Jagged / V shape Residuals vs Fitted plot

Question

What does a jagged / v-shaped residuals vs fitted plot mean? I am doing a multiple linear regression with three explanatory variables

I used a linear regression model with the data and indicators below, there are 217 data points: emissions.urban=lm(EN.ATM.METH.KT.CE~AG.LND.AGRI.K2+AG.LND.FRST.K2+SP.URB.TOTL,data=wbcc) summary(emissions.urban)

I've posted the data below AG.LND.AGRI.K2= Agricultural land (sq. km) AG.LND.FRST.K2=Forest area (sq. km) SP.URB.TOTL=Urban population EN.ATM.METH.KT.CE=Methane emissions (kt of CO2 equivalent)

iso3c	country	AG.LND.AGRI.K2	AG.LND.FRST.K2	SP.URB.TOTL	EN.ATM.METH.KT.CE
ABW	Aruba	20	4.2	46654	NA
AFG	Afghanistan	379190	12084.4	10131490	81510
AGO	Angola	69524.9	666073.8	21962884	35520
ALB	Albania	11740.81	7889	1762579	3160
AND	Andorra	188.3	160	67928	50
ARE	United Arab Emirates	3817.5	3173	8609395	52960
ARG	Argentina	1487680	285730	41796990	117850
ARM	Armenia	16773	3284.7	1876112	2430
ASM	American Samoa	49	171.3	48106	NA
ATG	Antigua and Barbuda	90	81.2	23927	200
AUS	Australia	3588950	1340051	22152761	139070
AUT	Austria	26528.3	38991.5	5238680	6660
AZE	Azerbaijan	47795	11317.7	5701802	43600
BDI	Burundi	20330	2796.4	1629988	2140
BEL	Belgium	13540	6893	11334006	8200
BEN	Benin	39500	31351.5	5869446	5530
BFA	Burkina Faso	121000	62164	6397866	15600
BGD	Bangladesh	92023	18834	62873466	83790
BGR	Bulgaria	50300	38930	5242987	6910
BHR	Bahrain	86	7	1523019	14850
BHS	Bahamas, The	140	5098.6	327359	230
BIH	Bosnia and Herzegovina	22110	21879.1	1608256	3380
BLR	Belarus	84530	87676	7470497	16730
BLZ	Belize	1720	12770.5	183005	530
BMU	Bermuda	3	10	63903	NA
BOL	Bolivia	377270	508337.6	8185478	25050
BRA	Brazil	2368788.01	4966196	185081854	416280
BRB	Barbados	100	63	89634	2350
BRN	Brunei Darussalam	144	3800	342330	8830
BTN	Bhutan	5130	27250.8	326515	860
BWA	Botswana	258616	152547	1666761	4560
CAF	Central African Republic	50800	223030	2038064	23200
CAN	Canada	581990	3469281	30997832	93980
CHE	Switzerland	15100.756	12691.1	6383962	4790
CHI	Channel Islands	92.2	10.2	53832	NA
CHL	Chile	156930	182107	16770077	13480
CHN	China	5285287	2199781.8	861289359	1238630
CIV	Cote d'Ivoire	212000	28367.1	13639151	6560
CMR	Cameroon	97500	203404.8	15279799	14960
COD	Congo, Dem. Rep.	315000	1261552.4	40874034	38620
COG	Congo, Rep.	106280	219460	3742867	3540
COL	Colombia	494920	591419.1	41431388	76700
COM	Comoros	1310	329.2	255487	270
CPV	Cabo Verde	790 457.2	370577	120
CRI	Costa Rica	17825	30348.7	4114567	4820
CUB	Cuba	63000	32420	8743468	12690
CUW	Curacao	NA	0.7	138059	NA
CYM	Cayman Islands	27	127.2	65720	NA
CYP	Cyprus	1309.47	1725.3	806771	740
CZE	Czech Republic	35230	26770.9	7923709	12430
DEU	Germany	166450	114190	64472284	53370
DJI	Djibouti	17020	58	771254	690
DMA	Dominica	250	478.7	51178	50
DNK	Denmark	26320	6284.4	5138400	7140
DOM	Dominican Republic	24290	21441	8953860	9200
DZA	Algeria	413588.47	19490	32332690	49550
ECU	Ecuador	54480	124978.3	11320846	20350
EGY	Egypt, Arab Rep.	38359.688	449.8	43781728	56870
ERI	Eritrea	75920	10552.6	1149669	3730
ESP	Spain	261833.239	185721.7	38264801	39940
EST	Estonia	10040	24384	921477	1050
ETH	Ethiopia	379030	170685	24941349	103110
FIN	Finland	22720	224090	4729705	4240
FJI	Fiji	4250	11400.2	513187	670
FRA	France	286601	172530	54570334	58340
FRO	Faroe Islands	30	0.8	20718	NA
FSM	Micronesia, Fed. Sts.	220	644.2	26378	60
GAB	Gabon	22126.4	235306	2005203	1120
GBR	United Kingdom	173508.616	31900	56395647	51210
GEO	Georgia	23718	28224	2208084	5210
GHA	Ghana	147827.4	79857.1	17820023	21350
GIB	Gibraltar	NA	0	33691	NA
GIN	Guinea	145000	61890	4842717	17830
GMB	Gambia, The	6050	2426.7	1512397	1700
GNB	Guinea-Bissau	8151.1	19800.1	869776	1500
GNQ	Equatorial Guinea	2840	24484.2	1025582	12230
GRC	Greece	61036	39018	8541900	9670
GRD	Grenada	80	177	41111	2030
GRL	Greenland	2431.1	2.2	49198	NA
GTM	Guatemala	38560	35278	8738685	11750
GUM	Guam	180	280	160239	NA
GUY	Guyana	12512.5	184153.4	210688	1540
HKG	Hong Kong SAR, China	50	NA	7481800	NA
HND	Honduras	33560	63592.6	5780230	8150
HRV	Croatia	14840	19391.1	2329285	3820
HTI	Haiti	18400	3473	6509478	4730
HUN	Hungary	52960	20530.1	7014174	7120
IDN	Indonesia	623000	921332	154926514	287500
IMN	Isle of Man	403	34.6	44980	NA
IND	India	1796740	721600	481980332	666510
IRL	Ireland	45160	7820.2	3179292	16820
IRN	Iran, Islamic Rep.	459540	107518.7	63728813	149690
IRQ	Iraq	92500	8250	28514939	16750
ISL	Iceland	18720	513.5	344066	530
ISR	Israel	6233	1400	8533651	12070
ITA	Italy	124050	95661.3	42306582	43670
JAM	Jamaica	4440	5968.9	1667459	780
JOR	Jordan	10218	975	9327507	6300
JPN	Japan	44200	249350	115494817	21110
KAZ	Kazakhstan	2160365	34546.8	10815873	41360
KEN	Kenya	276300	36110.9	15053275	40250
KGZ	Kyrgyz Republic	105413	13153.8	2429400	4990
KHM	Cambodia	55660	80683.7	4051341	20310
KIR	Kiribati	340	11.8	66405	20
KNA	St. Kitts and Nevis	60	110	16406	80
KOR	Korea, Rep.	16520	62870	42156641	25530
KWT	Kuwait	1500	62.5	4270563	6080
LAO	Lao PDR	23940	165955	2640299	7610
LBN	Lebanon	6580	1433.3	6069524	3250
LBR	Liberia	19540.4	76174.4	2634493	6210
LBY	Libya	153500	2170	5544510	37790
LCA	St. Lucia	106	207.7	34598	270
LIE	Liechtenstein	51.6	67	5498	20
LKA	Sri Lanka	28116	21130.2	4101702	10030
LSO	Lesotho	24333	345.2	621853	2320
LTU	Lithuania	29470	22010	1901682	3150
LUX	Luxembourg	1315.59	887	578234	540
LVA	Latvia	19380	34107.9	1299043	1940
MAC	Macao SAR, China	NA	NA	649342	NA
MAF	St. Martin (French part)	NA	12.4	NA	NA
MAR	Morocco	300690	57424.9	23450016	17670
MCO	Monaco	NA	0	39244	NA
MDA	Moldova	22571	3865	1121710	3310
MDG	Madagascar	408950	124298.1	10670457	17470
MDV	Maldives	79	8.2	219833	130
MEX	Mexico	1068910	656920.8	104088701	144610
MHL	Marshall Islands	86	94	46049	30
MKD	North Macedonia	12640	10014.9	1218402	2520
MLI	Mali	412010	132960	8891939	23290
MLT	Malta	103.8	4.6	497676	220
MMR	Myanmar	128890	285438.9	16943754	65790
MNE	Montenegro	2568	8270	419585	820
MNG	Mongolia	1134330	141727.8	2250777	17860
MNP	Northern Mariana Islands	30	243.6	52836	NA
MOZ	Mozambique	414138.32	367437.6	11587640	16850
MRT	Mauritania	396610	3128	2572517	6830
MUS	Mauritius	860	387.7	515916	1930
MWI	Malawi	56500	22417	3333777	11020
MYS	Malaysia	85710	191140.4	24973604	46580
NAM	Namibia	388100	66389	1322115	4510
NCL	New Caledonia	1840.3	8380.2	194500	NA
NER	Niger	466000	10797	4024595	29860
NGA	Nigeria	691234.5	216269.5	107106007	127900
NIC	Nicaragua	50650	34075.3	3909282	9830
NLD	Netherlands	18220	3695	16087009	17260
NOR	Norway	9862.97	121800	4463566	4850
NPL	Nepal	41210	59620.3	5995190	30800
NRU	Nauru	4	0	10834	0
NZL	New Zealand	104670	98925.9	4408037	32530
OMN	Oman	14588.9	25	4405789	5460
PAK	Pakistan	363000	37259	82094635	151020
PAN	Panama	22590	42138.4	2951905	5390
PER	Peru	236087	723303.7	25815966	31410
PHL	Philippines	124400	71885.9	51950201	67660
PLW	Palau	43	414.1	14652	20
PNG	Papua New Guinea	11900	358557.6	1193981	11310
POL	Poland	145120	94830	22786800	47540
PRI	Puerto Rico	1689	4963.3	2989009	NA
PRK	Korea, Dem. People's Rep.	26300	60300.9	16081083	18710
PRT	Portugal	35739.9	33120	6833619	11320
PRY	Paraguay	218190	161022.6	4435221	29070
PSE	West Bank and Gaza	2969.200134	101.4	3685020	NA
PYF	French Polynesia	455	1494.6	174090	NA
QAT	Qatar	670	0	2859020	8110
ROU	Romania	134140	69290.5	10451921	23780
RUS	Russian Federation	2154940	8153116	107723564	849570
RWA	Rwanda	18117	2760	2257829	2910
SAU	Saudi Arabia	1736290	9770	29343564	44170
SDN	Sudan	681861.6	183595.5	15458183	58850
SEN	Senegal	88780	80681.6	8057514	10750
SGP	Singapore	6.6	155.7	5685807	4150
SLB	Solomon Islands	1170	25229.7	169453	410
SLE	Sierra Leone	39490	25348.8	3423961	4610
SLV	El Salvador	14791.2	5838.8	4763725	3990
SMR	San Marino	23	10	33089	NA
SOM	Somalia	441250	59800	7333290	19430
SRB	Serbia	34640	27226.5	3899416	12400
SSD	South Sudan	285332	71570	2261021	34170
STP	Sao Tome and Principe	440	519	162955	30
SUR	Suriname	840	151962.9	388053	1370
SVK	Slovak Republic	18890	19259	2934665	4360
SVN	Slovenia	6120	12378.3	1157547	1980
SWE	Sweden	30090	279800	9108648	4580
SWZ	Eswatini	12220	4975.6	280423	1500
SXM	Sint Maarten (Dutch part)	NA	3.7	40812	NA
SYC	Seychelles	15.5	337	56661	90
SYR	Syrian Arab Republic	139210	5220.8	9708489	12770
TCA	Turks and Caicos Islands	10	105.2	36242	NA
TCD	Chad	502380	43130	3863362	53990
TGO	Togo	38200	12092.7	3543299	3450
THA	Thailand	221100	198730	35898129	84140
TJK	Tajikistan	47277	4238	2623424	5520
TKM	Turkmenistan	338380	41270	3167338	49580
TLS	Timor-Leste	3800	9211	412936	5280
TON	Tonga	350	89.5	24415	100
TTO	Trinidad and Tobago	540	2281.9	744725	1280
TUN	Tunisia	97430	7027.3	8221976	6420
TUR	Turkey	378020	222203.6	64186247	47400
TUV	Tuvalu	18	10	7549	10
TZA	Tanzania	396500	457450	21042571	62650
UGA	Uganda	144150	23379	11414209	33250
UKR	Ukraine	413290	96900	30721277	62950
URY	Uruguay	140159	20310	3317930	20940
USA	United States	4058103.538	3097950	272364755	622590
UZB	Uzbekistan	255777	36896.6	17258430	105930
VCT	St. Vincent and the Grenadines	70	285.4	58837	70
VEN	Venezuela, RB	215000	462309	25102966	72340
VGB	British Virgin Islands	70	36.2	14669	NA
VIR	Virgin Islands (U.S.)	40	199.1	101974	NA
VNM	Vietnam	121688	146430.9	36346227	87750
VUT	Vanuatu	1870	4423	78400	510
WSM	Samoa	757	1616.7	35494	300
XKX	Kosovo	5700	NA	NA	NA
YEM	Yemen, Rep.	233877	5490	11306428	8590
ZAF	South Africa	963410	170500.9	39946775	45140
ZMB	Zambia	238360	448140.3	8204576	17870
ZWE	Zimbabwe	162000	174445.8	4792105	11850

Answer 1

A dip (or hump) in the residuals curve is a sign of non-linearity. It means that you might bet a better model using some quadratic terms.

But that is not the most striking issue shown by your graph. The very non-uniform distribution of the predicted values is more important. The few very large values will make a big impact on the regression coefficients because of the use of least-squares as the loss function.

There are several ways of addressing this:

• ignore very large values as "outliers"
• use an L1 loss function
• transform the value to be predicted before making the model

In this case the third option looks best - if you make a linear regression for $\log y$ instead of $y$ you might get much better results.

Answer 2

I want to add to @chrishmorris's helpful answer, although I am never in favour of dropping outliers because they are awkward and much more in favour of working on logarithmic scale.

It is all too easy to over-interpret a "smooth" which for high fitted values is summarizing only a very few residuals and utterly dependent on what is seen within a smoothing window, even if a data point is down-weighted somehow as it enters or leaves a window. So, the V shape is probably dependent in the right-hand half of the plot on just about the 5 residuals visible as individuals.

Two rather different stories are broadly consistent with each other and the data, on this information:

1. You would be better off with transforming $y$ or perhaps even better off with a generalized linear model with logarithmic link, such as Poisson regression.

2. The regression is doing a fair job of following the main trend of the data, but uncertainty is bound to accompany severe skewness of the response, and there may be nonlinearity better modelled otherwise.

How many data points are there here? Is it possible to post the data?