lambda_1=(1+0j), ||x* - A x*||=1.5700924586837752e-16lambda_1=(1+0j), ||x* - A x*||=1.1102230246251565e-16rho(A) = 1.079128784747792
-x_200 = [3406689.32410674 6102361.18640517]check orthogonal: dot(x_1,x_2) approx 0: 0.0
-rho(A) = 1.0000000000000002
+rho(A) = 1.0
x_200 = [ 0.27639321 -0.17082039]condition(I) = 1.0
-condition(A) = 40000001.939191714, condition(A^(-1)) = 40000002.00307444error = 5.820766091346741e-11, error using inv(A) = 1.3387762010097504e-09
-cond(A) = 564652.3214074654error = 1.574562702444382e-11, error using inv(A) = 1.1932570487260818e-09
+cond(A) = 564652.3214000753error = 1.280568540096283e-09, error using inv(A) = 6.463262252509594e-06
-cond(A) = 4462817558043.969error = 5.334186425898224e-10, error using inv(A) = 6.22717197984457e-06
+cond(A) = 4462824600234.486error = 5.820766091346741e-10, error using inv(A) = 2460.8631448635133
-cond(A) = 5.771e+24error = 6.784830475226045e-10, error using inv(A) = 31732.823760853855
+cond(A) = 1.697e+24cond(A) = 3.64e+09, cond(A_monomial) = 1.694e+17cond(A) = 3.64e+09, cond(A_monomial) = 2.926e+18Correlation of Z_1 to Z_hat_1 = 0.9989685407197146
-Correlation of Z_2 to Z_hat_2 = -0.622136330126575Correlation of Z_1 to Z_hat_1 = 0.9987396670122092
+Correlation of Z_2 to Z_hat_2 = -0.6413512732572132
Correlation between k and k_hat:-0.83Correlation between k and k_hat:-0.93Correlation now 0.8300415031127608Correlation now 0.9300465034877902
[ 1.10000000e+00 -2.65486721e-09][ 1.10000000e+00 -2.65486733e-09]2.663274529925679e-092.663274500543771e-09norm = 2.3494397540428033e-10norm = 2.3494410875961204e-10Let’s see if these line up perfectly
diff --git a/lectures/lectures/latent_variables_files/figure-revealjs/cell-11-output-1.png b/lectures/lectures/latent_variables_files/figure-revealjs/cell-11-output-1.png index 31cccaf..257795a 100644 Binary files a/lectures/lectures/latent_variables_files/figure-revealjs/cell-11-output-1.png and b/lectures/lectures/latent_variables_files/figure-revealjs/cell-11-output-1.png differ diff --git a/lectures/lectures/latent_variables_files/figure-revealjs/cell-14-output-1.png b/lectures/lectures/latent_variables_files/figure-revealjs/cell-14-output-1.png index bb0119c..73c0823 100644 Binary files a/lectures/lectures/latent_variables_files/figure-revealjs/cell-14-output-1.png and b/lectures/lectures/latent_variables_files/figure-revealjs/cell-14-output-1.png differ diff --git a/lectures/lectures/latent_variables_files/figure-revealjs/cell-18-output-1.png b/lectures/lectures/latent_variables_files/figure-revealjs/cell-18-output-1.png index 389112a..3924a1e 100644 Binary files a/lectures/lectures/latent_variables_files/figure-revealjs/cell-18-output-1.png and b/lectures/lectures/latent_variables_files/figure-revealjs/cell-18-output-1.png differ diff --git a/lectures/lectures/latent_variables_files/figure-revealjs/cell-7-output-1.png b/lectures/lectures/latent_variables_files/figure-revealjs/cell-7-output-1.png index b253a54..649fb80 100644 Binary files a/lectures/lectures/latent_variables_files/figure-revealjs/cell-7-output-1.png and b/lectures/lectures/latent_variables_files/figure-revealjs/cell-7-output-1.png differ diff --git a/lectures/lectures/latent_variables_files/figure-revealjs/cell-9-output-1.png b/lectures/lectures/latent_variables_files/figure-revealjs/cell-9-output-1.png index 252cfd8..eca2dbb 100644 Binary files a/lectures/lectures/latent_variables_files/figure-revealjs/cell-9-output-1.png and b/lectures/lectures/latent_variables_files/figure-revealjs/cell-9-output-1.png differ diff --git a/lectures/lectures/linear_algebra_foundations.html b/lectures/lectures/linear_algebra_foundations.html index 04b9edc..32196a6 100644 --- a/lectures/lectures/linear_algebra_foundations.html +++ b/lectures/lectures/linear_algebra_foundations.html @@ -1121,9 +1121,9 @@beta =
- [-0.81011697 0.47576408 -0.08064125 -0.21201846 -1.02869662]
+ [ 1.27699654 1.24976206 -0.02314793 -0.1603755 -1.71006662]
beta_hat =
-[-0.81519622 0.47790174 -0.08893845 -0.21818441 -1.02778959]beta =
- [-0.81011697 0.47576408 -0.08064125 -0.21201846 -1.02869662]
+ [ 1.27699654 1.24976206 -0.02314793 -0.1603755 -1.71006662]
beta_hat =
-[-0.81519622 0.47790174 -0.08893845 -0.21818441 -1.02778959]cond(X'*X)=2819.3329786399063, cond(X_col'*X_col)=1.2999933999712892e+16cond(X'*X)=2819.332978639814, cond(X_col'*X_col)=1.1014450683078442e+16discrete_rvE(X) = 34500.0
E(f(X)) = 182.2474487139159
CDF(X) = [0.2 1. 0.1]
-Samples of X = [40000 15000 40000 40000 40000]Mean: 5.00
Variance: 2.50
-Draws of u: [3 5 5 6 6]
X_0 = 0, X_1 = 0
X_t indices =
- [0 0 0 0 0 0 0 0 0 0 0]Uncertainty Quantification in Applied Economics
+ + + +plot_data = (df
+ .assign(top_school = df["avg_score"] >= np.quantile(df["avg_score"], .99))
+ [["top_school", "number_of_students"]]
+ .query(f"number_of_students<{np.quantile(df['number_of_students'], .98)}")) # remove outliers
+
+plt.figure(figsize=(3,3))
+sns.boxplot(x="top_school", y="number_of_students", data=plot_data)
+plt.title("Number of Students of 1% Top Schools (Right)");
We can see that the top 1% of schools have a lot fewer students than the rest of the schools.
This might lead us to draw some conclusions, like “smaller schools are better”.
q_99 = np.quantile(df["avg_score"], .99)
+q_01 = np.quantile(df["avg_score"], .01)
+
+plot_data = (df
+ .sample(10000)
+ .assign(Group = lambda d: np.select([d["avg_score"] > q_99, d["avg_score"] < q_01],
+ ["Top", "Bottom"], "Middle")))
+plt.figure(figsize=(5,5))
+sns.scatterplot(y="avg_score", x="number_of_students", hue="Group", data=plot_data)
+plt.title("ENEM Score by Number of Students");
This captures exactly what we would expect if there was no relationship between school size and student performance.
Our false conclusion is a direct result of the formula \(\text{SE} = \frac{\sigma}{\sqrt{n}}\).
Since each observation in our dataset is a sample average, the schools with a lot of students give very precise estimates of the true mean, while the schools with fewer students give noisier estimates.
def run_experiment():
+ return np.random.normal(74, 2, 500)
+
+np.random.seed(42)
+
+plt.figure(figsize=(8,2.5))
+freq, bins, img = plt.hist([run_experiment().mean() for _ in range(10000)], bins=40, label="Experiment Means")
+plt.vlines(74, ymin=0, ymax=freq.max(), linestyles="dashed", label="True Mean", color="orange")
+plt.legend();
def run_experiment_unif():
+ return np.random.uniform(64, 84, 500)
+
+np.random.seed(42)
+
+plt.figure(figsize=(8,2.5))
+freq, bins, img = plt.hist([run_experiment_unif().mean() for _ in range(10000)], bins=40, label="Experiment Means")
+plt.vlines(74, ymin=0, ymax=freq.max(), linestyles="dashed", label="True Mean", color="orange")
+plt.legend();
np.random.seed(321)
+exp_data = run_experiment()
+exp_se = exp_data.std() / np.sqrt(len(exp_data))
+exp_mu = exp_data.mean()
+ci = (exp_mu - 2 * exp_se, exp_mu + 2 * exp_se)
+print(ci)
+x = np.linspace(exp_mu - 4*exp_se, exp_mu + 4*exp_se, 100)
+y = stats.norm.pdf(x, exp_mu, exp_se)
+plt.plot(x, y, color="orange")
+plt.vlines(ci[1], ymin=0, ymax=1)
+plt.vlines(ci[0], ymin=0, ymax=1, label="95% CI")
+plt.legend()
+plt.show()(73.82718114045632, 74.17341543460314)
def bias_sq(c,sigma=5,n=10):
+ return (c * (n-1)-1)**2 * sigma**4
+
+def mse(c,sigma=5,n=10):
+ return (c*(n-1)-1)**2 * sigma**4 + 2 * c**2 * sigma**4 * (n-1)
+
+n = 10
+sigma = 5
+c = np.linspace(0, 0.2, 100)
+plt.plot(c, bias_sq(c), label="Bias Squared")
+plt.plot(c, mse(c), label="MSE", color = "red")
+plt.vlines(1/(n-1), ymin=0, ymax=800, linestyles="dashed", label="1/(n-1)")
+plt.vlines(1/(n+1), ymin=0, ymax=800, linestyles="dashed", color = "red", label="1/(n+1)")
+
+plt.legend()
+plt.show()
Source: https://en.wikipedia.org/wiki/Bias_of_an_estimator
Source: https://www.shiksha.com/online-courses/articles/bias-and-variance/
+
