.8012413
500	5.4	8.9158284	-3.5158284
501	5.7	9.1307358	-3.4307358
502	9.2	9.3458763	-0.1458763
503	5.9	9.5611627	-3.6611627
504	9.3	9.7765079	-0.4765079
505	8.4	9.9918245	-1.5918245
506	6.0	10.2070253	-4.2070253
507	4.4	10.4220233	-6.0220233
508	8.7	10.6367312	-1.9367312
509	5.9	10.8510618	-4.9510618
510	11.3	11.0649279	0.2350721
511	9.3	11.2782423	-1.9782423
512	8.8	11.4909178	-2.6909178
513	9.4	11.7028672	-2.3028672
514	8.6	11.9140032	-3.3140032
515	9.7	12.1242388	-2.4242388
516	8.7	12.3334867	-3.6334867
517	12.9	12.5416597	0.3583403
518	10.0	12.7486706	-2.7486706
519	11.9	12.9544322	-1.0544322
520	11.7	13.1588573	-1.4588573
521	9.0	13.3618588	-4.3618588
522	10.6	13.5633493	-2.9633493
523	15.2	13.7632417	1.4367583
524	13.5	13.9614489	-0.4614489
525	14.6	14.1578835	0.4421165
526	14.7	14.3524585	0.3475415
527	13.5	14.5450866	-1.0450866
528	15.7	14.7356806	0.9643194
529	17.4	14.9241533	2.4758467
530	17.3	15.1104175	2.1895825
531	18.3	15.2943861	3.0056139
532	16.8	15.4759717	1.3240283
533	18.6	15.6550873	2.9449127
534	17.7	15.8316456	1.8683544
535	17.4	16.0055594	1.3944406
536	16.6	16.1767415	0.4232585
537	13.1	16.3451048	-3.2451048
538	14.4	16.5105619	-2.1105619
539	17.3	16.6730258	0.6269742
540	15.3	16.8324092	-1.5324092
541	15.9	16.9886249	-1.0886249
542	14.6	17.1415858	-2.5415858
543	12.8	17.2912046	-4.4912046
544	14.7	17.4373941	-2.7373941
545	13.9	17.5800671	-3.6800671
546	14.7	17.7191365	-3.0191365
547	13.8	17.8545150	-4.0545150
548	15.8	17.9861154	-2.1861154
549	17.5	18.1138505	-0.6138505
550	20.9	18.2376332	2.6623668
551	26.0	18.3573762	7.6426238
552	22.9	18.4729923	4.4270077
553	19.5	18.5843944	0.9156056
554	19.7	18.6914952	1.0085048
555	18.9	18.7942076	0.1057924
556	16.1	18.8924442	-2.7924442
557	16.6	18.9861180	-2.3861180
558	16.6	19.0751418	-2.4751418
559	19.6	19.1594283	0.4405717
560	18.1	19.2388903	-1.1388903
561	19.9	19.3134406	0.5865594
562	22.1	19.3829921	2.7170079
563	21.9	19.4474576	2.4525424
564	19.6	19.5067498	0.0932502
565	19.1	19.5607815	-0.4607815
566	24.0	19.6094655	4.3905345
567	24.1	19.6527148	4.4472852
568	21.6	19.6904761	1.9095239
569	19.1	19.7227943	-0.6227943
570	18.4	19.7497318	-1.3497318
571	18.7	19.7713509	-1.0713509
572	20.2	19.7877140	0.4122860
573	20.9	19.7988834	1.1011166
574	20.1	19.8049215	0.2950785
575	21.7	19.8058906	1.8941094
576	23.9	19.8018531	4.0981469
577	23.7	19.7928713	3.9071287
578	22.9	19.7790076	3.1209924
579	22.0	19.7603244	2.2396756
580	22.6	19.7368839	2.8631161
581	20.9	19.7087486	1.1912514
582	19.5	19.6759808	-0.1759808
583	20.6	19.6386428	0.9613572
584	19.4	19.5967971	-0.1967971
585	17.4	19.5505059	-2.1505059
586	20.2	19.4998316	0.7001684
587	18.7	19.4448366	-0.7448366
588	17.9	19.3855831	-1.4855831
589	21.0	19.3221337	1.6778663
590	20.3	19.2545506	1.0454494
591	19.8	19.1828961	0.6171039
592	20.1	19.1072327	0.9927673
593	18.9	19.0276226	-0.1276226
594	19.7	18.9441283	0.7558717
595	19.2	18.8568120	0.3431880
596	20.3	18.7657362	1.5342638
597	22.5	18.6709632	3.8290368
598	21.7	18.5725553	3.1274447
599	23.4	18.4705750	4.9294250
600	21.3	18.3650844	2.9349156
601	14.5	18.2561461	-3.7561461
602	14.6	18.1438223	-3.5438223
603	13.2	18.0281754	-4.8281754
604	16.2	17.9092677	-1.7092677
605	21.3	17.7871617	3.5128383
606	18.0	17.6619197	0.3380803
607	21.7	17.5336039	4.1663961
608	20.1	17.4022768	2.6977232
609	16.8	17.2680008	-0.4680008
610	15.9	17.1308381	-1.2308381
611	18.4	16.9908511	1.4091489
612	19.9	16.8481022	3.0518978
613	16.9	16.7026538	0.1973462
614	14.9	16.5545681	-1.6545681
615	12.9	16.4039075	-3.5039075
616	12.4	16.2507345	-3.8507345
617	14.9	16.0951113	-1.1951113
618	15.4	15.9371003	-0.5371003
619	15.2	15.7767638	-0.5767638
620	11.8	15.6141642	-3.8141642
621	11.6	15.4493639	-3.8493639
622	12.3	15.2824252	-2.9824252
623	17.7	15.1134104	2.5865896
624	14.8	14.9423819	-0.1423819
625	9.7	14.7694022	-5.0694022
626	8.4	14.5945334	-6.1945334
627	10.8	14.4178380	-3.6178380
628	14.1	14.2393783	-0.1393783
629	14.8	14.0592167	0.7407833
630	16.4	13.8774155	2.5225845
631	20.1	13.6940370	6.4059630
632	17.8	13.5091437	4.2908563
634	15.2	13.1350620	2.0649380
635	14.6	12.9459982	1.6540018
636	13.8	12.7556690	1.0443310
637	12.9	12.5641366	0.3358634
638	8.6	12.3714635	-3.7714635
639	10.2	12.1777120	-1.9777120
640	9.3	11.9829444	-2.6829444
641	6.1	11.7872232	-5.6872232
642	5.9	11.5906106	-5.6906106
643	6.7	11.3931690	-4.6931690
644	9.6	11.1949608	-1.5949608
645	16.4	10.9960483	5.4039517
646	12.6	10.7964939	1.8035061
647	10.2	10.5963598	-0.3963598
648	9.4	10.3957082	-0.9957082
649	10.5	10.1945947	0.3054053
650	11.6	9.9930693	1.6069307
651	10.2	9.7911820	0.4088180
652	10.3	9.5889828	0.7110172
653	12.3	9.3865216	2.9134784
654	9.9	9.1838484	0.7161516
655	11.7	8.9810130	2.7189870
656	10.0	8.7780655	1.2219345
657	7.8	8.5750558	-0.7750558
658	8.6	8.3720337	0.2279663
659	10.0	8.1690493	1.8309507
660	8.6	7.9661526	0.6338474
661	11.3	7.7633934	3.5366066
662	10.9	7.5608216	3.3391784
663	9.9	7.3584873	2.5415127
664	5.5	7.1564404	-1.6564404
665	5.1	6.9547308	-1.8547308
666	5.9	6.7534085	-0.8534085
667	6.4	6.5525233	-0.1525233
669	13.4	6.1522645	7.2477355
670	12.2	5.9529906	6.2470094
672	6.0	5.5564038	0.4435962
673	5.5	5.3591907	0.1408093
674	7.3	5.1627644	2.1372356
675	7.3	4.9671749	2.3328251
676	3.4	4.7724721	-1.3724721
678	-2.6	4.3859263	-6.9859263
679	2.4	4.1941832	-1.7941832
680	5.6	4.0035266	1.5964734
681	9.4	3.8140063	5.5859937
682	-1.0	3.6256725	-4.6256725
683	-3.9	3.4385749	-7.3385749
684	6.6	3.2527636	3.3472364
685	-1.2	3.0682884	-4.2682884
686	-3.9	2.8851994	-6.7851994
687	0.6	2.7035465	-2.1035465
688	3.6	2.5233795	1.0766205
689	1.7	2.3447486	-0.6447486
690	1.9	2.1677035	-0.2677035
691	4.9	1.9922942	2.9077058
692	0.7	1.8185708	-1.1185708
693	3.7	1.6465830	2.0534170
694	5.9	1.4763809	4.4236191
695	5.0	1.3080145	3.6919855
696	4.6	1.1415336	3.4584664
697	1.4	0.9769882	0.4230118
698	-1.6	0.8144282	-2.4144282
699	-3.1	0.6539037	-3.7539037
700	-6.6	0.4954644	-7.0954644
701	-4.1	0.3391605	-4.4391605
702	2.8	0.1850417	2.6149583
703	0.6	0.0331581	0.5668419
704	-0.3	-0.1164404	-0.1835596
705	-1.5	-0.2637039	-1.2362961
706	-6.1	-0.4085823	-5.6914177
707	-5.5	-0.5510259	-4.9489741
708	4.8	-0.6909845	5.4909845
709	11.6	-0.8284084	12.4284084
710	2.1	-0.9632475	3.0632475
711	-8.0	-1.0954518	-6.9045482
712	-5.3	-1.2249716	-4.0750284
713	7.5	-1.3517567	8.8517567
714	7.2	-1.4757574	8.6757574
715	-2.7	-1.5969235	-1.1030765
716	-4.0	-1.7152052	-2.2847948
717	-5.5	-1.8305526	-3.6694474
718	-8.7	-1.9429157	-6.7570843
719	-5.1	-2.0522445	-3.0477555
720	-3.7	-2.1584891	-1.5415109
721	-2.3	-2.2615996	-0.0384004
722	0.3	-2.3615260	2.6615260
723	-0.4	-2.4582184	2.0582184
724	-5.8	-2.5516268	-3.2483732
725	-7.9	-2.6417014	-5.2582986
726	-3.6	-2.7283920	-0.8716080
727	2.1	-2.8116489	4.9116489
728	-2.7	-2.8914221	0.1914221
729	-3.5	-2.9676616	-0.5323384
730	0.0	-3.0403174	3.0403174

Graph “smooth” term

hfx |> add_fitted(m1) |> select(year_day, mean_temp, .fitted) |>
  add_residuals(m1) |>
  ggplot(aes(year_day)) +
  geom_line(aes(y=.fitted), color="blue", size=2) + 
  geom_point(aes(y = mean_temp)) + my_theme

Graph “smooth” term

confint(m1, "s(year_day)", level = 0.95) |>
  ggplot(aes(year_day)) +
  geom_ribbon(aes(ymin = .lower_ci, ymax = .upper_ci), alpha = 0.25, fill="red") +
  geom_line(aes(y=.estimate), size=2, color="red") + 
  geom_point(aes(y = mean_temp - mean(mean_temp)), data = hfx) + 
  my_theme

Residuals

augment(m1, se_fit = TRUE) |> 
  ggplot(aes(x = .resid)) + 
  geom_histogram(binwidth = 1) + my_theme

LOESS smooths

m2 <- loess( mean_temp ~ year_day, data = hfx)
summary(m2)

Call:
loess(formula = mean_temp ~ year_day, data = hfx)

Number of Observations: 717 
Equivalent Number of Parameters: 4.36 
Residual Standard Error: 5.969 
Trace of smoother matrix: 4.74  (exact)

Control settings:
  span     :  0.75 
  degree   :  2 
  family   :  gaussian
  surface  :  interpolate     cell = 0.2
  normalize:  TRUE
 parametric:  FALSE
drop.square:  FALSE 

Predictions

hfx |> select(year_day, mean_temp) |>
  mutate(prediction = predict(m2),
         residuals = residuals(m2)) |> kable(digits = 2)

year_day	mean_temp	prediction	residuals
1	-11.5	-10.11	-1.39
2	-12.5	-9.88	-2.62
3	-9.0	-9.65	0.65
4	2.2	-9.43	11.63
5	-2.7	-9.20	6.50
6	-13.8	-8.97	-4.83
7	-14.1	-8.75	-5.35
8	-4.1	-8.53	4.43
9	-2.0	-8.31	6.31
10	-5.8	-8.09	2.29
11	1.3	-7.87	9.17
12	9.6	-7.65	17.25
13	6.7	-7.43	14.13
14	-5.0	-7.21	2.21
15	-6.4	-7.00	0.60
16	-5.0	-6.78	1.78
17	-4.3	-6.57	2.27
18	-4.3	-6.36	2.06
19	-9.1	-6.15	-2.95
20	-2.9	-5.94	3.04
21	-1.7	-5.73	4.03
22	-9.3	-5.52	-3.78
23	-0.4	-5.31	4.91
24	2.0	-5.10	7.10
25	-7.4	-4.90	-2.50
26	-10.8	-4.70	-6.10
27	-4.0	-4.49	0.49
28	4.3	-4.29	8.59
29	-3.0	-4.09	1.09
30	-7.0	-3.89	-3.11
31	-9.1	-3.69	-5.41
32	-2.7	-3.49	0.79
33	-2.9	-3.30	0.40
34	-14.0	-3.10	-10.90
36	3.2	-2.71	5.91
37	-4.8	-2.52	-2.28
38	0.7	-2.33	3.03
39	1.4	-2.14	3.54
40	-7.4	-1.95	-5.45
41	1.1	-1.76	2.86
42	3.2	-1.58	4.78
43	2.2	-1.39	3.59
44	-7.9	-1.21	-6.69
46	3.0	-0.84	3.84
47	1.7	-0.66	2.36
48	-5.8	-0.48	-5.32
49	-4.4	-0.30	-4.10
50	-1.7	-0.13	-1.57
51	2.7	0.05	2.65
52	5.6	0.23	5.37
53	-0.1	0.40	-0.50
54	-4.4	0.57	-4.97
55	0.7	0.75	-0.05
56	-3.5	0.92	-4.42
57	0.8	1.09	-0.29
58	0.4	1.26	-0.86
59	2.4	1.42	0.98
60	0.1	1.59	-1.49
61	3.0	1.75	1.25
62	4.3	1.92	2.38
63	0.8	2.08	-1.28
64	2.0	2.24	-0.24
65	1.0	2.40	-1.40
66	1.3	2.56	-1.26
67	1.4	2.72	-1.32
68	1.1	2.88	-1.78
69	0.6	3.03	-2.43
70	0.3	3.19	-2.89
71	-1.4	3.34	-4.74
72	-0.8	3.49	-4.29
73	1.8	3.65	-1.85
74	1.4	3.80	-2.40
75	-0.7	3.94	-4.64
76	-7.3	4.09	-11.39
77	-6.7	4.24	-10.94
78	-6.2	4.38	-10.58
79	-2.9	4.53	-7.43
80	-1.4	4.67	-6.07
81	0.5	4.81	-4.31
82	-0.5	4.95	-5.45
84	-3.5	5.23	-8.73
85	-2.0	5.37	-7.37
86	1.1	5.50	-4.40
87	3.4	5.64	-2.24
88	4.2	5.77	-1.57
89	5.7	5.90	-0.20
90	2.9	6.03	-3.13
91	3.1	6.16	-3.06
92	0.1	6.29	-6.19
93	2.2	6.42	-4.22
94	5.9	6.54	-0.64
95	2.3	6.67	-4.37
96	-1.5	6.79	-8.29
97	1.5	6.91	-5.41
98	-0.5	7.03	-7.53
99	-2.8	7.15	-9.95
100	-0.4	7.27	-7.67
101	2.2	7.39	-5.19
102	2.6	7.50	-4.90
103	5.7	7.61	-1.91
104	1.1	7.73	-6.63
105	-1.7	7.84	-9.54
106	-0.7	7.95	-8.65
107	5.8	8.06	-2.26
108	5.2	8.16	-2.96
109	5.3	8.27	-2.97
110	1.5	8.37	-6.87
111	3.8	8.48	-4.68
112	5.4	8.58	-3.18
113	7.9	8.68	-0.78
114	9.8	8.78	1.02
115	9.0	8.88	0.12
116	10.0	8.97	1.03
117	11.6	9.07	2.53
118	10.7	9.16	1.54
119	11.4	9.26	2.14
120	7.9	9.35	-1.45
121	6.4	9.44	-3.04
122	11.7	9.53	2.17
123	10.1	9.62	0.48
124	7.7	9.70	-2.00
126	14.5	9.88	4.62
127	8.8	9.96	-1.16
128	8.8	10.04	-1.24
130	7.5	10.20	-2.70
131	10.5	10.28	0.22
132	7.2	10.36	-3.16
133	10.8	10.44	0.36
134	9.7	10.52	-0.82
135	12.2	10.59	1.61
136	9.1	10.67	-1.57
137	12.8	10.74	2.06
138	9.1	10.81	-1.71
139	9.8	10.88	-1.08
140	14.5	10.95	3.55
141	14.2	11.02	3.18
142	11.4	11.09	0.31
143	10.4	11.15	-0.75
144	7.0	11.22	-4.22
145	12.5	11.28	1.22
146	13.7	11.35	2.35
147	8.4	11.41	-3.01
148	9.2	11.47	-2.27
149	13.9	11.53	2.37
150	12.2	11.59	0.61
151	17.5	11.65	5.85
152	21.9	11.71	10.19
153	12.2	11.77	0.43
154	5.0	11.82	-6.82
155	6.1	11.88	-5.78
156	5.4	11.93	-6.53
157	6.7	11.98	-5.28
158	11.0	12.04	-1.04
159	13.3	12.09	1.21
160	14.5	12.14	2.36
161	11.9	12.19	-0.29
162	9.1	12.24	-3.14
163	12.2	12.28	-0.08
164	17.3	12.33	4.97
165	8.2	12.38	-4.18
166	13.1	12.42	0.68
167	16.9	12.47	4.43
168	13.9	12.51	1.39
169	11.5	12.55	-1.05
170	17.2	12.60	4.60
171	18.8	12.64	6.16
173	14.7	12.72	1.98
174	14.6	12.76	1.84
175	18.0	12.79	5.21
176	11.2	12.83	-1.63
177	12.5	12.87	-0.37
178	18.6	12.90	5.70
179	16.6	12.94	3.66
180	20.7	12.97	7.73
181	22.2	13.01	9.19
182	19.5	13.04	6.46
183	17.3	13.07	4.23
184	21.7	13.10	8.60
185	25.2	13.13	12.07
186	23.8	13.16	10.64
187	23.3	13.19	10.11
188	17.0	13.22	3.78
189	19.4	13.24	6.16
190	20.1	13.26	6.84
191	18.9	13.28	5.62
192	19.3	13.30	6.00
193	18.3	13.32	4.98
194	17.4	13.33	4.07
195	15.6	13.34	2.26
196	19.8	13.35	6.45
197	22.1	13.36	8.74
198	20.9	13.37	7.53
199	20.2	13.37	6.83
200	19.0	13.37	5.63
201	20.0	13.38	6.62
202	18.0	13.37	4.63
203	20.0	13.37	6.63
204	24.1	13.37	10.73
205	24.2	13.36	10.84
206	24.4	13.35	11.05
207	22.3	13.34	8.96
208	22.3	13.33	8.97
209	21.9	13.32	8.58
210	22.7	13.30	9.40
211	22.4	13.29	9.11
212	22.7	13.27	9.43
213	23.6	13.25	10.35
214	21.7	13.23	8.47
215	25.2	13.21	11.99
216	24.4	13.19	11.21
217	20.8	13.16	7.64
218	23.3	13.14	10.16
219	25.4	13.11	12.29
220	22.8	13.08	9.72
221	21.2	13.05	8.15
222	21.9	13.02	8.88
223	20.6	12.99	7.61
224	22.2	12.96	9.24
225	21.3	12.92	8.38
226	20.9	12.89	8.01
227	22.4	12.85	9.55
228	21.2	12.81	8.39
229	18.8	12.77	6.03
230	17.2	12.73	4.47
231	18.5	12.69	5.81
232	17.4	12.65	4.75
233	16.7	12.61	4.09
234	19.5	12.57	6.93
235	19.9	12.52	7.38
236	19.6	12.48	7.12
237	20.5	12.43	8.07
238	18.3	12.38	5.92
239	21.0	12.34	8.66
240	22.9	12.29	10.61
241	23.6	12.24	11.36
242	20.5	12.19	8.31
243	17.2	12.14	5.06
244	16.5	12.09	4.41
245	18.5	12.04	6.46
246	19.9	11.99	7.91
247	19.2	11.93	7.27
248	19.9	11.88	8.02
249	22.1	11.83	10.27
250	16.7	11.78	4.92
251	13.2	11.72	1.48
252	12.5	11.67	0.83
253	12.1	11.61	0.49
254	15.3	11.56	3.74
255	16.5	11.50	5.00
256	17.8	11.45	6.35
257	19.0	11.39	7.61
258	20.4	11.34	9.06
259	19.2	11.28	7.92
260	21.0	11.22	9.78
261	19.3	11.17	8.13
262	11.3	11.11	0.19
263	10.5	11.05	-0.55
264	11.6	11.00	0.60
265	14.8	10.94	3.86
266	11.0	10.88	0.12
267	7.0	10.83	-3.83
268	7.4	10.77	-3.37
269	15.6	10.72	4.88
270	15.2	10.66	4.54
271	14.5	10.60	3.90
272	13.4	10.55	2.85
273	11.6	10.49	1.11
274	9.6	10.44	-0.84
275	7.7	10.38	-2.68
276	14.3	10.33	3.97
277	15.7	10.27	5.43
278	9.5	10.21	-0.71
279	8.9	10.14	-1.24
280	9.1	10.07	-0.97
281	6.7	9.99	-3.29
282	12.1	9.91	2.19
283	12.5	9.82	2.68
284	7.5	9.73	-2.23
285	10.6	9.64	0.96
286	6.2	9.54	-3.34
287	6.0	9.43	-3.43
288	8.8	9.32	-0.52
289	9.9	9.21	0.69
290	5.7	9.10	-3.40
291	2.5	8.98	-6.48
292	2.9	8.86	-5.96
293	10.9	8.73	2.17
294	7.0	8.60	-1.60
295	2.2	8.47	-6.27
296	4.1	8.34	-4.24
297	10.0	8.20	1.80
298	3.4	8.06	-4.66
299	0.8	7.92	-7.12
300	1.6	7.77	-6.17
301	11.4	7.62	3.78
302	12.1	7.47	4.63
303	8.5	7.32	1.18
304	5.7	7.16	-1.46
305	7.7	7.01	0.69
306	11.3	6.85	4.45
307	13.2	6.69	6.51
308	4.6	6.53	-1.93
309	3.1	6.37	-3.27
310	10.6	6.20	4.40
311	11.7	6.04	5.66
312	7.3	5.87	1.43
313	5.1	5.71	-0.61
314	6.8	5.54	1.26
315	0.8	5.37	-4.57
316	-0.7	5.20	-5.90
317	4.2	5.03	-0.83
318	1.2	4.87	-3.67
320	-1.9	4.53	-6.43
321	0.1	4.36	-4.26
322	-2.4	4.19	-6.59
323	-2.6	4.02	-6.62
324	0.8	3.85	-3.05
325	-0.1	3.68	-3.78
326	-7.6	3.52	-11.12
327	-9.1	3.35	-12.45
328	-1.4	3.19	-4.59
329	1.1	3.02	-1.92
330	0.3	2.86	-2.56
331	3.0	2.70	0.30
332	3.8	2.54	1.26
333	0.2	2.38	-2.18
334	-0.1	2.22	-2.32
335	-0.4	2.07	-2.47
336	2.4	1.92	0.48
337	6.6	1.77	4.83
338	-1.3	1.62	-2.92
339	-6.0	1.47	-7.47
341	-2.5	1.19	-3.69
342	-8.5	1.05	-9.55
343	-9.5	0.91	-10.41
344	-7.7	0.78	-8.48
345	-9.0	0.65	-9.65
346	-10.3	0.52	-10.82
347	-8.7	0.40	-9.10
348	-4.0	0.28	-4.28
349	2.8	0.16	2.64
350	-0.1	0.05	-0.15
351	-0.2	-0.06	-0.14
352	-2.4	-0.16	-2.24
353	-4.5	-0.26	-4.24
354	-2.3	-0.36	-1.94
355	4.2	-0.45	4.65
356	9.5	-0.54	10.04
357	-0.4	-0.62	0.22
358	-6.5	-0.70	-5.80
359	-10.3	-0.77	-9.53
360	-7.8	-0.84	-6.96
361	-10.7	-0.90	-9.80
362	-5.1	-0.96	-4.14
363	1.5	-1.02	2.52
364	-8.7	-1.06	-7.64
365	-8.3	-1.11	-7.19
366	-0.9	-1.14	0.24
367	-10.3	-1.17	-9.13
368	-10.6	-1.20	-9.40
369	-3.1	-1.22	-1.88
370	1.4	-1.24	2.64
371	-0.6	-1.26	0.66
372	-7.3	-1.28	-6.02
373	-6.7	-1.29	-5.41
374	2.4	-1.30	3.70
375	-0.4	-1.31	0.91
376	-6.9	-1.31	-5.59
377	-10.9	-1.32	-9.58
378	-9.7	-1.32	-8.38
379	-11.1	-1.31	-9.79
380	-7.9	-1.31	-6.59
381	-4.1	-1.30	-2.80
382	-8.3	-1.29	-7.01
383	-8.6	-1.28	-7.32
384	-7.5	-1.27	-6.23
385	-1.6	-1.25	-0.35
386	0.3	-1.24	1.54
387	-10.9	-1.22	-9.68
388	-4.4	-1.19	-3.21
389	7.5	-1.17	8.67
390	4.0	-1.15	5.15
391	-2.5	-1.12	-1.38
392	-0.7	-1.09	0.39
393	-3.4	-1.06	-2.34
394	-8.0	-1.03	-6.97
395	-1.9	-0.99	-0.91
396	-7.3	-0.96	-6.34
397	-11.4	-0.92	-10.48
398	-9.0	-0.88	-8.12
399	-8.5	-0.84	-7.66
400	-1.9	-0.80	-1.10
401	4.9	-0.75	5.65
402	-2.0	-0.71	-1.29
403	-4.3	-0.66	-3.64
404	1.7	-0.62	2.32
405	-5.0	-0.57	-4.43
406	-7.6	-0.52	-7.08
407	-10.3	-0.47	-9.83
408	-7.9	-0.41	-7.49
409	-3.4	-0.36	-3.04
410	-4.9	-0.31	-4.59
411	-3.5	-0.25	-3.25
412	2.4	-0.20	2.60
413	-5.7	-0.14	-5.56
414	-7.6	-0.08	-7.52
415	-9.9	-0.02	-9.88
416	-11.5	0.04	-11.54
417	-8.2	0.10	-8.30
418	-6.3	0.16	-6.46
419	-7.2	0.22	-7.42
420	-5.8	0.28	-6.08
421	-2.0	0.35	-2.35
422	-11.1	0.41	-11.51
423	-12.6	0.47	-13.07
424	-10.8	0.54	-11.34
425	-6.2	0.60	-6.80
426	-3.7	0.67	-4.37
427	-4.9	0.73	-5.63
428	-2.3	0.80	-3.10
429	-6.2	0.86	-7.06
430	-8.2	0.93	-9.13
431	-11.7	0.99	-12.69
432	-9.1	1.06	-10.16
433	-4.2	1.12	-5.32
434	-5.1	1.19	-6.29
435	2.9	1.26	1.64
436	-0.2	1.32	-1.52
437	-0.1	1.39	-1.49
438	1.8	1.45	0.35
439	3.1	1.52	1.58
440	4.1	1.58	2.52
441	-2.4	1.65	-4.05
442	-3.6	1.71	-5.31
443	-3.4	1.77	-5.17
444	-1.7	1.84	-3.54
445	0.8	1.90	-1.10
446	3.7	1.96	1.74
447	2.4	2.02	0.38
448	4.4	2.08	2.32
449	-0.1	2.14	-2.24
450	-2.4	2.20	-4.60
451	-3.3	2.26	-5.56
452	-1.5	2.32	-3.82
453	5.0	2.38	2.62
454	4.1	2.43	1.67
455	7.7	2.49	5.21
456	2.7	2.54	0.16
457	2.0	2.60	-0.60
458	3.7	2.65	1.05
460	0.3	2.77	-2.47
461	2.1	2.83	-0.73
462	3.5	2.90	0.60
463	-1.9	2.97	-4.87
464	-2.4	3.04	-5.44
465	-1.8	3.11	-4.91
466	3.2	3.18	0.02
467	6.7	3.26	3.44
468	8.1	3.34	4.76
469	11.8	3.42	8.38
470	7.8	3.50	4.30
471	3.2	3.59	-0.39
472	0.4	3.68	-3.28
473	3.6	3.77	-0.17
474	6.7	3.86	2.84
475	11.9	3.95	7.95
476	12.8	4.04	8.76
477	10.3	4.14	6.16
478	6.8	4.24	2.56
479	7.8	4.34	3.46
480	2.4	4.44	-2.04
481	5.7	4.54	1.16
482	5.7	4.64	1.06
483	7.7	4.74	2.96
484	5.0	4.85	0.15
485	5.1	4.95	0.15
486	5.6	5.06	0.54
487	6.4	5.16	1.24
488	7.3	5.27	2.03
489	4.8	5.38	-0.58
490	10.2	5.49	4.71
491	8.7	5.60	3.10
492	11.1	5.71	5.39
493	8.2	5.82	2.38
494	6.2	5.93	0.27
495	5.8	6.04	-0.24
496	9.2	6.15	3.05
497	7.1	6.26	0.84
498	6.3	6.37	-0.07
499	4.9	6.48	-1.58
500	5.4	6.59	-1.19
501	5.7	6.70	-1.00
502	9.2	6.80	2.40
503	5.9	6.91	-1.01
504	9.3	7.02	2.28
505	8.4	7.13	1.27
506	6.0	7.24	-1.24
507	4.4	7.34	-2.94
508	8.7	7.45	1.25
509	5.9	7.55	-1.65
510	11.3	7.66	3.64
511	9.3	7.76	1.54
512	8.8	7.86	0.94
513	9.4	7.96	1.44
514	8.6	8.06	0.54
515	9.7	8.16	1.54
516	8.7	8.26	0.44
517	12.9	8.35	4.55
518	10.0	8.44	1.56
519	11.9	8.54	3.36
520	11.7	8.63	3.07
521	9.0	8.71	0.29
522	10.6	8.80	1.80
523	15.2	8.89	6.31
524	13.5	8.97	4.53
525	14.6	9.05	5.55
526	14.7	9.13	5.57
527	13.5	9.20	4.30
528	15.7	9.28	6.42
529	17.4	9.35	8.05
530	17.3	9.42	7.88
531	18.3	9.49	8.81
532	16.8	9.55	7.25
533	18.6	9.61	8.99
534	17.7	9.67	8.03
535	17.4	9.73	7.67
536	16.6	9.78	6.82
537	13.1	9.83	3.27
538	14.4	9.88	4.52
539	17.3	9.92	7.38
540	15.3	9.96	5.34
541	15.9	10.00	5.90
542	14.6	10.03	4.57
543	12.8	10.06	2.74
544	14.7	10.09	4.61
545	13.9	10.11	3.79
546	14.7	10.13	4.57
547	13.8	10.15	3.65
548	15.8	10.17	5.63
549	17.5	10.18	7.32
550	20.9	10.19	10.71
551	26.0	10.21	15.79
552	22.9	10.22	12.68
553	19.5	10.24	9.26
554	19.7	10.25	9.45
555	18.9	10.27	8.63
556	16.1	10.28	5.82
557	16.6	10.30	6.30
558	16.6	10.31	6.29
559	19.6	10.32	9.28
560	18.1	10.34	7.76
561	19.9	10.35	9.55
562	22.1	10.36	11.74
563	21.9	10.38	11.52
564	19.6	10.39	9.21
565	19.1	10.40	8.70
566	24.0	10.42	13.58
567	24.1	10.43	13.67
568	21.6	10.44	11.16
569	19.1	10.45	8.65
570	18.4	10.46	7.94
571	18.7	10.47	8.23
572	20.2	10.49	9.71
573	20.9	10.50	10.40
574	20.1	10.51	9.59
575	21.7	10.52	11.18
576	23.9	10.52	13.38
577	23.7	10.53	13.17
578	22.9	10.54	12.36
579	22.0	10.55	11.45
580	22.6	10.56	12.04
581	20.9	10.56	10.34
582	19.5	10.57	8.93
583	20.6	10.58	10.02
584	19.4	10.58	8.82
585	17.4	10.59	6.81
586	20.2	10.59	9.61
587	18.7	10.60	8.10
588	17.9	10.60	7.30
589	21.0	10.60	10.40
590	20.3	10.60	9.70
591	19.8	10.61	9.19
592	20.1	10.61	9.49
593	18.9	10.61	8.29
594	19.7	10.61	9.09
595	19.2	10.61	8.59
596	20.3	10.60	9.70
597	22.5	10.60	11.90
598	21.7	10.60	11.10
599	23.4	10.59	12.81
600	21.3	10.59	10.71
601	14.5	10.58	3.92
602	14.6	10.58	4.02
603	13.2	10.57	2.63
604	16.2	10.56	5.64
605	21.3	10.55	10.75
606	18.0	10.54	7.46
607	21.7	10.53	11.17
608	20.1	10.52	9.58
609	16.8	10.51	6.29
610	15.9	10.50	5.40
611	18.4	10.48	7.92
612	19.9	10.47	9.43
613	16.9	10.45	6.45
614	14.9	10.43	4.47
615	12.9	10.42	2.48
616	12.4	10.40	2.00
617	14.9	10.38	4.52
618	15.4	10.36	5.04
619	15.2	10.33	4.87
620	11.8	10.31	1.49
621	11.6	10.29	1.31
622	12.3	10.26	2.04
623	17.7	10.23	7.47
624	14.8	10.21	4.59
625	9.7	10.18	-0.48
626	8.4	10.15	-1.75
627	10.8	10.12	0.68
628	14.1	10.08	4.02
629	14.8	10.05	4.75
630	16.4	10.02	6.38
631	20.1	9.98	10.12
632	17.8	9.94	7.86
634	15.2	9.86	5.34
635	14.6	9.82	4.78
636	13.8	9.78	4.02
637	12.9	9.74	3.16
638	8.6	9.69	-1.09
639	10.2	9.65	0.55
640	9.3	9.60	-0.30
641	6.1	9.55	-3.45
642	5.9	9.50	-3.60
643	6.7	9.45	-2.75
644	9.6	9.40	0.20
645	16.4	9.34	7.06
646	12.6	9.29	3.31
647	10.2	9.24	0.96
648	9.4	9.18	0.22
649	10.5	9.12	1.38
650	11.6	9.06	2.54
651	10.2	9.00	1.20
652	10.3	8.94	1.36
653	12.3	8.88	3.42
654	9.9	8.82	1.08
655	11.7	8.76	2.94
656	10.0	8.69	1.31
657	7.8	8.62	-0.82
658	8.6	8.56	0.04
659	10.0	8.49	1.51
660	8.6	8.42	0.18
661	11.3	8.35	2.95
662	10.9	8.28	2.62
663	9.9	8.21	1.69
664	5.5	8.13	-2.63
665	5.1	8.06	-2.96
666	5.9	7.99	-2.09
667	6.4	7.91	-1.51
669	13.4	7.75	5.65
670	12.2	7.68	4.52
672	6.0	7.52	-1.52
673	5.5	7.43	-1.93
674	7.3	7.35	-0.05
675	7.3	7.27	0.03
676	3.4	7.18	-3.78
678	-2.6	7.01	-9.61
679	2.4	6.92	-4.52
680	5.6	6.84	-1.24
681	9.4	6.75	2.65
682	-1.0	6.66	-7.66
683	-3.9	6.56	-10.46
684	6.6	6.47	0.13
685	-1.2	6.38	-7.58
686	-3.9	6.29	-10.19
687	0.6	6.19	-5.59
688	3.6	6.10	-2.50
689	1.7	6.00	-4.30
690	1.9	5.90	-4.00
691	4.9	5.81	-0.91
692	0.7	5.71	-5.01
693	3.7	5.61	-1.91
694	5.9	5.51	0.39
695	5.0	5.41	-0.41
696	4.6	5.30	-0.70
697	1.4	5.20	-3.80
698	-1.6	5.10	-6.70
699	-3.1	4.99	-8.09
700	-6.6	4.89	-11.49
701	-4.1	4.78	-8.88
702	2.8	4.67	-1.87
703	0.6	4.57	-3.97
704	-0.3	4.46	-4.76
705	-1.5	4.35	-5.85
706	-6.1	4.24	-10.34
707	-5.5	4.13	-9.63
708	4.8	4.02	0.78
709	11.6	3.90	7.70
710	2.1	3.79	-1.69
711	-8.0	3.68	-11.68
712	-5.3	3.56	-8.86
713	7.5	3.45	4.05
714	7.2	3.33	3.87
715	-2.7	3.21	-5.91
716	-4.0	3.10	-7.10
717	-5.5	2.98	-8.48
718	-8.7	2.86	-11.56
719	-5.1	2.74	-7.84
720	-3.7	2.62	-6.32
721	-2.3	2.50	-4.80
722	0.3	2.38	-2.08
723	-0.4	2.25	-2.65
724	-5.8	2.13	-7.93
725	-7.9	2.01	-9.91
726	-3.6	1.88	-5.48
727	2.1	1.76	0.34
728	-2.7	1.63	-4.33
729	-3.5	1.51	-5.01
730	0.0	1.38	-1.38

Predictions

augment(m2, se_fit = TRUE) |> kable(digits = 2)

mean_temp	year_day	.fitted	.se.fit	.resid
-11.5	1	-10.11	0.86	-1.39
-12.5	2	-9.88	0.85	-2.62
-9.0	3	-9.65	0.84	0.65
2.2	4	-9.43	0.83	11.63
-2.7	5	-9.20	0.83	6.50
-13.8	6	-8.97	0.82	-4.83
-14.1	7	-8.75	0.81	-5.35
-4.1	8	-8.53	0.80	4.43
-2.0	9	-8.31	0.80	6.31
-5.8	10	-8.09	0.79	2.29
1.3	11	-7.87	0.78	9.17
9.6	12	-7.65	0.78	17.25
6.7	13	-7.43	0.77	14.13
-5.0	14	-7.21	0.76	2.21
-6.4	15	-7.00	0.75	0.60
-5.0	16	-6.78	0.75	1.78
-4.3	17	-6.57	0.74	2.27
-4.3	18	-6.36	0.73	2.06
-9.1	19	-6.15	0.73	-2.95
-2.9	20	-5.94	0.72	3.04
-1.7	21	-5.73	0.71	4.03
-9.3	22	-5.52	0.71	-3.78
-0.4	23	-5.31	0.70	4.91
2.0	24	-5.10	0.69	7.10
-7.4	25	-4.90	0.69	-2.50
-10.8	26	-4.70	0.68	-6.10
-4.0	27	-4.49	0.67	0.49
4.3	28	-4.29	0.67	8.59
-3.0	29	-4.09	0.66	1.09
-7.0	30	-3.89	0.66	-3.11
-9.1	31	-3.69	0.65	-5.41
-2.7	32	-3.49	0.64	0.79
-2.9	33	-3.30	0.64	0.40
-14.0	34	-3.10	0.63	-10.90
3.2	36	-2.71	0.62	5.91
-4.8	37	-2.52	0.62	-2.28
0.7	38	-2.33	0.61	3.03
1.4	39	-2.14	0.60	3.54
-7.4	40	-1.95	0.60	-5.45
1.1	41	-1.76	0.59	2.86
3.2	42	-1.58	0.59	4.78
2.2	43	-1.39	0.58	3.59
-7.9	44	-1.21	0.58	-6.69
3.0	46	-0.84	0.57	3.84
1.7	47	-0.66	0.56	2.36
-5.8	48	-0.48	0.56	-5.32
-4.4	49	-0.30	0.55	-4.10
-1.7	50	-0.13	0.55	-1.57
2.7	51	0.05	0.54	2.65
5.6	52	0.23	0.54	5.37
-0.1	53	0.40	0.53	-0.50
-4.4	54	0.57	0.53	-4.97
0.7	55	0.75	0.52	-0.05
-3.5	56	0.92	0.52	-4.42
0.8	57	1.09	0.52	-0.29
0.4	58	1.26	0.51	-0.86
2.4	59	1.42	0.51	0.98
0.1	60	1.59	0.50	-1.49
3.0	61	1.75	0.50	1.25
4.3	62	1.92	0.50	2.38
0.8	63	2.08	0.49	-1.28
2.0	64	2.24	0.49	-0.24
1.0	65	2.40	0.48	-1.40
1.3	66	2.56	0.48	-1.26
1.4	67	2.72	0.48	-1.32
1.1	68	2.88	0.47	-1.78
0.6	69	3.03	0.47	-2.43
0.3	70	3.19	0.47	-2.89
-1.4	71	3.34	0.46	-4.74
-0.8	72	3.49	0.46	-4.29
1.8	73	3.65	0.46	-1.85
1.4	74	3.80	0.45	-2.40
-0.7	75	3.94	0.45	-4.64
-7.3	76	4.09	0.45	-11.39
-6.7	77	4.24	0.44	-10.94
-6.2	78	4.38	0.44	-10.58
-2.9	79	4.53	0.44	-7.43
-1.4	80	4.67	0.44	-6.07
0.5	81	4.81	0.43	-4.31
-0.5	82	4.95	0.43	-5.45
-3.5	84	5.23	0.43	-8.73
-2.0	85	5.37	0.42	-7.37
1.1	86	5.50	0.42	-4.40
3.4	87	5.64	0.42	-2.24
4.2	88	5.77	0.42	-1.57
5.7	89	5.90	0.42	-0.20
2.9	90	6.03	0.41	-3.13
3.1	91	6.16	0.41	-3.06
0.1	92	6.29	0.41	-6.19
2.2	93	6.42	0.41	-4.22
5.9	94	6.54	0.41	-0.64
2.3	95	6.67	0.40	-4.37
-1.5	96	6.79	0.40	-8.29
1.5	97	6.91	0.40	-5.41
-0.5	98	7.03	0.40	-7.53
-2.8	99	7.15	0.40	-9.95
-0.4	100	7.27	0.40	-7.67
2.2	101	7.39	0.40	-5.19
2.6	102	7.50	0.39	-4.90
5.7	103	7.61	0.39	-1.91
1.1	104	7.73	0.39	-6.63
-1.7	105	7.84	0.39	-9.54
-0.7	106	7.95	0.39	-8.65
5.8	107	8.06	0.39	-2.26
5.2	108	8.16	0.39	-2.96
5.3	109	8.27	0.39	-2.97
1.5	110	8.37	0.39	-6.87
3.8	111	8.48	0.38	-4.68
5.4	112	8.58	0.38	-3.18
7.9	113	8.68	0.38	-0.78
9.8	114	8.78	0.38	1.02
9.0	115	8.88	0.38	0.12
10.0	116	8.97	0.38	1.03
11.6	117	9.07	0.38	2.53
10.7	118	9.16	0.38	1.54
11.4	119	9.26	0.38	2.14
7.9	120	9.35	0.38	-1.45
6.4	121	9.44	0.38	-3.04
11.7	122	9.53	0.38	2.17
10.1	123	9.62	0.38	0.48
7.7	124	9.70	0.38	-2.00
14.5	126	9.88	0.38	4.62
8.8	127	9.96	0.38	-1.16
8.8	128	10.04	0.38	-1.24
7.5	130	10.20	0.38	-2.70
10.5	131	10.28	0.38	0.22
7.2	132	10.36	0.38	-3.16
10.8	133	10.44	0.38	0.36
9.7	134	10.52	0.38	-0.82
12.2	135	10.59	0.38	1.61
9.1	136	10.67	0.38	-1.57
12.8	137	10.74	0.38	2.06
9.1	138	10.81	0.38	-1.71
9.8	139	10.88	0.38	-1.08
14.5	140	10.95	0.38	3.55
14.2	141	11.02	0.38	3.18
11.4	142	11.09	0.38	0.31
10.4	143	11.15	0.38	-0.75
7.0	144	11.22	0.38	-4.22
12.5	145	11.28	0.38	1.22
13.7	146	11.35	0.38	2.35
8.4	147	11.41	0.38	-3.01
9.2	148	11.47	0.38	-2.27
13.9	149	11.53	0.38	2.37
12.2	150	11.59	0.38	0.61
17.5	151	11.65	0.38	5.85
21.9	152	11.71	0.38	10.19
12.2	153	11.77	0.38	0.43
5.0	154	11.82	0.38	-6.82
6.1	155	11.88	0.38	-5.78
5.4	156	11.93	0.38	-6.53
6.7	157	11.98	0.38	-5.28
11.0	158	12.04	0.38	-1.04
13.3	159	12.09	0.39	1.21
14.5	160	12.14	0.39	2.36
11.9	161	12.19	0.39	-0.29
9.1	162	12.24	0.39	-3.14
12.2	163	12.28	0.39	-0.08
17.3	164	12.33	0.39	4.97
8.2	165	12.38	0.39	-4.18
13.1	166	12.42	0.39	0.68
16.9	167	12.47	0.39	4.43
13.9	168	12.51	0.39	1.39
11.5	169	12.55	0.39	-1.05
17.2	170	12.60	0.39	4.60
18.8	171	12.64	0.39	6.16
14.7	173	12.72	0.39	1.98
14.6	174	12.76	0.39	1.84
18.0	175	12.79	0.39	5.21
11.2	176	12.83	0.39	-1.63
12.5	177	12.87	0.40	-0.37
18.6	178	12.90	0.40	5.70
16.6	179	12.94	0.40	3.66
20.7	180	12.97	0.40	7.73
22.2	181	13.01	0.40	9.19
19.5	182	13.04	0.40	6.46
17.3	183	13.07	0.40	4.23
21.7	184	13.10	0.40	8.60
25.2	185	13.13	0.40	12.07
23.8	186	13.16	0.40	10.64
23.3	187	13.19	0.40	10.11
17.0	188	13.22	0.40	3.78
19.4	189	13.24	0.40	6.16
20.1	190	13.26	0.40	6.84
18.9	191	13.28	0.40	5.62
19.3	192	13.30	0.40	6.00
18.3	193	13.32	0.40	4.98
17.4	194	13.33	0.40	4.07
15.6	195	13.34	0.40	2.26
19.8	196	13.35	0.40	6.45
22.1	197	13.36	0.41	8.74
20.9	198	13.37	0.41	7.53
20.2	199	13.37	0.41	6.83
19.0	200	13.37	0.41	5.63
20.0	201	13.38	0.41	6.62
18.0	202	13.37	0.41	4.63
20.0	203	13.37	0.41	6.63
24.1	204	13.37	0.41	10.73
24.2	205	13.36	0.41	10.84
24.4	206	13.35	0.41	11.05
22.3	207	13.34	0.41	8.96
22.3	208	13.33	0.41	8.97
21.9	209	13.32	0.41	8.58
22.7	210	13.30	0.41	9.40
22.4	211	13.29	0.41	9.11
22.7	212	13.27	0.41	9.43
23.6	213	13.25	0.41	10.35
21.7	214	13.23	0.41	8.47
25.2	215	13.21	0.41	11.99
24.4	216	13.19	0.41	11.21
20.8	217	13.16	0.41	7.64
23.3	218	13.14	0.41	10.16
25.4	219	13.11	0.41	12.29
22.8	220	13.08	0.41	9.72
21.2	221	13.05	0.41	8.15
21.9	222	13.02	0.41	8.88
20.6	223	12.99	0.41	7.61
22.2	224	12.96	0.41	9.24
21.3	225	12.92	0.41	8.38
20.9	226	12.89	0.41	8.01
22.4	227	12.85	0.41	9.55
21.2	228	12.81	0.41	8.39
18.8	229	12.77	0.41	6.03
17.2	230	12.73	0.41	4.47
18.5	231	12.69	0.41	5.81
17.4	232	12.65	0.41	4.75
16.7	233	12.61	0.41	4.09
19.5	234	12.57	0.42	6.93
19.9	235	12.52	0.42	7.38
19.6	236	12.48	0.42	7.12
20.5	237	12.43	0.42	8.07
18.3	238	12.38	0.42	5.92
21.0	239	12.34	0.42	8.66
22.9	240	12.29	0.42	10.61
23.6	241	12.24	0.42	11.36
20.5	242	12.19	0.42	8.31
17.2	243	12.14	0.42	5.06
16.5	244	12.09	0.42	4.41
18.5	245	12.04	0.42	6.46
19.9	246	11.99	0.42	7.91
19.2	247	11.93	0.42	7.27
19.9	248	11.88	0.42	8.02
22.1	249	11.83	0.42	10.27
16.7	250	11.78	0.42	4.92
13.2	251	11.72	0.42	1.48
12.5	252	11.67	0.42	0.83
12.1	253	11.61	0.43	0.49
15.3	254	11.56	0.43	3.74
16.5	255	11.50	0.43	5.00
17.8	256	11.45	0.43	6.35
19.0	257	11.39	0.43	7.61
20.4	258	11.34	0.43	9.06
19.2	259	11.28	0.43	7.92
21.0	260	11.22	0.43	9.78
19.3	261	11.17	0.43	8.13
11.3	262	11.11	0.43	0.19
10.5	263	11.05	0.43	-0.55
11.6	264	11.00	0.43	0.60
14.8	265	10.94	0.43	3.86
11.0	266	10.88	0.43	0.12
7.0	267	10.83	0.43	-3.83
7.4	268	10.77	0.43	-3.37
15.6	269	10.72	0.43	4.88
15.2	270	10.66	0.43	4.54
14.5	271	10.60	0.43	3.90
13.4	272	10.55	0.43	2.85
11.6	273	10.49	0.43	1.11
9.6	274	10.44	0.43	-0.84
7.7	275	10.38	0.43	-2.68
14.3	276	10.33	0.43	3.97
15.7	277	10.27	0.43	5.43
9.5	278	10.21	0.43	-0.71
8.9	279	10.14	0.43	-1.24
9.1	280	10.07	0.43	-0.97
6.7	281	9.99	0.43	-3.29
12.1	282	9.91	0.43	2.19
12.5	283	9.82	0.43	2.68
7.5	284	9.73	0.43	-2.23
10.6	285	9.64	0.43	0.96
6.2	286	9.54	0.43	-3.34
6.0	287	9.43	0.43	-3.43
8.8	288	9.32	0.43	-0.52
9.9	289	9.21	0.43	0.69
5.7	290	9.10	0.43	-3.40
2.5	291	8.98	0.43	-6.48
2.9	292	8.86	0.43	-5.96
10.9	293	8.73	0.43	2.17
7.0	294	8.60	0.42	-1.60
2.2	295	8.47	0.42	-6.27
4.1	296	8.34	0.42	-4.24
10.0	297	8.20	0.42	1.80
3.4	298	8.06	0.42	-4.66
0.8	299	7.92	0.42	-7.12
1.6	300	7.77	0.42	-6.17
11.4	301	7.62	0.42	3.78
12.1	302	7.47	0.42	4.63
8.5	303	7.32	0.42	1.18
5.7	304	7.16	0.42	-1.46
7.7	305	7.01	0.42	0.69
11.3	306	6.85	0.42	4.45
13.2	307	6.69	0.41	6.51
4.6	308	6.53	0.41	-1.93
3.1	309	6.37	0.41	-3.27
10.6	310	6.20	0.41	4.40
11.7	311	6.04	0.41	5.66
7.3	312	5.87	0.41	1.43
5.1	313	5.71	0.41	-0.61
6.8	314	5.54	0.41	1.26
0.8	315	5.37	0.41	-4.57
-0.7	316	5.20	0.41	-5.90
4.2	317	5.03	0.41	-0.83
1.2	318	4.87	0.41	-3.67
-1.9	320	4.53	0.41	-6.43
0.1	321	4.36	0.41	-4.26
-2.4	322	4.19	0.41	-6.59
-2.6	323	4.02	0.41	-6.62
0.8	324	3.85	0.41	-3.05
-0.1	325	3.68	0.41	-3.78
-7.6	326	3.52	0.41	-11.12
-9.1	327	3.35	0.41	-12.45
-1.4	328	3.19	0.41	-4.59
1.1	329	3.02	0.41	-1.92
0.3	330	2.86	0.41	-2.56
3.0	331	2.70	0.41	0.30
3.8	332	2.54	0.41	1.26
0.2	333	2.38	0.41	-2.18
-0.1	334	2.22	0.41	-2.32
-0.4	335	2.07	0.41	-2.47
2.4	336	1.92	0.42	0.48
6.6	337	1.77	0.42	4.83
-1.3	338	1.62	0.42	-2.92
-6.0	339	1.47	0.42	-7.47
-2.5	341	1.19	0.42	-3.69
-8.5	342	1.05	0.42	-9.55
-9.5	343	0.91	0.42	-10.41
-7.7	344	0.78	0.42	-8.48
-9.0	345	0.65	0.42	-9.65
-10.3	346	0.52	0.42	-10.82
-8.7	347	0.40	0.42	-9.10
-4.0	348	0.28	0.42	-4.28
2.8	349	0.16	0.43	2.64
-0.1	350	0.05	0.43	-0.15
-0.2	351	-0.06	0.43	-0.14
-2.4	352	-0.16	0.43	-2.24
-4.5	353	-0.26	0.43	-4.24
-2.3	354	-0.36	0.43	-1.94
4.2	355	-0.45	0.43	4.65
9.5	356	-0.54	0.43	10.04
-0.4	357	-0.62	0.43	0.22
-6.5	358	-0.70	0.43	-5.80
-10.3	359	-0.77	0.43	-9.53
-7.8	360	-0.84	0.43	-6.96
-10.7	361	-0.90	0.43	-9.80
-5.1	362	-0.96	0.43	-4.14
1.5	363	-1.02	0.43	2.52
-8.7	364	-1.06	0.43	-7.64
-8.3	365	-1.11	0.43	-7.19
-0.9	366	-1.14	0.43	0.24
-10.3	367	-1.17	0.44	-9.13
-10.6	368	-1.20	0.43	-9.40
-3.1	369	-1.22	0.43	-1.88
1.4	370	-1.24	0.43	2.64
-0.6	371	-1.26	0.43	0.66
-7.3	372	-1.28	0.43	-6.02
-6.7	373	-1.29	0.43	-5.41
2.4	374	-1.30	0.43	3.70
-0.4	375	-1.31	0.43	0.91
-6.9	376	-1.31	0.43	-5.59
-10.9	377	-1.32	0.43	-9.58
-9.7	378	-1.32	0.43	-8.38
-11.1	379	-1.31	0.43	-9.79
-7.9	380	-1.31	0.43	-6.59
-4.1	381	-1.30	0.43	-2.80
-8.3	382	-1.29	0.43	-7.01
-8.6	383	-1.28	0.43	-7.32
-7.5	384	-1.27	0.43	-6.23
-1.6	385	-1.25	0.43	-0.35
0.3	386	-1.24	0.43	1.54
-10.9	387	-1.22	0.42	-9.68
-4.4	388	-1.19	0.42	-3.21
7.5	389	-1.17	0.42	8.67
4.0	390	-1.15	0.42	5.15
-2.5	391	-1.12	0.42	-1.38
-0.7	392	-1.09	0.42	0.39
-3.4	393	-1.06	0.42	-2.34
-8.0	394	-1.03	0.42	-6.97
-1.9	395	-0.99	0.42	-0.91
-7.3	396	-0.96	0.42	-6.34
-11.4	397	-0.92	0.42	-10.48
-9.0	398	-0.88	0.42	-8.12
-8.5	399	-0.84	0.42	-7.66
-1.9	400	-0.80	0.41	-1.10
4.9	401	-0.75	0.41	5.65
-2.0	402	-0.71	0.41	-1.29
-4.3	403	-0.66	0.41	-3.64
1.7	404	-0.62	0.41	2.32
-5.0	405	-0.57	0.41	-4.43
-7.6	406	-0.52	0.41	-7.08
-10.3	407	-0.47	0.41	-9.83
-7.9	408	-0.41	0.41	-7.49
-3.4	409	-0.36	0.41	-3.04
-4.9	410	-0.31	0.41	-4.59
-3.5	411	-0.25	0.41	-3.25
2.4	412	-0.20	0.41	2.60
-5.7	413	-0.14	0.41	-5.56
-7.6	414	-0.08	0.41	-7.52
-9.9	415	-0.02	0.41	-9.88
-11.5	416	0.04	0.41	-11.54
-8.2	417	0.10	0.41	-8.30
-6.3	418	0.16	0.41	-6.46
-7.2	419	0.22	0.41	-7.42
-5.8	420	0.28	0.41	-6.08
-2.0	421	0.35	0.41	-2.35
-11.1	422	0.41	0.41	-11.51
-12.6	423	0.47	0.41	-13.07
-10.8	424	0.54	0.41	-11.34
-6.2	425	0.60	0.42	-6.80
-3.7	426	0.67	0.42	-4.37
-4.9	427	0.73	0.42	-5.63
-2.3	428	0.80	0.42	-3.10
-6.2	429	0.86	0.42	-7.06
-8.2	430	0.93	0.42	-9.13
-11.7	431	0.99	0.42	-12.69
-9.1	432	1.06	0.42	-10.16
-4.2	433	1.12	0.42	-5.32
-5.1	434	1.19	0.42	-6.29
2.9	435	1.26	0.42	1.64
-0.2	436	1.32	0.42	-1.52
-0.1	437	1.39	0.42	-1.49
1.8	438	1.45	0.42	0.35
3.1	439	1.52	0.43	1.58
4.1	440	1.58	0.43	2.52
-2.4	441	1.65	0.43	-4.05
-3.6	442	1.71	0.43	-5.31
-3.4	443	1.77	0.43	-5.17
-1.7	444	1.84	0.43	-3.54
0.8	445	1.90	0.43	-1.10
3.7	446	1.96	0.43	1.74
2.4	447	2.02	0.43	0.38
4.4	448	2.08	0.43	2.32
-0.1	449	2.14	0.43	-2.24
-2.4	450	2.20	0.43	-4.60
-3.3	451	2.26	0.43	-5.56
-1.5	452	2.32	0.43	-3.82
5.0	453	2.38	0.43	2.62
4.1	454	2.43	0.43	1.67
7.7	455	2.49	0.43	5.21
2.7	456	2.54	0.43	0.16
2.0	457	2.60	0.43	-0.60
3.7	458	2.65	0.43	1.05
0.3	460	2.77	0.43	-2.47
2.1	461	2.83	0.43	-0.73
3.5	462	2.90	0.43	0.60
-1.9	463	2.97	0.43	-4.87
-2.4	464	3.04	0.43	-5.44
-1.8	465	3.11	0.43	-4.91
3.2	466	3.18	0.43	0.02
6.7	467	3.26	0.43	3.44
8.1	468	3.34	0.43	4.76
11.8	469	3.42	0.43	8.38
7.8	470	3.50	0.43	4.30
3.2	471	3.59	0.43	-0.39
0.4	472	3.68	0.43	-3.28
3.6	473	3.77	0.43	-0.17
6.7	474	3.86	0.43	2.84
11.9	475	3.95	0.43	7.95
12.8	476	4.04	0.43	8.76
10.3	477	4.14	0.43	6.16
6.8	478	4.24	0.43	2.56
7.8	479	4.34	0.43	3.46
2.4	480	4.44	0.43	-2.04
5.7	481	4.54	0.42	1.16
5.7	482	4.64	0.42	1.06
7.7	483	4.74	0.42	2.96
5.0	484	4.85	0.42	0.15
5.1	485	4.95	0.42	0.15
5.6	486	5.06	0.42	0.54
6.4	487	5.16	0.42	1.24
7.3	488	5.27	0.42	2.03
4.8	489	5.38	0.42	-0.58
10.2	490	5.49	0.42	4.71
8.7	491	5.60	0.42	3.10
11.1	492	5.71	0.42	5.39
8.2	493	5.82	0.42	2.38
6.2	494	5.93	0.42	0.27
5.8	495	6.04	0.42	-0.24
9.2	496	6.15	0.42	3.05
7.1	497	6.26	0.42	0.84
6.3	498	6.37	0.41	-0.07
4.9	499	6.48	0.41	-1.58
5.4	500	6.59	0.41	-1.19
5.7	501	6.70	0.41	-1.00
9.2	502	6.80	0.41	2.40
5.9	503	6.91	0.41	-1.01
9.3	504	7.02	0.41	2.28
8.4	505	7.13	0.41	1.27
6.0	506	7.24	0.41	-1.24
4.4	507	7.34	0.41	-2.94
8.7	508	7.45	0.41	1.25
5.9	509	7.55	0.41	-1.65
11.3	510	7.66	0.41	3.64
9.3	511	7.76	0.41	1.54
8.8	512	7.86	0.41	0.94
9.4	513	7.96	0.41	1.44
8.6	514	8.06	0.41	0.54
9.7	515	8.16	0.41	1.54
8.7	516	8.26	0.41	0.44
12.9	517	8.35	0.41	4.55
10.0	518	8.44	0.41	1.56
11.9	519	8.54	0.41	3.36
11.7	520	8.63	0.41	3.07
9.0	521	8.71	0.41	0.29
10.6	522	8.80	0.41	1.80
15.2	523	8.89	0.41	6.31
13.5	524	8.97	0.41	4.53
14.6	525	9.05	0.41	5.55
14.7	526	9.13	0.41	5.57
13.5	527	9.20	0.41	4.30
15.7	528	9.28	0.41	6.42
17.4	529	9.35	0.41	8.05
17.3	530	9.42	0.41	7.88
18.3	531	9.49	0.40	8.81
16.8	532	9.55	0.40	7.25
18.6	533	9.61	0.40	8.99
17.7	534	9.67	0.40	8.03
17.4	535	9.73	0.40	7.67
16.6	536	9.78	0.40	6.82
13.1	537	9.83	0.40	3.27
14.4	538	9.88	0.40	4.52
17.3	539	9.92	0.40	7.38
15.3	540	9.96	0.40	5.34
15.9	541	10.00	0.40	5.90
14.6	542	10.03	0.40	4.57
12.8	543	10.06	0.40	2.74
14.7	544	10.09	0.40	4.61
13.9	545	10.11	0.40	3.79
14.7	546	10.13	0.40	4.57
13.8	547	10.15	0.40	3.65
15.8	548	10.17	0.40	5.63
17.5	549	10.18	0.40	7.32
20.9	550	10.19	0.40	10.71
26.0	551	10.21	0.40	15.79
22.9	552	10.22	0.40	12.68
19.5	553	10.24	0.40	9.26
19.7	554	10.25	0.39	9.45
18.9	555	10.27	0.39	8.63
16.1	556	10.28	0.39	5.82
16.6	557	10.30	0.39	6.30
16.6	558	10.31	0.39	6.29
19.6	559	10.32	0.39	9.28
18.1	560	10.34	0.39	7.76
19.9	561	10.35	0.39	9.55
22.1	562	10.36	0.39	11.74
21.9	563	10.38	0.39	11.52
19.6	564	10.39	0.39	9.21
19.1	565	10.40	0.39	8.70
24.0	566	10.42	0.39	13.58
24.1	567	10.43	0.39	13.67
21.6	568	10.44	0.39	11.16
19.1	569	10.45	0.39	8.65
18.4	570	10.46	0.39	7.94
18.7	571	10.47	0.39	8.23
20.2	572	10.49	0.38	9.71
20.9	573	10.50	0.38	10.40
20.1	574	10.51	0.38	9.59
21.7	575	10.52	0.38	11.18
23.9	576	10.52	0.38	13.38
23.7	577	10.53	0.38	13.17
22.9	578	10.54	0.38	12.36
22.0	579	10.55	0.38	11.45
22.6	580	10.56	0.38	12.04
20.9	581	10.56	0.38	10.34
19.5	582	10.57	0.38	8.93
20.6	583	10.58	0.38	10.02
19.4	584	10.58	0.38	8.82
17.4	585	10.59	0.38	6.81
20.2	586	10.59	0.38	9.61
18.7	587	10.60	0.38	8.10
17.9	588	10.60	0.38	7.30
21.0	589	10.60	0.38	10.40
20.3	590	10.60	0.38	9.70
19.8	591	10.61	0.38	9.19
20.1	592	10.61	0.38	9.49
18.9	593	10.61	0.38	8.29
19.7	594	10.61	0.38	9.09
19.2	595	10.61	0.38	8.59
20.3	596	10.60	0.38	9.70
22.5	597	10.60	0.38	11.90
21.7	598	10.60	0.38	11.10
23.4	599	10.59	0.38	12.81
21.3	600	10.59	0.38	10.71
14.5	601	10.58	0.38	3.92
14.6	602	10.58	0.38	4.02
13.2	603	10.57	0.38	2.63
16.2	604	10.56	0.38	5.64
21.3	605	10.55	0.38	10.75
18.0	606	10.54	0.38	7.46
21.7	607	10.53	0.38	11.17
20.1	608	10.52	0.38	9.58
16.8	609	10.51	0.38	6.29
15.9	610	10.50	0.38	5.40
18.4	611	10.48	0.38	7.92
19.9	612	10.47	0.38	9.43
16.9	613	10.45	0.38	6.45
14.9	614	10.43	0.38	4.47
12.9	615	10.42	0.38	2.48
12.4	616	10.40	0.38	2.00
14.9	617	10.38	0.38	4.52
15.4	618	10.36	0.38	5.04
15.2	619	10.33	0.38	4.87
11.8	620	10.31	0.38	1.49
11.6	621	10.29	0.39	1.31
12.3	622	10.26	0.39	2.04
17.7	623	10.23	0.39	7.47
14.8	624	10.21	0.39	4.59
9.7	625	10.18	0.39	-0.48
8.4	626	10.15	0.39	-1.75
10.8	627	10.12	0.39	0.68
14.1	628	10.08	0.39	4.02
14.8	629	10.05	0.39	4.75
16.4	630	10.02	0.39	6.38
20.1	631	9.98	0.40	10.12
17.8	632	9.94	0.40	7.86
15.2	634	9.86	0.40	5.34
14.6	635	9.82	0.40	4.78
13.8	636	9.78	0.40	4.02
12.9	637	9.74	0.41	3.16
8.6	638	9.69	0.41	-1.09
10.2	639	9.65	0.41	0.55
9.3	640	9.60	0.41	-0.30
6.1	641	9.55	0.41	-3.45
5.9	642	9.50	0.41	-3.60
6.7	643	9.45	0.42	-2.75
9.6	644	9.40	0.42	0.20
16.4	645	9.34	0.42	7.06
12.6	646	9.29	0.42	3.31
10.2	647	9.24	0.43	0.96
9.4	648	9.18	0.43	0.22
10.5	649	9.12	0.43	1.38
11.6	650	9.06	0.43	2.54
10.2	651	9.00	0.44	1.20
10.3	652	8.94	0.44	1.36
12.3	653	8.88	0.44	3.42
9.9	654	8.82	0.44	1.08
11.7	655	8.76	0.45	2.94
10.0	656	8.69	0.45	1.31
7.8	657	8.62	0.45	-0.82
8.6	658	8.56	0.46	0.04
10.0	659	8.49	0.46	1.51
8.6	660	8.42	0.46	0.18
11.3	661	8.35	0.47	2.95
10.9	662	8.28	0.47	2.62
9.9	663	8.21	0.47	1.69
5.5	664	8.13	0.48	-2.63
5.1	665	8.06	0.48	-2.96
5.9	666	7.99	0.48	-2.09
6.4	667	7.91	0.49	-1.51
13.4	669	7.75	0.49	5.65
12.2	670	7.68	0.50	4.52
6.0	672	7.52	0.51	-1.52
5.5	673	7.43	0.51	-1.93
7.3	674	7.35	0.52	-0.05
7.3	675	7.27	0.52	0.03
3.4	676	7.18	0.52	-3.78
-2.6	678	7.01	0.53	-9.61
2.4	679	6.92	0.54	-4.52
5.6	680	6.84	0.54	-1.24
9.4	681	6.75	0.55	2.65
-1.0	682	6.66	0.55	-7.66
-3.9	683	6.56	0.56	-10.46
6.6	684	6.47	0.56	0.13
-1.2	685	6.38	0.57	-7.58
-3.9	686	6.29	0.57	-10.19
0.6	687	6.19	0.58	-5.59
3.6	688	6.10	0.58	-2.50
1.7	689	6.00	0.59	-4.30
1.9	690	5.90	0.59	-4.00
4.9	691	5.81	0.60	-0.91
0.7	692	5.71	0.60	-5.01
3.7	693	5.61	0.61	-1.91
5.9	694	5.51	0.61	0.39
5.0	695	5.41	0.62	-0.41
4.6	696	5.30	0.63	-0.70
1.4	697	5.20	0.63	-3.80
-1.6	698	5.10	0.64	-6.70
-3.1	699	4.99	0.64	-8.09
-6.6	700	4.89	0.65	-11.49
-4.1	701	4.78	0.66	-8.88
2.8	702	4.67	0.66	-1.87
0.6	703	4.57	0.67	-3.97
-0.3	704	4.46	0.67	-4.76
-1.5	705	4.35	0.68	-5.85
-6.1	706	4.24	0.69	-10.34
-5.5	707	4.13	0.69	-9.63
4.8	708	4.02	0.70	0.78
11.6	709	3.90	0.71	7.70
2.1	710	3.79	0.71	-1.69
-8.0	711	3.68	0.72	-11.68
-5.3	712	3.56	0.73	-8.86
7.5	713	3.45	0.73	4.05
7.2	714	3.33	0.74	3.87
-2.7	715	3.21	0.75	-5.91
-4.0	716	3.10	0.75	-7.10
-5.5	717	2.98	0.76	-8.48
-8.7	718	2.86	0.77	-11.56
-5.1	719	2.74	0.78	-7.84
-3.7	720	2.62	0.78	-6.32
-2.3	721	2.50	0.79	-4.80
0.3	722	2.38	0.80	-2.08
-0.4	723	2.25	0.80	-2.65
-5.8	724	2.13	0.81	-7.93
-7.9	725	2.01	0.82	-9.91
-3.6	726	1.88	0.83	-5.48
2.1	727	1.76	0.83	0.34
-2.7	728	1.63	0.84	-4.33
-3.5	729	1.51	0.85	-5.01
0.0	730	1.38	0.86	-1.38

Residuals

augment(m2, se_fit = TRUE) |> 
  ggplot(aes(x = .resid)) + 
  geom_histogram(binwidth = 1) + my_theme

Predictions

new_data = tibble(year_day = seq(1:730))
augment(m2, newdata = new_data, se_fit = TRUE) |>
  ggplot(aes(x = year_day, y = .fitted)) + 
  geom_line(size = 2, color="blue") +
  geom_ribbon(aes(ymin = .fitted - 2*.se.fit,
                  ymax = .fitted + 2*.se.fit), alpha = 0.25, fill="blue") +
  geom_point(aes(y = mean_temp), data = hfx) + my_theme

LOESS with smaller span

m3 <- loess(mean_temp ~ year_day, data = hfx, 
            degree = 1, span = 0.25)
augment(m3, newdata = new_data, se_fit = TRUE) |>
  ggplot(aes(x = year_day, y = .fitted)) + 
  geom_line(size = 2, color="blue") +
  geom_ribbon(aes(ymin = .fitted - 2*.se.fit,
                  ymax = .fitted + 2*.se.fit), alpha = 0.25, fill="blue") +
  geom_point(aes(y = mean_temp), data = hfx) + my_theme

Residuals

augment(m3, se_fit = TRUE) |> 
  ggplot(aes(x = .resid)) + 
  geom_histogram(binwidth = 1) + my_theme

Summary

• Fitting GAM and LOESS smooths

• No model coefficients to extract

• Model statistics available with glance for GAM (but not LOESS)

• Computing predicted values with errors and residuals (compared to data)

Further reading

• Course notes

• Healy Chapter 6. Work with models.