GAM and LOESS models

Andrew Irwin, a.irwin@dal.ca

2024-02-15

GAM and LOESS models

• 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="cs"), data = hfx)
summary(m1)

Family: gaussian 
Link function: identity 

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

Parametric coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)   7.0590     0.1415   49.89   <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) 8.868      9 429.1  <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

R-sq.(adj) =  0.844   Deviance explained = 84.6%
GCV = 14.553  Scale est. = 14.353    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
9.87	-1967.44	3956.61	4006.33	10149.42	707.13	717

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

term	edf	ref.df	statistic	p.value
s(year_day)	8.87	9	429.13	0

Find predicted values and residuals

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

year_day	mean_temp	.value	.residual
1	-11.5	-4.6610701	-6.8389299
2	-12.5	-4.6207022	-7.8792978
3	-9.0	-4.5803070	-4.4196930
4	2.2	-4.5398571	6.7398571
5	-2.7	-4.4993251	1.7993251
6	-13.8	-4.4586838	-9.3413162
7	-14.1	-4.4179057	-9.6820943
8	-4.1	-4.3769635	0.2769635
9	-2.0	-4.3358298	2.3358298
10	-5.8	-4.2944774	-1.5055226
11	1.3	-4.2528789	5.5528789
12	9.6	-4.2110068	13.8110068
13	6.7	-4.1688340	10.8688340
14	-5.0	-4.1263329	-0.8736671
15	-6.4	-4.0834763	-2.3165237
16	-5.0	-4.0402368	-0.9597632
17	-4.3	-3.9965872	-0.3034128
18	-4.3	-3.9524999	-0.3475001
19	-9.1	-3.9079477	-5.1920523
20	-2.9	-3.8629032	0.9629032
21	-1.7	-3.8173391	2.1173391
22	-9.3	-3.7712281	-5.5287719
23	-0.4	-3.7245427	3.3245427
24	2.0	-3.6772557	5.6772557
25	-7.4	-3.6293396	-3.7706604
26	-10.8	-3.5807672	-7.2192328
27	-4.0	-3.5315110	-0.4684890
28	4.3	-3.4815438	7.7815438
29	-3.0	-3.4308382	0.4308382
30	-7.0	-3.3793669	-3.6206331
31	-9.1	-3.3271024	-5.7728976
32	-2.7	-3.2740175	0.5740175
33	-2.9	-3.2200847	0.3200847
34	-14.0	-3.1652769	-10.8347231
36	3.2	-3.0529263	6.2529263
37	-4.8	-2.9953288	-1.8046712
38	0.7	-2.9367468	3.6367468
39	1.4	-2.8771530	4.2771530
40	-7.4	-2.8165199	-4.5834801
41	1.1	-2.7548202	3.8548202
42	3.2	-2.6920265	5.8920265
43	2.2	-2.6281116	4.8281116
44	-7.9	-2.5630480	-5.3369520
46	3.0	-2.4293656	5.4293656
47	1.7	-2.3606920	4.0606920
48	-5.8	-2.2907604	-3.5092396
49	-4.4	-2.2195434	-2.1804566
50	-1.7	-2.1470137	0.4470137
51	2.7	-2.0731439	4.7731439
52	5.6	-1.9979067	7.5979067
53	-0.1	-1.9212747	1.8212747
54	-4.4	-1.8432206	-2.5567794
55	0.7	-1.7637170	2.4637170
56	-3.5	-1.6827366	-1.8172634
57	0.8	-1.6002520	2.4002520
58	0.4	-1.5162360	1.9162360
59	2.4	-1.4306610	3.8306610
60	0.1	-1.3434998	1.4434998
61	3.0	-1.2547251	4.2547251
62	4.3	-1.1643095	5.4643095
63	0.8	-1.0722256	1.8722256
64	2.0	-0.9784461	2.9784461
65	1.0	-0.8829436	1.8829436
66	1.3	-0.7856909	2.0856909
67	1.4	-0.6866605	2.0866605
68	1.1	-0.5858250	1.6858250
69	0.6	-0.4831573	1.0831573
70	0.3	-0.3786298	0.6786298
71	-1.4	-0.2722153	-1.1277847
72	-0.8	-0.1638864	-0.6361136
73	1.8	-0.0536158	1.8536158
74	1.4	0.0586240	1.3413760
75	-0.7	0.1728602	-0.8728602
76	-7.3	0.2891201	-7.5891201
77	-6.7	0.4074312	-7.1074312
78	-6.2	0.5278208	-6.7278208
79	-2.9	0.6503163	-3.5503163
80	-1.4	0.7749449	-2.1749449
81	0.5	0.9017340	-0.4017340
82	-0.5	1.0307110	-1.5307110
84	-3.5	1.2953295	-4.7953295
85	-2.0	1.4309602	-3.4309602
86	1.1	1.5687491	-0.4687491
87	3.4	1.7086500	1.6913500
88	4.2	1.8506165	2.3493835
89	5.7	1.9946024	3.7053976
90	2.9	2.1405615	0.7594385
91	3.1	2.2884475	0.8115525
92	0.1	2.4382141	-2.3382141
93	2.2	2.5898150	-0.3898150
94	5.9	2.7432040	3.1567960
95	2.3	2.8983348	-0.5983348
96	-1.5	3.0551611	-4.5551611
97	1.5	3.2136367	-1.7136367
98	-0.5	3.3737153	-3.8737153
99	-2.8	3.5353507	-6.3353507
100	-0.4	3.6984964	-4.0984964
101	2.2	3.8631064	-1.6631064
102	2.6	4.0291343	-1.4291343
103	5.7	4.1965338	1.5034662
104	1.1	4.3652588	-3.2652588
105	-1.7	4.5352628	-6.2352628
106	-0.7	4.7064997	-5.4064997
107	5.8	4.8789232	0.9210768
108	5.2	5.0524870	0.1475130
109	5.3	5.2271448	0.0728552
110	1.5	5.4028504	-3.9028504
111	3.8	5.5795575	-1.7795575
112	5.4	5.7572199	-0.3572199
113	7.9	5.9357912	1.9642088
114	9.8	6.1152252	3.6847748
115	9.0	6.2954756	2.7045244
116	10.0	6.4764962	3.5235038
117	11.6	6.6582407	4.9417593
118	10.7	6.8406629	3.8593371
119	11.4	7.0237163	4.3762837
120	7.9	7.2073549	0.6926451
121	6.4	7.3915323	-0.9915323
122	11.7	7.5762022	4.1237978
123	10.1	7.7613185	2.3386815
124	7.7	7.9468347	-0.2468347
126	14.5	8.3188821	6.1811179
127	8.8	8.5053208	0.2946792
128	8.8	8.6919744	0.1080256
130	7.5	9.0657413	-1.5657413
131	10.5	9.2527621	1.2472379
132	7.2	9.4398128	-2.2398128
133	10.8	9.6268471	1.1731529
134	9.7	9.8138187	-0.1138187
135	12.2	10.0006813	2.1993187
136	9.1	10.1873888	-1.0873888
137	12.8	10.3738947	2.4261053
138	9.1	10.5601530	-1.4601530
139	9.8	10.7461172	-0.9461172
140	14.5	10.9317411	3.5682589
141	14.2	11.1169784	3.0830216
142	11.4	11.3017830	0.0982170
143	10.4	11.4861084	-1.0861084
144	7.0	11.6699085	-4.6699085
145	12.5	11.8531370	0.6468630
146	13.7	12.0357476	1.6642524
147	8.4	12.2176940	-3.8176940
148	9.2	12.3989300	-3.1989300
149	13.9	12.5794093	1.3205907
150	12.2	12.7590856	-0.5590856
151	17.5	12.9379126	4.5620874
152	21.9	13.1158442	8.7841558
153	12.2	13.2928340	-1.0928340
154	5.0	13.4688357	-8.4688357
155	6.1	13.6438032	-7.5438032
156	5.4	13.8176900	-8.4176900
157	6.7	13.9904500	-7.2904500
158	11.0	14.1620369	-3.1620369
159	13.3	14.3324044	-1.0324044
160	14.5	14.5015062	-0.0015062
161	11.9	14.6692961	-2.7692961
162	9.1	14.8357278	-5.7357278
163	12.2	15.0007551	-2.8007551
164	17.3	15.1643316	2.1356684
165	8.2	15.3264112	-7.1264112
166	13.1	15.4869466	-2.3869466
167	16.9	15.6458867	1.2541133
168	13.9	15.8031786	-1.9031786
169	11.5	15.9587694	-4.4587694
170	17.2	16.1126062	1.0873938
171	18.8	16.2646363	2.5353637
173	14.7	16.5630646	-1.8630646
174	14.6	16.7093571	-2.1093571
175	18.0	16.8536315	1.1463685
176	11.2	16.9958348	-5.7958348
177	12.5	17.1359142	-4.6359142
178	18.6	17.2738168	1.3261832
179	16.6	17.4094899	-0.8094899
180	20.7	17.5428805	3.1571195
181	22.2	17.6739358	4.5260642
182	19.5	17.8026029	1.6973971
183	17.3	17.9288290	-0.6288290
184	21.7	18.0525613	3.6474387
185	25.2	18.1737468	7.0262532
186	23.8	18.2923328	5.5076672
187	23.3	18.4082664	4.8917336
188	17.0	18.5214947	-1.5214947
189	19.4	18.6319649	0.7680351
190	20.1	18.7396241	1.3603759
191	18.9	18.8444195	0.0555805
192	19.3	18.9462982	0.3537018
193	18.3	19.0452074	-0.7452074
194	17.4	19.1410943	-1.7410943
195	15.6	19.2339059	-3.6339059
196	19.8	19.3235894	0.4764106
197	22.1	19.4100920	2.6899080
198	20.9	19.4933609	1.4066391
199	20.2	19.5733431	0.6266569
200	19.0	19.6499858	-0.6499858
201	20.0	19.7232362	0.2767638
202	18.0	19.7930413	-1.7930413
203	20.0	19.8593485	0.1406515
204	24.1	19.9221048	4.1778952
205	24.2	19.9812573	4.2187427
206	24.4	20.0367533	4.3632467
207	22.3	20.0885398	2.2114602
208	22.3	20.1365640	2.1634360
209	21.9	20.1807730	1.7192270
210	22.7	20.2211141	2.4788859
211	22.4	20.2575343	2.1424657
212	22.7	20.2899809	2.4100191
213	23.6	20.3184008	3.2815992
214	21.7	20.3427414	1.3572586
215	25.2	20.3629497	4.8370503
216	24.4	20.3789730	4.0210270
217	20.8	20.3907582	0.4092418
218	23.3	20.3982527	2.9017473
219	25.4	20.4014035	4.9985965
220	22.8	20.4001578	2.3998422
221	21.2	20.3944627	0.8055373
222	21.9	20.3842654	1.5157346
223	20.6	20.3695131	0.2304869
224	22.2	20.3501528	1.8498472
225	21.3	20.3261318	0.9738682
226	20.9	20.2973972	0.6026028
227	22.4	20.2638961	2.1361039
228	21.2	20.2255756	0.9744244
229	18.8	20.1823830	-1.3823830
230	17.2	20.1342654	-2.9342654
231	18.5	20.0811699	-1.5811699
232	17.4	20.0230437	-2.6230437
233	16.7	19.9598339	-3.2598339
234	19.5	19.8914877	-0.3914877
235	19.9	19.8179522	0.0820478
236	19.6	19.7391746	-0.1391746
237	20.5	19.6551019	0.8448981
238	18.3	19.5656815	-1.2656815
239	21.0	19.4708604	1.5291396
240	22.9	19.3705857	3.5294143
241	23.6	19.2648046	4.3351954
242	20.5	19.1534643	1.3465357
243	17.2	19.0365120	-1.8365120
244	16.5	18.9138946	-2.4138946
245	18.5	18.7855595	-0.2855595
246	19.9	18.6514548	1.2485452
247	19.2	18.5115946	0.6884054
248	19.9	18.3660946	1.5339054
249	22.1	18.2150795	3.8849205
250	16.7	18.0586740	-1.3586740
251	13.2	17.8970025	-4.6970025
252	12.5	17.7301898	-5.2301898
253	12.1	17.5583605	-5.4583605
254	15.3	17.3816391	-2.0816391
255	16.5	17.2001503	-0.7001503
256	17.8	17.0140187	0.7859813
257	19.0	16.8233689	2.1766311
258	20.4	16.6283255	3.7716745
259	19.2	16.4290132	2.7709868
260	21.0	16.2255565	4.7744435
261	19.3	16.0180801	3.2819199
262	11.3	15.8067087	-4.5067087
263	10.5	15.5915667	-5.0915667
264	11.6	15.3727788	-3.7727788
265	14.8	15.1504697	-0.3504697
266	11.0	14.9247639	-3.9247639
267	7.0	14.6957861	-7.6957861
268	7.4	14.4636609	-7.0636609
269	15.6	14.2285129	1.3714871
270	15.2	13.9904667	1.2095333
271	14.5	13.7496470	0.7503530
272	13.4	13.5061783	-0.1061783
273	11.6	13.2601852	-1.6601852
274	9.6	13.0117924	-3.4117924
275	7.7	12.7611245	-5.0611245
276	14.3	12.5083062	1.7916938
277	15.7	12.2534619	3.4465381
278	9.5	11.9967164	-2.4967164
279	8.9	11.7381942	-2.8381942
280	9.1	11.4780200	-2.3780200
281	6.7	11.2163184	-4.5163184
282	12.1	10.9532140	1.1467860
283	12.5	10.6888315	1.8111685
284	7.5	10.4232953	-2.9232953
285	10.6	10.1567302	0.4432698
286	6.2	9.8892608	-3.6892608
287	6.0	9.6210116	-3.6210116
288	8.8	9.3521073	-0.5521073
289	9.9	9.0826726	0.8173274
290	5.7	8.8128319	-3.1128319
291	2.5	8.5427100	-6.0427100
292	2.9	8.2724315	-5.3724315
293	10.9	8.0021209	2.8978791
294	7.0	7.7319029	-0.7319029
295	2.2	7.4619022	-5.2619022
296	4.1	7.1922432	-3.0922432
297	10.0	6.9230507	3.0769493
298	3.4	6.6544492	-3.2544492
299	0.8	6.3865634	-5.5865634
300	1.6	6.1195178	-4.5195178
301	11.4	5.8534372	5.5465628
302	12.1	5.5884461	6.5115539
303	8.5	5.3246691	3.1753309
304	5.7	5.0622308	0.6377692
305	7.7	4.8012559	2.8987441
306	11.3	4.5418689	6.7581311
307	13.2	4.2841946	8.9158054
308	4.6	4.0283575	0.5716425
309	3.1	3.7744821	-0.6744821
310	10.6	3.5226932	7.0773068
311	11.7	3.2731154	8.4268846
312	7.3	3.0258732	4.2741268
313	5.1	2.7810913	2.3189087
314	6.8	2.5388943	4.2611057
315	0.8	2.2994068	-1.4994068
316	-0.7	2.0627534	-2.7627534
317	4.2	1.8290588	2.3709412
318	1.2	1.5984475	-0.3984475
320	-1.9	1.1469735	-3.0469735
321	0.1	0.9263600	-0.8263600
322	-2.4	0.7093284	-3.1093284
323	-2.6	0.4960031	-3.0960031
324	0.8	0.2865089	0.5134911
325	-0.1	0.0809704	-0.1809704
326	-7.6	-0.1204879	-7.4795121
327	-9.1	-0.3177530	-8.7822470
328	-1.4	-0.5108063	-0.8891937
329	1.1	-0.6996735	1.7996735
330	0.3	-0.8843809	1.1843809
331	3.0	-1.0649547	4.0649547
332	3.8	-1.2414210	5.0414210
333	0.2	-1.4138061	1.6138061
334	-0.1	-1.5821362	1.4821362
335	-0.4	-1.7464375	1.3464375
336	2.4	-1.9067360	4.3067360
337	6.6	-2.0630582	8.6630582
338	-1.3	-2.2154301	0.9154301
339	-6.0	-2.3638779	-3.6361221
341	-2.5	-2.6491062	0.1491062
342	-8.5	-2.7859390	-5.7140610
343	-9.5	-2.9189526	-6.5810474
344	-7.7	-3.0481731	-4.6518269
345	-9.0	-3.1736267	-5.8263733
346	-10.3	-3.2953397	-7.0046603
347	-8.7	-3.4133381	-5.2866619
348	-4.0	-3.5276483	-0.4723517
349	2.8	-3.6382963	6.4382963
350	-0.1	-3.7453085	3.6453085
351	-0.2	-3.8487110	3.6487110
352	-2.4	-3.9485299	1.5485299
353	-4.5	-4.0447916	-0.4552084
354	-2.3	-4.1375221	1.8375221
355	4.2	-4.2267478	8.4267478
356	9.5	-4.3124947	13.8124947
357	-0.4	-4.3947890	3.9947890
358	-6.5	-4.4736571	-2.0263429
359	-10.3	-4.5491250	-5.7508750
360	-7.8	-4.6212190	-3.1787810
361	-10.7	-4.6899652	-6.0100348
362	-5.1	-4.7553899	-0.3446101
363	1.5	-4.8175193	6.3175193
364	-8.7	-4.8763795	-3.8236205
365	-8.3	-4.9319967	-3.3680033
366	-0.9	-4.9843972	4.0843972
367	-10.3	-5.0336071	-5.2663929
368	-10.6	-5.0796527	-5.5203473
369	-3.1	-5.1225601	2.0225601
370	1.4	-5.1623555	6.5623555
371	-0.6	-5.1990651	4.5990651
372	-7.3	-5.2327152	-2.0672848
373	-6.7	-5.2633319	-1.4366681
374	2.4	-5.2909413	7.6909413
375	-0.4	-5.3155698	4.9155698
376	-6.9	-5.3372435	-1.5627565
377	-10.9	-5.3559886	-5.5440114
378	-9.7	-5.3718313	-4.3281687
379	-11.1	-5.3847978	-5.7152022
380	-7.9	-5.3949142	-2.5050858
381	-4.1	-5.4022069	1.3022069
382	-8.3	-5.4067019	-2.8932981
383	-8.6	-5.4084255	-3.1915745
384	-7.5	-5.4074039	-2.0925961
385	-1.6	-5.4036633	3.8036633
386	0.3	-5.3972298	5.6972298
387	-10.9	-5.3881296	-5.5118704
388	-4.4	-5.3763891	0.9763891
389	7.5	-5.3620343	12.8620343
390	4.0	-5.3450914	9.3450914
391	-2.5	-5.3255867	2.8255867
392	-0.7	-5.3035463	4.6035463
393	-3.4	-5.2789965	1.8789965
394	-8.0	-5.2519634	-2.7480366
395	-1.9	-5.2224732	3.3224732
396	-7.3	-5.1905522	-2.1094478
397	-11.4	-5.1562265	-6.2437735
398	-9.0	-5.1195223	-3.8804777
399	-8.5	-5.0804658	-3.4195342
400	-1.9	-5.0390832	3.1390832
401	4.9	-4.9954008	9.8954008
402	-2.0	-4.9494446	2.9494446
403	-4.3	-4.9012409	0.6012409
404	1.7	-4.8508160	6.5508160
405	-5.0	-4.7981959	-0.2018041
406	-7.6	-4.7434069	-2.8565931
407	-10.3	-4.6864752	-5.6135248
408	-7.9	-4.6274212	-3.2725788
409	-3.4	-4.5662536	1.1662536
410	-4.9	-4.5029794	-0.3970206
411	-3.5	-4.4376057	0.9376057
412	2.4	-4.3701396	6.7701396
413	-5.7	-4.3005883	-1.3994117
414	-7.6	-4.2289588	-3.3710412
415	-9.9	-4.1552582	-5.7447418
416	-11.5	-4.0794936	-7.4205064
417	-8.2	-4.0016722	-4.1983278
418	-6.3	-3.9218010	-2.3781990
419	-7.2	-3.8398870	-3.3601130
420	-5.8	-3.7559375	-2.0440625
421	-2.0	-3.6699595	1.6699595
422	-11.1	-3.5819601	-7.5180399
423	-12.6	-3.4919465	-9.1080535
424	-10.8	-3.3999256	-7.4000744
425	-6.2	-3.3059046	-2.8940954
426	-3.7	-3.2098906	-0.4901094
427	-4.9	-3.1118907	-1.7881093
428	-2.3	-3.0119120	0.7119120
429	-6.2	-2.9099616	-3.2900384
430	-8.2	-2.8060466	-5.3939534
431	-11.7	-2.7001740	-8.9998260
432	-9.1	-2.5923511	-6.5076489
433	-4.2	-2.4825848	-1.7174152
434	-5.1	-2.3708823	-2.7291177
435	2.9	-2.2572507	5.1572507
436	-0.2	-2.1416971	1.9416971
437	-0.1	-2.0242285	1.9242285
438	1.8	-1.9048522	3.7048522
439	3.1	-1.7835750	4.8835750
440	4.1	-1.6604043	5.7604043
441	-2.4	-1.5353470	-0.8646530
442	-3.6	-1.4084103	-2.1915897
443	-3.4	-1.2796012	-2.1203988
444	-1.7	-1.1489269	-0.5510731
445	0.8	-1.0163944	1.8163944
446	3.7	-0.8820109	4.5820109
447	2.4	-0.7457835	3.1457835
448	4.4	-0.6077192	5.0077192
449	-0.1	-0.4678251	0.3678251
450	-2.4	-0.3261084	-2.0738916
451	-3.3	-0.1825762	-3.1174238
452	-1.5	-0.0372354	-1.4627646
453	5.0	0.1099067	4.8900933
454	4.1	0.2588430	3.8411570
455	7.7	0.4095665	7.2904335
456	2.7	0.5620701	2.1379299
457	2.0	0.7163467	1.2836533
458	3.7	0.8723891	2.8276109
460	0.3	1.1897432	-0.8897432
461	2.1	1.3510407	0.7489593
462	3.5	1.5140756	1.9859244
463	-1.9	1.6788409	-3.5788409
464	-2.4	1.8453296	-4.2453296
465	-1.8	2.0135344	-3.8135344
466	3.2	2.1834483	1.0165517
467	6.7	2.3550642	4.3449358
468	8.1	2.5283750	5.5716250
469	11.8	2.7033735	9.0966265
470	7.8	2.8800528	4.9199472
471	3.2	3.0584057	0.1415943
472	0.4	3.2384251	-2.8384251
473	3.6	3.4201039	0.1798961
474	6.7	3.6034350	3.0965650
475	11.9	3.7884113	8.1115887
476	12.8	3.9750258	8.8249742
477	10.3	4.1632712	6.1367288
478	6.8	4.3531406	2.4468594
479	7.8	4.5446267	3.2553733
480	2.4	4.7377227	-2.3377227
481	5.7	4.9324212	0.7675788
482	5.7	5.1287152	0.5712848
483	7.7	5.3265977	2.3734023
484	5.0	5.5260615	-0.5260615
485	5.1	5.7270996	-0.6270996
486	5.6	5.9297048	-0.3297048
487	6.4	6.1338700	0.2661300
488	7.3	6.3395840	0.9604160
489	4.8	6.5467866	-1.7467866
490	10.2	6.7553865	3.4446135
491	8.7	6.9652914	1.7347086
492	11.1	7.1764095	3.9235905
493	8.2	7.3886487	0.8113513
494	6.2	7.6019169	-1.4019169
495	5.8	7.8161220	-2.0161220
496	9.2	8.0311722	1.1688278
497	7.1	8.2469752	-1.1469752
498	6.3	8.4634392	-2.1634392
499	4.9	8.6804720	-3.7804720
500	5.4	8.8979815	-3.4979815
501	5.7	9.1158759	-3.4158759
502	9.2	9.3340630	-0.1340630
503	5.9	9.5524508	-3.6524508
504	9.3	9.7709473	-0.4709473
505	8.4	9.9894604	-1.5894604
506	6.0	10.2078980	-4.2078980
507	4.4	10.4261682	-6.0261682
508	8.7	10.6441790	-1.9441790
509	5.9	10.8618382	-4.9618382
510	11.3	11.0790538	0.2209462
511	9.3	11.2957339	-1.9957339
512	8.8	11.5117863	-2.7117863
513	9.4	11.7271191	-2.3271191
514	8.6	11.9416401	-3.3416401
515	9.7	12.1552574	-2.4552574
516	8.7	12.3678789	-3.6678789
517	12.9	12.5794126	0.3205874
518	10.0	12.7897664	-2.7897664
519	11.9	12.9988484	-1.0988484
520	11.7	13.2065664	-1.5065664
521	9.0	13.4128284	-4.4128284
522	10.6	13.6175425	-3.0175425
523	15.2	13.8206165	1.3793835
524	13.5	14.0219584	-0.5219584
525	14.6	14.2214762	0.3785238
526	14.7	14.4190778	0.2809222
527	13.5	14.6146713	-1.1146713
528	15.7	14.8081645	0.8918355
529	17.4	14.9994655	2.4005345
530	17.3	15.1884822	2.1115178
531	18.3	15.3751225	2.9248775
532	16.8	15.5592945	1.2407055
533	18.6	15.7409060	2.8590940
534	17.7	15.9198651	1.7801349
535	17.4	16.0960797	1.3039203
536	16.6	16.2694578	0.3305422
537	13.1	16.4399074	-3.3399074
538	14.4	16.6073363	-2.2073363
539	17.3	16.7716526	0.5283474
540	15.3	16.9327643	-1.6327643
541	15.9	17.0905792	-1.1905792
542	14.6	17.2450054	-2.6450054
543	12.8	17.3959508	-4.5959508
544	14.7	17.5433234	-2.8433234
545	13.9	17.6870311	-3.7870311
546	14.7	17.8269819	-3.1269819
547	13.8	17.9630838	-4.1630838
548	15.8	18.0952448	-2.2952448
549	17.5	18.2233727	-0.7233727
550	20.9	18.3473756	2.5526244
551	26.0	18.4671615	7.5328385
552	22.9	18.5826382	4.3173618
553	19.5	18.6937137	0.8062863
554	19.7	18.8002961	0.8997039
555	18.9	18.9022933	-0.0022933
556	16.1	18.9996131	-2.8996131
557	16.6	19.0921637	-2.4921637
558	16.6	19.1798530	-2.5798530
559	19.6	19.2625889	0.3374111
560	18.1	19.3402793	-1.2402793
561	19.9	19.4128324	0.4871676
562	22.1	19.4801559	2.6198441
563	21.9	19.5421579	2.3578421
564	19.6	19.5987464	0.0012536
565	19.1	19.6498293	-0.5498293
566	24.0	19.6953145	4.3046855
567	24.1	19.7351101	4.3648899
568	21.6	19.7691619	1.8308381
569	19.1	19.7975246	-0.6975246
570	18.4	19.8202722	-1.4202722
571	18.7	19.8374788	-1.1374788
572	20.2	19.8492186	0.3507814
573	20.9	19.8555656	1.0444344
574	20.1	19.8565940	0.2434060
575	21.7	19.8523778	1.8476222
576	23.9	19.8429911	4.0570089
577	23.7	19.8285080	3.8714920
578	22.9	19.8090027	3.0909973
579	22.0	19.7845491	2.2154509
580	22.6	19.7552215	2.8447785
581	20.9	19.7210939	1.1789061
582	19.5	19.6822404	-0.1822404
583	20.6	19.6387351	0.9612649
584	19.4	19.5906520	-0.1906520
585	17.4	19.5380654	-2.1380654
586	20.2	19.4810493	0.7189507
587	18.7	19.4196777	-0.7196777
588	17.9	19.3540248	-1.4540248
589	21.0	19.2841647	1.7158353
590	20.3	19.2101715	1.0898285
591	19.8	19.1321193	0.6678807
592	20.1	19.0500821	1.0499179
593	18.9	18.9641341	-0.0641341
594	19.7	18.8743493	0.8256507
595	19.2	18.7808019	0.4191981
596	20.3	18.6835659	1.6164341
597	22.5	18.5827155	3.9172845
598	21.7	18.4783248	3.2216752
599	23.4	18.3704678	5.0295322
600	21.3	18.2592186	3.0407814
601	14.5	18.1446514	-3.6446514
602	14.6	18.0268401	-3.4268401
603	13.2	17.9058591	-4.7058591
604	16.2	17.7817822	-1.5817822
605	21.3	17.6546837	3.6453163
606	18.0	17.5246375	0.4753625
607	21.7	17.3917179	4.3082821
608	20.1	17.2559989	2.8440011
609	16.8	17.1175546	-0.3175546
610	15.9	16.9764591	-1.0764591
611	18.4	16.8327865	1.5672135
612	19.9	16.6866109	3.2133891
613	16.9	16.5380064	0.3619936
614	14.9	16.3870471	-1.4870471
615	12.9	16.2338071	-3.3338071
616	12.4	16.0783604	-3.6783604
617	14.9	15.9207812	-1.0207812
618	15.4	15.7611436	-0.3611436
619	15.2	15.5995217	-0.3995217
620	11.8	15.4359895	-3.6359895
621	11.6	15.2706212	-3.6706212
622	12.3	15.1034908	-2.8034908
623	17.7	14.9346725	2.7653275
624	14.8	14.7642404	0.0357596
625	9.7	14.5922685	-4.8922685
626	8.4	14.4188309	-6.0188309
627	10.8	14.2440018	-3.4440018
628	14.1	14.0678552	0.0321448
629	14.8	13.8904653	0.9095347
630	16.4	13.7119061	2.6880939
631	20.1	13.5322517	6.5677483
632	17.8	13.3515762	4.4484238
634	15.2	12.9874584	2.2125416
635	14.6	12.8041643	1.7958357
636	13.8	12.6201455	1.1798545
637	12.9	12.4354761	0.4645239
638	8.6	12.2502302	-3.6502302
639	10.2	12.0644819	-1.8644819
640	9.3	11.8783053	-2.5783053
641	6.1	11.6917744	-5.5917744
642	5.9	11.5049635	-5.6049635
643	6.7	11.3179465	-4.6179465
644	9.6	11.1307977	-1.5307977
645	16.4	10.9435910	5.4564090
646	12.6	10.7564005	1.8435995
647	10.2	10.5693005	-0.3693005
648	9.4	10.3823627	-0.9823627
649	10.5	10.1956198	0.3043802
650	11.6	10.0090707	1.5909293
651	10.2	9.8227128	0.3772872
652	10.3	9.6365438	0.6634562
653	12.3	9.4505613	2.8494387
654	9.9	9.2647630	0.6352370
655	11.7	9.0791463	2.6208537
656	10.0	8.8937090	1.1062910
657	7.8	8.7084486	-0.9084486
658	8.6	8.5233628	0.0766372
659	10.0	8.3384492	1.6615508
660	8.6	8.1537052	0.4462948
661	11.3	7.9691287	3.3308713
662	10.9	7.7847172	3.1152828
663	9.9	7.6004682	2.2995318
664	5.5	7.4163795	-1.9163795
665	5.1	7.2324485	-2.1324485
666	5.9	7.0486730	-1.1486730
667	6.4	6.8650505	-0.4650505
669	13.4	6.4982551	6.9017449
670	12.2	6.3150773	5.8849227
672	6.0	5.9491498	0.0508502
673	5.5	5.7663953	-0.2663953
674	7.3	5.5837771	1.7162229
675	7.3	5.4012927	1.8987073
676	3.4	5.2189399	-1.8189399
678	-2.6	4.8546192	-7.4546192
679	2.4	4.6726465	-2.2726465
680	5.6	4.4907958	1.1092042
681	9.4	4.3090647	5.0909353
682	-1.0	4.1274507	-5.1274507
683	-3.9	3.9459515	-7.8459515
684	6.6	3.7645646	2.8354354
685	-1.2	3.5832878	-4.7832878
686	-3.9	3.4021185	-7.3021185
687	0.6	3.2210545	-2.6210545
688	3.6	3.0400933	0.5599067
689	1.7	2.8592325	-1.1592325
690	1.9	2.6784697	-0.7784697
691	4.9	2.4978026	2.4021974
692	0.7	2.3172287	-1.6172287
693	3.7	2.1367457	1.5632543
694	5.9	1.9563511	3.9436489
695	5.0	1.7760427	3.2239573
696	4.6	1.5958179	3.0041821
697	1.4	1.4156744	-0.0156744
698	-1.6	1.2356098	-2.8356098
699	-3.1	1.0556218	-4.1556218
700	-6.6	0.8757078	-7.4757078
701	-4.1	0.6958656	-4.7958656
702	2.8	0.5160928	2.2839072
703	0.6	0.3363868	0.2636132
704	-0.3	0.1567455	-0.4567455
705	-1.5	-0.0228337	-1.4771663
706	-6.1	-0.2023531	-5.8976469
707	-5.5	-0.3818151	-5.1181849
708	4.8	-0.5612222	5.3612222
709	11.6	-0.7405766	12.3405766
710	2.1	-0.9198808	3.0198808
711	-8.0	-1.0991372	-6.9008628
712	-5.3	-1.2783482	-4.0216518
713	7.5	-1.4575161	8.9575161
714	7.2	-1.6366434	8.8366434
715	-2.7	-1.8157324	-0.8842676
716	-4.0	-1.9947855	-2.0052145
717	-5.5	-2.1738052	-3.3261948
718	-8.7	-2.3527938	-6.3472062
719	-5.1	-2.5317537	-2.5682463
720	-3.7	-2.7106873	-0.9893127
721	-2.3	-2.8895970	0.5895970
722	0.3	-3.0684852	3.3684852
723	-0.4	-3.2473542	2.8473542
724	-5.8	-3.4262065	-2.3737935
725	-7.9	-3.6050445	-4.2949555
726	-3.6	-3.7838705	0.1838705
727	2.1	-3.9626869	6.0626869
728	-2.7	-4.1414962	1.4414962
729	-3.5	-4.3203006	0.8203006
730	0.0	-4.4991027	4.4991027

Graph “smooth” term

hfx |> add_fitted(m1) |> select(year_day, mean_temp, .value) |>
  add_residuals(m1) |>
  ggplot(aes(year_day)) +
  geom_line(aes(y=.value), 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, ymax = upper), alpha = 0.25, fill="red") +
  geom_line(aes(y=est), size=2, color="red") + 
  geom_point(aes(y = mean_temp - mean(mean_temp)), data = hfx) + 
  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
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-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
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-9.7	378	-1.32	0.43	-8.38
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-7.9	380	-1.31	0.43	-6.59
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-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
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-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
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-8.0	394	-1.03	0.42	-6.97
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-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
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-4.3	403	-0.66	0.41	-3.64
1.7	404	-0.62	0.41	2.32
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-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
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-7.6	414	-0.08	0.41	-7.52
-9.9	415	-0.02	0.41	-9.88
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-7.2	419	0.22	0.41	-7.42
-5.8	420	0.28	0.41	-6.08
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-4.9	427	0.73	0.42	-5.63
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-8.2	430	0.93	0.42	-9.13
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-9.1	432	1.06	0.42	-10.16
-4.2	433	1.12	0.42	-5.32
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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
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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
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6.6	684	6.47	0.56	0.13
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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
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-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

Summary

• Fitting GAM and LOESS smooths

• No model coefficients to extract

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

Further reading

• Course notes

• Healy Chapter 6. Work with models.