typo in p1
This commit is contained in:
Morten Hjorth-Jensen
@@ -489,17 +489,17 @@ term which measures the deviation from the true data and the mean value of the m
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That is, show that
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</p>
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$$
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\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=(\mathrm{Bias}[\tilde{y}])^2+\mathrm{var}[\tilde{f}]+\sigma^2,
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\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\mathrm{Bias}[y]+\mathrm{var}[\tilde{y}]+\sigma^2,
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$$
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<p>with </p>
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$$
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(\mathrm{Bias}[\tilde{y}])^2=\left(\boldsymbol{y}-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right]\right)^2,
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\mathrm{Bias}[y]=\mathbb{E}\left[\left(\boldsymbol{y}-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right]\right)^2\right],
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$$
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<p>and </p>
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$$
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\mathrm{var}[\tilde{f}]=\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2.
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\mathrm{var}[\tilde{y}]=\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2.
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$$
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<p>The answer to this exercise should be included in the theory part of the report. This exercise is also part of the weekly exercises of week 37.
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@@ -489,17 +489,17 @@ term which measures the deviation from the true data and the mean value of the m
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That is, show that
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</p>
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$$
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\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=(\mathrm{Bias}[\tilde{y}])^2+\mathrm{var}[\tilde{f}]+\sigma^2,
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\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\mathrm{Bias}[y]+\mathrm{var}[\tilde{y}]+\sigma^2,
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$$
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<p>with </p>
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$$
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(\mathrm{Bias}[\tilde{y}])^2=\left(\boldsymbol{y}-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right]\right)^2,
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\mathrm{Bias}[y]=\mathbb{E}\left[\left(\boldsymbol{y}-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right]\right)^2\right],
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$$
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<p>and </p>
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$$
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\mathrm{var}[\tilde{f}]=\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2.
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\mathrm{var}[\tilde{y}]=\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2.
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$$
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<p>The answer to this exercise should be included in the theory part of the report. This exercise is also part of the weekly exercises of week 37.
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@@ -525,17 +525,17 @@ term which measures the deviation from the true data and the mean value of the m
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That is, show that
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</p>
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$$
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\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=(\mathrm{Bias}[\tilde{y}])^2+\mathrm{var}[\tilde{f}]+\sigma^2,
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\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\mathrm{Bias}[y]+\mathrm{var}[\tilde{y}]+\sigma^2,
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$$
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<p>with </p>
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$$
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(\mathrm{Bias}[\tilde{y}])^2=\left(\boldsymbol{y}-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right]\right)^2,
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\mathrm{Bias}[y]=\mathbb{E}\left[\left(\boldsymbol{y}-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right]\right)^2\right],
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$$
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<p>and </p>
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$$
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\mathrm{var}[\tilde{f}]=\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2.
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\mathrm{var}[\tilde{y}]=\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2.
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$$
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<p>The answer to this exercise should be included in the theory part of the report. This exercise is also part of the weekly exercises of week 37.
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@@ -2,7 +2,7 @@
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"cells": [
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{
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"cell_type": "markdown",
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"id": "6060626f",
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"id": "7424fea1",
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"metadata": {
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"editable": true
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},
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@@ -14,7 +14,7 @@
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},
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{
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"cell_type": "markdown",
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"id": "fd9f7098",
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"id": "c9ca1d90",
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"metadata": {
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"editable": true
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@@ -27,7 +27,7 @@
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},
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{
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"cell_type": "markdown",
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"id": "75f7ac9b",
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"id": "3e4164b6",
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"metadata": {
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"editable": true
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},
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@@ -63,7 +63,7 @@
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},
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{
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"cell_type": "markdown",
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"id": "988b2436",
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"id": "dd4dcdf5",
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"metadata": {
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"editable": true
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},
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@@ -85,7 +85,7 @@
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},
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{
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"cell_type": "markdown",
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"id": "0725f6b3",
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"id": "71ddc478",
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"metadata": {
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"editable": true
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@@ -100,7 +100,7 @@
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},
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{
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"cell_type": "markdown",
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"id": "de583594",
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"id": "d4184afb",
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"metadata": {
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"editable": true
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},
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@@ -133,7 +133,7 @@
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{
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"cell_type": "code",
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"execution_count": 1,
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"id": "130a5d79",
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"id": "62880fe6",
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"metadata": {
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"collapsed": false,
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"editable": true
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@@ -185,7 +185,7 @@
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},
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{
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"cell_type": "markdown",
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"id": "9bf030f4",
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"id": "6f7e7c97",
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"metadata": {
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"editable": true
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},
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@@ -207,7 +207,7 @@
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},
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{
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"cell_type": "markdown",
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"id": "3be47d8d",
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"id": "7377112f",
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"metadata": {
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"editable": true
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@@ -220,7 +220,7 @@
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},
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{
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"cell_type": "markdown",
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"id": "f7f14618",
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"id": "72f83fe7",
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"metadata": {
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"editable": true
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@@ -232,7 +232,7 @@
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},
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{
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"cell_type": "markdown",
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"id": "913d72b1",
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"id": "7a3578ff",
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"metadata": {
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"editable": true
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@@ -244,7 +244,7 @@
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},
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{
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"cell_type": "markdown",
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"id": "7a403d11",
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"id": "e0aef819",
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"metadata": {
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"editable": true
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@@ -254,7 +254,7 @@
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},
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{
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"cell_type": "markdown",
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"id": "108b7a27",
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"id": "c7f751af",
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"metadata": {
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"editable": true
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@@ -266,7 +266,7 @@
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},
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{
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"cell_type": "markdown",
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"id": "f1e64231",
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"id": "c026a4bd",
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"metadata": {
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"editable": true
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},
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@@ -295,7 +295,7 @@
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},
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{
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"cell_type": "markdown",
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"id": "30bca1e7",
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"id": "2635bcbd",
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"metadata": {
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"editable": true
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},
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@@ -313,7 +313,7 @@
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},
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{
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"cell_type": "markdown",
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"id": "ae6a3eeb",
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"id": "92892df2",
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"metadata": {
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"editable": true
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},
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@@ -330,7 +330,7 @@
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},
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{
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"cell_type": "markdown",
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"id": "5ba7dff9",
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"id": "35987aab",
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"metadata": {
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"editable": true
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},
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@@ -346,7 +346,7 @@
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},
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{
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"cell_type": "markdown",
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"id": "84ca72c8",
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"id": "e61048c2",
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"metadata": {
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"editable": true
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},
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@@ -358,7 +358,7 @@
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},
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{
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"cell_type": "markdown",
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"id": "0d734865",
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"id": "5ef456d4",
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"metadata": {
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"editable": true
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},
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@@ -369,7 +369,7 @@
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},
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{
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"cell_type": "markdown",
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"id": "1dac6c19",
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"id": "52e6fbad",
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"metadata": {
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"editable": true
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@@ -381,7 +381,7 @@
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},
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{
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"cell_type": "markdown",
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"id": "e04687a9",
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"id": "db6028a0",
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"metadata": {
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"editable": true
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@@ -393,7 +393,7 @@
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{
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"cell_type": "markdown",
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"id": "f99dbe69",
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"id": "9477ad5e",
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"metadata": {
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"editable": true
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@@ -405,7 +405,7 @@
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{
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"cell_type": "markdown",
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"id": "0b3e0564",
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"id": "8c2aeccd",
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"metadata": {
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"editable": true
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@@ -416,7 +416,7 @@
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{
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"cell_type": "markdown",
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"id": "f660901d",
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"id": "6a774b74",
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"metadata": {
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"editable": true
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@@ -428,7 +428,7 @@
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{
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"cell_type": "markdown",
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"id": "7a1c7f34",
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"id": "98fde9b7",
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"metadata": {
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"editable": true
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@@ -441,7 +441,7 @@
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{
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"cell_type": "markdown",
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"id": "becad040",
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"id": "87a4d11d",
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"metadata": {
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@@ -453,7 +453,7 @@
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{
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"cell_type": "markdown",
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"id": "242ede2f",
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"id": "b4844f1e",
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"metadata": {
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"editable": true
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@@ -463,7 +463,7 @@
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{
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"cell_type": "markdown",
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"id": "d4999c5f",
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"id": "f7662ade",
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"metadata": {
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"editable": true
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@@ -475,7 +475,7 @@
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},
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{
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"cell_type": "markdown",
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"id": "46386eef",
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"id": "1d61362e",
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"metadata": {
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"editable": true
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@@ -486,7 +486,7 @@
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},
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{
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"cell_type": "markdown",
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"id": "9f27e396",
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"id": "74bbb4ab",
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"metadata": {
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"editable": true
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@@ -519,7 +519,7 @@
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},
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{
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"cell_type": "markdown",
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"id": "902b759d",
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"id": "ac939724",
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"metadata": {
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"editable": true
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@@ -531,7 +531,7 @@
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},
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{
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"cell_type": "markdown",
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"id": "41a0b8d7",
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"id": "4066d012",
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"metadata": {
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"editable": true
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@@ -550,7 +550,7 @@
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},
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{
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"cell_type": "markdown",
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"id": "89cced85",
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"id": "7e5c5ef4",
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"metadata": {
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"editable": true
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},
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@@ -562,7 +562,7 @@
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},
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{
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"cell_type": "markdown",
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"id": "22bb8b90",
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"id": "690337b9",
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"metadata": {
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"editable": true
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@@ -576,19 +576,19 @@
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},
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{
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"cell_type": "markdown",
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"id": "e88ba22d",
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"id": "92dabf3c",
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"metadata": {
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"editable": true
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},
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"source": [
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"$$\n",
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"\\mathbb{E}\\left[(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2\\right]=(\\mathrm{Bias}[\\tilde{y}])^2+\\mathrm{var}[\\tilde{f}]+\\sigma^2,\n",
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"\\mathbb{E}\\left[(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2\\right]=\\mathrm{Bias}[y]+\\mathrm{var}[\\tilde{y}]+\\sigma^2,\n",
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"$$"
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]
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},
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{
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"cell_type": "markdown",
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"id": "e2156ab0",
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"id": "7da90966",
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"metadata": {
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"editable": true
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@@ -598,19 +598,19 @@
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},
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{
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"cell_type": "markdown",
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"id": "7d08fd36",
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"id": "33a81d54",
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"metadata": {
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"editable": true
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},
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"source": [
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"$$\n",
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"(\\mathrm{Bias}[\\tilde{y}])^2=\\left(\\boldsymbol{y}-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right]\\right)^2,\n",
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"\\mathrm{Bias}[y]=\\mathbb{E}\\left[\\left(\\boldsymbol{y}-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right]\\right)^2\\right],\n",
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"$$"
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]
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},
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{
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"cell_type": "markdown",
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"id": "5cdcf0c6",
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"id": "abb9bd28",
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"metadata": {
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"editable": true
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@@ -620,19 +620,19 @@
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},
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{
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"cell_type": "markdown",
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"id": "3235550b",
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"id": "5b4668fd",
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"metadata": {
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"editable": true
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},
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"source": [
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"$$\n",
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"\\mathrm{var}[\\tilde{f}]=\\frac{1}{n}\\sum_i(\\tilde{y}_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2.\n",
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"\\mathrm{var}[\\tilde{y}]=\\frac{1}{n}\\sum_i(\\tilde{y}_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2.\n",
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"$$"
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]
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},
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{
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"cell_type": "markdown",
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"id": "813e8659",
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"id": "b453f8fc",
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"metadata": {
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"editable": true
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@@ -651,7 +651,7 @@
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},
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{
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"cell_type": "markdown",
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"id": "adf1396f",
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"id": "6fe323b0",
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"metadata": {
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"editable": true
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@@ -676,7 +676,7 @@
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},
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{
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"cell_type": "markdown",
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"id": "a7cb0fd1",
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"id": "5fd2f486",
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"metadata": {
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"editable": true
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{
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"cell_type": "code",
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"execution_count": 2,
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"id": "3be6c174",
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"id": "e88e39ea",
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"metadata": {
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"collapsed": false,
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"editable": true
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@@ -716,7 +716,7 @@
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},
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{
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"cell_type": "markdown",
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"id": "9544f6e7",
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"id": "29817da1",
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"metadata": {
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"editable": true
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@@ -728,7 +728,7 @@
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{
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"cell_type": "code",
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"execution_count": 3,
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"id": "c2dc1c15",
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"id": "560fc9f4",
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"metadata": {
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"collapsed": false,
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"editable": true
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@@ -754,7 +754,7 @@
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},
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{
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"cell_type": "markdown",
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"id": "81854928",
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"id": "3fc05383",
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"metadata": {
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"editable": true
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},
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@@ -779,7 +779,7 @@
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},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "9a8a9dc4",
|
||||
"id": "bb21261e",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
@@ -793,7 +793,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "df82eef8",
|
||||
"id": "c5ab9dbf",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
@@ -823,7 +823,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "ba32d247",
|
||||
"id": "7d0dbb14",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
@@ -845,7 +845,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "cb0788db",
|
||||
"id": "e301049d",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
|
||||
Binary file not shown.
@@ -451,15 +451,15 @@ Show that you can rewrite this in terms of a term which contains the variance o
|
||||
term which measures the deviation from the true data and the mean value of the model (the bias term) and finally the variance of the noise.
|
||||
That is, show that
|
||||
\[
|
||||
\mathbb{E}\left[(\bm{y}-\bm{\tilde{y}})^2\right]=(\mathrm{Bias}[\tilde{y}])^2+\mathrm{var}[\tilde{f}]+\sigma^2,
|
||||
\mathbb{E}\left[(\bm{y}-\bm{\tilde{y}})^2\right]=\mathrm{Bias}[y]+\mathrm{var}[\tilde{y}]+\sigma^2,
|
||||
\]
|
||||
with
|
||||
\[
|
||||
(\mathrm{Bias}[\tilde{y}])^2=\left(\bm{y}-\mathbb{E}\left[\bm{\tilde{y}}\right]\right)^2,
|
||||
\mathrm{Bias}[y]=\mathbb{E}\left[\left(\bm{y}-\mathbb{E}\left[\bm{\tilde{y}}\right]\right)^2\right],
|
||||
\]
|
||||
and
|
||||
\[
|
||||
\mathrm{var}[\tilde{f}]=\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\bm{\tilde{y}}\right])^2.
|
||||
\mathrm{var}[\tilde{y}]=\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\bm{\tilde{y}}\right])^2.
|
||||
\]
|
||||
The answer to this exercise should be included in the theory part of the report. This exercise is also part of the weekly exercises of week 37.
|
||||
Explain what the terms mean and discuss their interpretations.
|
||||
|
||||
Binary file not shown.
@@ -421,15 +421,15 @@ Show that you can rewrite this in terms of a term which contains the variance o
|
||||
term which measures the deviation from the true data and the mean value of the model (the bias term) and finally the variance of the noise.
|
||||
That is, show that
|
||||
\[
|
||||
\mathbb{E}\left[(\bm{y}-\bm{\tilde{y}})^2\right]=(\mathrm{Bias}[\tilde{y}])^2+\mathrm{var}[\tilde{f}]+\sigma^2,
|
||||
\mathbb{E}\left[(\bm{y}-\bm{\tilde{y}})^2\right]=\mathrm{Bias}[y]+\mathrm{var}[\tilde{y}]+\sigma^2,
|
||||
\]
|
||||
with
|
||||
\[
|
||||
(\mathrm{Bias}[\tilde{y}])^2=\left(\bm{y}-\mathbb{E}\left[\bm{\tilde{y}}\right]\right)^2,
|
||||
\mathrm{Bias}[y]=\mathbb{E}\left[\left(\bm{y}-\mathbb{E}\left[\bm{\tilde{y}}\right]\right)^2\right],
|
||||
\]
|
||||
and
|
||||
\[
|
||||
\mathrm{var}[\tilde{f}]=\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\bm{\tilde{y}}\right])^2.
|
||||
\mathrm{var}[\tilde{y}]=\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\bm{\tilde{y}}\right])^2.
|
||||
\]
|
||||
The answer to this exercise should be included in the theory part of the report. This exercise is also part of the weekly exercises of week 37.
|
||||
Explain what the terms mean and discuss their interpretations.
|
||||
|
||||
+319
-647
File diff suppressed because one or more lines are too long
+209
-546
File diff suppressed because one or more lines are too long
@@ -324,19 +324,19 @@ term which measures the deviation from the true data and the mean value of the m
|
||||
That is, show that
|
||||
!bt
|
||||
\[
|
||||
\mathbb{E}\left[(\bm{y}-\bm{\tilde{y}})^2\right]=(\mathrm{Bias}[\tilde{y}])^2+\mathrm{var}[\tilde{f}]+\sigma^2,
|
||||
\mathbb{E}\left[(\bm{y}-\bm{\tilde{y}})^2\right]=\mathrm{Bias}[y]+\mathrm{var}[\tilde{y}]+\sigma^2,
|
||||
\]
|
||||
!et
|
||||
with
|
||||
!bt
|
||||
\[
|
||||
(\mathrm{Bias}[\tilde{y}])^2=\left(\bm{y}-\mathbb{E}\left[\bm{\tilde{y}}\right]\right)^2,
|
||||
\mathrm{Bias}[y]=\mathbb{E}\left[\left(\bm{y}-\mathbb{E}\left[\bm{\tilde{y}}\right]\right)^2\right],
|
||||
\]
|
||||
!et
|
||||
and
|
||||
!bt
|
||||
\[
|
||||
\mathrm{var}[\tilde{f}]=\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\bm{\tilde{y}}\right])^2.
|
||||
\mathrm{var}[\tilde{y}]=\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\bm{\tilde{y}}\right])^2.
|
||||
\]
|
||||
!et
|
||||
The answer to this exercise should be included in the theory part of the report. This exercise is also part of the weekly exercises of week 37.
|
||||
|
||||
Reference in New Issue
Block a user