From 07714779b68d45f46a6d02d77394a159af186b8b Mon Sep 17 00:00:00 2001 From: Claudio Maggioni Date: Fri, 7 May 2021 12:32:53 +0200 Subject: [PATCH] hw1: almost done, recheck for safety --- .../deliverable/linear_regression.pickle | Bin 601 -> 595 bytes .../nonlinear_model/saved_model.pb | Bin 189466 -> 87663 bytes .../variables/variables.data-00000-of-00001 | Bin 19722 -> 9529 bytes .../nonlinear_model/variables/variables.index | Bin 2850 -> 1669 bytes .../nonlinear_model_normalizers.pickle | Bin 278 -> 272 bytes assignment_1/deliverable/run_model.py | 68 +++++------ assignment_1/report_Maggioni_Claudio.md | 107 ++++++++++++++++-- assignment_1/report_Maggioni_Claudio.pdf | Bin 105195 -> 132411 bytes assignment_1/src/build_models.py | 72 ++++++------ assignment_1/src/statistical_test.py | 83 ++++++++++++++ assignment_1/src/test.npz | Bin 0 -> 10347 bytes 11 files changed, 250 insertions(+), 80 deletions(-) create mode 100644 assignment_1/src/statistical_test.py create mode 100644 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a/assignment_1/deliverable/nonlinear_model/variables/variables.index b/assignment_1/deliverable/nonlinear_model/variables/variables.index index 20b344777d2bd8cca8477249cfaa90042f910cab..c5d43a16a552302844b94142dab02aabed6a2c1f 100644 GIT binary patch delta 632 zcmZ1^*2+6UmurcJ##AL!sc&1ACz{(Zo#35#Q=jt)uLg@L<2T8?$%2e_ykZI)7&RhH zgY^~VrA;S4flTu3(;|)RK$j_j#1?0ZJuaDp7m*F!4i3u|(>;Q@Fe*5btmjYi(YI0&FNSuX1 zVGr-*mz;W>Yj`y{Ol!U3S5H>pGH^TqQk;1@)QABH<}))eFo75;cMQ5fG$V*O@M+-& p21e!rhV?vfF*eD2^S1DC9OU7j%*ZV{S%909Bm8$mw@Rt|Z2+P7ukQc= literal 2850 zcmb`JT}Tu`9KdJqw6-NSI`7UtuhA9LkB>`74SL*b;$6=*TPl=1BFy)O6CGkTk zR-X~FN>~wVrkc=CCL~p^Ib=v@v(>HfX}W6+{8QbgD%E7ih2Nv^6^qARkv z3R#RM4m8Q2qb+clLi_ohBhl>(9R{y2;NIfi?(z%89&dmP*D7*QJ}5=FNKh54eZdp7??wC>QXC=P5WAV9jStsN`M zRjG)9u?&Rx(a>nMWkR(K_1z=?#RdaU`3P`?HRoa* zc`9ur!#kPC$Wxb!PieN13|q@E%W{5@nT8Dp+en7P87M~a^;Slw*+v}LUrK<_?yjP+ z4PK=UJv?qhv-y+yq1&2m=;3e~X7zbCTrj1X)&42T*oGeVW}-Fx!@h>j#SXK!lQqt} zS%!J1g=2o~!miSV4z}A6%|AKQ^i{J99c(JctdGslY`|`wDVwY4Y9!Ol702m`et{j@Xnpwii3a&sB)M&GCjn+*E07_pAdQB63&7FhP3lsg$G>3F0ezuJ@oLE{{ G`1&_rWeBeT diff --git a/assignment_1/deliverable/nonlinear_model_normalizers.pickle b/assignment_1/deliverable/nonlinear_model_normalizers.pickle index 4e9916fc9deaf305a2cfe419f0bfc424cc25c1b5..54ad90359f97708204f1bfc912eb730169d318f0 100644 GIT binary patch delta 26 icmbQnG=Yhwfo1B}i7YKl8T=C`6a@fnb_ZMl delta 49 zcmbQhG>wU+fo1CMi7YJ~J*+7ul?AC&CU)gXh-8R&lqRKotjRfh;j_K*;sAjgk)Q2L F^#IU`6iNU9 diff --git a/assignment_1/deliverable/run_model.py b/assignment_1/deliverable/run_model.py index d4808f3..af2ae23 100755 --- a/assignment_1/deliverable/run_model.py +++ b/assignment_1/deliverable/run_model.py @@ -61,51 +61,39 @@ if __name__ == '__main__': # EDITABLE SECTION OF THE SCRIPT: if you need to edit the script, do it here ############################################################################ - ms = {"baseline", "linear", "nonlinear"} - - if len(sys.argv) != 2: - print("%s {model_name}" % sys.argv[0]) - sys.exit(1) - - if sys.argv[1] not in ms: - print("model name must be one of " + str(ms)) - sys.exit(2) - y_pred = None - if sys.argv[1] == "baseline": - # Load the trained model - baseline_model_path = './baseline_model.pickle' - baseline_model = load_model(baseline_model_path) - # Predict on the given samples - y_pred = baseline_model.predict(x) - else: - X = np.zeros([x.shape[0], 5]) - X[:, 0] = 1 - X[:, 1:3] = x - X[:, 3] = x[:, 0] * x[:, 1] - X[:, 4] = np.sin(x[:, 0]) + X = np.zeros([x.shape[0], 5]) + X[:, 0] = 1 + X[:, 1:3] = x + X[:, 3] = x[:, 0] * x[:, 1] + X[:, 4] = np.sin(x[:, 0]) - if sys.argv[1] == "linear": - # Load the linear model - linear_model_path = './linear_regression.pickle' - linear_model = load_model(linear_model_path) + # Load the linear model + linear_model_path = './linear_regression.pickle' + linear_model = load_model(linear_model_path) - # Predict on the given samples - y_pred = linear_model.predict(X) - else: - # Load saved Keras model - nonlinear_model = models.load_model('./nonlinear_model') - # Load saved normalizing factors for Xs - normalizers = load_model('./nonlinear_model_normalizers.pickle') - # Normalize Xs with normalizing factors - X = X[:, 1:3] - X -= normalizers["mean"] - X /= normalizers["std"] + # Predict on the given samples + y_pred = linear_model.predict(X) + # Load saved Keras model + nonlinear_model = models.load_model('./nonlinear_model') + # Load saved normalizing factors for Xs + normalizers = load_model('./nonlinear_model_normalizers.pickle') - # Run model on normalized data - y_pred = nonlinear_model.predict(X) - y_pred = y_pred[:, 0] + print("Linear model:") + mse = evaluate_predictions(y_pred, y) + print('MSE: {}'.format(mse)) + + # Normalize Xs with normalizing factors + X = X[:, 1:3] + X -= normalizers["mean"] + X /= normalizers["std"] + + # Run model on normalized data + y_pred = nonlinear_model.predict(X) + y_pred = y_pred[:, 0] + + print("Non-linear model:") ############################################################################ # STOP EDITABLE SECTION: do not modify anything below this point. diff --git a/assignment_1/report_Maggioni_Claudio.md b/assignment_1/report_Maggioni_Claudio.md index 2bd754b..c20b507 100644 --- a/assignment_1/report_Maggioni_Claudio.md +++ b/assignment_1/report_Maggioni_Claudio.md @@ -62,6 +62,21 @@ of your choice. Limit the assignment to 2500 words (formulas, tables, figures, etc., do not count as words) and do not include any code in the report. +## Premise on data splitting + +In order to perform a correct statistical analysis, I have splitted the given +datapoints in a training set, a validation set and a test set. The test set has +200 points (10% of the given datapoints), the validation set has 180 data points +(10% of the remaining 90% of the given datapoint) while the training set has +1620 datapoints (all the remaining datapoints). + +For the linear model, I will perform the linear regression using both the +training and the validation datapoints (since there is no need to choose +hyperparameters since the given family of models requires none). For the +nonlinear model, which is a feedforward NN in my case, I used the training set +to fit the model and I have used the validation set in order to choose the +architecture of the neural network. + ## Task 1 Use the family of models @@ -71,11 +86,16 @@ $f(\mathbf{x}, \boldsymbol{\theta}) = \theta_0 + \theta_1 \cdot x_1 + to fit the data. Write in the report the formula of the model substituting parameters $\theta_0, \ldots, \theta_4$ with the estimates you've found: -$$f(\mathbf{x}, \boldsymbol{\theta}) = \_ + \_ \cdot x_1 + \_ -\cdot x_2 + \_ \cdot x_1 \cdot x_2 + \_ \cdot \sin(x_1)$$ + +$$f(\mathbf{x}, \boldsymbol{\theta}) = 3.13696 -0.468921 \cdot x_1 +-0.766447 \cdot x_2 + 0.0390513 \cdot x_1 x_2 -0.918346 \cdot \sin(x_1)$$ + Evaluate the test performance of your model using the mean squared error as performance measure. +On the 200-datapoint test set described before, the MSE of this linear model is +$1.22265$. This result can be reproduced by running `src/statistical_tests.py`. + ## Task 2 Consider any family of non-linear models of your choice to address the @@ -84,6 +104,41 @@ using the mean squared error as performance measure. Compare your model with the linear regression of Task 1. Which one is **statistically** better? +The model I've chosen is a feed-forward neural network with 2 hidden layers: + +- one 22-neuron dense layer with hyperbolic tangent activation function; +- one 15-neuron dense layer with sigmoidal activation function; + +Finally, the output neuron has a linear activation function. As described +before, the validation set was used to manually tailor the NNs architecture. + +The fitted model has these final performance parameters: + +- Final MSE on validation set $\approx 0.0143344$ +- Final MSE on the test set $\approx 0.0107676$ + +I conclude the NN non-linear model is better than the linear +regression model because of the lower test set MSE. Since the test set is 200 +elements big, we can safely assume that the loss function applied on the test +set is a resonable approximation of the structural risk for both models. + +In order to motivate this intuitive argument, I will perform a student's T-test +to show that the non-linear model is indeed statistically better with 95% +confidence. The test was implemented in `src/statistical_tests.py` and it has +been performed by spitting the test set, assuming both resulting sets have an +expectation of the noise equal to 0, and by computing mean, variance and +T-score. + +The results of the test are the following: + +- T-test result: 5.37324 +- Linear model noise variance: 6.69426 +- Feedforward NN noise variance: $0.348799 \cdot 10^{-3}$ + +The test is clearly outside of the 95% confidence interval of $[-1.96,1.96]$ and +thus we reject the null hypothesis. Since the variance is lower for the NN we +conclude that this model is indeed the better one statistically speaking. + ## Task 3 (Bonus) In the [**Github repository of the @@ -95,6 +150,33 @@ performance of your model is better (i.e., the MSE is lower) than ours. Of course, you also have to tell us why you think that your model is better. +In order to understand if my model is better than the baseline model, I +performed another Student's T-test re-using the subset of the test set I've used +for my linear model. The results of the new test are the following: + +- T-test result: 1.1777 (for a $p$ value of 0.2417396) +- Baseline model noise variance: $0.366316 \cdot 10^{-3}$ +- Feedforward NN noise variance: $0.348799 \cdot 10^{-3}$ + +This test would accept my model as the better one (as the variance is lower) +with a 75% confidence interval, which is far from ideal but still can be +significant and could be a good chance at having a better model on the hidden +test set. + +To further analyze the performance of my NN w.r.t. the baseline model, I have +computed the MSE over my entire 200-datapoints test set for both models. Here +are the results: + +- MSE on entire test set for baseline model: 0.0151954 +- MSE on entire test set for feedforward NN model: 0.0107676 + +My model shows lower MSE for this particular test set, but even if this test +wasn't used to train both sets of course the performance may change for a +different test set like the hidden one. The T-test performed before suggests +that my model might have a decent chance at performing better than the baseline +model, but I cannot say that for sure or with a reasonable (like 90% or 95%) +confidence. + # Questions ## Q1. Training versus Validation @@ -238,7 +320,7 @@ Comment and compare how the (a.) training error, (b.) test error and problem was proved impossible to solve by a perceptron model by Minsky and Papert, then that would be quite a motivation in front of my boss. - On a morev general (and more serious) note, the perceptron model would be + On a more general (and more serious) note, the perceptron model would be unable to solve the problem in the picture since a perceptron can solve only linearly-separable classification problems, and even through a simple graphical argument we would be unable to find a line able able to separate @@ -248,11 +330,22 @@ Comment and compare how the (a.) training error, (b.) test error and 2. **Would you expect to have better luck with a neural network with activation function $h(x) = - x \cdot e^{-2}$ for the hidden units?** - The activation function is still linear and data is not linearly separable + First of all we notice that that activation function is still linear, even + if a $-e^{-2}$ scaling factor is applied to the output. Since the problem is + not linearly separable, we would still have bad luck since the new family of + models would still be inadequate for solving the problem. 3. **What are the main differences and similarities between the perceptron and the logistic regression neuron?** - - - + The perceptron neuron and the logistic regression neuron both are dense + neurons that recieve as parameters all values from the previous layer as + single scalars. Both require a number of parameters equal to the number of + inputs, which are used as scaling factors for the inputs recieved. 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