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I'm trying to teach myself machine learning and I have a similar question to this.
Is this correct:
For example, if I have an input matrix, where X1, X2 and X3 are three numerical features (e.g. say they are petal length, stem length, flower length, and I'm trying to label whether the sample is a particular flower species or not):
x1 x2 x3 label
5 1 2 yes
3 9 8 no
1 2 3 yes
9 9 9 no
That you take the vector of the first ROW (not column) of the table above to be inputted into the network like this:
i.e. there would be three neurons (1 for each value of the first table row), and then w1,w2 and w3 are randomly selected, then to calculate the first neuron in the next column, you do the multiplication I have described, and you add a randomly selected bias term. This gives the value of that node.
This is done for a set of nodes (i.e. each column actually will have four nodes (three + a bias), for simplicity, i removed the other three nodes from the second column), and then in the last node before the output, there is an activation function to transform the sum into a value (e.g. 0-1 for sigmoid) and that value tells you whether the classification is yes or no.
I'm sorry for how basic this is, I want to really understand the process, and I'm doing it from free resources. So therefore generally, you should select the number of nodes in your network to be a multiple of the number of features, e.g. in this case, it would make sense to write:
from keras.models import Sequential
from keras.models import Dense
model = Sequential()
model.add(Dense(6,input_dim=3,activation='relu'))
model.add(Dense(6,input_dim=3,activation='relu'))
model.add(Dense(3,activation='softmax'))
What I don't understand is why the keras model has an activation function in each layer of the network and not just at the end, which is why I'm wondering if my understanding is correct/why I added the picture.
Edit 1: Just a note I saw that in the bias neuron, I put on the edge 'b=1', that might be confusing, I know the bias doesn't have a weight, so that was just a reminder to myself that the weight of the bias node is 1.
Several issues here apart from the question in your title, but since this is not the time & place for full tutorials, I'll limit the discussion to some of your points, taking also into account that at least one more answer already exists.
So therefore generally, you should select the number of nodes in your network to be a multiple of the number of features,
No.
The number of features is passed in the input_dim argument, which is set only for the first layer of the model; the number of inputs for every layer except the first one is simply the number of outputs of the previous one. The Keras model you have written is not valid, and it will produce an error, since for your 2nd layer you ask for input_dim=3, while the previous one has clearly 6 outputs (nodes).
Beyond this input_dim argument, there is no other relationship whatsoever between the number of data features and the number of network nodes; and since it seems you have in mind the iris data (4 features), here is a simple reproducible example of applying a Keras model to them.
What is somewhat hidden in the Keras sequential API (which you use here) is that there is in fact an implicit input layer, and the number of its nodes is the dimensionality of the input; see own answer in Keras Sequential model input layer for details.
So, the model you have drawn in your pad actually corresponds to the following Keras model written using the sequential API:
model = Sequential()
model.add(Dense(1,input_dim=3,activation='linear'))
where in the functional API it would be written as:
inputs = Input(shape=(3,))
outputs = Dense(1, activation='linear')(inputs)
model = Model(inputs, outputs)
and that's all, i.e. it is actually just linear regression.
I know the bias doesn't have a weight
The bias does have a weight. Again, the useful analogy is with the constant term of linear (or logistic) regression: the bias "input" itself is always 1, and its corresponding coefficient (weight) is learned through the fitting process.
why the keras model has an activation function in each layer of the network and not just at the end
I trust this has been covered sufficiently in the other answer.
I'm sorry for how basic this is, I want to really understand the process, and I'm doing it from free resources.
We all did; no excuse though to not benefit from Andrew Ng's free & excellent Machine Learning MOOC at Coursera.
It seems your question is why there is a activation function for each layer instead of just the last layer. The simple answer is, if there are no non-linear activations in the middle, no matter how deep your network is, it can be boiled down to a single linear equation. Therefore, non-linear activation is one of the big enablers that enable deep networks to be actually "deep" and learn high-level features.
Take the following example, say you have 3 layer neural network without any non-linear activations in the middle, but a final softmax layer. The weights and biases for these layers are (W1, b1), (W2, b2) and (W3, b3). Then you can write the network's final output as follows.
h1 = W1.x + b1
h2 = W2.h1 + b2
h3 = Softmax(W3.h2 + b3)
Let's do some manipulations. We'll simply replace h3 as a function of x,
h3 = Softmax(W3.(W2.(W1.x + b1) + b2) + b3)
h3 = Softmax((W3.W2.W1) x + (W3.W2.b1 + W3.b2 + b3))
In other words, h3 is in the following format.
h3 = Softmax(W.x + b)
So, without the non-linear activations, our 3-layer networks has been squashed to a single layer network. That's is why non-linear activations are important.
Imagine, you have an activation layer only in the last layer (In your case, sigmoid. It can be something else too.. say softmax). The purpose of this is to convert real values to a 0 to 1 range for a classification sort of answer. But, the activation in the inner layers (hidden layers) has a different purpose altogether. This is to introduce nonlinearity. Without the activation (say ReLu, tanh etc.), what you get is a linear function. And how many ever, hidden layers you have, you still end up with a linear function. And finally, you convert this into a nonlinear function in the last layer. This might work in some simple nonlinear problems, but will not be able to capture a complex nonlinear function.
Each hidden unit (in each layer) comprises of activation function to incorporate nonlinearity.
I have started to learn ML, and am confused with make_friedman1. It highly improved my accuracy, and increased the data size. But the data isn't the same, it's changed after using this function. What does friedman! actually do?
If make_friedman1 asked here is the one in sklearn.datasets then it is the function which generates the “Friedman #1” regression problem. Here inputs are 10 independent variables uniformly distributed on the interval [0,1], only 5 out of these 10 are actually used. Outputs are created according to the formula::
y = 10 sin(π x1 x2) + 20 (x3 - 0.5)^2 + 10 x4 + 5 x5 + e
where e is N(0,sd)
Quoting from the Friedman's original paper, Multivariate Adaptive Regression Splines ::
A new method is presented for flexible regression modeling of high
dimensional data. The model takes the form of an expansion in product
spline basis functions, where the number of basis functions as well as
the parameters associated with each one (product degree and knot
locations) are automatically determined by the data. This procedure is
motivated by the recursive partitioning approach to regression and
shares its attractive properties. Unlike recursive partitioning,
however, this method produces continuous models with continuous
derivatives. It has more power and flexibility to model relationships
that are nearly additive or involve interactions in at most a few
variables
A spline is adding many polynomial curves end-to-end to make a new smooth curve.
I made a program that trains a decision tree built on the ID3 algorithm using an information gain function (Shanon entropy) for feature selection (split).
Once I trained a decision tree I tested it to classify unseen data and I realized that some data instances cannot be classified: there is no path on the tree that classifies the instance.
An example (this is an illustration example but I encounter the same problem with a larger and more complex data set):
Being f1 and f2 the predictor variables (features) and y the categorical variable, the values ranges are:
f1: [a1; a2; a3]
f2: [b1; b2; b3]
y : [y1; y2; y3]
Training data:
("a1", "b1", "y1");
("a1", "b2", "y2");
("a2", "b3", "y3");
("a3", "b3", "y1");
Trained tree:
[f2]
/ | \
b1 b2 b3
/ | \
y1 y2 [f1]
/ \
a2 a3
/ \
y3 y1
The instance ("a1", "b3") cannot be classified with the given tree.
Several questions came up to me:
Does this situation have a name? tree incompleteness or something like that?
Is there a way to know if a decision tree will cover all combinations of unknown instances (all features values combinations)?
Does the reason of this "incompleteness" lie on the topology of the data set or on the algorithm used to train the decision tree (ID3 in this case) (or other)?
Is there a method to classify these unclassifiable instances with the given decision tree? or one must use another tool (random forest, neural networks...)?
This situation cannot occur with the ID3 decision-tree learner---regardless of whether it uses information gain or some other heuristic for split selection. (See, for example, ID3 algorithm on Wikipedia.)
The "trained tree" in your example above could not have been returned by the ID3 decision-tree learning algorithm.
This is because when the algorithm selects a d-valued attribute (i.e. an attribute with d possible values) on which to split the given leaf, it will create d new children (one per attribute value). In particular, in your example above, the node [f1] would have three children, corresponding to attribute values a1,a2, and a3.
It follows from the previous paragraph (and, in general, from the way the ID3 algorithm works) that any well-formed vector---of the form (v1, v2, ..., vn, y), where vi is a value of i-th attribute and y is the class value---should be classifiable by the decision tree that the algorithm learns on a given train set.
Would you mind providing a link to the software you used to learn the "incomplete" trees?
To answer your questions:
Not that I know of. It doesn't make sense to learn such "incomplete trees." If we knew that some attribute values will never occur then we would not include them in the specification (the file where you list attributes and their values) in the first place.
With the ID3 algorithm, you can prove---as I sketched in the answer---that every tree returned by the algorithm will cover all possible combinations.
You're using the wrong algorithm. Data has nothing to do with it.
There is no such thing as an unclassifiable instance in decision-tree learning. One usually defines a decision-tree learning problem as follows. Given a train set S of examples x1,x2,...,xn of the form xi=(v1i,v2i,...,vni,yi) where vji is the value of the j-th attribute and yi is the class value in example xi, learn a function (represented by a decision tree) f: X -> Y, where X is the space of all possible well-formed vectors (i.e. all possible combinations of attribute values) and Y is the space of all possible class values, which minimizes an error function (e.g. the number of misclassified examples). From this definition, you can see that one requires that the function f is able to map any combination to a class value; thus, by definition, each possible instance is classifiable.
I am having a problem at hand where,
I need to classify the input data to one or more of the labels S1, S2, S3, S4
There is a relationship between the labels S1, S2, S3 and S4 which is,
If input is labelled Sn it must be labelled S1..Sn.
S1, S2, S3 and S4 are like different stages for an entity X to pass through. Based on input data X might get through one or many of the stages, X must pass through S1 to go to S2, S2 to go to S3 and so on
We want to ensure that only those X are allowed to pass which reach S3, so based on input data we decide whether to allow X to go through S1 or not
What machine learning models can we choose to predict if X reaches S3 if we have information like, input data and what stages X has passed for that input data
I am thinking in direction of a multi label classification There might be some relationship between input data stage S1 and S2
Update: I have to train with examples like
1. Input data is s1
2. Input data is s2
3. ..
4 ..
Some doubts
Your question is far from being clear, for example:
We want to optimize that most X reaches S3, so based on input data we decide whether to allow X to go through S1 or not
Actually suggest, that the best model would be "always answer yes" ,as it maximized number of objects reaching S3 (as it simply lets any object reach this point)
General ideas
I assume two possible interpretations:
You have a labels "pipeline", which simply means, that object cannot be labelled S_n if it has not been already labelled with all S_i for i < n
This does not seem to be the problem for one single model, you can pipeline models in a natural way, ie. train a model 1 which regognizes, if object x should have label S_1. Next, you train a model 2 on all data that has label S_1 in the training set and predict label S_2, and so on. During execution you simply ask each model i if it accepts (labels) the incoming object x, and stop when the first one says "no"
You have some more complex constraints on the labels, which may be strict or not.For such cases, you should try one of many methods of multi label classification with constraints, in particular there is a tech report regarding this aspect of ML.
Solution 1 - approximating test functions
If your problem can be described as:
You have data points X, such that for each of them you know the maximum number of some pipelineable tests T_i which x passes
You want to train a classifier able to predict, what is the maximum number of consequtive tests that your point x passes
You do not have access to actual tests T_i or they are very inefficient
Then the simplest way would be to apply the following training procedure instead of one classifier:
Take all your data points, label those with y=0 as 0 and those with y>=1 as 1 and train some binary classifier (for example SVM). So you simply temporarly relabel your data so it shows points that pass the first test and those who don't. Lets call this classifier cl_1
Now take your data points, label those with y=1 as 0 and those with y>=2 as 1 and again train binary classifier, and call it cl_2
Repest until all tests have their classifier, in general in we call the classifier cl_i when it can distinguish between points labeled with y=i-1 and those with y>=i.
Now, to classify your new point, you simply check iteratively all your cl_i for i=1,..,tests and answer with the largest such i that cl_i(x)=1. So you "simulate" your tests with classifiers, and simply say how many this tests' approximations it passed.
To sum up: each test can be approximated with one binary classifier, and then the question of "What is the biggest consequtive test number that our point passes" is approximated with "what is the biggest consequtive classifier number that out point is classified as true".
Solution 2 - simple regression
You can also simply apply regression from your input space into the number of tests it reaches. Regression actually has an imprinted assumption, that the output values are correlated. So if you train your data with pairs (x,y) where y is the number of last test passed by x, then you are actually using the fact, that the output y=3 is highly related to first getting y=2 in the computations. Such regression (non-linear!) could be simply done using neural networks (possibly regularized)
Is it a good practice to use sigmoid or tanh output layers in Neural networks directly to estimate probabilities?
i.e the probability of given input to occur is the output of sigmoid function in the NN
EDIT
I wanted to use neural network to learn and predict the probability of a given input to occur..
You may consider the input as State1-Action-State2 tuple.
Hence the output of NN is the probability that State2 happens when applying Action on State1..
I Hope that does clear things..
EDIT
When training NN, I do random Action on State1 and observe resultant State2; then teach NN that input State1-Action-State2 should result in output 1.0
First, just a couple of small points on the conventional MLP lexicon (might help for internet searches, etc.): 'sigmoid' and 'tanh' are not 'output layers' but functions, usually referred to as "activation functions". The return value of the activation function is indeed the output from each layer, but they are not the output layer themselves (nor do they calculate probabilities).
Additionally, your question recites a choice between two "alternatives" ("sigmoid and tanh"), but they are not actually alternatives, rather the term 'sigmoidal function' is a generic/informal term for a class of functions, which includes the hyperbolic tangent ('tanh') that you refer to.
The term 'sigmoidal' is probably due to the characteristic shape of the function--the return (y) values are constrained between two asymptotic values regardless of the x value. The function output is usually normalized so that these two values are -1 and 1 (or 0 and 1). (This output behavior, by the way, is obviously inspired by the biological neuron which either fires (+1) or it doesn't (-1)). A look at the key properties of sigmoidal functions and you can see why they are ideally suited as activation functions in feed-forward, backpropagating neural networks: (i) real-valued and differentiable, (ii) having exactly one inflection point, and (iii) having a pair of horizontal asymptotes.
In turn, the sigmoidal function is one category of functions used as the activation function (aka "squashing function") in FF neural networks solved using backprop. During training or prediction, the weighted sum of the inputs (for a given layer, one layer at a time) is passed in as an argument to the activation function which returns the output for that layer. Another group of functions apparently used as the activation function is piecewise linear function. The step function is the binary variant of a PLF:
def step_fn(x) :
if x <= 0 :
y = 0
if x > 0 :
y = 1
(On practical grounds, I doubt the step function is a plausible choice for the activation function, but perhaps it helps understand the purpose of the activation function in NN operation.)
I suppose there an unlimited number of possible activation functions, but in practice, you only see a handful; in fact just two account for the overwhelming majority of cases (both are sigmoidal). Here they are (in python) so you can experiment for yourself, given that the primary selection criterion is a practical one:
# logistic function
def sigmoid2(x) :
return 1 / (1 + e**(-x))
# hyperbolic tangent
def sigmoid1(x) :
return math.tanh(x)
what are the factors to consider in selecting an activation function?
First the function has to give the desired behavior (arising from or as evidenced by sigmoidal shape). Second, the function must be differentiable. This is a requirement for backpropagation, which is the optimization technique used during training to 'fill in' the values of the hidden layers.
For instance, the derivative of the hyperbolic tangent is (in terms of the output, which is how it is usually written) :
def dsigmoid(y) :
return 1.0 - y**2
Beyond those two requriements, what makes one function between than another is how efficiently it trains the network--i.e., which one causes convergence (reaching the local minimum error) in the fewest epochs?
#-------- Edit (see OP's comment below) ---------#
I am not quite sure i understood--sometimes it's difficult to communicate details of a NN, without the code, so i should probably just say that it's fine subject to this proviso: What you want the NN to predict must be the same as the dependent variable used during training. So for instance, if you train your NN using two states (e.g., 0, 1) as the single dependent variable (which is obviously missing from your testing/production data) then that's what your NN will return when run in "prediction mode" (post training, or with a competent weight matrix).
You should choose the right loss function to minimize.
The squared error does not lead to the maximum likelihood hypothesis here.
The squared error is derived from a model with Gaussian noise:
P(y|x,h) = k1 * e**-(k2 * (y - h(x))**2)
You estimate the probabilities directly. Your model is:
P(Y=1|x,h) = h(x)
P(Y=0|x,h) = 1 - h(x)
P(Y=1|x,h) is the probability that event Y=1 will happen after seeing x.
The maximum likelihood hypothesis for your model is:
h_max_likelihood = argmax_h product(
h(x)**y * (1-h(x))**(1-y) for x, y in examples)
This leads to the "cross entropy" loss function.
See chapter 6 in Mitchell's Machine Learning
for the loss function and its derivation.
There is one problem with this approach: if you have vectors from R^n and your network maps those vectors into the interval [0, 1], it will not be guaranteed that the network represents a valid probability density function, since the integral of the network is not guaranteed to equal 1.
E.g., a neural network could map any input form R^n to 1.0. But that is clearly not possible.
So the answer to your question is: no, you can't.
However, you can just say that your network never sees "unrealistic" code samples and thus ignore this fact. For a discussion of this (and also some more cool information on how to model PDFs with neural networks) see contrastive backprop.