Girosi King 2007 - Information Loss with One Principal Component

Summary

Lee-Carter keeps one principal component of the centred log-mortality age profiles. The share of variance it explains is a simple applicability diagnostic: for all-cause male mortality it is above 90% in 14 of the 24 countries tabulated (USA 93%), but for cause-specific series it is often below 50% (homicide, suicide, stomach cancer). There is no simple pattern that tells ex ante where the model works.

Overview

Read the age profile as a point in $R^{A}$ moving over time; Lee-Carter assumes it moves along a straight line. Percent variance explained by $k$ components: $Δ E_{k} = 1 - \sum_{t} ∥ ϵ_{t}^{(k)} ∥^{2} / \sum_{t} ∥ \tilde{m}_{t} ∥^{2}$ where $ϵ_{t}^{(k)} = \tilde{m}_{t} - \sum_{j \leq k} β_{j} γ_{j t}$ (footnote 4). Principal components in demography: Ledermann and Breas (1959); Bozik and Bell (1987); Bell and Monsell (1991).

Main Content

Percent of variance explained (Eq. 10 and footnote 4)

Full decomposition $m_{a t} = α_{a} + \sum_{j = 1}^{A} β_{aj} γ_{t j}$ is exact; Lee-Carter is the $j = 1$ term, error $= \sum_{j \geq 2} β_{aj} γ_{t j}$ . $Δ E_{k} \in [0, 1]$ measures how well $k$ components approximate the centred profiles.

Figure 1 (U.S. males 1950-2000, 3-D plots of first three PCs). All-cause: 93/97/99% for 1/2/3 components; cardiovascular 90/95/99%; Sweden lung cancer 38/56/69%; Sweden homicide 37/50/57%. In the latter two, one component ignores about two-thirds of the variation, so Lee-Carter “should not be counted on to perform as well”.

Table 1: percent of variance in male mortality explained by the first principal component (bold = 90% or higher)

Country All cause Cardio-vascular Lung cancer Homicide TB Respiratory Stomach cancer Other cancer Suicide Digestive
Japan 98 87 96 74 97 69 66 62 65 97
Chile 97 92 48 52 96 52 71 55 79 87
Canada 95 77 84 38 89 50 64 80 85 79
Australia 95 79 64 16 84 37 61 67 66 79
U.K 94 72 77 60 91 80 70 87 87 83
Mexico 93 81 77 64 97 82 32 71 63 97
USA 93 90 81 70 94 78 63 95 91 85
Austria 93 49 44 16 86 35 61 69 34 76
Italy 92 74 88 43 93 61 46 70 60 90
Finland 92 56 49 21 88 37 75 56 52 72
Belgium 92 70 78 40 85 47 57 67 80 70
Portugal 91 62 80 21 91 34 34 46 45 87
Switzerland 91 36 45 22 83 24 64 53 33 75
France 90 89 93 44 94 78 56 74 80 84
Sweden 90 41 55 37 82 29 69 62 38 64
Spain 89 91 93 66 94 74 54 74 73 94
Netherlands 89 40 78 41 81 39 61 70 72 70
Venezuela 86 64 46 50 93 37 52 43 23 93
Hungary 85 64 86 30 79 39 52 62 73 79
Norway 84 29 64 24 82 24 66 64 66 56
Ireland 83 57 68 28 86 37 59 43 70 71
New Zealand 82 57 40 19 79 40 40 34 59 61
Denmark 70 39 58 31 67 36 56 51 48 55
Iceland 57 53 42 40 54 20 55 18 18 32

Country	All cause	Cardio-vascular	Lung cancer	Homicide	TB	Respiratory	Stomach cancer	Other cancer	Suicide	Digestive
Japan	98	87	96	74	97	69	66	62	65	97
Chile	97	92	48	52	96	52	71	55	79	87
Canada	95	77	84	38	89	50	64	80	85	79
Australia	95	79	64	16	84	37	61	67	66	79
U.K	94	72	77	60	91	80	70	87	87	83
Mexico	93	81	77	64	97	82	32	71	63	97
USA	93	90	81	70	94	78	63	95	91	85
Austria	93	49	44	16	86	35	61	69	34	76
Italy	92	74	88	43	93	61	46	70	60	90
Finland	92	56	49	21	88	37	75	56	52	72
Belgium	92	70	78	40	85	47	57	67	80	70
Portugal	91	62	80	21	91	34	34	46	45	87
Switzerland	91	36	45	22	83	24	64	53	33	75
France	90	89	93	44	94	78	56	74	80	84
Sweden	90	41	55	37	82	29	69	62	38	64
Spain	89	91	93	66	94	74	54	74	73	94
Netherlands	89	40	78	41	81	39	61	70	72	70
Venezuela	86	64	46	50	93	37	52	43	23	93
Hungary	85	64	86	30	79	39	52	62	73	79
Norway	84	29	64	24	82	24	66	64	66	56
Ireland	83	57	68	28	86	37	59	43	70	71
New Zealand	82	57	40	19	79	40	40	34	59	61
Denmark	70	39	58	31	67	36	56	51	48	55
Iceland	57	53	42	40	54	20	55	18	18	32

Observations from Table 1

(1) Some causes (all-cause, TB, partly cardiovascular) are low-dimensional across countries, suggesting an underlying reason. (2) No clear relation with country size (Iceland is low, 57%, but other small countries are not). (3) Digestive and lung cancer vary widely across countries. (4) Homicide, respiratory, stomach cancer and suicide are rarely well summarised. (5) Within a country, variance explained varies strongly with cause. Conclusion: Lee-Carter “will usually be appropriate only for idiosyncratic choices of countries and causes of death, and there is no way to know ex ante which those are”.

Examples

Using $Δ E_{1}$ as a pre-check

Setup: a new country with all-cause male variance explained 84% (like Norway) and suicide 66% (like Norway too: 66%). Solution: apply a 90% rule of thumb as the bold entries suggest. Interpretation: all-cause Lee-Carter is borderline; cause-specific suicide forecasts would ignore a third of the motion. Prefer RWD, or add components, before trusting the fan.

import numpy as np
def pct_var_explained(m, k=1):
    mt = m - m.mean(1, keepdims=True); s = np.linalg.svd(mt, compute_uv=False)
    return (s[:k]**2).sum() / (s**2).sum()

Connections

Girosi King 2007 - Lee-Carter Model Estimation and Forecasting, Factor Analysis and PPCA, Model Comparison.

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Girosi King 2007 - Information Loss with One Principal Component

Girosi King 2007 - Information Loss with One Principal Component

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